FINAL MATH COOK / MATH PAPER PUBLISHER ASSESSMENT
October18,2026, America/Chicago

Mixed-state escape bounds and proper agnostic learning of stabilizer states in
odd-prime dimensions. Scott Schubr. Nine-page consolidated assessment draft.

VERDICT
Analytic validity: PASS. Full finite replay and package identities: PASS.
Originality/novelty: provisionally supported within the recorded current primary
literature and citation coverage. Value: a substantive, focused specialist
contribution; broader impact, practical efficiency and optimality remain unproved.
Readiness: mathematically and editorially ready for a publication decision in
the exact operation model, with nonblocking minor suggestions recorded below.
No material proof gap, required theorem repair or directly subsuming result was
identified. These are independent AI-assisted reviews, not external human
refereeing or formal proof-assistant verification.

This is the strongest current candidate in the SAME escape/learner family. It
contains the private qutrit theorem exactly at p3 and should supersede that
candidate editorially. Do not count or publish both as independent overlapping
works. The qutrit package remains preserved as a source checkpoint and prior
assessed edition. Subtree, theta and self-transport are different works.

Scott authorized preparation, delivery to Publisher, assessment and a direct
report to him. No public upload, submission, publication or Archive release was
authorized by this intake, and none was performed. Both sealed packages are
unchanged. Recommendations here do not constitute a manuscript revision.

SOURCE IDENTITY
Directory: D:/LUMINARY ODDITIES LLC/AI Research/odd_prime_learner_prepublication_Riemann_A_20261018
ZIP SHA256: 35e4dffd9a95256ba90cb957dc6e261e53aa99e9086ee3e8cea7ad7b051a18ef
TeX SHA256: 63f1a1a2400046d0083e8c654a5730a6239a713f09560fce83bcea06153b05b1
PDF SHA256: bff8ecbc813a36fed79653fe0420209f7e9bfa83c93b4dbc66e09e145b01b80b
GENERAL_PROOF SHA256: c041807f92005436abdfae2c10ec583f17453903c4acbc7d7ec0d5d2763b9ef8
28 unique manifest entries plus manifest =29 files; all29 unique ZIP members,
CRC values, file bytes and handoff hashes checked independently. ZIP190957bytes.
The qutrit predecessor's complete40 files and ZIP also passed fresh preservation
checks, retaining ZIP SHA2afda27faa4b26a86e96cd04da481075c093109115c67d0a498b246c62589e75.
Hashes establish identity and preservation, not mathematical correctness.

EXACT CLAIMS
For every odd prime p, n>=1 and arbitrary mixed density rho, let F be its maximum
overlap with a pure stabilizer, S the unsigned Lagrangian of ANY global optimizer,
and T any proper LINEAR subspace of S. An ordinary-copy physical sampling law B
satisfies:
 p=3mod4: B(S minus T)>=3(p-1)F^6/(4p^3), six copies per sample;
 p=1mod4: B(S minus T)>=(p-1)F^4/(2p^2), four copies per sample.
These are sufficient copy counts, not minimum-copy theorems. At p3 the first
bound is precisely F^6/18. No composite-dimension or approximate-local optimizer
claim is included.

For F>=tau>0, 0<epsilon,delta<1, the learner outputs a pure stabilizer tableau of
original-input overlap>=F-epsilon with confidence>=1-delta. Set L=ln(2/tau).
Copies, quantum gates and classical time are bounded conservatively by
 exp(O(p^2 L(L+ln p))) poly(n,p,1/epsilon,ln(2/delta)),
with an absolute constant in O. For EACH FIXED p this is quasipolynomial in
inverse overlap and polynomial in the other displayed parameters. It is not
polynomial in logp or a p-independent quasipolynomial bound. Taking tau=epsilon
removes the overlap promise and retains bounded runtime even when it fails.

The model uses ordinary independent copies, exact prime-dimensional Clifford
operations, coherent Weyl eigenspace measurements and finite-field arithmetic
with polynomial overhead in n,p. It assumes no purification, conjugated input,
matrix entries, maximizing-state oracle or free postselection. A separate
fixed finite universal gate-set approximation/robustness theorem is absent.

