ARBITRARY MIXED INPUTS IN EVERY ODD-PRIME LOCAL DIMENSION
Private research extension, Riemann A, October 18, 2026.
This is a proof checkpoint, not a manuscript or publication package.

Provenance and preservation
The user requests investigation, closest-result checks and independent verification,
while preserving the current qutrit paper package. The fixed source package is
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Nothing in it is edited. This note generalizes the escape and learner research
checkpoints in the two earlier riemann_A_mixed_qutrit_* directories.

1. STATEMENTS AND MODEL
Let p>=3 be prime, n>=1, D=p^n and V=F_p^(2n). Use phase-consistent Weyls
 W_(a,b)=omega^(-a.b/2) Z^a X^b, omega=exp(2pi i/p),
 [u,v]=a.b'-b.a', W_u W_v=omega^([u,v]/2) W_(u+v).
Let rho be any density operator on (C^p)^tensor n, chi(u)=Tr(rho W_u), and
F=max_s <s|rho|s> over pure n-qudit stabilizers. Choose ANY maximizing s and
write S for its unsigned Weyl Lagrangian. All subspaces below are LINEAR.

ESCAPE THEOREM. There is an efficiently sampleable distribution B_rho such that
for every proper T<S:
 p=3 mod4: B_rho(S\T)>=3(p-1) F^6/(4p^3), using six input copies;
 p=1 mod4: B_rho(S\T)>=(p-1) F^4/(2p^2), using four input copies.
The copy counts are sufficient counts, with no minimality assertion.
The law and arithmetic split use known skewed/tensor-Weyl measurement mechanisms.
The new-to-this-investigation increment is their mixed-input escape estimate.

LEARNER THEOREM. For every 0<tau<=1 and 0<epsilon,delta<1, given independent
copies of any such rho with F>=tau, a proper learner outputs a pure stabilizer
tableau with overlap >=F-epsilon with probability >=1-delta. Set L=ln(2/tau).
For fixed p its copies and gate/classical runtime are
 exp(O_p(L^2)) poly_p(n,1/epsilon,ln(2/delta)).
A conservative uniform-in-p bound is
 exp(O(p^2 L[L+ln p])) poly(n,p,1/epsilon,ln(2/delta)).
Constants in the second O are absolute in the exact-operation model below.
It is NOT a bound polynomial in log p, nor dimension-uniform quasipolynomial
in 1/tau with p-independent exponent. The explicit budgets in Section 8 are
the primary resource statement. No optimality or practical-cost claim is made.

Model: exact prime-dimensional Clifford/Fourier/SUM operations, coherent Weyl
eigenspace measurements, independent input copies and finite-field arithmetic,
with implementation cost polynomial in n and p. The constant-copy measurement
below is a stabilizer measurement on two or three copies. No purification,
conjugated rho, density-matrix access or free postselection is assumed. A separate
finite-universal-gate approximation analysis is not supplied by this theorem.

2. POSITIVE AUXILIARY LAW AND LOCAL PURITY ESTIMATE
Define A(x)=D^-1 Tr(rho W_x rho W_x^dagger). As in the qutrit proof, positivity
and Weyl twirling give A>=0 and sum_x A(x)=1. For the symplectic Fourier transform
hat A(u)=sum_x A(x) omega^[u,x], Weyl expansion gives hat A(u)=|chi(u)|^2.
A is a mathematical distribution; a physical two-copy A sampler is not assumed.

LOCAL LEMMA. If tau is a p-dimensional density matrix whose computational |0>
maximizes single-qudit stabilizer overlap f=tau_00, then
 g=sum_j tau_jj^2-Tr(tau^2)/p >= c_p f^2, c_p=(p-1)/(4p).
Proof: the p+1 stabilizer MUBs partition the p^2-1 nonidentity Weyl labels into
p+1 commuting lines. Weyl orthogonality gives sum_v <v|tau|v>=p+1 and
sum_v <v|tau|v>^2=1+P, where P=Tr(tau^2). Hence f>=1/p and
P<=min(1,(p+1)f-1). Also sum_j tau_jj^2>=f^2+(1-f)^2/(p-1).
Put h0=2/(p+1) and K0=(3p-1)(p+1)/(4p(p-1)). Exact subtraction gives
 1/p<=f<=h0:
 g-c_p f^2 >= K0(f-h0)(f-2(2p-1)/(3p-1)) >=0;
 h0<=f<=1:
 g-c_p f^2 >= K0(f-h0)(f-2/(3p-1)) >=0.
The first other root exceeds one; the second is <=h0. Endpoints are included.
This is the mixed extension of the same known MUB second-moment argument, not
a new MUB or design theorem. Sharpness of the estimate is not asserted.

