[petalas]

Summary

Scoped algebraic construction and sharp lower bound, with exact finite corroboration. For every positive odd axis n, including composite axes:

g = gcd(a,n), m = n/g
K_b[j,k] = exp(2 pi i (a(j-k)^2+b(j-k))/n)
K_b = phased replication * reduced m-dimensional chirp * phased fold
rank(K_b) = m
minimum number of phase-twisted K_b terms in an invertible sum = g
S = sum_(b=0)^(g-1) K_b:  S* S = n g I,  S^-1 = S*/(n g)

The factorization is optimal among all complex linear intermediate spaces. The g-term repair has condition number one. For n=9,a=3 the intermediate dimension and repair count are both three; for n=45,a=15 they are three and fifteen.

This adds one algebraic interface component for David Ganz / Dugongue's PR5564, Open next question 1. A complete composite-axis Gaussian caller remains missing. Character orthogonality and Gauss-sum algebra are classical; their global novelty is unclaimed. The explicit folding/replication interface, constrained repair construction and bounded reference implementation were produced by Nick Petalas. The predecessor's retained contributors and source credit remain unchanged.

Advance CLAIM. The delivered scope matches the claim. Only research/composite-chirp-phase-repair/ is added. No new kappa or selected-frontier change.

Evidence

• Before: PR5564 constructs prime-free coprime axes and exact CRT streams but leaves their numerical caller open.
• After: this package supplies a precise phase-only convolution contract, rank-optimal reduction and the sharp phase-repair count. It does not identify that operator with an existing real Gaussian resampler.
• Every a,b modulo every odd n through 31 passes: 5,456 factorization cases, 3,344,528 exact phase entries and 1,225 complete basis-vector checks through n=9.
• A separately represented oracle checks 127 literal finite-field matrix ranks at odd n<=15 and b=0,1; 16,255 complex Gram entries by integer cyclotomic reduction; every phase subset at n<=15 and g<=9, 890 subsets in total; and 8,128 inverse entries across all 64 repair families at odd n<=15. No floating-point rank or tolerance is used.
• All 17 enumerated domain/semantic controls pass, including omitted phases, duplicate phase residues, nonprimitive evaluations and the even-axis exception. Receipt JSON is generated from the checks and binds all eight package files.
• CPython 3.12.15 warning-strict normal and optimized commands and the registered wrapper pass locally. Direct normal and optimized runs each take under two seconds, measured separately as timeout checks, not speedup claims. Both registered timeouts are 120 seconds.
• Metadata/pins, exact committed bytes/modes, declared source history, official export integrity and normal/optimized read-only Git-free replay with empty executable PATH pass. AST/compilation inspection finds no acceptance assertions, recursion or function above sixty lines. No external linter/type checker was available. No predecessor scientific suite was rerun.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/composite-chirp-phase-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/composite-chirp-phase-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/composite-chirp-phase-repair
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/composite-chirp-phase-repair --check-source-history --check-modes

Any CPython 3.12 executable may replace the local absolute path. Expected markers are PASS_COMPOSITE_CHIRP_PHASE_REPAIR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. All registered scientific commands ran locally.

Limitations and unrun obligations

The proof covers complex linear maps and sums with a fixed quadratic coefficient a and arbitrary scalar weights/linear phase twists. A different coefficient, additional native operations or nonlinear encoding changes the model. Finite checks corroborate the written all-size proof; they do not replace it.

No actual composite Gaussian resampling caller, physical/native instruction trace, phase-table cost, rounded normalization, movement, owned restoration, storage/stock or complete supplier invoice is supplied. Matrix conditioning alone is not a rounded implementation theorem. No selected assumption is discharged; the selected cold replay, Lean, Docker and a complete multiplication theorem remain unrun/unclaimed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can an actual composite-axis consumer use this operator or its g-term repair with a favorable complete bill? First identify one literal kernel and integer coefficient a, prove its actual input/output contract, then price folds, phases, recursive calls, normalization and owned restoration. The missing producer, native implementation and complete invoice are the blockers. No continuation is reserved.

Related open work

• #5946 determines allowed second-input dimensions for unequal Fourier read layouts. This package instead factors a fixed modular quadratic convolution and counts phase terms; no shared numerical value is compared.

• #5949 optimizes the origin of a real Gaussian chirp matrix. This package uses a phase-only cyclic operator with integer quadratic coefficient and computes no origin optimum.

• #6000 constructs quadratic Walsh-Hadamard surrogates at fixed rational rank. This package exactly factors cyclic chirp matrices; the matrix families and objectives differ, so no shared numerical value is compared.

• #6001 proves elementary-operation costs for joint Gaussian-dyadic transitions. This package counts linear-phase terms in a modular quadratic convolution; neither result computes the other operation count.

The pre-publication main, merged-PR and live-claim refresh found no open PR computing this cyclic chirp rank or repair count. Search coverage does not exclude unpublished work or unread arbitrary diffs. PR5564 is a merged predecessor, cited above.

Completion refresh inspected the titles and result scopes of all 14 other open account PRs created since the advance CLAIM: #5997, #5998, #5999, #6000, #6001, #6003, #6004, #6005, #6006, #6007, #6008, #6009, #6010, #6011. They study thin-query capacity, precision selectors, scalar rewrites, Walsh-Hadamard agreement, Gaussian elementary transitions, residue-field determinants, connector words, observer error floors, support interning, group-gate ranks, characteristic-two jets, selected-invoice relaxations Pauli normalization and integer-shear words. None computes this modular quadratic-chirp rank or phase-repair count. No extra computations were run to check their results, and no conflicting shared numerical value was identified.

CI notes

At head a3c8b5644b6f95f6a33a0d3f764ed85cc982da59, run 40585019705 completed successfully. Both hosted checks passed: plan and replay (research/composite-chirp-phase-repair). The completed checks were read once, and the successful run head matches the PR head. The registered replay exercises both normal and optimized commands. This certifies the bounded automated replay; independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only one standalone research package and its opted-in replay are added. No shared code, selected record, frozen evidence or predecessor source changes.