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      "text": "# Long solutions to small equations — reproducibility package\n\nThis package accompanies the revised 24 October 2026 version by Bogdan Grechuk,\nEugene Go and Avraham Eisenberg. It contains the LaTeX source, BibTeX database,\nand computational material for the **38 proofs retained in the paper**.\nThe code, inputs, expected outputs and documentation for the two proofs replaced\nby citations to earlier papers have been removed. Shared dependencies needed by\nthe remaining cases are retained.\n\nThe introduction surveys 73 entries explicitly left open in the book:\n38 are proved here, 17 were completed after the book but before this paper,\n3 follow from older literature, and 15 remain open. The two equivalent\npresentations supported by the same earlier paper count as two table entries.\nThe 38 retained proofs consist of 15 affine cases, 5 generalized Fermat cases,\n8 plane quartics and 10 other homogeneous equations.\n\n## Archive layout\n\nThe ZIP root contains `longsmall.tex` and `references.bib`. All supporting\nmaterial is under `anc/`, including every original script, input, expected output,\nREADME and licence file. Original computational filenames and formats are\npreserved. The build script and package checker have been adjusted for this\nlayout, and `anc/SHA256SUMS` has been regenerated using paths relative to the ZIP\nroot. Run the commands below from that root; paths in each case's own README\nremain relative to its original location under `anc/code/`.\n\n## Build the paper\n\nFrom the ZIP root, run:\n\n```sh\nsh anc/build.sh\n```\n\nThe inputs are `longsmall.tex` and `references.bib`. The script runs pdfLaTeX,\nBibTeX, then repeats pdfLaTeX until references and table widths stabilize\n(at most five further passes). It writes intermediate files to `build/` and\ncopies `longsmall.pdf` to this directory. A normal TeX Live or MiKTeX installation with\nthe packages listed in the preamble and the standard `amsalpha` bibliography\nstyle is sufficient. No external figures, additional TeX source files, network\naccess or shell escape are required. The build was checked with pdfTeX 1.40.25\nand BibTeX 0.99d (TeX Live 2023/Debian).\n\nThe bibliography entry for the article marked “To appear” intentionally has no\npage range; BibTeX reports a missing-pages warning for that entry. No page range\nhas been invented. The final LaTeX pass has no unresolved references or citations,\nmissing characters or overfull boxes. The PDF is supplied separately and is not\nduplicated in this archive. PDF bytes can differ with TeX versions and timestamps.\n\n## Checks performed for this edition\n\nThis edition corrects the specific mathematical and editorial issues identified\nin feedback. The reported resultant factors and the elliptic-curve counterexample\nwere checked exactly; the recorded characteristic polynomials were reduced modulo\n2; and the doubled Prym map was checked against Bruin (2008), Proposition 6.2 and\nRemark 6.3. Proposition D now requires an odd prime unramified in the number field\nand a coordinate without poles on the entire residue disk over an algebraic\nclosure of each completion. The applications use\nsplit odd primes and record the necessary disk integrality.\n\n**The full proofs and computational certificate suites were not rerun.**\nAdditional checks cover:\n\n- A clean build using the supplied `anc/build.sh` and bibliography.\n- Citation and label resolution, theorem/table inventories and the 38-case count.\n- PDF metadata, page layout and table presentation, including the line in\n  Lemma 3.11.4 reported missing from an earlier PDF.\n- The current section-to-certificate map and removal of the two retired cases.\n- Python syntax, shell-wrapper syntax, and preservation of the retained code.\n- ZIP integrity and regenerated SHA-256 checksums for all distributed files,\n  including the latest LaTeX source and BibTeX database.\n\nTo repeat the targeted algebra checks (SymPy 1.14.0 was used), run:\n\n```sh\npython3 anc/check_corrections.py\n```\n\nThis checks the two resultants and the counterexample using exact arithmetic.\nFor Lemma 3.8.5 it checks the four polynomials in the historical output file;\nit does not recompute the underlying logarithms.\n\nTo repeat the non-mathematical package checks, run:\n\n```sh\npython3 anc/check_package.py\n```\n\nThis uses Python's standard library. It does not execute the certificates or\nbounded searches. `anc/SHA256SUMS` covers every distributed file except itself, with paths\nrelative to the ZIP root;\non systems with GNU coreutils it can also be checked with:\n\n```sh\nsha256sum -c anc/SHA256SUMS\n```\n\n## Reproduce computations\n\nThe retained optional lightweight suite is available as:\n\n```sh\npython3 anc/verify_quick.py\n```\n\nIt requires Python 3 and a C compiler (`cc`/`gcc`). It includes the 38-case\n`verify_statements.py` substitution and bounded-search checks, the first cubic\nsurface's certificate suite, the elementary quintic and trace checks, the shared\nnewspace-dimension check, and the retained finite-field and twist checks for\ncase 1.1.14. Bounded searches check transcription; they do not prove completeness.\nThe Fermat checker keeps the original certificate identifiers despite the gaps\nleft by the removed cases.\n\nThe earlier edition recorded successful lightweight checks on 22 October 2026.\nThose are historical results, not execution results for this edition. In\nparticular, the edited 38-case checker was checked for syntax and correspondence\nwith the retained cases, but its searches were not rerun. The full Magma/PARI/GP\nand large class-group computations were not rerun in these editorial passes.\nThe earlier attempted GP newform runs could not start because `gp` was unavailable.\n\nFull recomputation uses the software specified by each directory's README:\nPARI/GP 2.15.4 or 2.17.4; Python with cypari2, python-flint/Arb, SymPy, NumPy and\nmpmath; Magma (recorded runs V2.29-10); and a C/C++ compiler. Some cross-checks\nuse SageMath/passagemath. Do not run Python with `-O`: assertions are part of\nthe certificates. Compatibility with newer dependency versions was not tested.\n\nRun full certificates in the order documented locally. Large generated witness\nand root-count files must first be regenerated and then passed to the exact\ncheckers. In particular, `anc/code/1.3.8/classgroup/run_witnesses.sh` and\n`run_counts.sh` regenerate the large datasets for Section 5.8. The small fixed\ninputs and the original case 1.3.7 witness files are included.\n\nDirectories named `expected/` contain historical reference outputs, not new\nexecution logs. Dates, timings and phrases such as “run here” in the local\nREADMEs describe the supplied records. Reference outputs do not substitute for\nexecuting the assertions when a computational verification is requested.\n\n## Certificate map\n\nThe section numbers below match the final paper. Certificate IDs, filenames and\ninternal LaTeX labels retain their original identifiers. Each case's README maps\nindividual lettered certificates to scripts and fixed inputs.\n\n| Paper section | Certificate prefix / directory | Equation |\n|---|---|---|\n| 3.1 | `1.1.1` / `anc/code/1.1.1/` | $x^4-y^3-x+y=0$ |\n| 3.2 | `1.1.2` / `anc/code/1.1.2/` | $x^4-y^3+x-y=0$ |\n| 3.3 | `1.1.3` / `anc/code/1.1.3/` | $x^4+y^3+xy+1=0$ |\n| 3.4 | `1.1.4` / `anc/code/1.1.4/` | $x^4+y^3+2x-y=0$ |\n| 3.5 | `1.1.5` / `anc/code/1.1.5/` | $x^4+y^3+2x+y=0$ |\n| 3.6 | `1.1.6` / `anc/code/1.1.6/` | $x^4+y^3+y^2+x=0$ |\n| 3.7 | `1.1.7` / `anc/code/1.1.7/` | $x^4+y^3-y^2+x=0$ |\n| 3.8 | `1.1.8` / `anc/code/1.1.8/` | $x^4+y^3+2x-y+1=0$ |\n| 3.9 | `1.1.9` / `anc/code/1.1.9/` | $x^4+y^3+2x+y+1=0$ |\n| 3.10 | `1.1.10` / `anc/code/1.1.10/` | $x^4+y^3+2x+y-1=0$ |\n| 3.11 | `1.1.11` / `anc/code/1.1.11/` | $x^4+y^3+xy-y-1=0$ |\n| 3.12 | `1.1.12` / `anc/code/1.1.12/` | $x^4-y^3+xy+x-1=0$ |\n| 3.13 | `1.1.13` / `anc/code/1.1.13/` | $x^4+y^3+xy+y-1=0$ |\n| 3.14 | `1.1.14` / `anc/code/1.1.14/` | $x^4+y^3+xy+x+1=0$ |\n| 3.15 | `1.1.15` / `anc/code/1.1.15/` | $x^4-x^2+xy+y^3=0$ |\n| 4.1 | `1.2.1` / `anc/code/1.2.1/` | $x^4+3y^3+z^3=0$ |\n| 4.2 | `1.2.3` / `anc/code/1.2.3/` | $2x^4-y^4+z^3=0$ |\n| 4.3 | `1.2.4` / `anc/code/1.2.4/` | $x^5+y^4=2z^2$ |\n| 4.4 | `1.2.6` / `anc/code/1.2.6/` | $x^5+y^4=3z^2$ |\n| 4.5 | `1.2.7` / `anc/code/1.2.7/` | $x^5+y^4+3z^2=0$ |\n| 5.1 | `1.3.1` / `anc/code/1.3.1/` | $x^4+x^3y+xy^3=z^4$ |\n| 5.2 | `1.3.2` / `anc/code/1.3.2/` | $x^4-y^4+x^2yz+yz^3=0$ |\n| 5.3 | `1.3.3` / `anc/code/1.3.3/` | $x^4+x^2y^2+y^3z-yz^3=0$ |\n| 5.4 | `1.3.4` / `anc/code/1.3.4/` | $x^4+y^4+x^2yz-yz^3=0$ |\n| 5.5 | `1.3.5` / `anc/code/1.3.5/` | $x^4-x^2y^2+y^3z+yz^3=0$ |\n| 5.6 | `1.3.6` / `anc/code/1.3.6/` | $x^4+x^2y^2+y^3z+z^4=0$ |\n| 5.7 | `1.3.7` / `anc/code/1.3.7/` | $x^3z+y^3x+y^2z^2-z^3y=0$ |\n| 5.8 | `1.3.8` / `anc/code/1.3.8/` | $x^4-y^3z+xyz^2+xz^3=0$ |\n| 6.1 | `1.4.1` / `anc/code/1.4.1/` | $x^3+2x^2y+y^3+y^2z+xz^2+z^3+t^2x-t^2y-t^2z+tyz+t^3=0$ |\n| 6.2 | `1.4.2` / `anc/code/1.4.2/` | $2x^3+x^2y+xy^2+y^3+y^2z-2xz^2-z^3+t^2y+tz^2-t^3=0$ |\n| 6.3 | `1.4.3` / `anc/code/1.4.3/` | $2x^3+2x^2y+2y^3+y^2z-2xz^2-z^3+tz^2-t^3=0$ |\n| 6.4 | `1.4.4` / `anc/code/1.4.4/` | $8x^4+7y^4+2z^4-t^4=0$ |\n| 6.5 | `1.4.5` / `anc/code/1.4.5/` | $14x^5+5y^5+3z^5=0$ |\n| 6.6 | `1.4.6` / `anc/code/1.4.6/` | $21x^5+5y^5+4z^5=0$ |\n| 6.7 | `1.4.7` / `anc/code/1.4.7/` | $17x^5+6y^5+3z^5=0$ |\n| 6.8 | `1.4.8` / `anc/code/1.4.8/` | $17x^5+9y^5+5z^5=0$ |\n| 6.9 | `1.4.9` / `anc/code/1.4.9/` | $14x^5+13y^5+2z^5=0$ |\n| 6.10 | `1.4.10` / `anc/code/1.4.10/` | $4x^5+4y^5+11z^5=0$ |\n\nThe shared Frey certificates 1.4qa–c are in `anc/code/common/frey_quintic/`.\nOther shared directories provide class-group, descent, cyclic-cubic and Simon\nelliptic-descent helpers. In legacy code documentation, original shared result\nlabels A, B, C, G, O, S, T, U, AB, AD correspond respectively to the current\npaper's A, B, C, D, E, F, G, H, I, J.\n\n## Files and notices\n\nThe archive includes only the current paper's source, bibliography, build and\ncheck scripts, retained computational material and its shared dependencies.\nIt excludes old paper versions, source fragments, alternative proofs, old PDFs,\nLean formalizations, generated executables, Python caches and TeX intermediates.\nSubmission metadata and the distribution licence are supplied by the authors\nat deposit. The author byline is retained exactly as supplied in the final draft;\nno submission has been made.\n\nThe bundled Denis Simon GP scripts retain their original copyright and GPL\nnotices; GPL version 3 is included as `anc/code/common/simon_ell/COPYING`.\nOther supplied code retains its existing notices. This packaging update does\nnot assert a new blanket licence over the paper or research code.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.1: x^4 − y^3 − x + y = 0\n\nModel used in the section: `C: u^4 + v^3 + u − v = 0` with `(x, y) = (−u, −v)`; it is the source\ncurve `x^4 + y^3 + x − y = 0`, and all scripts use the source letters `(x, y)` for `(u, v)`.\nKnown points (model): `A=(−1,−1), B=(−1,0), C1=(−1,1), D=(0,−1), E=(0,0), F1=(0,1)`; basis `(E, B, D)`.\nBitangent field `K = Q[m]/(p_K)`, `p_K = m^27−480m^19−1344m^18−34560m^11+70656m^10−40704m^9+16384m^3−12288m−4096`.\n\nRequirements: PARI/GP ≥ 2.15 (`gp`); Python 3 with cypari2, python-flint (`nmod_poly`, `arb`),\nsympy, numpy; g++ (C++17); for 1.1.1j only, a Sage with function-field Jacobians\n(`from sage.all__sagemath_schemes import *`, e.g. passagemath). No Magma.\nRun everything from this folder; the programs taken from the source (1.1.1e, f, g) are run with `python3 -I`. \"Run here\" = run on 14 Oct 2026 (PARI 2.15.4, python-flint 0.9.0,\npartial passagemath for Sage), output in `expected/`. Scratch output goes to `work/` (not included).\n\n| id | command | time | run here |\n|---|---|---|---|\n| 1.1.1a | `gp -q 1.1.1a_geometry.gp`; `gp -q 1.1.1a_nodes.gp`; `gp -q 1.1.1a_bitangent.gp` | 1 s | yes, matches |\n| 1.1.1b | `gp -q 1.1.1b_dyadic.gp` | 5 s | yes, matches |\n| 1.1.1c | `python3 1.1.1c_code.py` | 10 s | yes, matches |\n| 1.1.1d | shared tools, `principal_bound` 10^7 (below) | ≈ 1 h | slice p ≤ 20000; full run from the replay log |\n| 1.1.1e | `python3 -I 1.1.1e_low_norm_census.py`; `python3 -I 1.1.1e_census_stream.py --verify` | 1 min; ≈ 7 min | low-norm census yes; census stream: slice |\n| 1.1.1f | `python3 -I 1.1.1f_strict_window.py --require-positive` | < 1 s | yes, matches |\n| 1.1.1g | `python3 -I 1.1.1g_kernel.py` | 3 s | yes, matches |\n| 1.1.1h | `gp -q 1.1.1h_qc3.gp` | 2 s | yes, matches |\n| 1.1.1i | `gp -q 1.1.1i_qc5.gp` | 1 s | yes, matches |\n| 1.1.1j | `1.1.1j_count.cpp` + `1.1.1j_jorder.py`; `1.1.1j_sieve_tables.py` (Sage); `python3 1.1.1j_sieve_logic.py` | 1 min; 1–3 min per prime; 1 s | yes, matches |\n\n## 1.1.1a — geometry and bitangent field\n* `1.1.1a_geometry.gp`: transport identity `P(−u,−v) = F(u,v)`; the six target points; affine singular\n  locus (resultant `157 · 64679`); the chart `X^4+Z+XZ^3−Z^3` at the hyperflex `O` is smooth.\n* `1.1.1a_nodes.gp`: at 157 and 64679 a unique singular point (an ordinary node), Hessian a unit,\n  critical value of valuation 1 (thickness 1).\n* `1.1.1a_bitangent.gp`: `Res_z(12m^5z − m^9 − 24mz^2 + 8m − 8, (12m^2z − m^6)^2 − 64(z^3 − z)) = 512 p_K(m)`;\n  `p_K` irreducible; proven factorization of `disc(p_K)`; `nfinit` with it: signature, `disc(K)`, decomposition\n  of 2, 3, 157, 64679, 333337; the bitangent identities `2ab = 3mη^2 − m + 1`, `b^2 = η^3 − η`, `a^2 − 4b ≠ 0`.\n\nExpected: `res affine sing = [157, 1; 64679, 1]`, `v_p(F(crit)) = 1  F(crit) mod p^4 = 572892686`\n(p = 157) and `11905259396277590310` (p = 64679), `R/lc == pK(m)? 1`, `irreducible: 1`,\n`disc(pK) = [2, 312; 3, 54; 157, 6; 64679, 6; 333337, 2; 172659335137, 2]`,\n`sig = [3, 12]  disc(K) = [2, 24; 157, 6; 64679, 6]`,\n`disc(K) value = 18394906743257121277769499404528301613431352459264`, `primes above 2: [[9, 3]]`,\n`p1(m,et)=0: 1   p2: 1`, identities `Mod(0, …)`, `a^2-4b !=0: 1`.\n(The real connectedness of `C(R)` is proved in the text; the code weights {12, 16} are 1.1.1c.)\n\n## 1.1.1b — dyadic descent images of the six points\n`1.1.1b_dyadic.gp`: for the unique prime `𝔭 | 2` (`e = 9, f = 3`) the class in `K_𝔭^×/K_𝔭^×2` (dimension 29)\nof `mη + η-intercept − v` at the six points: valuation parity and unit square class modulo `𝔭^19`\n(`idealstar`/`ideallog`); extra global check with 398 split-prime residue characters.\nExpected: `dyadic rank {A,B,D}: 3  {E,B,D}: 3  all six: 3`, `A==E mod squares (chars): 1`,\n`number of even cyclic factors of (O/P^19)^*: 28`.\n\n## 1.1.1c — orthogonal-code trace bounds κ(r)\n`1.1.1c_code.py`: `Q = v1v2+v3v4+v5v6+v5+v6` on `F_2^6`, the 27 nonzero isotropic vectors, all 2825\nsubspaces and 64 linear forms. Expected: `nonzero word weights: [12, 16]`,\n`num subspaces: 2825 Counter({3: 1395, 2: 651, 4: 651, 1: 63, 5: 63, 0: 1, 6: 1})`,\n`ell table: [(0, 0), (1, -1), (2, -2), (3, -3), (5, -3), (7, -5), (9, -3), (11, -5), (15, -5), (27, -5)]`,\n`universal min: -5`.\n\n## 1.1.1d — every prime ideal of norm ≤ 10^7 is principal\nShared tools (`../common/h31_classgroup/`, see its README) with `data/classgroup_params.json`\n(`principal_bound` 10^7; the window entries there are not used by this section, whose bound is 1.1.1f):\n```\nCG=../common/h31_classgroup\npython3 $CG/witnesses.py data/classgroup_params.json --workers 2 --out work/witnesses.jsonl.gz\npython3 $CG/check_witnesses.py data/classgroup_params.json work/witnesses.jsonl.gz\n```\n`witnesses.py` uses the rule of the source program `certify_3723_principals_fast.py`: the `idealmin`\nelement if its cofactor is strict (Lemma C), else a generator proposed by `bnfisprincipal`;\n`check_witnesses.py` enumerates the prime ideals itself and checks every witness exactly.\nFull run (≈ 1 h with 2 workers, not repeated here): `ALL 664464 PRIME IDEALS OF NORM <= 10000000 ...`;\nthe source program on the same rule (our replay, PARI 2.17.4) gave `664464 = 588850` strict `+ 75614`\ndirect, by residue degree 1, 2, 3, 4, 5, 6, 12: `663993, 426, 20, 9, 5, 9, 2`. (The strict/direct split\ndepends on PARI's `idealmin`; at bound 30000 both programs give `3246 = 1853 + 1393` with PARI 2.17.4.)\nRun here: slice `--lo 2 --hi 20000` (all 2665 prime ideals of norm ≤ 10^7 over p ≤ 20000, including all\n471 of residue degree ≥ 2): generated and all checked.\n\n## 1.1.1e — prime-ideal norm census and root-count census\n* `1.1.1e_low_norm_census.py` (defaults `--cutoff 10000000 --support 10000000000`; source program\n  `low_norm_census_3723.py`, defaults changed to this section's values): `a(q)` for all norms\n  `q ≤ 10^7` (exact for `q ≤ 10^5`, all powers `q^m < 10^10`; 9154 right-endpoint bins above), and all\n  prime ideals of residue degree ≥ 2 with norm in `(10^7, 10^10)`. Writes `data/3723_low_norm_census.json`,\n  `data/3723_higher_norms.json` unless `--output/--higher-output` are given.\n  Expected and obtained: `KNOWN_PRIME_IDEALS 664464 SMALL_EXACT_NORMS 5331 LARGE_POSITIVE_BINS 9154\n  HIGHER_UNKNOWN_NORMS 4180 HIGHER_UNKNOWN_IDEALS 9589 SHA256 5bf750889b7397e552157ebb9704e78827995d7c496abec0777d67cdf959634e`;\n  the regenerated files equal the ones in `data/`.\n* `1.1.1e_census_stream.py` (source `census_stream_3723.py`; defaults now `--start 10^7 --stop 1.1·10^8\n  --block 10^7 --workers 2 --output data/3723_census_blocks`): for every prime `10^7 ≤ p < 1.1·10^8`,\n  `r(p) = deg gcd(p_K, z^p − z)` (category 4 = three roots whose cube Frobenius also has three roots),\n  binned on the grid `edges(10^7)` (ratio 1.0005). `--verify` recomputes every stored block and compares\n  counts and stream hashes. Full run (≈ 7 min with 2 workers): `COMPLETE {\"start\": 10000000,\n  \"stop\": 110000000, \"block_size\": 10000000, \"completed_blocks\": 10, \"total_primes\": 5638730, ...}`;\n  root histogram `0: 2582146, 1: 1821326, 2: 563570, 3: 156431, 4: 278941, 5: 176800, 7: 29325,\n  9: 26194, 11: 0, 15: 3878, 27: 119` (printed by 1.1.1f).\n  Run here: slice `--start 10000000 --stop 11000000 --block 1000000 --workers 1 --output work/census_slice`\n  (61938 primes) and `python3 1.1.1e_compare_slice.py work/census_slice`:\n  `bins compared 190 identical 190 different 0`.\n* Cross-check with the shared tool: `python3 ../common/h31_classgroup/count_ideals.py data/classgroup_params.json\n  --support 10000001` counts all prime ideals of norm ≤ 10^7 and prints the SHA-256 of the same packed\n  `(q, a(q))` stream (`5bf75088…634e`, 664464 ideals).\n* `data/` holds the outputs of the full 1.1.1e runs (`3723_low_norm_census.json` 298 KB,\n  `3723_higher_norms.json` 61 KB, `3723_census_blocks/` 280 KB), so that 1.1.1f runs on its own.\n\n## 1.1.1f — explicit-formula lower bound 𝓑\n`1.1.1f_strict_window.py --require-positive` (source `verify_3723_strict_window.py`; reads\n`data/3723_strict_window.json` = the 128 integer heights, the census files above; writes\n`work/3723_strict_window_verified.json`): window with 128 equal cells on `[0, log 10^10]`, exact\nrational knots, triangle expansion of `𝒜(f)`, principal part `P_−`, higher-degree part `H`, census part\n`E` with the code bounds κ(r) of 1.1.1c, Dusart tail with `c = 36261/36260`, 224-bit Arb balls.\nExpected and obtained: `\"B_interval\": \"[0.02669177164848057422254271554072166630956050671295877265959286 +/- 2.69e-63]\"`,\n`\"rigorously_positive\": true`, `\"principal_ideal_count_required\": 664464`, `\"census_prime_count\": 5638730`,\n`\"higher_degree_nonprincipal_ideal_count\": 9589`, `\"principal_large_norm_bins\": 9154`, archimedean\n`[-1.0229409565916575887566… +/- 8.26e-66]`, `P_−` `[1.1435757488228141328217… +/- 1.50e-63]`.\n\n## 1.1.1g — height kernel identities\n`1.1.1g_kernel.py` (source `verify_3723_kernel.py`, without its file output): with sympy, the identity\n`(b−v)(v^2+bv+b^2−1) = (u−a)(u^3+u^2a+ua^2+a^3+1)` on `C`, the bidifferential identity, the cup matrix of the\nleading terms `(−t^4, −t, 1, 5t^−6, 2t^−3, −t^−2)` at `O`, the diagonal cancellation. (The relative order\n8 of the first Laurent correction is read off from `ρ^2(ρ−1) = −t^8 − t^9` in the text; the script records it.)\nExpected: `\"kernel_polynomial_identity_verified\": true`, `\"cup_matrix\"` = `[[0,−I],[I,0]]`,\n`\"diagonal_tangent_cancellation_verified\": true`.\n\n## 1.1.1h — quadratic Chabauty at p = 3; 1.1.1i — at p = 5\n`1.1.1h_qc3.gp`, `1.1.1i_qc5.gp` (independent reviewer code; input: the Riemann–Roch sections of the text).\nExpected (and obtained, identical to the reviewer's earlier output):\np = 3: `det = 3185`, `3M mod 729: [240, 62, 406; 62, 431, 692; 406, 692, 525]  min v(M) = -1`,\n`I_B/3 mod 3: [1, 1, 0; 0, 2, 2; 1, 1, 2]`, reductions and extra roots\n`A: 1274 [4511, 1997, 707]`, `A: 4680 [5946, 6246, 1629]`, `B: 5882 [1542, 2265, 5452]`,\n`C1: 2620 [1475, 1164, 2808]`, `D: 1442 [1405, 250, 6058]`, `F1: 5667 [6218, 5349, 6041]` (mod 3^8);\np = 5: `beta mod 5^8 = 309087`, `det R = 452`, `5^-1 I_B mod 5: [3, 3, 2; 2, 0, 0; 1, 3, 0]  v(det I_B) = 3`,\n`5M mod 5: [0, 3, 0; 3, 4, 0; 0, 0, 3]`, the seven extra roots `A 88061, B 51781, C1 291387, D 332159,\nF1 49757, P6 235554, P6 325535` with their coordinates mod 5^8.\n\n## 1.1.1j — finite fields, torsion, Mordell–Weil sieve\n1. `g++ -O2 -o work/count 1.1.1j_count.cpp`, then\n   `for q in 3 5 109 233 277; do for r in 1 2 3; do ./work/count $q $r; done; done > work/counts.txt`\n   and `python3 1.1.1j_jorder.py work/counts.txt` (Newton identities + functional equation):\n   `#J(F_3) = 91, #J(F_5) = 226, #J(F_109) = 1303500, #J(F_233) = 14102424, #J(F_277) = 23117400`;\n   counts `7, 13, 28`; `8, 34, 134`; `109, 12243, 1296898`; `259, 54447, 12648664`; `300, 77454, 21247200`.\n2. `python3 1.1.1j_sieve_tables.py q #J(F_q) [outdir]` (Sage, Hess-model Jacobian arithmetic) for\n   q = 109, 233, 277: the {2,3,5}-primary subgroups generated by the projections of E, B, D (Cayley tables)\n   and the primary coordinates of every point of `C(F_q)`; writes `outdir/sieve_data_q.json` (default `work/`).\n   The three tables are also in `data/sieve/` (6–12 KB each); the tables regenerated here (59 s, 93 s,\n   154 s) are identical to them.\n3. `python3 1.1.1j_sieve_logic.py [dir]` (default `data/sieve`): the 6 × 7 pairs of extra roots (3-adic,\n   5-adic coordinates of 1.1.1h/i), the odd-primary conditions at 109 and 277, then the joint 2-adic\n   condition at 109, 233, 277. Expected and obtained: subgroup orders `109: {2: 4, 3: 3, 5: 125}`,\n   `233: {2: 8, 3: 81}`, `277: {2: 8, 3: 27, 5: 25}`; compatible `j` per `i` at 109\n   `{0: [2, 6], 1: [0, 4, 5], 2: [0, 2], 3: [0, 4, 5], 4: [0, 2], 5: [0, 2]}` and at 277\n   `{0: [4], 1: [], 2: [2, 6], 3: [], 4: [4], 5: [4]}` (only (2,2) survives both), `joint survivors: []`.\n\n## Origin\nReviewer scripts written independently of the source (1.1.1a–c, h–j); the source's own programs for\n1.1.1e, f, g (`low_norm_census_3723.py`, `census_stream_3723.py`, `verify_3723_strict_window.py`,\n`verify_3723_kernel.py`; only defaults and output paths changed); shared tools for 1.1.1d.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.10: x^4 + y^3 + 2x + y - 1 = 0\n\nCoordinates: the target coordinates `(x,y)`; `O = [0:1:0]`, `A = (-1,1)`, `B = (1,-1)`.\n\nThe code of this section is the complete replay package of the solution (equation index 9111). Its\nscripts are included **unchanged and under their original names**, because the driver `verify_all.py`\ncalls them by these names and the full replay reported in the paper was made with exactly these files.\nThe small inputs the scripts read from their own folder are kept next to them:\n`9111_descent_data.json` (input of 1.1.10a), `9111_strict_window.json` (window of 1.1.10f) and\n`9111_strict_window_verified.json` (the recorded result of 1.1.10f; the driver reads two reference values\nfrom it). Not included (large data, regenerated by the code): the recorded principal certificates\n(`9111_principals_1m.json`, `9111_principals_15m.json`), the norm tables `9111_low_norm_census.json`,\n`9111_higher_norms.json` and the 30 census blocks `9111_census_blocks/`. Also not included: the slower\nalternative principal certifier `certify_principals_9111.py` and the TeX/SHA256 files of the package.\n\n**Note:** the scripts write their output JSON next to themselves (for example `9111_chabauty.json`).\nRun them in a copy of this folder.\n\n## Requirements\n\nPython 3 with cypari2 (PARI/GP 2.15 or later; the package was made with 2.17.2), SymPy and python-flint\n(Arb, `nmod_poly`); see `requirements.txt` (cypari2 2.2.4, sympy 1.14.0, python-flint 0.9.0). No Sage, no\nMagma. Here: Python 3.11, cypari2 with PARI 2.17.4, SymPy 1.14.0, python-flint 0.9.0.\n\n## Full replay (run here on 14–15 October 2026: about 2.7 hours with 2 workers; key lines in `expected/1.1.10_full_replay.txt`)\n\n```\npython3 verify_all.py --workers W\n```\nruns, in this order, 1.1.10a, b, h, g, then c (fresh from norm 1), d, e and f, and asserts the final\nnumbers (971,856 principal prime ideals; 26,759 higher-degree ideals; 38,465,791 census primes;\n`B > 7/1000`). The stages c, d, e are heavy (several CPU-hours on two cores); the full run is reported in\nthe paper and was not repeated here.\n\n## Codes\n\n| id | script | what it computes | run here |\n|---|---|---|---|\n| 1.1.10a | `verify_9111_descent.py` (+ `9111_descent_data.json`) | bad primes, nodes, bitangent field, the 17 classes, dyadic image, images of A and B | yes, 23 s; output identical to the recorded JSON |\n| 1.1.10b | `verify_9111_code_bound.py` | character-sum minima over all 2825 subspaces of `F_2^6` | yes, 1 s; identical |\n| 1.1.10c | `certify_9111_principals_fast.py` | principality of all prime ideals of norm `<= 1.5*10^7` | slice (`--bound 20000`); full run (all 971,856 ideals, 2.2 h) in `expected/1.1.10_full_replay.txt` |\n| 1.1.10d | `low_norm_census_9111.py` | norm tables for the explicit formula | slice (cutoff 20000, support `10^8`); full run: same SHA-256 as recorded |\n| 1.1.10e | `census_stream_9111.py` | `r(p) = deg gcd(f, z^p - z)` for all p in `[1.5*10^7, 7.65*10^8)` | slice (two blocks of 10^5); full run: 38,465,791 primes, as recorded |\n| 1.1.10f | `verify_9111_strict_window.py` | the strict-character lower bound B (224-bit Arb) | full run: identical to the recorded result, B > 7/1000 |\n| 1.1.10g | `9111_finite_sieve.py` | point counts, Jacobian arithmetic, saturation, sieve (standard library) | yes, 33 s; identical |\n| 1.1.10h | `9111_chabauty.py` | Chabauty-Coleman data at 3 and 5 | yes, < 1 s; identical |\n\n\"identical\" = the JSON written here equals the recorded JSON of the package apart from timing fields.\nThe JSON outputs of this run are in `expected/1.1.10{a,b,g,h}_*.json`, stdout in `expected/1.1.10{a,g,h}.txt`.\n\n### 1.1.10a  Geometry, bitangent field, dyadic descent data\n```\npython3 -I verify_9111_descent.py\n```\nKey output (`9111_descent_verified.json`): nodes p=31 `(22,14)` hessian 25; p=433 `(6,421)` hessian 72;\np=2341 `(2236,823)` hessian 1553; thickness 1 each; `\"bitangent_degree\": 27`, `\"signature\": [3, 12]`,\n`\"discriminant\": \"1033727079572205527447879379643917888211892763703115776\"`, `\"code_weights\": [12, 16]`,\n`\"known_independent_S_squareclasses\": 17`, dyadic dimensions 26 and 5, `\"local_dimension\": 3`,\n`\"union_rank\": 18`, `\"intersection_dimension\": 2`, `\"rational_point_image_rank\": 2`; stdout ends with\n`ALL EXACT GEOMETRY AND DYADIC ARITHMETIC ASSERTIONS PASSED; STRICT-CHARACTER RANK ARGUMENT IS SEPARATE`.\n\n### 1.1.10b  Character-sum bound on the 27 labels\n```\npython3 -I verify_9111_code_bound.py\n```\n`\"isotropic_nonzero_vectors\": 27`, `\"number_of_subspaces\": 2825` (1, 63, 651, 1395, 651, 63, 1), minima\nby r: 0:0, 1:-1, 2:-2, 3:-3, 5:-3, 7:-5, 9:-3, 11:-5, 15:-5, 27:-5; `\"universal_lower_bound\": -5`,\n`\"all_checks_passed\": true` (`expected/1.1.10b_verify_9111_code_bound.json`).\n\n### 1.1.10c  Principality of all prime ideals of norm <= 1.5*10^7\n```\npython3 -I certify_9111_principals_fast.py --bound 15000000 --workers W --output 9111_principals_replayed.json\n```\nFull-run expected values (recorded audit, `expected/1.1.10c_recorded.txt`): `\"all_passed\": true`,\n`\"prime_ideals_verified\": 971856`. Slice run here (`expected/1.1.10c_slice.txt`):\n`--bound 20000 --workers 1` -> `ALL PRIME IDEALS THROUGH NORM 20000 PROVED PRINCIPAL: 2246` (100 s).\n\n### 1.1.10d  Norm tables for the explicit formula\n```\npython3 -I low_norm_census_9111.py --cutoff 15000000 --support 100000000000 \\\n    --output 9111_low_norm_census_replayed.json --higher-output 9111_higher_norms_replayed.json\n```\nExpected (recorded, `expected/1.1.10d_recorded.txt`): `\"total_known_prime_ideals\": 971856` (27,293 exact\nsmall norms in 14,761 distinct values, 944,563 in 7,715 bins), `\"total_unknown_higher_degree_prime_ideals\": 26759`,\n`\"sha256_complete_known_norm_count_stream\": \"0c791ae6...a1a60b5\"`, polynomial discriminant\n`2^390 3^54 17^2 31^6 101^2 433^6 2341^6 2969^2 5227^2 924151^2`. Slice run here:\n`--cutoff 20000 --support 100000000` -> `KNOWN_PRIME_IDEALS 2246 ... HIGHER_UNKNOWN_IDEALS 1245` (consistent\nwith the 2246 ideals of 1.1.10c's slice).\n\n### 1.1.10e  Root-count census on [1.5*10^7, 7.65*10^8)\n```\npython3 -I census_stream_9111.py --start 15000000 --stop 765000000 --block 25000000 --workers W \\\n    --output 9111_census_blocks --verify\n```\nExpected (recorded, `expected/1.1.10e_recorded.txt`): 30 blocks, 38,465,791 primes; counts by r:\n0: 17610407, 1: 12425717, 2: 3851583, 3: 2971635, 5: 1200580, 7: 200672, 9: 177984, 15: 26479, 27: 734;\nthe 30 block hashes. Slice run here: `--start 15000000 --stop 15200000 --block 100000 --workers 1`\n-> 12,072 primes in two blocks.\n\n### 1.1.10f  The strict-character lower bound\n```\npython3 -I verify_9111_strict_window.py --spec 9111_strict_window.json \\\n    --low-norms 9111_low_norm_census_replayed.json --higher-norms 9111_higher_norms_replayed.json \\\n    --census 9111_census_blocks --output 9111_strict_window_replayed.json --require-positive\n```\nExpected (recorded, `expected/1.1.10f_recorded.txt` = `9111_strict_window_verified.json`):\n`archimedean_interval -1.42016127359146944673...`, `principal_sum_interval 1.53547434901545447756...`,\n`higher_degree_lower_interval -8.78024795564985661011...e-6`, `census_lower_interval -0.0590216892970131645604...`,\n`dusart_tail_lower_interval -0.0489055193247049610909...`,\n`B_interval [0.00737708655431125532657831344099130323480498928108842128134655 +/- 4.37e-63]`,\n`\"rigorously_positive\": true`, `\"rigorously_greater_than_7_over_1000\": true`.\n\n### 1.1.10g  Finite fields: point counts, Jacobian arithmetic, saturation, sieve\n```\npython3 -I 9111_finite_sieve.py\n```\nstdout (`expected/1.1.10g.txt`):\n```\nFIELD 3 COUNTS [4, 10, 28] JORDER 28\nFIELD 5 COUNTS [3, 23, 120] JORDER 64\nFIELD 11 COUNTS [18, 146, 1296] JORDER 2520\nFIELD 13 COUNTS [16, 216, 2128] JORDER 2912\nTWO_SATURATION [[1, 1], [0, 1]]\nMOD45_SURVIVOR [([3, 4], [(3, 13)])]\nMOD360_SURVIVOR [[228, 13]]\nSEVEN_SATURATION_AND_COEFFICIENTS [[1, 4, 2], [1, 3, 1]] [5, 1]\nPRIME13_ALL16_POINT_IMAGES_EXCLUDED\nALL FINITE SIEVE ASSERTIONS PASSED\n```\nJSON: `\"mod45_candidates\": [[33,3],[3,13]]`, the eight `mod360` candidates, `\"mod360_survivor\": [[228,13]]`,\n`\"seven_saturation_determinant\": 6`, `\"prime13_target_pairs\": [[68,29],[12,85],[180,141],[124,197]]`,\n`\"all_new_point_candidates_excluded\": true`.\n\n### 1.1.10h  Chabauty-Coleman at 3 and 5\n```\npython3 -I 9111_chabauty.py\n```\n`logs3 [[0,78,51],[78,15,66]]`, `logs5 [[80,60,120],[115,120,110]]`, `pairs_mod45 [[33,3],[3,13]]`,\n`ALL EXACT LOCAL ASSERTIONS PASSED; FINITE SIEVE AND RANK PREMISES REMAIN SEPARATE`; the full JSON\n(`expected/1.1.10h_9111_chabauty.json`) has the trace, annihilator, residual-trace and extra-candidate data\nlisted in the code manifest (e.g. `annihilator_mod27 (26,17,1)`, `annihilator_mod25 (11,0,1)`).\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.11: x^4 + y^3 + xy - y - 1 = 0\n\nCoordinates: `u = -x`, `v = y`, `C: G(u,v) = u^4 + v^3 - (u+1)v - 1 = 0`, `O = (0:1:0)`,\n`P+ = (1,0)`, `P- = (-1,0)` in `(u,v)`; bitangents `v = m u + b`, normalised `h_m = m u + b - v`;\n`K = Q[m]/(f)`, degree 27, `(r1,r2) = (3,12)`, `|D_K| = 2^28 13839869^6`. The descent script writes the\ncurve as `F = -G` with `(x,y) = (u,v)` and `h_m = eps (y - m x - b)`, `eps = -1`, which is the same function.\n\nRun every command from this folder.\n\n## Requirements\n\n- PARI/GP 2.15 or later (`gp`); tested with 2.15.4.\n- 1.1.11e reads `../common/h31_descent_lib/` (`descent_lib.gp`, `descent_points.gp`).\n- 1.1.11b-d: Python 3, cypari2, python-flint, numpy and `../common/h31_classgroup/`.\n\n## Codes\n\n| id | script | what it computes | run here |\n|---|---|---|---|\n| 1.1.11a | `1.1.11a_geometry_field.gp` | smoothness, bad primes 2 and `q0 = 13839869`, the node mod `q0` and its thickness, f, signature, discriminant, primes over 2 | yes, < 1 s |\n| 1.1.11b | `../common/h31_classgroup/count_ideals.py` | prime ideals of each norm `q < 5*10^8` | slice (`p <= 10^6`); full run ~1 CPU-hour |\n| 1.1.11c | `.../witnesses.py`, `.../check_witnesses.py` | Lemma C witnesses for all prime ideals of norm `<= 10^7` | slice (`p <= 3000`); full run several CPU-hours |\n| 1.1.11d | `.../bound.py` | `B^-` (64 cells, `T = log(5*10^8)`) | no (needs the full counts) |\n| 1.1.11e | `1.1.11e_descent.gp` | basis of `K(S,2)`, norm, good primes, dyadic image, rank <= 1 | yes, 17 s CPU |\n| 1.1.11f | `1.1.11f_logs.gp` + `data/1.1.11f_h.gp` | `div(h) = 21 P+ + E - 24 O`, 3-adic logarithm, unit coefficients | yes, 2 s |\n\n### 1.1.11a  Geometry, bad primes, field\n\n```\ngp -q 1.1.11a_geometry_field.gp < /dev/null\n```\nAffine singular points: `G_u = 0` gives `v = 4u^3`, then `G_v = 0` gives `48u^6 = u+1` (irreducible), and\nthe critical value is `-(5u^4+8u^3+3)/3`. Its factorisation in the critical algebra gives the node and its\nthickness; the singular locus mod 2, 3, q0 is computed from three resultants. Key lines:\n```\nRes(48u^6-u-1, 5u^4+8u^3+3) = [2, 6; 3, 1; 13839869, 1];  isprime(13839869): 1\n  bad p = 13839869: prime of the critical algebra with residue degree 1, valuation of the critical value 1, Hessian valuation 0  -> 1 ordinary node(s) of thickness 1\nnode mod q0: u = 950455 is a root of 48u^6-u-1 mod q0: 1;  v = 4u^3 mod q0 = 6126477\nintegral lift (u,v) = (950455, 4*950455^3): v_q0(G) = 1, G_u = 0, v_q0(G_v) = 1, v_q0(288u^5-1) = 0\np = 3: u-coordinates of singular points satisfy Mod(2, 3)          (no singular point: smooth at 3)\ndegree-27 factor == f: 1;  f irreducible: 1;  real roots: 3\nnfcertify (empty = maximal order certified): []\nsignature [3, 12];  disc 1886391169645809642134939302441035242972848781787136 = [2, 28; 13839869, 6]\nprimes over 2 (e,f): [[2, 1], [10, 1], [5, 1], [5, 2]]\n```\n\n### 1.1.11b, 1.1.11c, 1.1.11d  Class group\n\nParameters `data/classgroup_params.json`: f, `|D_K|`, `(3,12)`, `support = 5*10^8`,\n`principal_bound = 10^7`, the 64 window heights of the package file\n`008008_class_exact_input_500000000.json`, 224 bits. Full commands:\n```\nCG=../common/h31_classgroup; PRM=data/classgroup_params.json\npython3 -I $CG/count_ideals.py $PRM --workers W --out counts.npz                 # 1.1.11b\npython3 -I $CG/witnesses.py $PRM --workers W --out witnesses.jsonl.gz            # 1.1.11c\npython3 -I $CG/check_witnesses.py $PRM witnesses.jsonl.gz                        # 1.1.11c\npython3 -I $CG/bound.py $PRM counts.npz                                          # 1.1.11d\n```\n(Both witness scripts lift Python's 4300-digit limit for integer conversion themselves.) Slices run here (`expected/1.1.11b_slice.txt`, `1.1.11c_slice.txt`):\n```\nRANGE 2 1000000 SUPPORT 500000000 RATIONAL_PRIMES 78498 DISTINCT_NORMS 43726 PRIME_IDEALS 80457 ...\nALL 801 PRIME IDEALS OF NORM <= 10000000 OVER p IN [2, 3000] HAVE VALID WITNESSES; ...\n```\nExpected final numbers (paper and source TeX; the counts were also reproduced by an independent reviewer\ncount before this bundle was made): 26,350,721 prime ideals of norm `< 5*10^8`, of which 664,374 have norm\n`<= 10^7`; all 664,374 have witnesses; `B^- > 0.004909377030906924806352480651615121...`.\n\n### 1.1.11e  2-descent, rank <= 1\n\n```\ngp -q 1.1.11e_descent.gp < /dev/null\n```\nThe 19 classes are S-units proposed by `bnfinit`/`bnfsunit` (GRH is used only to propose them); the script\nchecks that their prime factors all lie over 2 and that they are independent (valuation parities and\nquadratic residue characters at auxiliary primes), so by Lemma 1.1.11class they form a basis of `K(S,2)`.\nLocal images at a good prime use all points of `C(F_p)`. At 2 the images of the four divisors of the text\nare computed with certified precision (`descent_points.gp`: every error term is bounded through the\nvaluations of all non-constant coefficients of the delta value as a polynomial in the error variables):\n(a) `P+ - O`, (b) `R1 - O` (`u = 5`, `v = 8s`), (c) `R2 - O` (`t = 2`), (d) `R3 + R3' - 2O` (`t = sqrt 2`).\nKey lines (`expected/1.1.11e.txt`):\n```\nnumber of S-unit classes (torsion, units, S-units): 19\nprimes above 2 (e,f): [[2, 1], [10, 1], [5, 1], [5, 2]]\nrank of the character matrix (independence): 19 of 19\nambient 19;  after the norm condition: 17\np = 487: ... -> 10       p = 491: ... -> 7       p = 499: ... -> 5       p = 503: ... -> 4\norbit sizes over 2: [2, 10, 5, 10]\nunions of orbits of size 12 or 16 (orbit indices, size): List([[[1, 2], 12], [[1, 4], 12]]);  the sum of the two has size 20 (not 12 or 16): dim J(Q_2)[2] <= 1, dim J(Q_2)/2J(Q_2) <= 4\ncertified square classes (ok, margin): (b) [1, 114]  (c) [1, 116]  (d) [1, 122]\nrank of the images of (a),(b),(c),(d): 4\nafter the dyadic condition: 1   => rank J(Q) <= 1\ndelta(P+ - O): nonzero 1;  in the final space 1\ndelta(P- - O): nonzero 0;  in the final space 1\n```\n\n### 1.1.11f  Principal divisor and 3-adic logarithm\n\n```\ngp -q 1.1.11f_logs.gp < /dev/null\n```\nInput `data/1.1.11f_h.gp`: the 22 coefficients of h, copied from the package file\n`008008_principal_n21.json` (fields `monomials`, `h_coefficients`). Key lines:\n```\nmonomials: 22;  maximal weight [24, 8, 0], next 23  => pole of order 24 at O\nRes_v(G,h) = c * [[u - 1, 21], [<irreducible cubic U>, 1]]\ngcd(G(1,v), h(1,v)) = 1/732305065375610740707623870238993*v;  G_v(P+) = -2\nG(E) = 0: 1,  h(E) = 0: 1;  21 + 3 = 24 zeros\nT(Z) = charpoly of t on E: 3-adic valuations [1, 2, 2, 0];  T mod 81 = Z^3 + 36*Z^2 + 9*Z + 60\ncharpoly of v on E: 3-adic valuations [-4, -1, 0, 0]  (one Newton edge: v3(v) = -4/3)\nomega/dt to degree 9: [-t^4 + 3*t^9 + O(t^10), t - 2*t^6 + 3*t^9 + O(t^10), 1 - 2*t^5 + 3*t^8 + O(t^10)]\npower sums p1..p9 mod 81: [45, 63, 63, 54, 27, 27, 0, 0, 0]\nTr(I0,I1,I2) mod 27: [0, 18, 9];  L = -Tr/21 mod 9: [0, 3, 6]\ntail: min over 10<=n<=26 of n/3 - v3(n) = 3; n >= 27: v3(n) <= log_3 n <= n/3 - 3\nwith 120 terms (tail valuation >= 40): v3(L) = [2, 1, 1],  L/3 mod 9 = [6, 4, 2]\nP+: omega/ds = [1/2, 1/2, 0], eta0 constant = 2 mod 3\nP-: omega/ds = [-1/4, 1/4, 0], eta0 constant = 2 mod 3\nO : omega/ds = [0, 0, 1], eta2 constant = 1 mod 3\n```\n\n## Sources\n\n1.1.11a and the dyadic part of 1.1.11e are new; the rest of 1.1.11e uses the reviewer descent library\n(`../common/h31_descent_lib/descent_lib.gp`); 1.1.11f combines reviewer checks with the data of the\npackage. 1.1.11b-d use the shared class-group tools.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.12: x^4 - y^3 + xy + x - 1 = 0\n\nCoordinates: `u = -x`, `v = y`, `C: G(u,v) = u^4 - v^3 - uv - u - 1 = 0`, `O = (0:1:0)`, `P = (0,-1)`.\nBitangents `v = m u + b`; slopes: `m = 0` (the rational bitangent `v = -1`), the roots of `f10` and of `f16`;\n`F10 = Q[m]/(f10)`, `F16 = Q[m]/(f16)`. The descent script writes the curve as `F = -G` with\n`(x,y) = (u,v)` and `h_m = y - m x - b` (= `v - m u - b`, the normalisation of the text).\n\nRun every command from this folder.\n\n## Requirements\n\n- PARI/GP 2.15 or later (`gp`); tested with 2.15.4. 1.1.12f reads `../common/h31_descent_lib/`.\n- 1.1.12b: Python 3 (standard library only).\n- 1.1.12c-e: Python 3, cypari2, python-flint, numpy and `../common/h31_classgroup/`.\n\n## Codes\n\n| id | script | what it computes | run here |\n|---|---|---|---|\n| 1.1.12a | `1.1.12a_geometry_fields.gp` | smoothness, bad primes, nodes, slope factorisation, F10/F16 invariants, primes 2, 3, 13 | yes, < 1 s |\n| 1.1.12b | `1.1.12b_orthogonal_group.py` | the group H (1920), orbits, code, invariants, `H -> S5`, Frobenius at 3, dyadic decomposition groups | yes, 22 s |\n| 1.1.12c | `../common/h31_classgroup/*` with `data/classgroup_params_F10.json` | F10, `T = 6`: all 94 prime ideals of norm `< e^6` principal; `B^-` | yes (full), 1 s |\n| 1.1.12d | `count_ideals.py` with `data/classgroup_params_F16.json --support 8900001` | F16: prime ideals of norm `<= 8.9*10^6` | yes (full), 60 s |\n| 1.1.12e | `count_ideals.py`, `witnesses.py`, `check_witnesses.py`, `bound.py` with `data/classgroup_params_F16.json` | F16, `T = 16`: all prime ideals of norm `<= 2*10^5` principal; `B^-` | yes (full): 60 s count, 16 min witnesses, 1 min check, 5 s bound |\n| 1.1.12f | `1.1.12f_descent.gp` | 2-descent with `S = {2,3}`: basis, norm, good primes, 2-adic and 3-adic images, final dimension | yes, 7 s CPU |\n| 1.1.12g | `1.1.12g_points_F2.gp` | `#C(F_2), #C(F_4), #C(F_8)`, zeta numerator, `#J(F_2)` | yes, < 1 s |\n\n### 1.1.12a  Geometry and the bitangent algebra\n```\ngp -q 1.1.12a_geometry_fields.gp < /dev/null\n```\nKey lines:\n```\ncritical algebra: ... = Mat([48*u^6 - 24*u^3 + u + 3, 1])\nfactor 48*u^6 - 24*u^3 + u + 3: norm of the critical value [2, -16; 3, -5; 5, 1; 313, 1; 14087, 1]; ...