ODD-PRIME MIXED LEARNER: INDEPENDENT CONDITIONAL BRIDGE REVIEW
2026-10-18. Private mathematical derivation; no publication/priority assessment.

Reviewed predecessor: ../riemann_A_mixed_qutrit_learner_20261018/LEARNER_PROOF.txt
SHA256 37C69E3F2658854756DFF84B0B1E5CB4A312B5E37EB026728A7BE22702A57B25.
Original qutrit evidence and manuscripts are unchanged. This review verifies the
learner bridge, CONDITIONALLY on the proposed odd-prime operational escape
laws and their independently audited constants. It does not certify those laws.

1. CONDITIONAL INPUT
Let p be an odd prime, n>=1, rho any n-qupit density operator, and let s be a
pure stabilizer maximizing F=<s|rho|s>, promised F>=tau>0. Write S for its
unsigned Lagrangian in V=F_p^(2n). Assume an operational r-copy distribution B
such that for every proper subspace T<S,
 B(S\T)>=kappa F^r.
Use (r,kappa)=(4,(p-1)/(2p^2)) for p=1 mod 4, or
(6,3(p-1)/(4p^3)) for p=3 mod 4, SUBJECT TO SEPARATE ESCAPE AUDIT.
The distribution and its physical implementation must hold for arbitrary mixed
states, including code-conditioned states. All later conclusions depend on this.
For arbitrary H!=S of dimension n, apply escape to T=S intersect H.

2. HIGH CORRELATIONS COMMUTE: COMPLETE GENERAL-p ARGUMENT
Put d=1-cos(2pi/p)>0, c(x)=|Tr(sigma W_x)|^2. If q_j are the spectral outcome
probabilities for a nonzero Weyl W_x, then
 1-c(x)=sum_(j,k) q_j q_k [1-cos(2pi(j-k)/p)]
       >=d(1-sum_j q_j^2).
Consequently c(x)>1-d/8 implies max_j q_j>=sum_j q_j^2>7/8.
For noncommuting Weyls U,V and spectral projectors E,G, EGE=E/p: every
nonidentity power V^k moves the U-eigenspace because their commutator is a
nontrivial pth root and p is prime. Thus ||EG||=p^(-1/2) and
 ||E+G||<=1+p^(-1/2)<=1+1/sqrt(3)<7/4.
Two such high correlations would give Tr(sigma(E+G))>7/4, a contradiction.
Hence all labels with c>1-d/8 commute, even for mixed sigma.
If sigma is supported on a specified stabilizer code C, any nonzero expectation
commutes with C by conjugation with its definite-eigenvalue Weyls.

3. EXPLICIT HIGH-SPAN PROCEDURE
Let a=kappa*tau^r/2, t=d/64, G={x:c(x)>1-d/16}, and
 m=ceil((4n^2 ln p+ln 8)/a).
Take m B samples; estimate every sampled c to error <=t, with simultaneous
failure <=1/8. Retain estimates >1-3d/32, include C and extend to a full
Lagrangian H; abort if nonisotropic. Every sampled G point is retained because
1-d/16-t=1-5d/64>1-6d/64. Every retained label has actual
c>1-7d/64>1-d/8, so the extension exists on the estimation event.
There are at most p^(4n^2) subspaces of V, by representing each as the column
span of a 2n by 2n matrix. For each fixed H with B(G\H)>a, missing all its
G\H labels has probability <=exp(-am). A union bound shows that every H
containing the sampled G labels satisfies B(G\H)<=a with failure <=1/8.
Thus the data-dependent H is good with probability >=3/4.
Spectral eigenvalue samples estimate chi, and ||z|^2-|chi|^2|<=2|z-chi|.
For example h=ceil(128*t^(-2)*ln(32m)) fresh copies per label suffice:
real and imaginary errors <=t/8 imply squared-modulus error <t/2, reserving
the remaining margin for deterministic finite-precision evaluation. Hoeffding
and a union bound give failure <=1/8. Root-of-unity evaluations can be bounded
by rational intervals; O(log(1/t)) absolute-precision bits suffice, in addition
to counter precision. No exact comparison oracle for transcendental numbers is
required: classify using the computed approximation, whose total error is <=t.
Exact cyclotomic arithmetic is another option. The set G is analytical only.

4. LOW CORRELATION AMPLIFIES THE TARGET
For ANY probability q_b of ANY spectral outcome of a Weyl,
 |chi|>=2q_b-1 by the reverse triangle inequality. Thus
 q_b <=(1+|chi|)/2.
If c<=1-d/16, concavity sqrt(1-u)<=1-u/2 gives
 q_b <=(1+sqrt(1-d/16))/2 <=1-d/64=:q<1.
This statement requires no p-specific Fourier inversion or purity assumption.
Conditioning on the eigenvalue that fixes s increases its overlap by >=1/q.
Such an eigenvalue occurs with positive probability on a target-preserving path.

5. EXACT GLOBAL-OPTIMIZER PRESERVATION
For a consistent stabilizer-code projector P fixing s, let w=Tr(P rho)>=F.
For each pure stabilizer t, P|t> is zero or sqrt(c0)*phase*|t'> with t' a
pure stabilizer and 0<c0<=1. This follows by successive prime-dimensional
stabilizer measurements: a noncommuting measurement has p equally likely
outcomes and each replaces a generator. Hence
 <t|P rho P|t>=c0<t'|rho|t'><=F,
while <s|P rho P|s>=F. The conditioned optimum is EXACTLY F/w, attained by s.
This argument applies to arbitrary mixed rho and tied maxima. It is essential:
a soft filter or separately measured physical tensor factors cannot replace P.

