Arithmetic Expression Geometry

This repository contains the manuscripts, notes, figures, bibliography, and build tooling for the Arithmetic Expression Geometry (AEG) research programme. AEG studies arithmetic expressions through their ordered evaluation histories rather than only through final numerical values.

The active manuscript series is now:

Arithmetic Expression Geometry 0
Arithmetic Expression Spaces: Expansion and Motion, Eikonal Equation, Horocycle Pullbacks, Commutativization, Contact Structure, and Tearing

Arithmetic Expression Geometry I: Foundations
Sequential Histories, Projective Semantics, and Zero Geometry

Arithmetic Expression Geometry II: Hyperbolic Real Function Theory
Horizontal Operators, Boundary Problems, and Arithmetic Holomorphicity

Arithmetic Expression Geometry III: Singular Zero Geometry and Tubes
Branched Pullbacks, Arithmetic Zero Networks, and Topological Transport

Arithmetic Expression Geometry IV: Projective Condensation and Computational Complexity
From Histories and Quotients to Representation and Cost

Arithmetic Expression Geometry V: Triadic Characteristics and Threaded Apertures
Three Readings, Observation Fibres, and the Syntax of Apertures

Arithmetic Expression Geometry VI: Process Geometry
Native Programs, Evaluation Orders, and the Arithmetic--Process Equivalence

Their canonical entry points are paper-0/aeg-paper-0.tex, paper-1/aeg-paper-1.tex, paper-2/aeg-paper-2.tex, paper-3/aeg-paper-3.tex, paper-4/aeg-paper-4.tex, paper-5/aeg-paper-5.tex, and paper-6/aeg-paper-6.tex. The governing scope, mathematical-status records, migration map, and acceptance criteria are under governance/. Paper 0 is introduced by the authoritative amendment governance/00a-paper-0-amendment.md and its present scope is fixed by governance/00b-paper-0-geometric-foundation-amendment.md, which makes Paper 0 the geometric foundation of the series and moves Paper I's geometric chapters into it.

An open research project — collaborators welcome

AEG is an open research project, and we are actively seeking collaborating researchers. The manuscripts are released as drafts precisely so that the programme can be entered at whichever point suits a new reader, not only at its beginning.

A newcomer does not have to accept the whole architecture in order to contribute. Each paper states its own scope, the status of each claim, and — deliberately — the places where the programme is unfinished. Those unfinished places are the intended entry points. Current examples:

• Paper IV — actual opposite-edge feature witnesses, faithful realizations of all multi-input gates by AES motions, and non-flat projective transport.
• Paper V — the derivation of an order relation from period–energy data, a non-decomposable triadic defect, the missing filler of the M_6 boundary, and a typed feedback operator. Paper V §17 lists these with their dependence order.

Ways to contribute include proving or refuting a stated conjecture, supplying a counterexample to a calibration, tightening a hypothesis, replacing an argument with a shorter one, or reporting an error in a claim's status. Refutations are as welcome as extensions: the series is organised so that an honest negative result is a contribution rather than a setback.

Please open an issue or a pull request in this repository. Because claim status is governed explicitly (see governance/), a proposed change should say which claim it touches and whether it strengthens, weakens, or refutes it.

Punctured and ported AES integration

The 2026-10-09 integration adds reusable finite-type geometry to Paper 0 and program substitution, exact partial evaluation, and feature sufficiency to Paper IV. Source and claim map. Actual opposite-edge feature witnesses and a faithful realization of all multi-input gates by AES motions remain open.

Draft status

All active manuscripts are drafts in the ordinary editorial and publication sense. Their conceptual organization, exposition, figures, and text--figure integration remain under development. This maturity label does not promote or weaken any theorem, proposition, conjecture, computation, or open problem; claim-level status is governed separately.

Version and DOI

The Zenodo DOI identifies an earlier archived version of the manuscript. It does not automatically identify any active restructured paper. A future release must receive explicit author approval before DOI metadata is changed.

Seven-paper architecture

0. Paper 0 — Arithmetic Expression Spaces. Arithmetic expansions under explicit operand-slot conventions, the distributive law as the rewrite relating distinct expansions, arithmetic expression spaces and the four operations as motions, the models E_0 and E_1, the continuous arithmetic motion, the dual reading called ripple geometry, affine cocycles, the accumulative commutative space with its weighted torsion--Stokes theorem, tearing, the contact structure of that defect, two obstructions to linear language, punctured spaces with a register of computed witnesses, finite-type compactification, arbitrary positive puncture counts and full-collar three-ended reconstruction. Paper 0 is the geometric foundation of the series; it does not revise Paper I's marked-history formalism.

1. Paper I — Foundations. Intrinsic sequential-tree classification, marked spinal histories, bilateral projective semantics, the affine sector, the q=4 Hecke arithmetic sublanguage, regular-zero rigidity, singular arithmetic expression spaces, and the interfaces to the later papers. The affine cocycles, continuous flow, hyperbolic model, ACS torsion, and contact curvature developed in earlier versions of this paper now belong to Paper 0.

2. Paper II — Hyperbolic Real Function Theory. Horizontal complex and analytic structures, operator domains, kernels, boundary problems, planar and cylindrical holomorphic pullbacks, and spectral questions.

