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VGSE-ENG-WP04 — Geometry-to-Quotient Coupling

Governed work ID: VGSE-GEOMETRY-QUOTIENT-COUPLING-001
Campaign: VGSE-001
Programme: MATH-PROGRAMME
Tracker: #1042
Research phase: fresh bounded successor to completed WP01–WP03

Primary question

What geometry-to-quotient correspondence is actually identifiable from the qualified structure and evidence, and what evidence distinguishes a structural law from arbitrary interpolation?

The target object is a possible coupling

[ \Phi:\mathcal G\rightarrow\mathcal Q, ]

where (\mathcal Q=(\mathbb R_{>0})^8) is the protected WP01/WP02 quotient coordinate space and (\mathcal G) is only the geometry family that the qualified C05 evidence actually supports.

Authority and live bind

This phase is authorized as a bounded mathematical research successor. It does not authorize physical, fabrication, source-correspondence, product, patent, or commercial claims.

Read-only reconnaissance on 2026-09-22 bound this admission to:

  • protected MATH-PROGRAMME/main: 598a13d8a390fcab722b84b4a494921fc50d0e56;
  • kickoff reference 7cfa047c820b08b9a7cd97dae5898bb25fa84b66, verified as the merge-base ancestor of current protected history;
  • terminal predecessor VGSE-ENGINEERING-DISCOVERY-001, still TERMINAL;
  • protected WP01/WP02/WP03 artifacts on current main;
  • protected MATHCERT head 2f27aed33b32b4caf6ff8622c87cfd6d40e97607;
  • protected MATHSOLVE head b0854bb7770296b610b655753bc62b27365b27bb.

No duplicate VGSE WP04 issue, branch, or PR implementing this objective was found before mutation.

Protected starting facts

WP01 removed eight internal-vertex gauge directions and fixed eight positive canonical quotient coordinates. WP02 established an exact global inverse from the positive projective response to those eight quotient coordinates. WP03 established ten exact feasibility relations describing the eight-dimensional positive response image inside the eighteen-dimensional response chart.

These are inputs, not work to reopen.

The C05 certificate is narrower than a generic source-geometry parameterization. It establishes that five retained algebraic witnesses extend to five distinct planar t-embeddings satisfying the prescribed rounded Figure-16 boundary and recorded geometric constraints. In particular, the protected C05 evidence records strict convexity/noncrossing, exact primitive closure, Kawasaki equalities, and certified boundary-angle inequalities.

The protected evidence does not establish VGSE-C06. The visible source geometry is not thereby tied to the pinned-C weight class. Figure-16-to-pinned-C correspondence remains blocked.

Research objects

Keep three levels separate:

  1. Geometry space (\mathcal G). Define exactly the variables, constraints, symmetries, scale status, relabeling/orientation conventions, degeneracies, connected components, and provenance of every admitted realization.
  2. Quotient space (\mathcal Q). Use only the eight canonical positive quotient coordinates except when raw weights are needed to demonstrate gauge equivalence.
  3. Response manifold (\mathcal R). Use the protected WP03 response map only for downstream validation. Matching a response does not prove a geometry coupling.

Do not silently promote a visible drawing or source-vector extraction into a certified parameterized geometry family.

First substantive obligation — evidence audit

Before proposing a preferred map, enumerate every authorized pair ((g,x)) and classify it as:

  • directly protected correspondence;
  • reconstructed correspondence;
  • convention-defined correspondence;
  • GCL-generated design pair;
  • inferred pair;
  • unsupported pairing.

The audit must answer how many genuinely independent geometry/quotient pairs exist. One pair is not evidence of a general law. A structural derivation may still succeed, but the derivation must supply the missing information.

Hypothesis ledger

At minimum test:

  • H0 — no identifiable coupling. Geometry and quotient response may be independent or the evidence may be insufficient.
  • H1 — direct Euclidean length rule. Deliberately test and attempt to falsify simple edge-length/ratio rules. Existing C06 evidence is an explicit adverse boundary.
  • H2 — dimensionless geometric invariants. Length ratios, angle combinations, normalized areas, projective/cross-ratio quantities where defined, local shape descriptors, and symmetry-invariant coordinates.
  • H3 — low-dimensional smooth coupling. If (x=\Phi(z)), measure rank, null directions, singular sets, and conditioning of (D\Phi).
  • H4 — bounded latent-variable coupling. If geometry alone is insufficient, test (x=\Phi(z,\lambda)) only with an explicit, bounded latent dimension and interpretable role.