6. COMPLETE CANDIDATE GENERATION AND SUCCESS PROBABILITY
Repeat the predecessor's procedure with eigenvalue guesses uniform over F_p.
At each code-conditioned stage construct H as above, measure ONE ORIGINAL rho
copy in H's full stabilizer eigenbasis and record its resulting tableau. Draw an
independent x from B. Terminate if x is in H or fails to commute with C;
otherwise guess its eigenvalue uniformly, extend P and continue. Rank-n code
termination records its uniquely specified stabilizer. Preserve prior candidates
on every abort. The algorithm never tests membership in S or low correlation.

For a target-preserving prefix, if good H=S the original-state measurement
records s with probability F>=tau. If good H!=S, escape and Section 3 give
 B({x in S\H:c(x)<=1-d/16})>=kappa F_sigma^r-a>=a.
These x commute with C subset S; a correct eigenvalue guess has probability 1/p.
Therefore progress (record s, or one target-preserving amplifying jump) has
conditional probability at least
 b=(3/4)min(tau,a/p)=3a/(4p).
The equality uses a/p<=tau for the candidate constants and r>=4.
Set J=ceil(ln(1/tau)/[-ln q])+1 and s0=b^J. J good jumps are impossible,
since tau*q^(-J)>1. Treat recording s as absorbing. Induction through at most
J stages yields probability >=s0 that an ideal run records s. Early rank-n
termination along such a path already specifies s. No p^(-n) penalty is used.

7. CAPPED PHYSICAL PREPARATION, VALIDATION, EXPLICIT RESOURCES
A conservative conditional-copy cap is Q=J(r*m+m*h+r+1). Each requested copy
of rho_P is produced by at most K=ceil(tau^(-1)ln(4Q/s0)) original-copy trials,
measuring the specified commuting code eigenspaces coherently and retaining a
trial only when all specified outcomes match. Abort after K failures. This
bounds resources even on wrong branches with zero or tiny code weight.
On target-preserving prefixes acceptance is w>=F>=tau. Coupling ONLY the
favorable witness process to the ideal run loses at most Q exp(-tau K)<=s0/4.
A capped run therefore records s with probability >=s0/2.
Use R=ceil((2/s0)ln(2/delta)) independent runs and N=R(J+1). Validate every
recorded candidate on v=ceil((2/epsilon^2)ln(4N/delta)) fresh original copies
using its rank-one stabilizer projector; select the largest empirical overlap.
Condition on the complete candidate list before validation. Independent
Hoeffding bounds imply returned overlap >=F-epsilon with probability >=1-delta.
An empty list returns a fixed stabilizer and is included in the failure event.
Total original copies <=R Q K+R J+N v.

8. FIXED-p VERSUS UNIFORM GROWING-p COMPLEXITY
Write L=ln(2/tau). Since d=Theta(p^-2), uniformly for p>=3,
 J=O(p^2 L), h=O(p^4 ln(32m)),
 m=O(kappa^-1 tau^-r (n^2 ln p+1)),
 ln(1/s0)=O(p^2 L [L+ln p]).
Indeed kappa^-1=O(p^2) in both residue classes and r<=6. The explicit formulas
in Section 7 are the primary bounds; a coarser uniform consequence is
 exp(O(p^2 L[L+ln p])) poly(n,p,1/epsilon,ln(2/delta)).
For each fixed p this is exp(O_p(L^2)) poly_p(n,1/epsilon,ln(2/delta)).
It is NOT a dimension-uniform quasipolynomial in 1/tau with p-independent
constants, nor a polynomial-time learner in log p. This argument pays p^2 in
conditioning depth and extra polynomial factors for correlation precision.
Taking tau=epsilon yields a promise-free additive learner: if F<epsilon every
stabilizer already meets the additive guarantee; otherwise the theorem applies.

9. OPERATIONAL MODEL AND AUDIT LIMITS
Gate/runtime claims require a specified efficient implementation of the assumed
r-copy sampler, coherent Weyl eigenspace measurements, prime-dimensional
Clifford synthesis and classical finite-field linear algebra. A model allowing
exact prime-p Clifford/Fourier/SUM gates and Weyl spectral measurements suffices;
classical costs polynomial in n and p can be absorbed in the displayed bound.
A finite universal gate-set formulation requires an additional approximation
analysis bounding accumulated errors, including rejection preparation; it is not
proved here by merely counting exact Clifford operations. Measuring all sites
and forgetting outcomes over-refines a Weyl eigenspace and is invalid.
The simultaneous tensor representation with coefficients (1,a,b), satisfying
1+a^2+b^2=0, is only part of the separate escape-law proof. This review does
not infer that its Fourier law, normalization or constants are correct solely
from that cancellation. Its efficient sampler must also be established.
No new literature novelty determination or external human refereeing is claimed.
Disposition: learner bridge passes as a conditional theorem with explicit
p-dependent constants; unconditional odd-prime theorem awaits the escape audit.
