# Paper 0 — Arithmetic Expression Spaces

The [formal derivation work](formal/README.md) starts chapter-by-chapter
verification of the current manuscript with an Adva-written equality checker,
separate Lean statements and an explicit coverage inventory. Initial evidence
covers named algebraic components, with a successor for Chapter 2's syntax
and guarded ordinary domains. Its first source finding, OQ-090, is resolved
by the marked-data/scalar-projection distinction in the amended proposition.
Consult the current mathematical-status register for each claim's scope.

Chapter 2 also restores the early concept of an evaluable expression and
separates partial semantics from effective rational evaluation and semantic
decision problems. The [historical discussion](../governance/discussions/evaluable-expressions-and-effective-presentations.md)
records the recovered drafts, the Richardson qualification, and the connection
to Adva's finite construction viewpoint.

Appendix C restores alternating expressions and their purely arithmetic
perturbation derivation before the general affine calculations. Its
[historical record](../governance/discussions/alternating-expressions-perturbation-acs-rg.md)
tracks the route to ACS and the later scale/RG interpretation, with exact
identities and interpretive claims kept at their respective scopes.
The [formal continuation](formal/README.md#appendix-c-induction-successor)
supplies finite-induction proofs and an explicit source map for the restored
expansion, perturbation and guarded difference quotient.

Appendix F collects [background on cuts and expression foundations](appendices/app-F-cuts-and-expression-foundations.tex):
Bauer--Taylor's observable two-sided cuts, Conway's recursive simplicity
construction, and the distinction between finite names, denotations and
observations. Adva's finite through adapter is a separate research precedent.
The proposed formal programme remains open and adds no certificate coverage.

**Provisional title:** *Arithmetic Expression Geometry 0*  
**Subtitle:** *Arithmetic Expression Spaces: Expansion and Motion, Eikonal
Equation, Horocycle Pullbacks, Commutativization, Contact Structure, and
Tearing*  
**MSC 2020:** 53D10 (primary); 53C22, 20F65; 30F45, 32G15, 11J70, 03F60

The subtitle was revised on 2026-10-29 to use, at conservative strength, only
two kinds of term: names this paper defines (expansion, motion,
commutativization, tearing) and standard mathematical names used literally
(eikonal equation, `def:regular-aes`; horocycle pullback, `thm:p0-ripple-pencil-formula`;
contact structure, `prop:contact-form`, non-degenerate exactly when
`mu*lambda != 0`).  Two earlier candidates were rejected as overstating the
proved content: *eikonal metrics*, since the metric is part of the tuple and it
is the identity `|grad a|^2 = mu^2 + lambda^2 a^2` that is eikonal, and *contact
curvature*, since `rem:p0-curvature-invariance` shows that the coefficient
`mu*lambda` is the curvature scalar of the declared arithmetic frame and
horizontal metric, not a contact-isomorphism invariant.  The former subtitle was
*Arithmetic Expression Spaces: Expansion, Motion, Ripple Duals, Commutativization,
and Tearing*.  The names *ripple geometry* and *tearing* are unchanged in the
body, where they keep their recorded statuses.  The front matter credits Mingli
Yuan as author and corresponding author and names DeepSeek Harness as
formalization and computational assistance; this matches Papers II--V, which
already carry a single human author.

Paper 0 is the geometric foundation of the series.  It develops:

1. arithmetic expansions: left- and right-expanded combs under the explicit
   operand-slot convention, one-hole sections, and the distributive law as the
   rewrite that relates distinct expansions of one expression;
2. arithmetic expression spaces: the regular AES axioms, the canonical
   arithmetic frame, the four operations as motions, the basic hyperbolic model
   `E_0`, the one-puncture model `E_1`, and the continuous arithmetic motion;
3. the projective dual reading — poles, horocycle pullback, reciprocals,
   infinity, fixed points, and the matrix synthesis — whose dual name is
   *ripple geometry* (descriptive and provisional);
4. the commutativization: affine cocycles, the relative defect, the
   accumulative commutative space (ACS), the weighted torsion--Stokes theorem,
   and the reading of that defect as *tearing*;
5. the contact structure of tearing: the contact form, the horizontal fields,
   the curvature bracket, finite open and closed defects, and the scalar
   horizontal differential;
6. two methodological obstructions to linear language and the structural
   proposals imported from process geometry;
7. punctured arithmetic expression spaces, the tangent cone of a puncture, and a
   register of six computed witnesses;
8. finite-type end conditions, arbitrary positive puncture-count examples,
   three-ended collar reconstruction, an exact four-puncture motion residue,
   and carefully separated circle and branched-sheet models.

The paper does **not** revise the marked spinal history terminology of Paper I.
Paper I remains responsible for the intrinsic sequential-tree classification,
mixed chirality, chronological marked histories, projective evaluation, the
Hecke `q=4` sublanguage, and the zero geometry.

