[petalas]

Summary

Scoped analytical proof with exact finite corroboration. Recover the sharp dimension-independent operator-norm error for unitary A,B from simultaneous tensor-square and tensor-cube comparisons. With a=arcsin(epsilon/2):

both tensor defects <= epsilon
  epsilon < 2 sin(pi/5)  -> ||A-B|| <= 2 sin(a/3)
  epsilon >= 2 sin(pi/5) -> ||A-B|| <= 2 sin(pi/3+a/3)

Scalar phases attain every value. At the threshold, exp(4pi i/5) attains error sqrt((5+sqrt(5))/2), while the left limit is 2 sin(pi/15). Thus the best possible error envelope jumps. The written proof covers every positive finite dimension and epsilon in [0,2]. This is a new quantitative application, with no general literature-priority claim.

Produced by Nick Petalas. Credit unpaidmoon #5837 for the finite-width phase example and question, DLamdata #5894 for separate selected phase-coverage work, and the retained native/caller contributors. Spectral reduction and trigonometry are classical. Advance CLAIM.

Evidence

Before: #5837 exhibits the width-two sign ambiguity, without this quantitative two-test envelope. After: the proof gives the exact envelope, equality branch and fifth-root witness. No predecessor result or MOVE census is replayed as the contribution.

CPython 3.12.15, standard library, local normal and optimized registered checks pass PASS_TENSOR_PHASE_STABILITY: 720 enumerated rational phases, 33 near-attaining fixtures, 16 dense unitary pairs and 11 controls. The threshold and jump witness are checked exactly in Q(sqrt(5)); dense tensor defects use literal Gram matrices. Deleting the false-bracket guard in a disposable probe is detected by its targeted rejection control. Independent human and formal proof-kernel review are unclaimed.

Exact reproduction commands from the repository root:

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

The two registered commands were timed locally at about 0.31 and 0.32 seconds on the committed, read-only Git-free export, well below their 120-second limits. These are check durations, not comparative performance claims. Metadata, exact committed bytes/source history, official export integrity, both exported replay modes and git diff --check pass. All eight exported files retain their bytes and modes. The generated receipt binds the proof, code and metadata by SHA256.

Limitations

The theorem assumes one fixed pair of exact unitary charts and absolute operator-norm comparisons. It does not cover projective checks, nonunitary operators or independently perturbed tensor factors. Approximate agreement at these two widths does not give the same error bound at every later width. No source MOVE admission, physical cost, new kappa, selected-frontier change or complete multiplication theorem is claimed. All five framework assumptions remain.

No selected cold replay, native census, physical tape implementation, Lean, Docker or proof-kernel check was run. The frozen catalogue and selected record are provenance; the standalone result imports no predecessor executable and needs no pending Git object. Local source baseline is canonical main de5ec2e, priority revision 8, refreshed before submission.

Open next questions

• Classify the sharp envelope for unequal width-two and width-three tolerances. Start with the two scalar feasible-arc families; their two-parameter first-contact classification remains open here.
• Apply the theorem to one actual fixed unitary chart pair with separately established tensor error premises, preserving existing absolute-phase coverage and pricing the physical error/tape contract.

Related open work

• Prove the selected #5599 scalar instructions lift at every column mul… #5837 proves the selected scalar-chart lift and poses the finite-width phase boundary. This result supplies a general quantitative two-test theorem, without auditing or extending that source census.
• Audit existing phase coverage for selected P104 MOVE maps #5894 reports existing absolute-phase coverage for selected MOVE maps. This result asserts no missing selected check and computes no MOVE census or source tariff.
• Prove odd-axis Clifford tensor Gram lower bound above one half #5883 bounds an odd-axis Clifford tensor Gram matrix. Its Gram eigenvalue quantity differs from the two-chart recovery error here; its result is neither assumed nor checked.
• Prove sharp zero-phase witness length 2n+2 for n=2 mod 4 (no κ) #5902 studies zero-nonconstant-phase tuples with full affine span. Its tuple-length quantity differs from the unitary operator-norm envelope here; no numerical comparison is made.

The completion refresh inspected the nine other open account PRs created since the CLAIM: #5909 through #5915, #5917 and #5918. They study target fees, sector workspace, colored batching, rounded-shear history, carry-state information, cyclic quotients, carry frequency, tree tensor degree and XOR synthesis. None studies this unitary operator-norm recovery envelope or computes a conflicting value. No additional numerical comparisons or predecessor replays were run.

CI notes

At commit f0d87ad28e09055fd44e5a93437be5eeba284491, run 40506634356 finished with plan failed; replay skipped.

The only failed step was 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 two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen-card/current-command routing (0 != 16), missing current/historical references (ValueError not raised, five entries), and sparse omission lists (two entries).

All ten failure/error headers and the final summary match PR #5915's plan log exactly; both logs were inspected. This confirms the reported shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a. Maintainers will repair it. No package, shared source, workflow or onboarding test was changed for this CI closeout.

The package replay was skipped. Hosted replay has not certified this package; only the local scientific and publication checks listed above have run. No hosted scientific pass or complete multiplication theorem is claimed.

Merge Danger

Door: two-way

Blast Radius: isolated

Seven new files under research/tensor-phase-stability. Existing source packages and the selected frontier retain their bytes.