[petalas]

Summary

Scoped proof with finite exact corroboration: classify when one invertible native address map can realize three ordered frame-subspace correspondences, and count every realizing map. This addresses the algebraic shared-G consistency question in Cao Yifan's merged #4610 package. The classical three-subspace normal form is credited as background, without a literature-priority claim.

compatible exactly when individual, pairwise-intersection,
triple-intersection and total-sum dimensions agree
realizing maps = q^R |GL_t(F_q)| product_S |GL_(m_S)(F_q)|
compatibility types in dimension n = sum_t binomial(n-2t+7,7)
dimension-nine types = 15,792

The proof gives the multiplicities and R explicitly, constructs the simultaneous decomposition over every field, and derives the finite-field stabilizer count from a block-triangular endomorphism algebra. This contribution was produced by Nick Petalas. Credit Cao Yifan, Cinna Davis, Dugongue, zxv, Jacob Alan Sussman and the retained native/compiler lineage. Independent human and formal proof-kernel review are unclaimed.

Evidence

• Before: matching each subspace dimension separately can admit impossible shared maps. After: the explicit criterion rejects both a coplanar/independent three-line mismatch and a triple-intersection mismatch with unchanged pairwise dimensions and total sum.
• Complete literal GL orbit enumeration covers all 4,958 ordered triples in F2 dimensions 0 through 3 and F3/F5 dimension 2. It yields 248 orbits and checks every orbit's realizing-map multiplicities. The F2 dimension-three portion has 4,096 triples and 128 orbits. The classifier and the literal matrix orbit construction use different acceptance routes; no predecessor executable was replayed.
• All 15,792 dimension-nine normal-form tuples are enumerated, reconstructed from their dimensions and bound to an ordered digest. This is not a literal nine-dimensional subspace or GL census.
• Both warning-strict registered checks pass locally on CPython 3.12.15/macOS arm64, approximately 0.46 seconds each. These are timeout measurements, not performance comparisons. Seven invalid-input controls and both incomplete-invariant examples pass.
• Package metadata and exact committed-byte/source-history preflight pass at 15e8433c431aaaf7fa3756407d44f4700fc8377a. The official committed export passes integrity checking; isolated optimized Git-free replay passes. Proof-byte drift, predecessor-source drift, Boolean receipt mutation and occupied receipt output reject. All nine exported files remain unchanged after the controls. RESULT.json was generated by the checker and binds six local source hashes.

Exact reproduction from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/three-frame-shared-map/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/three-frame-shared-map/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/three-frame-shared-map
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/check_research_commit.py --package research/three-frame-shared-map --check-source-history

Expected direct output is PASS_THREE_FRAME_SHARED_MAP with triples=4958, orbits=248, dimension_nine_orbits=15792. A clean CPython 3.12 executable can replace the local absolute path. The standard library is sufficient. SOURCES.json pins main 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7, and the predecessor README. The README is authenticated, not executed. The selected source-binding orientation passed; no selected supplier replay was run.

The current main guidance, live claims/open PRs and distinctive merged-package searches were checked before claiming and committing. Cached overlap covers open PR bodies and discussions but omits most sparse local notes; separate Git searches covered main READMEs. These searches establish coverage, not proof of novelty. The original card's fixed-decoder premise is explicitly changed to an algebraic interface side result in TASK.json.

Limitations and unrun obligations

The universal conclusion rests on the written proof. This is not a complete multiplication theorem or supplier admission. The theorem covers subspace-image equalities for at most three ordered subspaces. Prescribed vector actions, more subspaces, symplectic/dual constraints and absolute phases can add restrictions. No actual prelude is rebound, no native word or paid profile changes, and no new kappa is claimed. Physical realization, restoration, ownership, hosting, precision/tape, compiler and outer admission remain separate obligations. Lean, Docker, historical suites and the selected cold replay were not run. The selected frontier is unchanged.

Open next questions

1. Bind the criterion to one actual PR4610 prelude/late shared-G contract. First determine which contacts prescribe vector values and which prescribe only subspace images. Additional vector/phase requirements are the current blocker; a completed paid invoice is needed before claiming a gain.
2. Identify the relative-operator invariant needed for a four-subspace contract, starting with complementary planes and two graph subspaces. Three-subspace dimension data do not classify that domain, and no source-compatible four-contact contract is supplied here.

Related open work

• Add phase-exact native Clifford compiler and complete contact neighborhoods #4841 constructs phase-exact native Clifford compilation and chart neighborhoods. This result classifies unrestricted GL subspace correspondences and does not discharge its phase or compiler contracts.

• Prove exact four-frame concave subspace pricing (no new κ) #4866 optimizes a three-contact junction price in a schematic full-subspace metric. This result instead classifies common invertible maps and counts their realizers; no competing numerical quantity is compared.

• Prove exact cycle-constrained chart neighborhoods (no new κ) #4864 counts cycle-constrained simultaneous transvection chart outputs. Its symplectic axis fibers differ from unrestricted finite-field GL realizing maps.

• Count and index shortest binary chart words (no new κ) #4859 counts and indexes shortest axis words between binary Lagrangian charts. This package counts arbitrary realizing matrices for ordered subspace triples, without a shortest-word or phase claim.

• Prove exact orbit weights for ordered idempotents (no new κ) #4915 counts simultaneous centralizers and orbits of ordered idempotent operators. This result counts stabilizers and realizing maps for ordered subspace triples; the objects and enumerated domains differ, with no shared numerical quantity computed.

The completion refresh inspected all 81 open PRs on this account and the scopes of the nine other open submissions created since the CLAIM at 2026-10-22T23:39:58Z. The adjacent stabilizer-count result #4915 is distinguished above. The other new submissions concern bipartite/full-mask query capacity, tensor-error dimensions, fixed-basis routing, square-zero compilation, a quadratic inverse bound, ring-valued fanout and signed unitriangular tariffs. None reports a competing three-subspace realizing-map count. No additional scientific computations were run to audit those results.

CI notes

Both hosted checks passed at exact contribution head 15e8433c431aaaf7fa3756407d44f4700fc8377a in run 39874083898: planning in 4m47s and registered package replay in 1m11s. Run metadata confirms the matching head, completed status and successful conclusion. The checks were read once at completion. The registered replay executes both warning-strict normal and optimized commands.

Local package metadata/replay, committed-byte/source-history preflight, official export integrity and optimized Git-free replay also passed. Hosted CI certifies execution of these bounded checks; independent human and formal proof-kernel acceptance of the all-dimension theorem remain unclaimed. No CI failure or locally unrun registered check is claimed.

Merge danger

Door: two-way. Blast radius: package. Seven files are added in one new research directory. Shared code, predecessor packages and the selected frontier are unchanged.