3. Paper III — Singular Zero Geometry and Tubes. Multi-zero constructions, branched pullbacks, arithmetic zero networks, relative divisors, sign covers, threaded carriers, Lyashko--Looijenga forgetting, proper tubes, monodromy, braids, and the conditional knot-invariant programme.

4. Paper IV — Projective Condensation and Computational Complexity. The projective quotient tower, bivaluations, chronology-sensitive continuation, contextual residuals, minimal exact state, operational live configurations, fiber and entropy inequalities, rewrite obstructions, and fixed-model calibrations.

5. Paper V — Triadic Characteristics and Threaded Apertures. Observer contracts and observation fibres, characteristics as quotient plus residual, the quadratic scalar trichotomy and chirality as a retained central sign, the six-state carrier with its derived Clifford relation, triadic characteristic inference, typed multiplicity, through-crossings with resolution torsors and derived monodromy, a logic generated by a satisfaction form, the catalogue of non-faithful shadows, and — in its second part — typed apertures, threaded adequacy, ordered linear deduction with strong normalization, confluence and decidable derivability, and the six-port whole-cut kernel. Its scope is fixed by governance/00c-paper-V-triadic-aperture-scope.md.

6. Paper VI — Process Geometry. Native open programs with ordered holes and retained construction history, the three readings K/C/F, schedules as linear extensions of the construction order, the arity-one carrier and the recovery of the spine and comb orders, the canonical evaluation order for plane trees, the carousel correspondence and the planar screw module, and the typed dictionary and equivalence statements connecting process geometry with Papers I and V. Its scope is fixed by governance/00e-paper-VI-process-geometry-scope.md.

The dependency architecture is

Paper 0 -> Paper I -> Paper II
                   -> Paper III
                   -> Paper IV -> Paper V
Paper I -> Paper VI
Paper V -> Paper VI
Paper 0 -> Paper VI

Path--ripple research spine

Paper 0 activates the elementary interface

left-expanded comb
  -> affine one-hole maps
  -> arithmetic motion in E0
  -> matrices with pole at infinity

right-expanded comb
  -> reciprocal one-hole maps
  -> finite pole and ripple pencil
  -> general Möbius matrices

The same prefix matrix determines the propagated point, the operator pole, its fixed points, and the pullback of standard horocycles. Arithmetic reciprocals are also related to iteration: finite geometric-series and continued-fraction truncations are matrix products, while infinite values are asserted only under explicit formal or analytic convergence hypotheses. Ordinary division by zero remains inadmissible even when projective continuation reaches infinity.

Tearing spine

The order defect of arithmetic motion is developed in Paper 0 as a chain of exact statements rather than as a metaphor:

distributive expansion
  -> same operator, different charges
  -> affine cocycles and the relative defect
  -> commutativization: the accumulative commutative space
  -> weighted torsion--Stokes: the defect is an area
  -> contact structure: the same defect has density mu*lambda
  -> punctured spaces: tearing carried at a declared singular point

The names ripple geometry and tearing are descriptive; the computations they name are proved in Paper 0, and the reading of the computed hole witnesses is kept at the status COMPUTATIONALLY VERIFIED EXAMPLE.

Arithmetic--automorphic research spine

The later cross-paper development remains one-way:

Paper I arithmetic histories
  -> q=4 Hecke projective operators
  -> Paper II holomorphic pullback targets
  -> Paper III branched automorphic zero networks
  -> sign-cover geodesic knots and peripheral polynomial threads.

Each arrow has a different information level. Literal histories are not identified with operators, cells, endpoints, geometric sheets, or knots. The sextic Lyashko--Looijenga laboratory proves that one finite forgetting map can lose genuine genus-two moduli, but a general functor from marked histories to arithmetic prime divisors remains open.

Arithmetic--computational research spine

Paper IV activates a second conditional interface:

Paper I marked histories
  -> PGL2 operator evaluation
  -> projective quotient G/H
  -> future-relative contextual residual
  -> encoded online state
  -> operational live-configuration trace.

A quotient is an exact online state only when the future action descends and the observation factors through it. Residual cardinality gives a fixed-width state-selection bound; actual work, workspace, memory--time, and communication require a declared machine model. No hardness result is inferred merely from noncommutativity, negative curvature, group growth, or raw history-fiber size.

Building the manuscripts

The local build requires pdflatex and bibtex:

./build.sh

This builds Papers 0--VI. To build one manuscript only, run

./build.sh 0
./build.sh 1
./build.sh 2
./build.sh 3
./build.sh 4
./build.sh 5
./build.sh 6

The expected artifacts are paper-k/aeg-paper-k.pdf for k = 0,1,2,3,4,5,6.

A container build remains available for environments with Docker:

docker build -t aeg-paper .
docker run --rm -v "$(pwd):/work" aeg-paper

Paper IV's finite verification suite is:

python paper-4/scripts/verify-paper4.py

Repository authority

Not every note in this repository is current mathematics. When sources conflict, use the order specified in AGENTS.md, together with the Paper 0 amendment named above. In particular, notes/, images/sources/, archive/revision-*, archive/arxiv/, archive/paper4p/, and historical knot or thermal notes retain exploratory material and are not canonical paper sources unless a migration or source-audit record says otherwise.