[petalas]

Summary

Scoped written proof with exact finite corroboration. A new whole-bank Euclidean error guard reduces the sufficient grid of one complete four-dimensional, twelve-call orthogonal-witness detector by 46 fractional bits, using the same coarse endpoint separator:

- 228 sufficient fractional bits
+ 182 sufficient fractional bits

Every exact orthogonal apply has gain one in the maximum register Euclidean norm. Entry quantization and each final data truncation are still charged. For a side with exact gains beta_i, perturbation constants gamma_i, input radius R and s=ceil(sqrt(d)), put G=product beta_i, A=sum gamma_i/beta_i and K=G(RA+sm). At grid h with hA<1, its complete returned-bank error is at most hK/(1-hA). A rational two-side allocation makes every endpoint error at most eta/4 and funds threshold eta/2.

PROOF.md states the exact contract and all-size argument. The new standalone allocator and integer-grid interpreter implement the bound. No new kappa, optimum precision or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. Credit Ali Romman / #5339 for the complete detector, coarse separator and optimization question, and #5315 for the orthogonal realization; Krzysztof Wojt / #5262 for the inverse grammar and witness compiler; zxv / #5235 and the retained dirty-echo/tensor/phase lineage. Euclidean orthogonal invariance, Frobenius domination and forward-error propagation are classical. No global literature priority is claimed.

This changes accumulated rounding amplification. It holds the separation estimate fixed and does not pursue #5941's denominator-cancellation or native-binding questions. Predecessor scientific programs, counts and receipts are cited rather than audited or replayed. Advance CLAIM. Canonical base and refreshed main are 7c49fe3, priority revision 8, selected #6030.

Evidence

Before: the cited coordinate-norm guard charges dimension d at each exact apply. After: the new bound charges exact gain one and includes matrix perturbations through a finite product bound. The strict fixture executes at both 228 and 182 bits. Its exact selected endpoint is 33658561053084102656/87507831740087890625 and eta=12^-48. The sufficient inequality fails at 181 bits; this is not a lower bound on necessary precision.

Local CPython 3.12.15 standard-library checks pass:

• 1,218 complete program/input cases in dimensions 2..7 with parameters 1/4 and 1/7, including all independent input columns and 84 mixed dirty inputs. Seven program-pair patterns cover noncommuting orders, inverse words, aliased queries, repeated scalar sources, zeros, empty sides and scalar gains/denominators up to 2^40.
• 9,048 intermediate banks comprising 138,528 coordinates satisfy their own prefix Euclidean envelopes. Every final coordinate has its difference error and threshold decision checked against separate exact rational execution.
• 240 product inequalities, 256 exact precision-rounding cases, fourteen invalid-input controls and four semantic controls pass. The controls address omitted matrix quantization, wrong inverse tables, updated-value aliases and an unfunded one-bit reduction.
• Direct normal and optimized checks took 0.945s and 0.997s. The registered pair passed in 2.266s; each registered command has a 120-second cap. These are verification durations, not speedup measurements. Generated stdout receipts bind all eight package files.
• Committed bytes and declared source-history availability pass at 0f69b3a. Official eight-file export integrity and both read-only Git-free direct replays with empty PATH pass without changing the exported files.

Exact reproduction from the worktree root:

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

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

Limitations

The proof assumes exact real orthogonal operators, old-value instruction semantics, exact rational scalar arithmetic, once-per-output coordinate truncation, and an exact initial grid input with the stated norm radius. The allocator's coarse separator requires integer initial banks and matching Cayley operators. Nongrid input errors and noninteger-input separation need separate treatment. A zero at one tuple is not a universal identity certificate. The written theorem, rather than the finite cases alone, supplies the all-size statement.

This is a sufficient precision improvement for the specified numerical interface. It is not a runtime improvement, cheaper native call, least-precision theorem or complete supplier. Matrix preparation, scalar arithmetic, movement, ownership/restoration, native compatibility, stock, compiler, selected precision/tape policy and analytic recovery remain separate. No selected scientific replay, native suite, Lean or Docker build was run. The five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Can one native-compatible orthogonal witness use this guard with a fully charged caller? Pin its actual operators, grid and radius, then count preparation, arithmetic, movement and restoration. Native binding and the complete physical bill are missing.
• Can exact instruction norms reduce a scalar-heavy guard further? Start with one fixed linear or aliased-query instruction while retaining quantization and data truncation. No optimal-norm or least-precision theorem is supplied.

Related open work

The refreshed search includes this account's six open PRs created since the advance CLAIM. The five other submissions supply no competing result for this orthogonal-witness detector or its numerical error guard. No common numerical fixture is computed, so no numerical agreement or disagreement is asserted.

• Prove smaller bulk precision for cyclic-band solves (no κ) #6487 reduces bulk factor precision for cyclic-band solves. Its elimination residual and row grids differ from this orthogonal-register error bound.
• Admit uniform 28-bit Gaussian residues with centered packing (no κ) #6482 changes signed Gaussian residue packing and its payload precision. Its recovery lane and precision contract differ from this detector.
• Prove exact fixed-rank Gaussian inverse error (no κ) #6483 determines a fixed-rank Gaussian inverse approximation error. Its operator-norm asymptote differs from accumulated execution error here.
• Reduce complete dirty-phase scientific replay time by 17.87% (no κ) #6486 accelerates an exact dirty-phase replay and cites this PR as adjacent work. Its scalar-product and runtime quantities differ from this numerical guard.

The broader publication search covered 431 open PR bodies. Merged #5339, #5352 and #5941 retain their predecessor credit. Sparse historical notes, arbitrary diffs and unpublished work remain coverage limits. No additional mathematical computations were run to check another contribution.

CI notes

Both hosted checks passed at exact contribution head 0f69b3a6df0226ba279a75d6975476af12b13271 in run 41555088098: plan in 4m44s and registered package replay in 1m24s. Run metadata confirms this head and a completed successful run. The replay executes the registered warning-strict normal and optimized commands.

All required mathematical, metadata, committed-byte/source-history and read-only export checks also passed locally. No required check was left solely to CI. Hosted replay certifies the bounded registered checks; independent human and formal proof-kernel review of the written theorem remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated. Eight new files in one opt-in research package; no shared implementation or selected result changes.