For any hyperplane H<S, Clifford covariance and a computational GL(n,p) change
put s=|0^n>, S=<Z1,...,Zn>, H=<Z2,...,Zn>. Phase sums in computational PSD
principal blocks sigma_y=rho[(j,y),(k,y)] give
 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)].
Every summand in the second formula is nonnegative by
Tr(sigma^2)<=(Tr sigma)^2<=p sum_j sigma_jj^2. The target block has
sigma_0,00=F, so A(H)>=F^2/p. Normalize sigma_0 by t=Tr sigma_0>0.
Comparison with every ambient stabilizer v tensor |0^(n-1)> makes |0> a
single-qudit optimizer of tau=sigma_0/t with f=F/t. The local lemma yields
 A(S\H)>=c_p F^2.
No assumption that a conditioned block is pure is used.

3. PHYSICAL WEIGHTED-WEYL MEASUREMENT
Choose k nonzero scalars t_1,...,t_k in F_p with sum_i t_i^2=0. Then
 R_u=tensor_(i=1)^k W_(t_i u)
is an ordinary abelian unitary representation of V, as its Weyl phases cancel.
Consequently
 P_y=D^-2 sum_u omega^[y,u] R_u
are Hermitian orthogonal projectors summing to I. Tr P_y=D^(k-2), because
Tr R_u=0 for nonzero u and Tr R_0=D^k. For outcome Y on rho^tensor k,
 E omega^[u,Y]=product_i chi(t_i u).
Take two independent outcomes and subtract: B has Fourier transform
 hat B(u)=product_i |chi(t_i u)|^2.
If A_t(x)=A(t^-1 x), Fourier inversion gives B=A_(t1)*...*A_(tk).
Each scalar is nonzero, so this definition is normalized and well-defined.

Operational cost: the representation factors over n physical positions; for each
position, its two generating Paulis on k qudits commute since sum t_i^2=0.
Standard prime-p symplectic Clifford synthesis measures them jointly, with finite
polynomial overhead in p. It is important to retain the coherent eigenspaces.
This is a known weighted/skewed multi-copy stabilizer-measurement mechanism,
not a new primitive or an amplitude-tomography improvement.

4. ARITHMETIC SPLIT AND ESCAPE
If p=3 mod4, select a,b with a^2+b^2=-1. Existence follows by intersecting
the squares and minus-one minus the squares, each of size (p+1)/2. Both a,b
are nonzero because -1 is nonsquare. Use (t1,t2,t3)=(1,a,b), k=3. Since
nonzero scalar multiplication preserves H and S\H, each A_t has the same
masses on these sets as A. The three disjoint convolution events with exactly
one factor in S\H and two in H give
 B(S\H)>=3 A(H)^2 A(S\H)>=3(p-1) F^6/(4p^3).

If p=1 mod4, choose a with a^2=-1 and use (t1,t2)=(1,a), k=2. Two disjoint
events give
 B(S\H)>=2 A(H) A(S\H)>=(p-1) F^4/(2p^2).
Both cases extend to every proper T<S by enlarging T to a hyperplane H.
For p=5 no three-factor witness has BOTH a,b nonzero; the two-factor branch
handles it correctly. A zero coefficient is discarded, never inverted.
For p=3 the six-copy constant reduces to 1/18, preserving the qutrit theorem.

Pure-input relationship: A=p_rho=D^-1|chi|^2 when rho is pure. Qi equation (27)
with z=YR and R^2=-I has Y1=-a z1-b z2, giving exactly the weighted convolution
above. If b=0 the zero register disappears and the extra pure p-law sums to one.
The four/six-copy arithmetic split is already part of the closest known sampling
methods. These observations are not separate sampler or residue-class inventions.

