ODD-PRIME EXTENSION -- PROOF CHECKPOINT -- 2026-10-18

User scope: investigate extension, closest predecessors, independent verification;
preserve the current qutrit paper package. This is a research checkpoint, not a
new manuscript, Publisher handoff, publication decision, or release.

RESULT
The escape bound and proper additive agnostic stabilizer learner extend to every
odd prime p, for arbitrary mixed states on n p-dimensional systems, n>=1, in the
explicit exact prime-dimensional Clifford/coherent-measurement model.
Let F=max_s <s|rho|s>, S the unsigned Lagrangian of ANY maximizing stabilizer,
and T any proper linear subspace of S. A physically sampled difference law B has

 p = 3 mod 4: B(S minus T) >= 3(p-1)/(4p^3) F^6; six input copies per draw.
 p = 1 mod 4: B(S minus T) >= (p-1)/(2p^2) F^4; four input copies per draw.

These are sufficient copy counts, not minimum-copy claims. At p=3 the first
constant is 1/18, retaining the qutrit result as a special case. The arithmetic
split and weighted measurement are known ingredients, not a new primitive.

MECHANISM
A(x)=Tr(rho W_x rho W_x dagger)/p^n is a positive NORMALIZED distribution;
its Fourier transform is |Tr(rho W_u)|^2. In contrast, the characteristic
weight p_rho(u)=p^-n |Tr(rho W_u)|^2 has total mass Tr(rho^2).
The mixed local estimate is
 sum_j tau_jj^2 - Tr(tau^2)/p >= (p-1)/(4p) f^2,
when computational |0> maximizes stabilizer overlap f for tau. Complete MUB
second moments and two exact quadratic factorizations prove this. Hyperplane
blocks lift it to all n. Convolution of scaled A laws gives the displayed escape.
Stabilizer-code conditioning preserves a global optimizer and increases its
relative fidelity. Rational correlation thresholds, bounded rejection sampling,
repetition and fresh original-copy validation give the full learner.

For 0<epsilon,delta<1 the learner outputs a pure stabilizer s with
 <s|rho|s> >= F-epsilon with probability >=1-delta.
With a known lower promise F>=tau, L=ln(2/tau), the conservative resource bound 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)) times the corresponding polynomial.
When F is unknown, tau=epsilon suffices. This is not a p-independent or
polynomial-in-log-p guarantee; no practical resource advantage is demonstrated.

PROOF AND VERIFICATION
GENERAL_PROOF.txt SHA256:
c041807f92005436abdfae2c10ec583f17453903c4acbc7d7ec0d5d2763b9ef8
Separate assistant reviews examined escape, learner, complete combined proof,
verification code, and originality. FINAL_ODD_PRIME_PROOF_AUDIT.txt independently
read this exact combined source and found no fatal gap in the stated model.
These are internal assistant reviews, not external human refereeing or formal
proof-assistant verification. The learner review's earlier threshold version is
superseded for the final statement by the combined audit of rational thresholds.

Root diagnostics passed for 178 states in n=1, p=3,5,7,11,13,17,19 and n=2,
p=3,5. Thirty-six physical batch laws were checked. All optimizing stabilizer
Lagrangians were tested in these finite enumerations. Exact Fraction tests cover
1010 scalar grid identities; continuous validity rests on analytic factorization.
An independent implementation formed dense character projectors, without using
the affine-orbit construction, for p=5 two-copy and p=3 three-copy batches.
The complete outcome laws agreed within 2.8e-17; matrix residuals were <=5.62e-15.
The diagnostics use floating-point matrices, do not simulate the full learner,
and enumerate stabilizers only for their tested n=1,2. They do not prove all n.

LITERATURE AND SCOPED ORIGINALITY
Qi, Xu, Feng and Li, arXiv:2607.15559v1: pure-qudit escape and bootstrapping;
their mixed-input extension is an open direction. Chen, Gong, Ye and Zhang,
arXiv:2408.06967v3: mixed-qubit escape/complete learner and predecessor architecture.
Allcock, Doriguello, Ivanyos and Santha, arXiv:2510.06848v3: weighted/skewed
measurement, including the relevant arithmetic split; no new-sampler claim.
Hinsche, Eisert and Carrasco, arXiv:2505.15770: qudit measurements and explicit
conceptual expectation of an agnostic extension; their hidden-subgroup theorem
does not itself establish the arbitrary-mixed optimum guarantee here.
Noeller, Tran, Gachechiladze and Kueng, arXiv:2510.08070v2: multi-copy Weyl
measurement and amplitude-estimation hierarchy; a different output task.

Primary statements and relevant proof sections were checked, with actual Google
and Scholar discovery plus other web searches. No equivalent complete theorem
was identified in that coverage. This is a provisional originality judgment,
not worldwide priority or an exhaustive verified forward citation graph.
The likely increment is the all-prime mixed local inequality/escape and closing
the mixed-input learner analysis. Much of the learner composes known methods.
The earlier normalization error in the originality report was corrected there;
it was not an error in the proof or mathematical audits.

LIMITS AND NEXT DECISION
No composite-dimension theorem, optimal constants/copy counts, hardware gate
approximation robustness, efficient practical implementation or publication-ready
all-prime paper is claimed. No unresolved mathematical gap was found for the
stated theorem/model; independent human scrutiny remains appropriate.
The all-prime theorem belongs to the existing qutrit escape/learner family.
If manuscript revision is later authorized, consolidating it as the general
theorem with p=3 as a special case appears preferable to parallel overlapping
papers. The sealed qutrit source and ZIP remain intact; preservation is recorded
in PRESERVATION_RECEIPT.json. All new work is in this separate directory.
