[petalas]

Summary

Prove a sharp conditioning envelope beyond Ali Romman's PR5288. Claim type: scoped written proof with exact finite corroboration. No new kappa.

D_i^2=0, sum ||D_i||_2 <= rho, K factors, dimension n>=2 over R or C
Phi(t) = (sqrt(t^2+4)+t)/2
exact maximum norm and inverse norm = Phi(rho/K)^K
exact maximum condition number     = Phi(rho/K)^(2K)
attainment: alternating 2x2 upper/lower shears, embedded in dimension n

The new step is the sharp ordered-word/total-budget theorem and its two-dimensional attaining family. The single-shear singular values, SVD and Jensen's inequality are classical. Credit Ali Romman #5288 for the square-zero interaction contract, ywz65 #2701 for interaction coordinates, Vlado Vojdanovski #1673 and its retained Gaussian/Toeplitz lineage, and Nick Petalas #5153 for separate endpoint-refinement context. The new proof, attaining word and verifier were produced by Nick Petalas.

The uniform-transfer-audit card changes scope to a numerical-interface result. Canonical base is 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. Advance CLAIM. Only seven new package files are added; source pins are provenance and there are no external mathematical runtime inputs.

Evidence

• Before: Prove structure-dependent shear perturbation bounds (no new kappa) #5288 gives the sufficient Euclidean specialization product(1+d_i) <= exp(sum d_i). After: product Phi(d_i) <= Phi(rho/K)^K, with sharp length-independent norm supremum exp(rho/2). No small-error restriction is introduced. For two factors of norm 3/2, the attained norm and inverse norm are 4; the previous sufficient product is 25/4. These are analytic values, not runtime measurements.
• Exact checks cover 288 rational attaining word fixtures and 1,872 prefix observations, including 252 nonzero fixtures with strict bound improvement. Another 96 varied fixtures check 288 prefixes, 42 rank-two error factors, eight complex realification fixtures and 75 words with total error at least one. Domains overlap. All norm bounds use rational Gram principal minors; inverses use independent Gaussian elimination, and attainment uses exact Gram eigenvectors.
• All 64 declared two-slot smoothing pairs and 17 domain/semantic controls pass. Three disposable optimized mutations reject missing square-zero validation, changed product order and omitted norm validation. No predecessor scientific census or receipt is replayed as the deliverable.
• Both warning-strict normal and optimized commands pass locally on CPython 3.12.15, standard library only, with identical generated source-bound receipts. Their final separate timings are 1.50s and 1.51s, below each 120-second registered timeout. These are timeout observations, not a performance comparison. Registered package replay passes.
• Publication 99783830399fb818901c7c1e6000309627706a9d passes exact committed bytes, executable modes and source-history preflight. Official export/input-coverage checks pass. Both direct modes pass in a read-only Git-free export with empty PATH/PYTHONPATH; all eight exported files retain their hashes and modes. In-memory syntax compilation, function-size checks and staged whitespace pass. Separate lint/type analysis is unavailable and unrun.

From the worktree root:

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

Another CPython 3.12 path can replace the displayed local interpreter. Expected markers are PASS_SHARP_SQUARE_ZERO_CONDITIONING, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The sharp theorem concerns arbitrary square-zero interaction-error words in the Euclidean norm, rather than an additional fixed production circuit. Its physical-prefix corollary is sufficient. General errors changing the shear structure and arbitrary induced norms are outside this result. Finite tests corroborate the written universal proof; they do not replace it.

No actual coefficient storage/rounding policy, additive-noise or operator-defect optimum, reversible rounded-grid instruction, physical caller, native/compiler/tape/stock cost or complete signed supplier invoice is supplied. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected scientific cold replay are unrun. No required local check is deferred solely to CI.

Open next questions

• Apply the sharper suffix factors to one production word with a specified coefficient policy and additive-noise budget. First bind its Euclidean interaction errors and paid storage/access/restoration; a production prefix envelope and actual arithmetic kernel remain missing.
• Find the sharp operator-defect envelope at fixed K and total budget, starting with K=2. ||R-I|| is a separate quantity; the one-factor defect is d rather than Phi(d)-1. No follow-up claim is reserved.

Related open work

• PR6402 computes ordinary tensor-power operator defects for one square-zero endpoint error. This theorem studies same-space ordered noncommuting products, arbitrary factor sizes and total-budget conditioning attained in dimension two; no tensor-error value is compared or extended.

• PR6431 bounds integer coefficient and row-sum growth inside exactly restoring signed-unit ADD words. This result grants arbitrary square-zero coefficient errors and imposes no identity endpoint; it computes no Fibonacci or restoring-word value.

• PR6604 gives a precision minimum for a particular nested border under oracle errors. This result is a coefficient-error conditioning theorem and computes no nested-border precision minimum.

• PR6666 constructs approximate dyadic channels with a restoring four-block-shear word. It computes an infinity-norm approximation/capacity contract rather than this Euclidean worst-case conditioning envelope; no shared numerical value is compared.

Main README searches, merged/general-form searches, live open PR bodies, recent account work, handoffs and reservations were checked before commitment. The broad cached snapshot covered 613 open PRs and 6,716 Discussion records with complete remote pagination; sparse local coverage was 147 of 8,917 notes. Arbitrary historical diffs and unpublished work remain outside coverage. These checks are not a novelty certificate.

Completion refresh screened the eleven other open account PRs created since the advance CLAIM: #6664 through #6670 and #6672 through #6675. None computes a competing value for the fixed-K Euclidean square-zero conditioning envelope. PR6666 is adjacent work and is distinguished above. No additional scientific computation was run to check their results.

CI notes

Both hosted checks, plan and replay (research/sharp-square-zero-conditioning), passed at exact head 99783830399fb818901c7c1e6000309627706a9d in run 41670959724. The completed checks were read once with gh pr checks 6671 --repo CrocSwap/integer-mult-bounds; separate run metadata confirms completed success and the matching head.

Hosted replay confirms execution of the registered normal/optimized finite checks and package metadata validation. Local scientific, committed-byte/source-history/mode, syntax and read-only Git-free export checks also passed. No failed or waived check remains. Independent human or proof-kernel acceptance of the written theorem and the listed physical/supplier obligations remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Seven additive files in one opt-in research package.