[petalas]

Summary

Scoped proof with exact finite corroboration, produced by Nick Petalas. In the full formal binary group algebra of H(m,n), charge one call for any vector label x(u) or y(v) coupled to a rank-one binary register matrix. For arbitrary nonzero central target matrices Z_t on independent dirty roles, the exact minimum is:

K = sum_t rank(Z_t), P = number of nonzero targets
minimum calls = 2(K+P)
rank-two target I2: six calls, rather than eight separate commutator calls

The theorem allows free invertible scalar mixing, nilpotent rank-one couplings and every finite number of exactly restored dirty spectators. An explicit closed vector walk attains the bound in the full group; natural and contragredient block representations give matching lower bounds.

Credit Cinna Davis / 1337isnot1337 PR2258 for the coordinate-edge H(m,n) model and rank-nullity strategy, retaining the eumemic / woklerx and UT3/UT4 lineage. This independently posed question changes the alphabet to arbitrary unit-price vector labels and allows arbitrary central matrices. It does not audit or replay the predecessor. Rank factorization, unitriangular representations and rank-nullity are classical; no literature-wide priority claim is made.

Advance CLAIM. Only research/heisenberg-vector-central-tariff/ is added. Canonical main remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030.

Evidence

• Before, two separate rank-one commutator words cost eight calls for the rank-two central matrix. The new construction takes six, with a uniform lower bound proving optimality despite arbitrary restored dirty workspace.
• All 682 central matrices for 1<=m,n<=3 have exact Cayley minima matching the formula. The nine complete Cayley domains contain 37,448 elements. Independent minor ranks agree with constructive rank factorizations.
• The rank-two target with a restored second dirty role has 54 allowed generators. Full-formal BFS layers are 1, 54, 1,971 and 64,044. Every one of the 2,026 radius-two join queries has no radius-three hit, excluding all normalized words of length at most five. The six-call witness passes the same join adapter at split 3+3.
• Five four-role formal endpoint fixtures cover heterogeneous ranks through rank four, zero targets and dirty spectators. All 56 single-call omissions, four invalid matrix inputs and a chronology mutation are rejected. Reversal passes as an inverse control.
• Warning-strict normal and optimized direct replays pass in 0.65s and 0.70s on CPython 3.12.15, macOS arm64, standard library only. These are individual check durations, not performance claims. Generated source-hash records agree exactly. Both registered commands pass locally, each with a 120-second timeout.
• Package metadata/pins, committed-byte/mode/source-history preflight, official committed export integrity and a read-only Git-free optimized export replay with empty PATH pass. AST compilation, function-size inspection and whitespace checks pass. No local mathematical check is left solely to CI.

Reproduce from the repository root with the local interpreter below, or substitute any CPython 3.12 executable:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B -W error research/heisenberg-vector-central-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -B -W error research/heisenberg-vector-central-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/heisenberg-vector-central-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/heisenberg-vector-central-tariff --check-source-history --check-modes

Expected outputs include PASS_HEISENBERG_VECTOR_CENTRAL_TARIFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier prints a generated JSON record bound to the verifier and proof hashes; no receipt is hand-written. The universal claim rests on the written proof, with bounded finite corroboration.

Limitations and unrun obligations

An entire vector label costs one by hypothesis. Producing it from coordinate operations, physical scalar mixing, native realization, source access, ownership, hosting, scheduling, precision/tape, compiler closure, complete recursive/outer invoices and recovery remain unproved. No physical supplier saving, new kappa or selected-frontier change is claimed. All five selected framework assumptions remain.

Independent human and formal proof-kernel review are unclaimed. Lean, Docker, predecessor scientific suites and the selected cold replay were not run. No additional static analyzer was available; strict interpreter warnings and source compilation were run. Main-tree, live and cached overlap coverage is recorded in the README; arbitrary historical diffs and unpublished work remain unknown.

Open next questions

• Determine the exact tariff for support-weighted vector labels. Begin by charging every coordinate constituent of the six-call rank-two word; a native vector-label cost model is missing.
• Bind that word to an actual BIT source/consumer and price every dirty return and boundary operation. No physical label mapping or complete favorable signed bill is supplied here.

Related open work

• PR6410 prices central UT4 targets under a fixed three-letter alphabet using cubic coefficients. This result concerns class-two H(m,n), arbitrary vector labels and rank-dependent central matrices; it computes no UT4 tariff.
• PR6408 gives first-order vector-cocycle floors. Central targets can lie in that invariant's kernel; this result uses second-order central blocks with rank-nullity and computes none of its semidirect-product tariffs.

The completion refresh inspected the thirteen other account open PRs created since the advance CLAIM, #6649 through #6661. None studies central matrices in H(m,n) with unit-price vector labels or computes a shared target value. PR6655 classifies shortest formal braid words with a permutation endpoint; its paid alphabet and endpoint differ from this central-matrix tariff. No extra computations were run to compare other contributions.

CI notes

Both hosted checks passed at exact head db7e05ade23b7ff2a5176ca0dd4bb6556446d038 in run 41660389339: plan in 6m42s and replay (research/heisenberg-vector-central-tariff) in 3m29s. The completed check list was read once, and the PR still has that head.

Hosted replay confirms execution of the registered normal and optimized finite checks and package metadata validation. All local scientific, publication/source-history and export checks also passed. The universal written proof remains outside a formal kernel; physical label production and supplier obligations remain open. The delivered mathematical scope matches the advance CLAIM. No failed or waived check remains.

Merge danger

Two-way door. The change adds one isolated research package. Shared code, frozen predecessors and selected certificates are unchanged.