[petalas]

Summary

Scoped algebraic proof with finite corroboration: two binary Lagrangian charts in dimension 2n, at distance r, have exactly the following minimum-length words.

minimum length        r
chart paths           P(r) = product_(j=1)^r (2^j-1)
literal axis words    W(n,r) = 2^(r(n-1)) P(r)

An exact lexicographic rank/unrank bijection indexes every shortest literal word. A uniform integer index therefore gives a uniform word. The written proof covers all n>=1; the bounded reference implementation accepts 1<=n<=6. This adds shortest-word counting/indexing to Dugongue's #4841 single/paired-neighborhood work; no predecessor executable is replayed. The underlying finite symplectic geometry is classical, with no literature-priority claim. No new kappa is claimed.

This contribution is produced by Nick Petalas. Credit Dugongue #4841, Cinna Davis #4690, Cao Yifan #4610, Yiyang #4710, the retained native caller #4713/#4630/#4635, and their predecessor notices. Independent human and formal proof-kernel review are unclaimed.

Evidence

• Before: the cited single/paired-neighborhood result supplies one-step chart moves, without this package's shortest-word count and indexing contract.
• After: independent intersection-graph BFS covers every ordered pair of all 3, 15 and 135 charts at n=1,2,3, totaling 18,459 pairs. The literal checker exhausts 28,386 minimum-length axis tuples through n=2, including 1,644 successful words, and checks every successful word's exact rank/unrank. There are 58 larger round trips through n=6 and eleven rejection controls.
• Warning-strict normal and optimized registered checks pass locally on CPython 3.12.15, macOS arm64, in 5.39s and 5.41s. Both produce identical generated receipts bound to all six package files. Package metadata/replay and exact committed-byte/source-history preflight pass. Committed export integrity and a Git-free replay pass. An independent read-only proof/code review found no correctness gap.

From the repository root:

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

Expected status: PASS_SHORTEST_CHART_AXIS_WORDS. A clean Python 3.12 interpreter can replace the exact local path. Standard library only. Source provenance is canonical main 624de9dc66858af0945e5f8a0942488d62280d43, revision 7; submission fetch found no guidance/source change. The cached card scan covers 379 open PRs and 3,855 Discussion records; 6,194 tracked local note paths were outside its sparse coverage. Live open PRs, recent claims, merged-term searches and distinctive all-main README searches were checked again before committing. Search coverage is not a novelty certificate.

The historical card control raises KeyError: 'complex_profile' against the selected schema. The current research_selected_baseline.py source-binding control passes. This orientation failure does not enter the new theorem or verifier.

Limitations

The count concerns syntactically distinct nonzero-axis sequences and chart paths. Distinct full symplectic products and physical phase lifts can have different equivalence classes. This is not a native supplier, a complete multiplication theorem or a selected-frontier update. Paid profile, other incident edges, dirty restoration, ownership, precision, compiler and outer admission remain unrun. The code provides an exact indexing contract rather than a random-number generator. No Lean, Docker, historical suite or selected cold replay was run.

Open next questions

• Avoid scanning every ambient axis by constructing pairing-one quotient axes directly. Start with a canonical basis and an explicitly declared ordering; practical nine-bit indexing remains unimplemented.
• Classify distinct full symplectic products of shortest chart words. Start by enumerating product matrices for transverse n=2 endpoints; product/phase equivalence is outside the delivered contract.

Related open work

• Add phase-exact native Clifford compiler and complete contact neighborhoods #4841 supplies complete single/paired chart neighborhoods and the corrected one-step tariff; this contribution counts and indexes shortest paths/words between two endpoints.
• Prove exact cycle-constrained chart neighborhoods (no new κ) #4864 supplies exact cycle-constrained simultaneous multi-chart neighborhoods. This contribution instead counts minimum-length axis sequences between two fixed endpoints; the neighborhood cardinalities and shortest-word counts are different quantities.
• Prove exact four-frame concave subspace pricing (no new κ) #4866 optimizes a three-contact junction in the full-subspace dimension metric, with an explicit fixed-dimension/Lagrangian limitation. It supplies no competing shortest Lagrangian-word count.
• Prove exact mixed-direction Clifford batching criteria (no new κ) #4861 classifies mixed-direction full-space Clifford batching. Its tensor-child similarity and multiplicity conditions differ from chart-path and literal-axis-word multiplicities; no common numerical quantity is compared.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 22:15:35 UTC were refreshed. Ten open submissions, including this one, were inspected for overlap. The three nearby contributions above were read in full. The remaining submissions concern permutation workspace, parked volume, root-code floors, dirty-echo trees, cyclic checkpoints and XOR subsequences. No competing shortest-chart-word count was found, and no extra predecessor computations were run.

CI notes

Both hosted checks passed on contribution head 45352b400f986482a5f9cc16ce76292da5f6bcc3 in Registered research packages run 39864380398: plan in 3m41s and the registered research/shortest-chart-axis-words replay in 1m37s. Run metadata confirms the exact head, completed status and successful conclusion. The replay executes both warning-strict normal and optimized VERIFY.json commands. Local metadata/replay, exact committed-byte/source-history checks, official export integrity and read-only Git-free replay also passed. Hosted CI certifies execution of these bounded checks, not independent human or formal proof-kernel acceptance of the all-dimensional theorem.

Merge danger

Door: two-way. Blast radius: package. All changes are confined to one new research directory; frozen sources and the selected frontier are unchanged.