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VGSE-ENG-WP06 — Boundary Calibration Identifiability

Governed work ID: VGSE-BOUNDARY-CALIBRATION-IDENTIFIABILITY-001
Campaign: VGSE-001
Tracker: #1053
Protected start: 797139a7aef1a445461e24151493885be38159f1

Question

Are the five boundary calibration directions missing from WP05 recoverable from admitted embedding geometry alone?

WP06 must distinguish two very different statements:

  1. Fixed-response catalogue. The five protected C05 branches were generated from one already-fixed quotient/response. Identifying the branch from its geometry does not constitute recovery of the quotient.
  2. Structural inverse. Given the labeled embedding geometry and graph, but not the known quotient, algebraic branch witness, or source correspondence, determine whether the quotient is unique.

Only the second answers the geometry-to-quotient question.

Source-definition gate — TE3

The primary source defines the geometric edge weight of an embedding as the Euclidean length of the corresponding dual edge, and defines gauge equivalence using a positive vertex function fixed to (1) at every boundary vertex. A source-defined t-immersion/t-embedding must satisfy TE3: the original graph weights and these Euclidean geometric edge weights are gauge equivalent.

Therefore WP06 must test TE3 before interpreting boundary calibration as missing geometry.

For each retained C05 branch:

  1. compute all sixteen Euclidean dual-edge lengths from the protected embedding coordinates;
  2. solve the exact/log-linear internal-vertex gauge system with all six boundary gauges fixed to (1);
  3. report residuals edge by edge;
  4. compare against both the MATHSOLVE numerical representative and the exact MATHCERT representative (which are gauge-equivalent and must give the same quotient-level answer);
  5. determine whether C05's phrase “planar t-embeddings” is source-definition conformant.

The decision fork is strict:

  • if TE3 passes, geometry already determines the full weight quotient via the Euclidean length class, and the WP05 five-boundary-factor obstruction was an artifact of using the algebraic primitive factors rather than the source-defined geometric weight map;
  • if TE3 fails, the retained C05 objects are planar algebraic Kenyon–Smirnov realizations with the recorded convexity/Kawasaki properties, but the evidence currently does not justify calling them t-embeddings in the source-defined sense. In that case, open a correction path for C05 and analyze boundary rescaling only for the broader algebraic-realization class.

Do not preserve the existing label merely for continuity if TE3 falsifies it.

Candidate boundary-rescaling symmetry

For each degree-one boundary vertex (U_i), let (eta_i) denote its discrete-holomorphic boundary factor and let (k_{B_i}>0) be its boundary-edge weight.

Test the positive rescaling

[ eta_imapsto c_ieta_i,qquad k_{B_i}mapsto c_i^{-1}k_{B_i},qquad c_i>0, ]

with the complementary boundary datum rescaled inversely so that the prescribed boundary edge product remains fixed.

The first obligation is to determine exactly whether this transformation preserves:

  • every primitive edge increment;
  • every interior discrete-holomorphic equation;
  • the prescribed boundary geometry;
  • positivity and Kasteleyn signs;
  • the C05 mathematical embedding conditions that do not already encode the known quotient.

If it does, compute its induced action on the WP05 invariant basis.

Required quotient-rank test

WP05's five path basis rows have boundary exponents

[ (-B_1-B_2),quad (-B_1+B_3),quad (-B_1+B_4),quad (-B_1-B_5),quad (-B_1-B_6). ]

Under boundary rescaling, compute the exact linear map from

[ (log c_1,ldots,log c_6) ]

to the five logarithmic path coordinates.

Determine its rank and kernel exactly.

If the rank is five, then one fixed embedding admits a five-dimensional family of gauge-inequivalent quotient points unless an independently justified boundary normalization removes the symmetry.

Anti-circularity test

A proposed recovery of the boundary factors is inadmissible as geometry-only if it consumes any of:

  • the protected WP01 quotient values;
  • the pinned response matrix or its arrangement forms;
  • the algebraic branch witness (x,y);
  • a lookup table over the five retained branches;
  • source-to-pinned-C correspondence not independently established.

It is permitted to use these objects only to check a geometry-derived formula after that formula has been obtained independently.

Branch-coordinate trap

The five C05 embeddings are distinct, so an interior coordinate may identify which of the five algebraic roots generated a branch. WP06 must not confuse this with structural quotient recovery.

If the map “geometry → branch label → already-known quotient” is the only full reconstruction available, classify it as circular for the structural inverse question.

Terminal outcomes

Terminate at one of:

  • BOUNDARY_FACTORS_GEOMETRY_IDENTIFIABLE
  • BOUNDARY_FACTORS_IDENTIFIABLE_WITH_CANONICAL_NORMALIZATION
  • BOUNDARY_CALIBRATION_FIVE_DIMENSIONAL_OBSTRUCTION
  • BOUNDARY_CALIBRATION_PARTIAL_IDENTIFIABILITY
  • BOUNDARY_CALIBRATION_UNRESOLVED

A negative identifiability theorem is a successful outcome.

Required artifacts

Substantive branch research/vgse-eng-wp06 must retain at least:

  • BOUNDARY_SYMMETRY.json
  • QUOTIENT_ACTION.json
  • IDENTIFIABILITY_PROOF.md
  • NORMALIZATION_AUDIT.json
  • RESULTS.json
  • CLAIM_LEDGER.json
  • deterministic replay and tests.

Claim boundary

WP06 concerns mathematical identifiability only. It does not establish VGSE-C06, source correspondence, mechanics, rigid foldability, collision freedom, finite thickness, material behaviour, manufacturing, product performance, patentability, or commercial value.

Review and protection

Completion requires exact-head deterministic replay, distinct Adversary and Referee reviews, continuity rebind, required checks, native merge queue, protected-main readback, terminal checkpoint update, and tracker closure.