\n   p = 5: ... valuation of the critical value 1, Hessian valuation 0  -> 1 ordinary node(s) of thickness 1\n   p = 313: ... -> 1 ordinary node(s) of thickness 1\n   p = 14087: ... -> 1 ordinary node(s) of thickness 1\np = 2: no singular point (smooth fibre)\np = 3: u-coordinates of singular points: Mat([Mod(1, 3)*u, 1])\nRes_b factors (degree, multiplicity): [[1, 1], [10, 1], [16, 1]]\n   rational slope factor: m\nf10, f16 are factors: 1 1;  irreducible: 1 1;  27 = 1 + 10 + 16: 1\nF10: signature [2, 4], |disc| = [2, 8; 5, 2; 313, 2; 14087, 2];  nfcertify: []\n    p = 2: (e,f) = [[5, 1], [5, 1]]      p = 3: [[1, 1], [1, 1], [1, 4], [1, 4]]      p = 13: [[1, 1], [1, 1], [1, 2], [1, 6]]\nF16: signature [0, 8], |disc| = [2, 12; 5, 4; 313, 4; 14087, 4];  nfcertify: []\n    p = 2: (e,f) = [[5, 1], [1, 1], [5, 2]]      p = 3: [[1, 1], [1, 1], [1, 2], [1, 4], [1, 4], [1, 4]]\n```\n\n### 1.1.12b  Finite group computations\n```\npython3 -I 1.1.12b_orthogonal_group.py\n```\nUses only `F_2^6` and `q`. Key lines:\n```\n|H| = 1920\norbit sizes on 27: [1, 10, 16]\ncode weights: [(0, 1), (12, 36), (16, 27)]\ndim J2^H = 1  dim E^H = 3  dim (E/J2)^H = 2\n|image in S5| = 120\n|kernel N| = 16\nN elementary abelian: True\nevery nonzero element of N has normal closure N: True\n1.1.12 Frob3: ... : #elements 240  projected S5 types Counter({(1, 4): 240})  fixed dims on V Counter({3: 240})\ndyadic decomposition groups with orbits (5,5),(1,5,10): (order, dim V^D) counts: Counter({(20, 2): 96})\n```\n\n### 1.1.12c, 1.1.12d, 1.1.12e  Class groups of F10 and F16\n\nOne-cell windows (the triangle `f(x) = (1 - |x|/T)_+`); `bound.py --no-exp-terms` drops the two small\npositive exponential terms of the archimedean term, which gives the (slightly smaller) values stated in the\ntext. Commands:\n```\nCG=../common/h31_classgroup\n# 1.1.12c (F10, T = 6, principal_bound 403 >= every norm < e^6)\nP=data/classgroup_params_F10.json\npython3 -I $CG/count_ideals.py $P --out F10_counts.npz\npython3 -I $CG/witnesses.py $P --out F10_w.jsonl.gz\npython3 -I $CG/check_witnesses.py $P F10_w.jsonl.gz\npython3 -I $CG/bound.py $P F10_counts.npz --no-exp-terms\n# 1.1.12d (F16, all norms <= 8.9*10^6 > e^16)\nP=data/classgroup_params_F16.json\npython3 -I $CG/count_ideals.py $P --support 8900001 --out F16_counts_8900000.npz\n# 1.1.12e (F16, T = 16, principal_bound 2*10^5)\npython3 -I $CG/count_ideals.py $P --out F16_counts.npz                  # norms < e^16 (input of bound.py)\npython3 -I -X int_max_str_digits=0 $CG/witnesses.py $P --out F16_w.jsonl.gz\npython3 -I -X int_max_str_digits=0 $CG/check_witnesses.py $P F16_w.jsonl.gz\npython3 -I $CG/bound.py $P F16_counts.npz --no-exp-terms\n```\n(Some generators proposed by `bnfisprincipal` in F16 have more than 4300 digits; the witness scripts lift\nPython's conversion limit themselves, the flag `-X int_max_str_digits=0` is kept only for older copies.) Key lines (`expected/1.1.12c.txt`, `1.1.12d.txt`,\n`1.1.12e.txt`):\n```\nF10: RANGE 2 403 SUPPORT 404 ... PRIME_IDEALS 94\n     ALL 94 PRIME IDEALS OF NORM <= 403 OVER p IN [2, 403] HAVE VALID WITNESSES; ...\n     B_INTERVAL [0.08091779648991310992623240536831549010721 +/- 1.12e-42] POSITIVE True\nF16: RANGE 2 8900000 SUPPORT 8900001 RATIONAL_PRIMES 596222 DISTINCT_NORMS 293854 PRIME_IDEALS 596471 ...\n     RANGE 2 8886110 SUPPORT 8886111 ... PRIME_IDEALS 595650          (norms < e^16)\n     ALL 18074 PRIME IDEALS OF NORM <= 200000 ... HAVE VALID WITNESSES; ...\n     B_INTERVAL [0.07457215844456170870910920052554107746597 +/- 1.26e-42] POSITIVE True\n```\nThe text (Lemma 1.1.12class) states that the 94 ideals of F10 have explicit generators (cofactor 1); the\nshared tool also accepts strict Lemma C witnesses with cofactor > 1 (76 of the 94 here), which proves the\nsame statement.\n\n### 1.1.12f  2-descent, rank 0\n```\ngp -q 1.1.12f_descent.gp < /dev/null\n```\nS-unit bases proposed by `bnfinit` (GRH only proposes; support and independence are checked exactly).\nLocal points: at 2, `P` (u = 0), the 2-adic points with `u = 2`, `u = 4`, the point `t = 2` of the disk\nat O, and `R + R' - 2O` with `u(R) = sqrt 2`; at 3, `P`, `u = 1`, `u = 3`. All non-rational points are\nknown to finite precision; `descent_points.gp` certifies that the precision fixes the square classes.\nAt P the Q-component of `delta(P - O)` is `1/3` (leading coefficient of `v + 1 = u^4/3 + ...`). Key lines:\n```\nS-unit classes per component (Q, F10, F16): [3, 12, 17];  total 32\nrank of the character matrix (independence): 32 of 32\nambient 32;  after the norm condition: 29\np = 487 -> 26, 491 -> 19, 499 -> 17, 503 -> 12, 509 -> 12, 521 -> 11, 523 -> 9, 541 -> 8\ncertified (ok, margin): [[1, 58], [1, 54], [1, 58], [1, 55]]        (2-adic)  [[1, 36], [1, 35]]  (3-adic)\nrank of the five 2-adic images: 5  (= dim J(Q_2)/2J(Q_2) = 2 + 3 from 1.1.12b)\nrank of the three 3-adic images: 3  (= dim J(Q_3)[2] = 3 from 1.1.12b)\nafter the 2-adic condition: 3\nafter the 3-adic condition: 1   => rank J(Q) + dim J(Q)[2] <= 1\ndelta(P - O): nonzero 1;  in the final space 1\n```\n\n### 1.1.12g  Points over F_2, F_4, F_8\n```\ngp -q 1.1.12g_points_F2.gp < /dev/null\n```\n```\n#C(F_2), #C(F_4), #C(F_8) = [2, 4, 8]\nnumerator of Z(C/F_2, T) = 8*T^6 - 4*T^5 - T + 1;  #J(F_2) = L(1) = 4\nG(u,-1) = u^4 (4-fold contact of v = -1 at P);  G_v(P) = -3\nv + 1 = 1/3*u^4 - 1/9*u^5 + 1/27*u^6 - 1/81*u^7 + O(u^8)  at P (local parameter u)\n```\n\n## Sources\n\n1.1.12b and the descent library are reviewer code (independent review, scripts 07 and `descent_lib.gp`);\n1.1.12a, g and the point certification of 1.1.12f are new; 1.1.12c-e use the shared class-group tools.\nThe supplied package of this equation (case 3675/008114, Sage-based) is not used.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.13: x^4 + y^3 + xy + y - 1 = 0\n\nCoordinates: `v = -y`, `C: G(x,v) = x^4 - v^3 - (x+1)v - 1 = 0`, `O = (0:1:0)`,\n`P- = (-1,0)`, `P1 = (1,0)`, `P3 = (3,4)`, `Q+- = (1, alpha+-)` with `alpha^2 = -2`, `alpha+ = 1`,\n`alpha- = 2 (mod 3)`. Bitangents `v = m x + b`: the rational one `v = 2x - 2` (`m = 2`), the roots of `f10`\nand `f16`; `K10 = Q[m]/(f10)`, `K16 = Q[m]/(f16)`. Scripts d, e, f write the curve as `F = -G` with\n`y = v`; `h_m = y - m x - b`.\n\nRun every command from this folder.\n\n## Requirements\n\n- PARI/GP 2.15 or later (`gp`); tested with 2.15.4. 1.1.13d reads `../common/h31_descent_lib/`;\n  1.1.13e and 1.1.13f read `1.1.13ef_lib.gp`.\n- 1.1.13b: Python 3 (standard library). 1.1.13c: Python 3, cypari2, python-flint, numpy and\n  `../common/h31_classgroup/`.\n\n## Codes\n\n| id | script | what it computes | run here |\n|---|---|---|---|\n| 1.1.13a | `1.1.13a_geometry_fields.gp` | smoothness, bad primes, node thickness (2 at 13, 1 at 17, 43, 257), slope factorisation, K10/K16 invariants, primes 2, 3, 5, 13 | yes, < 1 s |\n| 1.1.13b | `1.1.13b_orthogonal_group.py` | S5 images of the Frobenius types at 3 and 5, dyadic decomposition groups (rest as 1.1.12b) | yes, 24 s |\n| 1.1.13c | `../common/h31_classgroup/*` with `data/classgroup_params_K10.json`, `_K16.json` | K10 (`T = 5`) and K16 (`T = 8`): all prime ideals of norm `< e^T` principal; `B^-` | yes (full), 3 s |\n| 1.1.13d | `1.1.13d_descent.gp` | 2-descent with `S = {2,13}`: 28 -> 25 -> 7 -> 2, rank <= 1 | yes, 12 s CPU |\n| 1.1.13e | `1.1.13e_principal_divisors.gp` | `h1`, `h+`, `h-` and their divisors | yes, < 1 s |\n| 1.1.13f | `1.1.13f_logs.gp` | traces of tiny integrals, log table, annihilators, disk coefficients | yes, 3 s CPU |\n\n### 1.1.13a  Geometry and the bitangent algebra\n```\ngp -q 1.1.13a_geometry_fields.gp < /dev/null\n```\nKey lines:\n```\ncritical algebra: -48*x^6 - x - 1 = Mat([48*x^6 + x + 1, 1])\nfactor 48*x^6 + x + 1: norm of the critical value [2, -10; 3, -9; 13, 2; 17, 1; 43, 1; 257, 1];  norm of the Hessian Mat([2236363, 1])\n   p = 13: ... valuation of the critical value 2, Hessian valuation 0  -> 1 ordinary node(s) of thickness 2\n   p = 17 / 43 / 257: ... -> 1 ordinary node(s) of thickness 1\np = 2: x-coordinates of singular points: Mat([Mod(1, 2)*x + Mod(1, 2), 1])\np = 3, 5, 7, 11: no singular point (smooth fibre)\np = 13: singular point (x,v) = [3, 4] mod 13;  v_13(G) at the 13-adic critical point = 2, Hessian mod 13 = 7  -> ordinary node of thickness 2\nRes_b factors (degree, multiplicity): [[1, 1], [10, 1], [16, 1]]\n   rational slope factor: m - 2\nK10: signature [2, 4], |disc| = [2, 8; 17, 2; 43, 2; 257, 2];  nfcertify: []\n    p = 2: [[5, 1], [5, 1]]   p = 3: [[1, 2], [1, 8]]   p = 5: [[1, 4], [1, 6]]   p = 13: [[1, 5], [1, 5]]\nK16: signature [0, 8], |disc| = [2, 12; 17, 4; 43, 4; 257, 4];  nfcertify: []\n    p = 2: [[5, 1], [1, 1], [5, 2]]   p = 3: [[1, 8], [1, 8]]   p = 5: [[1, 4], [1, 12]]   p = 13: [[1, 1], [1, 5], [1, 5], [1, 5]]\n```\n\n### 1.1.13b  Finite group computations\n```\npython3 -I 1.1.13b_orthogonal_group.py\n```\nPrints the same general facts as 1.1.12b (`|H| = 1920`, orbits 1, 10, 16, code weights 12 and 16,\ninvariants 1, 3, 2, `H -> S5` onto with kernel of order 16) and:\n```\n1.1.13 Frob3: K10 (2,8), K16 (8,8) ... projected S5 types Counter({(1, 4): 240}) ...\n1.1.13 Frob5: K10 (4,6), K16 (4,12) ... projected S5 types Counter({(2, 3): 160}) ...\ndyadic decomposition groups with orbits (5,5),(5,1,10): (order, dim V^D) counts: Counter({(20, 2): 96})\n```\n\n### 1.1.13c  Class groups of K10 and K16\n```\nCG=../common/h31_classgroup\nfor P in data/classgroup_params_K10.json data/classgroup_params_K16.json; do\n  python3 -I $CG/count_ideals.py $P --out counts.npz\n  python3 -I $CG/witnesses.py $P --out w.jsonl.gz\n  python3 -I $CG/check_witnesses.py $P w.jsonl.gz\n  python3 -I $CG/bound.py $P counts.npz --no-exp-terms\ndone\n```\nKey lines (`expected/1.1.13c.txt`):\n```\nK10: PRIME_IDEALS 32;  ALL 32 PRIME IDEALS OF NORM <= 148 ... HAVE VALID WITNESSES\n     B_INTERVAL [0.3436681640333619027158612633519258453768 +/- 4.00e-42] POSITIVE True\nK16: PRIME_IDEALS 465; ALL 465 PRIME IDEALS OF NORM <= 2980 ... HAVE VALID WITNESSES\n     B_INTERVAL [0.02282020346427122159168636691400815448944 +/- 4.93e-42] POSITIVE True\n```\n(`--no-exp-terms` reproduces the values of the text; with the exponential terms the bounds are a little\nlarger. As in 1.1.12c, some witnesses are strict Lemma C witnesses rather than generators.)\n\n### 1.1.13d  2-descent, rank <= 1\n```\ngp -q 1.1.13d_descent.gp < /dev/null\n```\nLocal points at 2: (a) `P1 - O` with the Q-component `1/((v+x-1)^2+3x^2-5x+4)(P1) = 1/2` (P1 lies on the\nrational bitangent; the identity is checked), (b) the 2-adic point with `x = -3`, (c) `t = 2` in the disk\nat O, (d), (e) the divisors cut out on `v = -5x-7` and `v = -5x-5` by the irreducible quadratic factor over\n`Q_2` (mod 2 the quartic is `(x+1)^2 (x^2+x+1)`, two coprime quadratics, so Hensel applies; the other\nquadratic splits over `Q_2` and gives the same class). Key lines:\n```\nS-unit classes per component (Q, K10, K16): [3, 10, 15];  total 28\nrank of the character matrix (independence): 28 of 28\nambient 28;  after the norm condition: 25\np = 487 -> 21, ..., p = 593 -> 7\nrank of the five 2-adic images: 5  (= dim J(Q_2)/2J(Q_2) = 2 + 3 from 1.1.13b)\nafter the 2-adic condition: 2   => rank J(Q) <= 2 - dim J(Q)[2] = 1\ndelta(P- - O): nonzero 0;  in the final space 1\ndelta(P1 - O): nonzero 1;  in the final space 1\ndelta(P1 + P3 - 2O): nonzero 1;  in the final space 1\n```\n\n### 1.1.13e  The principal divisors\n```\ngp -q 1.1.13e_principal_divisors.gp < /dev/null\n```\nThe functions are computed as the one-dimensional kernels of the vanishing conditions in `L(nO)` (they are\nunique up to a constant; the text takes their coefficients from the source), then every claim is checked\nexactly. Key lines:\n```\n== h1: 23 monomials, rank of the conditions 22 (kernel of dimension 1)\n   two largest weights [25, 24]  => pole of order exactly 25 at O\n   order at P- = 0,  at P1 = 22\n   Res_v(G,h) = c * [\"(x - 1)^22\"] * U,  deg U = 3;  U irreducible over Q(sqrt-2): 1;  U(1) != 0: 1,  U(-1) != 0: 1\n   residual points: F(E) = 0: 1,  h(E) = 0: 1\n== h+: ... pole of order exactly 23;  order at P- = 1,  at P1 = 18,  h(1, a) = Mod(0, a^2 + 2),  h(1, -a) = <nonzero>\n   Res_v(G,h) = c * [\"(x + 1)^1\", \"(x - 1)^19\"] * U, ...\n== h-: ... pole of order exactly 11;  order at P- = 2,  at P1 = 5,  h(1, a) = Mod(0, a^2 + 2),  h(1, -a) = <nonzero>\n   Res_v(G,h) = c * [\"(x + 1)^2\", \"(x - 1)^6\"] * U, ...\n```\n(`h-` vanishes at `(1, a)` as an element of `Q(a)[x,v]`; in 1.1.13f it is embedded with `a -> alpha-`,\nso that its zero is `Q-`.)\n\n### 1.1.13f  3-adic logarithms\n```\ngp -q 1.1.13f_logs.gp < /dev/null\n```\nKey lines:\n```\nE1: x-coordinates mod 3: Mat([x + 2, 3]);  ...\nE- (a -> alpha- = 2 mod 3): v3 of the charpoly coefficients of v: [1, 1, 1, 0]  (3-Eisenstein: v3(v) = 1/3)\nlog(E1 - 3O) mod 81 = [39, 48, 21]   (5 terms; omitted terms have valuation >= 5)\nlog(E+ - 3O) mod 81 = [69, 72, 69]   (5 terms)\nlog(E- - 3O) mod 81 = [51, 57, 33]   (18 terms; omitted terms have valuation >= 6)\nL = log(P1 - O) mod 27: [24, 15, 15];  v3(L) = [1, 1, 1]\nlog(Q+ - O) mod 27: [12, 9, 12];  log(Q- - O) mod 27: [18, 3, 0]\nA mod 9 = [5, 1, 0],  A mod 3 = [2, 1, 0],  B mod 3 = [2, 0, 1]\nA.log(Q+ - O) mod 27 = 15,  A.log(Q- - O) mod 27 = 12;  valuations 1, 1\nO  (s = t):   B.(0,0,-1) = -1 = 2 (mod 3)\nP- (s = v):   A.(1,-1,0)/(4x^3-v) = 2\nP1 (s = x-1): B.(1,1,0)/(3v^2+x+1) = 1\nP3 (s = x-3): A.(1,3,4)/(3v^2+x+1) = 2\nQ+- (s = x-1): A.(1,1,alpha)/(3alpha^2+2) mod 3 = 0, 0  (A omega = 0 mod 3 at x = 1)\n```\n\n## Sources\n\n1.1.13b, e, f and the descent library are reviewer code (independent review, scripts 07, 20, 21 and\n`descent_lib.gp`), with relative paths and the term numbers of the text (5 and 18); 1.1.13a and the point\ncertification of 1.1.13d are new; 1.1.13c uses the shared class-group tools. The supplied package of this\nequation (case 3664/008166) proves the result by a different route and is not used.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.14: x^4 + y^3 + xy + x + 1 = 0\n\nNotation of the section: v = y + 1; plane quartic C: q2^2 = q1 q3 with q1 = vz, q2 = x^2,\nq3 = -(v^2 - 3vz + xz + 3z^2); genus-5 curve D in P^4 (x:v:z:r:s) cut out by\nQ1 = vz + 3r^2, Q2 = x^2 + 3rs, Q3 = q3 + 3s^2; points P+- = (0:0:1:0:+-1);\ngenus-2 curve H: eta^2 = f(t) = 3t^6 - 18t^5 + 36t^3 + 6t, J = Jac(H), G = (t^2 - 24t, 1729/2 t) = [B0 - A0]\nwith A0 = (0,0), B0 = (24, 20748).\n\n## Software and how the outputs were obtained\n\n| Software | Used for | Command (run inside this folder) |\n|---|---|---|\n| Magma | `*.m` (codes 1.1.14b, c) | paste the file into Magma (or `magma -b file.m`) |\n| PARI/GP 2.15.4 | `*.gp` | `gp -q -f FILE.gp < /dev/null` |\n| Python 3.11 + SymPy 1.14 | `1.1.14a_symbolic.py`, `1.1.14c_check_f7.py`, `1.1.14c_check_twist.py`, `1.1.14f_series.py` | `python3 -I FILE.py` |\n| Sage (the modular `sagemath-schemes` distribution, Python 3.11) | `1.1.14b_check_smooth.py`, `1.1.14c_check_jac7.py` | `python3 -I FILE.py` with the Sage interpreter |\n\nThe Sage scripts start with `from sage.all__sagemath_schemes import *`; in a full Sage installation\n`from sage.all import *` can be used instead.\n\n**Magma.** Magma is not installed on the machine where this folder was assembled. The Magma scripts were run\non the public Magma calculator (V2.29-10) on 14 October 2026, each in under 60 s. The files in `expected/`\nfor Magma scripts are the outputs of those runs, or the original output shipped with the source, as stated\nfor each script below. In our own runs a first line `MAGMA_GETVERSION 2 29 10` and a last line `CPU ...`\nwere printed by two statements added at submission (`GetVersion()`, `Cputime()`); they are not part of\nthe scripts.\n\n**PARI, Python, Sage.** Every such script in this folder was run here (October 2026) and its output is in\n`expected/<script name>.txt`. Each run took at most about 45 s (most of it is the Sage import). Every output\nagrees with the expected values listed below; for the reviewer scripts it is identical, line by line, to\nthe log of the original review run.\n\n## Files\n\n| Code | File | Software | Run here |\n|---|---|---|---|\n| 1.1.14a | `1.1.14a_symbolic.py` | Python + SymPy | yes, pass |\n| 1.1.14a | `1.1.14a_extra.gp` | PARI/GP | yes, pass |\n| 1.1.14b | `1.1.14b_prym_lpoly.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.1.14b | `1.1.14b_check_smooth.py` (cross-check) | Sage | yes, pass |\n| 1.1.14c | `1.1.14c_certificate.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.1.14c | `1.1.14c_check_f7.py`, `1.1.14c_check_twist.py` (cross-checks) | Python | yes, pass |\n| 1.1.14c | `1.1.14c_check_jac7.py` (cross-check) | Sage | yes, pass |\n| 1.1.14e | `1.1.14e_beta.gp` | PARI/GP | yes, pass |\n| 1.1.14f | `1.1.14f_series.py` | Python + SymPy | yes, pass |\n\n\"Cross-check\" scripts are independent re-computations without Magma of facts that the Magma script of the\nsame code asserts. The proof uses the Magma script; the cross-checks are optional.\n\n## 1.1.14a: exact identities and local expansions (no Magma)\n\n`1.1.14a_symbolic.py` (from the source package, `TBTWO_25_symbolic_check.py`; a block of checks for two other\nequations was removed) asserts, in exact arithmetic:\n- F(x, v - 1) = x^4 + v(v^2 - 3v + x + 3);\n- the d = 9 identity 3AB - (4uw + 1) = 4 E9 with A = 2w^2 - 6u^4 - 1, B = 2w^2 + 6u^4 + 1,\n  E9 = 3w^4 - 27u^8 - 9u^4 - uw - 1;\n- 12 det(M(q1) - 2t M(q2) + t^2 M(q3)) = f(t) (3x3 symmetric matrices of q1, q2, q3);\n- f(24) = 20748^2, v_7(20748) = 1, f'(24) = 1 mod 7, (20748/7) mod 7 = 3;\n- the expansions of s, r, v at P+ in the parameter x, the coefficients a0, a1, a2 of the tangent\n  polynomial, the small root t1 = 2/3 x^2 + O(x^3), t1 + t2 = 24 + O(x^2), f(2x^2/3) = 4x^2 + O(x^3).\n\nExpected output (`expected/1.1.14a_symbolic.txt`, identical to the output shipped with the source):\n```\nALL SYMBOLIC CHECKS PASSED\nLocal expansions at P_+:\ns = x**3/432 - x**2/72 + x/6 + 1 + O(x^4)\nr = -x**4/72 + x**3/18 - x**2/3 + O(x^5)\nv = x**5/9 - x**4/3 + O(x^6)\nPrym polynomial: 3*t**6 - 18*t**5 + 36*t**3 + 6*t\n```\n\n`1.1.14a_extra.gp` adds: the factorisation and discriminant of f; P+ lies on D and w = (1,0,0,0,1/6) is\ntangent there; with the full 5x5 matrices of Q1, Q2, Q3 the tangent polynomial at P+ is t^2/12 - 2t;\ndisc(t^2 + t + 4) = 6 mod 7 is not a square; the Jacobian matrix of (Q1,Q2,Q3) with respect to (v,r,s) at P+\n(chart z = 1) has determinant 18; the fixed points of sigma; 1729 * 12 = 20748.\nExpected output (`expected/1.1.14a_extra.txt`):\n```\nfactor f: [t, 1; t^5 - 6*t^4 + 12*t^2 + 2, 1]\ndisc f: [-1, 1; 2, 6; 3, 10; 19, 1; 149, 1; 3461, 1]\nf(24) == 20748^2: 1  f'(24) mod 7: 1\n12*det(M(q1)-2tM(q2)+t^2M(q3)) == f: 1\nP+ on D: [0, 0, 0]  tangent: [0, 0, 0]\na_{P+}(t) = 1/12*t^2 - 2*t\ndisc(t^2+t+4) mod 7 = 6  is square mod 7: 0\nJacobian (v,r,s) at P+: [1, 0, 0; 0, 3, 0; 3, 0, 6]  det: 18\nsigma fixed points: r=s=0 gives [x^2, z*v, -z*x + (-v^2 + 3*z*v - 3*z^2)]\n1729*12 == 20748: 1\n```\n\n## 1.1.14b: smoothness of D and the Prym identification by L-polynomials (Magma)\n\n`1.1.14b_prym_lpoly.m` (our independent check, 14 Oct 2026) computes the determinant identity from the three\nquadrics alone and q1 q3 - q2^2 (minus the homogenised equation of C); then for p = 5, 7, 11, 13, 17 it checks\nthat D mod p and C mod p are nonsingular and Disc(f) != 0 mod p, that D has genus 5, and that\nL_p(D) = L_p(C) L_p(H). This equality confirms Prym(D/C) ~ J over Q including the quadratic twist.\nIt also prints #J_H(F_p) = L_H(1).\n\nExpected output (`expected/1.1.14b_prym_lpoly.txt`, the complete output of the 14 Oct 2026 calculator run):\n```\n12*det = 3*T^6 - 18*T^5 + 36*T^3 + 6*T\n equals f: true\nq1*q3-q2^2 = -x^4 - x*v*z^2 - v^3*z + 3*v^2*z^2 - 3*v*z^3\np 5 genus D 5  L(D)=L(C3)L(H): true  #J_H(F_p) = 23\np 7 genus D 5  L(D)=L(C3)L(H): true  #J_H(F_p) = 32\np 11 genus D 5  L(D)=L(C3)L(H): true  #J_H(F_p) = 163\np 13 genus D 5  L(D)=L(C3)L(H): true  #J_H(F_p) = 172\np 17 genus D 5  L(D)=L(C3)L(H): true  #J_H(F_p) = 292\n```\n(plus `Cputime` lines; total CPU 7.9 s.)\n\nCross-check without Magma: `1.1.14b_check_smooth.py` (Sage) shows that D is smooth over Q and over F_7 (the\nsingular locus of the affine cone is the vertex only) and that the Hilbert polynomial of D is 8t - 4\n(degree 8, genus 5). Expected output (`expected/1.1.14b_check_smooth.txt`):\n```\np = 7  dim(D affine cone) = 2   dim(singular locus cone) = 0 (<=0 means D smooth)\np = 0  dim(D affine cone) = 2   dim(singular locus cone) = 0 (<=0 means D smooth)\n genus check via Hilbert polynomial: 8*t - 4\n```\n\n## 1.1.14c: 2-Selmer group, finite Jacobians, sieve modulo 7 (Magma)\n\n`1.1.14c_certificate.m` (the certificate of the source package, `TBTWO_25_certificate.m`, unchanged) asserts:\n#TwoSelmerGroup(J) = 2 (class groups proved with the Minkowski bound via `SetClassGroupBoundMaps`, no GRH);\nG != 0; f(24) = 20748^2; v_7(20748) = 1; v_7(f'(24)) = 0; Disc(f) != 0 mod 5 and mod 7; #J(F_5) = 23;\n#J(F_7) = 32 (also by enumerating all points); G mod 7 has order 2; no T in J(F_7) with 2T = G mod 7;\nD mod 7 is nonsingular with 8 points. For each point of D(F_7) it takes a tangent vector w from the kernel\nof the 3x5 matrix, forms the monic tangent polynomial a(u) = w M1 w^T - 2u w M2 w^T + u^2 w M3 w^T with the\nfull 5x5 matrices, lists ALL points of J(F_7) with first Mumford polynomial a, and prints whether one of\nthem is 0 or G mod 7. It asserts that exactly two points (those with x = v = 0) are allowed.\n\nExpected output (`expected/1.1.14c_certificate.txt`; this is the transcription of the output shipped with\nthe source package, from a run on Magma V2.29-10 on 12 Sep 2026, one row per point; the order of the rows\nmay differ between runs, the set of rows may not). Our rerun on the public calculator (V2.29-10,\n14 Oct 2026, 0.6 s) passed every assertion and printed the same eight rows and the final line.\n```\n[ 0, 0, 1, 0, 1 ]    u^2 + 4*u       true\n[ 0, 0, 6, 0, 1 ]    u^2 + 4*u       true\n[ 2, 1, 4, 1, 1 ]    u^2 + u + 4     false\n[ 4, 3, 5, 4, 1 ]    u^2 + u + 4     false\n[ 3, 4, 2, 4, 1 ]    u^2 + u + 4     false\n[ 5, 6, 3, 1, 1 ]    u^2 + u + 4     false\n[ 0, 3, 6, 1, 0 ]    u^2 + u + 4     false\n[ 0, 4, 1, 1, 0 ]    u^2 + u + 4     false\nALL ARITHMETIC CHECKS PASSED\n```\nMeaning: the coordinates are (x, v, z, r, s) of the point of D(F_7); the polynomial is the monic tangent\npolynomial; `true` means that some point of J(F_7) with this first polynomial is 0 or G mod 7. Only P+ and\nP- (x = v = 0) give `true`.\n\nCross-checks without Magma (each independent of the others):\n- `1.1.14c_check_f7.py` (plain Python): the 8 points of D(F_7) (normalised with first nonzero coordinate 1),\n  smoothness at each, and the tangent polynomial for EVERY tangent vector (it is the same up to scalars).\n  Expected (`expected/1.1.14c_check_f7.txt`): `#D(F7) = 8`, two rows `{(1, 4, 0)}` (= u^2 + 4u, the points\n  P+-) and six rows `{(1, 1, 4)}` (= u^2 + u + 4).\n- `1.1.14c_check_jac7.py` (Sage): #J(F_7) = 32 by enumeration, G mod 7, 2G = 0 mod 7, no solution of\n  2T = G mod 7. It works in the odd-degree model t = 1/u, so the printed polynomials differ from Magma's.\n  Expected (`expected/1.1.14c_check_jac7.txt`):\n  ```\n  #J(F7) via frobenius poly: 32\n  G7 = (x + 2, y)  2*G7 = (1)\n  #enumerated = 32\n  solutions of 2T = G7 in J(F7): 0\n  2-torsion count: 8\n  ```\n- `1.1.14c_check_twist.py` (plain Python): at p = 11, 13, 17, 19, for every point of D(F_p) with a2 != 0,\n  the roots t_i of the tangent polynomial give values f(t_i) that are squares in F_p(t_i) (consistency of\n  the twist of H). Expected (`expected/1.1.14c_check_twist.txt`): the column \"inconsistent\" is 0 at every p.\n\n## 1.1.14e: the 7-adic logarithm of G and beta (PARI)\n\n`1.1.14e_beta.gp` computes the 7-adic roots of f, takes the root alpha = 3 mod 7, and integrates\nomega0 = dt/(2 eta) and omega1 = t dt/(2 eta) from W = (alpha, 0) to B0 = (24, 20748) in the local parameter\neta (series to 40 terms, 7-adic precision 7^25); then the valuations of l0, l1, beta = l1/l0, and the\n7-integrality of the series coefficients. Expected output (`expected/1.1.14e_beta.txt`):\n```\nf(24) == 20748^2: 1  v7(20748)=1  f'(24) mod 7 = 1\ndisc f mod 5,7: 4 6  factor disc: [-1, 1; 2, 6; 3, 10; 19, 1; 149, 1; 3461, 1]\n7-adic roots of f: [0, 37, 24, 27]~\nalpha mod 7^4 = 1641\nv7 of f'(alpha): 0\nv7(l0)=1  v7(l1)=1\nbeta mod 7^6 = 53924   beta mod 7 = 3\ncheck series coefficients of w0, w1 are 7-integral (min valuation): 0 0\nf(0)=0\n```\n(The roots are printed mod 49.)\n\n## 1.1.14f: exact expansions at P+- and the pulled-back differentials (Python)\n\n`1.1.14f_series.py` computes, for both branches s(0) = +1 and s(0) = -1, to order 18 in x with exact rationals:\nv, r, s; the tangent coefficients with w = (1, v', 0, r', s'); the two roots t1 (small) and t2 (near 24);\neta1/x and eta2; the four series S_i_k = t_i^k t_i'/(2 eta_i) (k = 0, 1) that make up phi^*omega0 and\nphi^*omega1; and the 7-adic valuations of the coefficients up to order 13 (99 means the coefficient is 0).\nExpected output (`expected/1.1.14f_series.txt`; branch -1 flips the signs of s and r and is otherwise the same):\n```\nbranch s(0)=1\n s: ['1', '1/6', '-1/72', '1/432', '1723/10368']\n r: ['0', '0', '-1/3', '1/18', '-1/72', '5/1296']\n v: ['0', '0', '0', '0', '-1/3', '1/9', '-1/27']\n min v7 of coeffs (v,r,s): 0\n a_p at x=0: 0 -2 1/12\n t1: ['0', '0', '2/3', '-1/9', '1/24']\n t2: ['24', '0', '0', '-1729/9', '12103/216']\n ...\n eta1/x: ['2', '-1/6', '1/18', '-1/72']  eta2: ['20748', '0', '0', '-18921601/36']\n  S1_0 first coeffs: ['1/3', '-1/18', '1/36']  v7: [0, 0, ...]\n  S1_1 first coeffs: ['0', '0', '2/9']  ...\n  S2_0 first coeffs: ['0', '0', '-1/72']  ...\n  S2_1 first coeffs: ['0', '0', '-1/3']  ...\n```\nAll listed valuations are >= 0. Meaning: phi^*omega0 has constant term +-1/3 and phi^*omega1 has constant\nterm 0, so phi^*(omega1 - beta omega0) has constant term -+beta/3, a 7-adic unit.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.15: x^4 - x^2 + xy + y^3 = 0\n\nNotation of the section: for x != 0, u = -y/x, v = -x; genus-2 curve C: v^2 + u^3 v = u + 1, equivalently\nw^2 = u^6 + 4u + 4 (w = 2v + u^3); O, E the two points at infinity (v ~ -u^3 at O, v ~ u^-2 at E);\nintegral model of U = C \\ {O}: y^2 = xz, x^3 - x + y(z+1) = 0, z^2 + z = y(1 - x^2);\nknown points A = (-1,-1,-1), B = (-1,0,0), C1 = (-1,1,-1), D = (0,0,0), E = (0,0,-1), F1 = (1,0,0),\nG = (2,-2,2), H = (4,-6,9) (the scripts call C1, F1 simply C, F); differentials\nomega1 = dx/(1+3z), omega2 = (y/x) omega1, eta1 = 2y omega1, eta2 = x omega1; p = 2.\n\nThe four files are the solver's scripts, unchanged and under their original names. The section's proof is\nthe original working notes (its file names `code/q9_*.m`, `code/q9_*.py` are the files of this folder).\n\n## Software and how the outputs were obtained\n\n| Software | Used for | Command (run inside this folder) |\n|---|---|---|\n| Magma V2.29-10 | `q9_rank_geometry.m` (1.1.15a), `q9_qc2.m` (1.1.15d) | paste the file into Magma (or `magma -b FILE.m`) |\n| Python 3 + SymPy | `q9_symbolic.py` (1.1.15b) | `python3 -I q9_symbolic.py` |\n| Python 3, standard library only | `q9_exact_mod2.py` (1.1.15c) | `python3 -I q9_exact_mod2.py` |\n\n**Magma.** Magma is not installed on the machine where this folder was assembled. The files\n`expected/q9_rank_geometry.txt` and `expected/q9_qc2.txt` are the recorded outputs of the solver's runs on\nthe Magma calculator (V2.29-10, 15 October 2026), with the solver's header and closing notes. Both scripts were\nrerun unchanged on the Magma calculator (V2.29-10) on 16 October 2026 and reproduced the recorded outputs\nexactly (0.36 s and 0.71 s).\n\n**Python.** Both Python scripts were run here on 16 October 2026 with `python3 -I` (Python 3.11, SymPy 1.14),\nfrom a separate working folder; their outputs `expected/q9_symbolic.json` and `expected/q9_exact_mod2.json`\nare byte-identical to the solver's recorded outputs. Run times: 1 s and 0.2 s.\n\n## Files\n\n| Code | File | Software | Run here |\n|---|---|---|---|\n| 1.1.15a | `q9_rank_geometry.m` | Magma | no (Magma calculator; recorded output, reproduced exactly on 16 Oct 2026) |\n| 1.1.15b | `q9_symbolic.py` | Python + SymPy | yes, identical |\n| 1.1.15c | `q9_exact_mod2.py` | Python (standard library) | yes, identical |\n| 1.1.15d | `q9_qc2.m` | Magma | no (Magma calculator; recorded output, reproduced exactly on 16 Oct 2026) |\n\n## 1.1.15a: model, bad fibre and rank bound (Magma)\n\n`q9_rank_geometry.m` sets both class-group proof-bound maps to the Minkowski bound\n(`SetClassGroupBoundMaps`), so the 2-descent does not depend on GRH, and prints: the discriminant of the\ngenus-2 model `HyperellipticCurve(t+1, t^3)` (that is, v^2 + u^3 v = u + 1); the upper bound\n`RankBound(Jacobian(HyperellipticCurve(t^6+4*t+4)))` (asserted to be 2); the genus; over F_8539,\ngcd(f, f') and gcd(f, f', f'') for f = u^6 + 4u + 4; #C(F_2) and #J(F_2). No search for points is made.\nExpected output (`expected/q9_rank_geometry.txt`):\n```\ndiscriminant -8539\nrank_upper_bound 2\ngenus 2\nbad_fibre_gcd u + 1709\nordinary_node_check 1\nF2_curve_points 5\nF2_Jacobian_order 13\n```\n\n## 1.1.15b: exact algebra (SymPy)\n\n`q9_symbolic.py` asserts: f(-v, uv) = v^2 (v^2 + u^3 v - u - 1); the eight known points satisfy the plane\nequation and the three relations of the integral model; the bidifferential identity of Step 3 (the numerator\nof LHS - RHS, with the curve relations for both points, has Groebner remainder 0); the orders of zero of the\nsix functions x, y, y - x, x + 1, x + y, z - 2x - 1 at the known points (from exact local expansions), and that\ntheir sums are the pole orders 3, 4, 4, 3, 4, 5 at O, which proves the six divisors; the residue vectors of\nlog(f / c t^k) eta at O. Expected output (`expected/q9_symbolic.json`), key entries:\n```\n\"kernel_groebner_remainder\": 0,\n\"principal_divisor_zero_orders\": x: D 1, E 2 | y: B 1, D 1, E 1, F 1 | y-x: A 1, D 2, E 1 |\n                                 x+1: A 1, B 1, C 1 | x+y: C 1, D 1, E 1, G 1 | z-2x-1: A 3, C 1, H 1\n\"eta_reciprocity_residues\": [[0,0],[0,0],[-1,-1],[0,0],[-1,1],[-4,0]],\n\"all_symbolic_checks_passed\": true\n```\n\n## 1.1.15c: the finite 2-adic certificate (standard library)\n\n`q9_exact_mod2.py` works in Z/2^40 Z with power series of length 64 (only 2-adic units are inverted; the\n2-part of n is split off before dividing by n). It builds the four residue disks (parameters y+1, x+1, x, x/y),\ncomputes the tiny integrals B -> F1 and D -> G of omega and eta, the period formulas\nI_E = (d1 + 2 d2)/13, I_F1 = (7 d1 + d2)/13, J_E = (e1 + 2 e2 + (4,0))/13, J_F1 = (7 e1 + e2 + (2,0))/13,\nthe values K(P, .) for P = D, F1 from the tiny integrals of nu_P with the poles subtracted, the matrix\nB_h (asserting its symmetry modulo 2^40), v_2(det L) = 3 (asserted), M = L^-1 B_h L^-T modulo 2^36, and the\ncoefficients c_1..c_7 of q(2u) on each disk modulo 2^16, asserting that 2^-e q(2u) reduces to u + u^2.\nIt checks finite arithmetic only: the height theory, the rank bound and the infinite tail are in the text.\nExpected output (`expected/q9_exact_mod2.json`), key entries:\n```\n\"I_E_mod64\": [2, 0], \"I_F_mod64\": [2, 22], \"J_E_mod64\": [60, 44], \"J_F_mod64\": [62, 30],\n\"M_mod64\": [[56, 19], [19, 53]], \"height_symmetry_mod_2N\": true, \"v2_log_matrix_determinant\": 3,\ndisc A: e 2, q_2u_coefficients_mod_32 [20, 28, 16, 16, 16, 0, 0], \"u + u^2\"\ndisc B: e 3, [24, 24, 0, 16, 0, 0, 0], \"u + u^2\"\ndisc D: e 2, [20, 12, 0, 16, 16, 0, 0], \"u + u^2\"\ndisc E: e 2, [20, 4, 8, 0, 16, 0, 0], \"u + u^2\"\n\"all_finite_checks_passed\": true\n```\n\n## 1.1.15d: the same computation in Magma (70 bits)\n\n`q9_qc2.m` repeats the computation of 1.1.15c in Magma's 2-adic field with precision 70 and Laurent series of\nlength 95, and prints I_E, I_F, J_E, J_F, the valuation of the asymmetry of B_h, v_2(det L), M, and for each\ndisk the minimal valuation e of the coefficients of q(2u), the reduction of 2^-e q(2u), the valuation of the\ntruncated tail and the valuation of q at the second known point. Expected output (`expected/q9_qc2.txt`):\n```\nsymmetry valuation 71\nlog determinant 3\nM\n[13126274611143697095*2^3 + O(2^68) -70808205619149878381 + O(2^68)]\n[-70808205619149878381 + O(2^68) -69287664107145450635 + O(2^68)]\ndisk 1 minimum 2 reduction u^2 + u        (disks 1-4 = A, B, D, E)\ndisk 2 minimum 3 reduction u^2 + u\ndisk 3 minimum 2 reduction u^2 + u\ndisk 4 minimum 2 reduction u^2 + u\nknown second root value 70                (on each disk)\n```\nThe printed `tail` lines concern only the truncated series; the bound for the infinite tail is a lemma of the\nsection. The values of I_E, I_F, J_E, J_F and M agree with 1.1.15c modulo 64 (and, in an independent\ncomputation modulo 2^320 made during the verification, to the full 68-72 bits printed here).\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.2: x^4 − y^3 + x − y = 0\n\nProof coordinates: `u = −y`, `z = x`, `C: u^3 + u + z^4 + z = 0` (the scripts write `F(u,z)`, or\n`y^3 + y + x^4 + x` with `(x, y) = (z, u)`). Bitangent field `K = Q[m]/(f)`,\n`f = m^27+480m^19−1344m^18−34560m^11−70656m^10−40704m^9−16384m^3+12288m−4096`.\n\nRequirements: PARI/GP ≥ 2.15 (`gp`); Python 3 with cypari2, python-flint, numpy (shared class-group\ntools `../common/h31_classgroup/`). No Sage, no Magma. Run everything from this folder.\n\"Run here\" = run on 14 Oct 2026 (PARI 2.15.4, python-flint 0.9.0), output in `expected/`.\n\n| id | command | time | run here |\n|---|---|---|---|\n| 1.1.2a | `gp -q 1.1.2a_bitangent.gp`; `gp -q 1.1.2a_field.gp`; `gp -q 1.1.2a_nodes.gp`; `gp -q 1.1.2a_descent.gp` | 10 s | yes, matches |\n| 1.1.2b | `1.1.2bcd_classgroup.sh` (stage b) | ≈ 20 min (1 worker) | slice p ≤ 2·10^6 |\n| 1.1.2c | `1.1.2bcd_classgroup.sh` (stage c) | ≈ 2 h (1 worker) | slice p ≤ 20000 |\n| 1.1.2d | `1.1.2bcd_classgroup.sh` (stage d) | 1 min | yes, on a full count (see below), matches |\n| 1.1.2e | `gp -q 1.1.2e_logs.gp`; `gp -q 1.1.2e_disks.gp` | 1 s | yes, matches |\n\n## 1.1.2a — curve, bitangent field, descent\n* `1.1.2a_bitangent.gp`: the slope resultant of `u = mz + b`; its degree-27 factor is `f`; `f`\n  irreducible with 3 real roots.\n* `1.1.2a_field.gp`: proven factorization of `disc(f)`; `nfinit` with it; `|D| = 2^24·277^6·66943^6`,\n  signature (3,12); one prime above 2, `(e,f) = (9,3)`; decompositions of 3, 277, 66943.\n* `1.1.2a_nodes.gp`: at 277 and 66943 a unique singular point (from `gcd` of the critical equations),\n  an ordinary node with unit Hessian and critical value of valuation 1 (thickness 1).\n* `1.1.2a_descent.gp` (reads `descent_common.gp`): `bnfinit`/`bnfsunit` only *propose* `−1`, 14 units and\n  one S-unit for `S = {2}`; the script checks integrality, norms ±1, support above 2, and\n  independence mod squares (valuation at 𝔭_2 plus 60 split-prime residue characters: rank 16).\n  Since `Cl(K)[2] = 0` (1.1.2b–d), `dim K(S,2) = 15 + 1 = 16`, so these 16 elements are a basis.\n  Then: norm-square condition (sign, v_2), local images at 487, 491, 499, 503 from all points of\n  `C(F_p)` (one-zero coordinates filled in from the norm; points with two zeros would abort), and the\n  dyadic image of 50 points of `C(Q_2)` (Hensel to 2^300) through unit square classes modulo `𝔭^19`.\n  (The source checks the independence with residue characters at 487, …, 541 instead; either suffices.)\n\nExpected and obtained:\n```\ndeg27 factor == TeX f: 1 / irreducible: 1 / real roots (signature r1): 3\nfield disc: [2, 24; 277, 6; 66943, 6]  sign 1 / TeX claim ... match |D|: 1 / signature: [3, 12] / primes above 2: [[9, 3]]\np=277 ... v_p(F(crit)) = thickness = 1 ;  p=66943 ... thickness = 1 ; gcd for singular z: linear\nrank of character matrix mod 2: 16 (need 16)\nafter norm-square: dim 14 ; candidate dim now 10, 6, 4, 3 (p = 487, 491, 499, 503)\n(O/pr^19)^* ... => dim K_v*/sq = 29 ; dim span of dyadic images = 3\nfinal Selmer upper bound dim (after norm, 487..503, dyadic): 2\n```\n\n## 1.1.2b, c, d — class group (shared tools)\n`data/classgroup_params.json`: `f`, `|D|`, (3,12), `principal_bound` 5·10^6, `support` 10^8, the 256\npositive integer window heights of the source (`001074_class_exact_input.json`).\n`./1.1.2bcd_classgroup.sh` (set `W` = number of workers) runs:\n* **b** `count_ideals.py`: all prime ideals of norm `< 10^8` (idealprimedec in the certified maximal order\n  for `p | disc f` or `p^2 < 10^8`, root counts `deg gcd(x^p − x, f)` otherwise). Expected:\n  `RATIONAL_PRIMES 5761455 ... PRIME_IDEALS 5763020`.\n* **c** `witnesses.py` + `check_witnesses.py`: a strict witness (Lemma C) for every prime ideal of norm\n  `≤ 5·10^6`, checked exactly against an independent enumeration. Expected:\n  `ALL 349551 PRIME IDEALS OF NORM <= 5000000 ... HAVE VALID WITNESSES`.\n* **d** `bound.py`: Theorem B with `T = log 10^8`, 256 cells, `k(q) = a(q)` for `q ≤ 5·10^6`, 224-bit\n  Arb balls. Expected: `B = [0.000875203229047890536437443663523468836... ]`, positive.\n\nRun here:\n* b, slice `count_ideals.py data/classgroup_params.json --hi 2000000`: `PRIME_IDEALS 149980` over the\n  148933 primes `p ≤ 2·10^6` (equal to our earlier full count restricted to these primes);\n* c, slice `--lo 2 --hi 20000` for both programs: `WITNESSES 2701` and `ALL 2701 PRIME IDEALS OF NORM <=\n  5000000 OVER p IN [2, 20000] HAVE VALID WITNESSES` (all prime ideals of residue degree ≥ 2 lie here);\n* d, full: `bound.py` on a complete count of all 5,763,020 prime ideals of norm < 10^8 made earlier\n  (14 Oct 2026) by our independent counter with the same method (root counts + idealprimedec),\n  converted to the `.npz` format: `B_INTERVAL [0.0008752032290478905364374436635234688366555 +/- 3.20e-44]\n  POSITIVE True`, `prime_ideals_proved_principal = 349551`. The same earlier checker also verified all\n  349,551 supplied witnesses of the source (176,756 + 172,795 in two halves).\n\n## 1.1.2e — principal divisors, 3-adic logarithms, annihilator, disks\n* `1.1.2e_logs.gp`: for `P_0 = (0,0)`, `P_1 = (0,−1)` the function `h_i ∈ L(31∞)` with order 28 at `P_i`\n  (kernel of a 28 × 29 linear system), `Res_u(F, h_i) = c_i (z − z(P_i))^28 d_i(z)` with `d_i` an\n  irreducible cubic, its 3-adic factorization, tiny integrals from the fibre point to each point of `E_i`\n  (70 terms, precision 3^60), trace, division by −28; cross product and primitive annihilator.\n* `1.1.2e_disks.gp`: `F_3`-points, the chart at infinity and its differential identity, and `λ·ω` at\n  the centres of the four disks for `λ = (2,1,1)`.\n\nExpected and obtained: `log(P0-oo) mod 3^10 = [57309, 48831, 7839]`,\n`log(P1-oo) mod 3^10 = [40674, 56691, 3162]`, `cross product valuations: [2, 2, 2]  primitive annihilator\nmod 3: [2, 1, 1]`; `lambda.omega const` `2, 2, 1` at the three affine disks and `2` at infinity.\n\n## Origin\nReviewer scripts written independently of the source (`.gp`); shared tools of this bundle for b–d.\n"
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    {
      "path": "anc/code/1.1.3/README.md",
      "size_bytes": 6302,
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.3: x^4 + y^3 + xy + 1 = 0\n\nCurve `C: F(x,y) = x^4 + y^3 + xy + 1 = 0` (no change of coordinates), `P0 = (0,-1)`, `P1 = (1,-1)`.\nBitangent field `K = Q[m]/(p_K)`, degree 27.\n\nRequirements: PARI/GP ≥ 2.15 (`gp`); Python 3 with cypari2, python-flint, numpy (for the shared\nclass-group tools in `../common/h31_classgroup/`). No Sage, no Magma.\nAll `.gp` files are run from this folder (`descent_common.gp` is read by two of them).\n\"Run here\" means: run on 14 Oct 2026 with PARI 2.15.4 / python-flint 0.9.0; output in `expected/`.\n\n| id | command | time | run here |\n|---|---|---|---|\n| 1.1.3a | `gp -q 1.1.3a_geometry.gp` ; `gp -q 1.1.3a_field.gp` | < 1 s | yes, matches |\n| 1.1.3b | `./1.1.3bc_classgroup.sh` (stages b) | 1 min | yes, matches |\n| 1.1.3c | `./1.1.3bc_classgroup.sh` (stage c) | 1 s | yes, matches |\n| 1.1.3d | `gp -q 1.1.3d_descent.gp` | 15 s | yes, matches |\n| 1.1.3e | `gp -q 1.1.3e_chabauty.gp` | < 1 s | yes, matches |\n\n## 1.1.3a — geometry and bitangent field\n`1.1.3a_geometry.gp`: the two points lie on `C`; no singular point in characteristic 0; the primes of\nbad reduction divide `3·144^4 − 5^5 = 17^2 · 4463467`; the chart at infinity `t^4 + w + t w^2 + w^4`\nis smooth at the hyperflex; slope polynomial `p_K` of the 27 bitangents `y = mx + b` by resultant\nelimination (`descent_common.gp`, function `slopepoly`), irreducible, 3 real roots.\n`1.1.3a_field.gp`: proven factorization of `disc(p_K)`, `nfinit` with it, field discriminant,\nsignature, decomposition of 2, 17 (and 3); unique ordinary node at 17 and 4463467 with unit Hessian and\ncritical value of valuation 2 resp. 1 (thickness).\n\nExpected key lines:\n```\n3*144^4-5^5 = 1289941963 = [17, 2; 4463467, 1]\nslope polynomial: m^27 - 48*m^22 + 144*m^17 + 17280*m^15 + 832*m^12 + 64512*m^10 - 2560*m^7 + 27648*m^5 - 110592*m^3 - 1024*m^2 + 4096\nirreducible: 1  real roots: 3\nfield disc: [2, 20; 4463467, 6]  matches 2^20*4463467^6: 1 sign 1\nsignature [3, 12]\nprimes above 2 (e,f): [[5, 1], [1, 1], [5, 1], [5, 1], [1, 1], [5, 2]]\nprimes above 17 (e,f): (9 primes)\np=17 ... thickness v_p(F(crit)) = 2 ;  p=4463467 ... thickness v_p(F(crit)) = 1\n```\n(`disc(p_K) = 2^312·17^12·4463467^6·212145823^2·219591733740230763014941^2`.)\n\n## 1.1.3b — prime-ideal counts and principal generators; 1.1.3c — the bound B^-(f)\nUses the shared tools (`../common/h31_classgroup/`, see its README) with `data/classgroup_params.json`\n(`p_K`, `|D| = 2^20·4463467^6`, signature (3,12), `C = 10^4`, support `10^6`, the 64 positive\ninteger window heights of the source). `./1.1.3bc_classgroup.sh` runs\n1. `count_ideals.py`: all prime ideals of norm `< 10^6` with their norms (idealprimedec in the\n   certified maximal order for `p ≤ 1000` and `p | disc(p_K)`, root counts of `p_K mod p` otherwise);\n2. `witnesses.py --direct`: for each prime ideal of norm `≤ 10^4` a generator proposed by\n   `bnfisprincipal` (only a proposer);\n3. `check_witnesses.py`: enumerates the prime ideals of norm `≤ 10^4` itself and checks for each a\n   generator: integral, in the ideal, `|N(g)| = N𝔭` exactly;\n4. `bound.py`: Theorem B with `n = 27`, `(3,12)`, `T = log 10^6`, the 64-cell window, `k(q) = a(q)` for\n   `q ≤ 10^4` and `0` above, in 224-bit Arb balls (exact rational knots, closed-form archimedean part).\n\nExpected (and obtained):\n```\nRANGE 2 999999 SUPPORT 1000000 ... PRIME_IDEALS 78493 ...\nWITNESSES 1260 STRICT 0 DIRECT 1260 ...\nALL 1260 PRIME IDEALS OF NORM <= 10000 ... HAVE VALID WITNESSES; COFACTOR_ONE 1260 ...\nprime_power_terms = 42743\narchimedean = \"[-0.790199426987521289824921086290 +/- 3.34e-31]\"\nprime_sum = \"[0.854417698530167955376961340582 +/- 4.70e-31]\"\nB_INTERVAL [0.06421827154264666555204025429213639168155 +/- 1.93e-42] POSITIVE True\n```\nThese agree with the source records (`verified_principal_ideals 1260`, `prime_ideals_in_support\n78493`, `PARTIAL_WINDOW_INTERVAL lower 0.0642182715426466655520402542921363916815...`).\nThe intermediate files are written to `work/` (about 1.5 MB; not included).\n\n## 1.1.3d — 2-descent, rank ≤ 2\n`1.1.3d_descent.gp`: `bnfinit`/`bnfsunit` only propose 15 units and 15 S-units for\n`S = {primes above 2, 17}`; the script checks integrality, norms ±1 and the support itself, and\nindependence mod squares (rank 30 of valuation parities + 80 residue characters). Then: norm-square\ncondition (sign, v_2, v_17), local images at 487, 491, 499, 503, 509, 521 from all points of\n`C(F_p)` (a single zero coordinate at a bitangent contact is filled in from the norm condition; points\nwith two zero coordinates would abort), dyadic image from 63 points of `C(Q_2)` (Hensel to 2^300) via\nunit square classes modulo `𝔭^{2e+1}`.\n(The class group is not taken from `bnfinit`: `Cl(K)[2] = 0` is 1.1.3b/c.)\n\nExpected (and obtained): `rank mod 2 of [valuations | characters]: 30 (need 30)`,\n`after norm-square: dim 27`, candidate dims `25, 22, 16, 14, 12, 7` (no multi-zero points),\n`dim span of dyadic images = 5`, `final 2-Selmer upper bound (norm, 6 good primes, dyadic): 2`.\n\n## 1.1.3e — 3-adic Chabauty–Coleman\n`1.1.3e_chabauty.gp`: the line `y = x/3 − 1` (`F = x^3 (27x+1)/27`, point `Q = (−1/27, −82/81)`,\n`t(Q) = 3/82`), tiny integral in the chart at infinity, `L0 = −log(Q)/3`; tangent line at `P1`\n(`F = (x−1)^2 (64x^2+101x+63)/64`), its 3-adic roots, tiny integrals `T` in the disks of `P0, P1`;\ncross product, `λ`, constant coefficients of `λ·ω` in the two disks; `F_3`-points.\n\nExpected (and obtained): `L0 mod 9 = [0, 6, 1]   v_3(L0) = [4, 1, 0]`, `a0 mod 9 = 0  a1 mod 9 = 7`,\n`s(A) mod 9 = 3  s(B) mod 9 = 6`, `T mod 9 = [0, 6, 0]`,\n`cross product L0 x T mod 9 = [3, 0, 0] valuations [1, 2, 3]`, `lambda mod 3 = [1, 0, 0]`,\n`lambda.omega constant coeff: P0 0  P1 0 (0 = unit)`, `affine F_3 points: List([[0, 2], [1, 2]])`.\n\n## Origin\nReviewer scripts written independently of the supplied package (`.gp` files); shared Python tools\nof this bundle for 1.1.3b/c. The supplied Sage scripts (`002304_*`) are not needed.\n"