Identifiability and normalization

For any surviving coupling, determine existence, uniqueness/equivalence class, Jacobian rank, null directions, symmetries, singular sets, conditioning, perturbation sensitivity, coordinate dependence, and held-out validity where evidence permits.

Do not confuse new geometric non-identifiability with the old weight gauge; WP01 already removed that gauge.

Normalization must treat translation, rotation, reflection if allowed, scale, boundary ordering, vertex relabeling, graph automorphisms, and orientation separately. Do not quotient scale unless the admitted mathematics makes it a nuisance symmetry. If normalization choice changes the inferred coupling, retain that dependence as evidence.

Minimum programme

  1. reconstruct the exact C05-admitted geometric object/family;
  2. define a canonical geometry representation;
  3. complete the paired-evidence audit;
  4. instantiate the hypothesis ledger;
  5. deliberately falsify the Euclidean-length hypothesis first;
  6. search for a smaller invariant geometry coordinate system;
  7. test existence and identifiability of maps into the eight quotient coordinates;
  8. compute Jacobian/sensitivity information for survivors;
  9. propagate surviving quotient predictions through protected WP02/WP03;
  10. use WP03 feasibility only as a consistency test;
  11. perform perturbation/held-out tests where independent realizations exist;
  12. preserve negative results.

Synthetic realizations are permitted only under a declared construction rule and must be labelled GCL DESIGN.

Required artifacts

The substantive branch research/vgse-eng-wp04 must retain at least:

  • README.md
  • GEOMETRY_MODEL.json
  • EVIDENCE_MATRIX.json
  • HYPOTHESIS_LEDGER.json
  • RESULTS.json
  • CLAIM_LEDGER.json
  • NEXT_ACTION.md
  • deterministic analysis entry point
  • replay tests
  • environment/requirements lock as appropriate

Exact symbolic maps must be retained separately from numerical approximations. A non-identifiability result should be executable or machine-checkable where possible.

Material acceptance

Green CI alone is insufficient. WP04 must produce an exact definition of the admitted geometry family, a complete pairing audit, at least one deliberately falsified coupling hypothesis, a determination of whether geometry alone identifies any quotient information, rank/conditioning analysis for any survivor, a precise latent-variable or non-identifiability result if geometry is insufficient, structural/held-out validation sufficient to distinguish discovery from interpolation when the evidence allows it, downstream WP02/WP03 consistency replay, and a claim ledger preserving all boundaries.

Terminal outcomes

The phase terminates at one evidence-supported disposition:

  • GEOMETRY_QUOTIENT_COUPLING_ESTABLISHED
  • GEOMETRY_QUOTIENT_COUPLING_PARTIAL
  • GEOMETRY_REQUIRES_LATENT_METRIC_VARIABLES
  • GEOMETRY_QUOTIENT_NON_IDENTIFIABLE
  • CANDIDATE_COUPLINGS_FALSIFIED

A decisive negative result is a successful research outcome. Do not search indefinitely for a positive map.

Claim boundary

WP04 may establish mathematical facts about admitted geometry, quotient variables, identifiability, and coupling within its declared model.

It does not establish Figure-16-to-pinned-C source correspondence, VGSE-C06, stiffness, mechanical advantage, rigid foldability, collision freedom, finite thickness, material/friction/hinge behaviour, manufacturability, tolerance, fatigue, product performance, novelty, priority, patentability, or commercial value.

A mathematical weight remains a mathematical weight unless a separately justified physical model establishes otherwise.

Review and protection

Substantive completion requires exact-head deterministic replay, distinct read-only Adversary and Referee reviews, required status checks, native merge-queue admission, and exact protected-main readback. Any substantive candidate-head mutation after review invalidates those reviews.

No WP05 is automatically authorized.