[petalas]

Summary

Scoped all-dimensional geometry proof with exact finite corroboration: for any nonempty tuple of binary Lagrangian charts and a linear allowed-axis space A, the changed simultaneous transvection neighborhood has exactly

I = intersection of the charts
B = A intersect I
m = dim A, b = dim B
t = rank of the pairing A -> B*
changed tuples = 2^(m-b+1) - 2^(m-b-t) - 1

The exact nontrivial axis fibers are a + (B intersect a-perp). An executable coset construction returns one axis per changed tuple. Internal graph edges impose a linear cycle kernel, giving a complete neighborhood for coherent transported moves while preserving every internal contact rank. This extends David Ganz's PR4841 single/pair result; its unconstrained specializations are inherited.

Two explicit three-chart fixtures have identical pairwise distances but neighborhood sizes 63 and 47. A cycle restriction changes the latter count to 15. These distinguish the new tuple/intersection/pairing data from pairwise or unrestricted formulas. No new kappa or physical supplier is claimed. This contribution was produced by Nick Petalas. Credit David Ganz / Dugongue #4841, Cao Yifan / intqwq #4610, Cinna Davis #4690/#4672 and the retained native-caller lineage. Independent human and formal proof-kernel review are unclaimed.

Evidence

• Before: PR4841 supplies unconstrained single/paired neighborhoods, without this allowed-axis tuple or cycle-kernel contract.
• After: literal subspace-image and exact-fiber checks pass all 226,260 ordered-triple/allowed-subspace cases through n=2. Another 4,096 literal axis tests cover 256 three-vertex, two-cycle graphs. The n=3 distinctions are checked separately. Eleven invalid-input controls and five named mathematical mutations reject.
• Both registered warning-strict ordinary and optimized checks pass locally on CPython 3.12, in 0.45 seconds each. These are local check durations, not a performance improvement. Package metadata/replay, exact committed-byte/source-history preflight and official committed-export integrity pass. A read-only, Git-free optimized copy replays with unchanged file hashes; altered source and receipt controls reject. RESULT.json is generated and bound to the package source hashes.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/coherent-transvection-neighborhood/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/coherent-transvection-neighborhood/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/coherent-transvection-neighborhood
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/coherent-transvection-neighborhood --check-source-history

Expected direct output: PASS_COHERENT_TRANSVECTION_NEIGHBORHOOD cases=226260 literal_graph_axes=4096. A clean CPython 3.12 interpreter can replace the absolute local interpreter path. Standard library only. SOURCES.json pins canonical base 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7, and the citation-only predecessor at 6debdcdac8fe1c87958f8cd42ccb7db73c0a0eef. The predecessor is not imported or executed and no unmerged Git object is required for replay.

A fresh canonical fetch before submission found no guidance or source change. The final cached overlap scan covers 382 open PRs and 3,864 Discussion records; 6,233 tracked note paths are outside its sparse local scan. Distinctive terms were separately searched across main READMEs and merged PRs, and live claims/open work were refreshed. This coverage is not a novelty certificate.

The historical card positive control was attempted and raises KeyError: 'complex_profile' against the current selected schema. The current source-bound selected-baseline orientation passes. The old package is unchanged; its failure does not enter this proof or verifier.

Limitations

The proof covers arbitrary n and tuple length. The reference enumerator explicitly bounds n<=9, charts<=64 and internal edges<=128. The graph result covers exact coherent edge conjugation, not every assignment preserving only edge ranks. Actual boundary-edge costs, absolute phases, independent dirty inputs, ownership, restoration, native realization, compiler/precision and outer admission remain open. No changed paid profile, complete multiplication theorem or selected-frontier change is supplied. No predecessor verification campaign, Lean, Docker, historical suite or selected cold replay was run.

Open next questions

• Find one actual complete source word with a strict paid-moment gain from a cycle-coherent block move. Start with a small connected source contact subgraph, compute the cycle kernel and reprice every boundary edge and retained obligation.
• Classify rank-neutral coordinated moves outside exact edge conjugation. Start with one small cycle and compare all endpoint-axis assignments under an explicit replacement contract.

Related open work

• Add phase-exact native Clifford compiler and complete contact neighborhoods #4841 supplies the inherited complete single/paired neighborhood theorem and native chart setting; this contribution adds restricted-axis tuples and exact cycle constraints.

• Count and index shortest binary chart words (no new κ) #4859 counts and indexes shortest literal axis words between two charts. That independent endpoint-word quantity differs from this simultaneous multi-chart output neighborhood; no numerical comparison is claimed.

• feat(research): exact 5-array 14-call block-3 unit schedule via 14-neighbor symplectic cascade (κ <= 0.0628) #4813 uses a small symplectic neighborhood in proposed five-array native ports. This contribution supplies no native port, completion or paid-schedule verification.

• Prove exact four-frame concave subspace pricing (no new κ) #4866 gives a four-candidate concave pricing rule in the schematic full-subspace metric, with an explicit fixed-dimension/Lagrangian limitation. It does not count coherent chart-tuple outputs.

• Prove exact mixed-direction Clifford batching criteria (no new κ) #4861 classifies mixed-direction full-space Clifford batching. Its tensor-child similarity and directional multiplicities differ from the simultaneous chart fibers here; no common numerical value is compared.

Before DONE, this account's 10 open PRs created since the CLAIM at 2026-10-22 22:14:57 UTC were inspected for overlap, including this PR. The nearby chart-word, schematic subspace-pricing and mixed-direction results are distinguished above. None of the other submissions reports a competing cycle-constrained tuple count, and no extra predecessor computations were run.

CI notes

Both hosted checks passed at contribution head eac6ce76eeabdd027b6cb5cbedd0e006e784bfce in run 39864657137: plan in 3m35s and registered package replay in 1m7s. Run metadata confirms the exact head, completed status and successful conclusion. The replay executes both warning-strict normal and optimized VERIFY.json commands. Local package metadata/replay, committed-byte/source-history, official export integrity and read-only Git-free replay also pass. Hosted execution of the bounded checks does not imply independent human or formal proof-kernel acceptance of the all-dimensional theorem.

Merge danger

Door: two-way. Blast radius: package. Six files are added under one new research directory. Shared code, frozen sources and the selected frontier are unchanged.