5. COHERENT CONDITIONING PRESERVES THE GLOBAL OPTIMIZER
For a consistent product P of commuting stabilizer eigenprojectors fixing s,
w=Tr(P rho)>=F and rho_P=P rho P/w. Every pure stabilizer t has P|t>=0 or
sqrt(c)*phase*|t'>, t' stabilizer, 0<c<=1, by standard prime-p stabilizer
measurement closure. Thus <t|P rho P|t><=cF<=F, with equality for s. The exact
conditioned optimum is F/w and is still attained by s. This proves the hypothesis
needed to reapply escape on every favorable prefix, including tied optimizers.
In particular the cumulative acceptance on ORIGINAL copies is always >=F>=tau;
there is no nested product of unrelated inverse-F preparation penalties.

6. RATIONAL THRESHOLDS FOR GENERAL-p MIXED CORRELATIONS
Put eta=1/(4p^2), q=1-eta/4=1-1/(16p^2), c(x)=|Tr(sigma W_x)|^2.
For a nonzero Weyl with spectral probabilities z_j, write Delta=1-cos(2pi/p).
Then
 1-c(x)=sum_(j,l) z_j z_l [1-cos(2pi(j-l)/p)]
       >=Delta(1-sum_j z_j^2).
Since sin(pi/p)>=2/p, Delta=2sin^2(pi/p)>=8/p^2. Therefore c>1-3eta implies
max_j z_j>=sum_j z_j^2>1-3eta/Delta>=29/32>7/8.
Spectral projectors E,G of noncommuting Weyls obey EGE=E/p and
||E+G||<=1+1/sqrt(p)<=1+1/sqrt(3)<7/4. Two such high coefficients would
give Tr(sigma(E+G))>7/4, a contradiction. Thus labels with c>1-3eta commute.
For a code-supported sigma, any label with nonzero expectation commutes with
its code labels, by conjugation with definite-eigenvalue code Weyls.

For ANY spectral outcome of probability z_b, reverse triangle gives
|chi|>=2z_b-1 and hence z_b<=(1+|chi|)/2. If c<=1-eta, concavity of square
root implies z_b<=(1+sqrt(1-eta))/2<=1-eta/4=q. Correct target-eigenvalue
projection therefore amplifies target overlap by >=1/q, while Section 5 keeps
the target globally optimal. The projector has positive weight on a favorable
path. These proofs assume neither purity nor a qubit embedding.

7. ROBUST HIGH-SPAN CAPTURE
Write r=2k and kappa for the residue-class escape constant in Section 1.
Set a=kappa*tau^r/2, t0=eta/4, and
 m=ceil((4n^2 ln p+ln8)/a), h=ceil(128*t0^-2 ln(32m)).
Take m fresh B_sigma samples. For every label estimate c to error <=t0,
simultaneously with failure <=1/8, using h copies per label. Keep estimates
>1-2eta, include the current code span C, and extend to a full Lagrangian H;
abort only on observed nonisotropy or capped-preparation failure.
The procedure does NOT test the statistical mass-capture guarantee.

On the estimation event all sampled G={c>1-eta} labels are retained, and every
retained label has c>1-(9/4)eta>1-3eta. Section 6 makes the span isotropic.
There are <=p^(4n^2) subspaces of F_p^(2n), by the matrix-column-span overcount.
For any fixed H with B(G\H)>a, missing G\H on all m samples has probability
<=exp(-am). Union over all subspaces gives failure <=1/8 and makes the guarantee
uniform over the data-dependent output H. Thus with probability >=3/4,
 B_sigma(G\H)<=a and H is a full Lagrangian containing C.

To implement estimates: average h spectral eigenvalues to obtain z. Hoeffding
for bounded real and imaginary parts at tolerance t0/8 gives combined tail
<=4 exp(-h*t0^2/128)<=1/(8m). Squared-modulus stochastic error is at most
2sqrt(2)t0/8<t0/2. Evaluate the empirical squared modulus with deterministic
absolute error <=t0/2 to include classical rounding. Cyclotomic arithmetic or
rational enclosures for the fixed roots of unity with O(log(p)) extra precision
bits achieve this in polynomial time; no exact comparison with real numbers is
required. The retention and analysis thresholds themselves are rational.