    },
    {
      "path": "anc/code/1.1.4/README.md",
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.4: x^4 + y^3 + 2x − y = 0\n\nModel used in the section: `C: v^4 − u^3 + u + 2v = 0` with `(x, y) = (v, −u)`; the scripts use the\nsource letters `y^4 − x^3 + x + 2y` (`x = u`, `y = v`). Points (model): `P0 = (0,0)`, `P1 = (1,0)`,\n`P−1 = (−1,0)`, `P* = (16,−8)`. Bitangent lines `u = m v + η`; field `K = Q[m]/(f1)`,\n`f1 = m^27−480m^19+2688m^18−34560m^11−141312m^10−162816m^9+16384m^3−49152m+32768` (`polys.gp`),\nintercept `η = eta1` (`eta.gp`). Source tag of the equation: 4230.\n\nRequirements: PARI/GP ≥ 2.15 (`gp`); Python 3 with cypari2, python-flint, numpy (shared class-group tools\n`../common/h31_classgroup/`). No Sage, no Magma. Run everything from this folder.\n\"Run here\" = run on 14 Oct 2026 (PARI 2.15.4, python-flint 0.9.0), output in `expected/`.\nThe four `1.1.4a_*.gp` scripts are reviewer scripts that treat this equation and that of 1.1.5\n(source tag 4227, polynomial `g2`) together; the lines for 4230/`f1`/`K1` are the ones for this section.\n\n| id | command | time | run here |\n|---|---|---|---|\n| 1.1.4a | `gp -q 1.1.4a_basic.gp`; `gp -q 1.1.4a_bitangent.gp`; `gp -q 1.1.4a_field.gp`; `gp -q 1.1.4a_nodes.gp` | 3 s | yes, matches |\n| 1.1.4b | `1.1.4bcd_classgroup.sh` (stage b) | ≈ 3 h (1 worker) | slice p ≤ 2·10^6 |\n| 1.1.4c | `1.1.4bcd_classgroup.sh` (stage c) | ≈ 1.5 h (1 worker) | slice p ≤ 20000 |\n| 1.1.4d | `1.1.4bcd_classgroup.sh` (stage d) | 4 min | yes, on a full count (see below), matches |\n| 1.1.4e | `gp -q 1.1.4e_selmer.gp`; `gp -q 1.1.4e_dyadic_points.gp` | 5 s | yes, matches |\n| 1.1.4f | `gp -q 1.1.4f_divisors.gp`; `gp -q 1.1.4f_residual.gp`; `gp -q 1.1.4fg_logs_even.gp` | 2 s | yes, matches |\n| 1.1.4g | `gp -q 1.1.4fg_logs_even.gp` (writes `work/logs4230.txt`), then `gp -q 1.1.4g_odd.gp` | 2 s | yes, matches |\n\n## 1.1.4a — geometry and bitangent field\n`1.1.4a_basic.gp`: transport identity, target points, `Res = 14332523 = 83·172681` (bad primes);\n`1.1.4a_bitangent.gp`: the bitangent conditions `8D − 4BC + B^3 = 0`, `64E − (4C − B^2)^2 = 0` for\n`x = my + n`, resultant in `n` = `f1`, irreducible, 3 real roots; `1.1.4a_field.gp`: `nfinit` with the proven\nprime factors of `disc(f1)`, `nfcertify = []`, `disc(K1) = 2^28·83^6·172681^6`, signature, primes over 2,\n83, 172681; `1.1.4a_nodes.gp`: the singular point `(2187/128, −243/32)`, `F = −3482803089/2097152`\nof valuation 1 at 83 and 172681, unit Hessian, a unique singular point mod p (thickness 1).\nExpected and obtained: `N1=14332523 [83, 1; 172681, 1]`, `4230 deg27 factor matches f: 1`,\n`irreducible f1: 1`, `polsturm f1 real roots: 3`, `K1 sig [3, 12] disc [2, 28; 83, 6; 172681, 6]\n2326889515985953907282417842991419362237373569040384`, `certify K1: []`,\n`K1 primes over 2 (e,f): [[10, 1], [16, 1], [1, 1]]`, `p=83: v(F)=1 ... v(Hess)=0`, same at 172681,\n`gcd degree in y at 83: 1` and at 172681.\n\n## 1.1.4b, c, d — class group (shared tools)\n`data/classgroup_params.json`: `f1`, `|D|`, (3,12), `principal_bound` 2,020,000, `support` 5·10^8, the 64\npositive integer window heights of the source (`4230_original_window.json`).\n`./1.1.4bcd_classgroup.sh` (set `W` = number of workers) runs `count_ideals.py` (b),\n`witnesses.py` + `check_witnesses.py` (c) and `bound.py` (d) (see `../common/h31_classgroup/README.md`).\nExpected full results: b `RATIONAL_PRIMES 26355867 ... PRIME_IDEALS 26362584`; c\n`ALL 150219 PRIME IDEALS OF NORM <= 2020000 ... HAVE VALID WITNESSES`; d\n`B = [0.000100160932045204380585899904968878437 +/- ...]`, i.e. `1.00157·10^-4 < B^- < 1.00165·10^-4`.\n\nRun here:\n* b, slice `--hi 2000000`: `PRIME_IDEALS 151209` over the primes `p ≤ 2·10^6` (= our earlier full count\n  restricted to these primes);\n* c, slice `--lo 2 --hi 20000`: `WITNESSES 2430`, `ALL 2430 PRIME IDEALS OF NORM <= 2020000 OVER p IN\n  [2, 20000] HAVE VALID WITNESSES` (this range contains all prime ideals of residue degree ≥ 2);\n* d, full: `bound.py` on a complete count of all 26,362,584 prime ideals of norm < 5·10^8 made earlier\n  (14 Oct 2026) by our independent counter with the same method, converted to `.npz`:\n  `B_INTERVAL [0.0001001609320452043805858999049688784370622 +/- 4.35e-44] POSITIVE True`, inside the\n  source's interval `[0.00010015717418099256, 0.0001001646903988629]`. The same earlier checker verified\n  all 150,219 supplied witnesses of the source (`... 150219 prime ideals proved principal by single\n  witnesses; 150219 witnesses all used`).\n\n## 1.1.4e — 2-descent, rank ≤ 2; dyadic independence of G0, G1\n`1.1.4e_selmer.gp` (reads `polys.gp`, `eta.gp`, `selmer_lib.gp`, and `data/4230_S_unit_representatives.gp`\n= the 18 elements of the source's descent input; no `bnfinit`): support of every element above 2,\nindependence (rank 18 of the dyadic square classes in the three completions above 2, total dimension\n33); since `Cl(K)[2] = 0`, `dim K(S,2) = 15 + 3 = 18` and they are a basis. Then the norm condition, local\nimages at 487, …, 523 from all points of `C(F_p)` (one-zero points completed from the norm), the dyadic\nclasses of the local points `(0,0), (1,0), (−1,0), (16,−8)`, `(x_A, 2)`, the three points with `y = 4`, and\ntwo points in the chart at infinity (Hensel to 2^80), unit groups modulo `𝔭^{2e+1}`.\n`1.1.4e_dyadic_points.gp`: the four points of the text (`x_A`, `w_A` mod 2^36) and their rank.\nExpected and obtained: `basis elements supported on the primes above 2: 1`, `K(S,2) basis size: 18`,\n`independence rank of basis via dyadic classes: 18`, `after norm: dim 16`, global dims\n`12, 9, 6, 4, 4, 4, 4` (p = 487, …, 523), `dyadic image rank: 4`, `FINAL fake Selmer dim (norm+good primes+2): 2`,\n`(0,0) ... cumulative rank 1`, `(1,0) ... cumulative rank 2`;\n`xA mod 2^36 = 29760316588`, `wA mod 2^36 = 59134464016`, `rank of four source points: 4`, `rank of G0,G1: 2`.\n\n## 1.1.4f — 2-adic logarithms; 1.1.4g — chart coefficients\n`1.1.4f_divisors.gp`: `div(x+2y)`, `div(y)`, the function `g10 ∈ L(10O)` with order 7 at `P1` and its residual\ncubic `c(y)`; `1.1.4f_residual.gp`: `c` irreducible, Newton polygon at 2, 2-adic factorization (one linear,\none quadratic factor), `x` as a function of `y` on `c`; `1.1.4fg_logs_even.gp`: even chart (`v = 2t`)\ncoefficients and tail bound, `log G0`, the points `Q1` (odd chart of `P1`) and `Q2, Q3` (chart at infinity),\n`log G1`, the annihilator mod 8 and `Δ`; `1.1.4g_odd.gp`: the two odd charts `ε = ±1`.\nExpected and obtained: `f10 = -13043*x^2 + (27304*y^2 + 22862*y + 19496)*x + (-5780*y^3 - 47008*y^2 - 16272*y - 6453)`,\n`valuation at P1: 7`, `newton polygon slopes 2: [3, -9/2, -9/2]`; `even chart t-coeffs (deg1..4) for omega_0,1,2:\n[2, 0, 32, 0] [0, 2, 0, 48] [0, -4, 0, -128]`, `even tail bound n-v(n) holds: 1`, `v2(log G0) = [1, 3, 4]`,\n`v2(x(Q1)-1) = 3`, `inf forms integral: 1`, `v2(log G1) = [-1, 1, 1]`, `a/a1 mod 8 = [4, 1, 2]  (valuations [2, 0, 1])`,\n`v2(Delta) = 2`; for `eps = ±1`: `v bound j-1: 1  Dinv bound j+1: 1`, `v(C_eps) = 1`,\n`vals of [t^1..t^8]H: [0, 0, 5, 4, 5, 8, 9, 9]`, `I0/(-2) = t + (coeffs >=1): 1   I2/(-2) = eps t + (>=1): 1`.\n\n## Origin\nReviewer scripts written independently of the source; the 18 descent elements are the source's\n(checked here); shared tools of this bundle for b–d.\n"
    },
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      "path": "anc/code/1.1.5/README.md",
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.5: x^4 + y^3 + 2x + y = 0\n\nProof coordinates: `u = y`, `z = x`, `C: F(u,z) = u^3 + u + z^4 + 2z = 0`, point at infinity `O`; solutions\n`(u,z) = (0,0), (−16,8)`. The reviewer scripts use the letters `X = u`, `Y = z` (`X^3 + X + Y^4 + 2Y`) or\n`x = u`, `y = z`. Bitangent field `K = Q[m]/(g2)`,\n`g2 = m^27+480m^19−2688m^18−34560m^11−141312m^10−162816m^9−16384m^3+49152m−32768` (`polys.gp`),\nintercept `eta2` (`eta.gp`). Source tag of the equation: 4227.\n\nRequirements: PARI/GP ≥ 2.15 (`gp`); Python 3 with cypari2, python-flint, numpy (shared class-group tools\n`../common/h31_classgroup/`). No Sage, no Magma. Run everything from this folder.\n\"Run here\" = run on 14 Oct 2026 (PARI 2.15.4, python-flint 0.9.0), output in `expected/`.\nThe four `1.1.5a_{basic,bitangent,field,nodes}.gp` scripts treat this equation and that of 1.1.4\n(source tag 4230) together; the lines for 4227/`g2`/`K2` are the ones for this section.\n\n| id | command | time | run here |\n|---|---|---|---|\n| 1.1.5a | `gp -q 1.1.5a_basic.gp`; `gp -q 1.1.5a_bitangent.gp`; `gp -q 1.1.5a_field.gp`; `gp -q 1.1.5a_nodes.gp`; `python3 1.1.5a_code_weights.py`; `python3 1.1.5a_quadpts.py`; `gp -q 1.1.5a_selmer.gp` | 10 s | yes, matches |\n| 1.1.5b | `1.1.5bcd_classgroup.sh` (stage b) | ≈ 20 min (1 worker) | slice p ≤ 2·10^6 |\n| 1.1.5c | `1.1.5bcd_classgroup.sh` (stage c) | ≈ 3 h (1 worker) | slice p ≤ 20000 |\n| 1.1.5d | `1.1.5bcd_classgroup.sh` (stage d) | 1 min | yes, on a full count (see below), matches |\n| 1.1.5e | `gp -q 1.1.5ef_logs.gp`; `gp -q 1.1.5e_hP_check.gp` | 1 s | yes, matches |\n| 1.1.5f | `gp -q 1.1.5ef_logs.gp` (first part) | 1 s | yes, matches |\n\n## 1.1.5a — geometry, bitangent field, bad prime, descent dimensions\n* `1.1.5a_basic.gp`: transport `x^4+y^3+2x+y` at `(x,y) = (z,u)` equals `F(u,z)`, both points lie on `C`,\n  `N2 = 14365291` (= `243^3 + 2^14`, prime) is the only bad odd prime.\n* `1.1.5a_bitangent.gp`: the degree-27 factor of the bitangent resultant is `g2`; irreducible; 3 real roots.\n* `1.1.5a_field.gp`: `nfinit` with the proven prime factors of `disc(g2)`, `nfcertify = []`,\n  `disc = 2^28·14365291^6`, signature (3,12), primes over 2 `(e,f) = (10,1),(16,1),(1,1)`.\n* `1.1.5a_nodes.gp`: identity `12F − 4uF_u − 3zF_z = 8u + 18z`; at `(−2187/128, 243/32)` the valuations of\n  `(F, F_u, F_z, Hess)` at `q = 14365291` are `(1,1,1,0)`; the singular point mod q is unique.\n* `1.1.5a_code_weights.py`: `Q = v1v2+v3v4+v5v6+v5+v6`, 27 nonzero isotropic vectors, word weights {12, 16},\n  the code map is injective.\n* `1.1.5a_quadpts.py`: Hensel lifts (mod 2^60) of points of `C` over quadratic rings `Z_2[s]`, written to\n  `work/quadpts4227.gp` (one lift, `s^2 = −1`, fails by design: non-unit derivative).\n* `1.1.5a_selmer.gp` (reads `polys.gp`, `eta.gp`, `selmer_lib.gp`, `data/4227_S_unit_representatives.gp`\n  = the 18 representatives of the source's descent input, and `work/quadpts4227.gp`; no `bnfinit`):\n  support of each representative above 2, independence (rank 18 of the dyadic square classes, total\n  dimension 33; with `Cl(K)[2] = 0` this makes them a basis of `K(S,2)`), norm condition, local images at\n  487, …, 523 from all points of `C(F_p)`, dyadic classes of rational, chart-at-infinity and quadratic\n  trace points (unit groups modulo `𝔭^{2e+1}`).\n\nExpected and obtained: `N2=14365291 Mat([14365291, 1])`, `4227 deg27 factor matches g: 1`,\n`irreducible ... g2: 1`, `polsturm ... g2: 3`, `K2 sig [3, 12] disc [2, 28; 14365291, 6]`, `certify K2: []`,\n`K2 primes over 2 (e,f): [[10, 1], [16, 1], [1, 1]]`, `4227: ... v_q(G)=1 v(Gu)=1 v(Gz)=1 v(Hess)=0`,\n`identity 12F-4uFu-3zFz = 8*x + 18*y`, `gcd degree: 1`, `weights: [12, 16] injective: True`;\n`K(S,2) basis size: 18`, `basis elements supported on the primes above 2: 1`,\n`independence rank of basis via dyadic classes: 18`, `after norm: dim 16`, global dims `11, 9, 6, 4`\n(p = 487, 491, 499, 503; then 4, 4, 4 at 509, 521, 523), `dyadic image rank: 4`,\n`FINAL fake Selmer dim (norm+good primes+2): 2`.\n\n## 1.1.5b, c, d — class group (shared tools)\n`data/classgroup_params.json`: `g2`, `|D|`, (3,12), `principal_bound` 6·10^6, `support` 10^8, the 64\npositive integer window heights of the source (`4227_original_window.json`).\n`./1.1.5bcd_classgroup.sh` (set `W` = number of workers) runs `count_ideals.py` (b),\n`witnesses.py` + `check_witnesses.py` (c) and `bound.py` (d).\nExpected full results: b `RATIONAL_PRIMES 5761455 ... PRIME_IDEALS 5764533`; c\n`ALL 412122 PRIME IDEALS OF NORM <= 6000000 ... HAVE VALID WITNESSES`; d\n`B = [0.000226857116386139171700439051770555690... ]` (inside the source's interval\n`[0.0002268562956377895, 0.00022685793705901072]`).\n\nRun here:\n* b, slice `--hi 2000000`: `PRIME_IDEALS 149426` over the primes `p ≤ 2·10^6` (= our earlier full count\n  restricted to these primes);\n* c, slice `--lo 2 --hi 20000`: `WITNESSES 2468`, `ALL 2468 PRIME IDEALS OF NORM <= 6000000 OVER p IN\n  [2, 20000] HAVE VALID WITNESSES` (this range contains all prime ideals of residue degree ≥ 2);\n* d, full: `bound.py` on a complete count of all 5,764,533 prime ideals of norm < 10^8 made earlier\n  (14 Oct 2026) by our independent counter with the same method, converted to `.npz`:\n  `B_INTERVAL [0.0002268571163861391717004390517705556908477 +/- 2.06e-44] POSITIVE True`.\n  The same earlier checker verified all 412,122 supplied witnesses of the source.\n\n## 1.1.5e — principal divisors, 2-adic logarithms, annihilator; 1.1.5f — the integer disk\n`1.1.5ef_logs.gp`: the `P`-disk series `u(z) ∈ Z_2[[z]]` and the coefficients of `∫(1,z,u)dz/(3u^2+1)`\nin `t` (`z = 2t`) with the tail bound `v_2 ≥ n − v_2(n)` (1.1.5f); `div(u+2z) = 3P + R − 4O`; `L_P`;\n`h_D ∈ L(31O)` (29 monomials, 1-dimensional kernel over `Q(ζ)`), `Res_u(F, h_D) = c(z^2−z+1)^14 q_D(z)`,\n`q_D` irreducible with root valuations `2, 1, −6`; `L_D` from the tiny integrals; the annihilator.\n`1.1.5e_hP_check.gp`: `L_P` once more by the source's route `div(h_P) = 28P + E_P − 31O` (chart over `Q_2(i)`).\nExpected and obtained: `omega_0 t-coeffs 1..4: [2, 0, -32, 0]`, `omega_1 ...: [0, 2, 0, -48]`,\n`omega_2 ...: [0, -4, 0, 128]`, `tail bound n-v(n): 1`, `u+2z restricted: [y - 8, 1; y, 3]`,\n`v2(L_P) = [1, 3, 4]   L_P mod 2^16: [510, 3064, 32784]`, `dim L(31O) basis: 29`, `kernel dim: 1`,\n`Res factors: [[2, 14], [3, 1]]`, `irreducible: 1`, `qD root valuations (newton): [2, 1, -6]`, `gcd deg 1`,\n`v2(L_D) = [0, 0, 2]   L_D mod 2^16: [12669, 59033, 38772]`, `annihilator normalized a2=1, mod 8: [0, 4, 1]`;\n`L_P via h_P route mod 2^16: [510, 3064, 32784]  (u+2z route gave [510, 3064, 32784])`.\n(The zeros `t = 0` and `t = 4`, i.e. `z = 0, 8`, are the two known points; with `λ ≡ (0,4,1) mod 8`\nonly the `t^2` coefficient has valuation 2, as stated in the text.)\n\n## Origin\nReviewer scripts written independently of the source; the 18 descent representatives are the\nsource's (checked here); shared tools of this bundle for b–d.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.6: x^4 + y^3 + y^2 + x = 0\n\nProof coordinates: `u = y`, `z = x`, `C: F(u,z) = u^3 + u^2 + z^4 + z = 0`, point at infinity.\nRational points `(u,z)`: `∞, P0 = (0,0), P1 = (0,−1), A = (−1,0), B = (−1,−1), R = (−3,2)`.\nBitangent field `K = Q[m]/(f)`,\n`f = m^27−160m^19−1344m^18+1280m^15−3840m^11+23552m^10−40704m^9+4096m^7−8192m^6−4096m−4096`.\nSource tag of the equation: 4564 (internal case 001457).\n\nRequirements: PARI/GP ≥ 2.15 (`gp`); Python 3 with cypari2, python-flint, numpy; for 1.1.6f a Sage with\nfunction-field Jacobians (`from sage.all__sagemath_schemes import *`, e.g. passagemath). No Magma.\nRun everything from this folder. \"Run here\" = run on 14 Oct 2026 (PARI 2.15.4, python-flint 0.9.0,\npartial passagemath), output in `expected/`. The reviewer scripts `1.1.6a_{field,transport,bitangent}.gp`\nand `count.gp` treat this equation and that of 1.1.7 (source tag 4569) together.\n\n| id | command | time | run here |\n|---|---|---|---|\n| 1.1.6a | `python3 1.1.6a_selmer.py 4564 data/001457_descent_exact_input.json`; `gp -q 1.1.6a_field.gp`; `gp -q 1.1.6a_transport.gp`; `gp -q 1.1.6a_bitangent.gp` | 20 s | yes, matches |\n| 1.1.6b | `1.1.6bc_classgroup.sh` (stage b) | ≈ 30 min (1 worker) | slice p ≤ 20000 |\n| 1.1.6c | `1.1.6bc_classgroup.sh` (stage c); `gp -q 1.1.6c_B0.gp` | 10 s | yes, matches |\n| 1.1.6d | `gp -q 1.1.6d_logs.gp`; `gp -q 1.1.6df_counts.gp` | 3 s | yes, matches |\n| 1.1.6e | `gp -q 1.1.6e_zero_tree.gp` | 3 s | yes, matches |\n| 1.1.6f | `python3 1.1.6f_sieve23.py 4564 23` (Sage); `gp -q 1.1.6df_counts.gp` | 1 min | yes, matches |\n\n## 1.1.6a — geometry, bitangent field, bad primes, descent dimensions\n* `1.1.6a_transport.gp`: transport `(x,y) = (z,u)`, the five target points; lines `u = −z−1`, `z = 0`,\n  `z = −1`; `Res(81z+16, 4z^3+1) = 515057 = 601·857` (bad primes).\n* `1.1.6a_bitangent.gp`: bitangent conditions for `u = mz + n`, resultant in `n`; its degree-27 factor is `f`\n  (case 4564), irreducible, 3 real roots.\n* `1.1.6a_field.gp`: proven factorization of `disc(f)`, `nfinit`, `disc(K) = 2^24·3^6·601^6·857^6`, (3,12),\n  `nfcertify = []`.\n* `1.1.6a_selmer.py` (cypari2; reads `data/001457_descent_exact_input.json` = the source's field polynomial,\n  intercept `η` and 24 S-unit candidates, all rechecked; no `bnfinit`): exact bitangent identities at `η`;\n  support of the 24 candidates on the 9 primes above {2,3}; rank 24 under valuations and residue\n  characters; norm condition → 21; local images at 487, …, 523 from all points of `C(F_p)` → 18, 15, 12,\n  10, 8, 6, 5; the unique prime above 2 `(e,f) = (9,3)`, three 2-adic points (Hensel to 2^60), square\n  classes on `(O_K/𝔭^19)^*` → final dimension 2; the known points lie in the final space.\n  It writes `work/selmer_4564_G.json` (the 5 remaining basis vectors).\n\nExpected and obtained: `bitangent identities exact zero: True True`, `candidates 24`,\n`primes above S: 9`, `all candidates supported above S: OK`, `rank ...: 24 of 24`,\n`after norm conditions: dim 21`, `remaining dim 18, 15, 12, 10, 8, 6, 5`, `primes above 2: [(9, 3)]`,\n`local point images: rank 3`, `dimension after 2-adic condition: 2`,\n`images of P0 P1 independent mod squares: True  final dim: 2`; `nf disc = [2, 24; 3, 6; 601, 6; 857, 6]\nsign=[3, 12]`, `nfcertify: []`; `4564: F(v,u) == M(u,v): 1`, `Res(81z+16,4z^3+1)=515057`;\n`4564 deg27 factor == claimed f: 1`, `irreducible: 1`, `signature (real roots): 3`.\n\n## 1.1.6b — generators of all prime ideals of norm ≤ 450000; 1.1.6c — the bound B_0 (T = 13)\nShared tools (`../common/h31_classgroup/`) with `data/classgroup_params.json` (`f`, `|D|`, (3,12),\n`principal_bound` 450000, `T` = 13, one cell `[1]`, 192 bits). `./1.1.6bc_classgroup.sh` runs\n`witnesses.py --direct` (generators proposed by `bnfisprincipal`) and `check_witnesses.py` (b: the prime\nideals of norm ≤ 450000 enumerated independently, each generator integral, in the ideal, `|N(g)| = N𝔭`),\nthen `count_ideals.py` (all prime ideals of norm `< e^13`) and `bound.py --no-exp-terms` (c):\n`B_0 = γ + log 4π + α − π²/(4T) − 2α log 2/T − log|D|/27 + (4/27) Σ_{N𝔭^m < e^13} log N𝔭/(N𝔭^m+1) (1 − m log N𝔭/T)`,\nin Arb balls (the condition `N𝔭^m < e^13 = 442413.39…` is decided exactly).\n`1.1.6c_B0.gp` is a short independent PARI check of the same number (60 digits).\n\nExpected: b `ALL 37489 PRIME IDEALS OF NORM <= 450000 ... HAVE VALID WITNESSES; COFACTOR_ONE 37489`;\nc `PRIME_IDEALS 36932` (norm < e^13), `prime_power_terms_with_multiplicity = 37100`,\n`B_INTERVAL [0.02648130591267555881624709231690509647393 +/- ...] POSITIVE True`;\n`1.1.6c_B0.gp`: `ideals<=450000: 37489 terms 37100 B0 = 0.026481305912675558816247092316905032306`.\nRun here: c in full (above values obtained); b on the slice `--lo 2 --hi 20000`: all 2410 prime ideals\nof norm ≤ 450000 over these primes (all those of residue degree ≥ 2 among them) have checked generators\n(`expected/1.1.6b.txt`; the full run, ≈ 37489 × 50 ms here, was not repeated). Our earlier independent checker also verified all\n37,489 generators supplied by the source (`checked 37489 missing/invalid 0`).\n\n## 1.1.6d — principal divisors, tiny integrals, 2-adic logarithms, #J(F_2)\n`1.1.6d_logs.gp` (reads `tinylog_run.gp`, `tinylog.gp`): for `P_i = (0,0), (0,−1)` the function\n`h_i ∈ L(38∞)` (36 monomials) with order 35 at `P_i`, `Res_u(F, h_i) = c_i (u − u(P_i))^35 p_i(u)` with\n`p_i` an irreducible cubic, its 2-adic factorization and disk centres, tiny integrals (power sums of the\nroots, 160 terms), `log D_i = −(trace)/35`. `1.1.6df_counts.gp` (reads `count.gp`): `#C(F_{2^k})`, `k ≤ 3`,\n`L_2(T)`, `#J(F_2)`; and the same at 23 (for 1.1.6f).\nExpected and obtained: `log D (mod 2^30): [703361398, 924118332, 517389630]` and\n`[516396498, 887777362, 803609192]` (≡ the source's `LOG [0, 0] [15495542, 1371452, 14073150]`,\n`LOG [0, -1] [13080018, 15362130, 15080040]` mod 2^24), `4564 p=2: [5, 9, 5] #J=35`.\n\n## 1.1.6e — dyadic zero tree\n`1.1.6e_zero_tree.gp` (reads `tinylog.gp`, `tinylog_run.gp`, `chab2.gp`): cross product of the two logs\n(common valuation 2) and the primitive annihilator; in the five disks (centres `P0, P1, A, B, ∞`; `t = 2s`)\nthe series of the annihilating integral, the ball partition (each ball has 0 or exactly 1 zero), the\nHensel roots `s mod 2^14` and the coefficient pairs `(c0, c1) mod 128` from the logs at the roots.\nExpected and obtained: `cross product common valuation: 2`, annihilator `[1, 13277403, 4930309]` mod 2^24\n(= the source's `[699041, 3772347, 3371045]` mod 2^22 after normalization), `half-log vectors mod 2:\n[1, 0, 1] [1, 1, 0]`; zeros `P0: s = 0 (1,0), 11613 (106,48)`; `P1: 0 (0,1), 10130 (124,35)`;\n`A: 0 (126,0), 7586 (64,104), 4595 (120,79), 16383 (4,127)`; `B: 0 (0,126)`; `∞: 0 (0,0), 529 (5,65)`;\n`total zeros: 11`. The roots `s = 0` are the centres `P0, P1, A, B, ∞`, and `s = 16383 = −1` in the disk\nof `A` is `R = (−3, 2)`; the five new zeros `11613, 10130, 7586, 4595, 529` have\n`70c0 + c1 mod 128 = 44, 11, 104, 31, 31`.\n\n## 1.1.6f — sieve modulo 23\n`gp -q 1.1.6df_counts.gp`: `4564 p=23: [22, 538, 12616] L=12167*T^6 - 1058*T^5 + 138*T^4 + 140*T^3 + 6*T^2\n- 2*T + 1 #J=11392 = [2, 7; 89, 1]`. `1.1.6f_sieve23.py 4564 23` (Sage, function-field Jacobian over\n`F_23`): `H1 = 89·[P1 − ∞]` has order 128, `H0 = 89·[P0 − ∞] = 70·H1`, the set `𝓡` of `k` with\n`89·[Q − ∞] = k·H1` over the 22 points `Q ∈ C(F_23)` equals the source's 19-element set\n`{0,1,2,4,6,8,23,24,33,35,52,58,70,99,101,116,117,126,127}`, and the values `44, 11, 31, 104` of the extra\nzeros are not in it. Expected: `order of H1 = 89*D1: 128`, `H0 = k*H1 with k = [70]`,\n`matches supplied R: True`, `extra-zero value 44 in R: False` (same for 11, 31, 104).\n\n## Origin\nReviewer scripts written independently of the source; the descent candidates and `η` are the\nsource's (rechecked); shared tools of this bundle for b, c.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.7: x^4 + y^3 − y^2 + x = 0\n\nProof coordinates: `u = −y`, `z = −x`, `C: F(u,z) = u^3 + u^2 − z^4 + z = 0` (= −(target polynomial)),\npoint at infinity `O`. Rational points `(u,z)`: `O, (0,0), (0,1), (−1,0), (−1,1), (1,−1), (6,4)`;\n`G1 = [(0,1) − O]`, `G2 = [(1,−1) − O]`. Bitangent field `K = Q[m]/(f)`,\n`f = m^27−160m^19−1344m^18−1280m^15−3840m^11+23552m^10−40704m^9−4096m^7+8192m^6−4096m−4096`.\nSource tag of the equation: 4569 (internal case 001448). All numbers are in `(u,z)`.\n\nRequirements: PARI/GP ≥ 2.15 (`gp`); Python 3 with cypari2, python-flint, numpy; a C compiler; for\n1.1.7h a Sage with function-field Jacobians (`from sage.all__sagemath_schemes import *`). No Magma.\nRun everything from this folder. \"Run here\" = run on 14 Oct 2026 (PARI 2.15.4, python-flint 0.9.0, partial\npassagemath), output in `expected/`. The reviewer scripts `1.1.7a_{field,transport,bitangent}.gp` treat\nthis equation and that of 1.1.6 (source tag 4564) together.\n\n| id | command | time | run here |\n|---|---|---|---|\n| 1.1.7a | `python3 1.1.7a_selmer.py 4569 data/001448_descent_exact_input.json '[[1,0],[2,0],[0,1]]'`; `python3 1.1.7a_normdim.py data/001448_descent_exact_input.json 2,3,5`; `gp -q 1.1.7a_field.gp`; `gp -q 1.1.7a_transport.gp`; `gp -q 1.1.7a_bitangent.gp` | 30 s | yes, matches |\n| 1.1.7b | `1.1.7bc_classgroup.sh` (stage b) | 25 s | yes, matches |\n| 1.1.7c | `1.1.7bc_classgroup.sh` (stage c); `gp -q 1.1.7c_B0.gp` | 10 s | yes, matches |\n| 1.1.7d | `python3 1.1.7d_o6minus.py` | 40 s | yes, matches |\n| 1.1.7e | `gp -q 1.1.7e_logs2.gp`; counts (below) and `gp -q 1.1.7eh_lpoly.gp` (`expected/1.1.7eh_counts.txt`) | 10 s | yes, matches |\n| 1.1.7f | `gp -q 1.1.7f_zero_tree2.gp` | 10 s | yes, matches |\n| 1.1.7g | `gp -q 1.1.7g_logs3.gp`; `gp -q 1.1.7g_zero_tree3.gp` | 1 min | yes, matches |\n| 1.1.7h | counts (below); `gp -q 1.1.7eh_lpoly.gp`; `python3 1.1.7h_sieve.py 509` and `... 293` (Sage) | counts 2 min; 7 min + 3 min | yes, matches |\n\nPoint counts (1.1.7e, h): `gcc -O2 -o work/count3 1.1.7eh_count.c`, then\n`./work/count3 p k -1` prints `#C(F_{p^k})` for `p = 2, 293, 509`, `k = 1, 2, 3` (histograms of\n`u ↦ u^3+u^2` and `z ↦ z^4 − z`). `1.1.7eh_lpoly.gp` turns the counts (typed into it) into `L_p(T)` and `#J(F_p)`.\n\n## 1.1.7a — geometry, bitangent field, bad primes, descent dimensions\n* `1.1.7a_transport.gp`: `x^4+y^3−y^2+x` at `(x,y) = (−z,−u)` equals `−F(u,z)`; the six target points; the\n  lines `u = −z`, `z = 0`, `z = 1`; `|Res(81z+16, 4z^3−1)| = 547825 = 5^2·17·1289`.\n* `1.1.7a_bitangent.gp`: the degree-27 factor of the bitangent resultant is `f` (case 4569); irreducible; 3 real roots.\n* `1.1.7a_field.gp`: proven factorization of `disc(f)`, `disc(K) = 2^24·3^6·17^6·1289^6`, (3,12), `nfcertify = []`.\n* `1.1.7a_selmer.py 4569 ... '[[1,0],[2,0],[0,1]]'` (cypari2; reads the source's descent input: field\n  polynomial, `η`, 25 S-unit candidates, all rechecked; no `bnfinit`): exact bitangent identities; support\n  of the candidates on the 10 primes above {2,3,5}; rank 25; norm condition → 21; local images at\n  487, …, 523 → 18, 15, 13, 10, 7, 6, 6; dyadic images of three 2-adic points `(u,z) ≈ (1,0), (2,0), (0,1)`\n  (Hensel to 2^60) → 3; the 5-adic point `X_5` (`u = 1`, `z ≡ 4 mod 25`) and its character vector at the five\n  primes above 5 → 2. (The third argument gives the 2-adic starting points; without it the script uses\n  those of 1.1.6.) Writes `work/selmer_4569_G.json`.\n* `1.1.7a_normdim.py`: the norm-condition rank (sign, v_2, v_3, v_5) of the 25 candidates.\n\nExpected and obtained: `candidates 25`, `primes above S: 10`, `all candidates supported above S: OK`,\n`rank ...: 25 of 25`, `after norm conditions: dim 21`, remaining dims `18, 15, 13, 10, 7, 6, 6`,\n`dimension after 2-adic condition: 3`, `primes above 5: [(1, 3), (1, 6), (1, 6), (1, 6), (1, 6)]`,\n`dimension after 5-adic condition (image = line of u=1 point): 2`,\n`images of (0,1) (1,-1) independent mod squares: True  final dim: 2`;\n`n gens 25 norm-condition rank 4 kernel dim 21`; `4569: F(-v,-u) == -M(u,v): 1`,\n`Res(81z+16,4z^3-1)=-547825 [5, 2; 17, 1; 1289, 1]`; `nf disc = [2, 24; 3, 6; 17, 6; 1289, 6] sign=[3, 12]`.\n(The node at 5, thickness 2 and component group of order 2, is treated in the text; the local character of `X_5` is\ncomputed in `1.1.7a_selmer.py` (5-adic condition).)\n\n## 1.1.7b — generators of all prime ideals of norm ≤ 3000; 1.1.7c — the bound B_0 (T = 8)\nShared tools (`../common/h31_classgroup/`) with `data/classgroup_params.json` (`f`, `|D|`, (3,12),\n`principal_bound` 3000, `T` = 8, one cell, 192 bits). `./1.1.7bc_classgroup.sh` runs `witnesses.py --direct`\nand `check_witnesses.py` (b), `count_ideals.py` and `bound.py --no-exp-terms` (c); `1.1.7c_B0.gp` is a short\nindependent PARI check (50 digits) that also prints the decompositions of 2, 3, 5.\nExpected and obtained: `ALL 413 PRIME IDEALS OF NORM <= 3000 ... HAVE VALID WITNESSES; COFACTOR_ONE 413`;\n`PRIME_IDEALS 413` of norm `< e^8`, `prime_power_terms_with_multiplicity = 428`,\n`B_INTERVAL [0.08498996217526250100534057122993185132083 +/- 1.07e-42] POSITIVE True`;\n`prime ideals norm<=3000: 413  B0 = 0.084989962175262501005340571229931851166`,\n`2: [[9, 3]]`, `3: [[1, 3], [1, 6], [1, 6], [2, 6]]`, `5: [[1, 3], [1, 6], [1, 6], [1, 6], [1, 6]]`.\n\n## 1.1.7d — the O_6^-(2) catalogue\n`1.1.7d_o6minus.py` (pure Python): `q(v) = v1v2+v3v4+v5v6+v5+v6` on `F_2^6`, the orthogonal group generated\nby the 36 reflections (order 51840), its action on the 27 nonzero isotropic vectors, and for each element\nof cycle type (3,6,6,6,6) the pair (dim of the fixed space, rank of the coinvariant map).\nExpected and obtained: `nonzero isotropic: 27  nonsingular (reflection) vectors: 36`,\n`code weights of nonzero v: {16: 27, 12: 36}`, `group order 51840`,\n`... {(1, 1): 4320, (2, 0): 1440, (0, 0): 720}`. (The character vector of `X_5` is printed by 1.1.7a.)\n\n## 1.1.7e — 2-adic principal divisors, logarithms, #J(F_2)\n`1.1.7e_logs2.gp` (reads `logs.gp`): `h_X ∈ L(38O)` with order 35 at `X = (0,0), (0,1), (1,−1)`, residual\ncubics and their disk centres, tiny integrals (144 terms mod 2^60), the logs mod 2^40, the relation\n`D(0,0) = 2G1 − 2G2`, half-logs. Expected and obtained:\n`log D(0,0) mod 2^40: [714153886116, 445000321592, 1048252576132]`,\n`log G1 mod 2^40: [578889133504, 570949811744, 721623895966]`,\n`log G2 mod 2^40: [771568004334, 348449650948, 747253421788]` (each equal to the source's value),\n`relation D(0,0) = 2G1 - 2G2: valuation of difference [87, 85, 87]`, `half-log vectors mod 2: G1 [0, 0, 1] G2 [1, 0, 0]`;\ncounts `2 1 5`, `2 2 9`, `2 3 5` and `[8*T^6 + 8*T^5 + 8*T^4 + 4*T^3 + 4*T^2 + 2*T + 1, 35, [5, 1; 7, 1]]`.\n\n## 1.1.7f — dyadic zero tree\n`1.1.7f_zero_tree2.gp` (reads `logs.gp`, `balls.gp`): annihilator (normalized by its odd middle coefficient),\nthe five disks, ball partition, roots mod 2^20 and coefficients mod 2^16. Expected and obtained:\n`disk (0,0)`: `s = 0 (2,65534)`, `s = 3 (65525,9)` [point (6,4)]; `disk (0,1)`: `s = 0 (1,0)`;\n`disk (-1,0)`: `s = 0 (65532,4)`, `s = 902770 (18122,35366)` [`Q_2`]; `disk (-1,1)`: `s = 0 (65534,0)`,\n`s = 1 (0,1)` [point (1,−1)]; `disk O`: `s = 0 (0,0)`, `s = 607280 (38704,39424)` [`R_2`, `(304,0) mod 512`];\n`total zeros 9`.\n\n## 1.1.7g — 3-adic logarithms, ternary zero tree with the cusp chart\n`1.1.7g_logs3.gp`: `h_X ∈ L(66O)` with order 63 at `X = (0,0), (1,−1)`, `3·log D_X mod 3^24` (precision\nchecked by recomputing with fewer terms). `1.1.7g_zero_tree3.gp`: cross product (common valuation 2),\nannihilator, the five smooth disks (`t = 3s`) and the cusp chart (`u = 3a`, `z = 1 + 3b(a)`; coefficient\nbound checked for `n < 260`), ball trees, roots and the coefficients `(a,b) mod 3^8`.\nExpected and obtained: `3 log D(0,0) mod 3^24: [265295843469, 256546487256, 167647527990]`,\n`3 log D(1,-1) mod 3^24: [247088970603, 249160065058, 89859469233]`, `reduction mod 3 of supplied: [2, 0, 2]`,\n`det valuation: 2`; zeros `(0,0): (2,6559)`, `(1,−1): (0,1)`, `(−1,0): 0 (6557,4)` and `273546 (2705,832)` [`Q_3`,\n`(32,22) mod 81`], `(−1,1): (6559,0)`, `O: (0,0)`, cusp: `a = 0 (1,0)`, `397615 (1055,2942)` [`R_3`, `(2,26) mod 81`],\n`2 (6550,9)` [point (6,4)]; `min valuation of coefficients n in [121,259]: 32` (≥ 28); `total ternary zeros 9`.\n\n## 1.1.7h — finite comparisons at 509 and 293\nCounts `293: 284, 85720, 25164212`, `509: 510, 259528, 131838318`; `1.1.7eh_lpoly.gp`:\n`#J(F_293) = 24294816 = 2^5·3^4·7·13·103`, `#J(F_509) = 131974656 = 2^9·3·11·73·107`.\n`1.1.7h_sieve.py p` (Sage, function-field Jacobian over `F_p`):\np = 509: compares `257763·(304·G1)` with `257763·[Q − O]` for all 510 points `Q` — no match;\np = 293: the points `Q` with `759213·[Q − O] = 759213·(10·G1 + 6·G2)` are exactly the nine\n`(u,z) = (0,0),(1,20),(47,88),(60,82),(68,67),(119,277),(124,164),(162,64),(197,60)`, and for none of them is\n`299936·[Q − O]` equal to `299936·(32·G1 + 22·G2)` (`Q_3`) or `299936·(2·G1 + 26·G2)` (`R_3`).\nExpected: `R_2 (304,0) mod 512: matching residue points: []`; the nine points above with\n`equals Q_3 class: False  equals R_3 class: False`; sanity lines for (6,4), (0,0), (−1,0) all `True`.\n\n## Origin\nReviewer scripts written independently of the source (only the stack limit was added to the two 3-adic\nscripts); the descent candidates and `η` are the source's (rechecked); shared tools for b, c.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.8: x^4 + y^3 + 2x - y + 1 = 0\n\nCoordinates used by all scripts: `u = -y`, `z = x`, so the curve is\n`C: G(u,z) = -u^3 + u + z^4 + 2z + 1 = 0`, with the rational hyperflex `O = [U:Z:W] = [1:0:0]`\nand the points `P0 = (0,-1)`, `P1 = (1,-1)` in `(u,z)`. The bitangents are `u = m z + eta`; `f` is the\ndegree-27 slope polynomial and `K = Q[m]/(f)`.\n\nRun every command from this folder. `work/` is created by 1.1.8f and read by 1.1.8g.\n\n## Requirements\n\n- PARI/GP 2.15 or later (`gp`); tested with 2.15.4.\n- For 1.1.8b-d: Python 3 with cypari2, python-flint (>= 0.6) and numpy, and the shared tools in\n  `../common/h31_classgroup/` (see the README there). Tested with cypari2 / PARI 2.15.4, python-flint 0.9.0.\n\n## Codes\n\n| id | script | what it computes | run here |\n|---|---|---|---|\n| 1.1.8a | `1.1.8a_geometry_field.gp` | transport, smoothness and hyperflex at O, bad primes and nodes, bitangent polynomial, field K | yes, < 1 s |\n| 1.1.8b | `../common/h31_classgroup/count_ideals.py` | number of prime ideals of each norm `q < 10^7` | yes (full), 96 s |\n| 1.1.8c | `../common/h31_classgroup/witnesses.py`, `check_witnesses.py` | Lemma C witnesses for all prime ideals of norm `<= 10^5`, exact check | yes (full), 150 s + 19 s |\n| 1.1.8d | `../common/h31_classgroup/bound.py` | `B^-(f)` of Theorem B (48 cells, `T = log 10^7`) | yes (full), 6 s |\n| 1.1.8e | `1.1.8e_descent.gp` + `data/1.1.8e_selmer_elements.gp` | the 2-descent: S-classes, local conditions, rank <= 2 | yes, 15 s |\n| 1.1.8f | `1.1.8f_logs.gp` | principal divisors `div(h_i) = 16 P_i + E_i - 19 O`, 2-adic logarithms, annihilator | yes, 45 s CPU |\n| 1.1.8g | `1.1.8g_charts.gp` | low coefficients of the primitives on the two integral 2-adic charts | yes, 15 s CPU |\n\nAll outputs in `expected/` are from runs on 14-15 Oct 2026 and agree with the values of the paper\nand of the code manifest.\n\n### 1.1.8a  Geometry, bitangent field, bad primes\n\n```\ngp -q 1.1.8a_geometry_field.gp < /dev/null\n```\nChecks `F(z,-u) = G(u,z)`; the chart at O (`d/dW = -1`, line `W = 0` meets C in `Z^4`); the critical\nalgebra `u^2 = 1/3`, `z^3 = -1/2` (field `Q(sqrt 3, 4^(1/3))`), where the critical value is\n`(4u+9z+6)/6`; its prime factorisation gives the odd bad primes and the node thickness; own elimination of\nthe bitangent conditions; the eta identity; `nfinit` + `nfcertify` of K. Key lines:\n```\ncritical_value_norm = -7100725/5038848 = [-1, 1; 5, 2; 431, 1; 659, 1] / [2, 8; 3, 9]\n  bad p = 5: prime of L with residue degree 2, valuation of critical value 1  -> 2 geometric node(s) of thickness 1\n  bad p = 431: prime of L with residue degree 1, valuation of critical value 1  -> 1 geometric node(s) of thickness 1\n  bad p = 659: prime of L with residue degree 1, valuation of critical value 1  -> 1 geometric node(s) of thickness 1\ndegree-27 factor == f: 1;  f irreducible: 1;  real roots: 7\nidentity F(m z+eta, z) = (z^2 - m^3 z/2 - 3 m^2 eta/2 - m^6/8)^2: 1\nnfcertify (empty = maximal order certified): []\nsignature [7, 10];  disc 5505237022261200366725146950702947368960000000000 = [2, 30; 5, 10; 431, 6; 659, 6]\nprimes over 2 (e,f): [[8, 3], [1, 3]]\n```\n\n### 1.1.8b, 1.1.8c, 1.1.8d  Class group (Theorem B, Lemma C)\n\nParameters: `data/classgroup_params.json` (f, `|D_K|`, signature `[7,10]`, `support = 10^7`,\n`principal_bound = 10^5`, the 48 window heights of the package file `011135_window_exact_discovery.json`,\n224 bits). Commands (the count and witness files are a few MB and are not included):\n```\nCG=../common/h31_classgroup; PRM=data/classgroup_params.json\npython3 -I $CG/count_ideals.py $PRM --out counts.npz                    # 1.1.8b\npython3 -I $CG/witnesses.py $PRM --out witnesses.jsonl.gz               # 1.1.8c (proposes)\npython3 -I $CG/check_witnesses.py $PRM witnesses.jsonl.gz               # 1.1.8c (exact check)\npython3 -I $CG/bound.py $PRM counts.npz                                 # 1.1.8d\n```\nKey lines (`expected/1.1.8b.txt`, `1.1.8c.txt`, `1.1.8d.txt`):\n```\nRANGE 2 9999999 SUPPORT 10000000 RATIONAL_PRIMES 664579 DISTINCT_NORMS 360422 PRIME_IDEALS 665664 SHA256 dc0aa91d...\nALL 9432 PRIME IDEALS OF NORM <= 100000 OVER p IN [2, 100000] HAVE VALID WITNESSES; COFACTOR_ONE 1965 COFACTOR_GT_ONE 7467 ...\narchimedean = \"[-0.873380190425780748703324857763 +/- 3.38e-31]\"\nprime_sum = \"[0.966424801163642608925660142701 +/- 3.51e-31]\"\nB_INTERVAL [0.09304461073786186022233528493868798942962 +/- 3.19e-42] POSITIVE True\n```\nThe archimedean term equals `normalized_base - correction_integral` of the package\n(`-0.78915950623... - 0.08422068419...`), and `B` agrees with the package value\n`0.0930446107378618602223352849386879894296...`. Difference from the package: the package checked the\n9,432 single generators of its witness file `011135_principals_100000.jsonl.gz` (1.4 MB, not included);\nhere the witnesses are regenerated (`bnfinit` only proposes) and 7,467 of them are strict Lemma C witnesses\nwith cofactor > 1 instead of generators. The conclusion (every prime ideal of norm `<= 10^5` is principal) is\nthe same.\n\n### 1.1.8e  2-descent\n\n```\ngp -q 1.1.8e_descent.gp < /dev/null\n```\nInput `data/1.1.8e_selmer_elements.gp`: the 85 atomic elements (`EL`, power-basis coordinates) and the 19\nclasses as exponent lists (`GEN`), transcribed from the package file\n`research/session7_frontier/011135_membership_witnesses.json` (fields `elements`,\n`generators[].compact_full_generator`; the transcription was compared with the JSON and is identical).\nThe script factors the atomic elements itself, checks that the 19 classes have even valuation away from 2,\ntheir independence, the norm rows, the characters at the primes over 7, 11, 59, 83, the dyadic\nsquare-class coordinates (units modulo `p^(2e+3)`) and the images of P0, P1 and the point `t = 2` of\nthe disk at O. Key lines:\n```\ndistinct primes in atomic divisors: 61\nall 19 classes have even valuation away from 2 (S-classes): 1\nrank of the 19 classes via 138 residue characters: 19\ndimension after norm conditions: 17\n   dimension after p=7: 15  ... p=11: 13  ... p=59: 11  ... p=83: 9\nunit square-class dims: 25 + 4; with valuation parity: 26 + 5\nrank of the three local point images: 3\nSelmer (fake) dimension: 2\nkernel basis (1-based class support; span = <e3+e14+e19, e5+e14>): [[3, 5, 19], [5, 14]]\n```\n\n### 1.1.8f  Principal divisors, 2-adic logarithms, annihilator\n\n```\ngp -q 1.1.8f_logs.gp < /dev/null        # writes work/1.1.8f_annihilator.txt, work/1.1.8f_logs.txt\n```\nConstructs `h_i` in `L(19 O)` with a zero of order 16 at `P_i` (one-dimensional kernel), checks the order,\n`Res_u(G, h_i) = c (z+1)^16 * squarefree cubic`, the characteristic polynomials of `t = z/u` and\n`w = 1/u` on `E_i` (disk of O), and computes `log G_i = -(1/16) Tr int_0^t omega` with 300 terms. Key lines:\n```\n  order at P_0: 16          2-vals coeffs t: [1, 3, 3, 0]   w: [4, 7, 4, 0]\n  order at P_1: 16          2-vals coeffs t: [6, 2, 3, 0]   w: [24, 8, 5, 0]\n4 log G0 mod 2^12: [3044, 434, 1012]   4 log G1 mod 2^12: [4064, 481, 1506]\nannihilator mod 16: [12, 14, 9]   COFACTORVAL -2\nannihilator (integers mod 2^40): [265386006124, 122174830158, 9]\n```\n\n### 1.1.8g  The two integral charts\n\n```\ngp -q 1.1.8g_charts.gp < /dev/null      # reads work/1.1.8f_annihilator.txt\n```\nKey lines:\n```\neven chart: v2([t^n]H_e) n=1..12: [0, 4, 2, 2, 3, 8, 10, 6, 8, 8, 12, 12]\nodd chart: v2([t^n]H_o) n=1..14: [0, 0, 4, 1, 3, 3, 4, 3, 7, 4, 6, 5, 6, 6]\n```\n(The script also checks the stated all-order lower bounds on the computed range; the bounds themselves are\nproved in the text.)\n\n## Sources\n\n1.1.8a, e, f, g are reviewer scripts of the independent audit (t1-t4, t17-t18,\nt5, t6) merged and given relative paths; the critical-algebra node check of 1.1.8a is new. 1.1.8b-d use the\nshared class-group tools. The original package (`h31_3355`, checkpoint 43, Sage/passagemath) is not\nneeded.