[JosieElliston]

Summary

Independent audit of #6134 (Nick Petalas, head b7889209ef72f7086d49024cd7c35c1d55ed61d8), the exact stochastic rank-power frontier inf omega = k/(bR) - 1. No error found. No new κ, supplier or selected-frontier change.

• Every step of Prove sharp random query cost versus error frontier (no new κ) #6134's PROOF.md checks for complex matrices, 1 <= R <= k, 0 < t < 1: trace/rank bound, probabilistic Cauchy-Schwarz, strict concave price, worst-input lambda_max >= trace/k, attainment, R = k uniqueness and the epsilon^2/(1+epsilon^2) saving (1/4097 at 2^-6). Prove sharp random query cost versus error frontier (no new κ) #6134's verifier replays PASS, normal and -O.
• F1 (strengthening): for a proper cap R < k, the optimal laws are exactly 0 with probability 1-b, else (k/(bR)) P with P a random rank-R orthogonal projection with E P = (R/k) I. Prove sharp random query cost versus error frontier (no new κ) #6134 states this family only as sufficient; the converse is proved in README.md.
• F2 (precision): equality in (3) alone allows mixed ranks with one common scale (exact k = 3 example). Endpoint ranks come only from strict concavity, so t < 1 is essential for uniqueness. Prove sharp random query cost versus error frontier (no new κ) #6134 uses this correctly.
• F3 (wording): "restrict [a dirty auxiliary state] to zero" is not an available step. The conclusion still holds because averaging x and -x gives the linear error plus E||c||^2; an exact example shows an offset can help at one input only.
• F4 (coverage): Prove sharp random query cost versus error frontier (no new κ) #6134's "only endpoint ranks attain" check is tautological on its own laws, and it never tests a non-attaining law. This package adds 2,631 random unbiased laws for the trace steps.

Evidence

research/audit-pr6134-query-power-frontier/verify.py (stdlib, exact rationals, imports nothing from #6134):

• 2,631 seeded unbiased laws A_i = M_i M^{-1} with oblique, nonsymmetric and mixed-rank outcomes satisfy (3) and tr C/k >= k/E r - 1, and on each one a literal basis input already has error at least k/E r - 1. The elementary price step E r <= R^(1-t) E r^t is not evaluated. All 1,343 exact equalities have the classified shape. Most are trivial (A = I or k = 1): four have k >= 2 with two distinct nonzero outcomes, and one has mixed ranks.
• 13 non-coordinate attainers (rank-one tetrahedral, Pythagorean and Hadamard frames; a rank-two complement law) have exactly isotropic covariance (k/(bR) - 1) I.
• Necessity controls: oblique-idempotent, unequal-scale and mixed-scale laws are strict.
• Exact check of the ±x auxiliary-offset identity.

Run python3.12 -I -W error -B research/audit-pr6134-query-power-frontier/verify.py (and -O). It prints PASS_AUDIT_PR6134 in about 0.3 s and also passes on CPython 3.10. The real-matrix checks do not cover the complex case, which rests on the written proof.

Scope and remaining obligations

This covers #6134's head above only. #6134's own open bridge is unchanged: the response-rank-to-price map is not established for any supplier. Open next question: does the frontier survive a constant-factor price bridge?

Credit

The theorem, attaining law and corollary are Nick Petalas's (#6134). The trace/rank inequality and coordinate sparsification are classical (Safaryan, Shulgin and Richtárik, as #6134 credits).

Produced by Josie Elliston with Claude Opus 5.5 (Anthropic) assistance, in a donated session.

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