Arithmetic Expression Geometry

This repository contains the manuscripts, notes, figures, and build tooling for the Arithmetic Expression Geometry (AEG) research programme. AEG studies arithmetic expressions through their ordered evaluation histories. An unmarked sequential tree first requires a choice of accumulator; the resulting fully marked planar tree is equivalent to a bounded marked history. From that history, operator evaluation and charge/geometric shadows form separate branches, and only an operator together with an initial value determines an endpoint. The repository keeps these levels distinct.

The active manuscripts are:

Arithmetic Expression Geometry I: Foundations
Sequential Histories, Affine Flow, Torsion, and Contact 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

Their canonical entry points are paper-1/aeg-paper-1.tex, paper-2/aeg-paper-2.tex, paper-3/aeg-paper-3.tex, and paper-4/aeg-paper-4.tex. The governing scope, mathematical-status register, migration map, and acceptance criteria are under governance/.

Draft status

All four active manuscripts are drafts in the ordinary editorial and publication sense. Their conceptual organization, exposition, figures, and text--figure integration remain under development and are not yet presented as release-quality replacements for the archived legacy manuscript. In particular, archive/legacy-root-manuscript/ may currently offer a more polished reading and visual experience even where the active series has the newer mathematical organization.

Here draft is a maturity label for the manuscripts as complete scholarly and visual works. It is not a mathematical claim-status label in the governance system and does not weaken or promote any theorem, proposition, conjecture, computation, or open problem. Claim-level status remains controlled separately by governance/05-mathematical-status.md.

Version and DOI

The Zenodo DOI above identifies an earlier archived version of the manuscript. It does not automatically identify any active restructured paper. The active sources state their own dates and review status; a future release must receive explicit author approval before DOI metadata is changed.

Four-paper architecture

1. Paper I — Foundations. Sequential histories, bilateral projective semantics, the affine sector, the q = 4 Hecke arithmetic sublanguage, cocycles, continuous flow, the basic hyperbolic model, complete regular-zero rigidity, global torsion, and contact curvature.
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, holomorphic branch singularities, the order-four Hecke zero network, arithmetic register correspondences, the sign-cover realization of the T(2,4) link complement, its marked toric and logarithmic-tangent polynomial normal form u^2=t^4, polynomially threaded carrier surfaces, a four-strand central extension and group-cohomological residue calibration, the sextic Lyashko--Looijenga laboratory with full six-braid and genus-two monodromy, Hecke periodic-orbit knots, discriminants, proper tubes, monodromy, and the conditional knot-invariant programme.
4. Paper IV — Projective Condensation and Computational Complexity. The projective quotient tower, rank-one bivaluations, chronology-sensitive continuation descent, contextual residuals and minimal exact state, operational live-configuration geometry, fiber and entropy inequalities, rewrite exactness obstructions, and fixed-model calibrations from Horner, OBDDs, transforms, matrix chains, and checkpointing.

Papers I--IV are draft manuscripts under mathematical review. Paper III separates proved regular and conditional topological results from structural proposals and open problems. Paper IV separates proved quotient/residual results and fixed-model complexity calibrations from the open multi-wire, approximate, and machine-robust programmes.

Paper entry points and generated PDFs follow the uniform convention paper-k/aeg-paper-k.tex and paper-k/aeg-paper-k.pdf.

Arithmetic--automorphic research spine

The cross-paper development activates the following one-way interface:

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 toric thread
  -> logarithmic polynomial cone, threaded carrier, and braid-center residue.

Each arrow has a different information level. Literal histories are not identified with projective operators, group elements, cells, endpoints, or geometric sheets. The exact Hecke example reaches its automorphic zero graph and geodesic-flow knots at the operator-quotient level. Finite typed registers give explicit relative divisors, Frobenius actions, and a tagged trace that detects some domain history lost by the terminal operator. The family z^2=t^m gives an exact cross-slice comparison among arithmetic components, carrier topology, discriminant winding, braid exponent, and linking, while its coefficient path is still supplied rather than history-derived. The fixed pencil x^6-x-t upgrades this to nonabelian transport: one event polynomial controls real carrier walls, all six-braids, all genus-two mapping classes, full symplectic monodromy, and a finite 1296-sheet LL forgetting problem. Two explicit sheets in distinct residual source-rotation orbits have the same event polynomial but different genus-two Igusa moduli at t=1; thus this forgetting provably loses genuine complex geometry. On the 216-sheet quotient cover, the moduli observable has a single-valued spectral polynomial, trace--norm descent, a canonical constant/zero-sum splitting, and a monodromy-invariant forgetting variance with an exact positive lower bound at Q_0; it need not be constant along arbitrary open paths in the event-polynomial base. The two explicit slice pencils have ratio ((t-beta)/t)^5 and balanced regular-fiber divisor charge 5[beta]-5[0], a logarithmic charge rather than a finite Dirichlet energy. The full 216-orbit census, arithmetic/period comparison, marked-monodromy refinement, and Hodge/Siegel-energy interpretation remain open. Neither the LL sheet nor its parameter loop is yet selected by a general arithmetic history. A general functor from marked histories to arithmetic prime divisors remains open. The active synthesis and claim ledger are recorded in governance/discussions/arithmetic-automorphic-zero-networks.md.

Arithmetic--computational research spine

Paper IV activates a second one-way interface:

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

The arrows are conditional. A quotient is an exact online state only when the future action descends and the observation factors through it. A residual cardinality gives a fixed-width state-selection bound, while actual work, workspace, memory--time, and communication require a declared machine model. The paper proves left/right continuation criteria, the minimal deterministic residual theorem, finite fiber and entropy inequalities, and exact case-study counts. It does not infer hardness from noncommutativity, negative curvature, exponential group growth, or raw history-fiber size. The detailed decisions, source audit, red-team report, and closure record are under governance/.

Building the manuscripts

The local build requires pdflatex and bibtex:

./build.sh

This builds Papers I--IV. To build one manuscript only, run ./build.sh 1, ./build.sh 2, ./build.sh 3, or ./build.sh 4. The expected artifacts are paper-k/aeg-paper-k.pdf for k = 1,2,3,4.

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

The restructuring audit records baseline and current build results in governance/audit-report.md and the per-paper closure reports.

Repository authority

Not every note in this repository is current mathematics. When sources conflict, use the order specified in AGENTS.md. In particular, notes/, images/sources/, archive/revision-*, archive/arxiv/, archive/paper4p/, and notes/knots-and-loops/ retain historical or exploratory material and are not canonical paper sources unless the migration log or a per-paper source audit says otherwise.

The subject classification and per-directory filename ordering for research notes are documented in notes/README.md.