[petalas]

Summary

Prove the exact best COPY approximation from the constant-weight phase-history layer. This is a scoped written mathematical proof with exact finite corroboration, not a new kappa or physical supplier.

For S=diag(1,i), let U_A apply S on the k axes of A among n axes. Every admitted operation is source-preserving y += c_A U_A^(tensor f) x, with arbitrary independent initial x,y and arbitrary complex exterior coefficients. For every n>=1, 1<=k<=n and integer f>=1:

L = sum_{j=0}^{min(k,n-k)} binom(k,j) binom(n-k,j) 2^(-fj)
min ||I - sum_{|A|=k} c_A U_A^(tensor f)||_HS^2
  = 1 - binom(n,k) 2^(-fk)/L > 0
unique attaining coefficient: c_A = ((1-i)/2)^(kf)/L

The norm is normalized by response dimension 2^(nf). For the complete two-register endpoint normalized by twice that dimension, the minimum is half this value. For k=1 the response error is (2^f-1)/(2^f+n-1). At n=8,k=4 it is 251/321 for f=1 and 1851/1921 for f=2. No exact COPY is possible within this fixed history contract.

This contribution was produced by Nick Petalas. Credit Ali Romman’s #5727 at 8b74ecd0eea26d95b400e56435d35fbd261dc49b for the explicit finite-span application question and exact projection contract. Its implementation and predecessor results are cited, not audited or replayed. Constant-weight intersection counts, tensor independence and Hilbert-space normal equations are classical. The new step is this explicit history family’s closed optimum, attaining coefficients and full independent endpoint. Retain the broader native/controller lineage credited by #5727. Apache-2.0.

Evidence

Before, #5727 supplied an exact finite-span engine and left a named-history application open. This package gives an all-parameter written formula, with a standalone standard-library corroborator that imports no predecessor executable.

Local CPython 3.12.15 checks pass:

• All 51 literal interfaces in 1<=n<=6, 1<=k<=n, 1<=f<=min(8,floor(10/n)); 49,538 source diagonal entries and 1,653 independently computed Gram entries.
• Exact elimination of 398 real/imaginary normal equations; all 23,484 independent source/dirty-accumulator endpoint columns; literal squared residuals agree with the formula.
• 160 larger formula evaluations through n=128,f=256, nine semantic controls and ten malformed/range controls. Larger matrices are not enumerated.
• Warning-strict normal and optimized direct checks, both registered checks, metadata/pins and committed-byte/source-history preflight pass. Final registered-command timings are 0.627 and 0.611 seconds locally, below the job limit.
• Official six-file committed export and declared-input integrity checks pass. Both direct modes pass in a sealed read-only Git-free export with empty PATH, retaining every exported byte and mode. Generated stdout receipts bind all six package hashes; no receipt is handwritten.

From the repository root:

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

Expected markers are PASS_CONSTANT_WEIGHT_PHASE_COPY_ERROR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. A local Python 3.12 can replace the absolute interpreter path elsewhere.

Canonical main was refreshed before submission at de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. The selected metadata orientation passes. Live claims, open PRs, this account’s recent work, the overlap helper, main text and merged PRs were checked. The helper covered 448 open PRs and 5,379 Discussion records but only 192 of 7,845 local HEAD notes; direct Git searches supplemented that sparse coverage. This is bounded overlap evidence, not a universal novelty certificate. The advance CLAIM has the delivered scope.

Limitations and unrun obligations

The fixed history alphabet and source-preserving accumulator contract are essential. Source feedback, other history weights and inclusion of the identity history require a new analysis. Attaining complex rational coefficients can have odd denominators. Their Gaussian-dyadic realization is unclaimed.

Native physical delivery, coefficient preparation, physical source restoration, precision, movement, ownership, stock, storage and the complete recursive invoice remain unproved and unpriced. The evaluator caps n at 128 and f at 256; the universal statement rests on the written proof. No selected supplier replay, Lean build, complete multiplication theorem, independent human review or formal proof-kernel review is claimed. Only the isolated six-file package is changed.

Related open work

• Prove odd-axis Clifford tensor Gram lower bound above one half #5883 bounds the full odd-axis Clifford tensor Gram spectrum; this contribution computes the attained COPY error for one constant-weight subgroup layer, with no new full-group constant.

• Prove exact additive cost for two anticommuting query types (no new κ) #5877 prices two anticommuting query types; this contribution concerns a commuting phase-history span and normalized approximation error, without a paid query tariff.

• Prove sharp all-Clifford helper-detour bounds and correction budgets #5779 bounds all-Clifford helper-detour tariffs; this contribution fixes the available histories and studies their response projection, without chart-path or complete-invoice claims.

• Prove sharp zero-phase witness length 2n+2 for n=2 mod 4 (no κ) #5902 studies zero-phase full-affine-span tuples and first-visible tensor widths; this contribution fixes a constant-weight phase-history alphabet and computes its COPY projection error.

• Prove sharp Gaussian Newton residual prefix depth (no new κ) #5900 studies Gaussian Newton residual prefix precision; this contribution computes a response error without a precision schedule or a common residual-tree quantity.

• Prove unbounded dyadic acquisition at rational rank one (no new κ) #5904 studies exact dyadic decoder acquisition at rational rank one; this contribution permits complex exterior coefficients and leaves their Gaussian-dyadic realization open.

No other open PR was found computing this constant-weight COPY error. The eight other account PRs opened since the CLAIM were inspected: #5897, #5898, #5899, #5900, #5901, #5902, #5904 and #5905. No same-instance numerical claim was identified; no additional mathematical computations were run to validate those contributions or compare different quantities.

Open next questions

• Admit two positive history weights together and solve the phase-sensitive invariant two-layer normal equations. This tests whether the obstruction survives a larger caller alphabet. No multi-layer optimum is supplied.
• Implement one attaining coefficient vector with a fully charged Gaussian-dyadic caller, or price an approximation. Coefficient preparation, native delivery, restoration and the complete physical bill are blockers.

Merge Danger

Door: two-way. The contribution adds one isolated research package.

Blast Radius: research. Shared implementations, frozen sources and the selected frontier are unchanged.

CI notes

At commit aedf15cadfc10d566d7811936019a21207c99545, run 40501261851 has plan failed; replay skipped. The only failed plan step is Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight assertion failures concern frozen-card routing, missing current/historical references and sparse-reference expectations.

All ten failure/error headers and the final failure summary match the same step in PR #5901 run 40501075176, confirming the reported shared main de5ec2e7a onboarding failure. Maintainers will handle the repair. No package, shared implementation or test files were changed in response.

Hosted replay has not certified this package. Only the local scientific, registered, publication and export checks described above have run. The replay job was skipped; it did not pass. Independent human and formal proof-kernel review remain unclaimed.