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.1.9: x^4 + y^3 + 2x + y + 1 = 0\n\nCoordinates: the scripts use the package coordinates `u = y`, `z = x`, so\n`C: G(u,z) = u^3 + u + z^4 + 2z + 1 = 0`; the bitangent lines `u = m z + eta` are `y = m x + eta`.\n`O = [0:1:0]` (`[U:Z:W] = [1:0:0]`), `P0 = (x,y) = (-1,0)`,\n`D_eff = {x^2 - 2x + 3 = 0, y = x - 2}`, `G0 = [P0 - O]`, `D = [D_eff - 2O]`. `K = Q[m]/(f)`, degree 27.\n\nRun every command from this folder. `work/` is created by 1.1.9e and 1.1.9f; 1.1.9g reads it.\n\n## Requirements\n\n- PARI/GP 2.15 or later (`gp`).\n- 1.1.9b-d: Python 3, cypari2, python-flint, numpy and `../common/h31_classgroup/`.\n- 1.1.9e and 1.1.9h (sieve part): Sage or passagemath (`from sage.all__sagemath_schemes import *`);\n  tested with a partial passagemath install (`sage.all__sagemath_schemes`, `__sagemath_flint`,\n  `__sagemath_pari`) on Python 3.11.\n\n## Codes\n\n| id | script | what it computes | run here |\n|---|---|---|---|\n| 1.1.9a | `1.1.9a_geometry_field.gp` | smoothness and hyperflex, bad primes and the node, bitangent polynomial and eta, field K | yes, < 1 s |\n| 1.1.9b | `../common/h31_classgroup/count_ideals.py` | prime ideals of each norm `q < 10^9` | slice only (`p <= 10^6`); full run ~2.5 CPU-hours |\n| 1.1.9c | `.../witnesses.py`, `.../check_witnesses.py` | Lemma C witnesses for all prime ideals of norm `<= 1.35*10^7`, exact check | slice only (`p <= 3000`); full run several CPU-hours |\n| 1.1.9d | `.../bound.py` | `B^-(f)` (96 cells, `T = log 10^9`) | no (needs the full counts) |\n| 1.1.9e | `1.1.9e_global_rank.py` + `data/1.1.9e_*.json` | 2-descent: 17 classes, good primes, dyadic image, rank 2 | yes, 30 s |\n| 1.1.9f | `1.1.9f_logs.gp` | `div(h0)`, `div(hD)`, 2-adic logs, annihilator | yes, 50 s CPU |\n| 1.1.9g | `1.1.9g_chart.gp` | the integral chart, the extra zero `t_*` and its coordinates | yes, 18 s CPU |\n| 1.1.9h | `1.1.9h_counts.gp`, `1.1.9h_sieve23.py` | point counts at 23, `#J(F_23)`, sieve at 23 | yes, 1 s and 30 s |\n\n### 1.1.9a  Geometry, bitangent field, bad primes\n\n```\ngp -q 1.1.9a_geometry_field.gp < /dev/null\n```\nSame checks as 1.1.8a for `G = u^3+u+z^4+2z+1` (critical algebra `u^2 = -1/3`, `z^3 = -1/2`,\ncritical value `(4u+9z+6)/6`). Key lines:\n```\ncritical_value_norm = 12527563/5038848 = Mat([12527563, 1]) / [2, 8; 3, 9]\n  bad p = 12527563: prime of L with residue degree 1, valuation of critical value 1  -> 1 geometric node(s) of thickness 1\n     node mod p: (u,z) = [4469046, 2189611]\ndegree-27 factor == f: 1;  f irreducible: 1;  real roots: 3\nidentity F(x, m x+eta) = (x^2 + m^3 x/2 + 3 m^2 eta/2 - m^6/8)^2: 1\nnfcertify (empty = maximal order certified): []\nsignature [3, 12];  disc 4150490676440774655225550489056357993175100440969216 = [2, 30; 12527563, 6]\nprimes over 2 (e,f): [[1, 3], [8, 3]]\n```\n(The node is `(x,y) = (2189611, 4469046)`; 1.1.9e prints its Hessian `4898948`.)\n\n### 1.1.9b, 1.1.9c, 1.1.9d  Class group\n\nParameters `data/classgroup_params.json`: f, `|D_K| = 2^30 12527563^6`, `(3,12)`, `support = 10^9`,\n`principal_bound = 13500000`, the 96 heights of the package file `011197_analytic_class_premises.json`,\n288 bits. Full commands (use `--lo/--hi` slices and `--workers`; count files for `10^9` are ~400 MB):\n```\nCG=../common/h31_classgroup; PRM=data/classgroup_params.json\npython3 -I $CG/count_ideals.py $PRM --workers W --out counts.npz                 # 1.1.9b\npython3 -I $CG/witnesses.py $PRM --workers W --out witnesses.jsonl.gz            # 1.1.9c\npython3 -I $CG/check_witnesses.py $PRM witnesses.jsonl.gz                        # 1.1.9c\npython3 -I $CG/bound.py $PRM counts.npz                                          # 1.1.9d\n```\nRun here only on slices (`expected/1.1.9b_slice.txt`, `1.1.9c_slice.txt`):\n```\nRANGE 2 1000000 SUPPORT 1000000000 RATIONAL_PRIMES 78498 DISTINCT_NORMS 44159 PRIME_IDEALS 81695 ...\nALL 829 PRIME IDEALS OF NORM <= 13500000 OVER p IN [2, 3000] HAVE VALID WITNESSES; ...\n```\nExpected final numbers (from the package records and the independent reviewer count, not run here):\n50,851,991 prime ideals of norm `< 10^9` over 50,847,534 rational primes; 879,491 prime ideals of norm\n`<= 1.35*10^7`, all principal; archimedean correction `0.06310199540718684684...`;\n`B^- = [3.35410151812786874392878630930131137834921189405101738829e-5 +/- 8.12e-62]`.\nThe package proved 150 of the 879,491 ideals with product witnesses `prod alpha_j^e_j`; the shared tools\nregenerate a witness for every ideal (a strict Lemma C witness or a generator proposed by `bnfisprincipal`,\nchecked by its exact norm), so no product witnesses are needed.\nLarge `bnfisprincipal` generators can exceed Python's default limit of 4300 digits for integer/string\nconversion; the witness scripts lift that limit themselves.\n\n### 1.1.9e  2-descent: rank 2 and odd index of <G0, D>\n\n```\npython3 -I 1.1.9e_global_rank.py       # Sage/passagemath python; writes work/1.1.9e_global_rank.json\n```\nThis is the package script `verify_011197_global_rank.py` with `orthogonal_frobenius_bounds.py` inlined\nand paths, output and source-hash recording changed (no mathematical change). Inputs:\n`data/1.1.9e_rank_input.json` = the fields `field, eta, gens, signature, discriminant, membership` of the\npackage file `011197_rank_plain_input.json` (re-serialized compactly), `data/1.1.9e_field_input.json` =\npart of `011197_class_probe_input.json`. It rebuilds every prime ideal from the field, checks the 161\natomic factorisations over 141 prime ideals, enumerates the orthogonal group (51,840 elements) for the\ngood-prime bounds, lifts all `F_p`-points, computes the dyadic characters from `(O_K/p^(2e+3))^*`, the\nthree dyadic witnesses with exact error bounds, the characters of `delta(G0)`, `delta(D)` at `m = -1 (5)`,\n`m = -21 (37)`, and the bad fibre at 12527563. Key lines (`expected/1.1.9e.txt`):\n```\nFULL_COMPACT_MEMBERSHIP_PASS 161 141\nGOOD_PRIME_FULL_IMAGE 13 dimension 0 remaining 13\nGOOD_PRIME_FULL_IMAGE 17 dimension 0 remaining 11\nGOOD_PRIME_FULL_IMAGE 43 dimension 0 remaining 9\nGOOD_PRIME_FULL_IMAGE 53 dimension 0 remaining 7\nGOOD_PRIME_FULL_IMAGE 59 dimension 2 remaining 4\nGOOD_PRIME_FULL_IMAGE 67 dimension 1 remaining 3        (73, 97, 101, 103, 107, 109: remaining 3)\npre_dyadic_dimension 3\ndyadic_local_metadata ... \"dimension\": 26 ... \"dimension\": 5\nfull_dyadic_image_dimension 3\nfinal_global_rows [[0, 0, 1, 0, 1, 0, ...], [0, 0, 0, 1, 1, 1, 0, ..., 1]]      (= e3+e5, e4+e5+e6+e17)\nknown_point_independence_characters [{\"p\": 5, ..., \"point_characters\": [1, 1]}, {\"p\": 37, ..., \"point_characters\": [1, 0]}]\nrank_upper_bound 2\nknown_subgroup_rank_lower_bound 2\ncritical_value_norm \"12527563/5038848\"\nbad_fiber_geometry [{\"p\": 12527563, ..., \"singular_u\": 4469046, \"singular_z\": 2189611, \"critical_value_valuation\": 1, \"ordinary_node_thickness\": 1, \"hessian_determinant\": 4898948}]\nFRESH_CLASS_CONDITIONAL_GLOBAL_RANK_PASS\n```\n\n### 1.1.9f  Principal divisors, 2-adic logarithms, annihilator\n\n```\ngp -q 1.1.9f_logs.gp < /dev/null       # writes work/1.1.9f_annihilator.txt, work/1.1.9f_logs.txt\n```\nKey lines:\n```\n  E0 norm factorization degrees/mults: [[1, 16], [3, 1]]      2-vals charpoly t: [1, 3, 3, 0]   w: [4, 7, 4, 0]\norder of h_D at Q: 8\n  ED norm factorization degrees/mults: [[2, 8], [3, 1]]       2-vals charpoly t: [2, 2, 3, 0]   w: [8, 9, 5, 0]\n4 log G0 mod 2^12: [3988, 2194, 1012]   4 log D mod 2^12: [1848, 3542, 3892]\nannihilator normalized a2=9 mod 16: [6, 10, 9]   COFACTORVAL -1\nannihilator (integers mod 2^50): [857957383579798, 646672900410122, 9]\n```\n\n### 1.1.9g  The integral chart, the extra zero and its coordinates\n\n```\ngp -q 1.1.9g_chart.gp < /dev/null      # reads work/1.1.9f_*.txt\n```\nKey lines:\n```\nv2 [t^n]H, n=1..12: [0, 0, 1, 1, 2, 5, 7, 5, 8, 7, 11, 11]\n[t^4]H vs 36a0-42a1-14a2: +oo                      (i.e. equal)\nt* mod 2^32: 3366055599   v2(G(t*)) = 45\nminor [1, 2] v2(det)=-1 c0 mod 2^24 = 10965931  cD mod 2^24 = 8513226   mod 32: [11, 10]\n```\n\n### 1.1.9h  Mordell-Weil sieve at 23\n\n```\ngp -q 1.1.9h_counts.gp < /dev/null\npython3 -I 1.1.9h_sieve23.py           # Sage/passagemath python\n```\nKey lines:\n```\np = 23: #C(F_23^k), k=1,2,3: [26, 622, 12044]\n#J(F_23) = 14432 = [2, 5; 11, 1; 41, 1]\nhess matches among 26 points: []\npoints with 451[Q-O] - target principal (RR check): 0 of 26\n```\n(`target = 451(11 G0 + 10 D)`; the Riemann-Roch test decides equality of classes independently of the\ngroup-law implementation.)\n\n## Sources\n\n1.1.9a: like 1.1.8a (own); 1.1.9e: package script (see above); 1.1.9f, g, h: reviewer scripts\n(t7, t8, t9, t10/t11 of the independent audit) with relative paths; only the Hess model of Sage's\nfunction-field Jacobian is used in 1.1.9h. 1.1.9b-d: shared class-group tools.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.2.1: x^4 + 3y^3 + z^3 = 0\n\nPrimitive solutions: (±1, 0, −1) and (±895, −6202, 4199). Source: Avi Eisenberg's proof (`five.pdf`,\nchapters 5 and 6). The source's own programs are not available; every code below is an independent\nreviewer script or a small variant written by the paper's writers. Each script starts with a\ncomment line naming its code id and origin.\n\n## Software\n- PARI/GP 2.15.4 (`gp`), run here on 14 Oct 2026.\n- Python 3 with SymPy (1.14) and cypari2 (PARI 2.15.4).\n- Denis Simon's `ell.gp` (+ `qfsolve.gp`, `resultant3.gp`) from `../common/simon_ell/` (only 1.2.1a).\n\nRun every command from this folder. All gp scripts that are run directly end with `quit` (`M_def.gp` and\n`M_points.gp` are only read by the others).\n\n## Run everything (about 2 minutes)\n```\ngp -q 1.2.1a_cubic_field_K.gp\npython3 -I 1.2.1b_sieve577.py;  gp -q 1.2.1b_chabauty577.gp;  python3 -I 1.2.1b_smooth577.py\ngp -q 1.2.1c_odd_algebra.gp;  gp -q 1.2.1c_odd_norms.gp;  gp -q 1.2.1c_mod8_forms.gp\ngp -q 1.2.1d_sextic_L.gp;  gp -q 1.2.1d_units_L.gp\ngp -q 1.2.1e_field_M_identities.gp;  gp -q 1.2.1e_points_M.gp\ngp -q 1.2.1f_classgroup_M.gp\ngp -q 1.2.1g_selmer_M.gp;  gp -q 1.2.1g_independence.gp           # selmer: about 1 min\ngp -q 1.2.1h_sieve661_data.gp          # writes c5_661.txt\npython3 -I 1.2.1h_sieve661.py          # reads c5_661.txt, writes c5_surv.json\npython3 -I 1.2.1h_sieve661_primary.py  # reads c5_661.txt\ngp -q 1.2.1i_logs661.gp                # writes c5_log661.txt\npython3 -I 1.2.1i_chabauty661.py       # reads c5_661.txt, c5_log661.txt, c5_surv.json\n```\nThe order matters only for 1.2.1h and 1.2.1i. The three intermediate files are generated in seconds and\nare not shipped. `M_def.gp` (the field M = Q[y]/(y^12 − 36y^10 + 432y^8 − 2352y^6 + 5184y^4 − 15552), θ,\nR0, d, the basis D) and `M_points.gp` (the points of the source's appendix 5.F; R, S, U, V are G1..G4 of\nthe section) are inputs read by the scripts of 1.2.1e–i.\n\nAll codes were run here; `expected/<id>.txt` is the output of that run (stdout and stderr, one `### command`\nheader per script). Every expected line of the section was reproduced.\n\n## Codes\n\n### 1.2.1a — K = Q(∛3); E_0, E_2 over K\n`gp -q 1.2.1a_cubic_field_K.gp` (1 s). Class number and certification, fundamental unit, Simon's full\n2-descent (`bnfellrank`) of E_0: V^2 = U^3 − 36U and E_2: V^2 = U^3 − 36ε^2U, torsion, the points A, B,\nindependence of A, B, T0, T1 modulo 2E_0(K), and whether ±ε^3/2 are squares. The last line (added for the\nbundle, see \"Changes\") lists the elements a + bθ + cθ^2 of norm 1 in the box 0≤a≤4, 0≤b≤3, 0≤c≤2.\nKey output:\n```\nK: disc [-1, 1; 3, 5] h 1 certify 1 fu [Mod(y^2 - 2, y^3 - 3)]\nE0 over K: [r,s]=[2, 4] ...\nE2 over K: [r,s]=[0, 2]\ntorsion E0 [4, [2, 2]] E2 [4, [2, 2]]\nA,B on E0: 11\nA,B,T0,T1 independence mod 2 checked        (no \"dependent\" line)\neps^3/2 square? 0 -eps^3/2 square? 0\nnorm-1 elements in the unit box: [[1, 0, 0], [4, 3, 2]]      (1 and eps = 4+3θ+2θ^2)\n```\nDisc of y^3 − 3 is −243 = disc K, so O_K = Z[θ].\n\n### 1.2.1b — sieve at 577 on C_0 and 577-adic Chabauty\n`python3 -I 1.2.1b_sieve577.py` (SymPy; a few seconds): roots of θ^3 − 3 mod 577, E_0(F_577) (a basis of\n(Z/24)^2 is built in each factor), H = ⟨A, B, T0, T1⟩ mod 577, [8]A, [8]B in the 3-torsion, all points of\nP^2(F_577) on C_0 and their images.\n`gp -q 1.2.1b_chabauty577.gp`: 577-adic logarithms (via `ellformallog` on the first multiple in the formal\ngroup), the annihilator, and its pairing with the pull-back coefficients θ, −θ^2/3 at (1:0:0), (0:1:0).\n`python3 -I 1.2.1b_smooth577.py` (added for the bundle): C_0 is smooth mod 577 (Gröbner basis [1] in the\nthree affine charts) and the dependent-coordinate derivatives at the two centres are 4 and 12.\nKey output:\n```\n#C0(F_577) = 572\n#E(F_577)= 576\n|H| = 576\n[8]A coords: ((8, 0), (8, 0), (8, 0)) [8]B: ((8, 8), (0, 0), (16, 16))\nsurviving points: [(1, 0, 0), (0, 1, 0)]\nlog A/p mod p^2: [4351, 4351, 4351]\nlog B/p mod p^2: [55622, 183530, 93777]\nannihilator mod p: [28, 304, 245]\nsum c_i theta_i = 496   sum c_i(-theta_i^2/3) = 241\nC0 Groebner bases mod 577 in charts a=1,b=1,c=1: [[1], [1], [1]]\ndf/dc at (1,0,0): 4   at (0,1,0): 12\n```\n\n### 1.2.1c — algebra of the odd cases\n`gp -q 1.2.1c_odd_algebra.gp`, `gp -q 1.2.1c_odd_norms.gp`, `gp -q 1.2.1c_mod8_forms.gp` (each < 1 s).\nIdentities Z^3 + 3Y^3 = P^2, (θ^2 − 2)(h_A + h_Bθ + h_Cθ^2)^2 = Z + θY, the conic, N(h) = −P,\nUY + VZ = 3n^7, P(k+2l, 2k+3l) = l^6 f(k/l); f irreducible with its discriminant; the j = 1 case has no\nprimitive solution mod 8 (and mod 16); the j = 1 form mod 2 and, for b even, mod 8; the classification of the\nnormalised norm mod 8. Key output:\n```\nZ^3+3Y^3 - P^2 = 0\nP(2,7)=801025 =895^2? 1 Y(2,7)=6202 Z(2,7)=-4199\n(th^2-2)(...)^2 == Z+th*Y: 1\nconic at (hA,hB,hC): 0\nnorm(hA+hB th+hC th^2) + P = 0\nU*Y+V*Z - 3n^7 = 0\nP(k+2l,2k+3l) vs l^6 f(k/l): 0\nf irreducible: 1 disc f: [2, 18; 3, 25]\nj=1: primitive (a,b,c) mod 16 with coefficient = 0 mod 16: 0\nN(h)+P = 0\nobserved [n mod2, m mod2, gcd, N(h/g) mod 8, -N(h/g) mod 8]: [[0, 1, 2, 2, 6], [0, 1, 2, 6, 2], [1, 0, 1, 7, 1], [1, 1, 1, 5, 3]]~\nj=1 primitive solutions mod 8: 0\nj=1 coeff mod 2: (Mod(1, 2)*b^4 + ...)\nj=1 coeff with b=2b1 mod 8: Mod(2, 8)*a^4 + ... (Mod(6, 8)*c^2 ...)*a^2 + ... (Mod(4, 8)*c^4 ...)\n```\nThe last lines of `1.2.1c_mod8_forms.gp` print data used in 1.2.1f: θ ≡ 3 and ε ≡ 7 (mod 8) 2-adically, and the\ntwo real values 19.077… and −1.308… of W^2.\n\n### 1.2.1d — sextic field L and the 2-adic comparison\n`gp -q 1.2.1d_sextic_L.gp`, `gp -q 1.2.1d_units_L.gp` (< 1 s). L = Q(ξ), ξ^6 − 9ξ^4 + 27ξ^2 − 3 = 0:\ndiscriminant, class number (certified), f(r) = 0, the primes over 2 and the valuations of π0, π1, unit square\nclasses, 3/ξ^6 = (θ^2 − 2)(52 + 36θ + 25θ^2)^2, 2/(π0π1^2) a unit, the valuations at q = 0, 2, and the local\ncomparison of −r and 2 − r with the eight candidates times ±1, ±2, ±5, ±10 (by `nfislocalpower`).\nKey output:\n```\nL: disc [2, 4; 3, 11] sign [2, 2] h 1 certify 1\nf(r)=Mod(0, x^6 - 9*x^4 + 27*x^2 - 3)\nR(xi)==r 1  Pi(xi)==pi1 1\nprimes over 2: [[2, 2], [1, 2]]\nval pi0: [0, 1] val pi1: [1, 0]\nunits independent mod squares: 1\n3/xi^6 == (th^2-2)(52+36th+25th^2)^2: 1\n2/(pi0 pi1^2) is unit: 1\nq=0 v(q-r) at P2: [-1, -1]  v(4/(q-r)): [5, 3] v(2): [2, 1]\nq=2 v(q-r) at P2: [-1, -1]  v(4/(q-r)): [5, 3] v(2): [2, 1]\nq in 0+4Z2: matches global candidate [u2,u3,pi1] bits [0, 0, 1] with scalar 2\nq in 0+4Z2: matches global candidate [u2,u3,pi1] bits [0, 0, 1] with scalar 10\nunits -1,th^2-2,u2,u3 independent mod squares: 1\nunit rank 3 torsion 2\n```\nNo line for q ∈ 2 + 4Z_2: no candidate matches there. Valuations are listed in the order (π1, π0) of PARI's\nprime list; the section writes them in the order (π0, π1).\n\n### 1.2.1e — degree-12 field M, covering quartic, map, points\n`gp -q 1.2.1e_field_M_identities.gp`, `gp -q 1.2.1e_points_M.gp` (< 1 s). Discriminant and signature of M,\nR0^2 = θ^2ε, the relative quartic of d, N_M(d), c4, Δ, j of E, the identity\n(T^2 − 3 + 2θ)(T^2 − WT + R0)(T^2 + WT + R0) = T^6 − 9T^4 + 27T^2 − 3, r1r2 = θ^2D*/4, Q(0) = (2θ^2)^2, the\nWeierstrass, curve and differential identities (exact polynomial reduction modulo V^2 − Q(q)), the points on\nE and E', the dual isogeny, and the images of q = 0 and q = −8/3. Key output:\n```\nnfdisc(M) = [-1, 1; 2, 14; 3, 21]  sign [2, 5] ...\nN_M(d) = -5184\nc4 + 32 d^2 = Mod(0, ...)  disc + 256 d^6 = Mod(0, ...)  j = Mod(128, ...)\n(5.9): 0\nr1 r2 - th^2 D*/4 = Mod(0, ...)\nf divisible by (q-r1)(q-r2): 1\nQ(0) - (2th^2)^2 = Mod(0, ...)  deg Q = 4\nWeierstrass identity holds: 1\nE identity holds: 1\ndifferential identity holds: 1\non E: R 1 S 1 U 1 V 1 P 1 Q 1 P1 1\non E': Q0 1 D' 1\ndual(Q0)==P 1  dual(D')==Q 1\nq->0, V->-e limit == P: [1, 1]  valuations [0, 0]\nQ(-8/3) square in M: 2\n image on E: 1  ==P1: 0  ==P-P1: 1\n image on E: 1  ==P1: 1  ==P-P1: 0\nV1 formula image == P1: 1  V1^2 == Q(-8/3): 1\n```\n\n### 1.2.1f — class group of M\n`gp -q 1.2.1f_classgroup_M.gp` (< 1 s): `bnfinit` + `bnfcertify`, so h(M) = 1 without GRH; in particular\nCl(M)[2] = 0. Output: `h=1` and `bnfcertify(M) = 1  time ...` (the time in ms varies).\n\n### 1.2.1g — 2-isogeny Selmer groups over M; rank 6\n`gp -q 1.2.1g_selmer_M.gp` (about 1 minute here). Candidate space from the units and the five primes over 2\nand 3 (12 generators); at each of these primes the local image is spanned by Kummer images of random local\npoints (seed `setrand(2024)`), with `nfhilbert` keys, until the two dimensions reach the exact total; then the\nSelmer dimensions and the cumulative counts. `gp -q 1.2.1g_independence.gp` (< 1 s): the classes α(T), α(R),\nα(S), α(U), α(V) and α'(T'), α'(Q0), α'(D') are independent; torsion of E(M). Key output:\n```\np=2 e,f=[4, 1] dim loc grp 6  dim alpha_v(E) 4 (44 pts)  dim alpha_v(E') 2 (77 pts)  sum 6 required 6\np=2 e,f=[2, 2] dim loc grp 6  dim alpha_v(E) 4 (17 pts)  dim alpha_v(E') 2 (85 pts)  sum 6 required 6\np=2 e,f=[2, 2] dim loc grp 6  dim alpha_v(E) 4 (5 pts)  dim alpha_v(E') 2 (86 pts)  sum 6 required 6\np=3 e,f=[6, 1] dim loc grp 2  dim alpha_v(E) 1 (1 pts)  dim alpha_v(E') 1 (1 pts)  sum 2 required 2\np=3 e,f=[3, 2] dim loc grp 2  dim alpha_v(E) 1 (92 pts)  dim alpha_v(E') 1 (5 pts)  sum 2 required 2\ndim Sel(alpha_E) = 5   dim Sel(alpha_E') = 3   => rank <= 6\ncumulative counts E: [1024, 256, 64, 64, 64, 32]   E': [256, 16, 8, 8, 8]\nalpha_E classes of T,R,S,U,V independent: 1\nalpha_E' classes of T',Q0,D' independent: 1\ntorsion E(M): [2, [2]]\n```\n(dimensions 4, 2 at the dyadic places are the local image sizes 16, 4; at the places over 3 they are 2, 2.)\n\n### 1.2.1h — sieve at 661\n`gp -q 1.2.1h_sieve661_data.gp` (< 1 s; writes `c5_661.txt`), then `python3 -I 1.2.1h_sieve661.py`\n(a few seconds; writes `c5_surv.json`) and `python3 -I 1.2.1h_sieve661_primary.py` (a few seconds).\nTwelve embeddings (θ, d) mod 661, group orders, the index of H in the product (Hermite normal form), ℓ-ranks,\nand the sieve over all q ∈ P^1(F_661) and all ordinate signs. Key output:\n```\norders: [628, 696, 628, 696, 696, 628, 696, 696, 696, 696, 696, 628] ...\nindex [prod:H] = 31746686976 [2, 22; 3, 2; 29, 2]\nresidues with points in all 12: 27  patterns: 106512  survivors: 64\nsurvivor q residues: Counter({0: 32, 438: 32})\nl= 3  factors: 8  rank mod l: 6\nl= 29  factors: 8  rank mod l: 6\nl= 157  factors: 4  rank mod l: 4\ntotal 106512 after 2: 6146 after 3: 1152 after 29: 64 after 157: 64 final 64\n```\nThese scripts list the embeddings in the reviewer's order (as returned by `polrootsmod`), not in the order of\n(5.23) of the paper; the counts do not depend on it. 438 ≡ −8/3 (mod 661).\n\n### 1.2.1i — 661-adic logarithms and the 64 local systems\n`gp -q 1.2.1i_logs661.gp` (< 1 s; writes `c5_log661.txt`, copied into `expected/1.2.1i.txt`), then\n`python3 -I 1.2.1i_chabauty661.py` (< 1 s). The second script sorts the embeddings as in the paper, prints the\nnormalised logarithms (rows R, S, U, V, P, Q, P1), a kernel basis of the 6 × 12 matrix mod 661 with identity\nin the last six columns, and, for each of the 64 survivors, solves c·ℓ + T(c·D) ≡ 0 (mod 661). Key output:\n```\nR [189, 58, 628, 482, 557, 620, 588, 182, 605, 80, 113, 637]\n...\nP1 [585, 485, 301, 601, 451, 626, 376, 530, 362, 38, 644, 412]\nkernel rows:\n[74, 2, 659, 439, 29, 256, 1, 0, 0, 0, 0, 0]\n...\nconsistent: 4\n(0, '++++++++++++', [0], 592)\n(0, '------------', [0], 69)\n(438, '------------', [0], 395)\n(438, '++++++++++++', [0], 266)\n```\n(the last number is c^(1)·D, the derivative residue for the first annihilator.)\n\n## Changes made to the reviewer scripts (no change to the mathematics)\n- All: the path to Simon's scripts now points to `../common/simon_ell/`; the shared inputs `m_def.gp`,\n  `m_pts.gp` are renamed `M_def.gp`, `M_points.gp`; a header comment and a final `quit` were added.\n- `1.2.1c_odd_algebra.gp` (reviewer script `c5a.gp`): the line printing `norm(hA+hB*th+hC*th^2) + P` failed in PARI 2.15\n  (`norm` of a t_POLMOD whose coefficients contain free variables). It now computes the same norm as\n  `polresultant(t^3-3, hA+hB*t+hC*t^2, t)`, as `c5a2.gp` does.\n- `1.2.1d_sextic_L.gp` (reviewer script `c5b.gp`): two loops called `nfroots(nf, 'X^2 - ...)` with the field in the\n  variable x; PARI requires a variable of higher priority than x. They now use `Xh = varhigher(\"X\")`. With\n  this fix the script also prints the unit-independence line and the line `2/(pi0 pi1^2) ~ unit mask 5`.\n- `1.2.1a_cubic_field_K.gp`: one line added at the end (the unit-box enumeration cited by the section).\n- `1.2.1b_smooth577.py`: new 15-line SymPy check (smoothness of C_0 mod 577 and the derivatives 4, 12),\n  adapted from the original `smooth127.py` check; the adapted script is self-contained.\n\n## Not reproduced by a script\n- 1.2.1a: the local image sizes 8, 16 (above 2) and 4 (above 3) and the counts 16 / 4 of surviving pairs of the\n  source's full 2-descent (Simon's program does its own local computation; only its result is compared).\n- 1.2.1d: the source's extra test against the local image of J(Q_2) (the section uses q ∈ 2Z_2 directly).\n- 1.2.1e: the explicit order of the source's appendix 5.C (the section needs only disc M, given by `nfdisc`).\n- 1.2.1f: the individual tower facts of the section's proof (ramification of F/K, Hilbert symbols); h(M) = 1\n  certified implies the lemma.\n- 1.2.1g: the torsion reductions at (23,12,9), (83,11,12) of the source (`elltors` over M is used instead).\n- 1.2.1i: the source's second algorithm for the logarithms (division polynomials).\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.2.3: 2x^4 − y^4 + z^3 = 0\n\nPrimitive solutions: (0, ±1, 1), (±1, ±1, −1). Source: Avi Eisenberg's proof (`five.pdf`, chapter 1). The\nsource's programs are not available; all codes are independent reviewer scripts (renamed, two of them\nsplit or trimmed), plus two short checks added for the bundle. Each script starts with a comment naming its\ncode id and origin.\n\nNotation: f(A,B) = A^3 + 6A^2B + 6AB^2 + 4B^3, g(A,B) = A^3 + 3A^2B + 6AB^2 + 2B^3, F = f(t,1)g(t,1);\nK = Q(h), h^3 + 3h − 2 = 0, e = 1 − h − h^2, ρ = h − 1, π = −3 − h − h^2, d = 2 − h + h^2;\nE: y^2 = x(x − 12d)(x + 4d) over K. Mask bits 1, 2, 4, 8, 16 stand for (−1, e, ρ, h, π).\n\n## Software\n- PARI/GP 2.15.4 (`gp`).\n- Python 3 with cypari2 (PARI 2.15.4) for `1.2.3c_two_cover_2adic.py`, `1.2.3e_selmer.py` (the former also\n  imports numpy, unused); `1.2.3a_E6_congruences.py` and `1.2.3f_sieve29.py` are pure Python.\n\nRun every command from this folder. All gp scripts end with `quit`.\n\n## Run everything (about 15 seconds)\n```\ngp -q 1.2.3a_descent_E6.gp;  python3 -I 1.2.3a_E6_congruences.py\ngp -q 1.2.3b_cubic_field_K.gp\npython3 -I 1.2.3c_two_cover_2adic.py\ngp -q 1.2.3d_map_identities.gp;  gp -q 1.2.3d_infinity.gp\npython3 -I 1.2.3e_selmer.py;  gp -q 1.2.3e_kummer_points.gp\npython3 -I 1.2.3f_sieve29.py\ngp -q 1.2.3g_logs29.gp;  gp -q 1.2.3g_table_check.gp\n```\nAll codes were run here on 14 Oct 2026; `expected/<id>.txt` holds the output (one `### command` header per\nscript). Every expected line of the section was reproduced.\n\n## Codes\n\n### 1.2.3a — cubic descent over Q(√2); E_6\n`gp -q 1.2.3a_descent_E6.gp` (first part of reviewer script `basic.gp`; < 1 s): the expansions of\n(1 + √2)^{±1}(a + b√2)^3 (printed with w = √2), F and disc F, and `ellrank`, `elltors` of E_6: Y^2 = X^3 + 6X.\n`python3 -I 1.2.3a_E6_congruences.py` (added; < 1 s): the elementary 2-isogeny certificate quoted in the\nsection. Key output:\n```\nF=t^6 + 9*t^5 + 30*t^4 + 60*t^3 + 60*t^2 + 36*t + 8\ndisc F=[2, 26; 3, 6]\nE6 ellrank:[0, 0, 0, []]\ntors:[2, [2], [[0, 0]]]\nW^2 = 2U^4 + 3V^4: values mod 16 for (U,V) not both even [2, 3, 5]; none is a square mod 16: True\nW^2 = -U^4 + 24V^4: values mod 16 for (U,V) not both even [7, 8, 15]; none is a square mod 16: True\nW^2 = -2U^4 + 12V^4: values mod 16 for (U,V) not both even [10, 12, 14]; none is a square mod 16: True\nW^2 = 2U^4 - 12V^4: values mod 32 for (U,V) not both even [2, 20, 22]; none is a square mod 32: True\n#E6(F_5) = 4\n#E6(F_7) = 8\n#E6(F_13) = 10\n```\n\n### 1.2.3b — the cubic field K\n`gp -q 1.2.3b_cubic_field_K.gp` (second part of reviewer script `basic.gp` plus the unit-box line; < 1 s). Key output:\n```\ndisc:-216 sign:[1, 1] clgp:[1, [], []] fu:[Mod(-h^2 - h + 1, h^3 + 3*h - 2)] tu:[2, -1]\ncertify:1\nindex:1 zk:[1, h, h^2 + 2]\nN(e)=1 1/e=5*h^2 + 3*h + 17\n2-prime 1 e=1 f=1 isprinc h:1 rho:0\n2-prime 2 e=2 f=1 isprinc h:0 rho:1\n3: val pi:1 N(pi)=-3 N(h)=2 N(rho)=-2\nf(rf)=Mod(0, h^3 + 3*h - 2) g(rg)=Mod(0, h^3 + 3*h - 2) rf==2/(h-1):1\ndisc f:-864 disc g:-216 res:-512\n2 = h rho^2/e ? 1\n(1-h+h^2)^2 = -3 pi e ? 1\nnorm-1 elements a+bh+ch^2 in the box: List([[1, 0, 0], [17, 3, 5]])\n```\n(the box is 0≤a≤18, 0≤b≤3, 0≤c≤5; the two elements are 1 and e^{−1}.)\n\n### 1.2.3c — 2-adic two-cover descent for C_±\n`python3 -I 1.2.3c_two_cover_2adic.py` (reviewer script `lem11.py`; about 6 s). It labels square classes of\nK ⊗ Q_2 by Hilbert symbols against a spanning set (Gram rank = local dimension is asserted), partitions Z_2 into\nballs on which the classes of t − r_f, t − r_g are constant (plus two root balls, 2 + 8Z_2 and 5 + 8Z_2, where it\nsamples points, and four balls of the chart u = 1/t ∈ 2Z_2), and compares the local images, modulo the common\nQ_2 scalars, with the global candidates. Key output:\n```\nprime 1 Gram rank 3 expected dim 3\nprime 2 Gram rank 4 expected dim 4\naffine balls: stable 16 root 2\ninfinity-chart balls 4\nsigma 1 local image size (mod Q2 diag): 4\nsigma -1 local image size (mod Q2 diag): 4\nsigma 1 #pairs (before global diag quotient) surviving: 8  classes: [(0, 0)]\nsigma -1 #pairs (before global diag quotient) surviving: 0  classes: []\n```\nThis is the source's conclusion (only the trivial pair for σ = +1, nothing for σ = −1) by a different encoding\nand ball partition: the source's 42 balls and its 7-bit / 14-bit labels ((41,49,122,13,11) and the allowed\nlabel sets {0,1032,9328,9715}, {1,1160,9329,9714}) are NOT reproduced by any script.\n\n### 1.2.3d — quartic, map to E, differential, points at infinity\n`gp -q 1.2.3d_map_identities.gp` (reviewer script `ell.gp` without its last loop; < 1 s) and `gp -q 1.2.3d_infinity.gp`\n(reviewer script `inf.gp`; < 1 s). All identities are checked modulo h^3 + 3h − 2 and v^2 = q(t). Key output:\n```\n(t-rf)(t-rg)q - F = 0\ncurve eq residual: 0\ndifferential check Y - (q' + 2v(2t+k)) = 0\nP on E: 0  Q on E: 0\n1 + k u + m u^2 + n/2 u^3 ? (531/128*u^4 + ...)      [the difference starts at u^4]\nsign 1: xE=(-8*u^-1 + ...)*h^2 + ... + (16*u^-2 + ...)   [pole: image O]; dx/2y constant term -1/4\nsign -1: xE=(-3 + ...)*h^2 + (-6 + ...)*h + (-6*u + ...)  yE=(21 + ...)*h^2 + ...   [image Q]; dx/2y constant term 1/4\n```\n\n### 1.2.3e — 2-Selmer group of E over K\n`python3 -I 1.2.3e_selmer.py` (reviewer script `sel2E.py`; about 5 s): local images at the two places over 2, the place\nover 3 and the real place, spanned by T0, T1 and local points found by a seeded random search (seed 1; the\nsource's three local points are not used), then the 1024 global pairs. `gp -q 1.2.3e_kummer_points.gp`\n(reviewer script `kumglob.gp`): the mask pairs of T0, T1, P, Q. Key output:\n```\npr1 local image size 8 expected 8\npr2 local image size 16 expected 16\npr3 local image size 4 expected 4\nreal local image size 2 expected 2\nSelmer survivors: 16 [(0, 0), (0, 2), (4, 9), (4, 11), (10, 5), (10, 7), (14, 12), (14, 14), (18, 29), (18, 31), (22, 20), (22, 22), (24, 24), (24, 26), (28, 17), (28, 19)]\nT0: [18, 29]\nT1: [28, 19]\nP: [24, 26]\nQ: [24, 24]\n```\n(the last line of `kumglob.gp`, \"R check P\", is a leftover sanity print of the reviewer.)\n\n### 1.2.3f — reductions at 13 and 29; subgroup, 3-saturation, character test, infinity tuples\n`python3 -I 1.2.3f_sieve29.py` (reviewer script `sieve29.py`; < 1 s). Key output:\n```\np 13 h 9 #E 16\np 29 h 6 #E 36\nroots mod 29: [6, 9, 14]\nh 6 d 3 #E 36 exponent 18 ord P 18 ord Q 9  [exp/3]P (16, 28)  [exp/3]Q (16, 28)\nh 9 d 16 #E 24 exponent 12 ord P 12 ord Q 3  [exp/3]P (22, 17)  [exp/3]Q (22, 12)\nh 14 d 10 #E 36 exponent 18 ord P 3 ord Q 9  [exp/3]P None  [exp/3]Q (5, 15)\nsuccessive subgroup sizes [36, 324, 648, 1296]  product order 31104  index 24\nrf residues [12, 22, 18] rg residues [5, 8, 13]\nt mod 29 passing character test: []\n(0, 0, 0) True\n(1, 1, 1) True\n(the six mixed tuples) False\n```\n\n### 1.2.3g — 29-adic logarithms and annihilator\n`gp -q 1.2.3g_logs29.gp` (reviewer script `logs29.gp`) and `gp -q 1.2.3g_table_check.gp` (reviewer script `tab1F.gp`, which checks\nthe source's table of τ([n]G) mod 29^3 against these logarithms); < 1 s each. Key output:\n```\nroots mod 29^3: [13172, 14132, 21474]~\nh=6 pt P N=18 v(tau)=1 log/29 mod 29^2 = 647\nh=6 pt Q N=9 v(tau)=1 log/29 mod 29^2 = 672\nh=9 pt P N=12 v(tau)=1 log/29 mod 29^2 = 368\nh=9 pt Q N=3 v(tau)=1 log/29 mod 29^2 = 302\nh=14 pt P N=3 v(tau)=1 log/29 mod 29^2 = 410\nh=14 pt Q N=9 v(tau)=1 log/29 mod 29^2 = 764\nc mod 29: [7, 17, 8]  sum mod 29: 3\n[20677, 18, 647]  -> 647      (and the five other rows reproduce 368, 410, 672, 302, 764)\n```\n`logs29.gp` uses the first multiple of G in the formal group (N = 18, 12, 3 for P and 9, 3, 9 for Q); the source\nuses 18, 12, 18; the logarithms agree. The source's second algorithm (division polynomials) is not scripted.\n\n## Changes made to the reviewer scripts (no change to the mathematics)\n- `basic.gp` split into `1.2.3a_descent_E6.gp` and `1.2.3b_cubic_field_K.gp` (the definitions of f, g repeated\n  in the second); the concatenated output equals that of the original. One line added to the second: the\n  unit-box enumeration cited by the section.\n- `ell.gp` → `1.2.3d_map_identities.gp`: its last loop (images of infinity) did not parse in PARI 2.15 (a `\\\\`\n  comment inside an unbraced multi-line `for`) and was deleted; `1.2.3d_infinity.gp` does that computation.\n- Added `1.2.3a_E6_congruences.py` (the congruences mod 16/32 and #E_6(F_p), p = 5, 7, 13).\n- Header comments and final `quit` lines.\n\n## Not reproduced by a script\n- 1.2.3c: the source's literal 7-bit labels of the basis, its 42-ball table and its allowed label sets (see\n  above); only the conclusion is reproduced.\n- 1.2.3e: the source's three local points x = −6−7h−5h^2, −7−6h−7h^2, −7−6h+3h^2 (the script finds its own).\n- 1.2.3g: the division-polynomial algorithm of the source.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.2.4: x^5 + y^4 = 2z^2\n\nPrimitive solutions: (−1, ±1, 0), (1, ±1, ±1). Source: Avi Eisenberg's proof (`five.pdf`, chapters 4 and 6).\nThe source's programs are not available; all codes are independent reviewer scripts (renamed; `c4a.gp`\nsplit in two), plus short checks added for the bundle (marked \"added\"). Each script starts with a comment\nnaming its code id and origin.\n\nNotation: L = Q(θ), g = θ^5 + 5θ^3 + 5θ − 2; u1 = 1 − θ − θ^3, u2 = 1 − 3θ + θ^2 − θ^3, q = −1 + 2θ + 2θ^2 + θ^3,\nℓ = 1 + θ^2; E: ζ^2 = ξ^3 − 5θξ^2 + 5θ^2ξ and E': ζ^2 = ξ^3 + 10θξ^2 + 5θ^2ξ over L.\n\n## Software\n- PARI/GP 2.15.4 (`gp`); Denis Simon's `ell.gp`, `qfsolve.gp`, `resultant3.gp` from `../common/simon_ell/`\n  (only `1.2.4e_rank_simon.gp`).\n- Python 3 (pure Python for the `.py` scripts).\n\nRun every command from this folder. All gp scripts end with `quit`.\n\n## Run everything (about 15 seconds)\n```\ngp -q 1.2.4a_descent_j0.gp;  python3 -I 1.2.4a_congruences.py\ngp -q 1.2.4b_quintic_field.gp;  gp -q 1.2.4b_small_primes.gp\npython3 -I 1.2.4c_mod8.py\ngp -q 1.2.4d_covering_map.gp\ngp -q 1.2.4e_points.gp;  gp -q 1.2.4e_rank_simon.gp\npython3 -I 1.2.4f_sieve751.py\ngp -q 1.2.4g_logs751.gp\n```\nAll codes were run here on 14 Oct 2026; `expected/<id>.txt` holds the output (one `### command` header per\nscript). Every expected line of the section was reproduced.\n\n## Codes\n\n### 1.2.4a — fifth-power descent and the case j = 0\n`gp -q 1.2.4a_descent_j0.gp` (first lines of reviewer script `c4a.gp`): rank and torsion of Y^2 = X^3 − 10X^2 + 20X;\n(a + b√2)^5 = F + G√2; the rational parts F + 2G, 3F + 4G for j = 1, 2. `python3 -I 1.2.4a_congruences.py`\n(added): the elementary congruences of the section. Key output:\n```\nLemma 4.1 curve: conductor 1600 ellrank [0, 0] analytic rank 0 torsion [2, [2]]\n(a+b sqrt2)^5 check: 1 1\nj=1: 1  j=2: 1\nF mod 16 for a = 5u^2, u odd: [5, 13]  odd squares mod 16: [1, 9]\n5r^4-10r^2s^2+4s^4 mod 16, (r,s) not both even: [4, 5, 12, 13, 15]  fourth powers mod 16: [0, 1]\nE : ... d=2:  W^2 = 2U^4-10U^2V^2+10V^4: values mod 25 ...: [2, 7, 10, 12, 17, 22]; none is a square mod 25: True\nE': ... d=2:  W^2 = 2U^4+20U^2V^2+10V^4: ... none is a square mod 25: True\nE': d=-2: W^2 = -2U^4+20U^2V^2-10V^4: ... none is a square mod 25: True\nE': d=-1: W^2 = -U^4+20U^2V^2-20V^4: values mod 16 ...: [12, 15]; none is a square mod 16: True\n#E(F_3) = 2\n#E(F_7) = 10\n```\n\n### 1.2.4b — the quintic field L and the valuation lemma\n`gp -q 1.2.4b_quintic_field.gp` (rest of reviewer script `c4a.gp`; < 1 s) and `gp -q 1.2.4b_small_primes.gp` (added; the\nprime ideals of norm < 29 with the generators listed in the section, and the quadratic characters of u1, u2).\nKey output:\n```\ndisc(g)=[2, 6; 5, 5] disc(L)=[2, 6; 5, 5] index=1 sign=[1, 2] h=1 certify=1\nnorms u1,u2,q,l,de,chi: [1, 1, 4, 5, 81, -1296]\nbnfisunit u1,u2: [-1, 0, 1]~[0, 1, 1]~\nminpoly et: x^5 + 10*x^4 + 20*x^3 + 40*x^2 + 20*x + 8 disc [2, 22; 5, 5]\nminpoly chi: x^5 + 20*x^4 + 180*x^3 + 720*x^2 + 1620*x + 1296 disc [2, 22; 3, 12; 5, 5]\nN(a-et b)-F1 = 0   N(3a-chi b)-81F2 = 0\nprimes over 3: [[1, 1], [1, 4]]\nval de at P3: [0, 1] val chi at P3: [0, 1]\nprimes over 2: [[1, 1], [2, 2]] val theta: [1, 0] val q: [0, 1]\nprimes over 5: [[5, 1]] val l: [1]\nF1(2,1)=560 F2(3,1)=4885\nF1 mod 5 == (a-2b)^5: 1  F2 mod 5 == 3(a-3b)^5: 1\ng mod 3 factor: [x + 2, 1; x^4 + x^3 + 2, 1]\n   prime ideal of norm 2 (e=1) generated by listed element(s) [x]\n   ... (one line for each prime ideal of norm 2, 4, 3, 5, 7, 11, 13, 17, 23; none of norm 19)\ncharacter rows (u1,u2) at (7,theta-5): [0, 1]  at (17,theta-3): [1, 0]\n```\n(`bnfisunit` gives the exponents on PARI's fundamental units and −1: u1, u2 and −1 generate the units modulo\nsquares.)\n\n### 1.2.4c — the computation modulo 8\n`python3 -I 1.2.4c_mod8.py` (reviewer script `c4mod8.py`; < 1 s). Key output:\n```\nsquares: 336\n   (i,k)= (0, 0)  residues: [(1, 0, 0, 0, 0), (1, 0, 4, 4, 4)]\ng1  #lhs= 32  intersections (1,u1,u2,u1u2): [2, 0, 0, 0]\ng2  #lhs= 32  intersections (1,u1,u2,u1u2): [0, 0, 0, 0]\n```\n\n### 1.2.4d — the elliptic covering\n`gp -q 1.2.4d_covering_map.gp` (reviewer script `c4map.gp`; < 1 s; identities reduced modulo v^2 = q(t)). Key output:\n```\nq(t)*(t-eta) == h1(t): 1\ncurve identity (should be 0): 1\ndU/dt = V/v identity: 1\nxi at - branch -> Mod(0, ...)  zeta -> Mod(0, ...)\nvaluations (leading) xs:0 zs:0\n```\n(the lines ending in `: 1` test `... == 0`, so 1 means the identity holds.)\n\n### 1.2.4e — rank 2 over L\n`gp -q 1.2.4e_rank_simon.gp` (reviewer script `c4rank.gp`; about 6 s): Simon's `bnfell2descent_viaisog`. The script first requires `bnfcertify(K) = 1` (line added in the bundle), so the class group and units that\nthe descent uses are proved and the result does not depend on GRH.\n`gp -q 1.2.4e_points.gp` (reviewer script `c4pts.gp` plus one added line; < 1 s). Key output:\n```\n#S^(phi')(E'/K)     = 4\n#S^(phi)(E/K)       = 4\nrank                = 2\nRESULT [rank_lower, Selmer-based bound + tors, pts] = [2, 3]  time ms ...\non curves: P 1 Q 1 S on E' 1\nphihat(S) on E: 1  phihat(S) == P+T0: 1\n[a,b,c] divisible by 2? 0          (all seven nonzero combinations of P, Q, T0)\ntorsion E: [2, [2]]  E': [2, [2]]\nsquares in L: 5th^2/l 1  xi(Q)/(u1u2l) 1  xi(S)/(u1u2l) 1\n```\n\n### 1.2.4f — sieve at 751\n`python3 -I 1.2.4f_sieve751.py` (reviewer script `c4sieve.py`; 1–2 s). Key output:\n```\nroots of g mod 751: [65, 168, 348, 386, 535]\ntheta=65 #E=768 gen=(7, 285) T=(327, 0) coordP=(2, 0) coordQ=(173, 1) T0=(192, 0)\ntheta=168 #E=768 gen=(3, 240) T=(353, 0) coordP=(82, 0) coordQ=(295, 1) T0=(192, 0)\ntheta=348 #E=768 gen=(2, 293) T=(463, 0) coordP=(302, 0) coordQ=(357, 0) T0=(192, 0)\ntheta=386 #E=768 gen=(4, 72) T=(69, 0) coordP=(314, 0) coordQ=(85, 0) T0=(192, 0)\ntheta=535 #E=768 gen=(1, 7) T=(554, 0) coordP=(342, 0) coordQ=(379, 0) T0=(192, 0)\n|H|= 36864  index= 7247757312\nP 3-part (coef of order-3 elt): [1, 2, 1, 1, 0]  raw mod 3: [2, 1, 2, 2, 0]\nQ 3-part (coef of order-3 elt): [1, 2, 0, 2, 2]  raw mod 3: [2, 1, 0, 1, 1]\nresidues with points in all factors: 57  product points tested: 1674  survivors: [('inf', (None, None, None, None, None)), ('inf', ((236, 412), ...))]\n```\n(7247757312 = 2^28·3^3; the second survivor is (P_1, …, P_5) at t = ∞; the \"raw mod 3\" rows are the matrix\nprinted in the section.)\n\n### 1.2.4g — 751-adic logarithms and annihilator\n`gp -q 1.2.4g_logs751.gp` (reviewer script `c4log.gp`; < 1 s). Key output:\n```\nroots mod p^3: [30368252, 51283705, 229643381, 365409950, 170424214]~\norders P: [192, 192, 192, 192, 64]  Q: [384, 384, 128, 384, 384]\nlog P / p mod p^2: [423722, 191178, 352807, 294331, 252200]\nlog Q / p mod p^2: [330447, 456587, 117975, 48293, 396869]\nc mod p: [626, 131, 473, 0, 0]  sum c mod p: 479\ncheck c.logP, c.logQ (mod p^2 approx): 0 0\nrank mod p of [LP;LQ] first 3 coords: 2\n```\n(−479 ≡ 272 mod 751 for the other branch. The script uses `ellformallog` on the first multiple in the formal\ngroup; the source's division-polynomial algorithm is not scripted.)\n\n## Changes made to the reviewer scripts (no change to the mathematics)\n- `c4a.gp` split into `1.2.4a_descent_j0.gp` and `1.2.4b_quintic_field.gp` (F, G redefined in the second); the\n  concatenated output equals that of the original.\n- `c4rank.gp`: Simon's files loaded from `../common/simon_ell/` (same parameters as the reviewer's header).\n- Added: `1.2.4a_congruences.py`, `1.2.4b_small_primes.gp`, and the last line of `1.2.4e_points.gp` (the square\n  classes 5θ^2/ℓ, ξ(Q)/(u1u2ℓ), ξ(S)/(u1u2ℓ)).\n- Header comments and final `quit` lines.\n\n## Not reproduced by a script\n- 1.2.4e: the source's own local computation (counts 16/32 → 16/16 → 4/4 at the real place and 5, at the linear\n  and the quartic dyadic place; the dyadic completions mod 256 and mod 16 with 65536 elements, 49152 units, 1648\n  squares) and its surviving masks 0, 6, 32, 38. Simon's program reaches rank 2 independently, and the added\n  line checks the classes 32, 38 quoted in the section (5θ^2/ℓ is the class of T0 and of T0'; Q and S give 38).\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.2.6: x^5 + y^4 = 3z^2\n\nPrimitive solutions: (−1, ±1, 0), (11, ±29, ±538). Source: Avi Eisenberg's proof (`five.pdf`, chapter 3).\nThe source prints no verifier code for this chapter. All codes are independent reviewer scripts or small\nhelpers of the paper's writers (W8). Each script starts with a comment naming its code id and origin.\n\nNotation: L = Q(ϑ), g = ϑ^5 − 5ϑ^3 + 5ϑ − 4; K = Q(√5) = Q(e), e^2 = e + 1, d = 2e − 1 = √5; ρ = −21 + 12e,\nτ = −3 + 6e; E0, E1, E2 over K as in the section (E1: ξ(ξ + 2eρ)(ξ − 2eτ), E2: ξ(ξ − eρ)(ξ − eτ)).\n\n## Software\n- PARI/GP 2.15.4 (`gp`); Denis Simon's `ell.gp`, `qfsolve.gp`, `resultant3.gp` from `../common/simon_ell/`\n  (only `1.2.6d_rank_simon.gp`).\n- Python 3 (pure Python; SymPy 1.14 for `1.2.6b_congruences.py`).\n\nRun every command from this folder. All gp scripts end with `quit`.\n\n## Run everything (about 10 seconds)\n```\ngp -q 1.2.6a_identities.gp;  python3 -I 1.2.6a_mod8.py;  gp -q 1.2.6a_quintic_field.gp\npython3 -I 1.2.6b_congruences.py\npython3 -I 1.2.6c_local_test.py\ngp -q 1.2.6d_rank_simon.gp;  gp -q 1.2.6d_kummer.gp;  gp -q 1.2.6d_reductions.gp\npython3 -I 1.2.6e_coset_sieves.py\ngp -q 1.2.6f_logs.gp;  python3 -I 1.2.6f_constants.py\n```\nAll codes were run here on 14 Oct 2026; `expected/<id>.txt` holds the output (one `### command` header per\nscript). Every expected line of the section was reproduced.\n\n## Codes\n\n### 1.2.6a — the quintic field L and the mod-8 obstruction\n`gp -q 1.2.6a_identities.gp` (reviewer script `c3a.gp`; identities and minimal polynomials; `et`, `chi` are μ1, μ2 of the\nsection), `python3 -I 1.2.6a_mod8.py` (reviewer script `c3a_mod8.py`; enumeration in R = (Z/8)[T]/(g)),\n`gp -q 1.2.6a_quintic_field.gp` (helper `quint.gp`: `bnfinit` + `bnfcertify`, units, primes over 2 and 5,\nsquare characters of e1, e2). Each < 1 s. Key output:\n```\nirreducible: 1  disc(g)=[2, 4; 3, 2; 5, 5]\nminpoly et: x^5 + 15*x^4 + 120*x^3 + 360*x^2 + 720*x + 432  chi: x^5 - 60*x^3 + 360*x^2 - 900*x + 864\ndisc h1: [2, 24; 3, 10; 5, 5] disc h2: [2, 16; 3, 10; 5, 5]\nN(2a-et b) - 16F1: 0\nN(2a-chi(a+b)) - 8F2: 0\nj=1 form: 1\nj=2 form: 1\ndistinct squares mod 8: 336\nj= 1  #lhs= 16  intersections: [0, 0, 0, 0, 0, 0, 0, 0]\nj= 2  #lhs= 16  intersections: [0, 0, 0, 0, 0, 0, 0, 0]\ndisc 450000 sign [1, 2] h 1 zk [1, x, x^2 - 2, x^3 - x^2 - 3*x + 2, x^4 - 4*x^2 - x + 2]\ncertify 1\nN(e1),N(e2),N(th-2),N(q)=[1, 1, 2, 4]\nisunit e1,e2: [1, 0, 0]~[0, -1, 0]~\nprimes over 5 [e,f]: [[5, 1]]\nresidues at (7,th-3): [4, 6] at (13,th-9): [6, 6]\n```\n(`zk` is a PARI basis of Z[ϑ], index 1; 450000 = 2^4·3^2·5^5. `fact 2` shows (2) = 𝔭_1𝔭_2^2 with residue degrees\n1, 2. Residues 4 = square, 6 = non-square mod 7 and mod 13 give the independent character rows of e1, e2.)