# Lara Rodríguez–Thakur higher-genus zeta-like conjectures: status

2026-10-23, America/New_York. Written by an AI agent (claude-opus-5-5, Claude
Code fork) for Taylor; saved to disk by the parent agent because the fork
harness blocked the fork's own write. Not human-reviewed.

## Status

Both numbered conjectures of LRT20 (arXiv:2003.12910; JTNB 33 (2021) 553–581)
are **PROVED** by K. Chung, T. Ngo Dac, F. Pellarin, "Universal families of
Eulerian multiple zeta values in positive characteristic", arXiv:2111.06973,
Adv. Math. 422 (2023) 109003 [CNP].

- **Conj 3.1** (ζ(q^n−1, (q−1)q^n, …, (q−1)q^{n+k}) zetalike for every class
  number one A): CNP Theorem 2 ("main theorem") gives
  ζ_A(q^n−1,(q−1)q^n,…,(q−1)q^{n+k−1}) = α_{n,k} ζ_A(q^{n+k}−1) with
  α_{n,k} ∈ H (Hilbert class field) for every A, n ≥ 1, k ≥ 0; class number one
  gives H = K. Explicitly
  α_{n,k} = (−1)^k λ_{n+k}^{−1} Σ_{i=0}^{k} λ_{n+i} ε_{k−i}^{q^{n+i}}.
- **Conj 3.3** (explicit R_n for rings (i)–(iv)): CNP's class-number-one
  section contains a theorem stated as "Conjecture [LRT20, 3.3] holds", via the
  shtuka-function formula
  α_{n,1} = −(ν^{(n+1)}/δ^{(n+2)}|_Ξ · (f|_{Ξ^{(1)}})^{−q^n} + 1).
- CNP abstract: "In particular, we prove a conjecture of Lara Rodriguez and
  Thakur." (CNP TeX lines 276–296, 769–790, 891–933.)

## Verification performed

1. **Exact algebraic check** (`src/algebraic_check.sage`, in the function field
   of each curve). LRT Conj 3.3 copied verbatim from the LRT TeX (lines
   299–316), CNP's restatement of it (RTEg1–4) and CNP's proved formulas
   (CNPEg1–4) were compared. LRT's formula equals CNP's proved formula exactly
   for case (i) n ≤ 6, (ii) n ≤ 4, (iii) n ≤ 3, (iv) n ≤ 6. LRT's n = 1
   theorems (Thm 3.2, Appendix A–C) also agree with both.
2. **Independent numerical recomputation**, cases (i) and (iv), q = 2.
   S_d(k) is summed over σ-orbits (y ↦ y+1), with N(a) = f² + fg + g²P(x) and
   the trace recursion p_k = g p_{k−1} + N p_{k−2}, in F_2((1/x)) to t-adic
   precision 700. Truncation: D = 11 for case (i), D = 13 for case (iv). The
   tail check (D vs D−1) agrees to ≥ 700; at D = 11 case (iv) agreed only to
   about 514, so the D = 13 run is used. For n = 1..4,
   ζ(2^n−1, 2^n)/ζ(2^{n+1}−1) matches the prediction to precision 700 in both
   cases. Control: substituting the n = 1 prediction for the n = 2 ratio fails
   at valuation 2. In case (i), (x²+x+1)ζ(1,2) − ζ(3) = O(t^698), which
   re-verifies LRT Thm 3.2.
3. **Depth 3–4 members of the family are zetalike.** Rational reconstruction
   finds low-degree ratios for ζ(1,2,4)/ζ(7), ζ(3,4,8)/ζ(15) and
   ζ(1,2,4,8)/ζ(15) in both cases. New explicit identity, case (i), numerical
   only (to precision ≥ 600, degree-(2, 9) reconstruction from 600
   coefficients): x(x+1)(x⁸+x⁶+x⁵+x³+1)·ζ(1,2,4) = (x²+x+1)·ζ(7). CNP's theorem
   implies such an identity exists; this explicit form was not compared against
   CNP's closed formula for α.
4. **Negative evidence.** Reconstruction with degree bound about 305 (640
   coefficients) finds no rational ratio, in either case (i) or case (iv), for:
   ζ(1,1)/ζ(2), ζ(1,3)/ζ(4), ζ(3,1)/ζ(4), ζ(2,1)/ζ(3), ζ(1,5)/ζ(6),
   ζ(5,1)/ζ(6), ζ(1,6)/ζ(7), ζ(1,2,1)/ζ(4), ζ(2,2,3)/ζ(7), ζ(3,4,1)/ζ(8).
   This is consistent with LRT's statement that ζ(1, q−1) is not zetalike in
   case (i). It rules out only ratios of bounded height.

