FINAL INDEPENDENT ADVERSARIAL AUDIT: ODD-PRIME EXTENSION
Date: 2026-10-18
Audited source: GENERAL_PROOF.txt
Source SHA256: c041807f92005436abdfae2c10ec583f17453903c4acbc7d7ec0d5d2763b9ef8
Scope: complete mathematical proof and stated exact-operation resource model.
This audit independently read the whole source, rather than adopting previous
review conclusions. No source, qutrit baseline, sealed package or registry was
edited. This is internal mathematical review, not formal or human refereeing.
No fresh literature originality assessment or gate-library approximation proof
is included.

VERDICT
No fatal gap found. The complete escape bound and proper additive learner are
supported for arbitrary mixed n-qudit states, every odd prime p, n>=1, in the
explicit exact prime-dimensional Clifford/measurement model. Fixed-p and stated
conservative growing-p bounds follow from the explicit budgets. This finding
does not establish novelty, publication readiness, practical efficiency, or a
finite-universal-gate complexity theorem.

INDEPENDENT CHECKS AND OBJECTION DISPOSITIONS

1. Auxiliary law and local MUB inequality.
Positivity follows from Tr(A B)>=0 for positive operators; Weyl twirling gives
total mass one. The Weyl expansion/Fourier identity yields |chi|^2, consistent
with the stated symplectic signs. MUB second moments give 1+P, and bounding each
basis probability by the optimizer f gives P<=(p+1)f-1. The diagonal Cauchy bound
and P<=1 yield exactly the two claimed quadratic lower bounds. Their common
leading coefficient is
 p/(p-1)-(p-1)/(4p)=(3p-1)(p+1)/(4p(p-1)).
The first second-root 2(2p-1)/(3p-1)>1 for p>1; the second 2/(3p-1)
is at most 2/(p+1). The signs are correct on both closed intervals.
The block decomposition works for every n, including n=1 with one block.
The normalized target block inherits optimality by ambient product-stabilizer
comparison. Thus A(H)>=F^2/p and A(S minus H)>=(p-1)F^2/(4p).
Disposition: no purity assumption or endpoint gap found.

2. Physical PVM and convolution.
The t_i^2 cancellation, rather than sum t_i cancellation, is the necessary
condition for an ordinary representation. It gives R_u R_v=R_(u+v),
R_u dagger=R_(-u), projector orthogonality and completeness. For k=2 the
projectors have rank one; for k=3 rank D. Nonzero coefficients make every
nonidentity tensor trace vanish. With the chosen P_y sign,
E omega^[u,Y]=product chi(t_i u); subtraction of independent outcomes gives
the squared-modulus product. The pushforward A_t(x)=A(t^-1 x) has Fourier
transform hat A(tu), with no missing p or D factor because the underlying
finite-vector-space scalar map is bijective.
The representation factors into commuting physical-position Pauli generators;
joint stabilizer measurement is available in the stated model. Coherence inside
eigenspaces is essential and is expressly retained.
Disposition: no normalization, sign, rank, or operational-sampling gap found.

3. Arithmetic and all-n escape.
The squares-set intersection proves a^2+b^2=-1. When p=3 mod4 neither
coefficient can vanish. When p=1 mod4 the two-factor choice (1,a) with
a^2=-1 is nonzero. In particular p=5 is correctly routed through k=2;
a three-factor all-nonzero witness is unavailable there.
Scalar maps preserve every linear H and S minus H. The designated convolution
events are disjoint in factor space and each sums into S minus H, even though
additional configurations may also do so. This proves precisely
3(p-1)F^6/(4p^3) or (p-1)F^4/(2p^2). Enlarging any proper T to a hyperplane
only weakens the event and handles arbitrary codimension. The p=3 constant
is 1/18.
Disposition: zero-coefficient inversion and all-n lifting objections resolved.

4. Global optimization after conditioning.
For each stabilizer t, P t is zero or a scalar multiple of a stabilizer t'.
Its norm squared c is at most one, so its unnormalized rho overlap is <=cF.
The original optimizer s remains unchanged under P, attaining F. Thus the
conditioned optimum is exactly F/w, with w>=F, including ties. A fresh original
copy accepts all specified commuting constraints with probability w, rather
than requiring sequential independently compounded preparation costs.
Disposition: valid for coherent projector conditioning; no wrong-branch
weight lower bound is asserted or required.

