INDEPENDENT PUBLISHER PROOF AND CLAIM AUDIT
Mixed-state escape bounds and proper agnostic learning of stabilizer states
in odd-prime dimensions
Review date: October 18, 2026, America/Chicago

VERDICT

PASS for the two stated analytic theorems and unknown-optimum corollary, in
the explicitly stated exact Clifford/coherent-measurement model. I found no
material proof gap or incorrect claimed constant. This is an independent
assistant proof audit, not formal proof-assistant verification, external human
refereeing, a worldwide-priority finding, or publication authorization.

SOURCE AND SCOPE

Read the complete 356-line sealed manuscript and the complete GENERAL_PROOF
development proof. The source reviewed is:
 D:/LUMINARY ODDITIES LLC/AI Research/odd_prime_learner_prepublication_Riemann_A_20261018/manuscript/main.tex
 SHA256 63f1a1a2400046d0083e8c654a5730a6239a713f09560fce83bcea06153b05b1
The source hash was freshly read and matched. The development proof is:
 D:/LUMINARY ODDITIES LLC/AI Research/odd_prime_learner_prepublication_Riemann_A_20261018/proof/GENERAL_PROOF.txt
 SHA256 c041807f92005436abdfae2c10ec583f17453903c4acbc7d7ec0d5d2763b9ef8
Both source hashes were freshly rechecked after writing this report.
The final manuscript, rather than any producer's PASS label or prior qutrit
assessment, supplies the claims and argument audited here.

No source file or frozen package was edited. This report is outside the package.
Numerical suites and package integrity are assigned to a separate reviewer;
no numerical receipt is a premise of the analytic conclusions here.

EXACT CLAIM LEDGER

1. Every odd prime p >= 3; every n >= 1; every density operator rho on
   (C^p)^{tensor n}; F is the maximum squared overlap with a pure stabilizer.
   For the unsigned Lagrangian S of ANY global optimizing stabilizer and every
   proper LINEAR T < S, the physically sampleable law B_rho obeys:
      p = 3 mod 4: B_rho(S\T) >= 3(p-1) F^6 / (4 p^3), six copies;
      p = 1 mod 4: B_rho(S\T) >= (p-1) F^4 / (2 p^2), four copies.
   The claim is existence of sufficient sampling constructions, with polynomial
   overhead in n,p; not a minimum-copy or sharp-constant theorem.

2. For 0 < tau <= 1 and 0 < epsilon,delta < 1, with F >= tau, a proper learner
   returns a pure stabilizer tableau of true overlap >= F-epsilon with success
   probability >= 1-delta. For L=ln(2/tau), its copies and operation count are
       exp(O(p^2 L(L+ln p))) poly(n,p,epsilon^{-1},ln(2/delta)),
   with absolute constants. Fixed p gives quasipolynomial inverse overlap.
   Inputs are independent original copies. The model explicitly allows exact
   prime-dimensional Clifford operations and coherent Weyl eigenspace
   measurements. It does not assume purification, conjugated copies, an
   optimizer oracle, matrix entries, or uncharged postselection.

3. Taking tau=epsilon gives a proper additive learner without an overlap
   promise. If F<epsilon every proper output already meets the guarantee.

ANALYTIC AUDIT

A. Weyl conventions and mixed positive overlap law (manuscript lines 74-99)

The phase convention W_(a,b)=omega^{-a.b/2} Z^a X^b is consistent with
ZX=omega XZ. Direct multiplication gives the displayed [u,v]/2 phase;
there is no switched conjugation sign in the subsequent transform.

For A(x)=D^{-1}Tr(rho W_x rho W_x^dagger), positivity follows because the
trace of a product of two positive operators is nonnegative. Weyl twirling
gives sum_x A(x)=1, including mixed rho. Independent Weyl expansion gives
 A(x)=D^{-2} sum_v |chi(v)|^2 omega^{[x,v]},
and with the manuscript's plus-sign Fourier convention this yields
hat A(u)=|chi(u)|^2, since [u,x]+[x,v]=[u-v,x].

