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Arithmetic Expression Spaces: Expansion and Motion, Eikonal Equation, Horocycle Pullbacks, Commutativization, Contact Structure, and Tearing

A mix of human-written and AI-generated textHuman understanding: all partsmath.GM — General Mathematicsmath.DG — Differential Geometry

Contributed by Mingli Yuan ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

An arithmetic expression is a word of operations; a motion is a flow on a manifold. For expressions whose evaluation has a single internal spine, the two are one structure: affine for addition and multiplication, projective once the second operand slot is divided, contact-geometric in the order defect. A regular arithmetic expression space (AES) is an oriented Riemannian surface (M,g)(M,g) with a smooth assignment a ⁣:M→Ra\colon M\to\mathbb R satisfying the eikonal equation ∣∇a∣g2=μ2+λ2a2|\nabla a|_g^2=\mu^2+\lambda^2a^2; the four operations then act as motions translating and scaling aa. In the basic model E0\mathfrak E_0, the upper half-plane with the left-invariant metric of Aff+(1,R)\mathrm{Aff}^+(1,\mathbb R) (curvature −λ2-\lambda^2), addition and positive multiplication are affine translations and dilations obeying a Baumslag--Solitar law, and aa is a Laplace--Beltrami eigenfunction of eigenvalue 2λ22\lambda^2. Left-expanded words stay in the affine Borel cell fixing ∞\infty; the right-expanded language, once second-slot division is admitted, generates all of PGL2(R)\mathrm{PGL}_2(\mathbb R) through the rank-one Bruhat decomposition, and pulling standard horocycles back along one operator word gives the dual reading, ripple geometry. Commutativizing the motions by total charge yields the accumulative commutative space, where the order defect of two histories is an exact weighted area, the tearing. The same defect is the constant μλ\mu\lambda in the bracket [Du,Dv]=μλ∂a[D_u,D_v]=\mu\lambda\partial_a of the horizontal fields of α=da−μ du−λa dv\alpha=da-\mu\,du-\lambda a\,dv, a contact form whenever μλ≠0\mu\lambda\ne0, and it is the curvature of that horizontal connection; for E0\mathfrak E_0 the basic model is proved to be the Reeb quotient of the contact model, and fixed-charge tearing and fixed-state Reeb drift are Legendre dual readings of the same curvature. The generators ∂a\partial_a, a∂aa\partial_a, and alog⁡a ∂aa\log a\,\partial_a span an infinite-dimensional Lie algebra, so no finite-dimensional Lie group carries all three ranks, and no observable of the charge endpoint detects tearing. For every finite puncture count a conformally compactifiable punctured AES is constructed, with an explicit four-circle zero locus and exact reconstruction from three-ended pants blocks. All results are elementary, planar, or local to a three-dimensional contact model.