\n\n### 1.2.6b — congruences and identities of the reduction to C and H\n`python3 -I 1.2.6b_congruences.py` (section \"1.2.6\" of the helper `w8_checks.py`; SymPy; 2 s). No reviewer\nscript exists for these (P2 checked them by hand). Key output:\n```\n(a+b sqrt3)^5 = a*(a**4 + 30*a**2*b**2 + 45*b**4)  + sqrt3* b*(5*a**4 + 30*a**2*b**2 + 9*b**4)\n  j=1 rational part = 2*F + 3*G ? True\n  j=2 rational part = 7*F + 12*G ? True\nu even, b odd: c^2 mod 16 = [13]  squares mod 16: [0,1,4,9]\n  (3.1) RHS mod 16, r,s odd           : [7, 15]\n  (3.1) RHS mod 16, r=0 mod 4, s odd  : [13]\n  (3.1) RHS mod 16, r=2 mod 4, s odd  : [1]\n  (3.1) RHS mod 16, r odd, s=2 mod 4  : [5]\n  (3.1) RHS mod 16, r odd, 4|s        : [1]\n  (5r^4+9s^4) mod 3 (3!|rs): [2]\nN(alpha) - (r^4-15r^2s^2+45s^4): 0\n  sigma=1: Y^2 - f(X) for (3.11): 0\n  sigma=-1: Y^2 - f(X) for (3.11): 0\nf(-9) = 46656 = 216^2 = 46656\n  (u,r,s)=(1, 1, 0) -> (x,y,z)=(-1,1,0), x^5+y^4-3z^2=0\n  (u,r,s)=(1, 2, 1) -> (x,y,z)=(11,-29,538), x^5+y^4-3z^2=0\n```\n\n### 1.2.6c — pointwise local test for X = σq^2\n`python3 -I 1.2.6c_local_test.py` (reviewer script `c3local.py`; pure Python, 2 s). Square classes in K_2 (unit part mod 8\nup to unit squares) and K_3 (residue in F_9), q sampled mod 2^10 (and 2^6 as a check), 3^6, and the reciprocal\nballs; candidate pairs with basis (−1, e, 2, 3, √5), mask bits 1, 2, 4, 8, 16. Key output:\n```\nsigma 1 local image sizes at 2: 4 (coarse 4 ) at 3: 2\n  norm cond: 256  after 2: 4  after 3: 1  survivors: [(0, 0)] [((1, 0), (1, 0))]\nsigma -1 local image sizes at 2: 6 (coarse 6 ) at 3: 2\n  norm cond: 256  after 2: 8  after 3: 2  survivors: [(11, 13), (26, 25)] [((0, -3), (-6, 0)), ((6, 3), (3, -6))]\n```\n(mask 11 = −3e, 13 = −6, 26 = 3e√5 = 6 + 3e, 25 = −3√5 = 3 − 6e. The script samples q more finely than the\nsection's balls mod 16; by the precision argument of the section this is enough.)\n\n### 1.2.6d — ranks, torsion and Kummer classes of E0, E1, E2 over K\n`gp -q 1.2.6d_rank_simon.gp` (reviewer script `c3rank.gp`; Simon's `bnfellrank`, PARI's `elltors`; 1 s; it first requires\n`bnfcertify(K) = 1`, a line added in the bundle, so the descent is unconditional),\n`gp -q 1.2.6d_kummer.gp` (reviewer script `c3kum.gp`), `gp -q 1.2.6d_reductions.gp` (helper `c3tors.gp`: orders modulo\nthe primes above 11 and 19). Key output:\n```\nE0: [r,s]=[0, 2]  pts=[]\nE1: [r,s]=[1, 3]  pts=[..., [Mod(18*y, y^2 - y - 1), Mod(72, y^2 - y - 1)]]\nE2: [r,s]=[1, 3]  pts=[[Mod(-2*y + 1, y^2 - y - 1), Mod(-5*y + 5, y^2 - y - 1)], ...]\nE0 torsion: [4, [2, 2]]\nE1 torsion: [4, [2, 2]]\nE2 torsion: [4, [2, 2]]\nP1 on E1: 1  P2 on E2: 1\nE1: independence check done; ...          (no \"dependent combination\" line)\nE2: independence check done; ...\nE1: delta(P1)*(2e,6) squares? [1, 1]\nE2: delta(T0)*(-e,3d) squares? [1, 1]\nE0 p=11 roots [4, 8] orders [8, 12]\nE0 p=19 roots [5, 15] orders [16, 12]\nE1 p=11 roots [4, 8] orders [8, 12]\nE1 p=19 roots [5, 15] orders [16, 28]\nE2 p=11 roots [4, 8] orders [12, 16]\nE2 p=19 roots [5, 15] orders [28, 24]\n```\nSimon's descent replaces the source's full 2-descent; the source's 2-adic points (ξ = −1−e, −4+4e on E0; −2−e,\n1+e on E1; e, 1−2e on E2) and its intermediate counts 64, 16, 4, 4, 4 / 64, 16, 8, 8, 8 are not recomputed.\n\n### 1.2.6e — coset sieves (E1 at 89, E2 at 11)\n`python3 -I 1.2.6e_coset_sieves.py` (reviewer script `c3sieve.py`; pure Python, < 1 s). Key output:\n```\nE1 p=89 orders=[80, 96] |H|=480 index=16 |coset|=60 (O,O) in coset: False\n  survivors: [(80, (2, 17), (16, 17)), (80, (2, 72), (16, 72))]\nE2 p=11 orders=[12, 16] |H|=96 index=2 |coset|=12 (O,O) in coset: False\n  survivors: [(0, (0, 0), (0, 0))]\n```\n\n### 1.2.6f — p-adic logarithms, annihilators, derivative constants\n`gp -q 1.2.6f_logs.gp` (reviewer script `c3log.gp`; `ellformallog`, precision p^8) and `python3 -I 1.2.6f_constants.py`\n(last part of the helper `w8_checks.py`). Key output:\n```\nE1 at 89: [e mod p^3, n, log/p mod p^2] = [[215123, 20, 2490], [489847, 24, 2948]]\nE2 at 11: [[37, 6, 117], [1295, 8, 76]]\nE1 at 89: lambda1/144 in the two embeddings = [32, 78] ; c1.(.) = 63 26\nE2 at 11: c2.(-e/W) for W=(1,1),(1,-1),(-1,1),(-1,-1): [5, 3, 8, 6]\nc1 from (2490,2948): (11, 2)  c2 from (117,76): (10, 4)\n```\n(the source's formal parameters t_n mod p^3, 202386, 657176 at 89 and 1067, 33 at 11, are not printed.)\n\n## Changes made to the scripts (no change to the mathematics)\n- `c3a.gp` → `1.2.6a_identities.gp`: its first line `K = bnfinit(g,1); nf=K.nf;` ran before g was defined\n  (so it built the trivial field) and was never used; deleted. Output unchanged.\n- `c3rank.gp` → `1.2.6d_rank_simon.gp`: Simon's files loaded from `../common/simon_ell/`; the stray line\n  ` simon_hdr.gp` (an error message only) deleted; two comment lines began with a single `\\`, which gp reads as\n  the metacommands `\\t` and `\\c` (they printed the list of PARI types and of all functions, and waited for\n  RETURN); they now begin with `\\\\`. The computed lines are unchanged.\n- `quint.gp` → `1.2.6a_quintic_field.gp`: one line added (the primes over 5).\n- `w8_checks.py` split: section \"1.2.6\" → `1.2.6b_congruences.py` (with the two symbols `d = sqrt(5)`, `tt` that\n  the original defined in its 1.2.7 part), Chabauty constants → `1.2.6f_constants.py`.\n- Header comments and final `quit` lines.\n\n## Not reproduced here\n- 1.2.6d: the source's own 2-descent data (see above).\n- The optional brute-force search of a reviewer (`brute.py`; not cited by the section).\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.2.7: x^5 + y^4 + 3z^2 = 0\n\nPrimitive solutions: (−1, ±1, 0). Source: Avi Eisenberg's proof (`five.pdf`, chapter 2). The source prints no\nverifier code for this chapter. All codes are independent reviewer scripts or a small helper of the\npaper's writers (W8). Each script starts with a comment naming its code id and origin.\n\nNotation: H: Y^2 = f(X) = X(X^2 − 30X + 45)(X^2 − 45), J = Jac(H); K = Q(√5), e = (1 + √5)/2,\nθ2 = 15 + 6√5, θ3 = 3√5.\n\n## Software\n- PARI/GP 2.15.4 (`gp`).\n- Python 3: `1.2.7a_mod16.py` pure Python; `1.2.7a_identities.py` SymPy 1.14; `1.2.7b_selmer_J.py` cypari2\n  (PARI 2.15.4).\n\nRun every command from this folder.\n\n## Run everything (about 10 seconds)\n```\npython3 -I 1.2.7a_mod16.py;  python3 -I 1.2.7a_identities.py\npython3 -I 1.2.7b_selmer_J.py\ngp -q 1.2.7c_point_counts.gp\n```\nAll codes were run here on 15 Oct 2026; `expected/<id>.txt` holds the output (one `### command` header per\nscript). Every expected line of the section was reproduced.\n\n## Codes\n\n### 1.2.7a — congruences mod 16 and algebraic identities of the descent\n`python3 -I 1.2.7a_mod16.py` (reviewer script `mod16.py`; < 1 s) and `python3 -I 1.2.7a_identities.py` (section \"1.2.7\"\nof the helper `w8_checks.py`; 3 s). Key output:\n```\nF mod16 with a=-5u^2, opposite parity: [1, 9, 13]  -5c^2 mod16: [0, 3, 11, 12]\nF mod16 general opposite parity: [1, 5, 9, 13]\nc^2=F, u even, b odd: [13]  squares: [0, 1, 4, 9]\nneg case RHS mod16: [3, 7, 11]  u^4 odd: [1]\nparity r,s 1 1 RHS mod16 [5, 13]\nparity r,s 0 1 RHS mod16 [9, 13]\nparity r,s 1 0 RHS mod16 [1, 13]\nr odd, s=2 mod 4: [13]\n(a+b sqrt(-3))^5 = a*(a**4 - 30*a**2*b**2 + 45*b**4)  + sqrt(-3)* b*(5*a**4 - 30*a**2*b**2 + 9*b**4)\nF mod 16 when a=-5u^2, a+b odd: [1, 9, 13]  -5c^2 mod 16: [0, 3, 11, 12]\n  (a even forces u even, hence 4|a: True )\nN(alpha) - (r^4+15r^2s^2+45s^4): 0\nY^2 - f(X) for (2.11): 0\nf(-1) = 3344  f(-1/3) = 200384/243\n```\nThe line \"general opposite parity: [1, 5, 9, 13]\" shows that the value set {1, 9, 13} needs a = −5u^2. With\n\"parity r,s 1 0\" = {1, 13} and \"r odd, s = 2 mod 4\" = {13}, the case r odd, 4 | s gives 1.\n\n### 1.2.7b — 2-descent on J = Jac(H)\n`python3 -I 1.2.7b_selmer_J.py` (reviewer script `sel2J.py`; about 7 s). Square classes of Q_p × K_p × K_p are labelled by\nHilbert symbols against spanning sets (the Gram ranks are asserted); the local image at p = 2, 3, 5 is spanned by\nthe images v0, v1 of the 2-torsion and of the points X = −1, −1/3 (and further random points, seed 2, if\nneeded); the script asserts the sizes 16, 4, 4 = 2^{dim J(Q_p)/2J(Q_p)} and tests the 1024 global candidates\n(q, g_a, g_b), g in the span of (−1, e, 2, 3, √5) with masks of bits 1, 2, 4, 8, 16 and q fixed by the norm\ncondition. Key output:\n```\ndisc f = [2, 14; 3, 20; 5, 10]\nK: h = 1  fu = [Mod(-1/2*y + 1/2, y^2 - 5)]\np 2 local image size 16 expected 16\np 3 local image size 4 expected 4\np 5 local image size 4 expected 4\nSelmer survivors (a,b,q): [(0, 0, 1), (2, 26, -5), (25, 3, 5), (27, 25, -1)]\n```\nThese four are the source's (0,0), (2,26), (25,3), (27,25), the images of 0, T0, T1, T0 + T1. The method (Hilbert\nsymbols) differs from the source's mod-8 labels; the source's intermediate counts 64, 16 are not printed.\n\n### 1.2.7c — point counts and #J(F_p)\n`gp -q 1.2.7c_point_counts.gp` (reviewer script `jcount.gp` plus added lines; < 1 s). Key output:\n```\np=7 #J(F_p)=20  #H(F_p)=2\np=11 #J(F_p)=144  #H(F_p)=12\np=13 #J(F_p)=196  #H(F_p)=14\ndirect #H(F_7)=2\ndirect #H(F_11)=12\ndirect #H(F_13)=14\np=7 #H(F_p)=2 #H(F_p^2)=50  (h1^2+h2)/2-p=20  #J(F_p) from charpoly=20\np=11 #H(F_p)=12 #H(F_p^2)=166  (h1^2+h2)/2-p=144  #J(F_p) from charpoly=144\np=13 #H(F_p)=14 #H(F_p^2)=222  (h1^2+h2)/2-p=196  #J(F_p) from charpoly=196\n```\n\n## Changes made to the scripts (no change to the mathematics)\n- `jcount.gp` → `1.2.7c_point_counts.gp`: added a direct count of #H(F_{p^2}) and the comparison of\n  (#H(F_p)^2 + #H(F_{p^2}))/2 − p with the characteristic-polynomial value (a helper script used the\n  values 50, 166, 222 as inputs; they are now computed).\n- `w8_checks.py` → `1.2.7a_identities.py`: only its section \"1.2.7\" is kept, without that final formula loop.\n- Header comments.\n\n## Not reproduced here\n- 1.2.7b: the source's mod-8 label tables and its intermediate counts 64, 16 (optional).\n- The optional brute-force search of a reviewer (`search.cpp`; not cited by the section).\n"
    },
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.3.1: x^4 + x^3y + xy^3 = z^4\n\nNo solution with xyz ≠ 0. Source: Avi Eisenberg's proof `t9t6.pdf`, part (a) and appendix C.1. The source\ndescribes a reproduction package that is not in our files and prints no verifier code; its appendix C.1 prints\nc_P and a_P of the map E1 → G1 but not b_P. All codes are independent reviewer scripts (renamed, a few\nlines fixed) plus one fast identity check added for the bundle. Each script starts with a comment naming its\ncode id and origin.\n\nNotation: F = Q(α), α^3 + α + 1 = 0; g(t) = t^2 + αt + α^2 + 1, so (t − α)g(t) = t^3 + t + 1;\nE0: y^2 = x^3 + (3α^2 + 4)x with P = (−α, 2); E1: W^2 = X^3 + k1X, k1 = bd − 4e (b, d, e from the quartic of\n1.3.1c), with the point R1 of 1.3.1c; G0, G1 the genus-one curves of the section.\n\n## Software\n- PARI/GP 2.15.4 (`gp`); Denis Simon's `ell.gp` from `../common/simon_ell/` (only `1.3.1d_rank_simon.gp`).\n- Python 3 with cypari2 (PARI 2.15.4) for `1.3.1c_mapcheck.py`, `1.3.1d_*.py`, `1.3.1e_*.py`,\n  `1.3.1f_strassmann.py`; `1.3.1b_local_exclusion.py` is pure Python.\n\nRun every command from this folder. All gp scripts end with `quit`.\n\n## Run everything (about 10 seconds)\n```\ngp -q 1.3.1a_field_F.gp;  gp -q 1.3.1a_powers.gp\npython3 -I 1.3.1b_local_exclusion.py\ngp -q 1.3.1c_conic_quartic.gp;  gp -q 1.3.1c_identities.gp;  python3 -I 1.3.1c_mapcheck.py;  gp -q 1.3.1c_point_R1.gp\ngp -q 1.3.1d_rank_simon.gp;  python3 -I 1.3.1d_index.py;  python3 -I 1.3.1d_index_place11.py\npython3 -I 1.3.1e_sieve47.py;  python3 -I 1.3.1e_minus_R1.py\npython3 -I 1.3.1f_strassmann.py\n```\nAll codes were run here on 15 Oct 2026; `expected/<id>.txt` holds the output (one `### command` header per\nscript; ANSI colour codes removed). Every expected line of the section was reproduced (for 1.3.1c the\nslow script `map2.gp` is replaced, see below).\n\n## Sign convention for R1 (read before 1.3.1c–f)\nThe map E1 → G1 used by all scripts is the one P4 derived, t = −2ND/(V0N^2 + 2m0ND + D^2) with\nN = W − bX/2 − d, D = 2X − b^2/2 (the source does not print b_P). With this map the point of E1 with t = 0\n(besides O) is (X1, W) with X1 = b^2/4 = (264α^2 − 324α − 212)/961 and W = −(23104α^2 + 33848α + 49008)/29791,\ni.e. the NEGATIVE of the point R1 printed in the section; the section's R1 is the point where this formula reads\n0/0 (handled by `1.3.1e_minus_R1.py`). The section's sign corresponds to composing the map with [−1] on E1.\nThe scripts call their point \"R1\"; all statements of 1.3.1d–f (halving, index, sieve, Strassmann) are\nunchanged under R1 ↦ −R1, so only the label differs. `1.3.1c_point_R1.gp` prints this comparison.\n\n## Codes\n\n### 1.3.1a — the cubic field F\n`gp -q 1.3.1a_field_F.gp` (reviewer script `a1.gp` without its last lines) and `gp -q 1.3.1a_powers.gp` (reviewer script `a1b.gp`\nvia `nfroots`). Key output:\n```\ndisc -31 sign [1, 1] h 1 cyc []\nbnfcertify 1\nfu [Mod(x, x^3 + x + 1)] tu [2, -1]\nnorm alpha -1\nfact 31 [[31, [-14, 0, 1]~, 2, 1, ...], 2; [31, [-3, 0, 1]~, 1, 1, ...], 1]\nN(A)=31 fact(A) Mat([[31, [-14, 0, 1]~, 2, 1, ...], 1])\nN(alpha-3)=-31 N(3a^2+4)=31 N(3a^2+a+1)=31\nidentity: 0\nG(X, aX) = 3*X^2*x^2 + X^2\n(t-a)g(t) = (t^3 + t + 1)\ns=1 k=0 square? 1 4th? 1          (all other (s,k): 4th? 0)\npairs equal mod 4th powers: 0\nroots of g in F: []\n-(3a^2+4) sq: 0 (3a^2+4) sq: 0\n-(3a^2+a+1) sq: 0 (3a^2+a+1) sq: 0\n```\n(A = 3α^2 + 1 generates the ramified prime above 31; G(X, αX) = (3α^2 + 1)X^2.)\n\n### 1.3.1b — local exclusion of k = 3 (at 3) and k = 2 (at 13)\n`python3 -I 1.3.1b_local_exclusion.py` (reviewer script `loc.py`; < 1 s; any command-line argument adds the slower case\n(13, 2, 2)). Key output:\n```\n3 1 k= 3 {1: 0, -1: 0} {1: [], -1: []}\n3 2 k= 3 {1: 0, -1: 0} {1: [], -1: []}\n13 1 k= 2 {1: 0, -1: 0} {1: [], -1: []}\ncontrol 3 2 k= 0 {1: 3, -1: 3}\ncontrol 3 2 k= 1 {1: 9, -1: 9}\ncontrol 13 1 k= 0 {1: 6, -1: 6}\ncontrol 13 1 k= 1 {1: 3, -1: 3}\ncontrol 3 2 k= 2 {1: 3, -1: 3}\ncontrol 13 1 k= 3 {1: 9, -1: 9}\n```\n(Without an argument the line for (13, 1, 2) is printed twice.)\n\n### 1.3.1c — the maps G0 → E0 and G1 → E1\n- `gp -q 1.3.1c_conic_quartic.gp` (reviewer script `map1.gp`, coefficient extraction fixed): conic parametrisation,\n  Q(m0) = V0^2, the quartic in s, a2 = a6 = 0, k1 = bd − 4e and k1/(3α^2+α+1) = ((40α^2−4α+28)/31)^4.\n- `gp -q 1.3.1c_identities.gp` (added; < 1 s): the same formulas as exact identities on both sides — the map\n  quartic → E1 lands on E1, the inverse E1 → quartic lands on the quartic (reduced modulo W^2 = cubic), both\n  composites are the identity, and quartic → G1 lands on G1 (Y^4 + αg(t) = 0, reduced modulo yq^2 = quartic).\n- `python3 -I 1.3.1c_mapcheck.py` (reviewer script `mapcheck.py`): 300 random points modulo the prime q = 1000000000039.\n- `gp -q 1.3.1c_point_R1.gp` (reviewer script `sieve47.gp`, which also computed data for its own sieve): the point R1,\n  t1(R1) = 0, t1(−R1) = 0/0, and the sign comparison with the section's R1.\nKey output:\n```\nconic check: Mod(0, y^3 + y + 1)\nQ(m0)-V0^2 = Mod(0, y^3 + y + 1)  t(m0)=Mod(0, y^3 + y + 1)\nb=-12/31*y^2 - 36/31*y + 4/31 c=0 d=448/961*y^2 + 848/961*y + 1504/961 e=-656/961*y^2 + 1504/961*y - 1104/961\na2=Mod(0, y^3 + y + 1) a4=39936/29791*y^2 - 205568/29791*y + 169216/29791 a6=Mod(0, y^3 + y + 1)\na4 - k1 = Mod(0, y^3 + y + 1)\nk1/(3a^2+a+1) = 3280128/923521*y^2 - ... vs ((40a^2-4a+28)/31)^4 = 3280128/923521*y^2 - ...\nquartic -> E1 lands on E1: 1\nE1 -> quartic lands on the quartic: 1\nquartic -> E1 -> quartic is the identity: [1, 1]\nE1 -> quartic -> E1 is the identity: [1, 1]\nquartic -> G1 lands on G1 (Y^4 + al g(t) = 0): 1\nq= 1000000000039  points: 300  on E: 300  roundtrip: 300  Y^4=-a g(t): 300  a6 check c= 0\nR1 on E1: 1 t1(R1) num/den: [0, ...]\n t1(-R1) num/den: [0, 0]\nR1 == paper's R1: 0   -R1 == paper's R1: 1\n```\nThe map G0 → E0 (t = (3α^2 + 4 − 2αx − x^2)/(4x), Y = y/(2x)) is a hand identity in the section; it is used, not\nre-derived, by `1.3.1e_sieve47.py`.\n\n### 1.3.1d — ranks, torsion, index\n`gp -q 1.3.1d_rank_simon.gp` (reviewer script `rk.gp`; Simon's `bnfell2descent_viaisog`, 2 s), `python3 -I 1.3.1d_index.py`\n(reviewer script `index.py`), `python3 -I 1.3.1d_index_place11.py` (reviewer script `idx2.py`). Key output:\n```\n=== E0\n#S^(phi')(E'/K)     = 4\n#S^(phi)(E/K)       = 2\nrank                = 1\n=== E1\n#S^(phi')(E'/K)     = 2\n#S^(phi)(E/K)       = 4\nrank                = 1\nE0 P halving quartic roots in F: []  x(Pt)= -y\nE0 P+T halving quartic roots in F: []  x(Pt)= 4*y^2 - 3*y + 4\nE0 cards at 47 [48, 48, 48]\nE0 cards at 67 [68, 68, 68]\nE0 l= 3  certificate place (p, r, N): (11, 2, 12)\nE0 l= 47  certificate place (p, r, N): (1879, 342, 1880)\nE1 P halving quartic roots in F: []  x(Pt)= 264/961*y^2 - 324/961*y - 212/961\nE1 P+T halving quartic roots in F: []  x(Pt)= 1152/31*y^2 - 512/31*y + 1600/31\nE1 cards at 47 [48, 48, 48]\nE1 cards at 67 [68, 68, 68]\nE1 l= 3  certificate place (p, r, N): (11, 2, 12)\nE1 l= 47  certificate place (p, r, N): (1879, 342, 1880)\n```\n(also l = 11: (43, 38, 44), 17: (67, 13, 68), 67: (1607, 1154, 1608), 131: (523, 114, 524) for both curves; the\nsection uses l = 3 and 47. `idx2.py` prints #E = 12 at (11, α − 2) and [4]P ≠ O.)\n\n### 1.3.1e — Mordell–Weil sieve at 47\n`python3 -I 1.3.1e_sieve47.py` (reviewer script `sieve47.py`) and `python3 -I 1.3.1e_minus_R1.py` (reviewer script `negR1.py`). Key\noutput:\n```\nE0 place alpha=25: #E=48, ord(P)=24, #pts listed=48, 0/0 count=0, A mod v=Mod(46, 47)\nE0 place alpha=34: #E=48, ord(P)=4, #pts listed=48, 0/0 count=0, A mod v=Mod(41, 47)\nE0 place alpha=35: #E=48, ord(P)=12, #pts listed=48, 0/0 count=0, A mod v=Mod(13, 47)\nE0 survivors (n,e,t): [(0, 0, 'inf'), (0, 1, 'inf')]\nE1 place alpha=25: #E=48, ord(P)=24, #pts listed=48, 0/0 count=1, A mod v=Mod(37, 47)\nE1 place alpha=34: #E=48, ord(P)=6, #pts listed=48, 0/0 count=1, A mod v=Mod(32, 47)\nE1 place alpha=35: #E=48, ord(P)=6, #pts listed=48, 0/0 count=1, A mod v=Mod(41, 47)\nE1 survivors (n,e,t): [(0, 0, 0), (1, 0, 0), (23, 0, '0/0')]\n25 ... t(-R1) mod v: Mod(2, 47)\n34 ... t(-R1) mod v: Mod(30, 47)\n35 ... t(-R1) mod v: Mod(45, 47)\n```\nThe entry (23, 0, '0/0') is −R̄1 (in the scripts' labelling); its three t values computed by the second expression\ndiffer, so it is not a survivor.\n\n### 1.3.1f — Strassmann data at 47\n`python3 -I 1.3.1f_strassmann.py` (reviewer script `strass.py`; 47-adic precision 47^30; < 1 s). Key output:\n```\nE0 R0=O: v(lambda_j)= [1, 1, 1]  h=f1-f2 coeff valuations a0..a6: ['inf/0', 'inf/0', 2, 'inf/0', 5, 'inf/0', 6]  h-valuation-in-S: 2\nE0 R0=T: v(lambda_j)= [1, 1, 1]  h=f1-f2 coeff valuations a0..a6: ['inf/0', 'inf/0', 2, 'inf/0', 5, 'inf/0', 6]  h-valuation-in-S: 2\nE1 R0=O: v(lambda_j)= [1, 1, 1]  h=f1-f2 coeff valuations a0..a6: ['inf/0', 1, 2, 3, 4, 5, 6]  h-valuation-in-S: 1\nE1 R0=R1: v(lambda_j)= [1, 1, 1]  h=f1-f2 coeff valuations a0..a6: ['inf/0', 1, 2, 3, 4, 5, 6]  h-valuation-in-S: 1\n```\n('inf/0' marks a coefficient that is exactly 0.) The source's values λ mod 47^6 (661714269, 973712630,\n9275659150) are not printed.\n\n## Changes made to the reviewer scripts (no change to the mathematics)\n- `a1.gp` → `1.3.1a_field_F.gp`: its last lines (squares and fourth powers via `ispower` on t_POLMOD, and\n  `nfroots(nf, t^2 + x*t + ...)` with the field in x) fail in PARI 2.15 and were removed; `a1b.gp` does these\n  checks correctly.\n- `map1.gp` → `1.3.1c_conic_quartic.gp`: `polcoeff(qs, k, s)` was applied to a t_POLMOD in y whose coefficients\n  contain s and returned wrong values (b = c = d = 0 and a garbled \"a4 − k1\" line). The coefficients are now\n  taken from `lift(qs)` and wrapped in `Mod(·, f)`, exactly as in `map2.gp`; the line now reads `a4 - k1 = 0`.\n- `map2.gp` (symbolic check on the function field of E1; needs `-s 400000000` and many minutes here) is\n  replaced by the added `1.3.1c_identities.gp`. Besides being slow, its line \"quartic identity residual\" did not\n  reduce modulo W^2 = cubic (the modulus had main variable y, not W) and printed `0` (= false) followed by an\n  unreduced expression; reducing that printed expression correctly (W^2 → X^3 + k1X, then modulo y^3 + y + 1)\n  gives 0, so the identity holds. Its last line `Y^4 = -a g(t) residual: ` prints `n2==0`, i.e. 1 when the identity\n  holds; it uses the same faulty reduction (an earlier run quoted in the helper notes printed `residual: 0`, i.e. the\n  unreduced test returned false; it is not evidence either way). The identity is proved by `1.3.1c_identities.gp`\n  and agrees with the 300/300 random points of `1.3.1c_mapcheck.py`.\n- `sieve47.gp` → `1.3.1c_point_R1.gp`: its final `writebin(\"pts.bin\", …)` (input of reviewer script `sieve47b.gp`, which\n  `1.3.1e_sieve47.py` replaces) removed; one comparison line added.\n- `rk.gp`: Simon's `ell.gp` loaded from `../common/simon_ell/`; its last line (a print \"E1 model … skip\")\n  removed.\n- `strass.py`: `P.polcoeff` → `P.polcoef` (same PARI function; avoids a deprecation warning).\n- Header comments and final `quit` lines.\n\n## Not reproduced here\n- The source's reproduction package and its own numbers (e.g. λ mod 47^6).\n- No brute-force search for this equation exists in the reviewer folders.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.3.2: x^4 - y^4 + x^2yz + yz^3 = 0\n\nNotation of the section: chart x != 0, r = y/x, t = z/x, F(r,t) = 1 + rt + rt^3 - r^4;\nV = (r(t-r) : 1 : r^2+rt+t^2 : r : t) on the three quadrics Q1, Q2, Q3 in P^4 (5x5 matrices m1, m2, m3 in the\nscripts); genus-2 curve H: eta^2 = h(s) = 6s^6 + 22s^5 + 15s^4 - 14s^3 - 7s^2 + 6s - 1, J = Jac(H);\nT = (s^2 + s - 1/2, 0), G = (s^2 + 2s - 1, 2s - 1); Gamma = {Phi1 = Phi2 = 0} on the Kummer surface\n(the two quartics f1, f2 are written out in the scripts).\n\n## Software and how the outputs were obtained\n\n| Software | Used for | Command (run inside this folder) |\n|---|---|---|\n| Magma | `*.m` (codes 1.3.2a-e) | paste the file into Magma (or `magma -b FILE.m`) |\n| PARI/GP 2.15.4 | `1.3.2a_extra.gp` | `gp -q -f 1.3.2a_extra.gp < /dev/null` |\n| Sage (the modular `sagemath-schemes` distribution, Python 3.11) | `1.3.2*_check_*.py` (they import `jac_e02.py`) | `python3 -I FILE.py` with the Sage interpreter |\n\nThe Sage scripts start with `from sage.all__sagemath_schemes import *` (directly or through `jac_e02.py`);\nin a full Sage installation `from sage.all import *` can be used instead. `jac_e02.py` is an independent\ngenus-2 Jacobian arithmetic with Cassels-Flynn Kummer coordinates, written by the reviewer; it is not run\non its own.\n\n**Magma.** Magma is not installed on the machine where this folder was assembled. The Magma scripts were run\non the public Magma calculator (V2.29-10) on 14 October 2026, each in under 60 s (CPU times: 1.3.2a 8.6 s,\n1.3.2b 2.0 s, 1.3.2c 0.0 s, 1.3.2e 0.2 s, 1.3.2d below 12 s). The four scripts 1.3.2a, b, c, e are the\ncertificates of the source package, unchanged. Their files in `expected/` are the logs shipped with that\npackage (Magma V2.29-8, hence the first line `MAGMA_GETVERSION 2 29 8`); our 14 Oct 2026 run on V2.29-10\npassed every assertion and printed the same output line by line apart from that version line. The first line\n`MAGMA_GETVERSION ...` and the last two lines `PKG_OK`, `PKG_MAGMA_END` were printed by statements that the\nsubmission tool of the package adds around the script (`print \"MAGMA_GETVERSION\", GetVersion();` before,\n`print \"PKG_OK\"; print \"PKG_MAGMA_END\";` after). A failed assertion stops the script before `PKG_OK`.\nThe two 1.3.2d scripts are our own; see 1.3.2d.\n\n**PARI, Sage.** Every such script in this folder was run here (October 2026); its output is in\n`expected/<script name>.txt`. The longest run took 84 s (`1.3.2a_check_sieve_local.py`). Every output agrees\nwith the expected values listed below and, except for the timing lines and `1.3.2b_check_generic.py` (see\nthere), is identical to the log of the original review run.\n\n## Files\n\n| Code | File | Software | Run here |\n|---|---|---|---|\n| 1.3.2a | `1.3.2a_sieve_certificate.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.2a | `1.3.2a_extra.gp` (supplement) | PARI/GP | yes, pass |\n| 1.3.2a | `1.3.2a_check_exact.py`, `1.3.2a_check_sieve_local.py`, `1.3.2a_check_lifts.py`, `1.3.2a_check_p113.py` (cross-checks) | Sage | yes, pass |\n| 1.3.2b | `1.3.2b_symbolic_map.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.2b | `1.3.2b_check_generic.py` (cross-check) | Sage | yes, pass |\n| 1.3.2c | `1.3.2c_zero_fiber.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.2c, e | `1.3.2c_check_fibers.py` (cross-check) | Sage | yes, pass |\n| 1.3.2d | `1.3.2d_mordell_weil.m`, `1.3.2d_deficiency.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.2e | `1.3.2e_nonzero_fiber.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n\n\"Cross-check\" scripts are independent re-computations without Magma (reviewer code) of facts that the Magma\nscripts assert. The proof uses the Magma scripts; the cross-checks are optional.\n\n## 1.3.2a: Mordell-Weil group, sieves, 29-adic lifts, p = 113, local data (Magma)\n\n`1.3.2a_sieve_certificate.m` asserts: H nonsingular; `MordellWeilGroupGenus2(J)` gives finite_index, proved,\nrank_upper = 1, invariants [2, 0], generators T, G; Kummer(0) = (0,0,0,1). Sieve: for p = 5 and 29,\nord(G mod p) = 31 and the list of all (n mod 31, e) with Kummer(nG + eT) on Gamma mod p. 29-adic Kummer\npseudo-addition (precision 29^30): first lifts n = n0 + 31d (d = 0..28) with v(Phi_i) >= 2, second lifts\nn = n0 + 899d with v(Phi_i) >= 3. p = 113: ord(G mod 113) = 6728 = 8 * 841, and no n in [0, 6727] with\nn mod 841 in {291, 550} has Kummer(nG) on Gamma mod 113. Local data at 29: valuations of the primitive Kummer\nvector of Q = 31G, the vector (k1,k2,k3)/29^2 mod 29, (L1, L2) at it; base point K(2G) in the patch k1 = 1,\ndirection (K(33G) - K(2G))/29 mod 29, directional derivatives of Phi1, Phi2. Also: K(2G) lies on Gamma; the\ndeterminant identity h = -4 det(m1' - 2x m2' + x^2 m3') for the 3x3 blocks; the tangent quadratics at (+-1:1:0)\nnormalise to x^2 + 4x - 13/8 and `RationalPoints(J, ., 2)` = {2G, -2G}; the function-field check of 1.3.2b and\nthe identity-fibre audit of 1.3.2c are repeated.\n\nExpected output (`expected/1.3.2a_sieve_certificate.txt`):\n```\nfull Mordell-Weil group\nAbelian Group isomorphic to Z/2 + Z\nDefined on 2 generators\nRelations:\n    2*A.1 = 0\nfinite index true proved true rank upper bound 1\nfinite sieves p=5,p=29 [ <0, 0>, <2, 0>, <8, 0>, <11, 0>, <12, 0>, <14, 0>, <15,\n0>, <16, 0>, <17, 0>, <19, 0>, <20, 0>, <23, 0>, <29, 0> ]\n[ <0, 0>, <2, 0>, <14, 0>, <17, 0>, <29, 0> ]\n29-adic necessary residues [ 0, 31, 62, 93, 124, 155, 186, 217, 248, 279, 310,\n341, 372, 403, 434, 465, 496, 527, 558, 589, 620, 651, 682, 713, 744, 775, 806,\n837, 868, 2, 231, 668, 897 ]\n[ 2, 18211, 7860, 26069 ]\np=113 eliminates residues 291 and 550 mod 841 true\nformal local data at 0 and +/-2 [ 9, 10, 6 ]\n[ 13, 24 ]\n[ 1, 25, 2, 27 ]\n[ 0, 17, 9, 15 ]\n[ 23, 9 ]\nonly possible MW classes are 0 and +/-2G; +/-2 come from (+/-1:1:0) true\nfunction-field substitution in image and Kummer equations true\nall tangent/base-point exceptions and the identity fiber are excluded true\n```\nMeaning: a pair <n, e> in a sieve list means that Kummer(n G + e T) satisfies Phi1 = Phi2 = 0 mod p. The\n29-adic lists are the n for which v(Phi_i(Kummer(nG))) >= 2 (first lifts) or >= 3 (second lifts). All 29\nlifts of 0 mod 31 survive because v(Phi_i(Kummer([m]Q))) >= 4 always; that disc is handled by the isolation\nargument of the section instead.\n\nSupplement `1.3.2a_extra.gp` (PARI): Q1(V) = Q2(V) = -F, Q3(V) = 2F; h and its factorisation\n(2s^2+2s-1)(3s^4+8s^3+s^2-4s+1); Delta - (rt+2) = -F with Delta = r^4 - rt^3 + 1; F(+-1,t) = +-(t^3+t);\nr^6 + r^2 + 8 has no real root; the partial derivatives used for smoothness and F_y = -866 z^3 at\nx^2 = -3z^2, y = 6z; K(2G) -> (1 : -4 : -13/8). Expected output (`expected/1.3.2a_extra.txt`):\n```\nQ1(V)+F, Q2(V)+F, Q3(V)-2F: [0, 0, 0]\nh = 6*s^6 + 22*s^5 + 15*s^4 - 14*s^3 - 7*s^2 + 6*s - 1\nfactor h: [2*s^2 + 2*s - 1, 1; 3*s^4 + 8*s^3 + s^2 - 4*s + 1, 1]\nDelta-(rt+2) = r^4 + (-t^3 - t)*r - 1  (= -F: 1)\nF(1,t) = t^3 + t   F(-1,t) = -t^3 - t\nrt=-2 => -r^2*F = Mat([r^6 + r^2 + 8, 1])  real roots of r^6+r^2+8: 0\npartials: Fx Fy Fz = [4*X^3 + 2*Z*Y*X, Z*X^2 + (-4*Y^3 + Z^3), Y*X^2 + 3*Z^2*Y]\nF_y at x^2=-3z^2, y=6z: -866*Z^3\nK(2G) -> (1 : -4 : -13/8): [1, -4, -13/8]\n```\n\nCross-checks without Magma (Sage; reviewer code with its own Jacobian arithmetic `jac_e02.py`):\n- `1.3.2a_check_exact.py`: K(2G) = [32, -128, -52, -441], K(G), K(T); 2G lies on Gamma, G, 3G, 5G, 7G do not.\n- `1.3.2a_check_sieve_local.py`: both sieves (p = 5, 29), the local data at 29, and direct valuations of\n  Phi at 31mG (all 4) and at (31m +- 2)G (all 1). It normalises the local vector by k4 and therefore prints\n  `avec: [4, 27, 22]` and `L(avec): [4, 23]`; with the normalisation of the Magma script the values are\n  (9, 10, 6) and (13, 24), also printed (`L(9,10,6): [13, 24]`). Both are nonzero, as needed.\n  It writes `Kn.pkl` (about 1 MB, the Kummer points K(nG), n <= 126) into the current folder; this file is\n  generated, not included.\n- `1.3.2a_check_lifts.py` (run after the previous script; reads `Kn.pkl`): the 29-adic lifts\n  14 -> 231 -> 18211, 17 -> 668 -> 7860, 2 -> 2 -> 2, 29 -> 897 -> 26069.\n- `1.3.2a_check_p113.py`: the p = 113 elimination, with the line\n  `n mod 6728 with n mod 841 in {291,550} on Gamma mod 113: []  unknown: []`. It also redoes the 5 and 29\n  sieves with a finite-field implementation whose Kummer formula degenerates at some points (marked `?`);\n  those sieves are fully checked by `1.3.2a_check_sieve_local.py`.\n\nKey expected lines (`expected/1.3.2a_check_*.txt`):\n```\nK(2G) = [32, -128, -52, -441]\np=5: n in 1..31 with nG=0 mod p: [31]\n  sieve list: [(0, 0), (2, 0), (8, 0), (11, 0), (12, 0), (14, 0), (15, 0), (16, 0), (17, 0), (19, 0), (20, 0), (23, 0), (29, 0)]\np=29: n in 1..31 with nG=0 mod p: [31]\n  sieve list: [(0, 0), (2, 0), (14, 0), (17, 0), (29, 0)]\nval K(31G): [2, 2, 2, 0]\nbase2 [1, 25, 2, 27] dir2 [0, 17, 9, 15] ...\nderivs [23, 9]\n   level2 m mod 841: [587] -> n mod 26071: [18211]  claim 18211\n   level2 m mod 841: [253] -> n mod 26071: [7860]  claim 7860\nord(G mod 113) = 6728 factor 2^3 * 29^2 ...\nn mod 6728 with n mod 841 in {291,550} on Gamma mod 113: []  unknown: []\n```\n\n## 1.3.2b: the map C -> J over the function field Q(C) (Magma)\n\n`1.3.2b_symbolic_map.m` asserts, over K = Q(C) with C: 1 + rt + rt^3 - r^4 = 0: V lies on the three quadrics;\nthe tangent kernel has dimension 2; with a complement w, the tangent quadratic q1 - 2x q2 + x^2 q3 is\nirreducible of degree 2; the gcd of Phi1, Phi2 (as polynomials in k4 after substituting\n(1, 2q2/q3, q1/q3, k4)) has degree 1; its root satisfies Phi1, Phi2 and the Kummer equation;\n`RationalPoints(J over K, quadratic, 2)` has exactly 2 elements, with the same Kummer point, equal to\n(1, 2q2/q3, q1/q3, k4). Expected output (`expected/1.3.2b_symbolic_map.txt`):\n```\ncommon Kummer coordinate factor degree 1\nsymbolic substitution in both image quartics and Kummer quartic true\nk4 numerator degree data 26 24\n```\nMeaning: the tangent quadratic defines a point of J(Q(C)) (two opposite points, one Kummer point), so it\ngives a morphism C -> J over Q, and the Kummer image of every point of C lies on Gamma.\n\nCross-check (Sage) `1.3.2b_check_generic.py`: at about 400 random points of C over F_10007 the tangent\nconstruction lands on Gamma. Expected (`expected/1.3.2b_check_generic.txt`): `irr_bad`, `split_bad`,\n`phi_irr_bad` and `phi_split_none` are 0:\n```\n{'irr_ok': 191, 'irr_bad': 0, 'split_ok': 210, 'split_bad': 0, 'phi_irr_ok': 191, 'phi_irr_bad': 0, 'phi_split_onegood': 210, 'phi_split_none': 0}\n```\n(We added `set_random_seed(1)` so that the sample is reproducible; the original review run used an\nunseeded sample and printed 192/208 instead of 191/210, also with all zeros.)\n\n## 1.3.2c: tangent data, the q2 = q3 = 0 locus, the discriminant resultant (Magma)\n\n`1.3.2c_zero_fiber.m` asserts in Q[r,t] (lex): the left 3x3 minor of the tangent matrix equals Delta/4,\nDelta = r^4 - rt^3 + 1; the explicit w (formula (16) of the source, also printed in the section) satisfies\nV^T M_i w = 0; (r-1)^2(r+1)^2(r^2+1)^2 and t - r^7 + r^3 lie in (F, num q2, num q3); Delta - (rt+2) lies in (F);\nRes_t(F, num(q2^2 - q1 q3)) = r^4(r-1)^4(r+1)^4(r^2+1)^4 A8 B8 C24 / 64 with the displayed A8, B8, C24, all\nfactors of degree > 1 irreducible; F(1,t) = t(t^2+1), F(-1,t) = -t(t^2+1).\nExpected output (`expected/1.3.2c_zero_fiber.txt`):\n```\nq2=q3 simultaneous-vanishing elimination polynomials [\n    <r - 1, 2>,\n    <r + 1, 2>,\n    <r^2 + 1, 2>\n]\n-r^7 + r^3 + t\ndenominator identity modulo curve true\nresultant scalar relative to displayed factorization 1/64\ndiscriminant resultant factor degrees/exponents [ <1, 4>, <1, 4>, <1, 4>, <2,\n4>, <8, 1>, <8, 1>, <24, 1> ]\nno rational discriminant-zero point with r*t nonzero true\n```\n\nCross-check (Sage) `1.3.2c_check_fibers.py` (also for 1.3.2e). Key expected lines\n(`expected/1.3.2c_check_fibers.txt`):\n```\nminor == den/4: True\nden-(rt+2) in (F): True\nnonzero-fiber GB: [r^4 - r*t - 1, r^2*t^2 - t^2, t^3]\nresultant factorization: [(1, 4), (1, 4), (1, 4), (2, 4), (8, 1), (8, 1), (24, 1)] ...\n(r^2-1)^2(r^2+1)^2 in I: True  t-r^7+r^3 in I: True\n```\nand the only rational roots of the resultant factors are r = 0, 1, -1.\n\n## 1.3.2d: independent computation of J(Q) and of the deficient places (Magma)\n\nOur own scripts (14 Oct 2026), with all class groups forced to the Minkowski bound (`SetClassGroupBoundMaps`,\nno GRH):\n- `1.3.2d_mordell_weil.m`: `RankBounds`, `#TwoSelmerGroup`, `TorsionSubgroup`, `MordellWeilGroupGenus2`,\n  the height of the free generator, and the position of the claimed T, G.\n- `1.3.2d_deficiency.m`: the places where H is deficient (no Q_v-rational divisor of degree 1; `0` denotes\n  the infinite place), the 2-Selmer size and #J[2](Q), and a search for points on H of height <= 1000.\n\nThey are the one-curve copies (entry \"E02\") of our scripts `m1_jacobians.m` (six curves in one calculator\nsession, total CPU 11.4 s) and `m1b_deficiency.m` (three curves, total CPU 0.8 s); the computations for\ndifferent curves do not interact. The files in `expected/` contain the version line and the E02 lines of\nthose runs:\n```\n== E02 factors [ <2, 1>, <4, 1> ]\n   rank bounds 1 1  #Sel2 8  torsion [ 2 ]\n   MW [ 2, 0 ]\n finite_index true  proved true  rank_upper 1\n   gen 1 order 2 (x^2 + x - 1/2, 0, 2)\n   gen 2 order 0 (x^2 + 2*x - 1, 2*x - 1, 2)\n   height 1.3968654622486495785068973342381\nE02 claimed point (x^2 + x - 1/2, 0, 2) = MW.1\nE02 claimed point (x^2 + 2*x - 1, 2*x - 1, 2) = MW.2\n```\n```\nE02 bad primes [ 2, 3, 433 ]\ndeficient (0 = infinity) [ <2, true>, <3, false>, <433, false>, <0, false> ]\n   #deficient places 1\n   #TwoSelmerGroup 8  J[2](Q) size 2\n   small points on H {@ @}\n```\nMeaning: J(Q) = Z/2 + Z generated by T and G (the generators chosen by Magma may differ by sign or by\nadding T between runs). #Sel^(2) = 8 alone gives rank <= 2; H is deficient at exactly one place (p = 2), so\nby Poonen-Stoll Sha(J)[2] is nontrivial, which lowers the bound to rank <= 1.\n\n## 1.3.2e: the fibre over K(+-2G) (Magma)\n\n`1.3.2e_nonzero_fiber.m` asserts: K(2G) = (32, -128, -52, -441); with the explicit w, the lex Groebner basis\nof (F, num(2q2 + 4q3), num(8q1 + 13q3)) in Q[r,t] is [r^4 - rt - 1, r^2t^2 - t^2, t^3]; Delta - (rt+2) lies\nin (F); Delta(+-1, 0) != 0. Expected output (`expected/1.3.2e_nonzero_fiber.txt`):\n```\nfiber Groebner basis [\n    r^4 - r*t - 1,\n    r^2*t^2 - t^2,\n    t^3\n]\nfiber points [ <-1, 0>, <1, 0> ]\n```\nCross-check: `1.3.2c_check_fibers.py` prints `nonzero-fiber GB: [r^4 - r*t - 1, r^2*t^2 - t^2, t^3]`.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.3.3: x^4 + x^2y^2 + y^3z - yz^3 = 0\n\nNotation of the section: C: x^4 + x^2y^2 + y^3z - yz^3 = 0, involution iota(x:y:z) = (-x:y:z), quotient\nE: W^2 = y^4 - 4y^3z + 4yz^3 (the conductor-37 curve 37a1, minimal model [0,0,1,-1,0], also\nY^2 = X^3 - 16X + 16); Ritzenthaler-Romagny data with A = [0,1,0; -1,0,0; 1,0,-1];\ngenus-2 curve H: eta^2 = -(s^2+1)(s^4+2s^3+2s^2+1), J_H = Jac(H).\n\n## Software and how the outputs were obtained\n\n| Software | Used for | Command (run inside this folder) |\n|---|---|---|\n| Magma | `*.m` (codes 1.3.3a, b, c) | paste the file into Magma (or `magma -b FILE.m`) |\n| PARI/GP 2.15.4 | `*.gp` | `gp -q -f FILE.gp < /dev/null` |\n\n**Magma.** Magma is not installed on the machine where this folder was assembled. The Magma scripts were run\non the public Magma calculator (V2.29-10) on 14 October 2026, each in under 60 s. The files in `expected/`\nfor Magma scripts are the outputs of those runs, as stated for each script below. In our own runs a first\nline `MAGMA_GETVERSION 2 29 10` was printed by a statement added at submission (`GetVersion()`); it is not\npart of the scripts.\n\n**PARI.** Both PARI scripts were run here (October 2026; under 2 s each); their outputs are in\n`expected/<script name>.txt` and agree with the expected values below.\n\n## Files\n\n| Code | File | Software | Run here |\n|---|---|---|---|\n| 1.3.3a | `1.3.3a_certificate.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.3a | `1.3.3a_extra.gp` (supplement) | PARI/GP | yes, pass |\n| 1.3.3b | `1.3.3b_prym_lpoly.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.3b | `1.3.3b_check_counts.gp` (cross-check) | PARI/GP | yes, pass |\n| 1.3.3c | `1.3.3c_mordell_weil.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n\n## 1.3.3a: smoothness, the Prym curve, rank of J_H, J_C(F_3), J_C(F_5) (Magma)\n\n`1.3.3a_certificate.m` (the certificate `t6_4_proof_certificate.m` of the source, unchanged) asserts:\nC is nonsingular of genus 3; A^{-1} = [0,-1,0; 1,0,0; 0,-1,-1]; a(s) = 2s, b(s) = -(s^2+1), c(s) = -s^2\n(a(s) = Ai[1,1] + 2 Ai[2,1] s + Ai[3,1] s^2, etc.); b(b^2 - ac) = -(s^2+1)(s^4+2s^3+2s^2+1), squarefree;\n`RankBounds(J_H)` = (0, 0); #TorsionSubgroup(J_H) = 2; for p = 3, 5: C mod p nonsingular,\n`ClassGroupAbelianInvariants` = [2,14,0] (p = 3) and [2,176,0] (p = 5), and L_p(C) = L_p(E) L_p(H) with\nE: y^2 = x^3 - 16x + 16 and\nL_3(C) = 1 + 27T^6, L_3(E) = 1 + 3T + 3T^2, L_3(H) = 1 - 3T + 6T^2 - 9T^3 + 9T^4,\nL_5(C) = 1 + 4T + 15T^2 + 32T^3 + 75T^4 + 100T^5 + 125T^6, L_5(E) = 1 + 2T + 5T^2,\nL_5(H) = 1 + 2T + 6T^2 + 10T^3 + 25T^4.\nThe script does not call `SetClassGroupBounds(\"GRH\")`; a fresh Magma session proves class groups\nunconditionally.\n\nExpected output (`expected/1.3.3a_certificate.txt`; the line printed by the 14 Oct 2026 run, CPU 1.2 s, after\nall assertions passed):\n```\nCERTIFIED: all exact assertions passed\n```\nMeaning of [2,14,0]: Pic(C over F_3) = Z/2 + Z/14 + Z; since C has an F_3-point, the degree map is onto and\nJ_C(F_3) = Z/2 + Z/14 (order 28 = L_3(C)(1)); likewise J_C(F_5) has order 352.\n\nSupplement `1.3.3a_extra.gp` (PARI): A^{-1}, a, b, c, the polynomial of H, the quotient identity\n(2X + y^2)^2 - (y^4 - 4y^3 + 4y) = 4(X^2 + y^2 X + y^3 - y) (z = 1, X = x^2), E has conductor 37 and the same\nminimal model [0,0,1,-1,0] as Y^2 = X^3 - 16X + 16, L_3(1) = 28, L_5(1) = 352, gcd 4.\nExpected output (`expected/1.3.3a_extra.txt`):\n```\nA^-1 = [0, -1, 0; 1, 0, 0; 0, -1, -1]\na,b,c = [2*s, -s^2 - 1, -s^2]\nb(b^2-ac) = -s^6 - 2*s^5 - 3*s^4 - 2*s^3 - 3*s^2 - 1  == -(s^2+1)(s^4+2s^3+2s^2+1): 1\n(2X+y^2)^2-(y^4-4y^3+4y) - 4*(X^2+y^2*X+y^3-y) = 0\nE: conductor 37  j 110592/37 ; Y^2=X^3-16X+16: conductor 37  j 110592/37  same minimal model: 1 [0, 0, 1, -1, 0]\nL3(1), L5(1) = 28, 352   gcd = 4\n```\n\n## 1.3.3b: Prym identification at all good primes up to 79 (Magma)\n\n`1.3.3b_prym_lpoly.m` builds the quotient W^2 = q^2 - 4r (q = y^2, r = y^3z - yz^3; in s = y/z:\ns^4 - 4s^3 + 4s) from the equation alone and, for every prime 3 <= p <= 79 where C, the quotient and H all\nhave good reduction, compares LPolynomial(C) with LPolynomial(C/iota) * LPolynomial(H). It prints the primes\nwhere the equality holds, fails, or was skipped. A wrong quadratic twist of H would fail at about half of\nthe primes.\n\nIt is the one-curve copy (entry \"E04\") of our script `m2_prym_lpoly.m`, which treated five quartics in one\ncalculator session (14 Oct 2026, total CPU 39.9 s); the computations for different curves do not interact.\nExpected output (`expected/1.3.3b_prym_lpoly.txt`: the version line and the E04 lines of that run; Magma\nprints the variable of the quotient polynomial as `x`):\n```\nE04 quotient W^2 = x^4 - 4*x^3 + 4*x\n   L(C)=L(C/iota)L(H) holds at [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79 ]\n   FAILS at []\n  skipped (bad reduction) [ 37 ]\n```\n\nCross-check without Magma: `1.3.3b_check_counts.gp` (PARI; one-curve copy, entry \"E04\", of our script\n`prym_counts.gp`) counts the points of C, of the quotient and of H over F_p, F_{p^2}, F_{p^3} for\np in {3,5,7,11,13,17,19,23} and checks #C - (q+1) = [#(C/iota) - (q+1)] + [#H - (q+1)] for q = p, p^2, p^3,\nwhich is equivalent to L_p(C) = L_p(C/iota) L_p(H) (genus 3 = 1 + 2). Expected output\n(`expected/1.3.3b_check_counts.txt`; 1 = holds):\n```\nE04 [[3, 1], [5, 1], [7, 1], [11, 1], [13, 1], [17, 1], [19, 1], [23, 1]]\n```\n\n## 1.3.3c: independent computation of J_H(Q) (Magma)\n\n`1.3.3c_mordell_weil.m` runs `RankBounds`, `TwoSelmerGroup`, `TorsionSubgroup` and `MordellWeilGroupGenus2`\nwith all class groups forced to the Minkowski bound (`SetClassGroupBoundMaps`, no GRH). It is the one-curve\ncopy (entry \"E04\") of our script `m1_jacobians.m` (six curves in one calculator session, 14 Oct 2026, total\nCPU 11.4 s). Expected output (`expected/1.3.3c_mordell_weil.txt`: the version line and the E04 lines of that\nrun):\n```\n== E04 factors [ <2, 1>, <4, 1> ]\n   rank bounds 0 0  #Sel2 2  torsion [ 2 ]\n   MW [ 2 ]\n finite_index true  proved true  rank_upper 0\n   gen 1 order 2 (x^2 + 1, 0, 2)\n```\nSo J_H(Q) = Z/2, generated by (s^2 + 1, 0); #Sel^(2) = 2 = #J_H(Q)[2].