ANALYTIC REVIEW
Root read the complete final TeX and all nine rendered PDF pages. The separate
proof reviewer independently read and derived the entire manuscript and frozen
GENERAL_PROOF checkpoint. Neither promoted the qutrit verdict or producer PASS
labels into proof of the broader claim.

The complete dependency chain passes: positive normalized Weyl overlap kernel;
its correct mixed-state Fourier transform; complete-MUB purity-sensitive local
estimate; PSD principal-block hyperplane lift; nonzero scaled convolution;
known arithmetic split; physical character projectors and their ranks/signs;
global-optimizer preservation under coherent code projections; p-dependent
uncertainty/correlation thresholds; finite classical precision; data-dependent
capture via a uniform subspace union bound; favorable branch amplification;
stopped-witness cap coupling; original-state candidate validation and total cost.

The local estimate is
 sum_j tau_jj^2 - Tr(tau^2)/p >= ((p-1)/(4p)) f^2,
when computational|0> maximizes local stabilizer overlap f. Its two factorizations
and closed endpoint ranges were rederived exactly. The all-system lift retains
mixed purity instead of using an invalid pure-state fidelity floor. The four-
or six-copy laws are positive convolutions of scalar-pushforwards of this
auxiliary overlap kernel; no sampler for the auxiliary law is assumed.

The delicate algorithmic point is correctly handled. Until the favorable witness
fails or first records the fixed original optimizer, its code fixes that optimizer
and original-copy acceptance is>=F>=tau. Only those active requests pay
 Q exp(-tau K)<=s_0/4.
No acceptance lower bound is imposed on wrong prefixes, whose costs are still
deterministically capped. Recorded candidates survive later aborts. Validation
uses fresh ORIGINAL rho copies. The n1 and tau1 endpoints, tied optimizers and
zero-weight wrong branches are covered. Root and proof reviewer independently
obtain J=O(p^2L), kappa^-1<=2p^2 and log(1/s_0)=O(p^2L(L+lnp)), with absolute
constants; the remaining budget RQK+RJ+Nv has the asserted dependence.

FRESH COMPUTATIONAL REVIEW
The integrity, --dense and --full documented modes all freshly passed in an
isolated byte-identical package copy using bundled Python3.12.14/NumPy2.3.5,
Python-B, enabled assertions and local BLAS thread limits. The full mode ran
the complete finite suite and dense check without relying on the producer's
earlier partial/memory-limited receipt.

The suite covers178 density inputs in the specified n1/n2 cases and primes,
36 raw physical batch-law checks (first four states per case, NOT all178), and
1010 exact Fraction grid identities. Grid checks do not prove continuous sign;
finite complex128 checks do not prove arbitrary n. The supplied distinct dense
character-projector implementation freshly passed its p3/p5 cases. Publisher
added a separate p7,n1,(1,2,3)-weight dense PVM check in joint dimension343:
all49 outcome probabilities agreed with the affine implementation to2.78e-17.
All-sector probability/character checks and the stated selected-sector matrix
checks passed. No full bootstrap learner, practical sample complexity, hardware
execution or arbitrary-n enumeration is claimed.

Source inspection and corruption controls establish that ordinary documented
checks are substantive. The producer integrity checker now rejects changed,
missing and extra files with explicit failures. Two residual robustness limits
are noted below. Independent exact manifest/ZIP auditing was stricter than the
producer checker. Both originals and both ZIPs remained byte-identical afterward.