The distinction from D^{-1}|chi(u)|^2 is essential and correctly retained:
the latter has total mass Tr(rho^2), whereas A has total mass one. Pure inputs
make them equal. No physical two-copy sampler for A is assumed.

B. Local mixed inequality, including all endpoints (lines 101-131)

For the complete p+1 stabilizer MUBs, write z_v for outcome probabilities.
The partition into Weyl lines gives sum z_v=p+1 and sum z_v^2=1+P with
P=Tr(tau^2). Thus global single-system optimality at |0> implies
 f>=1/p, P<=min(1,(p+1)f-1).
Independently, the remaining computational probabilities imply
 sum_j tau_jj^2 >= f^2+(1-f)^2/(p-1).

Subtracting c_p f^2, c_p=(p-1)/(4p), leaves quadratic leading coefficient
 K_0=p/(p-1)-(p-1)/(4p)
    =(3p-1)(p+1)/(4p(p-1)).
With h_0=2/(p+1), the exact two resulting factorizations are:
 low-purity branch:
 K_0 (f-h_0) (f-2(2p-1)/(3p-1));
 purity-at-most-one branch:
 K_0 (f-h_0) (f-2/(3p-1)).
These follow from the indicated upper bounds on P, in the correct inequality
direction because P enters with coefficient -1/p. The first other root
exceeds 1 for p>1; the second is below h_0. Both products have the claimed
signs on [1/p,h_0] and [h_0,1], with equality at the shared endpoint allowed.
There is no purity assumption or omitted low-f region.

C. Lift to hyperplane masses (lines 133-153)

Clifford covariance puts an arbitrary optimizer at |0^n>. An additional
computational linear change fixes that state while putting any hyperplane of
S at span(Z_2,...,Z_n). In positive principal blocks sigma_y on the first
coordinate, direct phase summation gives
 A(H)=p^{-1} sum_y Tr(sigma_y^2),
 A(S\H)=sum_y [sum_j sigma_y,jj^2-p^{-1}Tr(sigma_y^2)].
The normalization p^{-1} is correct: |H|/D=p^{n-1}/p^n.

Every block contribution to the second sum is nonnegative because
 Tr(sigma^2)<=(Tr sigma)^2<=p sum_j sigma_jj^2.
The block sigma_0 contains diagonal entry F and so gives A(H)>=F^2/p.
Normalizing that block is legitimate since Tr(sigma_0)>=F>0. A better
single-system stabilizer than |0> would, tensored with the remaining zeros,
improve the original global optimum. Consequently the local lemma applies to
sigma_0/Tr(sigma_0), and its trace cancels exactly to give c_p F^2.
For n=1 there is just one block and H={0}; no separate exceptional case fails.
Ties between original optimizing stabilizers do not affect this reasoning.

D. Operational distribution and arithmetic split (lines 155-192)

For nonzero weights t_i with sum_i t_i^2=0, R_u=tensor_i W_(t_i u) is an
ordinary representation of the additive group V: the projective phases cancel.
The displayed P_y are orthogonal Hermitian projectors by character
orthogonality, sum to I, and have trace/rank D^{k-2}. For k=2 they are
rank-one projectors; for k=3 they have rank D. No nonexistent rank is used.

With the plus-sign projector definition, E omega^{[u,Y]}=product_i chi(t_i u).
Two independent outcomes and subtraction therefore give exactly
 hat B(u)=product_i |chi(t_i u)|^2.
For A_t(x)=A(t^{-1}x), its Fourier transform is hat A(tu), without an extra
Jacobian on the finite field. Hence B is the convolution of the A_t laws.
The nonzero-weight condition ensures normalization and legitimate inversion.

