INDEPENDENT PUBLISHER ORIGINALITY, VALUE AND OVERLAP REVIEW
Consolidated odd-prime mixed-state escape / proper agnostic learner
October 18, 2026 (America/Chicago); online refresh completed after midnight UTC.

SOURCE AND REVIEW SCOPE

Title: Mixed-state escape bounds and proper agnostic learning of stabilizer
states in odd-prime dimensions. Author: Scott Schubr.
Sealed manuscript read in full:
D:/LUMINARY ODDITIES LLC/AI Research/odd_prime_learner_prepublication_Riemann_A_20261018/manuscript/main.tex
Fresh SHA256: 63f1a1a2400046d0083e8c654a5730a6239a713f09560fce83bcea06153b05b1.

This is a fresh all-prime originality/mathematical-value review. It reads and
extends the earlier Publisher qutrit literature comparison rather than treating
that verdict as evidence that a materially broader theorem is automatically
correct. Full validity, computational replay, manuscript layout and package
integrity are separate assigned reviews. No numerical scripts were run in this
literature subreview. No sealed source was changed, uploaded or published.

VERDICT

Subject to independent analytic validity and evidence review, the manuscript
has a credible, provisionally original and worthwhile specialist research
contribution. It extends a pure-input odd-prime learner to arbitrary mixed
inputs for every odd prime, while preserving a proper pure-stabilizer output
and fixed-prime quasipolynomial inverse-overlap resource class. This is a
meaningful removal of a stated restriction of the nearest theorem. It is
stronger and more coherent as one paper than separately publishing the private
qutrit candidate followed immediately by its containing all-prime extension.

The novelty case is specific. The new claim is the purity-sensitive local
estimate, its operational mixed-input escape consequence and the complete
mixed learner composition. Fourier overlap kernels, MUB second moments,
weighted Weyl measurements, the mod-4 arithmetic and bootstrapping are known
ingredients. Most algorithmic architecture is adaptation and careful analysis.
There is no new general learning paradigm, practical implementation or improved
inverse-overlap complexity class. Mathematical value is substantive within the
specialist problem, with broader impact and optimality unestablished.

No equivalent arbitrary-mixed, proper odd-prime learner with these guarantees
was found in the sources and citation coverage below. This supports provisional
novelty with moderate confidence; it is not exhaustive worldwide priority.

EXACT CLAIM BEING COMPARED

Input: arbitrary density rho on (C^p)^(tensor n), n>=1, every odd prime p.
F=max_s <s|rho|s>, with pure stabilizer comparators. The overlap convention is
squared fidelity with the pure comparator. For the unsigned label Lagrangian S
of ANY exact global optimizer and each proper LINEAR subspace T<S:

  B_rho(S\T) >= 3(p-1) F^6/(4p^3),  p=3 mod4;
  B_rho(S\T) >= (p-1) F^4/(2p^2),    p=1 mod4.

B is the difference of independent outcomes of known weighted commuting-Weyl
measurements. A draw consumes six copies in the first branch and four in the
second. These are sufficient counts, not copy lower bounds or optimal counts.
The auxiliary A(x)=p^(-n)Tr(rho W_x rho W_x^dagger) is a normalized positive
mathematical law with Fourier transform |Tr(rho W_u)|^2. A physical two-copy
sampler for A is not assumed. In particular A must not be confused with the
mixed, subnormalized characteristic weight p^(-n)|Tr(rho W_x)|^2.

Output: a classical pure stabilizer tableau s of overlap >=F-epsilon with
probability >=1-delta. With F>=tau>0 and L=log(2/tau), the stated resources are
exp(O(p^2 L(L+log p))) poly(n,p,epsilon^(-1),log(2/delta)), with an absolute
constant in the displayed O. Fixed-p resources are quasipolynomial in 1/tau
and polynomial in n, inverse accuracy and log confidence. Tau=epsilon removes
the overlap promise, absorbing polynomial inverse accuracy in the exponential.

The model allows ordinary independent copies, exact prime Clifford operations,
coherent Weyl eigenspace measurement and finite-field arithmetic. It supplies
no purification, conjugated copies, matrix entries, target oracle or free
postselection. It does not prove composite local dimensions, approximate-local
maximizer list recovery, finite-universal-gate compilation robustness, uniform
polynomial log-p cost, or an efficiently practical implementation.

PRIMARY COMPARISONS AND CURRENT VERSION REFRESH

The numbered entries are paraphrases, with sections indicating what was read.
Current arXiv abstract histories and relevant HTML passages were reopened.

1. Wentao Qi, Boyan Xu, Shiguang Feng, Lvzhou Li,
Agnostic learning of qudit stabilizer states, arXiv:2607.15559v1, July 17, 2026.
Current record still lists only v1.
https://arxiv.org/abs/2607.15559
https://arxiv.org/html/2607.15559v1
Theorems 1/13 give proper agnostic learning for PURE inputs in every odd-prime
dimension; Theorem 7 gives (p-1)F^6/(4p^3) proper-subspace escape. Equation 27,
Sections III-A/III-B and IV supply the closest distribution, local MUB argument
and bootstrap. Section VI item 2 expressly identifies mixed-input distribution
and anticoncentration as further obstacles. This is the main statement/method
predecessor. Its purity assumption cannot simply be omitted. The proposed
mixed local inequality and operational law address precisely this restriction.
The growing-p asymptotic class is comparable; no complexity-class improvement
over that pure-only baseline is claimed.

