[petalas]

Summary

Scoped algebraic proof with exact finite corroboration: the best Euclidean conditioning of every equal-gain unitary observation cycle, including all equality cases.

A = I + r S, (Sx)_j = U_j x_(j+1),  W>=2, d>=1, 0<r<1
H = U_0 ... U_(W-1),  a=1+r², b=2r, c=cos(π/W)

W even: min condition = sqrt((a+bc)/(a-bc)); equality iff H=-I
W odd:  min condition = (1+r)/sqrt(a-bc);   equality iff H= I

Three banks, r=1/2: exact optimum sqrt(3), attained over the rationals.

The theorem permits changed couplings. It does not replace the constant 3 that Clear1nsight #4842 proves for the unchanged proposed cascades. Bankwise unitary reencoding only conjugates H and cannot perform the required change for free. The classical cycle-spectrum method is the proof technique; no literature-priority claim is intended.

This contribution was produced by Nick Petalas. Credit Chris R / Clear1nsight #4842 for the complete-cascade setting, hexapode #4818/#4819/#4826 for the schedule context, and the retained Dugongue #4743, Yiyang / Lyy-iiis #4710, lee-phys #4688 and Jacob Alan Sussman #4628 lineage. No predecessor scientific suite is replayed as the deliverable. Only five files in research/unitary-cycle-optimal-conditioning/ are added. No new κ or selected-frontier change.

Evidence

Before, the cited work gives sharp conditioning for fixed source matrices. After, this package gives a quantified optimum over all unitary couplings, a necessary and sufficient ordered-product criterion, and an explicit attaining closing edge.

The independent literal rational-matrix path computes complete Gram characteristic polynomials and compares them with Chebyshev predictions. It checks 756 signed scalar cycles, including 378 attaining products; 576 two-dimensional signed-permutation cycles, including 72 exact attainers; 42 noncommuting rational gauge fixtures; and 11 boundary controls. Exact extremal-root brackets prove the finite nonattaining gaps at W=2,3. Larger W and arbitrary complex dimension are covered by the written proof, with finite spectral corroboration through W=13,d=2.

From the repository root, using the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/unitary-cycle-optimal-conditioning/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/unitary-cycle-optimal-conditioning/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/unitary-cycle-optimal-conditioning
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/unitary-cycle-optimal-conditioning --check-source-history

Any CPython 3.12 executable can replace the displayed interpreter. Expected markers are PASS_UNITARY_CYCLE_OPTIMAL_CONDITIONING, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

Both warning-strict registered commands pass locally, timed individually at 3.18 and 3.21 seconds. Metadata/source pins and the five-file committed-byte/source-history preflight pass at dea483c4aa02a823dbdf351641382fbc1b2d62f6. The official committed export and integrity check pass. An optimized Git-free replay reproduces the generated source-hash-bound receipt and leaves all six export files unchanged. There are no runtime predecessor imports or unmerged source-object dependencies.

Advance CLAIM. Canonical main refreshed unchanged at 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Before committing, the complete remote snapshot covered 397 open PRs and 3,897 Discussion records; recent handoffs, local reservations, distinctive main-tree formulas and merged searches were also checked. Sparse local-note coverage is incomplete; searches are not a novelty certificate.

Limitations and unrun obligations

The theorem assumes equal edge magnitude r, unitary factors and Euclidean bank norms. It prices the completed observation matrix only. It does not optimize unequal gains, weighted bank norms, nonunitary couplings or intermediate execution norms.

Source-compatible native emission, independent target delivery, dirty-bank ownership/restoration, full paid caller invoice, stock, compiler, precision/tape and analytic recovery remain open and unpriced. No supplier or complete multiplication theorem is admitted. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, predecessor scientific replays and selected-frontier replay were unrun. Every registered package command ran locally; none is deferred to CI.

Open next questions

• Realize the attaining ordered product in one actual three-bank native word, retaining all dirty inputs and independent targets. Start with a literal closing port and a complete paid invoice. Its source compatibility and restoration are missing.
• Fix a physically priced unequal-gain interface before optimizing its conditioning. Equal-gain normality does not settle that problem; a normalization/precision ledger is needed before expansion.

Related open work

• Certify complete block-cascade conditioning and audit paid-word admission (no new κ) #4842 studies the unchanged complete cascades and reports constant 3; it is cited as context. This result changes couplings and does not compute or contest any source matrix's constant.
• feat(research): exact 3-array 10-call block-4 unit schedule on C^16 (κ = 0.1315 > 0.10) #4818 proposes a block-four three-bank schedule. The new optimum applies to the abstract changed cycle class, with no admission of its paid schedule.
• feat(research): exact 13-array 41-call block-4 boundary schedule on C^16 (κ = 0.1714) #4819 proposes a thirteen-bank block-four cascade. The W=13 formula is an algebraic design target; no original thirteen-bank value is recomputed.
• feat(research): exact 3-array 9-call block-5 unit schedule on C^32 (κ = 0.3174) #4826 proposes a three-bank block-five cascade. The sqrt(3) fixture changes its coupling contract and is not a new nine-call unit.
• feat(research): exact 5-array 14-call block-3 unit schedule via 14-neighbor symplectic cascade (κ <= 0.0628) #4813 proposes a five-bank native-port schedule. This package supplies a cycle-conditioning theorem, not the missing chronological realization or restoration.
• Emit shared-port fusion and bound its free five-target completion (no new κ) #4856 emits shared native ports with paid fusion and leaves completion open. The present endpoint-only theorem supplies no completion word or paid-call reduction.
• Prove sharp conditioning of the complete complex port (no new κ) #4828 computes sharp conditioning of a distinct complete two-bank port. Its operator and numerical quantity differ from the cyclic observation matrix here.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T22:39:38Z were refreshed. Ten entries were returned, including this PR. The nine others, #4873/#4874/#4875/#4876/#4878/#4879/#4881/#4882/#4883, concern compiler separators, shared Fourier cancellation, same-phase shear costs, dyadic rotation tariffs, mixed-precision density, Schur ordering, affine-box precision, UT4 coefficient relaxations and packed tape access. None studies the unitary-cycle conditioning optimum or supplies a competing value on its fixtures. No additional same-quantity numerical comparison or disagreement was found in these scopes; no extra computations were run to check their results.

CI notes

Both hosted checks passed on head dea483c4aa02a823dbdf351641382fbc1b2d62f6 in Registered research packages run 39867379419: planning in 5m11s and the registered package replay in 1m41s. Run metadata confirms that exact head, completed status and success. The checks were read once at completion. The registered replay executes both warning-strict ordinary and optimized commands. All registered local checks, committed-byte/source-history preflight, official export integrity and optimized Git-free replay also passed. Hosted CI certifies execution of these scoped checks; independent human and formal proof-kernel acceptance of the quantified theorem remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated. Only one new research package is added; shared implementation, frozen sources and the selected record are unchanged.