5. Rational correlation thresholds and finite precision.
Delta=2 sin^2(pi/p)>=8/p^2 and eta=1/(4p^2) give
3 eta/Delta<=3/32. Hence c>1-3eta implies a spectral probability >29/32,
in particular >7/8. Noncommuting Weyl spectral projectors have overlap
EGE=E/p, including degenerate n-qudit eigenspaces, so their sum has norm
1+1/sqrt(p)<7/4. This excludes two such high noncommuting labels.
Code-supported states annihilate expectations of labels not commuting with C.
For any spectral probability z, the reverse triangle inequality gives
z<=(1+sqrt(c))/2. At c<=1-eta this is <=q=1-eta/4, proving favorable-jump
amplification. Strict and weak inequalities used at thresholds are compatible.
For estimates, empirical and true characteristic means have modulus <=1;
coordinate errors <=t0/8 imply squared-modulus error <=sqrt(2)t0/4<t0/2.
The stated Hoeffding coefficient 128 is sufficient. Rounding budget t0/2
leaves total error <=t0. Root-of-unity enclosure precision requires only
O(log p) accuracy bits, plus ordinary accumulator/index bit lengths.
Disposition: rational thresholds remove real-number comparison dependence;
no finite-precision threshold gap found in this exact-operation model.

6. Adaptive high-span capture.
On the estimation event every sampled G label is retained and all retained
labels commute, including their compatibility with C. Extending their span to
a Lagrangian is valid. The p^(4n^2) matrix-column-span overcount covers every
possible output subspace, so the union bound applies despite data dependence.
The chosen m gives failure <=1/8. Combining this with estimation failure
<=1/8 gives success >=3/4. Capture is an analytical guarantee and need not
be tested by the algorithm.
Disposition: adaptive-choice and hidden-good-event objections resolved.

7. Progress, stopped witness, and capped preparation.
When H=S the fresh original-copy basis sample returns s with probability F.
When H differs from S, escape on S intersect H supplies mass >=2a outside H.
Removing G outside H loses <=a; the remaining labels have c<=1-eta and
commute with C. The correct eigenvalue guess succeeds with probability 1/p.
Every outside-H jump is independent of C since C lies in H.
Thus the uniform good-stage witness probability is at least
(3/4)min(tau,a/p)=b0. It is legitimate to count recording s as absorbing even
if the actual algorithm continues; the candidate is preserved.
J favorable nonabsorbing jumps would imply tau q^-J>1, so iteration gives
ideal witness probability >=b0^J even when rank or recording causes early stop.
The stopped favorable witness alone has preparation acceptance >=tau.
Q bounds all conditioned requests in a run; coupling loses at most
Q exp(-tau K)<=s0/4, leaving more than s0/2. Unfavorable paths can have zero
acceptance and remain bounded by their caps. No coupling claim about all
unrestricted ideal successes is used.
Disposition: no hidden free-postselection or wrong-prefix acceptance gap found.

8. Selection, unknown F, and resources.
Independent repetitions miss s with probability <=delta/2. Conditioning on
the completed candidate list is legitimate because validation uses fresh
original copies. At most N candidates and the chosen v give simultaneous
epsilon/2 overlap accuracy with failure <=delta/2. Maximizing estimates then
loses at most epsilon. All outputs are pure stabilizer tableaus.
The deterministic budget RQK+RJ+Nv covers preparation attempts, original
basis samples and validation. J=O(p^2 L), kappa^-1=O(p^2), and
log(1/s0)=O(p^2 L(L+log p)). The remaining factors and logarithms fit the
stated polynomial overhead and conservative exponential. At fixed p this
reduces to exp(O_p(L^2)) times the displayed polynomial. For growing p it
does not become polynomial in log p. Exact Clifford implementation with
polynomial cost in p is an explicit hypothesis, not a derived hardware claim.
Taking tau=epsilon is valid for 0<epsilon<1; when F<epsilon every output
already satisfies the additive target. If tau=1 then J=1 and the same witness
argument still works.
Disposition: no endpoint, confidence-allocation or asymptotic resource gap found.

REMAINING LIMITS
The mathematical audit treats prime-dimensional stabilizer synthesis, measurement
closure and the basic Weyl/MUB identities as standard algebraic ingredients;
it checked their actual roles and compatibility rather than executing a
proof assistant or synthesizing hardware circuits. Literature provenance,
particularly exact overlap with the cited Chen/Qi architecture and mixed-input
predecessors, remains a separate assessment. The sealed qutrit package identity
quoted in the source was not freshly recomputed by this audit. None of these
limits supplies a detected counterexample to the stated theorem.

Source-bound disposition: PASS internal mathematical audit, within the stated
scope and exact-operation model; no requested proof correction identified.