\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.3.4: x^4 + y^4 + x^2yz - yz^3 = 0\n\nNotation of the section: C: X^4 + Y^4 + X^2YZ - YZ^3 = 0, C(Q) = {(0:0:1), (0:1:1)}; elliptic quotient\nE: v^2 = w^3 + w^2 - 64; Ritzenthaler-Romagny matrix A = [1,-1,0; 0,-1,0; 1,1,1];\ngenus-2 curve H: y^2 = f_H(s) = 10s^6 - 26s^5 + 9s^4 + 18s^3 - 5s^2 - 6s - 1, J_H = Jac(H).\n\n## Software and how the outputs were obtained\n\n| Software | Used for | Command (run inside this folder) |\n|---|---|---|\n| Magma | `*.m` (codes 1.3.4a, b) | paste the file into Magma (or `magma -b FILE.m`) |\n| PARI/GP 2.15.4 | `*.gp` | `gp -q -f FILE.gp < /dev/null` |\n| Python 3.11 + SymPy 1.14 | `1.3.4a_check_rr.py` | `python3 -I 1.3.4a_check_rr.py` |\n\n**Magma.** Magma is not installed on the machine where this folder was assembled. The Magma scripts were run\non the public Magma calculator (V2.29-10) on 14 October 2026, each in under 60 s. The files in `expected/`\nfor Magma scripts are the outputs of those runs, or the log shipped with the source, as stated for each\nscript below.\n\n**PARI, Python.** Every such script in this folder was run here (October 2026; each under 10 s); its output\nis in `expected/<script name>.txt` and agrees with the expected values below.\n\n## Files\n\n| Code | File | Software | Run here |\n|---|---|---|---|\n| 1.3.4a | `1.3.4a_certificate.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.4a | `1.3.4a_check_counts.gp` (cross-check) | PARI/GP | yes, pass |\n| 1.3.4a | `1.3.4a_check_rr.py` (cross-check) | Python + SymPy | yes, pass |\n| 1.3.4b | `1.3.4b_deficiency.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.4b | `1.3.4b_check_q2.gp` (illustration) | PARI/GP | yes, pass |\n\n\"Cross-check\" scripts are independent re-computations without Magma of facts that the Magma scripts assert.\nThe proof uses the Magma scripts; the cross-checks are optional.\n\n## 1.3.4a: smoothness, quotient, Prym model, J_H, reductions at 5, 7, 11 (Magma)\n\n`1.3.4a_certificate.m` (the certificate `t6_5_proof_certificate.m` of the source package, unchanged) asserts:\n- C is nonsingular over Q, genus 3;\n- the quotient identity on Y = 1 (u = X/Y, t = Z/Y): v^2 - (w^3 + w^2 - 64) = 64(u^4 + 1 + u^2 t - t^3) for\n  w = 4t, v = 4(2u^2 + t);\n- A^{-1} = [1,-1,0; 0,-1,0; -1,2,1]; a = 1 - s^2, b = 2s^2 - 2s - 1, c = s^2;\n  f_H = b(b^2 - ac) = 10s^6 - 26s^5 + 9s^4 + 18s^3 - 5s^2 - 6s - 1;\n- J_H: `RankBounds` = (0, 0); torsion invariants [2]; `MordellWeilGroupGenus2`: Z/2 with finite_index =\n  proved = true, rank_upper = 0;\n- p = 5, 7, 11: C mod p nonsingular; `ClassGroupAbelianInvariants(C mod p)` = [110,0], [2,140,0], [17,68,0];\n  LPolynomial(C) = LPolynomial(E) * LPolynomial(H).\n\nExpected output (`expected/1.3.4a_certificate.txt`). This is the log shipped with the source package\n(Magma V2.29-8, hence the first line `MAGMA_GETVERSION 2 29 8`); our 14 Oct 2026 run on V2.29-10 (CPU 1.9 s)\npassed every assertion and printed the same output line by line apart from that version line. The first line\nand the last two lines (`PKG_OK`, `PKG_MAGMA_END`) were printed by statements that the submission tool of the\npackage adds around the script; a failed assertion stops the script before `PKG_OK`.\n```\nC_Q nonsingular true genus 3\nA\n[ 1 -1  0]\n[ 0 -1  0]\n[ 1  1  1]\nA inverse\n[ 1 -1  0]\n[ 0 -1  0]\n[-1  2  1]\na,b,c -s^2 + 1\n2*s^2 - 2*s - 1\ns^2\nPrym genus-2 polynomial 10*s^6 - 26*s^5 + 9*s^4 + 18*s^3 - 5*s^2 - 6*s - 1\nH genus 2 JH rank bounds 0 0 JH torsion invariants [ 2 ]\ncertified Mordell-Weil\nAbelian Group isomorphic to Z/2\nDefined on 1 generator\nRelations:\n    2*MW.1 = 0\ntrue true 0\np 5 C nonsingular true class invariants [ 110, 0 ]\nL(C)=L(E)L(H) true 125*x^6 - 50*x^5 + 40*x^4 - 12*x^3 + 8*x^2 - 2*x + 1\n5*x^2 - x + 1\n25*x^4 - 5*x^3 + 2*x^2 - x + 1\np 7 C nonsingular true class invariants [ 2, 140, 0 ]\nL(C)=L(E)L(H) true 343*x^6 - 98*x^5 + 63*x^4 - 36*x^3 + 9*x^2 - 2*x + 1\n7*x^2 + 2*x + 1\n49*x^4 - 28*x^3 + 10*x^2 - 4*x + 1\np 11 C nonsingular true class invariants [ 17, 68, 0 ]\nL(C)=L(E)L(H) true 1331*x^6 - 242*x^5 + 66*x^4 - 4*x^3 + 6*x^2 - 2*x + 1\n11*x^2 + 5*x + 1\n121*x^4 - 77*x^3 + 30*x^2 - 7*x + 1\nPKG_OK\nPKG_MAGMA_END\n```\nMeaning: the last entry 0 of the class-group invariants is the degree part Z, the entries before it give\nJ_C(F_p) = Pic^0(C over F_p): Z/110, Z/2 + Z/140, Z/17 + Z/68 (orders L_C(1) = 110, 280, 1156).\nL_p(C) = L_p(E) L_p(H) means that J_C and E x J_H have the same number of points over every finite\nextension of F_p.\n\nCross-checks without Magma:\n- `1.3.4a_check_counts.gp` (PARI; one-curve copy, entry \"E05\", of our script `prym_counts.gp`): counts the\n  points of C, of the quotient C/iota and of H over F_p, F_{p^2}, F_{p^3} for p = 5, ..., 23 and checks\n  #C - (q+1) = [#(C/iota) - (q+1)] + [#H - (q+1)] for q = p, p^2, p^3 (equivalent to\n  L_p(C) = L_p(C/iota) L_p(H)). As a negative control it runs the same test for the quadratic twist of H by -1,\n  which must fail at some primes. Expected output (`expected/1.3.4a_check_counts.txt`; 1 = holds):\n  ```\n  E05 [[5, 1], [7, 1], [11, 1], [13, 1], [17, 1], [19, 1], [23, 1]]\n  E05-twist(-1) [[5, 1], [7, 0], [11, 0], [13, 1], [17, 1], [19, 0], [23, 0]]\n  ```\n- `1.3.4a_check_rr.py` (SymPy; the parts for this equation of a helper script `chk2.py`, with the\n  smoothness test run at the three primes of the certificate): A^{-1}, a, b, c, f_H and its factorisation,\n  and smoothness of C mod 5, 7, 11 (the Groebner basis of F and its partial derivatives is [1] in every\n  affine chart). Expected output (`expected/1.3.4a_check_rr.txt`):\n  ```\n  E05 Ai Matrix([[1, -1, 0], [0, -1, 0], [-1, 2, 1]])\n   a,b,c= 1 - s**2 2*s**2 - 2*s - 1 s**2  f_H= 10*s**6 - 26*s**5 + 9*s**4 + 18*s**3 - 5*s**2 - 6*s - 1 (2*s**2 - 2*s - 1)*(5*s**4 - 8*s**3 - s**2 + 4*s + 1)\n  E05 smooth mod 5 True\n  E05 smooth mod 7 True\n  E05 smooth mod 11 True\n  ```\n\n## 1.3.4b: 2-Selmer group and deficient places of H (Magma)\n\n`1.3.4b_deficiency.m` (our own script, 14 Oct 2026; class groups forced to the Minkowski bound with\n`SetClassGroupBoundMaps`, no GRH) prints the bad primes of H, for each bad prime and for the infinite place\n(`0`) whether H is deficient there (no Q_v-rational divisor of degree g - 1 = 1), the number of deficient\nplaces, #Sel^(2)(Q, J_H) and #J_H[2](Q), and a search for points on H of height <= 1000. It is the one-curve\ncopy (entry \"E05\") of our script `m1b_deficiency.m` (three curves in one calculator session, total CPU\n0.8 s); the computations for different curves do not interact. Expected output\n(`expected/1.3.4b_deficiency.txt`: the version line and the E05 lines of that run):\n```\nE05 bad primes [ 2, 3, 431 ]\ndeficient (0 = infinity) [ <2, true>, <3, false>, <431, false>, <0, false> ]\n   #deficient places 1\n   #TwoSelmerGroup 4  J[2](Q) size 2\n   small points on H {@ @}\n```\nMeaning: #Sel^(2) = 4, so dim Sel^(2) = 2; the rational 2-torsion point takes one dimension, so the Selmer\ngroup alone gives rank <= 1. H is deficient at exactly one place (2), an odd number, so by Poonen-Stoll\nSha(J_H)[2] is nontrivial and takes the other dimension: the rank is 0, in agreement with\n`RankBounds` = (0, 0) printed by 1.3.4a.\n\nIllustration of the deficiency at 2 (PARI; a helper script `chk5.gp`) `1.3.4b_check_q2.gp`: for s in\nZ_2, f_H(s) mod 8 takes only the values 3 and 7, so f_H(s) is never a square in Q_2 (odd squares are 1 mod 8);\nfor s = 1/t with t in 2Z_2, s^-6 f_H(s) = t^6 f_H(1/t) is 2 mod 4, of odd valuation, again not a square.\nSo H(Q_2) is empty. Expected output (`expected/1.3.4b_check_q2.txt`):\n```\nf mod 8 on Z_2: [3, 7]\ns^-6 f mod 4 near infinity: [2]\n```\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.3.5: x^4 - x^2y^2 + y^3z + yz^3 = 0\n\nNotation of the section: C: X^4 - X^2Y^2 + Y^3Z + YZ^3 = 0, C(Q) = {(0:0:1), (0:1:0), (+-1:1:0)};\nRitzenthaler-Romagny matrix A = [0,1,0; 1,0,0; 1,0,1], quadrics q1 = b^2 + ac, q2 = ab, q3 = -b^2 - ac + bc,\ncover D = {q1 = u^2, q2 = uv, q3 = v^2} in P^4; genus-2 curve H: y^2 = -s^6 - 2s^5 + 3s^4 + 2s^3 - 3s^2 + 1,\nJ_H = Jac(H), T = (s^2 - 1, 0), G = (s^2 + s, -s - 1); the two quartic forms Phi1 (= F1), Phi2 (= F2) on the\nKummer surface are written out in the scripts. The function-field computations use the affine chart Y = 1,\nr = X/Y, t = Z/Y, where C: r^4 - r^2 + t + t^3 = 0; R = r^2; the tangent quadratic is proportional to\nA2 s^2 + B s + A0 (A2, B, A0 polynomials in R, t given in the section and in the scripts).\n\n## Software and how the outputs were obtained\n\n| Software | Used for | Command (run inside this folder) |\n|---|---|---|\n| Magma | `*.m` (codes 1.3.5a-e) | paste the file into Magma (or `magma -b FILE.m`) |\n| Python 3.11 + SymPy 1.14 | `1.3.5a_check_smooth.py`, `1.3.5b_check_tangent.py`, `1.3.5b_check_determinant.py`, `1.3.5e_check_fiber.py` | `python3 -I FILE.py` |\n| Python 3.11 + python-flint 0.9 | `1.3.5c_check_sieve.py`, `1.3.5d_check_valuations.py` (both import `g2_e07.py`) | `python3 -I FILE.py` |\n\n`g2_e07.py` is the reviewer's independent genus-2 Jacobian arithmetic (Cantor's algorithm in a quintic model,\nCassels-Flynn Kummer coordinates of the sextic model, exact rationals); it is not run on its own.\n\n**Magma.** Magma is not installed on the machine where this folder was assembled. The Magma scripts were run\non the public Magma calculator (V2.29-10) on 14 October 2026, each in under 60 s (CPU times: 1.3.5a 2.8 s,\n1.3.5b 2.0 s, 1.3.5c 3.7 s, 1.3.5d 1.0 s, 1.3.5e 0.2 s). The five scripts 1.3.5a-e are the certificates of the\nsource package, unchanged. Their files in `expected/` are the logs shipped with that package (Magma V2.29-8,\nhence the first line `MAGMA_GETVERSION 2 29 8`); our 14 Oct 2026 run on V2.29-10 passed every assertion and\nprinted the same output line by line apart from that version line. The first line and the last two lines\n(`PKG_OK`, `PKG_MAGMA_END`) were printed by statements that the submission tool of the package adds around\nthe script; a failed assertion stops the script before `PKG_OK`. The two Magma cross-checks of 1.3.5b are our\nown scripts; see there.\n\n**Python.** Every Python script in this folder was run here (October 2026; each under 15 s); its output is in\n`expected/<script name>.txt` and agrees with the expected values below (for `1.3.5d_check_valuations.py` it\nis identical to the log of the original review run, apart from timing lines).\n\n## Files\n\n| Code | File | Software | Run here |\n|---|---|---|---|\n| 1.3.5a | `1.3.5a_reductions.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.5a | `1.3.5a_check_smooth.py` (cross-check) | SymPy | yes, pass |\n| 1.3.5b | `1.3.5b_core_certificate.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.5b | `1.3.5b_check_mordell_weil.m`, `1.3.5b_check_prym_lpoly.m` (cross-checks) | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.5b | `1.3.5b_check_tangent.py`, `1.3.5b_check_determinant.py` (cross-checks) | SymPy | yes, pass |\n| 1.3.5c | `1.3.5c_kummer_sieve.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.5c, d | `1.3.5c_check_sieve.py` (cross-check) | python-flint | yes, pass |\n| 1.3.5d | `1.3.5d_mahler.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.5d | `1.3.5d_check_valuations.py` (cross-check) | python-flint | yes, pass |\n| 1.3.5e | `1.3.5e_fiber.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.5e | `1.3.5e_check_fiber.py` (cross-check) | SymPy | yes, pass |\n\n\"Cross-check\" scripts are independent re-computations of facts that the certificate of the same code asserts.\nThe proof uses the certificates; the cross-checks are optional.\n\n## 1.3.5a: smoothness of C mod p and class groups (Magma)\n\n`1.3.5a_reductions.m` asserts that C mod p is nonsingular for p = 3, 5, 11, 17, 19, 53 and prints the\nclass-group invariants. The section uses only p = 3: C mod 3 nonsingular implies that C is smooth over Q\n(and J_C(F_3) = Z/40). Expected output (`expected/1.3.5a_reductions.txt`):\n```\np 3 smooth true invariants [ 40, 0 ]\np 5 smooth true invariants [ 6, 72, 0 ]\np 11 smooth true invariants [ 3, 468, 0 ]\np 17 smooth true invariants [ 4, 1316, 0 ]\np 19 smooth true invariants [ 6240, 0 ]\np 53 smooth true invariants [ 2, 87570, 0 ]\nPKG_OK\nPKG_MAGMA_END\n```\nCross-check `1.3.5a_check_smooth.py` (SymPy; the smoothness part of a helper script `chk2.py`, run at the\nsix primes above): the Groebner basis of F and its partial derivatives mod p is [1] in every affine chart.\nExpected (`expected/1.3.5a_check_smooth.txt`): `E07 smooth mod p True` for p = 3, 5, 11, 17, 19, 53.\n\n## 1.3.5b: Prym model, J_H(Q), generic tangent map and Kummer identities (Magma)\n\n`1.3.5b_core_certificate.m` asserts:\n- q1 = b^2 + ac, q2 = ab, q3 = -b^2 - ac + bc (from A); -4 det(N1 - 2s N2 + s^2 N3) =\n  -s^6 - 2s^5 + 3s^4 + 2s^3 - 3s^2 + 1; H nonsingular of genus 2;\n- `MordellWeilGroupGenus2(J_H)`: invariants [2, 0], finite_index = proved = true, rank_upper = 1;\n  generators T = (s^2 - 1, 0), G = (s^2 + s, -s - 1); Kummer(4G) = (32 : 16 : -12 : 73);\n- at the point (0:1:1:1:0) of D over (1:1:0): the J_H(Q)-lifts of its tangent quadratic are {4G, -4G};\n- over Q(C), C: r^4 - r^2 + t + t^3 = 0: the generic point (t : r^2 : 1 + t^2 : r : rt) lies on D; the tangent\n  kernel has dimension 2; the tangent quadratic has degree 2, is irreducible and is proportional to\n  A2 s^2 + B s + A0 (formulas in the section); `RationalPoints(J_H over Q(C), xpol, 2)` has exactly 2 elements\n  with equal Kummer points; for that Kummer point K: K[2] A2 + K[1] B = 0, K[3] A2 - K[1] A0 = 0,\n  F1(K) = F2(K) = 0 and the Kummer equation holds.\n\nExpected output (`expected/1.3.5b_core_certificate.txt`):\n```\nCERTIFIED T6(7) CORE true true 1 known classes [\n    -4*MW.2,\n    4*MW.2\n]\ngeneric image identities true\nPKG_OK\nPKG_MAGMA_END\n```\nMeaning: over Q(C) exactly two points of J_H have the tangent quadratic as Mumford polynomial and they are\nopposite; so the tangent quadratic defines a point of J_H(Q(C)), i.e. a morphism C -> J_H over Q, and rational\npoints of C go to J_H(Q). Phi1 = Phi2 = 0 at the generic Kummer point is a closed condition, so it holds at every\npoint of C. The two cross-product identities say that the first three Kummer coordinates are proportional\nto (A2 : -B : A0).\n\nCross-checks:\n- `1.3.5b_check_mordell_weil.m` (Magma, our own script; one-curve copy, entry \"E07\", of `m1_jacobians.m`,\n  run with five other curves in one calculator session on 14 Oct 2026, total CPU 11.4 s): Mordell-Weil group\n  with all class groups forced to the Minkowski bound (no GRH) and the position of the claimed T, G.\n  Expected (`expected/1.3.5b_check_mordell_weil.txt`: the version line and the E07 lines of that run):\n  ```\n  == E07 factors [ <1, 1>, <1, 1>, <4, 1> ]\n     rank bounds 1 1  #Sel2 4  torsion [ 2 ]\n     MW [ 2, 0 ]\n   finite_index true  proved true  rank_upper 1\n     gen 1 order 2 (x^2 - 1, 0, 2)\n     gen 2 order 0 (x^2 + x, -x - 1, 2)\n     height 0.26930252756556990003493641314842\n  E07 claimed point (x^2 - 1, 0, 2) = MW.1\n  E07 claimed point (x^2 + x, -x - 1, 2) = MW.2\n  ```\n- `1.3.5b_check_prym_lpoly.m` (Magma, our own script; one-curve copy, entry \"E07\", of `m2_prym_lpoly.m`, run\n  with four other quartics in one calculator session on 14 Oct 2026, total CPU 39.9 s): builds the quotient\n  C/iota: W^2 = q^2 - 4r from the equation alone and checks L_p(C) = L_p(C/iota) L_p(H) at every good prime\n  3 <= p <= 79 (this confirms the quadratic twist of H). Expected (`expected/1.3.5b_check_prym_lpoly.txt`:\n  the version line and the E07 lines of that run; Magma prints the variable as `x`):\n  ```\n  E07 quotient W^2 = x^4 - 4*x^3 - 4*x\n     L(C)=L(C/iota)L(H) holds at [ 3, 5, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79 ]\n     FAILS at []\n    skipped (bad reduction) [ 7, 13 ]\n  ```\n- `1.3.5b_check_tangent.py` (SymPy, reviewer): the generic point lies on the three quadrics modulo the curve\n  equation, and for two different choices of the tangent vector w the tangent quadratic c1 - 2c2 s + c3 s^2\n  is proportional to A2 s^2 + B s + A0 (both cross-product expressions reduce to 0). Expected\n  (`expected/1.3.5b_check_tangent.txt`):\n  ```\n  point on quadrics mod rel: [0, 0, 0]\n  w with w_0=0: c3*B+2*c2*A2 = 0 ; c3*A0-c1*A2 = 0 ; c3 mod rel nonzero: True\n  w with w_2=0: c3*B+2*c2*A2 = 0 ; c3*A0-c1*A2 = 0 ; c3 mod rel nonzero: True\n  ```\n- `1.3.5b_check_determinant.py` (SymPy, reviewer): the 3x5 tangent matrix at the generic point has rank 3 on C\n  except where r = 0 and t^3 + t = 0 (Groebner basis of the curve and all 3x3 minors), and the determinant\n  formula for H. Expected (`expected/1.3.5b_check_determinant.txt`):\n  ```\n  GB(curve, 3x3 minors of tangent matrix): [r, t**3 + t]\n  Fpol = -x**6 - 2*x**5 + 3*x**4 + 2*x**3 - 3*x**2 + 1\n  ```\n\n## 1.3.5c: Mordell-Weil sieve mod p (Magma)\n\n`1.3.5c_kummer_sieve.m`: for each p in {3, 5, 7, ..., 97} with H mod p nonsingular, it computes ord(G mod p)\nand, for e in {0, 1} and 0 <= n < ord(G mod p), lists the n with Kummer(eT + nG) satisfying F1 = F2 = 0 mod p.\nOnly p = 3 is used: ord(G mod 3) = 5, survivors e = 0, n in {0, 1, 4}, none with e = 1.\nThe loop runs over the 24 primes 3..97; 7 and 13 are skipped (H has bad reduction there), so 22 blocks are\nprinted. Expected output (`expected/1.3.5c_kummer_sieve.txt`; first and last block):\n```\np 3 ordG 5 surv [\n    [ 0, 1, 4 ],\n    []\n]\n...\np 97 ordG 87 surv [\n    [ 0, 4, 15, 72, 83 ],\n    []\n]\nPKG_OK\nPKG_MAGMA_END\n```\nCross-check `1.3.5c_check_sieve.py` (python-flint, reviewer; exact arithmetic over Q, reducing\nKummer(eT + nG) mod 3). It also computes the Mahler data of 1.3.5d. Expected\n(`expected/1.3.5c_check_sieve.txt`):\n```\nsieve mod 3 (reduce K(eT+nG) over Q):\n e 0 surv [0, 1, 4]\n e 1 surv []\n periodicity mod 5 of reductions OK\nID Mahler (v,res): [(100, 0), (5, 1), (5, 2), (7, 2), (8, 2), (7, 1), (7, 1)]\nB Mahler (v,res): [(100, 0), (2, 2), (2, 2), (3, 1), (4, 2), (5, 2), (7, 1), (8, 1)]\nv3 h(r), r=2,5,8: [3, 3, 3]\n```\n\n## 1.3.5d: 3-adic data: Mahler coefficients and valuations (Magma)\n\n`1.3.5d_mahler.m` computes, with exact Kummer arithmetic over Q:\n- S = 5G, f(m) = F1(K([m]S)) in the chart k4 = 1 for m = 0..6, its Mahler coefficients (forward differences)\n  as (v_3, residue): (inf), (5,1), (5,2), (7,2), (8,2), (7,1), (7,1); reduced at 3^5: [0,1,2,0,0,0,0];\n- the affine coordinates of K(S): valuations (4,3,2), residues (2,2,1); of K(3S): valuations (6,5,4),\n  residues (1,1,1);\n- B = 4G, h(m) = F1(K(B + [m]S)) in the chart k1 = 1: Mahler a0 = 0, (v, res) of a1, a2, a3 = (2,2), (2,2),\n  (3,1); F1(K(+-4G)) = 0; v_3(h(2)) = v_3(h(5)) = v_3(h(8)) = 3; among m mod 27 only m = 0 has v_3(h(m)) >= 5.\n(`<100, 0>` encodes the value 0; (v, res) means 3^v exactly divides the value and the quotient is res mod 3.)\nExpected output (`expected/1.3.5d_mahler.txt`):\n```\nIDENTITY VALUES [ <100, 0>, <5, 1>, <5, 1>, <9, 2>, <5, 1>, <5, 1>, <9, 2> ]\nIDENTITY MAHLER [ <100, 0>, <5, 1>, <5, 2>, <7, 2>, <8, 2>, <7, 1>, <7, 1> ]\nIDENTITY REDUCED AT 3^5 [ 0, 1, 2, 0, 0, 0, 0 ]\nDEEP ID 1 VALUES [ <100, 0>, <9, 2>, <9, 2>, <13, 2>, <9, 2>, <9, 2>, <13, 2> ]\nDEEP ID 1 MAHLER [ <100, 0>, <9, 2>, <9, 1>, <11, 2>, <10, 1>, <13, 1>, <13, 1>\n]\nDEEP ID 1 REDUCED [ 0, 2, 1, 0, 0, 0, 0 ]\nID COORD 0 [ <4, 2>, <3, 2>, <2, 1>, <0, 1> ]\nID COORD 1 [ <6, 1>, <5, 1>, <4, 1>, <0, 1> ]\nB VALUES [ <100, 0>, <2, 2>, <3, 2>, <3, 2>, <2, 2> ]\nB MAHLER [ <100, 0>, <2, 2>, <2, 2>, <3, 1>, <4, 2> ]\nB REDUCED AT 3^2 [ 0, 2, 2, 0, 0 ]\nB2 VALUES [ <100, 0>, <3, 2>, <4, 2>, <4, 2>, <3, 2>, <4, 2>, <4, 1> ]\nB2 MAHLER [ <100, 0>, <3, 2>, <3, 2>, <4, 1>, <5, 2>, <6, 2>, <8, 1> ]\nGLOBAL KUMMER ZERO 0 0\nB ALL MOD27 [ <0, 100>, <1, 2>, <2, 3>, <3, 3>, <4, 2>, <5, 3>, <6, 3>, <7, 2>,\n<8, 3>, <9, 4>, <10, 2>, <11, 3>, <12, 3>, <13, 2>, <14, 3>, <15, 3>, <16, 2>,\n<17, 3>, <18, 4>, <19, 2>, <20, 3>, <21, 3>, <22, 2>, <23, 3>, <24, 3>, <25, 2>,\n<26, 3> ]\nB HENSEL MOD27 [ <0, 100> ]\nPKG_OK\nPKG_MAGMA_END\n```\nUsed in the section: IDENTITY MAHLER, ID COORD 1, B MAHLER, and h(2), h(5), h(8) (the entries 2, 5, 8 of\nB ALL MOD27).\n\nCross-check `1.3.5d_check_valuations.py` (python-flint, reviewer). It first runs `1.3.5c_check_sieve.py`\n(same Mahler data), then: for m = 3^e a (e = 0..3) the identity-disc valuations\nv(u) = (>= 2e+4, 2e+3, 2e+2) with residues (1,1,1) for e >= 1 and v(F1) = 4e + 5; in the 4G disc\nv(h(m)) = e + 2 and h(m)/(9m) = 2 mod 3 for e = 1, 2, 3. Key expected lines\n(`expected/1.3.5d_check_valuations.txt`):\n```\n e=1 a=1 m=3: v(u)=[6, 5, 4] res=[1, 1, 1] vPhi=9 OK\n e=2 a=1 m=9: v(u)=[8, 7, 6] res=[1, 1, 1] vPhi=13 OK\n e=3 a=1 m=27: v(u)=[10, 9, 8] res=[1, 1, 1] vPhi=17 OK\n e=1 a=1: v(h)=3 res(h/9m)=2 v(h/9m)=0\n e=2 a=1: v(h)=4 res(h/9m)=2 v(h/9m)=0\n e=3 a=1: v(h)=5 res(h/9m)=2 v(h/9m)=0\n```\nand every identity-disc row ends with `OK`.\n\n## 1.3.5e: fibre over K(+-4G) by Groebner basis; common zeros of A2, B (Magma)\n\n`1.3.5e_fiber.m` computes in Q[R,t] (lex): GB(R^2 - R + t + t^3, 2B + A2, 8A0 + 3A2) = [R - 1, t] with\nA2(1,0), B(1,0), A0(1,0) = -8, 4, 3; GB(R^2 - R + t + t^3, A2, B) = [R, t] with A0(0,0) = -1; and (asserted,\nnot used in the section) Res_R(R^2 - R + t + t^3, A2) = t q(t) with q of degree 11 irreducible.\nExpected output (`expected/1.3.5e_fiber.txt`):\n```\nGROEBNER BASIS [\n    R - 1,\n    t\n]\nSPECIALIZED COEFFICIENTS -8 4 3\nDEGREE-ZERO TANGENT SUBCASE GROEBNER BASIS [\n    R,\n    t\n]\nCONSTANT DIVISOR LIFTS {@ @}\nPKG_OK\nPKG_MAGMA_END\n```\nCross-check `1.3.5e_check_fiber.py` (SymPy, reviewer). Expected (`expected/1.3.5e_check_fiber.txt`):\n```\nGB fiber K(4G): [R - 1, t]\nGB A2=B=0: [R, t]\nRes_R(curve,A2) factors: (1, [(t, 1), (t**11 + 16*t**10 - ... + 40*t + 32, 1)])\n```\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.3.6: x^4 + x^2y^2 + y^3z + z^4 = 0\n\nNotation of the section: C: X^4 + X^2Y^2 + Y^3Z + Z^4 = 0, C(Q) = {(0:1:0), (0:1:-1)};\nquadrics q1 = -(b^2 + ac), q2 = (ab - b^2 - ac - bc)/2, q3 = (ab + b^2 + ac + bc)/2, cover\nD = {q1 = u^2, q2 = uv, q3 = v^2} in P^4; genus-2 curve H: y^2 = -16 det(N1 - 2s N2 + s^2 N3)\n= 2s^5 + 6s^4 - 4s^3 - 10s^2 + 12s - 4, J_H = Jac(H), T = (s^2 + 2s - 2, 0), G = (s + 5/2, -9/4)\n(degree-1 Mumford form); Phi1, Phi2 = the two quartic forms on the Kummer surface (printed by the script);\nR = 28G; elliptic quotient E: y^2 = x^3 + 16x - 64.\n\n## Software and how the outputs were obtained\n\n| Software | Used for | Command (run inside this folder) |\n|---|---|---|\n| Magma | `1.3.6a_kummer_image.m` | paste the file into Magma (or `magma -b 1.3.6a_kummer_image.m`) |\n| Python 3.11 + python-flint 0.9 | `1.3.6a_check_sieve47.py`, `1.3.6a_check_padic47.py` (both import `jac_e09.py`) | `python3 -I FILE.py` |\n| Python 3.11 + SymPy 1.14 | `1.3.6a_check_smooth.py`, `1.3.6a_check_model.py` | `python3 -I FILE.py` |\n| PARI/GP 2.15.4 | `1.3.6a_check_lpolys.gp` | `gp -q -f 1.3.6a_check_lpolys.gp < /dev/null` |\n\n`jac_e09.py` is the reviewer's independent genus-2 Jacobian arithmetic (Cantor's algorithm with python-flint,\nFlynn's Kummer map for a quintic model; it contains Phi1, Phi2 and the Kummer equation as printed by the\nMagma log). It is not run on its own.\n\n**Magma.** Magma is not installed on the machine where this folder was assembled. The Magma script was run on\nthe public Magma calculator (V2.29-10) on 14 October 2026, in under 60 s (CPU 7.9 s). It is the certificate\n`t6_9_kummer_image.m` of the source package, unchanged. Its file in `expected/` is the log shipped with that\npackage (Magma V2.29-8, hence the first line `MAGMA_GETVERSION 2 29 8`); our 14 Oct 2026 run on V2.29-10\npassed every assertion and printed the same output line by line apart from that version line. The first line\nand the last two lines (`PKG_OK`, `PKG_MAGMA_END`) were printed by statements that the submission tool of the\npackage adds around the script; a failed assertion stops the script before `PKG_OK`.\n\n**Python, PARI.** Every such script in this folder was run here (October 2026; each under 10 s); its output is\nin `expected/<script name>.txt` and agrees with the expected values below (for `1.3.6a_check_padic47.py` it is\nidentical to the log of the original review run).\n\n## Files\n\n| Code | File | Software | Run here |\n|---|---|---|---|\n| 1.3.6a | `1.3.6a_kummer_image.m` | Magma | no (Magma calculator, 14 Oct 2026) |\n| 1.3.6a | `1.3.6a_check_sieve47.py`, `1.3.6a_check_padic47.py` (cross-checks) | python-flint | yes, pass |\n| 1.3.6a | `1.3.6a_check_smooth.py`, `1.3.6a_check_model.py` (cross-checks) | SymPy | yes, pass |\n| 1.3.6a | `1.3.6a_check_lpolys.gp` (cross-check) | PARI/GP | yes, pass |\n\n\"Cross-check\" scripts are independent re-computations without Magma of facts that the certificate asserts.\nThe proof uses the certificate; the cross-checks are optional.\n\n## 1.3.6a: the certificate (Magma)\n\n`1.3.6a_kummer_image.m` asserts:\n- H: y^2 = -16 det(N1 - 2s N2 + s^2 N3) = 2s^5 + 6s^4 - 4s^3 - 10s^2 + 12s - 4, nonsingular of genus 2;\n  `RankBounds(J_H)` = (1, 1); torsion Z/2;\n- it finds two quartic forms Phi1, Phi2 on the Kummer surface by interpolation through Kummer images of points\n  of D over number fields. The interpolation proves nothing by itself;\n- over Q(C), at the generic point (z(1+z)/x : x : (1-z+z^2)/x : 1 : z) of D (x = X/Y, z = Z/Y): the tangent\n  quadratic has degree 2 and is irreducible; there are exactly 2 lifts in J_H(Q(C)); they have equal Kummer\n  points; and Phi1 = Phi2 = 0 there. This identity is the proof-relevant output\n  (`GENERIC KUMMER IMAGE IDENTITIES [ true, true ]`);\n- `MordellWeilGroupGenus2`: Z/2 + Z, finite_index = proved = true, rank_upper = 1; T = (s^2+2s-2, 0, 2),\n  G = (s + 5/2, -9/4, 1); (a transcription check, not used) the points of J_H with Mumford polynomial\n  s^2 - 6s + 4 are T + G and T - G;\n- in the affine chart k4 = 1 the Phi_i have no constant or linear part; their quadratic parts are\n  Q1 = 48k1^2 - 4k1k3 + k2^2 + 24k2k3, Q2 = -48k1^2 + 240k1k3 - 60k2^2 - 24k2k3, and\n  (Q1, Q2)(24, 7, 1) = (39, 8) mod 47;\n- for j = 1..5 the primitive integer Kummer coordinates of [j]R (R = 28G) have 47-adic valuations\n  (2, 2, 2, 0); the normalised direction of R is (24, 7, 1); Phi1, Phi2 at [j]R have valuation 4;\n- sieve mod 47: ord T = 2, ord G = 28; among eT + nG (e in {0,1}, 0 <= n < 28) only (0, 0) satisfies\n  Phi1 = Phi2 = 0 mod 47;\n- p = 5, 7, 11, 13, 47: C and H nonsingular mod p; class-group invariants of C mod p;\n  L_p(C) = L_p(E) L_p(H).\n\nExpected output: `expected/1.3.6a_kummer_image.txt` (the full transcript, including the 28-term\ncoefficients of Phi1, Phi2 and the Kummer quartic). Key lines:\n```\nH polynomial 2*s^5 + 6*s^4 - 4*s^3 - 10*s^2 + 12*s - 4\nrank 1 1 torsion\nAbelian Group isomorphic to Z/2\n...\nmatrix dimensions 40 35\nkernel dimension 2\nKummer image equations [\n    944*x1^4 + 3360*x1^3*x3 - 368*x1^3*x4 + ...\n    3776*x1^3*x2 - 9024*x1^3*x3 - 576*x1^3*x4 + ...\n]\n...\nGENERIC xpol degree 2 number of J lifts 2\nGENERIC KUMMER IMAGE IDENTITIES [ true, true ]\nMW certification\nAbelian Group isomorphic to Z/2 + Z\nDefined on 2 generators\nRelations:\n    2*MW.1 = 0\ntrue true 1\nT,G (x^2 + 2*x - 2, 0, 2) (x + 5/2, -9/4, 1)\nLOCAL QUADRATIC FORMS [\n    48*r1^2 - 4*r1*r3 + r2^2 + 24*r2*r3,\n    -48*r1^2 + 240*r1*r3 - 60*r2^2 - 24*r2*r3\n]\nat MW tangent mod47 [ 39, 8 ]\nP47 LIFT 1 coordinate valuations [ 2, 2, 2, 0 ]\nequation valuations [ 4, 4 ]\ntangent direction raw [ 14, 8, 28 ]\n28 normalized [ 24, 7, 1 ]\nleading equation residues [ 33, 14 ]\n...                                     (LIFT 2-5: same valuations; normalized [ 2, 28, 4 ],\n                                         [ 28, 16, 9 ], [ 8, 18, 16 ], [ 36, 34, 25 ])\nSIEVE 47 ordT 2 ordG 28 allowed [ <0, 0> ]\nREDUCTION CERT 5 smooth C,H true true class invariants [ 198, 0 ]\nL factor true\nREDUCTION CERT 7 smooth C,H true true class invariants [ 2, 198, 0 ]\nL factor true\nREDUCTION CERT 11 smooth C,H true true class invariants [ 2, 2, 320, 0 ]\nL factor true\nREDUCTION CERT 13 smooth C,H true true class invariants [ 12, 288, 0 ]\nL factor true\nREDUCTION CERT 47 smooth C,H true true class invariants [ 2, 2, 8, 2520, 0 ]\nL factor true\nPKG_OK\nPKG_MAGMA_END\n```\nMeaning: the \"direction\" of [j]R is the residue mod 47 of (k1, k2, k3)/47^2 divided by k4. For j = 1..5 it is\nj^2 (24, 7, 1) mod 47 and the Phi_i have valuation 4, matching the m^2 scaling used in the section.\n\nCross-checks without Magma:\n- `1.3.6a_check_sieve47.py` (python-flint, reviewer): ord T mod 47 = 2, ord G mod 47 = 28, T is not in <G>\n  mod 47, and only (e, n) = (0, 0) lies on Phi1 = Phi2 = 0. Expected (`expected/1.3.6a_check_sieve47.txt`):\n  ```\n  order T mod 47: 2  order G mod 47: 28\n  Kummer points lie on Magma's printed Kummer quartic: ok\n  T in <G> mod 47: False\n  allowed (e,n): [(0, 0)]\n  PHI1 only: [(0, 0)]\n  PHI2 only: [(0, 0)]\n  ```\n  (`PHI1 only` lists the (e, n) with Phi1 = 0 mod 47, ignoring Phi2; likewise `PHI2 only`.)\n- `1.3.6a_check_padic47.py` (python-flint, reviewer; an optional deeper test at m = 47, which did not finish\n  in the review run, was removed): exact Kummer coordinates of [m]R for m = 1..5, their 47-adic valuations,\n  normalised directions and the valuations of Phi1, Phi2, and (Q1, Q2)(24, 7, 1) mod 47. Expected\n  (`expected/1.3.6a_check_padic47.txt`):\n  ```\n  m=1  coord vals [2, 2, 2, 0]  normalized dir [24, 7, 1]  v(PHI1)=4 v(PHI2)=4  approx_digits=861\n  m=2  coord vals [2, 2, 2, 0]  normalized dir [2, 28, 4]  v(PHI1)=4 v(PHI2)=4  approx_digits=3445\n  m=3  coord vals [2, 2, 2, 0]  normalized dir [28, 16, 9]  v(PHI1)=4 v(PHI2)=4  approx_digits=7751\n  m=4  coord vals [2, 2, 2, 0]  normalized dir [8, 18, 16]  v(PHI1)=4 v(PHI2)=4  approx_digits=13780\n  m=5  coord vals [2, 2, 2, 0]  normalized dir [36, 34, 25]  v(PHI1)=4 v(PHI2)=4  approx_digits=21532\n  expected m^2*(24,7,1): [[24, 7, 1], [2, 28, 4], [28, 16, 9], [8, 18, 16], [36, 34, 25]]\n  Q1,Q2 at (24,7,1) mod 47: 39 8\n  ```\n- `1.3.6a_check_smooth.py` (SymPy, reviewer): C is smooth mod 5, 7, 11, 13, 47 (Groebner basis of F and its\n  partial derivatives is [1] in every affine chart). Expected (`expected/1.3.6a_check_smooth.txt`):\n  `p smooth in all charts: True` for p = 5, 7, 11, 13, 47.\n- `1.3.6a_check_lpolys.gp` (PARI, reviewer): #J_C(F_p) from point counts of C over F_p, F_{p^2}, F_{p^3}\n  equals #E(F_p) * #J_H(F_p) for p = 5, 7, 11, 13, 47 (and equals the product of the Magma class-group\n  invariants). Expected (`expected/1.3.6a_check_lpolys.txt`):\n  ```\n  p=5 #J_C(F_p)=198 = [2, 1; 3, 2; 11, 1] ; #E*#J_H=198\n  p=7 #J_C(F_p)=396 = [2, 2; 3, 2; 11, 1] ; #E*#J_H=396\n  p=11 #J_C(F_p)=1280 = [2, 8; 5, 1] ; #E*#J_H=1280\n  p=13 #J_C(F_p)=3456 = [2, 7; 3, 3] ; #E*#J_H=3456\n  p=47 #J_C(F_p)=80640 = [2, 8; 3, 2; 5, 1; 7, 1] ; #E*#J_H=80640\n  ```\n- `1.3.6a_check_model.py` (SymPy; the parts for this equation of a helper script `chk2.py`):\n  H = -16 det(...) = 2(s^2+2s-2)(s^3+s^2-2s+1); the point (Z(Y+Z), X^2, Y^2-YZ+Z^2, XY, XZ) satisfies\n  q_i - (u^2, uv, v^2) = (-1, -1/2, 1/2) F, so pi(C) lies in D; and the Ritzenthaler-Romagny model\n  y^2 = -(s^2-2s-2)(s^3-s^2-2s-1)/2, which becomes H under s -> -s, y -> y/2 (H(s) = 4 RR(-s)).\n  Expected (`expected/1.3.6a_check_model.txt`):\n  ```\n  E09 H: 2*s**5 + 6*s**4 - 4*s**3 - 10*s**2 + 12*s - 4  factored: 2*(s**2 + 2*s - 2)*(s**3 + s**2 - 2*s + 1)\n    E09 on D: -X**4 - X**2*Y**2 - Y**3*Z - Z**4  ratio to F: -1\n    E09 on D: -(X**4 + X**2*Y**2 + Y**3*Z + Z**4)/2  ratio to F: -1/2\n    E09 on D: (X**4 + X**2*Y**2 + Y**3*Z + Z**4)/2  ratio to F: 1/2\n  E09 Ai Matrix([[0, -1, 0], [1/2, -1/2, -1/2], [1/2, 1/2, 1/2]]) det 2\n   a,b,c= s**2/2 + s s**2/2 - s - 1 s**2/2 - s  RR f= -(s**2 - 2*s - 2)*(s**3 - s**2 - 2*s - 1)/2\n  ```\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.3.7: x^3 z + y^3 x + y^2 z^2 - z^3 y = 0\n\nNotation of the section: F = x^3 z + y^3 x + y^2 z^2 - z^3 y; points P1 = (1:0:0), P2 = (0:1:0), P3 = (0:0:1),\nP4 = (0:1:1); G = [P3 - P1]; bitangent field K = Q(w), g(w) = 0 (`data/g.txt`), degree 28, signature (4,12);\nthe bitangent y = a x + n z with a in K (`data/a.txt`) and n = -1/w^2 (`data/n.txt`); S = the 13 prime ideals\nof K above 2, 11, 41269.\n\nThe scripts are those of the verification of this solution (step ids s01-s09 in the original working notes,\nTable 1), with the retained scripts under their original names. The map to the code ids of the section is in the table below.\n\n## Software\n\nPython 3 with cypari2 (PARI 2.17.4), python-flint 0.9.0 (Arb), numpy, sympy (s03, and the Sturm sequences in\ns08); `gp` (PARI/GP 2.15.4) for s05c. No Magma, no Sage. Let `PY` be a Python with cypari2 and python-flint.\n\n**No result depends on GRH.** `bnfinit` is called only in s05a, to *propose* generators of prime ideals; s05b\nand s05c accept a generator only after an exact check.\n\n**PARI versions.** Each generator in `data/witnesses/` is stored as 28 integer coordinates on the fixed Z-basis of\nO_K in `data/zk.txt` (the basis returned by PARI 2.17.4's `nfinit` for g; SHA-256\n`eb575120fdad42a7686adbdb15ed11e193eabb453981b6d3176e9cee89772ed0`). PARI 2.15.4 returns a different integral\nbasis, so the files are always decoded through `data/zk.txt`, never through the running `nf.zk`; s05a, s05b and\ns05c all do this. s05c repeats the exact check as an independent GP script, run with GP 2.15.4.\n\n**Running.** Run from this folder, e.g. `$PY s01_curve.py`. The scripts import the local module `t62lib.py`;\nto run them with `python3 -I`, put this folder on `sys.path` explicitly, for example\n`$PY -I -c 'import sys,runpy; sys.path.insert(0,\".\"); sys.argv=sys.argv[1:]; runpy.run_path(sys.argv[0], run_name=\"__main__\")' s01_curve.py`.\n`PY=python3 ./run_all.sh` reruns every step except s05a (about 8 minutes) and compares with `expected/`;\n`./run_all.sh --full` also regenerates the witnesses (13 more minutes, one worker). Order: s04 and s06 must\nprecede s08, because they write `data/ks2_basis.json` and `data/local2.json`.\n\n## Files and code ids\n\n| Code | Step | Script | What it checks | Time | Run here |\n|---|---|---|---|---|---|\n| 1.3.7a | s01 | `s01_curve.py` | bad primes 2, 11, 41269 (eliminants in the three charts, singular points over F_p-bar); one F_p-rational ordinary node at each (split at 2, 41269, non-split at 11); v_p(F(lift)) = 1; points with a zero coordinate; divisors of x, y, z; #C(F_{p^k}), L-polynomials, #J(F_p) = 31, 141, 531, 10238 (p = 3, 5, 7, 17); J(Q)_tors = 0 | 1 s | yes, identical |\n| 1.3.7b | s02 | `s02_field.py` | eliminant m^24 B(m); B irreducible, squarefree; the bitangent identity over K; `nfcertify`; d_K = 2^42 11^6 41269^6, signature (4,12); prime ideals above 2, 11, 41269; -n a square, a a unit; Frobenius cycle types (orders 15, 7, 9); no quadratic subfield | 1 s | yes, identical |\n| 1.3.7c | s03 | `s03_global_kernel.py` | V = F_2^6 with the 28 odd quadratic forms: supports of size 12, pairwise distinct; transvections generate a group of order 1451520; (M/V)^G = 0; J[2]^G = 0 | 5 s | yes, identical |\n| 1.3.7d | s05a, s05b, s05c | `run_s05a.sh` -> `s05a_principal_witnesses.py`; `s05b_check_classgroup.py`; `wit2gp.py` + `s05c_check_witnesses.gp` | generators for the 37,155 prime ideals of norm < e^13 (proposals); exact check; B_0(13) > 0 in ball arithmetic | 13 min; 4 min; 3 min | yes: s05a regenerated all nine witness files, identical to `data/witnesses/`; s05b, s05c identical |\n| 1.3.7e | s04 | `s04_sunits.py` | -1, u_1..u_15 (units), pi_1..pi_13 (generators of the 13 prime ideals of S) from the source; printed norms; independence (rank 29 on 550 quadratic characters; 29 characters suffice). Writes `data/ks2_basis.json` | 1 s | yes, identical |\n| 1.3.7f | s06 | `s06_local2.py` | (K (x) Q_2)^x / squares = F_2^36 via `ideallog` modulo P^(2e+1); 82 points of C(Q_2) (Hensel with proved precision); W_2 of dimension 4 modulo Q_2^x; the four points of the source give the same W_2. Writes `data/local2.json` | 1 s | yes, identical |\n| 1.3.7h | s08 | `s08_selmer.py` | The paper uses the final dimension calculation with only p=2 and p=11. The script also records unused consistency checks: the fake Selmer group in K(2,S)/Q(2,S) cut out by (N), (11), (41269), (2), (real) has dimension 1, spanned by [a] = delta(G); real signs on certified isolating intervals; the variants with conditions removed | 1 s | yes, identical |\n| 1.3.7i | s09 | `s09_chabauty.py 3` | Chabauty-Coleman at 3: integral expansions; E ~ P1 + P2 + P3 + 31G; log(31G), checked with 62G; Strassmann bounds; the extra zero U* in the disc of P3 | 1 s | yes, identical |\n\n\"Run here\" = rerun on 16 October 2026 with `python3 -I` as above (PARI 2.17.4; s05c with GP 2.15.4), from a scratch\ncopy of this folder; outputs compared with `expected/` after removing timing lines.\n\nData files (`data/`): `g.txt`, `a.txt`, `n.txt` (from the source, verified in s02), `paper_C2.json` (transcription),\n`ks2_basis.json` (written by s04), `local2.json` (written by s06), `zk.txt` (the fixed integral basis),\n`witnesses/wit_*.jsonl.gz` (1.1 MB; written by s05a, checked by s05b and s05c).\n\n## 1.3.7a-c: the curve, the field, the global kernel\n\nExpected (`expected/s01_curve.out`, `s02_field.out`, `s03_global_kernel.out`), key lines:\n```\nchart ('x', 'y') gcd of eliminants = [2, 1; 11, 1; 41269, 1]          (the same in the other two charts)\nPASS: p=11: the singular point (1, 1, 7) is an ordinary double point (non-split) ...\nPASS: #J(F_p) = 31, 141, 531, 10238 for p = 3, 5, 7, 17\nPASS: gcd(#J(F_5), #J(F_17)) = 1, so J(Q)_tors = 0 (torsion injects at good odd p)\nRes_n(e1,e2) factors (degree in m, multiplicity): [(1, 24), (28, 1)]\nPASS: d_K = 2^42 * 11^6 * 41269^6  ([2, 42; 11, 6; 41269, 6])\n   primes above 2: (e,f) = [(4, 1), (2, 3), (2, 3), (4, 3)]  local degrees [4, 6, 6, 12]\n   order divisible by 5: Frob_13 has cycle type [3, 5, 5, 15] (order 15) ... Frob_29 [7, 7, 7, 7] ... Frob_3 [1, 9, 9, 9]\nPASS: the transvections generate a group of order 1451520 = |Sp_6(F_2)| acting on the 28 odd forms\nPASS: (M/V)^G = W/U = 0: the global kernel is 0, so delta is injective on J(Q)/2J(Q)\n```\n(One informational line of s01 says that two components of a reducible plane quartic meet with total\nintersection number at least 3; the section proves irreducibility of the special fibres with the branches at the\nnode, and no PASS line depends on that sentence.)\n\n## 1.3.7d: the class group of K\n\n* s05a (proposals): `PY=$PY ./run_s05a.sh OUTDIR` runs `s05a_principal_witnesses.py LO HI OUT` on nine ranges of\n  rational primes (one worker, about 13 minutes) and writes `wit_LO_HI.jsonl.gz`; log in\n  `expected/s05a_principal_witnesses.out`. Rerun here (16 October 2026, PARI 2.17.4, 15 minutes): the nine\n  regenerated files have the same content as those in `data/witnesses/`.\n* s05b (exact check and bound): `$PY s05b_check_classgroup.py data/witnesses/wit_*.jsonl.gz`. Expected\n  (`expected/s05b_check_classgroup.out`):\n```\nindex [O_K : Z[w]] = [2, 24; 717344179, 1; 9625055309383932097, 1]\nprime ideals of norm <= 442413: 37155\nPASS: all 37155 prime ideals of norm < e^13 are principal (exact check of 37155 generators, 237 s)\nprime-power terms: 25333 distinct pairs (N(P), m); 37333 pairs (P, m)\narchimedean+discriminant part: [-0.78489255426850599306 +/- 3.50e-21]\nB_0(13) = [0.0303690492767119765536576158681 +/- 1.39e-32]\nPASS: B_0(13) > 0 (rigorous ball arithmetic), hence Cl(K)[2] = 0 by Theorem B; h_K is odd\n```\n* s05c (independent GP check): `mkdir -p out; python3 wit2gp.py out/wit.gp data/witnesses/wit_*.jsonl.gz`, then\n  `echo 'WIT=\"out/wit.gp\"; read(\"s05c_check_witnesses.gp\")' | gp -q -s 400000000 -D parisizemax=2000000000`.\n  Expected (`expected/s05c_check_witnesses_gp2154.out`): `ALL 37155 prime ideals of norm <= 442413 matched\n  one-to-one to exact generators`, and B_0(13) = 0.030369049276711976553657615868... in floating point.\n\n## 1.3.7e-h: S-units, local images, the Selmer group\n\nExpected (`expected/s04_sunits.out`, `s06_local2.out`, `s08_selmer.out`), key lines:\n```\nPASS: the ideals (pi_1),...,(pi_13) are pairwise distinct: they are the 13 primes of S\nPASS: the 29 elements are independent in K^x/K^x2 (character matrix of rank 29)\nPASS: dim (K (x) Q_2)^x / squares = 36\nPASS: the classes l(Q) a of these points span a space W_2 of dimension 4 modulo Q_2^x\nPASS: the paper's four points span the same W_2 (dimension 4 mod Q_2^x)\n   dim of {kappa in K(2,S) satisfying all conditions} = 5\nPASS: dim Sel_fake = dim V - dim Q(2,S) = 1\nPASS: [a] is not in Q(2,S): delta(G) != 0, and Sel_fake = <[a]>\n   without condition p=11 (e mod 2 = [1, 1, 1, 1, 0]): dim Sel would be 2\n   without condition p=2                         : dim Sel would be 14\n   with the conditions p=2 and p=11 only        : dim Sel would be 1\n```\n\n## 1.3.7i: Chabauty-Coleman\n\nExpected (`expected/s09_chabauty3.out`), key lines:\n```\nE for 31G: x-cubic 80621568*x^3 + 19551871*x^2 - 576258756*x + 551007360 ;  y = 27648/10517*x^2 + 154341055/30667572*x - 31989800/2555631\n   L_x = log(31G) . omega = 642322739790 (mod 3^25), valuation 2\n   L_y = log(31G) . omega = 657631671201 (mod 3^25), valuation 4\n   L_z = log(31G) . omega = 233976154908 (mod 3^25), valuation 1\nPASS: log(62G) = 2 log(31G) to the working precision (independent Riemann-Roch + integration check)\nPASS: Hensel: J1(s0) has valuation 43 > 2 v(J1'(s0)) = 6, so U* = s0 + O(3^40) is a zero of eta1\n   U* = 589926054506 (mod 3^25), t = 3U*;  int eta2 at U* = 168750803736 (mod 3^25)\nCONCLUSION: C(Q) = {P1, P2, P3, P4}\n```\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.3.8: x^4 - y^3 z + x y z^2 + x z^3 = 0\n\nNotation of the section: F = x^4 - y^3 z + x y z^2 + x z^3, f = F(x,y,1) = x^4 - y^3 + xy + x;\npoints oo = (0:1:0) (a rational hyperflex, tangent z = 0), P0 = (0:0:1), P1 = (-1:0:1);\nT = [P0 - oo] (order 3), G1 = [P1 - oo]; the other 27 bitangents y = theta x + k(theta) z with B(theta) = 0,\nB(m) = m^27 - 48m^22 + 1344m^18 + 144m^17 - 3264m^13 + 832m^12 - 40704m^9 - 4608m^8 - 2560m^7 - 39936m^4\n- 15360m^3 - 1024m^2 + 4096; K = Q(theta) (degree 27, signature (3,12)); S = the four prime ideals above 2.