Provenance statement

# Authorship and provenance This register separates human research direction and final responsibility from model formalization, implementation, verification and drafting. GitHub account ownership, CI success and mergeability are not independent review. ## Historical weekly provenance register: 2026-10-12 through 2026-10-18 The dispositions below are snapshots at 2026-10-18; later acceptance is recorded separately, without rewriting the historical findings. ### P-2026-10-18-07 — Finite process spectra and cyclic-prefix separation - **Human origin/direction:** Mingli Yuan selected the process-spectrum, Alexandrov reconstruction and finite-observation questions and retains final acceptance and release authority. - **Model assistance:** Codex (OpenAI) supplied the finite constructions, proofs, enumeration, executable receivers, mutation controls and prose. - **Objectification:** Draft PR [#17](https://github.com/mountain/aeg-paper/pull/17), head `e4d51425ccfa3575c87f5bd7a9d66cbb4128ee16`. - **Result:** the three-event chain and fork have equal undirected cover Laplacian spectra but different causal orders and Alexandrov lower opens; all 19 labelled three-element posets reconstruct in the declared finite route. The guarded loop/stopping pair shares each prescribed finite packet prefix, while terminal status separates them. - **Boundary:** no general rejection of spectral methods, infinite-run certificate or Adva anytime theorem. - **Human disposition:** retained Draft/Open; explicit approval of the final reviewed head and independent human mathematical review are absent. - **Status:** `draft-bounded-result`. ### P-2026-10-18-08 — Process-representation gap programme - **Human origin/direction:** Mingli Yuan selected the representation-gap work plan, process-geometric questions and final publication boundary. - **Model assistance:** AI systems implemented the six frozen experiments, replay and red-team review. The original producing-model identity is not consistently fixed in the initial commit record and remains a provenance gap; the review is model cross-checking, not independent human review. - **Objectification:** PR [#18](https://github.com/mountain/aeg-paper/pull/18), reviewed head `1556eaedaee0946b2e395320739234dc59ec846c`. - **Reproduced finite evidence:** braid representation counts `5461 -> 1457 -> 577 -> 6`, the bounded separating probe and 20 moving-obstacle interval certificates. - **Blocking corrections:** both self-embedding checkers accept a triangle that collapses at `t=1/2`; the committed R2 fixture violates necessary planar Gauss parity; the strict-torus nontrivial-knot exclusion is false; and the freeze guard can refresh existing contract digests. Green replay CI therefore does not establish mathematical correctness. - **Review disposition at 2026-10-18:** model review required changes; no merge or theorem promotion was recorded. - **Status:** reproducible experiment with explicit `rejected-or-unknown` mathematical/verification blockers. ## Subsequent acceptance and repairs — 2026-10-19 UTC Mingli Yuan explicitly requested merging all open aeg-paper PRs into main in this session. This authorizes repository integration; it does not establish independent human mathematical review or promote a finite result to a theorem. - **PR #17:** merged as `b75b0cbe17fc8a7417a8db9c392f33109f04a80a` after normal/optimized producer and independent-receiver replay matched frozen evidence. The exact reviewed head's Paper 0/I/IV integration and PDF build workflow succeeded. Its bounded-result and native-Adva limitations remain. - **PR #18 repair direction:** restrict positive motion certification to verified rigid translations; retain the historical R1/R2-named operations as abstract signed-word calculations; withdraw planar Reidemeister and knot-process faithfulness claims. Correct sphere/torus and framing statements. - **PR #18 model contribution:** Codex (OpenAI) implemented the two narrowed embedding routes, geometry and integrity regressions, immutable historical contract pins, explicit replay inventory, temporary-output replay and fresh raw-table comparison, and drafted the corrected reports. The original producing-model provenance gap remains recorded above. - **Repair identities:** v1.1 contract `7600707e3b9a98a37cfde325464b260b9a2c36a8`; implementation `0697b1ca0eb85f4383811f23534e78af19191d8e`; integrated candidate `fdead22a64f3ece04778c598f32df92001e8031e`. - **Preservation:** original v1 contracts and rejection-review artifacts remain unchanged. Historical reports, evidence and fixtures are indexed by exact commit/blob/SHA-256 in `research/process-representation-gap/reviews/2026-10-19-repair-history.json`. Current exp6 evidence is an explicit v1.1 successor. - **Acceptance boundary:** documentary research integration only. General motion certification, planar local-move certification and native Adva continuation/learning integration remain open; no manuscript theorem changes. ## Contribution interpretation Mingli Yuan is the human research director, acceptance and release authority, and final human scientific responsibility holder. Model contributions must be named per commit/PR when known and must not be reassigned to the GitHub account holder. Model cross-review is not independent human review.

References

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  7. Yuan, Mingli. Geometry of Arithmetic Expressions: I. Basics and Problems. 2022. Historical draft, 23 December 2022, commit 339e0ce; arithmetic-expression subsection defines partial evaluation and evaluability.link
  8. Yuan, Mingli. Geometry of Arithmetic Expressions: Alternating Threadlike Expressions. 2023. Historical draft, 21 February 2023; alternating words, distributive expansion and exact arithmetic propagation of an initial perturbation.link
  9. Yuan, Mingli. Geometry of Arithmetic Expressions: I. Basic Concepts and Unsolved Problems. 2023. Historical draft, 2 January 2023; the Foundational problem connects rational arithmetic, division guards and semantic decision questions.link
  10. Yuan, Mingli. Arithmetic Expression Geometry: Flow, Accumulation, Dual Timescales, and a Minimalist Realization of Renormalization. 2025. Exploratory note, 20 May 2025, Section 3; cited for the historical interpretation of perturbation gain as accumulated scale, not as a proof of a general renormalization construction.link
  11. Yuan, Mingli. Arithmetic Expression Geometry I: Foundations–Sequential Histories, Projective Semantics, and Zero Geometry. 2026. Mathematical-review manuscript
  12. Yuan, Mingli. Arithmetic Expression Geometry IV: Projective Condensation and Computational Complexity. 2026. Companion manuscript, repository path \textttpaper-4/.
  13. Yuan, Mingli. The Inevitability of Holes: A Ledger of Six Computed Obstructions. 2026. Exploration register, retained as \textttnotes/foundations-and-geometry/06-hole-obstructions-ledger.md in the AEG paper repository. Non-authoritative working record; not a paper claim. English rendering of a Chinese working note of the same content.
  14. Yuan, Mingli. Process Geometry Mathematical Core: Why the Bolza Surface Appears. 2026. Section 2.6, pinned at commit c47c96f. Product-sign quotient after an independently declared metric square-root layer; not a canonical compactification of the hyperbolic plane.link
  15. Yuan, Mingli. Technical Report: The Program-Process Core of Adva. 2026. Pinned research report at commit cce7300; finite program substitution and program/value distinction, Sections 1.2–1.3.link
  16. Yuan, Mingli; Process Geometry contributors. Process Geometry. 2026. Research software and literate calibrations, version 0.0.4; pre-alpha research preview. Cited here only for the research-local calibrations named in the text, which carry the repository's own maturity labels. https://github.com/mountain/process-geometrylink

Version history

  1. v1Submitted by Mingli YuanInitial depositCurrentOct 30, 2026