For p=3 mod4, the two square sets of size (p+1)/2 intersect, yielding
a^2+b^2=-1. Neither coefficient can vanish because -1 is nonsquare.
Weights (1,a,b) give k=3. For p=1 mod4, take a^2=-1 and k=2 with weights
(1,a), where a is nonzero. This also avoids the p=5 zero-coefficient trap.

Scalar multiplication preserves a linear hyperplane H and S\H. The three
events with one convolution input in S\H and the other two in H are disjoint
and all sum into S\H. The two analogous events in the k=2 branch have the
same property. Thus the constants are exactly
 3 (F^2/p)^2 c_p F^2=3(p-1)F^6/(4p^3),
 2 (F^2/p) c_p F^2=(p-1)F^4/(2p^2).
Enlarging a proper linear T to H gives S\H subset S\T, in the needed direction.
At p=3 the first constant is 1/18, matching rather than changing the baseline.

The measurement is operational in the declared model. The two generators on
each position's k registers commute and are independent, and standard
symplectic Clifford synthesis measures their joint eigenspaces. This retains
the joint projectors rather than incorrectly measuring tensor factors and
discarding their coherences. Construction and choice of field weights can be
performed with polynomial overhead in p, as permitted by the theorem.

E. Exact global-optimizer preservation (lines 194-227)

Let a consistent stabilizer code projector P fix the chosen original optimizer
s. Its weight is w>=F. For every pure stabilizer t, stabilizer measurement
closure gives P|t>=0 or sqrt(c) times another normalized stabilizer t', where
0<c<=1. Therefore <t|P rho P|t>=c<t'|rho|t'><=F, and s attains F.
After normalization the ambient stabilizer optimum is exactly F/w, attained
by the same s. This controls ALL competing stabilizers, not merely states
inside the code. Tied optimizers are permitted.

A correct new eigenspace projection has probability at most q when its label
is low correlation, so the overlap increases by at least 1/q. The cumulative
code acceptance on ORIGINAL copies stays at least the ORIGINAL F. It is not a
product of separate inverse-conditional-overlap costs.

F. General-p high correlations and amplification (lines 204-225)

With eta=1/(4p^2), the minimum separation of distinct eigenvalue phases is
Delta=1-cos(2pi/p)=2 sin^2(pi/p)>=8/p^2. Thus c(x)>1-3eta forces a spectral
outcome of probability >1-3eta/Delta>=29/32>7/8. For noncommuting Weyls,
the relevant spectral projectors satisfy EGE=E/p, GEG=G/p; this follows by
averaging the nontrivial Weyl commutator character. Hence
||E+G||<=1+1/sqrt(p)<7/4. Two high noncommuting labels would violate this
operator norm bound. The zero label commutes automatically.

A code-supported state has zero expectation for a label not commuting with its
code, by conjugation with a definite-eigenvalue code Weyl. Finally the reverse
triangle inequality for the spectral probability z_b gives
 z_b<=(1+|chi|)/2<=(1+sqrt(1-eta))/2<=1-eta/4=q
for c<=1-eta. No Hermitian-qubit reduction is assumed.

G. Finite-precision estimation and data-dependent capture (lines 229-252)

Spectral measurements of the generally non-Hermitian Weyl give unit-modulus
complex eigenvalues. Hoeffding separately for their real and imaginary parts
at tolerance t_0/8 gives tail <=4 exp(-h t_0^2/128), using their [-1,1] ranges.
The selected h makes this <=1/(8m). On the good event, the complex mean error
is at most sqrt(2)t_0/8, and the squared-modulus error is at most
sqrt(2)t_0/4<t_0/2. Deterministic root-of-unity approximation with error budget
t_0/2 therefore gives total error <=t_0. Averaging rounded eigenvalues does not
amplify their error by h; O(log p) extra precision beyond counters suffices.