## Canonical source closure

```text
paper-0/
  aeg-paper-0.tex
  sections/
    01-introduction.tex
    02-expansion.tex
    02a-evaluation-and-effectivity.tex
    03-aes-and-motion.tex
    04-ripple-geometry.tex
    05-reciprocals-fixed-points.tex
    06-projective-unification.tex
    07-affine-cocycles.tex
    08-acs-tearing.tex
    09-contact-of-tearing.tex
    10-beyond-linear.tex
    11-holed-aes.tex
    12-compactification.tex
    12-interface.tex
  appendices/
    app-A-calculations.tex
    app-B-hyperbolic-calculations.tex
    app-C-affine-cocycles.tex
    app-D-acs-contact.tex
    app-E-circle-surgery.tex
    app-F-cuts-and-expression-foundations.tex
```

Sections 7--9 and appendices B--D were moved from Paper I by the amendment
`governance/00b-paper-0-geometric-foundation-amendment.md`; they keep their
labels, hypotheses, and statuses.  The paper uses the shared bibliography at
`bibliography/aeg-paper.bib`.

## Claim boundary

Proved in the manuscript, at the statuses recorded in
`governance/05b-paper-0-status-register.md`:

- both pure combs have a unique internal evaluation order;
- right combs become left combs after passing to opposite operations;
- distributivity preserves the operator while changing the charge of an
  expansion;
- the basic hyperbolic model is a complete regular AES of constant curvature;
- the four operations act as motions, with the Baumslag--Solitar-type transport
  relation for the additive ruler;
- the pullback of a horizontal horocycle by a real orientation-preserving
  Möbius map is a line or a circle tangent at the operator pole;
- geometric-series and continued-fraction truncations satisfy the stated
  recursions and matrix formulas, with convergence claimed only under explicit
  hypotheses;
- translations, nonzero scalings, and inversion generate `PGL_2(K)`;
- the ACS evaluation formula and the weighted torsion--Stokes theorem;
- the history group is the free product of the two step families, the charge
  endpoint is its abelianization, and of the two charges only the multiplicative
  one descends to the motion (`thm:p0-descent`);
- the contact form is non-degenerate exactly when `mu*lambda != 0`, and the
  horizontal fields satisfy `[D_u,D_v] = mu*lambda*d_a`; the bracket defect is a
  canonical 2-form on the distribution, and `mu*lambda` is its value on the
  orthonormal frame of the declared horizontal metric;
- the basic model is the Reeb quotient of the contact model, with the horizontal
  fields descending to the canonical frame and `d(alpha)` equal to
  `-mu*lambda` times the hyperbolic area form;
- the arithmetic generators do not close into a finite-dimensional Lie algebra
  (so no finite-dimensional Lie-group action carries all three ranks), and no
  observable of the commutativized charges detects tearing;
- in a regular AES the assignment has no critical point, so every level set is a
  smooth curve;
- the four-circle configuration is the zero locus of an explicit punctured AES
  whose metric is forced by the eikonal identity, with the puncture
  non-extendable and at finite distance (`thm:p0-four-circle-model`);
- for every `k >= 1` there is a conformally compactifiable punctured AES with
  exactly `k` essential punctures (`thm:p0-arbitrary-k-punctures`); this is an
  existence result, not a classification by puncture number;
- for a punctured space whose assignment extends smoothly, and whose given metric
  is conformal near each puncture to a background metric extending smoothly and
  non-degenerately, the puncture set is exactly the critical set of the
  assignment, hence canonical and unique, and the number of holes equals the
  number of critical points (`thm:p0-punctures-are-critical`,
  `cor:p0-puncture-count`; `PROVED WITH STATED HYPOTHESES` in section 8 of
  `governance/05b-paper-0-status-register.md`).  The unrestricted September 22
  form of the same statements, without the conformal-background hypothesis,
  remains refuted and is `UNSUPPORTED AND EXCLUDED` in section 6 of that
  register; the abstract of this paper claims neither form;
- the tearing is puncture-independent, no canonical holonomy of a puncture
  exists, and in every regular AES the two orderings of a rectangle end at points
  with different assignments, so the ACS comparison is canonical rather than a
  workaround for holes;
- in the four-circle model a small circle about the puncture meets the zero locus
  in eight sign-changing points, the conformal class extends while the metric does
  not, and the puncture lies at finite distance;
- the tangent cone of that zero locus at the puncture is `(xy)^2 = 0`.

Labelled as proposals rather than theorems: the name *ripple geometry*, the
language of *tearing*, the model-label convention `E_k`, and the vocabulary
imported from process geometry.  Open extensions include an intrinsic projective
AES for ripple geometry, quotient-stable path/ripple invariants, a canonical
choice of punctures, multi-wire generalization, and a general theory of infinite
arithmetic expressions.

## Compactification integration of 2026-10-09

The geometric basis for ported AES is Section 12. Geometric punctures, conformal
point filling, boundary circles, and computational inputs remain different objects.
The critical-puncture theorem now includes the extending-conformal-background
hypothesis required by the September 29 review.

Run the newly authored, standard-library exact finite checks with:

```bash
python paper-0/scripts/verify-compactification.py
```

These checks supplement the manuscript proofs. They are not a rerun of the
original 39-check research archive, whose bytes were unavailable during this
integration. See `governance/h2-integration-2026-10-09.md` for source coverage,
mathematical status, and the remaining original-artifact recovery task.