## Still open from LRT20 (informal speculations, not numbered conjectures)

1. This family is the only primitive zetalike family for the four rings
   (numerical evidence only).
2. Zetalike ⇔ Eulerian (rational versus algebraic ratio) in higher genus.
3. Higher class number: when α_{n,k} ∈ H lies in K; whether H-linear
   combinations over ideal classes are Eulerian. CNP already show the family is
   Eulerian for every A.
4. Solvability of the Frobenius-difference equations in LRT Remarks 4(2)/(III),
   i.e. a finite-level mechanism. CNP bypass this.
5. Dedekind-type relative zeta (LRT §6): no zetalike examples are known.

No paper after CNP addressing 1–5 was found. The search was not exhaustive:
Semantic Scholar citations of LRT20 plus two web searches.

## Refutations and corrected expectations

- **R1.** It was natural to expect the LRT higher-genus conjectures to be open;
  in fact both are theorems of CNP (evidence above; re-verified exactly in
  Verification §1). The parent's "open conjectures" list was wrong on this item.
- **R2 (erratum in CNP).** CNP display (RTEg2), restating LRT Conj 3.3(ii),
  drops the constant θ⁴. It writes "(X−θ)(Y−η)² − X² + (−θ³+1)X + θ² − θ"; the
  correct expansion has "+ θ⁴ + θ² − θ". The displayed formula differs from both
  LRT's and CNP's proved formula for n = 1..4, while LRT's equals CNP's proved
  formula. CNP's later proof line "X⁴ − θX³ + X² − X − θ² + θ" (TeX line about
  1059) matches the correct formula, so the slip is cosmetic and the theorem
  stands.
- **R3.** LRT stated the family only for class number one and doubted single
  zetalike values in higher class number. CNP show the family is Eulerian for
  every A with ratio in H. Class number one was needed only to get the ratio
  into K, not for Eulerianity.
- **R4.** The control in Verification §2 failed as intended, so the numerical
  test is not vacuous.
- **Carried forward:** one prior negative claim (LRT: ζ(1,1) not zetalike in
  case (i)), reproduced numerically. Excluded: 0.

## Sources

- `sources/arxiv-2003.12910/`: LRT20 TeX source and PDF (from arXiv).
- `sources/arxiv-2111.06973/`: CNP TeX source and PDF (from arXiv).

## Limitations

- MZVs were recomputed numerically only for cases (i) and (iv); cases (ii) and
  (iii) were checked algebraically against CNP's formulas only.
- CNP statements and proof structure were read, not every proof line.
- No human review.

## Provenance (for ai-usage)

- Agent: fork `claude-opus-5-5`, Claude Code fork harness; no descendants.
- Time: about 2026-10-23T14:15 to 14:23 −04:00.
- Tools: WebSearch (standard and extended), Semantic Scholar get_citations,
  `curl` downloads of arXiv source and PDFs, Sage 10.8.
- Script hashes: algebraic_check.sage 56db11da…, compare.sage da907e88…,
  numeric_case_i.sage 077ef6c6…, sanity.sage 58626690….
- Token counts are not exposed by the harness; the parent's notification
  reports about 190k subagent tokens and 26 tool uses.