\n\nThe scripts are those of the verification of this solution (computation ids C1-C14 in the original working notes,\nSection 8), unchanged except for the output folder of `classgroup/run_counts.sh` (now `work/counts/` inside\nthis folder). The map to the code ids of the section is in the table below.\n\n## Software\n\nPython 3 with cypari2 (PARI 2.17.4; the exact checks C1-C4 and C8 were also run with PARI 2.15.4),\npython-flint 0.9.0 (FLINT/Arb), numpy, sympy, mpmath; `gp` (PARI/GP 2.15.4) for the `.gp` files; gcc for the\ntwo C programs. No Magma, no Sage. Let `PY` be a Python with cypari2 and python-flint.\n\nPARI's `bnfinit`, `bnfsunit`, `bnfisprincipal` (which assume GRH) are used only to *propose* units, S-units and\ngenerators of prime ideals; every proposal is verified exactly by other code, so no result depends on GRH. The\nmaximal order is certified by `nfcertify`, which is unconditional. Different PARI versions return different\nintegral bases, so no data file depends on one: generators are stored as polynomials in theta.\n\nScripts that import a module of their own folder (`kfield.py`, `bound_data.py`) insert that folder into\n`sys.path` themselves, so they also run with `python3 -I`.\n\n## Files and code ids\n\n| Code | Computation (id of the source) | File(s) | Command (from this folder) | Time | Run here |\n|---|---|---|---|---|---|\n| 1.3.8a | curve (C2) | `curve/curve.py`, `curve/count_points.c` | `gcc -O2 -o curve/count_points curve/count_points.c && (cd curve && python3 curve.py)` | 15 s | yes, identical |\n| 1.3.8b | bitangents (C3) | `descent/bitangents.gp`, `descent/k_paper.gp` | `cd descent && gp -q < bitangents.gp` | 1 s | yes, identical |\n| 1.3.8c | field K (C1) | `field/field.gp` | `gp -q < field/field.gp` | 1 s | yes, identical |\n| 1.3.8d | J[2] in F_2^27 (C4) | `descent/theta.py` | `python3 descent/theta.py` | 1 s | yes, identical |\n| 1.3.8e | principal prime ideals (C7, C8) | `classgroup/gen_witnesses.py`, `classgroup/run_witnesses.sh`, `classgroup/check_witnesses.py`, `classgroup/kfield.py` | see 1.3.8e | 17 min + 2 min | yes: generation reproduced all 20 files identically; check identical (PARI 2.17.4 and 2.15.4) |\n| 1.3.8f | prime ideal counts (C5, C6, C9) | `classgroup/nroots.c`, `classgroup/run_counts.sh`, `classgroup/smallprimes.py`, `classgroup/crosscheck_counts.py`, `classgroup/data/small_25000.json` | see 1.3.8f | 23 min + 1 min + 25 s | C6, C9: yes, identical; C5: count files of the original run, SHA-256 checked |\n| 1.3.8g | fixed window and bound (C11) | `classgroup/bound.py`, `classgroup/bound_data.py`, `classgroup/classgroup_params.json` | see 1.3.8g | 40 s | yes, identical |\n| 1.3.8h | S-units, local image at 2, Selmer group (C12) | `descent/selmer.py` | `cd descent && $PY selmer.py` | 5 s | yes, identical |\n| 1.3.8i | the fixed basis of K(2,S) in paper_sunits.json (C13) | `descent/paper_sunits.py`, `descent/paper_sunits.json` | `cd descent && $PY paper_sunits.py` | 1 s | yes, identical |\n| 1.3.8j | Chabauty-Coleman at 3 (C14) | `chabauty/chabauty3.py` | `$PY chabauty/chabauty3.py` | 4 s | yes, identical |\n\nExpected outputs are in the `expected/` subfolder next to each script. \"Run here\" = rerun on 16 October 2026\n(Python scripts with `python3 -I`, from a scratch copy of this folder), compared with `expected/` after removing\ntiming lines.\n\n## Data not included (distributed separately or regenerated)\n\nTwo large data sets are not in this folder:\n\n* **Generator files of 1.3.8e** (`classgroup/data/witnesses/w_*.jsonl.gz`, 20 files, 34 MB): one proposed\n  generator for each of the 77,813 prime ideals of norm <= 10^6, as `num/den` polynomials in theta. Regenerate these files with\n  `PY=$PY classgroup/run_witnesses.sh` (about 20 minutes, one worker). Since PARI only proposes the\n  generators, another PARI version may propose other generators; the checker accepts any valid ones.\n* **Root-count files of 1.3.8f** (`work/counts/r_*.bin`, five files, 26,355,867 bytes = one byte per prime\n  p < 5*10^8): regenerated by `classgroup/run_counts.sh` (about 23 minutes). The SHA-256 of the five files\n  concatenated in increasing order of p is\n  `f1faf51bbcf2562ad0f2ffb8c095dd6855701a15a49c0f82644e582cd0cccbaa`; the per-range lines printed by\n  `nroots` are in `classgroup/expected/nroots_*.out`.\n\n## 1.3.8a: the curve (C2)\n\n`curve/curve.py` (with the brute-force point counter `curve/count_points.c`): the points with a zero\ncoordinate; F, F_x, F_y and the elimination showing that the only odd bad primes are 11 and 652081 (Groebner basis\n{1} over F_3 and F_5; y = -9/5, x = 3y^2; 108y^6 + y + 1 = 2^3 * 11 * 652081 / 5^6); at 11 and 652081 the node,\nits Hessian and F(lift) modulo p^2 (regular model); N_k = #C(F_{p^k}), L-polynomials and #J(F_p) for\np = 3, 5, 7, 17, 29; C(F_3); the discriminant of the cubic in y, which shows that C(R) is connected.\nExpected (`curve/expected/curve.out`), key lines:\n```\n(c) p=11: singular point (x,y)=(4,7) mod p; ... Hessian det mod p = 9 ...; F(lift) mod p^2 = 66 ...\n(c) p=652081: singular point (x,y)=(573841,521663) mod p; ... Hessian det mod p = 217362 ...; F(lift) mod p^2 = 91909512788 ...\n(d) p= 3  N=[3, 9, 27]  L=[1, -1, 0, 0, 0, -9, 27]  #J(F_p)=18\n(d) p= 5  N=[5, 29, 119]  ... #J(F_p)=108      (d) p= 7 ... #J(F_p)=360\n(d) p=17  N=[29, 313, 4856] ... #J(F_p)=9729   (d) p=29 ... #J(F_p)=22224\n(e) #J(F_17) odd: True ; gcd(#J(F_p), p=5,7,17,29) = 3\n(f) C(F_3) = ['(0:1:0)', '(0:0:1)', '(2:0:1)']\n(g) disc_y = ... = x^2 (4x - 27(x^3+1)^2); real roots of the second factor: 0 ; value at 0: -27\nALL CURVE CHECKS PASSED\n```\n\n## 1.3.8b: the bitangents (C3)\n\n`descent/bitangents.gp`: Res_k(E1, E2) = B/4096 (PARI's `factor` drops the constant and prints `Mat([B,1])`);\nB irreducible and squarefree; over K the gcd of E1(theta,k), E2(theta,k) is linear; f(x, theta x + k(theta))\n= (x^2+ux+v)^2 in K[x]; k(theta) equals the element given in the source (`k_paper.gp`); norms of\ntheta - k(theta) and -k(theta) are squares; B over Q_2 (degrees 2, 5, 10, 10), mod 3 and mod 113.\nExpected (`descent/expected/bitangents.out`): `B is a factor of Res_k: 1`, `B irreducible: 1   squarefree: 1`,\n`deg gcd_k(E1,E2) over K = 1`, `G(x, theta x + k(theta), 1) == (x^2+ux+v)^2 in K[x]: 1`,\n`k(theta) equals the paper's k(theta): 1`, `B over Q_2: degrees [2, 5, 10, 10]`.\n\n## 1.3.8c: the field K (C1)\n\n`field/field.gp`: B irreducible; disc(B) with its prime factors proved prime; `nfcertify` (maximal order,\nunconditional); d_K, signature, index; the prime ideals above 2, 11, 652081; root discriminant.\nExpected (`field/expected/field.out`): `disc(B) = [2, 318; 11, 6; 41, 2; 198839, 2; 652081, 6;\n17743299257028546229859, 2]`, `nfcertify: []`, `d_K = [2, 34; 11, 6; 652081, 6]  sign = [3, 12]`,\n`primes above 2 (e,f): [[2, 1], [10, 1], [5, 1], [5, 2]]`, `rd = 79.90...`.\n\n## 1.3.8d: J[2] inside F_2^27 (C4)\n\n`descent/theta.py`: the quadratic form q(a) = a1a2 + a3a4 + a5^2 + a5a6 + a6^2 of Arf invariant 1 on F_2^6, the\n27 nonzero singular vectors, the weights of the image of J[2] in F_2^27, and the orbit argument at 2 (orbit\nsizes 2, 5, 10, 10). Expected (`descent/expected/theta.out`): `weights of nonzero image vectors: {12: 36, 16: 27}`,\n`unions of weight 12: [[2, 10], [2, 10]] -> their sum has weight 20 not in {12,16}: dim J(Q_2)[2] <= 1`,\n`THETA MODEL CHECKS PASSED`.\n\n## 1.3.8e: every prime ideal of norm <= 10^6 is principal (C7, C8)\n\nGeneration (proposals only): `PY=$PY classgroup/run_witnesses.sh` writes `classgroup/data/witnesses/w_*.jsonl.gz`\n(resumable blocks of 50,000 rational primes; `bnfinit` with `setrand(1)`); log in\n`classgroup/expected/gen_witnesses.log` (`DONE`, 77,813 ideals, about 16 min). Rerun here (16 October 2026,\nPARI 2.17.4, 17 minutes): all 20 regenerated files have the same content as the separately distributed ones.\n\nCheck (exact, no `bnf`): `cd classgroup && $PY check_witnesses.py 1000000 data/witnesses ../work/counts/r_2_100000000.bin`\n(and the same with a Python whose cypari2 uses PARI 2.15.4). It enumerates the prime ideals itself (roots of\nB mod p for p > 1000 not dividing disc(B), cross-checked against the byte file of 1.3.8f; `idealprimedec` in the\ncertified maximal order otherwise) and checks for each generator g: integer coordinates on the certified basis\nof the running PARI, g in the prime ideal, |N(g)| = N(p) with N(g) = Res(B, num)/den^27 computed with FLINT;\neach record is used exactly once. Expected (`classgroup/expected/check_witnesses.log`,\n`classgroup/expected/check_witnesses_pari2.15.4.log`):\n```\nPARI 2.17.4: ALL 77813 PRIME IDEALS OF NORM <= 1000000 ARE PRINCIPAL (generators checked exactly); by degree {\"1\": 77583, \"2\": 193, \"3\": 18, \"5\": 15, \"6\": 3, \"10\": 1}\nPARI 2.15.4: ALL 77813 PRIME IDEALS OF NORM <= 1000000 ARE PRINCIPAL ...\n```\n\n## 1.3.8f: prime ideal counts up to 5*10^8 (C5, C6, C9)\n\n* C5: `gcc -O3 -march=native -o classgroup/nroots classgroup/nroots.c -lm && classgroup/run_counts.sh`\n  computes deg gcd(B, x^p - x) over F_p by exact 128-bit integer arithmetic for every prime p < 5*10^8 (one byte\n  per prime; the marker 255 at the primes 2, 11, 41, 198839, 652081 dividing disc(B)). Expected\n  (`classgroup/expected/nroots_*.out`): `RANGE 2 100000000 PRIMES 5761455 SUMROOTS 5757919 FNV 023369452f99ad80`\n  and four similar lines; SHA-256 of the concatenation as above.\n* C6: `$PY classgroup/smallprimes.py 25000 classgroup/data/small_25000.json`: all prime ideals over p <= 25000 and\n  over p | disc(B) (`idealprimedec`; 2764 rational primes, 16987 prime ideals). The file regenerated here is\n  identical to the one in this folder.\n* C9: `cd classgroup && $PY crosscheck_counts.py ../work/counts`: `root counts of nroots.c confirmed for 299996\n  primes with FLINT (5905 also with PARI polrootsmod)`.\n\n## 1.3.8g: the fixed window and the bound (C11)\n\nThe fixed window is in `classgroup/classgroup_params.json`; the optimizer is unnecessary and omitted.\n\n* C11: `cd classgroup && $PY bound.py classgroup_params.json ../work/counts data/small_25000.json` evaluates\n  B^-(f) of Theorem B for this window in Arb ball arithmetic. Expected (`classgroup/expected/bound.out`):\n```\nprime ideals of norm < R: 26365548 (degree-one ideals over p > sqrt(R): 26360341; over small/bad p: 5207)\nsha256 of the root-count bytes: f1faf51bbcf2562ad0f2ffb8c095dd6855701a15a49c0f82644e582cd0cccbaa\nprime ideals of norm <= C (all proved principal by check_witnesses.py): 77813\nC_inf = [-1.1614608353053368460 +/- 2.71e-20]\narchimedean part C_inf - int(1-f)W = [-1.215796113312597138688253 +/- 4.43e-25]\nprime part = [1.268980762763662101585279 +/- 1.73e-25]\nB^-(f) = [0.0531846494510649628970266145944 +/- 4.60e-32]\nB^-(f) > 0 RIGOROUSLY: True\n```\n\n## 1.3.8h: S-units, local image at 2, Selmer group (C12)\n\n`descent/selmer.py`: the 19 S-units (proposed by `bnfinit`/`bnfsunit`) checked exactly (units integral of norm\n+-1, the others of norm +-2^k), independence by quadratic characters at 400 degree-one prime ideals (rank 19),\nreal signs (Arb, = `nfeltsign`), square-class coordinates at the four prime ideals above 2 (`idealstar`/`ideallog`\nmodulo p^(2e+1); checked against `nfislocalpower`), the local image W_2 from four points of C(Q_2), the Selmer\ngroup, and the bound from the condition at 2 alone. Expected (`descent/expected/selmer.out`):\n```\nrank of the 19 x 400 quadratic-character matrix over F_2: 19 (primes q <= 2791)\nlocal image W_2 spanned by delta_2(P - oo) at P = (x, y mod 2^60): [(-40, 177036671367769442), (-39, 271477316397691054), (-37, 408928458825745180), (-35, 1073474074041454754)]  dim 4\ndim Sel_fake = 1  elements: [['-1', 'u1', 'u2', 'u3', 'u4']]\ndim with the condition at 2 only (no sign, no norm condition): 1  elements: [['-1', 'u1', 'u2', 'u3', 'u4']]\ntheta - k(theta) == product of ['-1', 'u1', 'u2', 'u3', 'u4'] mod squares (characters agree at 399 primes)\ndelta(T) = -k(theta) is a square in K: True\nSELMER CHECKS PASSED: rank J(Q) <= 1\n```\n(The script's docstring mentions a Hilbert pairing; the square classes are in fact computed with\n`idealstar`/`ideallog`, as stated above. The basis generated by `selmer.py` differs from the fixed basis in\n`paper_sunits.json`: the Selmer generator is -u1u2u3u4 in the former and\nu1u2u3u4 (epsilon1 epsilon2 epsilon3 epsilon4 in the paper) in the latter; see 1.3.8i.)\n\n## 1.3.8i: the fixed basis of K(2,S) in paper_sunits.json (C13)\n\n`descent/paper_sunits.py` checks the fixed units and S-units in\n`descent/paper_sunits.json` (u1, ..., u14, pi1, ..., pi4 in the data file correspond\nto epsilon1, ..., epsilon14, pi1, ..., pi4 in the lemma labelled `lem:1.3.8-sunits` in the paper):\nexact norms as printed, integrality, rank 19 modulo squares, and u1u2u3u4 = theta - k(theta) modulo squares. Expected (`descent/expected/paper_sunits.out`): `19 printed elements independent mod squares\n(rank 19);  u1u2u3u4 == theta - k(theta) mod squares: True`.\n\n## 1.3.8j: Chabauty-Coleman at 3 (C14)\n\n`chabauty/chabauty3.py`: the integral expansions of w_z, w_x, w_y at oo; the function g in L(14 oo) with\nord_P1 g = 9; the divisor E of degree 4 (roots of valuation 3/4); log(-9 G1) as a trace of one tiny integral,\nand log(-18 G1) as a check; the annihilating differentials on the three residue disks. Expected\n(`chabauty/expected/chabauty3.out`), key lines:\n```\nlog(-9G1): valuations (z,x,y) = (5,2,6); primitive vector / x-coordinate mod 3^6 = (432, 1, 81)\ncheck: log(-18 G1) = 2 log(-9 G1) to precision 3^114\n   oo: ... a0 = 1 (unit: True)     P0: ... a0 = 59048 (unit: True)     P1: ... a0 = 41849 (unit: True)\nStrassmann: each residue disc contains exactly one rational point; C(Q) = {oo, P0, P1}\n```\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.4.1: x^3+2x^2y+y^3+y^2z+xz^2+z^3+t^2x-t^2y-t^2z+tyz+t^3 = 0\n\nOnly the zero solution (the cubic surface S: F = 0 has no rational point), by an order-3 Brauer-Manin\nobstruction. Notation of the section: K = Q(s), s^2 = -23; h(T) = T^3 - T + 1; L = K(a), h(a) = 0;\ntau(a) = -(23a + s(6a^2+9a-4))/46; G = G0 + s*G1 the cubic form of the section (coefficient 288 at x y^2);\nfor a place w of K above a split prime p, sigma_w is the image of s in Z_p and G^(w) = G0 + sigma_w G1;\nw+ and w- are the places above 3 with sigma = 2 and sigma = -2 mod 27.\n\nThe fixed input `data/cubic_a_quadrics.json` contains the coefficient tables of\nq0, q1 and q2 over L used in the paper (SHA-256\n`4068055166c54e52e52c77b66286d557748e76113e8d26adf164617b03a0d128`).\nCertificate `1.4.1b_curve.py` reads these tables and checks the identities derived\nfrom them by exact arithmetic. No external working notes are required.\n\n## Requirements\n\nPython 3.10 or later, standard library only (no SymPy, PARI, Sage, Magma), and a C compiler (`cc`, or set\n`CC`) for `local_residues.c` (C library only). The Python scripts import `common141.py` from their own folder\nthemselves, so they run from any directory and under `python3 -I`. Do not use `python -O` (the checks are\n`assert`s; `local_residues` returns a nonzero exit status on failure).\n\n## Run\n\n`python3 run_all.py` runs every code id (about 2 seconds in total), compiles `local_residues.c` into a\ntemporary directory, and compares each output with `expected/<id>.txt`; `python3 run_all.py 1.4.1c` runs\none id; `python3 run_all.py --write` regenerates `expected/`. The files in `expected/` were produced on\n17 October 2026 with Python 3.11.15 and GCC 13.3.0 (`python3 -I run_all.py --write`) and then checked again\n(all identical).\n\n| id | file | what it computes | command | time |\n|---|---|---|---|---|\n| 1.4.1a | `1.4.1a_fields.py` | field facts of Lemma lem:1.4.1-field | `python3 -I -B 1.4.1a_fields.py` | < 1 s |\n| 1.4.1b | `1.4.1b_curve.py`, `data/cubic_a_quadrics.json` | the twisted cubic C and div_S(G) (Lemma lem:1.4.1-curve) | `python3 -I -B 1.4.1b_curve.py` | < 1 s |\n| 1.4.1c | `1.4.1c_bad_primes.py` | reduction of S modulo primes (Lemma lem:1.4.1-bad) | `python3 -I -B 1.4.1c_bad_primes.py` | < 1 s |\n| 1.4.1d | `local_residues.c` | 2-adic valuations of G on S (Lemma lem:1.4.1-zero) | `cc -O2 -o local_residues local_residues.c && ./local_residues 2 5 && ./local_residues 2 6` | < 1 s |\n| 1.4.1e | `local_residues.c` | 3-adic valuations of G on S (Lemma lem:1.4.1-three) | `./local_residues 3 3 && ./local_residues 3 4` | < 1 s |\n| 1.4.1f | `1.4.1f_local_points.py` | local points of S (remark after the proof) | `python3 -I -B 1.4.1f_local_points.py` | < 1 s |\n\nHelper: `common141.py` (exact arithmetic in L = Q(a, s) as 6-tuples of Fractions, sparse multivariate\npolynomials, the forms F and G, integer factorization by trial division).\n\n\n## 1.4.1a - the fields\n`h(+-1) != 0`, `disc h = -23`; `tau(a)` is a root of h, `tau(a) != a`, `tau^2(a) := P(P(a,s),s)` equals\n`P(a,-s)` (tau(a) with s -> -s), `tau^3(a) = a`; splitting of p = 2, 3, 7, 23, 29, 31, 59, 337, 9013 in K\n(Legendre symbol, and -23 = 1 mod 8 for p = 2) and the roots of h mod p; `h = (T-20)(T-13)^2 mod 23`,\n`(20-13)^2 = 3 mod 23`; in F_27 = F_3[T]/(h): `P(T,2) = T^3` and `P(T,-2) = T^9`; the p-adic square roots of\n-23 modulo 27, 81, 32, 64, 31, 59; a consistency loop over p < 2000 (0 or 3 roots of h mod p if (-23/p) = 1,\none root if -1). Expected: `PASS 1.4.1a`, with the table\n`2, 3, 29, 31: split, h irreducible; 59: split, roots [17, 46, 55]; 7, 337, 9013: inert; 23: ramified`,\n`Frob(w+) = tau, Frob(w-) = tau^2`.\n\n## 1.4.1b - the twisted cubic\nEvery monomial of q0 and q1 contains x or y; on the plane (x,y) = (lam w, mu w), q_k = w (A_k w + B_k z + C_k t);\n(W, Z, T) = cross product of the coefficient vectors (degrees 2, 3, 3); phi = (lam W, mu W, Z, T) (printed).\nChecks: `phi^*q0 = phi^*q1 = phi^*q2 = phi^*F = phi^*G = 0`; the 4x4 coefficient determinant of the four binary\ncubics is nonzero (printed); `q_k(tau^j phi) != 0` for j = 1, 2 and all k; `Res(W, Z - aT) = Res(W, Z - tau(a)T) = 0`,\n`Res(W, Z - tau^2(a)T) != 0`, `Res(W, T) != 0`; x^3-coefficients 1 (F) and 0 (G), x y^2-coefficient 288 (G).\nAdditional checks not needed in the proof: q_k(P0) = q_k(P1) = 0 and\n`q2(P2) = a + 3 + s(4 - 9a - 6a^2)/23`; grevlex leading monomials x^2, xy, y^2 of q0, q1, q2 - c q1; the reductions\nof q0, q1, q2 at a = 17, s = 6 mod 59 agree with the fixed reference coefficients in `1.4.1b_curve.py`, and phi mod 59 is a twisted cubic\non them (determinant 40 mod 59). Expected: `PASS 1.4.1b`.\n\n## 1.4.1c - reduction modulo primes\nMacaulay matrices (56 x 56) of the partial derivatives of F in degree 5 for the variable orders (x,y,z,t) and\n(x,y,t,z), determinants by fraction-free elimination:\n`2^6 * 3^17 * 7 * 23 * 29 * 31 * 59^2 * 337 * 9013^2` and `2^6 * 3^19 * 23 * 31 * 59^3 * 73 * 9013^2`,\ngcd `2^6 * 3^17 * 23 * 31 * 59^2 * 9013^2`. Ranks modulo p of the 90 x 56 matrix of the degree-5 map\n(g0, ..., g4) -> g0 F + sum g_i dF/dv_i: `{2: 50, 3: 56, 23: 55, 29: 56, 31: 55, 59: 55, 9013: 55}`.\nSingular points: (1,18,3,10) mod 23, (1,23,28,12) mod 31, (1,47,12,14) mod 59, (1,7636,7996,3944) mod 9013,\nand (1 : r^3+r : r^5 : r) over F_8 (r^3+r^2+1 = 0) mod 2. `G(P31) = 6, 16 mod 31` at s = 15, 16. Consistency:\n(1,23,28,12) is the only singular F_31-point (exhaustive). No Groebner basis is used.\nExpected: `PASS 1.4.1c`.\n\n## 1.4.1d, 1.4.1e - 2-adic and 3-adic residues\n`local_residues p k` visits all primitive (x,y,z,t) mod p^k with F = 0 mod p^k and evaluates G at both\nsquare roots of -23 mod p^k (those that lift to Z_p). Expected:\n- `2 5`: 12288 solutions, `(v at s=13, v at s=19) = (1, 4)` for all (13 = 5 mod 8, 19 = 3 mod 8);\n- `2 6`: 98304 solutions, `(v at s=19, v at s=45) = (4, 1)` for all;\n- `3 3`: 10206 solutions, `(v at s=2, v at s=25) = (0, 1)` for all (s = 2 is w+);\n- `3 4`: 275562 solutions, `(v at s=25, v at s=56) = (1, 0)` for all.\nAll valuations are below k, hence they hold at every p-adic point of S.\n\n## 1.4.1f - local points\nThe points (1,0,1,1), (1,0,0,6), (1,0,0,20), (1,0,1,10), (1,0,0,8363) are zeros of F mod 2, 23, 31, 59, 9013\nwith a nonzero partial derivative; S mod 3 has 7 points, among them (1,0,0,1) (gradient (1,1,2,2)); a real\npoint on F(1,0,0,t) = t^3 + t^2 + 1; the 3-adic points with x = 0 from F(0,1,z,0) = z^3 + z + 1, z = 1 mod 3.\nExpected: `PASS 1.4.1f`.\n"
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      "path": "anc/code/1.4.10/README.md",
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      "sha256": "d32426a590d6528e7157b371164215ae969aa854f50cd7abc375b36f7a224f88",
      "mode": 420,
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for Section 1.4.10: 4x^5 + 4y^5 + 11z^5 = 0\n\nInteger solutions exactly (n, −n, 0). Proof: z = 2w, x^5 + y^5 + 88w^5 = 0; a zero coordinate forces w = 0; for\nxyw ≠ 0 the Frey curve (u, v, w') = ε(x^5, 88w^5, y^5), εx ≡ 3 mod 4, has level 88 (w odd, type III^* at 2) or 22\n(w even); E[5] irreducible by Mazur's theorem; S_2(Γ_0(22))^new = 0, and at level 88 the prime 3 (T_3 = {0,1,4})\nexcludes both orbits. Source: the original working notes (Section 11 of the consolidated document; original packet\n`target_quintic_proof.tex`).\n\nAll scripts were written for the paper; they use `../common/frey_quintic/fq.py` and `fqsieve.gp`. The solver's\ncertificates (`quintic_4411_2adic.gp`, `quintic_4411_newforms.gp`, `quintic_4411_trace_sets.py`,\n`verify_small_newforms.gp`) were read, not run.\n\n## Requirements\nPython 3.10+ (standard library); PARI/GP 2.15+ (run here with 2.15.4).\n\n## Codes\n| id | script | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.10a | `1.4.10a_elementary.py` | H = 608; zero coordinates; 2 ∣ z and the new equation; hypotheses of Lemma lem:1.4:terms; r = 3 + 5v_2(w); Tate data and N_5; the points over F_3, the trace of Y^2 = X^3 − X, T_3 (two implementations); the sets modulo 25 (remark) | `python3 -I 1.4.10a_elementary.py` | < 1 s | yes |\n| 1.4.10b | `1.4.10b_newforms.gp` | `elllocalred` on the Frey curve for x ≡ 1 mod 4 (ε = −1), all w mod 2^6; S_2(Γ_0(22))^new and S_2(Γ_0(88))^new, minimal polynomials of a_3 and their values at 0, 1, 4 mod 5; the sieve with T_3 | `gp -q 1.4.10b_newforms.gp` | < 1 s | yes |\n\n`python3 run_all.py` runs both (about 1 s) and compares with `expected/`. Run here on 17 October 2026.\n\nKey output:\n```\n1.4.10a even term 88w^5, v_2(w) = 0: r = 3, type III*, f_2 = 3, v_2(Delta_min) = 10; N_5 = 88\n        even term 88w^5, v_2(w) = 1: r = 8, type I8,   f_2 = 1, v_2(Delta_min) =  8; N_5 = 22\n        q = 3: 4 points, 3 with a zero coordinate; traces {0}; T_q = {0,1,4}\n1.4.10b level 22: dim S_2^new = 0\n        level 88: dim S_2^new = 3, orbits of dimensions [1, 2]; minimal polynomials of a_3: [x + 3, x^2 - x - 4]\n        values at T = 0, 1, 4 modulo 5: [[3, 4, 2], [1, 1, 3]]\n```\n\n## Not used by the proof\n- The congruence modulo 25 (5 ∣ w): printed by 1.4.10a for the remark of the section; the level does not depend on it.\n"
    },
    {
      "path": "anc/code/1.4.2/README.md",
      "size_bytes": 5385,
      "sha256": "927630005ce91f9224edbe45fc2b13ed945938b6738f4bdf86156d5e1caaf14a",
      "mode": 420,
      "encoding": "utf-8",
      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.4.2: 2x^3 + x^2y + xy^2 + y^3 + y^2z − 2xz^2 − z^3 + t^2y + tz^2 − t^3 = 0\n\nOnly the zero solution. Proof: the cyclic cubic extension L = K(q), K = Q(√−23), h(q) = q^3 − q + 1; the\nplane section ℓ_+ = (−1+s)x + 4y = 0 splits into three lines over L; for a rational point the local invariants of\nψ = ℓ_+/ℓ_− vanish except at the two places above 2, where they are equal and nonzero (reciprocity, Theorem AC).\n\nAll scripts were written for this bundle (agent S2, 17 Oct 2026). The source's certificate scripts\n(`verify_tritangent_factors.py`, `verify_fields.gp`, `local_residue_b.py`, `verify_local_points_bc.py`,\n`steiner_exact_bc.py`, `fano_*.py` in the section 8.7 reproducibility archive) were read as references and were\nNOT run. No PARI/GP, Magma or Sage is used: the source's PARI field checks (`verify_fields.gp`) are replaced by\nthe elementary arguments of Lemma 1.4.2-field and code 1.4.2a.\n\nNotation: `p` in the scripts (as in the source) is the element ρ = −((s−1)q^2 + 8)/(4(3q^2−1)) of the text;\nthe scripts write u for q.\n\nRequirements: Python 3.10+; SymPy 1.14 (codes a, b; the scripts set `SYMPY_GROUND_TYPES=python`); a C99\ncompiler (`cc`, or `$CC`) for the C part of 1.4.2c. Shared helpers: `../common/cyclic_cubic/` (`cubicfield.py`,\n`restree.py`, `brute_residues.c`). Do not use `python -O` (the checks are `assert`s).\n\nRun everything: `python3 run_all.py` (about 7 s); one id: `python3 run_all.py 1.4.2c`.\n\"Run here\" = run on 17 Oct 2026 with Python 3.11.15, SymPy 1.14.0, gcc 13.3 (`python3 -I run_all.py <id>`, started\nfrom another directory); the output is in `expected/<id>.txt`.\n\n| id | script(s) | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.2a | `1.4.2a_fields.py` | h: rational roots, disc = −23; h mod 23 = (q−13)^2(q−20); no root mod 2, 3; 2, 3 split in K; s2 = 45 mod 256, s3 = 25 mod 81 (187 mod 243); N(3q^2−1) = 23; the case distinction of Lemma 1.4.2-field for all p < 3000; reduced forms of disc −23 | `python3 run_all.py 1.4.2a` | 0.5 s | yes |\n| 1.4.2b | `1.4.2b_plane.py` | τ(q) = (−q + s/h'(q))/2 has order 3; F(x,(1−s)x/4,z,t) = −∏(t − τ^i(p)x − τ^i(q)z) (directly and by a resultant); det = (s−1)/2, norm 6; the conjugate identity; F(0,0,z,t) | `python3 run_all.py 1.4.2b` | 1.1 s | yes |\n| 1.4.2c | `1.4.2c_residues.py`, `../common/cyclic_cubic/brute_residues.c` (case B) | complete residue trees mod 2^8 and 3^5, the valuation pairs and the precision at which they are determined; the congruences of the proofs by hand; direct enumeration of all primitive solutions mod 2^6 and 3^4 | `python3 run_all.py 1.4.2c` | 2.9 s | yes |\n| 1.4.2d | `1.4.2d_split_primes.py` | confirmation of Lemma 1.4.2-good for the 32 primes 5 ≤ p < 500 that split in K with h irreducible mod p: the plane cubic has no F_p-point | `python3 run_all.py 1.4.2d` | 2.6 s | yes |\n\n## Expected key lines\n1.4.2a:\n```\nh = q^3 - q + 1: no rational root; disc(h) = -23 (squarefree)\nh mod 23 = (q-13)^2 (q-20); simple root 20\n2 and 3 split in K; s2 = 617517 mod 2^20 (= 45 mod 256), s3 = 8935 mod 3^10 (= 25 mod 81, 187 mod 243)\nRes(h, 3q^2-1) = N(3q^2-1) = 23\nprimes 3 <= p < 3000, p != 23: (behaviour in K, number of roots of h) -> [(('inert', 1), 217), (('split', 0), 144), (('split', 3), 67)]\nreduced forms of discriminant -23: [(1, 1, 6), (2, -1, 3), (2, 1, 3)]\nPASS 1.4.2a\n```\n1.4.2b:\n```\ntau(q) = -3*s*u**2/23 - 9*s*u/46 + 2*s/23 - u/2\nF(x,(1-s)x/4,z,t) = -prod_i (t - tau^i(p) x - tau^i(q) z): verified\nresultant form of the identity: verified\ndet = s/2 - 1/2  N(det) = 6\nPASS 1.4.2b\n```\n(u stands for q in the printed expressions.)\n1.4.2c:\n```\np=2, mod 2^8: level sizes [3, 12, 48, 192, 768, 3072, 12288, 49152]  pairs [((2, 1), 16384), ((2, 4), 16384), ((5, 1), 16384)]\n  every pair determined from 2^6 on\np=3, mod 3^5: level sizes [4, 36, 324, 2916, 26244]  pairs [((0, 0), 19683), ((3, 0), 6561)]\n  every pair determined from 3^4 on\nat (1,0,0,-1) mod 3, x = 1: y = 3Y with Y mod 27 in [7, 16]\nB mod 2^6 (s=45): 98304 primitive solutions; pairs (v(l+),v(l-)):count (2,1):32768 (2,4):32768 (5,1):32768\nB mod 3^4 (s=25): 157464 primitive solutions; pairs (v(l+),v(l-)):count (0,0):118098 (3,0):39366\nPASS brute_residues B\n```\n1.4.2d: `split primes with h irreducible, 5 <= p < 500: 32 primes, from 13 to 499`, `PASS 1.4.2d`.\n\n## Correspondence with the source's checks\n- `verify_tritangent_factors.py` (identity and determinant by Gröbner reduction) ↔ 1.4.2b (explicit generator τ\n  and a resultant; the Cayley–Salmon norm identities of the original packet are not needed and not checked).\n- `verify_fields.gp` (PARI `nfdisc`, factorization mod 2, 3) ↔ 1.4.2a (root tests, the case distinction).\n- `local_residue_b.py` (trees mod 2^8, 3^5) ↔ 1.4.2c (own tree, plus a direct enumeration in C and the steps of\n  the proofs by hand).\n- `verify_local_points_bc.py`, `steiner_exact_bc.py`, `fano_*.py`, `surface_discriminant.py` (local solubility,\n  Steiner trihedra, 27 lines, bad primes): not needed by the proof; not reproduced.\n"
    },
    {
      "path": "anc/code/1.4.3/README.md",
      "size_bytes": 4503,
      "sha256": "0a249d3341ec0e8a8d668a49f2143f2e454ee52fd33d3f17f883a44ec7f12205",
      "mode": 420,
      "encoding": "utf-8",
      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.4.3: 2x^3 + 2x^2y + 2y^3 + y^2z − 2xz^2 − z^3 + tz^2 − t^3 = 0\n\nOnly the zero solution. Proof as in 1.4.2 with K = Q(√−11), L = K(u), g(u) = u^3 − 4u − 4 (u = 2q,\n2q^3 − 2q − 1 = 0; the same field as h_C(v) = v^3 − v^2 − v − 1), the plane ℓ_+ = (2+s)y + 3t = 0, and the\ndecisive places above 3.\n\nAll scripts were written for this bundle (agent S2, 17 Oct 2026). The source's certificate scripts\n(`verify_tritangent_factors.py`, `verify_fields.gp`, `local_residue_c.py`, `verify_additional_checks.py`,\n`verify_local_points_bc.py`, `steiner_exact_bc.py`, `fano_*.py`) were read as references and were NOT run. No\nPARI/GP, Magma or Sage is used.\n\nNotation: `p` in the scripts (as in the source) is the element ρ = (s + 2 − 6q^2)/(6(3q^2−1)) of the text, q = u/2.\n\nRequirements: Python 3.10+; SymPy 1.14 (codes a, b); a C99 compiler (`cc`, or `$CC`) for the C part of 1.4.3c.\nShared helpers: `../common/cyclic_cubic/`. Do not use `python -O`.\n\nRun everything: `python3 run_all.py` (about 5 s); one id: `python3 run_all.py 1.4.3c`.\n\"Run here\" = run on 17 Oct 2026 with Python 3.11.15, SymPy 1.14.0, gcc 13.3; output in `expected/<id>.txt`.\n\n| id | script(s) | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.3a | `1.4.3a_fields.py` | g: rational roots, disc(g) = −176, disc(h_C) = −44, u = v^2 − 1 and v = u^2/2 − 1, h_C(v+1) = v^3 + 2v^2 − 2 (Eisenstein at 2); g mod 11 = (u−4)^2(u−3); no root mod 3; 3 split, 2 inert in K; s3 = 31 mod 81; ω = (1+s)/2; N(3u^2−4) = 176; the case distinction for all p < 3000 | `python3 run_all.py 1.4.3a` | 0.5 s | yes |\n| 1.4.3b | `1.4.3b_plane.py` | τ(u) = (−u + 4s/g'(u))/2 has order 3; F(x,y,z,−(2+s)y/3) = 2∏(x − τ^i(p)y − τ^i(q)z) (directly and by a resultant); det = 2; the conjugate identity; F(x,0,z,0) | `python3 run_all.py 1.4.3b` | 0.9 s | yes |\n| 1.4.3c | `1.4.3c_residues.py`, `../common/cyclic_cubic/brute_residues.c` (case C) | complete residue tree mod 3^4 with the valuation pairs; all primitive solutions mod 8; the congruences of the proofs by hand; direct enumeration of all primitive solutions mod 3^4 and mod 8 | `python3 run_all.py 1.4.3c` | 1.4 s | yes |\n| 1.4.3d | `1.4.3d_split_primes.py` | confirmation of Lemma 1.4.3-good for the 32 primes 5 ≤ p < 500 that split in K with g irreducible mod p (including the source's 5, 31, 59) | `python3 run_all.py 1.4.3d` | 2.5 s | yes |\n\n## Expected key lines\n1.4.3a:\n```\ng = u^3 - 4u - 4: no rational root; disc(g) = -176 = -2^4*11;  disc(h_C) = -44\nh_C(v+1) = v**3 + 2*v**2 - 2 (Eisenstein at 2)\ng mod 11 = (u-4)^2 (u-3); simple root 3\n3 splits in K, s3 = 3190 mod 3^8 (= 31 mod 81); 2 is inert in K\nRes(g, 3u^2-4) = N(3u^2-4) = 176\nprimes 3 <= p < 3000, p != 11: (behaviour in K, number of roots of g) -> [(('inert', 1), 219), (('split', 0), 144), (('split', 3), 65)]\nPASS 1.4.3a\n```\n1.4.3b:\n```\ntau(u) = -3*s*u**2/11 + 9*s*u/22 + 8*s/11 - u/2\np = -s*u**2/11 + 3*s*u/22 + 8*s/33 - 1/3\nF(x,y,z,-(2+s)y/3) = 2 prod_i (x - tau^i(p) y - tau^i(q) z): verified\ndet = 2\nPASS 1.4.3b\n```\n1.4.3c:\n```\np=3, mod 3^4: level sizes [4, 36, 324, 2916]  pairs [((1, 0), 2187), ((1, 3), 729)]\nclass of (-1,0,0,1), t = 1: y = 3Y with Y mod 27 in [5, 23] ; v(1-29Y) = 2, v(33Y+1) = 0\nmod 8: 256 primitive solutions, all with y odd and t even\nC mod 3^4 (s=31): 157464 primitive solutions; pairs (v(l+),v(l-)):count (1,0):118098 (1,3):39366\nC mod 8: 256 primitive solutions, 0 with y even or t odd\nPASS brute_residues C\n```\n1.4.3d: `split primes with g irreducible, 5 <= p < 500: 32 primes, from 5 to 487`, `PASS 1.4.3d`.\n\n## Correspondence with the source's checks\n- `verify_tritangent_factors.py` ↔ 1.4.3b; `verify_fields.gp` (`nfdisc` = −44, irreducibility mod 3) ↔ 1.4.3a.\n- `local_residue_c.py` (tree mod 3^4; no F_p-point on the planes at 5, 31, 59) ↔ 1.4.3c and 1.4.3d.\n- `verify_additional_checks.py`, part \"cubic C\" (y odd and t even mod 8) ↔ 1.4.3c.\n- The tame criterion at 2 (unit norms in a tamely totally ramified cubic extension) is not needed: ψ ≡ 1 mod 2 is\n  a cube by Hensel's lemma (Lemma 1.4.3-two); the Eisenstein shift is checked in 1.4.3a for the remark only.\n"
    },
    {
      "path": "anc/code/1.4.4/README.md",
      "size_bytes": 3357,
      "sha256": "e7d96f5c8f332cc56e1aed773a21b95672b3a138c4388981607facb6300524c6",
      "mode": 420,
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for section 1.4.4: 8x^4 + 7y^4 + 2z^4 − t^4 = 0\n\nOnly the zero solution. Proof: for a primitive solution, f = 7y^2 + 2z^2 − 3t^2 and A_+ = 2x^2 + 2xz + z^2 satisfy\n(−7, f)_p (7, A_+)_p = 1 at every prime p and = −1 at ∞, contradicting Hilbert's product formula (Section 1.4.4).\n\nAll scripts were written for this bundle (agent S2, 17 Oct 2026). The source's script `verify_additional_checks.py`\n(part \"quartic\") was read as a reference and NOT run. No PARI/GP, Magma or Sage.\n\nRequirements: Python 3.10+; SymPy 1.14 (code 1.4.4a only); a C99 compiler for the C part of 1.4.4c. Shared\nhelpers: `../common/cyclic_cubic/` (`restree.py`, `brute_residues.c`). Do not use `python -O`.\n\nRun everything: `python3 run_all.py` (about 1 s). \"Run here\" = run on 17 Oct 2026 with Python 3.11.15,\nSymPy 1.14.0, gcc 13.3; output in `expected/<id>.txt`.\n\n| id | script(s) | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.4a | `1.4.4a_identities.py` | −fg − 72x^4 − 14(y^2−z^2)^2 = −9Q, t^4 − 7y^4 − 2A_+A_− = −Q, A_+A_− = 4x^4 + z^4, the two norm forms, A_+ = x^2 + (x+z)^2, A_+ ± A_−, g − f = 6t^2; H = 288 | `python3 run_all.py 1.4.4a` | 0.4 s | yes |\n| 1.4.4b | `1.4.4b_residues.py` | all tuples mod 16 (only x,y,z,t odd); A_+ mod 8 ∈ {1,5}, f ≡ 6 mod 8; mod 3 (f ≡ 0 only for x ≡ t ≡ 0); units w mod 81 with 7 + 2w^4 ≡ 0: v_3(7+2w^2) = 2; all (x,z,t) mod 7: the table of Lemma 1.4.4-seven; the Legendre symbols used | `python3 run_all.py 1.4.4b` | 0.1 s | yes |\n| 1.4.4c | `1.4.4c_symbols.py`, `../common/cyclic_cubic/brute_residues.c` (case Q) | the two Hilbert symbols (Serre's formulas) on the complete residue trees of primitive points over Z_2, Z_3, Z_7 (mod 2^5, 3^4, 7^2); direct enumeration of all primitive solutions mod 3^4 (v_3(f)) and 7^2 ((f/7)(A_+/7)) | `python3 run_all.py 1.4.4c` | 0.4 s | yes |\n\n## Expected key lines\n```\nidentity -fg - 72x^4 - 14(y^2-z^2)^2 = -9Q : ok\nidentity t^4 - 7y^4 - 2A+A- = -Q : ok\nH = 288\nmod 16: primitive solutions have parity pattern [(1, 1, 1, 1)]\nx, z odd: A_+ mod 8 in [1, 5] ;  y, z, t odd: f mod 8 in [6]\nmod 3: Q = f = 0 only at [(0, 1, 1, 0), (0, 1, 2, 0), (0, 2, 1, 0), (0, 2, 2, 0)]\nmod 81: units w with 7 + 2w^4 = 0 have w^2 = 1 mod 3 and v_3(7 + 2w^2) = 2\nmod 7 (z != 0): (r, s, A_+/z^2, f/z^2) in [(0, 2, 1, 4), (0, 5, 1, 4), (2, 3, 6, 3), (2, 4, 6, 3), (5, 3, 5, 3), (5, 4, 5, 3)] ; for z = 0 both are squares\np=2, mod 2^5: ... classes [((1, 0, 1, 1), 2048)]\np=3, mod 3^4: ... classes [((0, '>=4', 1, 1), 162), ((0, 0, 1, 1), 4374), ((0, 2, 1, 1), 1296), ((2, 0, 1, 1), 1458)]\np=7, mod 7^2: ... classes [((0, 0, -1, -1), 1372), ((0, 0, 1, 1), 1372)]\nQ mod 3^4: 393660 primitive solutions; v_3(f):count 0:314928 2:78732\nQ mod 7^2: 115248 primitive solutions; f, A+ units at 7 with (f/7)(A+/7) = 1: 115248, = -1: 0, f or A+ divisible by 7: 0\n```\n(In 1.4.4c the tuple is (v_p(f), v_p(A_+), (−7,f)_p, (7,A_+)_p); `'>=4'`: A_+ ≡ 0 mod 3^4 on that class, which does\nnot matter because 7 is a square in Q_3.)\n\n"
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      "mode": 420,
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for Section 1.4.5: 14x^5 + 5y^5 + 3z^5 = 0\n\nOnly integer solution (0, 0, 0). Proof: Frey curve E = E_{u,v}, (u, v, w) = ε(3z^5, 14x^5, 5y^5) with\nu ≡ −1 mod 4; E[5] irreducible (T_31 = {0,1,4}); level 210 (x even) or 3360 (x odd); newform sieve at twelve\nprimes; two rational survivors at level 3360 = the newforms of G± = [0,±1,0,−890,∓10200], excluded by the\nTate–Kummer pairing at 3 and 5. Source: the original working notes (Section 6 of the consolidated document; original\npacket `table9_problem5_solutions_standalone_corrected.tex`).\n\nAll scripts were written for the paper; they use the libraries `fq.py` and `fqsieve.gp` of\n`../common/frey_quintic/` (see its README). The solver's certificates (`verify_quintic_sieves.gp`,\n`table9_pair12_trace_sets.py`, `verify_trace_assertions.py`, `verify_tate_models.gp`) were read, not run.\n\n## Requirements\nPython 3.10+ (standard library); PARI/GP 2.15+ (`gp`; run here with 2.15.4). Peak memory about 0.3 GB.\n\n## Codes\n| id | script | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.5a | `1.4.5a_elementary.py` | H = 704; zero coordinates; hypotheses of Lemma lem:1.4:terms; even term 14x^5, r = 1 + 5v_2(x); Tate data and N_5 (Lemmas lem:1.4:frey, lem:1.4:level) | `python3 -I 1.4.5a_elementary.py` | < 1 s | yes |\n| 1.4.5b | `1.4.5b_traces.py` | T_q(14,5,3) for q = 11, 13, 17, 31, 41, 61, 71, 101, 131, 151, 181, 191, two implementations | `python3 -I 1.4.5b_traces.py` | 18 s | yes |\n| 1.4.5c | `1.4.5c_sieve.gp` | `elllocalred` on the Frey curve for all x, odd z mod 2^6; the sieve at levels 210 and 3360 | `gp -q 1.4.5c_sieve.gp` | 30 s | yes |\n| 1.4.5d | `1.4.5d_survivors.gp` | the two survivors = newforms of G± (Sturm bound 1536); conductor, Δ_min, c_4, types at 3, 5; square classes; isogeny class | `gp -q 1.4.5d_survivors.gp` | 43 s | yes |\n\n`python3 run_all.py` runs all four (about 1.5 min), compares with `expected/` (timing lines ignored) and checks that\nthe sets T_q hard-coded in `1.4.5c_sieve.gp` are those asserted by `1.4.5b_traces.py`. \"Run here\" = run on\n17 October 2026 (Python 3.12, GP 2.15.4, `nice -n 10`).\n\nKey output:\n```\n1.4.5a  even term 14x^5,  v_2(x) = 0: r =  1, type III  f_2 = 5, v_2(Delta_min) =  6, 2 in N_5; N_5 = 3360\n        even term 14x^5,  v_2(x) = 1: r =  6, type I4   f_2 = 1, v_2(Delta_min) =  4, 2 in N_5; N_5 = 210\n1.4.5b  q =  31:  50 points,  0 with a zero coordinate ...; traces a_q at the others: {-4,0,4}   T_q = {0,1,4}\n1.4.5c  level 210: dim S_2^new = 5, orbits = 5 ... after q=11: 2; ... after q=61: 1; after q=71: 0\n        level 3360: dim S_2^new = 48, orbits = 36 ... pairs (orbit, lambda): 38; after q=11: 12; after q=13: 8;\n        after q=17: 6; after q=31: 2; ...   survivors: orbits 3 and 12 (rational)\n1.4.5d  orbit 3: a_0..a_1536 equal those of G+;  orbit 12: a_0..a_1536 equal those of G-\n        Delta_min(G+-) = 2^6 3^6 5^2 7^4; ratio at 3: 6/2 = 3 mod 5 (nonsquare); ratio at 5: 2/2 = 1 (square)\n```\nThe orbit numbers are those of GP 2.15.4 (`mfeigenbasis`); other versions may order the orbits differently.\n\n## Not used by the proof\n- The solver's original proof also used the level 1680 (another sign normalization); with u ≡ −1 mod 4 it does\n  not occur, and it is not computed here.\n"
    },
    {
      "path": "anc/code/1.4.6/README.md",
      "size_bytes": 2234,
      "sha256": "689e804accbbe02249a9c2deb19f0c93cb39f61138c2ea827c0c5d2db7c3e15a",
      "mode": 420,
      "encoding": "utf-8",
      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for Section 1.4.6: 21x^5 + 5y^5 + 4z^5 = 0\n\nOnly integer solution (0, 0, 0). Proof: Frey curve (u, v, w) = ε(21x^5, 4z^5, 5y^5), εx ≡ 3 mod 4; E[5]\nirreducible (T_31); level 840 (z odd, type I_1^* at 2) or 210 (z even); the sieve excludes every newform.\nSource: the original working notes (Section 7 of the consolidated document; original packet\n`table9_problem5_solutions_standalone_corrected.tex`).\n\nAll scripts were written for the paper; they use `../common/frey_quintic/fq.py` and `fqsieve.gp`. The solver's\ncertificates were read, not run.\n\n## Requirements\nPython 3.10+ (standard library); PARI/GP 2.15+ (run here with 2.15.4).\n\n## Codes\n| id | script | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.6a | `1.4.6a_elementary.py` | H = 960; zero coordinates; hypotheses of Lemma lem:1.4:terms; r = 2 + 5v_2(z); Tate data and N_5 | `python3 -I 1.4.6a_elementary.py` | < 1 s | yes |\n| 1.4.6b | `1.4.6b_traces.py` | T_q(21,5,4) for q = 11, 13, 17, 31, 41, 61, 101, 131, 151, 181, 211, two implementations | `python3 -I 1.4.6b_traces.py` | 16 s | yes |\n| 1.4.6c | `1.4.6c_sieve.gp` | `elllocalred` on the Frey curve for x ≡ 3 mod 4, all z mod 2^6; the sieve at levels 210 and 840 | `gp -q 1.4.6c_sieve.gp` | 2 s | yes |\n\n`python3 run_all.py` runs all three (about 20 s) and compares with `expected/`. Run here on 17 October 2026.\n\nKey output:\n```\n1.4.6a  even term 4z^5, v_2(z) = 0: r = 2, type I1*, f_2 = 3, v_2(Delta_min) = 8; N_5 = 840\n        even term 4z^5, v_2(z) = 1: r = 7, type I6,  f_2 = 1, v_2(Delta_min) = 6; N_5 = 210\n1.4.6c  level 210: ... after q=11: 3; ... after q=31: 1; after q=41: 1; after q=61: 0\n        level 840: dim 12, orbits 11 (field degrees 1 x 10, 2) ... after q=11: 6; after q=31: 5; after q=41: 3; after q=61: 0\n```\n\n## Not used by the proof\n- The level 1680 of the solver's original normalization (it does not occur with u ≡ −1 mod 4).\n"
    },
    {
      "path": "anc/code/1.4.7/README.md",
      "size_bytes": 2579,
      "sha256": "4c3b3820a72a61dbe8a4542950785d9b2148cc39554bb7bd68799603692f0447",
      "mode": 420,
      "encoding": "utf-8",
