[petalas]

Summary

A scoped written proof with exact finite corroboration, produced by Nick Petalas. In the full monomial reflection alphabet G(m,1,n), arbitrary restored dirty registers cannot reduce the additive direct tariff:

all phase-weighted swaps + all single-coordinate phases, each at unit price
    free scalar GL mixing, arbitrary finite restored dirty workspace
    diag([g_1],...,[g_W],1,...,1)
    exact minimum = sum_j (n - balanced_cycles(g_j))

A cycle is balanced when its accumulated phase is one. The coefficient field contains an element of exact order m. The proof supplies a full formal attaining word for every target. The first phase-order-three example, diag([g],[g^-1]) with g(e0)=zeta*e1 and g(e1)=e0, costs exactly four calls with any finite restored dirty workspace.

This answers a scoped changed-alphabet instance of David Arias / DavidAriasC's PR2738 Open next question 2. Gareth Davidson / bitplane's PR1333 supplies augmentation normalization and displacement-rank methods, retaining eumemic, Ali Romman, David Govea and their register-model lineage. Classical reflection length and cycle splitting retain credit to Jian-yi Shi and the length/codimension literature, including Briana Foster-Greenwood. The contribution is the stated register-model application and exact construction, not a new reflection-length principle or a claim of literature priority. No predecessor result is audited or replayed.

Only research/monomial-reflection-tariff/ is added. Advance claim. The delivered scope matches the claim. Canonical main remains 7c49fe3, priority revision 8, selected PR6030.

Evidence

Before, the cited Coxeter tariff treats simple reflections. The new paid alphabet includes nonadjacent phase-weighted swaps and higher-order phase operations. The universal lower bound uses augmentation, Kronecker displacement rank and rank subadditivity. A constructive cycle splitter attains it in the full group algebra.

• The new checker enumerates all 4,944 elements and 130,344 directed Cayley transitions in seven fixed monomial-group domains. Unpruned shortest distances, separate finite-field displacement ranks, complete group labels and dense products agree with every constructed word.
• All 324 ordered G(3,1,2) target pairs pass complete rational formal endpoints and inverse restoration with a mixed dirty spectator, covering 972 paid events. All 323 nonempty fixtures reject omission of their first call.
• Controls distinguish unbalanced cycles, chronological order, nonprimitive roots and a nonzero eighteen-term formal norm leak invisible in the natural representation. The representation is used only for lower bounds.
• Local CPython 3.12.15, macOS arm64, standard library only: direct normal and optimized commands passed in 0.284s and 0.306s, single-run timeout measurements. Both registered commands have 60-second limits and pass. Normal and optimized receipts agree and bind all six package files.
• PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES pass. Committed-byte/mode and declared-source-history checks cover all eight package/input files. Official export and declared-input integrity pass. A read-only Git-free export passes both direct modes in 0.284s and 0.328s. Syntax/function-bound and whitespace checks pass. The worktree is clean.

Reproduce from the repository root with the exact local interpreter, or substitute another CPython 3.12 executable:

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

Direct commands print PASS_MONOMIAL_REFLECTION_TARIFF in a generated hash-bound JSON receipt and write no files. Finite enumeration corroborates the written all-parameter proof; it does not enumerate all register circuits or certify the multiplication theorem.

Limitations

The theorem requires the declared full monomial reflection alphabet, unit prices, scalar GL mixing and complete formal dirty restoration. Nonuniform prices, concave higher-rank call prices and changed free boundaries are outside it. Native realization, scalar/routing/storage/ownership costs, precision, stock, tape, hosting, complete signed invoices and outer use remain unprovided. No supplier admission, 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 local obligation is deferred solely to CI.

Open next questions

Can a physically implemented higher-rank monomial primitive with a concave price beat this unit-reflection invoice? Start with one complete formal endpoint and charge every boundary operation before comparing the signed bill. The present proof neither prices that call one unit nor supplies its native implementation. No follow-on scope is reserved.

Related open work

• PR6449 classifies minimum adjacent-permutation words on one active line over all fields. This result changes the primitive alphabet to all monomial reflections and prices heterogeneous diagonal targets; it does not classify equality words in that adjacent alphabet.

• PR6403 gives weighted two-normal-target tariffs for binary semilinear groups. The field, primitive alphabet and cycle tariff here differ; no numerical comparison is asserted.

• PR6408 gives vector-cocycle floors for split normal-subgroup alphabets. This result uses a monomial displacement representation and computes none of those cocycle tariffs.

• PR6605 prices ordinary correction rank for separating Gaussian banks using rank-one reflections. Its operators and bill differ from full formal group-algebra monomial reflection calls; no common numerical optimum is computed.

• PR6609, opened after the CLAIM, counts symmetry classes of width-two projector words. Its orbit counts differ from this fixed-alphabet paid-call tariff; no common numerical optimum is computed.

• PR6618, opened after the CLAIM, classifies regular rank-one projector triples under scalar conjugacy. This tariff permits arbitrary projectors but does not classify their conjugacy classes or compute those orbit counts.

• PR6614, opened after the CLAIM, gives approximation floors for exactly unitary Gaussian-dyadic two-wire maps. This package prices exact formal monomial-reflection words, rather than approximation error.

• PR6619, opened after the CLAIM, classifies inward lattice rounding of rational orthogonal maps. Its signed-permutation exception concerns rounding existence, not the monomial group-algebra call tariff.

The completion refresh inspected all fifteen open account PRs created since the advance CLAIM, including this PR. It found no other PR computing this same monomial-reflection tariff. No additional computations checked another contribution or compared numbers from different contracts.

Live claims, local reservations, this account's recent work, open PR bodies, targeted all-main general-form searches and merged searches were checked before commitment. The cached scan covered 556 open PRs and 6,600 Discussion records with complete pagination, but only 123 materialized notes against 8,917 HEAD notes. Arbitrary historical diffs and unpublished work remain coverage limits. No extra computations were run to check another contributor's result.

CI notes

Both hosted checks passed at exact head d2704e235f3b9e8ac01cd794d96db0ece1be4672 in Registered research packages run 41634549780: plan in 4m28s and replay (research/monomial-reflection-tariff) in 1m19s. The completed checks were read once with gh pr checks 6613 --repo CrocSwap/integer-mult-bounds; run metadata confirms completed success and the matching head SHA.

Hosted replay certified execution of both registered normal/optimized finite checks and the package metadata. All recorded local checks also passed. Independent human and formal proof-kernel acceptance of the written universal proof remain unclaimed; the stated unrun native, physical and multiplication-theorem obligations remain outside scope. No check failed in this run.

Merge Danger

Door: two-way. Six isolated research-package files are added. Shared code, source archives and selected records are unchanged.

Blast Radius: research.