All sampled labels in G={c>1-eta} are retained. A retained label has
c>1-9eta/4>1-3eta, so retained labels and C span an isotropic space.
For a fixed H with B(G\H)>a, all m samples missing G\H has probability
<=exp(-am). There are <=p^{4n^2} subspaces, using column spans of square
2n by 2n matrices as an overcount. The stated m makes the union bound <=1/8.
Consequently even the DATA-DEPENDENT completed H obeys the capture bound,
except on this event or the <=1/8 estimation error event. Capture success
>=3/4 is justified; there is no independence assumption between H and samples.
The algorithm need not recognize the hidden mass event.

H. Favorable witness and logarithmic depth (lines 254-289)

On a favorable prefix C subset S and all eigenvalue choices fix s. Its
conditioned optimum is >=tau. Given good capture, if H=S the independent
ORIGINAL basis measurement records s with probability F>=tau. If H differs
from S, then S intersect H is proper, and escape gives mass >=2a outside H
within S. Removing G\H costs at most a, leaving low-correlation target-label
mass >=a. Such a label commutes with C, is outside H and therefore independent
of C; its correct eigenvalue is guessed with probability 1/p.

Partitioning over good H gives stage probability >=(3/4)min(tau,a/p)=b_0.
Here a/p<=tau follows from r>=4, kappa<=1 and tau<=1. Candidate success and
favorable progress need not be disjoint for this lower bound. Wrong guesses
are not presumed detectable or cheap.

Every favorable nonabsorbing jump amplifies overlap by >=1/q. The choice
J=ceil(ln(1/tau)/[-ln q])+1 gives tau q^{-J}>1. Thus J favorable jumps without
an earlier recorded target are impossible. Rank-n favorable termination also
records s. Treating recording as absorbing and inducting through at most J
stages proves ideal witness probability >=b_0^J=s_0. This is logarithmic in
inverse overlap (with p^2 dependence), rather than n target-eigenvalue guesses.

I. Capped preparation, wrong branches and stopping (lines 276-295)

For each conditional-copy request the specified current code is tested on at
most K fresh original copies. On a favorable prefix its cumulative acceptance
weight is >=F>=tau, so cap failure <=(1-tau)^K<=exp(-tau K). The normalized
accepted state is exactly rho_P, independently for fresh requests; finite caps
do not bias the accepted state when they succeed.

The coupling need only follow the target-fixing witness UNTIL s is first
recorded. There are at most Q requests before that stopping time. A union bound
therefore loses at most Q exp(-tau K)<=s_0/4. This yields actual target-listing
probability >=3s_0/4, and the manuscript safely uses only s_0/2.

No acceptance lower bound is imposed on wrong prefixes. Their weight may be
arbitrarily small or zero; the physical algorithm still terminates within the
same cap. Later aborts do not erase already recorded candidates. This is why
it is legitimate to stop the analytic coupling upon first recording s, even
though the physical run can continue. An ideal conditional state on a zero-
weight wrong prefix need never be defined in this stopped favorable process.
There is no free-postselection assumption or charged-success-only runtime.

J. Original-state validation and deterministic budget (lines 297-318)

Independent repetition R makes total failure to list s <=exp(-Rs_0/2)<=delta/2.
At most J basis samples plus one final code-state candidate occur per run,
so N=R(J+1) is valid even with aborts. Conditional on the entire random list,
validation on NEW ORIGINAL rho copies is independent. Bernoulli Hoeffding
with v=ceil(2 epsilon^{-2} ln(4N/delta)) and a union bound gives simultaneous
overlap error <=epsilon/2 with failure <=delta/2. Maximizing estimates then
gives true overlap >=F-epsilon. Validation does not accidentally compare
fidelities in the final conditioned state.

There are at most J(rm+mh+r) conditional-copy requests in a run: r copies for
each of m capture draws, h for each estimated sampled label, r for each progress
draw. Q includes additional slack. All rejection trials, successful or not,
are charged in RQK; original basis measurements cost at most RJ; final overlap
validation costs at most Nv. Thus RQK+RJ+Nv is a deterministic original-copy
budget on favorable and unfavorable branches alike.

