ODD-PRIME ESCAPE INDEPENDENT REVIEW
2026-10-18. Private research audit; no publication authorization.
Reviewer scope: local mixed lemma, all-n block reduction, physical sampler, convolution escape. Sealed qutrit paper and baseline unchanged.
Baseline read: riemann_A_mixed_qutrit_escape_20261018_213904/INTEGRATED_PROOF.txt.

VERDICT
The proposed mixed escape extension is analytically valid with the conventions below. This audit establishes the escape primitive, not the end-to-end odd-prime learner or worldwide originality. No counterexample found by analytic hostile review; this note contains no new numerical state search.

LOCAL LEMMA
Let p be odd prime, tau a p-dimensional density matrix, and f=tau_00 its largest single-qudit stabilizer overlap. Put P=Tr(tau^2), g=sum_j tau_jj^2-P/p, c=(p-1)/(4p).
The p+1 stabilizer MUBs have sum of squared probabilities 1+P, by Weyl-line partition and Parseval. Since each probability <=f and total sum p+1, P<=(p+1)f-1; also P<=1. Computational diagonal Cauchy yields sum tau_jj^2>=f^2+(1-f)^2/(p-1). In particular f>=1/p.
Put h=2/(p+1) and K=(3p-1)(p+1)/(4p(p-1)).
For 1/p<=f<=h,
 g-c f^2 >= K(f-h)(f-2(2p-1)/(3p-1)) >=0.
For h<=f<=1,
 g-c f^2 >= K(f-h)(f-2/(3p-1)) >=0.
The first other root exceeds 1; the second is at most h. Endpoints included. Thus g>=c f^2. At p=3 this exactly reproduces c=1/6. Optimality is not established.

ALL-n REDUCTION
D=p^n, rho any density matrix, F maximum pure stabilizer overlap, S any maximizing stabilizer Lagrangian, H any hyperplane of S. A Clifford and displacement send maximizing state to |0^n> and H to span(Z_2,...,Z_n). Define A(x)=D^-1 Tr(rho W_x rho W_x^dagger). Weyl twirling gives A>=0 and sum A=1; its symplectic Fourier transform is |chi(u)|^2.
With PSD blocks sigma_y=rho[(j,y),(k,y)],
 A(H)=p^-1 sum_y Tr(sigma_y^2), A(S)=sum_y sum_j sigma_y,jj^2.
Every block difference is nonnegative since Tr(sigma^2)<=(Tr sigma)^2<=p sum diag^2. The y=0 block gives A(H)>=F^2/p. Let t=Tr sigma_0 and tau=sigma_0/t. Ambient optimality against v tensor |0^(n-1)> establishes local stabilizer maximum f=F/t, so the local lemma gives A(S\H)>=c F^2. This handles mixed postselection correctly, and all tied optimizers.

PHYSICAL SAMPLING AND SCALAR CONVENTIONS
Use W_u W_v=omega^([u,v]/2)W_(u+v), W_u^dagger=W_-u. Any consistently reversed symplectic sign gives the same law after label reversal. Choose nonzero coefficients c_1,...,c_m with sum c_i^2=0. Then R_u=tensor_i W_(c_i u) is an ordinary representation of additive V, since Weyl phases cancel. P_y=|V|^-1 sum_u omega^([y,u])R_u are orthogonal character projectors, sum P_y=I. For u!=0 at least one Weyl trace vanishes; hence Tr P_y=D^(m-2). No sectors are missing for m=2 or m=3.
Measure P on rho^tensor m twice independently and subtract outcomes. Its characteristic function is product_i |chi(c_i u)|^2. If A_c(x)=A(c^-1 x), then hat A_c(u)=hat A(cu); thus B=A_(c1)*...*A_(cm). This is a physical 2m-copy law. A itself is auxiliary: no physical two-copy sampling assertion is made. The representation factors over physical positions, and generators are commuting Paulis, permitting standard stabilizer measurement synthesis.

ARITHMETIC BRANCHES AND ESCAPE
If p=3 mod4, choose a,b with a^2+b^2=-1 and coefficients (1,a,b). Existence follows by intersecting squares and -1 minus squares (each has (p+1)/2 elements); neither coefficient is zero because -1 is nonsquare. Thus six copies suffice. Each scaled distribution preserves masses of H and S\H. Exactly one of three independent convolution inputs in S\H, the other two in H, gives three disjoint successful events:
 B(S\H)>=3 c F^6/p^2=3(p-1)F^6/(4p^3).
If p=1 mod4 choose a^2=-1 and coefficients (1,a). Four copies suffice, and two disjoint events give
 B(S\H)>=2c F^4/p=(p-1)F^4/(2p^2).
For any proper linear T<S extend to H and use S\H subset S\T. These bounds therefore hold for every such T. Affine T is not claimed.
Important caveat: p=5 has no solution with a,b both nonzero in the three-factor form. Do not assert that arbitrary two-square witnesses have both nonzero coefficients. The four-copy branch resolves this and is stronger there. A zero-coefficient register contributes W_0=I and can simply be discarded; it must not be assigned an invertible scaled A law.

PRIMARY-LITERATURE COMPARISON, LIMITED SCOPE
Read via direct primary arXiv HTML, not a Google discovery search:
https://arxiv.org/html/2607.15559v1 (Qi et al., version 1, 17 July 2026): stated learner input is pure; Proposition 3 equation (27) is the relevant pure marginal; single-qudit MUB identities in Lemmas 6-7 are direct antecedents. For z=YR, R^2=-I gives Y1=-a z1-b z2, turning equation (27) into p_psi * (p_psi)_a * (p_psi)_b; for pure rho, A=p_psi, so our law agrees. This does not independently certify their mixed-input circuit or extend their theorem merely by citation.
https://arxiv.org/html/2510.06848v3 (Allcock et al., version 3, 25 August 2026): Result 1 and subsequent special-prime discussion already use the sum-of-squares/skew transformation, explicitly distinguishing p=3 mod4 from p=1 mod4. The arithmetic split and multi-copy measurement mechanism are prior methods. The present comparison does not claim their invention.
https://arxiv.org/html/2510.08070v2 (An infinite hierarchy of multi-copy quantum learning tasks): primary HTML retrieved and Weyl conventions located, but its full measurement hierarchy was not audited in this focused review. Therefore no novelty or minimum-copy claim against this source is cleared.
These targeted readings are not a completed literature survey. Parent research should reconcile full prior results before originality conclusions.

LIMITS
Exact analytic audit; floating diagnostics and independent implementation remain separate tasks. No optimal constants or copy lower bounds. No mixed odd-prime learner resource theorem proved here. No external human refereeing, publication readiness, or release permission implied. Preserve one result-family extension relationship to the qutrit escape and learner results.