8. EXPLICIT LEARNER, FAILURE BUDGET AND RESOURCES
Use the established qutrit bootstrap architecture with p possible eigenvalues.
Let
 J=ceil(ln(1/tau)/[-ln q])+1,
 b0=3a/(4p), s0=b0^J,
 Q=J(r*m+m*h+r+1), K=ceil(tau^-1 ln(4Q/s0)),
 R=ceil((2/s0)ln(2/delta)), N=R(J+1),
 v=ceil((2/epsilon^2)ln(4N/delta)).

One run maintains a coherent stabilizer-code projector P and its unsigned span C.
For at most J stages, prepare sigma=rho_P using capped original-copy requests,
construct H as in Section 7, and measure one fresh ORIGINAL rho copy in H's
stabilizer basis, recording that candidate. Independently draw x~B_sigma. Abort
if x lies in H or fails to commute with C; otherwise choose its eigenvalue
uniformly in F_p and extend the code. If rank reaches n, record the uniquely
specified code state and stop. Preserve candidates on every abort. A zero-weight
code exhausts its preparation cap; the physical algorithm does not recognize
zero weight or the hidden target and does not test whether the sampled x is low.

Fix one original global optimizer s and call a jump favorable when it is a
low-correlation target label with the correct eigenvalue. On a favorable prefix,
Section 5 preserves global optimality and F_sigma>=tau. For a good H=S, the
original basis measurement records s with probability F>=tau. For good H!=S,
escape applied to S intersect H gives B_sigma(S\H)>=2a; removing high mass
outside H costs at most a, so a favorable label occurs with mass >=a and its
correct eigenvalue guess costs 1/p. These labels commute with C subset S.
Partitioning over the random good H therefore gives stage progress probability
>=(3/4)min(tau,a/p)=b0. The equality uses r>=4 and the stated kappa<=1.
Recording s is absorbing. Induction gives ideal witness success >=s0, because
J favorable jumps would force tau*q^-J>1; early rank-n favorable termination
already records s. This is O(p^2 log(2/tau)) jumps, rather than n guesses.

Each conditioned-copy request allows at most K fresh original trials. A trial
coherently measures the specified commuting code eigenvalues and retains only
the joint desired outcome. Use Clifford conjugation to single-coordinate Z
measurements and undo; separately measuring physical tensor factors is invalid.
Wrong branches have bounded cost, even if their code weight is zero or tiny.
On the favorable witness process BEFORE s is recorded, acceptance is >=F>=tau.
Coupling that stopped witness to capped preparation loses probability at most
Q exp(-tau K)<=s0/4. No control of unrestricted wrong-prefix ideal successes is
needed. A physical run thus records s with probability >=s0/2.

Repeat R times independently. At most N candidates occur and the probability of
missing s throughout is <=delta/2. Conditional on the completed list, measure
each candidate's rank-one stabilizer projector on v fresh ORIGINAL rho copies.
Hoeffding and union bounds give simultaneous overlap error <=epsilon/2 with
failure <=delta/2. Return the candidate with largest estimate, or any fixed
stabilizer if no candidate exists. This proves the proper additive guarantee.

The deterministic original-copy budget is RQK+RJ+Nv. All tableau, measurement
and Clifford overheads are polynomial in n,p and the listed parameter counts.
Uniformly, with L=ln(2/tau):
 J=O(p^2 L), kappa^-1=O(p^2), r<=6,
 m=O(kappa^-1 tau^-r(n^2 ln p+1)), h=O(p^4 ln(32m)),
 ln(1/s0)=O(p^2 L[L+ln p]).
These yield the two complexity statements in Section 1. For unknown F choose
tau=epsilon: if F<epsilon every stabilizer already meets the additive guarantee;
otherwise the theorem applies. No promise about a specific noise channel is used.

9. DEPENDENCIES AND LIMITS
The all-p theorem extends the qutrit result within the SAME research family.
Known ingredients: odd-prime stabilizer measurement algebra and synthesis, MUB
second moments, weighted/skewed measurement arithmetic, and Chen/Qi bootstrapping.
The independently reviewed local mixed estimate and operational escape are the
principal increments. Lower bounds for general Weyl-amplitude tomography address
a different task; this learner does not reconstruct all amplitudes or rho.
Finite matrix experiments are corroboration only, not proof dependencies.
See separate proof, diagnostic and originality reports for their actual scope.
No human refereeing, formal proof-assistant verification, optimal resources,
composite-dimension extension, publication readiness or priority claim is implied.