      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for Section 1.4.7: 17x^5 + 6y^5 + 3z^5 = 0\n\nOnly integer solution (0, 0, 0). Proof: x = 3X, 1377X^5 + 2y^5 + z^5 = 0 (1377 = 3^4·17); Frey curve\n(u, v, w) = ε(1377X^5, 2y^5, z^5), εX ≡ 3 mod 4; E[5] irreducible (T_31, also T_41); level 1632 (y odd) or 102\n(y even); the primes 41, 11, 131 exclude every newform. Source: the original working notes (Section 8 of the\nconsolidated document; original packet `table9_problem5_solutions_standalone_corrected.tex`).\n\nAll scripts were written for the paper; they use `../common/frey_quintic/fq.py` and `fqsieve.gp`. The solver's\ncertificates (`table9_e3_newforms.gp`, `table9_pair34_trace_sets.py`, `table9_pair34_tate2.gp`) were read, not run.\n\n## Requirements\nPython 3.10+ (standard library); PARI/GP 2.15+ (run here with 2.15.4).\n\n## Codes\n| id | script | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.7a | `1.4.7a_elementary.py` | H = 832; zero coordinates; 3 ∣ x and the new equation; hypotheses of Lemma lem:1.4:terms; r = 1 + 5v_2(y); Tate data and N_5; solutions modulo 25 (remark) | `python3 -I 1.4.7a_elementary.py` | < 1 s | yes |\n| 1.4.7b | `1.4.7b_traces.py` | T_q(1377,2,1) for q = 11, 31, 41, 131, 241, two implementations | `python3 -I 1.4.7b_traces.py` | 9 s | yes |\n| 1.4.7c | `1.4.7c_sieve.gp` | `elllocalred` on the Frey curve for X ≡ 3 mod 4, all y mod 2^6; the sieve at levels 102 and 1632 (order 41, 11, 131, 31, 241) | `gp -q 1.4.7c_sieve.gp` | 4 s | yes |\n\n`python3 run_all.py` runs all three (about 15 s) and compares with `expected/`. Run here on 17 October 2026.\n\nKey output:\n```\n1.4.7a  solutions (X, y, z) mod 5 of 1377X^5 + 2y^5 + z^5 = 0 mod 25: [(0,0,0), (1,4,0), (2,3,0), (3,2,0), (4,1,0)]\n1.4.7b  q = 41: 25 points, 0 with a zero coordinate; traces {-10,10}; T_q = {0}\n1.4.7c  level 102: ... after q=41: 2; after q=11: 0\n        level 1632: dim 32, orbits 20, pairs 20; after q=41: 8; after q=11: 2; after q=131: 0\n        orbit 4: (a_41, a_11, a_131) = [-5, 3, 9];  orbit 8: [-5, -3, -9]\n```\n\n## Not used by the proof\n- The level 816 of the solver's original normalization (it does not occur with u ≡ −1 mod 4).\n- The congruence modulo 25 (5 ∣ z): printed by 1.4.7a for the remark of the section; the level does not depend on it.\n"
    },
    {
      "path": "anc/code/1.4.8/README.md",
      "size_bytes": 2308,
      "sha256": "ef5ceec7e7140b1e32efbd71d4c5ebaf96343d1c24af6d307f73487f3df1aa99",
      "mode": 420,
      "encoding": "utf-8",
      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for Section 1.4.8: 17x^5 + 9y^5 + 5z^5 = 0\n\nOnly integer solution (0, 0, 0). Proof: exactly one variable even; Frey curve with v the even term and u the first\nodd term (order 17x^5, 9y^5, 5z^5), u ≡ −1 mod 4; E semistable, level 510 only; E[5] irreducible (T_41 = {0});\nthe primes 41 and 11 exclude every newform. Source: the original working notes (Section 9 of the consolidated document;\noriginal packet `table9_problem5_solutions_standalone_corrected.tex`).\n\nAll scripts were written for the paper; they use `../common/frey_quintic/fq.py` and `fqsieve.gp`. The solver's\ncertificates (`table9_e4_newforms.gp`, `table9_pair34_trace_sets.py`, `table9_pair34_tate2.gp`) were read, not run.\n\n## Requirements\nPython 3.10+ (standard library); PARI/GP 2.15+ (run here with 2.15.4).\n\n## Codes\n| id | script | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.8a | `1.4.8a_elementary.py` | H = 992; zero coordinates; hypotheses of Lemma lem:1.4:terms; the three possible even terms, r = 5v_2; Tate data and N_5 = 510 | `python3 -I 1.4.8a_elementary.py` | < 1 s | yes |\n| 1.4.8b | `1.4.8b_traces.py` | T_q(17,9,5) for q = 11, 31, 41, 71, two implementations | `python3 -I 1.4.8b_traces.py` | < 1 s | yes |\n| 1.4.8c | `1.4.8c_sieve.gp` | `elllocalred` on the Frey curve for each choice of the even variable (3072 classes); the sieve at level 510 (order 41, 11, 31, 71) | `gp -q 1.4.8c_sieve.gp` | < 1 s | yes |\n\n`python3 run_all.py` runs all three (about 2 s) and compares with `expected/`. Run here on 17 October 2026.\n\nKey output:\n```\n1.4.8b  q = 41: 25 points, 0 with a zero coordinate; traces {-10,10}; T_q = {0}\n1.4.8c  level 510: dim 9, orbits 8 (field degrees 1 x 7, 2), pairs 9; after q=41: 3; after q=11: 0\n        (survivors of q=41: orbits 2, 3 with a_11 = 0, 4, and orbit 8 (y^2 - 6) at one prime above 5 with a_11 = 0)\n```\n\n## Not used by the proof\n- The level 4080 of the solver's original normalization (it does not occur with u ≡ −1 mod 4).\n"
    },
    {
      "path": "anc/code/1.4.9/README.md",
      "size_bytes": 2302,
      "sha256": "0aa2d40e274e89b496237c1b5a9b22ec6936657903621742e7fde7b9c63f41db",
      "mode": 420,
      "encoding": "utf-8",
      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Code for Section 1.4.9: 14x^5 + 13y^5 + 2z^5 = 0\n\nOnly integer solution (0, 0, 0). Proof: y = 2Y, 7x^5 + 208Y^5 + z^5 = 0 (208 = 2^4·13); Frey curve\n(u, v, w) = ε(7x^5, 208Y^5, z^5), εx ≡ 1 mod 4; good or multiplicative at 2 with 5 ∣ v_2(Δ_min), level 91;\nE[5] irreducible by Mazur's theorem; modulo 11 every point has x = 0, so a_11(f) ≡ ±12 is required, and no\nnewform of level 91 has it. Source: the original working notes (Section 10 of the consolidated document; original packet\n`table9_problem5_solutions_standalone_corrected.tex`).\n\nAll scripts were written for the paper; they use `../common/frey_quintic/fq.py` and `fqsieve.gp`. The solver's\ncertificates (`table9_e5_newforms.gp`, `verify_small_newforms.gp`) were read, not run.\n\n## Requirements\nPython 3.10+ (standard library); PARI/GP 2.15+ (run here with 2.15.4).\n\n## Codes\n| id | script | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4.9a | `1.4.9a_elementary.py` | H = 928; zero coordinates; 2 ∣ y and the new equation; hypotheses of Lemma lem:1.4:terms; r = 4 + 5v_2(Y); Tate data and N_5 = 91; the points over F_11 and T_11 (two implementations) | `python3 -I 1.4.9a_elementary.py` | < 1 s | yes |\n| 1.4.9b | `1.4.9b_newforms.gp` | `elllocalred` on the Frey curve for x ≡ 1 mod 4, all Y mod 2^6; S_2(Γ_0(91))^new, minimal polynomials of a_11 and their values at 2, 3 mod 5; the sieve with T_11 = {2, 3} | `gp -q 1.4.9b_newforms.gp` | < 1 s | yes |\n\n`python3 run_all.py` runs both (about 1 s) and compares with `expected/`. Run here on 17 October 2026.\n\nKey output:\n```\n1.4.9a  projective points of 7x^5 + 208Y^5 + z^5 = 0 over F_11 (5): (0,1,1), (0,1,3), (0,1,4), (0,1,5), (0,1,9)\n        q = 11: 5 points, 5 with a zero coordinate; T_q = {2,3}\n1.4.9b  level 91: dim S_2^new = 7, orbits of dimensions [1, 1, 2, 3];\n        minimal polynomials of a_11: [x, x + 6, x^2 - 18, x^3 - 2*x^2 - 6*x + 8]\n        values at T = 2, 3 modulo 5: [[2, 3], [3, 4], [1, 1], [1, 4]]\n```\n"
    },
    {
      "path": "anc/code/common/cyclic_cubic/README.md",
      "size_bytes": 2325,
      "sha256": "727eec51fe24b03070a595d2fda71472f7097d3f9c11e5099ede4b74d85954e4",
      "mode": 420,
      "encoding": "utf-8",
      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Shared helpers for Sections 1.4.2, 1.4.3, 1.4.4 (cubic surfaces B, C and the diagonal quartic Q)\n\nWritten for this bundle (agent S2, 17 Oct 2026). These files have no code id of their own; they are used by the\ncodes 1.4.2a–d, 1.4.3a–d and 1.4.4c (see the READMEs of `../../1.4.2/`, `../../1.4.3/`, `../../1.4.4/`).\nNo PARI/GP, Magma or Sage. The scripts that use them add this folder to `sys.path` themselves.\n\n| file | language | what it provides |\n|---|---|---|\n| `cubicfield.py` | Python 3 + SymPy | exact arithmetic in Q[u,s]/(g(u), s^2 − D) for a monic cubic g with disc(g) = D·m^2: normal forms, inverses, the generator τ(u) = (−u + m·s/g'(u))/2 of Gal(L/K) (checked: g(τ(u)) = 0, τ^3 = id, τ ≠ id), orbits, norms |\n| `restree.py` | Python 3 (standard library) | complete residue trees of normalized primitive p-adic points of F = 0 (the first coordinate prime to p is 1; level k+1 = all p^3 digit extensions of level k that satisfy F ≡ 0 mod p^(k+1)), valuations modulo p^k, 2-adic and p-adic square roots, roots modulo p, Legendre symbols |\n| `brute_residues.c` | C99, C library only | direct enumeration of ALL primitive tuples modulo p^k (no normalization, no tree): case `B` (mod 2^6, 3^4), `C` (mod 3^4, 8), `Q` (mod 3^4, 7^2); it checks the expected sets of valuation pairs / symbols and prints `PASS brute_residues <case>` |\n\nWhy the trees are complete: every primitive point over Z_p, divided by its first coordinate that is a p-adic unit,\nis a normalized primitive point, and its reduction modulo p^k satisfies the conditions defining level k. The\nvaluation of a linear form on a class of level k is determined when it is smaller than k; the scripts assert\nthat this is the case on every class of the last level.\n\n`brute_residues.c` is compiled by the `run_all.py` of each section into a temporary directory\n(`$CC -O2 -std=c99`, default `cc`); by hand: `cc -O2 -std=c99 -o brute brute_residues.c && ./brute B`.\nRun times here (2 cores, nice 10): B 1.8 s, C 1.1 s, Q 0.2 s.\n"
    },
    {
      "path": "anc/code/common/frey_quintic/README.md",
      "size_bytes": 5331,
      "sha256": "eb0817b8321ee192a3ca65a0d268c69b3656d20f96c6163106f726e1fa0a14a0",
      "mode": 420,
      "encoding": "utf-8",
      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Common code for Sections 1.4.5–1.4.10 (the six diagonal quintics A x^5 + B y^5 + C z^5 = 0)\n\nCodes 1.4qa–1.4qc of the common framework of Section 1.4 (Lemmas lem:1.4:terms … lem:1.4:pairing), and the two\nlibraries used by the codes of `../../1.4.5/` … `../../1.4.10/`. All scripts were written for the paper. The\nsolver's certificates (`verify_quintic_sieves.gp`, `verify_standard_tate.gp`, `verify_tate_models.gp`,\n`verify_small_newforms.gp`, `verify_trace_assertions.py`, `table9_*.py`, `table9_e*_newforms.gp`, `quintic_4411_*`\nin `section87_reproducibility/certificates/`) were read as references but not run; their claims are recomputed here.\n\n## Requirements\n- Python 3.10 or later, standard library only (`fq.py`, `runner.py`, `1.4qb_newdims.py`, the `*a_*.py`, `*b_*.py`\n  scripts of the sections). Exact integer arithmetic, no floating point.\n- PARI/GP 2.15 or later (`gp`; run here with GP 2.15.4) for the `.gp` scripts: `mfinit`, `mfeigenbasis`, `mffields`,\n  `nfinit`, `nfcertify`, `idealprimedec`, `idealval`, `elllocalred`, `ellglobalred`, `msinit`, `msnew`, `mshecke`.\n  `1.4qc` needs `-D nbthreads=1 -D threadsizemax=1000000000` (set by `run_all.py`). Peak memory about 0.8 GB.\n\n## Files\n| file | contents |\n|---|---|\n| `fq.py` | Python library: valuations, factorization, Grechuk's H; the hypotheses of Lemma lem:1.4:terms and the level table of Lemmas lem:1.4:frey, lem:1.4:level (`report_levels`); the projective points of A a^5 + B b^5 + C c^5 = 0 over F_q and the trace sets T_q by two implementations (`trace_set`: character sums for (u, v) = (A a^5, B b^5); `trace_set_variants`: point counting with a table of square roots for all twelve signed orderings of the three terms and all scalings λ ∈ F_q^×, with the assertion that only the two traces ±a_q occur) |\n| `fqsieve.gp` | GP library: `fq_check2` (f_2 and v_2(Δ_min) of E_{u,v} by `elllocalred`); `fq_sieve`/`fq_print` (the newform sieve: every Galois orbit of S_2(Γ_0(N))^new, its coefficient field Q[y]/(P), the maximal order by `nfinit` certified by `nfcertify`, every prime ideal λ above 5 by `idealprimedec`; a pair (orbit, λ) is excluded at the first q with `idealval(a_q − t, λ) = 0` for all t ∈ T_q) |\n| `runner.py` | the harness used by every `run_all.py` (output to `out/`, comparison with `expected/`, timing lines ignored) |\n| `1.4qa_tate2.gp`, `1.4qb_newdims.py`, `1.4qc_modsym_check.gp` | the codes below |\n| `run_all.py` | runs 1.4qa–1.4qc; `--sections` also runs the six section folders |\n| `expected/` | the outputs of this run (`python3 run_all.py --update`) |\n\n## Codes\n| id | script | what it computes | command | time | run here |\n|---|---|---|---|---|---|\n| 1.4qa | `1.4qa_tate2.gp` | E_{u,v}: Y^2 = X(X−u)(X+v), u ≡ −1 mod 4, r = v_2(v): (1) the model S^2 + RS = R^3 + ((v−u−1)/4)R^2 − (uv/16)R for r ≥ 4 and its c_4, Δ, as polynomial identities; (2) the congruences of the hand run of Tate's algorithm (r = 2: I_1^*, r = 3: III^*) on all classes of u and v/2^r modulo 2^7; (3) `elllocalred` on the same classes for r = 1…12 | `gp -q 1.4qa_tate2.gp` | < 1 s | yes |\n| 1.4qb | `1.4qb_newdims.py` | dim S_2(Γ_0(N))^new at N = 210, 3360, 840, 102, 1632, 510, 91, 88, 22 from the genus of X_0(M), M ∣ N (no PARI) | `python3 -I 1.4qb_newdims.py` | < 1 s | yes |\n| 1.4qc | `1.4qc_modsym_check.gp` | for the nine levels and their 53 test primes: charpoly of T_q on the new modular symbols with sign +1 = product over the `mf` orbits of the charpolys of a_q | `gp -q -D nbthreads=1 -D threadsizemax=1000000000 1.4qc_modsym_check.gp` | 5 min (level 3360: 4.8 min) | yes |\n\n\"Run here\" = run on 17 October 2026 with Python 3.12 and GP 2.15.4 (`nice -n 10`, one process), outputs in\n`expected/`.\n\nKey output (1.4qa):\n```\n(3) r=1: type III, f_2=5, v_2(Delta_min)=6   [for u = 1 mod 4: f_2 in [5]]\n(3) r=2: type I1*, f_2=3, v_2(Delta_min)=8   [for u = 1 mod 4: f_2 in [4]]\n(3) r=3: type III*, f_2=3, v_2(Delta_min)=10   [for u = 1 mod 4: f_2 in [4]]\n(3) r=4: type I0, f_2=0, v_2(Delta_min)=0   [for u = 1 mod 4: f_2 in [4]]\n(3) r=5: type I2, f_2=1, v_2(Delta_min)=2   [for u = 1 mod 4: f_2 in [4]]\n...\nPASS 1.4qa: Tate table at 2 for u = -1 mod 4 (24576 curves, r = 1..12, u and v/2^r modulo 2^7)\n```\nKey output (1.4qb): `dim S_2^new = 5, 48, 12, 3, 32, 9, 7, 3, 0` (genus of X_0(N) = 41, 737, 177, 15, 273, 101, 7, 9, 2).\nKey output (1.4qc): one line per level, e.g.\n`level 3360: dim new = 48 (modular symbols) = 48 (mf); charpoly of T_q agrees for q in [11, 13, …, 191]`.\n\n## Notes\n- The proof of the Tate table (Lemma lem:1.4:frey) is the hand run of Tate's algorithm in the paper; every\n  congruence used there is a polynomial congruence modulo at most 2^5, so part (2) of 1.4qa checks it for all\n  integers. Part (3) is an independent confirmation with PARI's implementation.\n- 1.4qc confirms the Hecke data level by level with a second algorithm (modular symbols against the trace formula);\n  the reduction modulo the prime ideals above 5 is done in the section codes only.\n"
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      "path": "anc/code/common/h31_classgroup/README.md",
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# h31_classgroup — class-group certificates for the bitangent fields\n\nGeneric tools for the step \"Cl(K)[2] = 0\" of the two-variable equations of chapter 1.1\n(K = the degree-27 bitangent field). The proof combines\n\n* **Theorem B** (explicit formula): if `B^-(f) > 0` then `Cl(K)[2] = 0`. It needs the number\n  `a(q)` of prime ideals of every norm `q < e^T`, and the number `k(q)` of those proved principal.\n* **Lemma C** (strong norm induction): if every prime ideal `p` with `N(p) <= C` has a witness\n  `0 != g in p` with `|N(g)|/N(p) = prod r^e`, every `r^e < N(p)`, then all of them are principal;\n  so `k(q) = a(q)` for `q <= C` (and `k(q) = 0` is used above `C`).\n\nThe tools are driven by one JSON parameter file per equation, `code/<section>/data/classgroup_params.json`.\nThey are based on the replay pipeline of equation 1.1.1 (`certify_3723_principals_fast.py`,\n`census_stream_3723.py`, `verify_3723_strict_window.py`) and on our independent checkers\n(`count_ideals.py`, `wcheck.py`, `Bgen.py`); here they are generic and self-contained.\n\n## Files\n\n| file | does |\n|---|---|\n| `cgfield.py` | loads the parameter file; builds `K` with `nfinit([f, primes of disc f])` and requires `nfcertify = []` (maximal order proved), the stated discriminant and signature |\n| `count_ideals.py` | `a(q)` for all norms `q < R` (`R = support`, or `floor(e^T)+1`); output `.npz` (+ printed SHA-256 of the packed `(q, a(q))` stream) |\n| `witnesses.py` | proposes a witness for every prime ideal of norm `<= C`: `idealmin` element if its cofactor is strict, otherwise a generator proposed by `bnfisprincipal` (bnfinit is only a proposer); output `.jsonl.gz` |\n| `check_witnesses.py` | exact check: enumerates all prime ideals of norm `<= C` itself and matches each to an unused witness (integrality, membership, exact norm, proved primes, `r^e < N(p)`); every witness must be used |\n| `bound.py` | `B^-(f)` in Arb ball arithmetic (python-flint): exact rational knots of the autocorrelation, closed-form archimedean term via the triangle expansion, all prime powers `q^m < e^T` |\n\n## Parameter file\n\n```json\n{\n \"equation\": \"x^4+y^3+xy+1=0\",\n \"section\": \"1.1.3\",\n \"polynomial\": \"x^27 - 48*x^22 + ... + 4096\",\n \"field_discriminant\": \"8291514845387242623105868334206933045051654144\",\n \"signature\": [3, 12],\n \"principal_bound\": 10000,\n \"support\": 1000000,\n \"window_heights\": [348713, 439422, ...],\n \"precision_bits\": 224\n}\n```\n\n* `polynomial`: monic integral defining polynomial of `K` in the variable `x` (PARI syntax).\n* `field_discriminant`, `signature`: checked against the certified maximal order.\n* `principal_bound` = `C` of Lemma C (inclusive: norms `<= C`).\n* exactly one of `support` (integer `R`, `T = log R`, sum over `q^m < R`) and `T` (a decimal string;\n  sum over `q^m < e^T`, decided in ball arithmetic; counts are then made for norms `<= floor(e^T)`).\n* `window_heights`: the positive integer heights of the step function `v` on `N` equal cells of\n  `[0, T]`; `[1]` is the plain triangle `f(x) = 1 - |x|/T`.\n* `precision_bits`: Arb precision of `bound.py` (default 224).\n\nExample files: `code/1.1.3/data/classgroup_params.json` (64 cells), `code/1.1.6/...` and\n`code/1.1.7/...` (one cell, `T` = 13 and 8).\n\n## Commands\n\nRun from any folder (the scripts find `cgfield.py` themselves; `python3 -I` also works):\n\n```\nCG=../common/h31_classgroup            # from code/<section>/\nPRM=data/classgroup_params.json\npython3 $CG/count_ideals.py   $PRM --out counts.npz [--workers W] [--lo LO --hi HI]\npython3 $CG/witnesses.py      $PRM --out witnesses.jsonl.gz [--workers W] [--lo LO --hi HI] [--direct]\npython3 $CG/check_witnesses.py $PRM witnesses.jsonl.gz [more files] [--lo LO --hi HI]\npython3 $CG/bound.py          $PRM counts.npz [more .npz] [--no-exp-terms]\n```\n\n* Slicing: `--lo/--hi` restrict the rational primes. Count files with disjoint ranges tiling\n  `[2, R-1]` can be given together to `bound.py` (it checks the tiling). Witness files can be produced\n  and checked in slices; the checker requires a witness for every prime ideal over `p` in its own\n  `--lo..--hi` range, so the slices of a complete check must tile `[2, C]`.\n* Key output lines: `count_ideals.py` → `RANGE lo hi SUPPORT R ... PRIME_IDEALS N SHA256 ...`;\n  `witnesses.py` → `WITNESSES N STRICT s DIRECT d BY_DEGREE {...}`; `check_witnesses.py` →\n  `ALL N PRIME IDEALS OF NORM <= C ... HAVE VALID WITNESSES ...`; `bound.py` → `key = value` lines\n  (counts, archimedean part, prime sum, `B`) and `B_INTERVAL [...] POSITIVE True`.\n* `STRICT`/`DIRECT` in the output of `witnesses.py` count the accepted `idealmin` elements and the\n  `bnfisprincipal` generators; `--direct` uses generators only (`|N(g)| = N(p)`), as sections 1.1.3,\n  1.1.6 and 1.1.7 state their certificates. Either kind proves principality by Lemma C.\n* `--no-exp-terms` drops the two small positive exponential terms of the closed-form archimedean\n  term (still a valid lower bound). Sections 1.1.6 and 1.1.7 state their `B_0` in this form.\n\n## Tested here (14 Oct 2026; PARI 2.15.4 via cypari2, python-flint 0.9.0; machine shared with other jobs)\n\nEnd-to-end test on the small certificate of 1.1.3 (`code/1.1.3/1.1.3bc_classgroup.sh`, about one minute):\n\n| step | result |\n|---|---|\n| `count_ideals.py` (R = 10^6) | `PRIME_IDEALS 78493` (12 s) |\n| `witnesses.py --direct` (C = 10^4) | `WITNESSES 1260 STRICT 0 DIRECT 1260` (49 s); without `--direct`: `STRICT 887 DIRECT 373` |\n| `check_witnesses.py` | `ALL 1260 PRIME IDEALS OF NORM <= 10000 ... HAVE VALID WITNESSES` (3 s) |\n| `bound.py` | `B = [0.06421827154264666555204025429213639168155 +/- 1.93e-42]`, 42743 terms — the source's 224-bit value |\n\nFurther runs (outputs in the `expected/` folders of the sections):\n\n| section | run | result |\n|---|---|---|\n| 1.1.7 | all four tools, full (C = 3000, T = 8, `--no-exp-terms`) | 413 ideals with generators; `B_0 = 0.0849899621752625010053405712299318513208...` |\n| 1.1.6 | count + `bound.py --no-exp-terms` full (T = 13); generators on the slice p ≤ 20000 | 36932 ideals of norm < e^13; `B_0 = 0.0264813059126755588162470923169050964739...`; 2410 generators checked |\n| 1.1.2 | `bound.py` on a complete count of norms < 10^8 (made earlier by our independent counter, same method) | `B = 0.000875203229047890536437...` (83 s); slices: count p ≤ 2·10^6, witnesses p ≤ 20000 |\n| 1.1.5 | same, norms < 10^8 | `B = 0.000226857116386139171700...` (80 s); slices as above |\n| 1.1.4 | same, norms < 5·10^8 (14.3 million distinct norms) | `B = 0.000100160932045204380585...` (≈ 4 min); slices as above |\n| 1.1.1 | `count_ideals.py --support 10000001`; witnesses on p ≤ 20000 (C = 10^7) | 664464 ideals, SHA-256 `5bf75088…634e` = the source's low-norm stream hash; 2665 witnesses checked |\n\nWitness generation agrees with the source program of 1.1.1 (same rule): at C = 30000, both give\n`3246 = 1853 strict + 1393 direct` with PARI 2.17.4. The checker rejects a witness file with one changed\ncoordinate or one missing record (tested).\n\n## Commands for the larger certificates (not run here in full)\n\nRuntimes on this (loaded) machine, per worker: counting 0.2–0.4 ms per rational prime (`10^8`: ≈ 20 min,\n`5·10^8`: ≈ 2 h, `10^9`: ≈ 4 h); witnesses 20–50 ms per prime ideal (`C = 5·10^6`: ≈ 2–3 h); checking\n≈ 2 ms per prime ideal; `bound.py` ≈ 15 µs per distinct norm (`10^8`: ≈ 1.5 min, `5·10^8`: ≈ 4 min). Use\n`--workers`. Count files take ≈ 12 bytes per distinct norm (`10^8`: ≈ 40 MB, `10^9`: ≈ 400 MB); witness\nfiles ≈ 30–60 bytes per prime ideal (gzip). Memory: `count_ideals.py` and `bound.py` hold the norm arrays\n(≈ 12 bytes per norm); `check_witnesses.py` streams the witness files (they must be sorted by `p`, as\n`witnesses.py` writes them).\n\n| section | `support` / `T` | `principal_bound` | expected counts |\n|---|---|---|---|\n| 1.1.1 | (strict-character criterion, see `code/1.1.1/README.md`) | 10^7 | 664,464 prime ideals of norm ≤ 10^7 |\n| 1.1.2 | 10^8 (256 cells) | 5·10^6 | 5,763,020 of norm < 10^8; 349,551 of norm ≤ 5·10^6; `B^- = 0.000875203229...` |\n| 1.1.3 | 10^6 (64 cells) | 10^4 | 78,493; 1,260; `B^- = 0.0642182715426466...` |\n| 1.1.4 | 5·10^8 (64 cells) | 2,020,000 | 26,362,584; 150,219; `B^- ∈ [1.00157, 1.00165]·10^-4` |\n| 1.1.5 | 10^8 (64 cells) | 6·10^6 | 5,764,533; 412,122; `B^- ∈ [2.2685629, 2.2685794]·10^-4` |\n| 1.1.6 | T = 13 (1 cell) | 450,000 | 36,932 (37,489 of norm ≤ 450,000); `B_0 = 0.0264813059126...` |\n| 1.1.7 | T = 8 (1 cell) | 3,000 | 413; 413; `B_0 = 0.08498996217...` |\n\n## Adding an equation\n\nWrite `code/<section>/data/classgroup_params.json` with the bitangent polynomial (any monic integral\ndefining polynomial of `K`), `|disc K|`, the signature, `C`, `support` (or `T`) and the integer window\nheights of the source (for a one-cell triangle window: `[1]`), then run the four commands above. If\n`principal_bound ≥ e^T`, every prime ideal in the sum is principal and `c_m(q) = a(q)` for all `m`.\nSections 1.1.8–1.1.13 do this in their own folders.\n\n## Requirements\n\nPython 3 with cypari2 (PARI/GP ≥ 2.15), python-flint ≥ 0.6 (`nmod_poly`, `arb`), numpy. No Sage,\nno Magma, no GRH: `bnfinit` only proposes elements, and every accepted witness is verified exactly.\n"
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      "path": "anc/code/common/h31_descent_lib/README.md",
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# h31_descent_lib: marked-bitangent 2-descent in PARI/GP\n\nUsed by `code/1.1.11/1.1.11e_descent.gp`, `code/1.1.12/1.1.12f_descent.gp` and\n`code/1.1.13/1.1.13d_descent.gp` (read with `read(\"../common/h31_descent_lib/descent_lib.gp\")` and\n`read(\"../common/h31_descent_lib/descent_points.gp\")`). PARI/GP 2.15 or later; tested with 2.15.4.\n\nThe curve is given as `F(x,y) = 0`, a plane quartic with a rational hyperflex `O` at infinity, and the\nfinite bitangents `y = m x + b` grouped into components `L = prod K_i`, `K_i = Q[m]/(f_i)`. The descent\nmap sends a point `P` to `(eps (y(P) - m_i x(P) - b_i))_i` in `prod K_i^*/K_i^*2`.\n\n## descent_lib.gp (reviewer code of the independent audit)\n\n| function | does |\n|---|---|\n| `mkcomp(F, f, Sp, \"\")` | one component: `bnfinit` of `Q[aa]/(f)` (GRH; only used to *propose* S-units), the slope `m` and intercept `b` in `K_i`, a check of the square identity `F(x, m x + b) = lead * (x^2 + c1 x + c0)^2`, the primes over `Sp`, and the proposed generators of `K_i(S,2)` (torsion, units, S-units) |\n| `auxchars`, `globcharmat`, `charvec`, `express` | valuation parities at S and quadratic residue characters at auxiliary primes; the rank of this matrix proves the independence of the generators |\n| `normmat` | rows of the norm condition (sign and parities of `v_p(N)` for `p` in S) |\n| `goodplace(comps, eps, F, p, N)` | local images at a good prime `p` from all points of `C(F_p)` (Hensel lifts), as a constraint |\n| `mkplace`, `globmat`, `locimage`, `constraint`, `seldim`, `f2ker` | local square-class coordinates (residue characters, or `ideallog` in `(O/p^(2e+1))^*` over 2), images of the global basis, and the F_2 linear algebra |\n| `delta`, `deltaline`, `dline_cert`, `linepts`, `infpts` | delta values of rational points, of divisors cut out by lines (with a precision check), and of points of the disk at infinity |\n\nEvery proposed S-unit is checked by the calling scripts (`idealfactor`: support in S), and the\ndimension of `K_i(S,2)` follows from `Cl(K_i)[2] = 0` (proved separately), so the GRH-conditional\n`bnfinit` is never used as a proof.\n\n## descent_points.gp (added for the bundle)\n\nCertified local images of the specific p-adic divisors named in the text:\n\n| function | divisor |\n|---|---|\n| `ptimg(comps, eps, p, x0, ypad)` | `P - O`, `P = (x0, y)`, `x0` exact, `y` a p-adic number (e.g. from `polrootspadic`) |\n| `ptimgx(...)` | the same with `y0` exact and `x` p-adic |\n| `infimg(comps, eps, p, t0, wpad)` | `P - O` for the point `t = t0` of the disk at O (`w` p-adic) |\n| `quadimg(comps, eps, p, d, X, a, c, M)` | `R + R' - 2O` for an affine point over `Q_p(sqrt d)` with `x = X` exact and `y = a + c sqrt d` known mod `p^M` |\n| `quadinfimg(...)` | the same for a point of the disk at O with `t = T` exact and `w = a + c sqrt d` |\n| `newtonr(g, W, W0, d, p, N)` | Newton iteration in `Z_p[sqrt d]` (derivative a unit); returns `a, c` and the residual valuation `M` |\n\nThe delta value is written as a polynomial in error variables (each error lies in `p^N Z_p`); `certpoly`\nrequires, at every prime `pr | p` of every component, that the minimum over all non-constant monomials of\n`v_pr(coefficient) + N e(pr) deg` exceeds `v_pr(value) + 2 e(pr) + 1`, so the error cannot change the square\nclass. The returned `margin` is the smallest excess.\n"
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      "text": "                    GNU GENERAL PUBLIC LICENSE\n                       Version 3, 29 June 2007\n\n Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>\n Everyone is permitted to copy and distribute verbatim copies\n of this license document, but changing it is not allowed.\n\n                            Preamble\n\n  The GNU General Public License is a free, copyleft license for\nsoftware and other kinds of works.\n\n  The licenses for most software and other practical works are designed\nto take away your freedom to share and change the works.  By contrast,\nthe GNU General Public License is intended to guarantee your freedom to\nshare and change all versions of a program--to make sure it remains free\nsoftware for all its users.  We, the Free Software Foundation, use the\nGNU General Public License for most of our software; it applies also to\nany other work released this way by its authors.  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Definitions.\n\n  \"This License\" refers to version 3 of the GNU General Public License.\n\n  \"Copyright\" also means copyright-like laws that apply to other kinds of\nworks, such as semiconductor masks.\n\n  \"The Program\" refers to any copyrightable work licensed under this\nLicense.  Each licensee is addressed as \"you\".  \"Licensees\" and\n\"recipients\" may be individuals or organizations.\n\n  To \"modify\" a work means to copy from or adapt all or part of the work\nin a fashion requiring copyright permission, other than the making of an\nexact copy.  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But this requirement does not apply\nif neither you nor any third party retains the ability to install\nmodified object code on the User Product (for example, the work has\nbeen installed in ROM).\n\n  The requirement to provide Installation Information does not include a\nrequirement to continue to provide support service, warranty, or updates\nfor a work that has been modified or installed by the recipient, or for\nthe User Product in which it has been modified or installed.  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If your rights have been terminated and not permanently\nreinstated, you do not qualify to receive new licenses for the same\nmaterial under section 10.\n\n  9. Acceptance Not Required for Having Copies.\n\n  You are not required to accept this License in order to receive or\nrun a copy of the Program.  Ancillary propagation of a covered work\noccurring solely as a consequence of using peer-to-peer transmission\nto receive a copy likewise does not require acceptance.  However,\nnothing other than this License grants you permission to propagate or\nmodify any covered work.  These actions infringe copyright if you do\nnot accept this License.  Therefore, by modifying or propagating a\ncovered work, you indicate your acceptance of this License to do so.\n\n  10. Automatic Licensing of Downstream Recipients.\n\n  Each time you convey a covered work, the recipient automatically\nreceives a license from the original licensors, to run, modify and\npropagate that work, subject to this License.  You are not responsible\nfor enforcing compliance by third parties with this License.\n\n  An \"entity transaction\" is a transaction transferring control of an\norganization, or substantially all assets of one, or subdividing an\norganization, or merging organizations.  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Patents.\n\n  A \"contributor\" is a copyright holder who authorizes use under this\nLicense of the Program or a work on which the Program is based.  The\nwork thus licensed is called the contributor's \"contributor version\".\n\n  A contributor's \"essential patent claims\" are all patent claims\nowned or controlled by the contributor, whether already acquired or\nhereafter acquired, that would be infringed by some manner, permitted\nby this License, of making, using, or selling its contributor version,\nbut do not include claims that would be infringed only as a\nconsequence of further modification of the contributor version.  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To \"grant\" such a patent license to a\nparty means to make such an agreement or commitment not to enforce a\npatent against the party.\n\n  If you convey a covered work, knowingly relying on a patent license,\nand the Corresponding Source of the work is not available for anyone\nto copy, free of charge and under the terms of this License, through a\npublicly available network server or other readily accessible means,\nthen you must either (1) cause the Corresponding Source to be so\navailable, or (2) arrange to deprive yourself of the benefit of the\npatent license for this particular work, or (3) arrange, in a manner\nconsistent with the requirements of this License, to extend the patent\nlicense to downstream recipients.  \"Knowingly relying\" means you have\nactual knowledge that, but for the patent license, your conveying the\ncovered work in a country, or your recipient's use of the covered work\nin a country, would infringe one or more identifiable patents in that\ncountry that you have reason to believe are valid.\n\n  If, pursuant to or in connection with a single transaction or\narrangement, you convey, or propagate by procuring conveyance of, a\ncovered work, and grant a patent license to some of the parties\nreceiving the covered work authorizing them to use, propagate, modify\nor convey a specific copy of the covered work, then the patent license\nyou grant is automatically extended to all recipients of the covered\nwork and works based on it.\n\n  A patent license is \"discriminatory\" if it does not include within\nthe scope of its coverage, prohibits the exercise of, or is\nconditioned on the non-exercise of one or more of the rights that are\nspecifically granted under this License.  You may not convey a covered\nwork if you are a party to an arrangement with a third party that is\nin the business of distributing software, under which you make payment\nto the third party based on the extent of your activity of conveying\nthe work, and under which the third party grants, to any of the\nparties who would receive the covered work from you, a discriminatory\npatent license (a) in connection with copies of the covered work\nconveyed by you (or copies made from those copies), or (b) primarily\nfor and in connection with specific products or compilations that\ncontain the covered work, unless you entered into that arrangement,\nor that patent license was granted, prior to 28 March 2007.\n\n  Nothing in this License shall be construed as excluding or limiting\nany implied license or other defenses to infringement that may\notherwise be available to you under applicable patent law.\n\n  12. No Surrender of Others' Freedom.\n\n  If conditions are imposed on you (whether by court order, agreement or\notherwise) that contradict the conditions of this License, they do not\nexcuse you from the conditions of this License.  If you cannot convey a\ncovered work so as to satisfy simultaneously your obligations under this\nLicense and any other pertinent obligations, then as a consequence you may\nnot convey it at all.  For example, if you agree to terms that obligate you\nto collect a royalty for further conveying from those to whom you convey\nthe Program, the only way you could satisfy both those terms and this\nLicense would be to refrain entirely from conveying the Program.\n\n  13. Use with the GNU Affero General Public License.\n\n  Notwithstanding any other provision of this License, you have\npermission to link or combine any covered work with a work licensed\nunder version 3 of the GNU Affero General Public License into a single\ncombined work, and to convey the resulting work.  The terms of this\nLicense will continue to apply to the part which is the covered work,\nbut the special requirements of the GNU Affero General Public License,\nsection 13, concerning interaction through a network will apply to the\ncombination as such.\n\n  14. Revised Versions of this License.\n\n  The Free Software Foundation may publish revised and/or new versions of\nthe GNU General Public License from time to time.  Such new versions will\nbe similar in spirit to the present version, but may differ in detail to\naddress new problems or concerns.\n\n  Each version is given a distinguishing version number.  If the\nProgram specifies that a certain numbered version of the GNU General\nPublic License \"or any later version\" applies to it, you have the\noption of following the terms and conditions either of that numbered\nversion or of any later version published by the Free Software\nFoundation.  If the Program does not specify a version number of the\nGNU General Public License, you may choose any version ever published\nby the Free Software Foundation.\n\n  If the Program specifies that a proxy can decide which future\nversions of the GNU General Public License can be used, that proxy's\npublic statement of acceptance of a version permanently authorizes you\nto choose that version for the Program.\n\n  Later license versions may give you additional or different\npermissions.  However, no additional obligations are imposed on any\nauthor or copyright holder as a result of your choosing to follow a\nlater version.\n\n  15. Disclaimer of Warranty.\n\n  THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY\nAPPLICABLE LAW.  EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT\nHOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM \"AS IS\" WITHOUT WARRANTY\nOF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,\nTHE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR\nPURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM\nIS WITH YOU.  SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF\nALL NECESSARY SERVICING, REPAIR OR CORRECTION.\n\n  16. Limitation of Liability.\n\n  IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING\nWILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS\nTHE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY\nGENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE\nUSE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF\nDATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD\nPARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),\nEVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF\nSUCH DAMAGES.\n\n  17. Interpretation of Sections 15 and 16.\n\n  If the disclaimer of warranty and limitation of liability provided\nabove cannot be given local legal effect according to their terms,\nreviewing courts shall apply local law that most closely approximates\nan absolute waiver of all civil liability in connection with the\nProgram, unless a warranty or assumption of liability accompanies a\ncopy of the Program in return for a fee.\n\n                     END OF TERMS AND CONDITIONS\n\n            How to Apply These Terms to Your New Programs\n\n  If you develop a new program, and you want it to be of the greatest\npossible use to the public, the best way to achieve this is to make it\nfree software which everyone can redistribute and change under these terms.\n\n  To do so, attach the following notices to the program.  It is safest\nto attach them to the start of each source file to most effectively\nstate the exclusion of warranty; and each file should have at least\nthe \"copyright\" line and a pointer to where the full notice is found.\n\n    <one line to give the program's name and a brief idea of what it does.>\n    Copyright (C) <year>  <name of author>\n\n    This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU General Public License as published by\n    the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n    GNU General Public License for more details.\n\n    You should have received a copy of the GNU General Public License\n    along with this program.  If not, see <https://www.gnu.org/licenses/>.\n\nAlso add information on how to contact you by electronic and paper mail.\n\n  If the program does terminal interaction, make it output a short\nnotice like this when it starts in an interactive mode:\n\n    <program>  Copyright (C) <year>  <name of author>\n    This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.\n    This is free software, and you are welcome to redistribute it\n    under certain conditions; type `show c' for details.\n\nThe hypothetical commands `show w' and `show c' should show the appropriate\nparts of the General Public License.  Of course, your program's commands\nmight be different; for a GUI interface, you would use an \"about box\".\n\n  You should also get your employer (if you work as a programmer) or school,\nif any, to sign a \"copyright disclaimer\" for the program, if necessary.\nFor more information on this, and how to apply and follow the GNU GPL, see\n<https://www.gnu.org/licenses/>.\n\n  The GNU General Public License does not permit incorporating your program\ninto proprietary programs.  If your program is a subroutine library, you\nmay consider it more useful to permit linking proprietary applications with\nthe library.  If this is what you want to do, use the GNU Lesser General\nPublic License instead of this License.  But first, please read\n<https://www.gnu.org/licenses/why-not-lgpl.html>.\n"
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      "text": "> **Provenance.** Certificate IDs retain the original numbering; use the map in the top-level README for the shorter paper. Dates, timings and statements such as “run here” below describe the supplied October 2026 archive, not new runs for this edition. New validation is reported only in the top-level README.\n\n# Denis Simon's 2-descent scripts for PARI/GP\n\nUnmodified copies of three GP scripts by Denis Simon, as distributed with SageMath\n(`sage/ext_data/pari/simon/` of the passagemath 10.8.12 wheels):\n\n| file | version (from its header) | SHA-256 |\n|---|---|---|\n| `ell.gp` | 06/04/2011 (Sage's copy) | f137f61c596b4d86c72c294b46bccb8743beac55e5373d74ec7265baaf708800 |\n| `qfsolve.gp` | 09/01/2014 | 739b6e06cb2c4b3e3adc7318bd327f3f5a3d9acef5fa2b52e68c34b85a0ac998 |\n| `resultant3.gp` | 13/01/2014 | a195c027027ffd7e1134fbe1dfe88be2375e442a810c3ba51264701e885893b8 |\n\nCopyright (C) 2011, 2014 Denis Simon, distributed under the GNU General Public License (see the header of\neach file; original address www.math.unicaen.fr/~simon/). The files are included unchanged so that the\nscripts below run without Sage.\n\n`ell.gp` provides `bnfellrank` (2-descent for a general curve over a number field),\n`bnfell2descent_viaisog` (descent via a 2-isogeny) and `bnfell2descent_complete` (curves with full rational\n2-torsion). The number field must be given in the variable `y`. The search parameters are the global\nvariables `DEBUGLEVEL_ell, LIM1, LIM3, LIMTRIV, MAXPROB, LIMBIGPRIME`; each calling script sets them (directly or\nwith `default_ell(...)`) before use. The reviewers' scripts load `qfsolve.gp` and `resultant3.gp` as well.\n\n## Used by\nRun from the section folder, the scripts load the files with relative paths, e.g.\n```\n\\r ../common/simon_ell/qfsolve.gp\n\\r ../common/simon_ell/resultant3.gp\n\\r ../common/simon_ell/ell.gp\n```\n- 1.2.1: `1.2.1a_cubic_field_K.gp`\n- 1.2.4: `1.2.4e_rank_simon.gp`\n- 1.2.6: `1.2.6d_rank_simon.gp`\n- 1.3.1: `1.3.1d_rank_simon.gp` (loads `ell.gp` only)\n\nTested with PARI/GP 2.15.4 on 14–15 Oct 2026.\n"
    }
  ]
}