ORIGINALITY AND VALUE
The closest prior theorem is Qi-Xu-Feng-Li2607.15559v1: pure-input odd-prime
proper agnostic learning and escape. Its conclusion VI explicitly leaves the
mixed-input distribution/anticoncentration extension as a future obstacle.
The present mixed local gap and operational escape remove that restriction and
support a complete proper learner across all odd primes.
https://arxiv.org/html/2607.15559v1

Chen-Gong-Ye-Zhang2408.06967v3 already prove mixed-QUBIT proper agnostic learning
and supply the main bootstrap architecture, including stronger local/list results
in that different setting. Allcock et al.2510.06848v3 supply weighted/skewed
measurements and the residue-class arithmetic; Noller et al.2510.08070v2 supply
related tensor-Weyl measurements. These methods are accurately credited.
https://arxiv.org/html/2408.06967v3
https://arxiv.org/html/2510.06848v3
https://arxiv.org/html/2510.08070v2

Hinsche-Eisert-Carrasco2505.15770v3 explicitly anticipate qudit agnostic extensions;
the proposed direction was therefore not invented here. Their mixed exact-
symmetry/gap task does not itself establish this best-comparator theorem.
https://arxiv.org/html/2505.15770v3

Recent Holt-Subramanian2609.38085v2 exact mixed stabilizer-group learning, bounded-
bond MPS/TTN proper learning, high-fidelity mixed-qubit single-copy learning in
Bittel et al.2610.02031v1, and the inspected pure-qubit improper/PFR or sample-
optimal algorithms do not subsume this arbitrary-mixed, proper odd-prime target.
The separate originality report records16 primary leads, exact task/model
differences, actual Google/Scholar and bounded forward-citation coverage, and
current versions. No equivalent full theorem was located in that coverage.
This is provisional novelty with moderate confidence, not worldwide priority.
https://arxiv.org/html/2609.38085v2
https://arxiv.org/html/2609.30148v1
https://arxiv.org/html/2610.02031v1

The strongest positive case is removal of a real input-class restriction in
the nearest predecessor, with an analytic mechanism and an actual operational
learning consequence. It is more than a date/format/constant change. The strongest
skeptical reading is that much of the mathematics and algorithmic architecture
is inherited and the local inequality is elementary. This limits conceptual
breadth but does not remove the identifiable new mixed-input theorem. A focused
specialist research paper/preprint is proportionate. No evidence supports a
general new paradigm, optimal resources, practical efficiency or broad complexity
breakthrough. Inverse-overlap complexity class is not improved over the matched
prior pure-input/qubit architectures.

MINOR RECOMMENDATIONS AND READINESS
No mandatory mathematical correction was found. The nine-page PDF is coherent,
legible, fully referenced and explicit about scope/AI assistance. Recommended
nonblocking improvements for a separately authorized revision:
1. Optional related-work citations to Iyer-Liang for the earlier mixed positive
   Fourier-dual overlap kernel, Holt-Subramanian for recent mixed-qudit exact-group
   and general difference sampling, and Bittel et al. for the single-copy
   high-fidelity/exact-pure alternatives. The existing paper does not falsely
   claim their ingredients as inventions.
2. Replace numerical diagnostic asserts with explicit failures, or reject an
   optimized Python environment. Negative controls show a deliberately false
   escape coefficient can emit PASS when optimization removes the assertions.
   Enabled-assertion ordinary replay passed honestly. The theorem is analytic.
3. Reject duplicate and noncanonical manifest paths explicitly. The actual
   sealed manifest is unique/canonical and was independently checked, but a
   duplicate entry is accepted by the supplied checker.
4. Optional proof exposition: explicitly kill the auxiliary favorable witness on
   first unfavorable jump as well as absorbing it on first recorded optimizer.

Do not silently patch the sealed package. Any requested changes should preserve
this edition, bind new sources/hashes, and rerun the checks affected by changes.
Human specialist scrutiny would be useful; internal assistant assessments do
not become external refereeing by being numerous or agreeing.

SUPPORTING REVIEW RECORDS
proof-and-claims.txt: complete independent analytic derivation and claim ledger.
code-and-replay.txt: full fresh replay, independent p7 check and negative controls.
originality-and-value.txt: current sources, citation coverage, novelty/value axes.
publisher-independent-reading.txt: root's independent proof and source reading.
pdf_review/presentation-review.txt: all-page inspection with source identity.
independent_replay/: copied sources, commands/logs, outputs, controls and final
preservation receipt. These files remain outside both original packages.

Assessment and reporting complete. Current publication candidate is the single
consolidated all-prime paper; private qutrit predecessor remains preserved.