For an empty candidate list the returned fixed stabilizer is still proper.
If the promise F>=tau fails after setting tau=epsilon, the unknown-optimum
corollary remains valid because F<epsilon makes every pure stabilizer's
nonnegative overlap already exceed F-epsilon. Runtime caps never used F.

K. Uniform-in-p complexity and endpoints (lines 303-310)

Uniformly for p>=3, kappa^{-1}<=2p^2, r<=6, -ln q>=1/(16p^2), and
J<=16p^2 ln(1/tau)+2=O(p^2 L), with L>=ln2. Furthermore
 b_0^{-1}=8p/(3 kappa tau^r),
 ln(1/b_0)=O(ln p+L)
with absolute constants. Therefore
 ln(1/s_0)=J ln(1/b_0)=O(p^2 L(L+ln p)).
The remaining m,h,Q,K,R,N,v factors in the copy budget are products of fixed
polynomial powers in n,p,epsilon^{-1},ln(2/delta), logarithms of these
quantities, and exp(O(L)) or exp(O(ln(1/s_0))). They are absorbed by the stated
uniform bound. Log n in h and K is absorbed by a fixed polynomial in n; it
does not leak into the exponent multiplying L. Conventional counter and
finite-field bit costs likewise have polynomial overhead.

At tau=1, J=1 exactly and the proof is sound: a favorable nonabsorbing jump
would make overlap >1, so successful target recording must occur. The bound
remains conservative. For n=1, rank-one completion immediately identifies
the correct target on a favorable extension. For p=3, the baseline constant
and proof are retained. The p=5 arithmetic exception is explicitly addressed.
The confidence range 0<delta<1 and accuracy range 0<epsilon<1 do not produce
zero budgets. Conservative computable upper bounds can replace logarithmic
ceilings while recomputing dependent budgets consistently; no exact comparison
with a transcendental logarithm is required.

STANDARD DEPENDENCIES AND EXTERNAL CROSS-CHECK

The proof uses standard odd-prime stabilizer projection closure and Clifford
symplectic synthesis, Weyl orthogonality/twirling, and complete stabilizer MUB
second moments. These are attributed. As a targeted primary-source check I
read Qi et al., arXiv:2607.15559v1, Theorem 4/Appendix A and Section III-A,
and its mixed-input discussion in Section VI:
 https://arxiv.org/html/2607.15559v1
That source supplies the standard isotropic-space synthesis fact and pure-input
MUB geometry; it does not substitute for the mixed local proof above. Hostens
et al.'s original Clifford/stabilizer framework was identified at:
 https://arxiv.org/abs/quant-ph/0408190
The substantive novelty/priority search is assigned separately. This targeted
dependency check is not described as a comprehensive literature search.

NONBLOCKING PRESENTATION SUGGESTIONS

1. A reader may benefit from one explicit sentence that the ideal witness is
   killed when it ceases to be target-fixing and is stopped once the target is
   first recorded; zero-weight wrong-prefix conditional states are not needed
   for its definition. The present stopped-coupling argument already supports
   this interpretation and is valid; this is a clarity suggestion, not repair.
2. The local quadratic coefficient identity and the elementary bound
   kappa^{-1}<=2p^2 could be included in an appendix if a more checkable algebra
   or uniform-constant derivation is desired. Both are already reconstructible.

REMAINING LIMITS

No material analytic obligation remains for the exact stated theorems after
this audit. Full publication readiness still depends on the separate code,
package, presentation and closest-literature assessments. This report does
not claim sharp constants, minimum numbers of copies, practical efficiency,
dimension-polylogarithmic cost, composite-dimensional extension, an
approximate-local-optimizer result, or finite-universal-gate robustness.
Numerical diagnostics do not implement the full learner and cannot prove the
all-n result. The preserved qutrit theorem is a special case of this same
consolidated family, not another independent paper counted by this audit.