2. Sitan Chen, Weiyuan Gong, Qi Ye, Zhihan Zhang,
Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic
estimation, arXiv:2408.06967v3, December 5, 2024; current version v3.
https://arxiv.org/html/2408.06967v3
Lemma 5.3, Theorem 5.5, Section 6 and Corollary 6.3 establish arbitrary-mixed
QUBIT escape and proper agnostic learning. The paper also treats approximate
local maximizers/list recovery, a separate strength absent from this manuscript.
The learning architecture, amplification and candidate validation are substantial
prior work. This qubit theorem does not itself supply a proper p-ary learner.
The new manuscript accurately credits this framework.

3. Jonathan Allcock, Joao F. Doriguello, Gabor Ivanyos, Miklos Santha,
Reconquering Bell sampling on qudits: stabilizer learning and testing, quantum
pseudorandomness bounds, and more, arXiv:2510.06848v3, August 25, 2026;
current version v3.
https://arxiv.org/html/2510.06848v3
Introduction, Theorem 43, Corollary 44 and distribution discussion were read.
Skewed sampling, register mixing and sum-of-squares construction are prior.
Corollary 44 already distinguishes p=1 mod4 from p=3 mod4 using square roots
of -1. Thus the arithmetic split is not new. Realizable learning and the stated
pure-input testing guarantees do not supply arbitrary-mixed proper additive
optimization. The manuscript's explicit attribution is appropriate.

4. Marcel Hinsche, Jens Eisert, Jose Carrasco,
The abelian state hidden subgroup problem: Learning stabilizer groups and beyond,
arXiv:2505.15770v3, May 27, 2026; PRX Quantum 7, 020337 (2026).
Current arXiv version v3; journal DOI 10.1103/6frk-891j.
https://arxiv.org/html/2505.15770v3
Introduction and Section II.4 were read. The introduction explicitly anticipates
qudit agnostic extensions from its Fourier routine, so that direction was already
proposed. Its mixed working algorithm requires exact expectation-one symmetries
and a gap, a different task from a best pure stabilizer comparator. Crediting
the conceptual precedent, as this manuscript does, avoids a false first-idea claim.

5. Jan Noeller, Viet T. Tran, Mariami Gachechiladze, Richard Kueng,
An infinite hierarchy of multi-copy quantum learning tasks,
arXiv:2510.08070v2, September 1, 2026; current version v2.
https://arxiv.org/html/2510.08070v2
Section 4, its tensor-Weyl measurement and Theorem 10 were read. Commuting
tensor-power measurements on arbitrary mixed inputs are prior; the qutrit
three-copy primitive is a special case. Independent-outcome subtraction is
standard Fourier processing. Amplitude-estimation copy lower bounds address a
different output task and cannot establish minimum-copy requirements here.

6. Vishnu Iyer, Daniel Liang,
Tolerant Testing of Stabilizer States with Mixed State Inputs,
arXiv:2411.08765v2, May 12, 2025; current version v2.
https://arxiv.org/html/2411.08765v2
Remark 2.13, Lemma 2.12, Facts 2.14--15 and the testing construction were read.
The positive Fourier-dual overlap law for mixed qubits is prior, with a different
normalization. The manuscript's adjoint-Weyl odd-prime identity is a standard
extension of this geometry, not its main originality. No mixed odd-prime proper
learner or optimizer escape is supplied. An additional citation would improve
the Section 2 context, but the present text does not claim the kernel as new.

7. Isaac Holt, Sathyawageeswar Subramanian,
Learning quantum symmetries, arXiv:2609.38085v2, September 30, 2026;
current version v2.
https://arxiv.org/html/2609.38085v2
Definition 9.1, Sections 9.1--9.3 and Theorem 9.27 were read. Their group is
defined by |Tr(rho W_x)|=1, and their Fourier-difference linearisation includes
the weighted sampling mechanism. Learning exact mixed-state stabilizers is
not maximizing a pure stabilizer comparator. For instance full-rank mixtures
can have trivial exact group yet nontrivial optimizing pure comparators. This
recent close sampling/learning source merits an additional related-work citation;
it does not remove the proposed optimization increment.

8. Lennart Bittel, Jens Eisert, Weiyuan Gong, Antonio Anna Mele, Louis Schatzki,
Adaptivity is all you need: Optimal stabilizer learning using just single-copy
measurements, arXiv:2610.02031v1, October 1, 2026; current record only v1.
https://arxiv.org/html/2610.02031
Freshly discovered through Qi's Scholar-related list. Theorem III.1 gives a
proper mixed-QUBIT learner when a known optimum lower bound exceeds sqrt(2/3).
Appendix A, Theorem A.1 extends EXACT PURE stabilizer learning to prime qudits.
Neither proves arbitrary-mixed odd-prime agnosticism for all positive tau.
This is useful recent context and an optional citation; it prevents implying
multi-copy necessity or the absence of mixed high-fidelity alternatives.

9. Constantin Cedillo Vayson de Pradenne, Jordan Cotler,
Proper Agnostic Learning of Matrix Product States and Tree Tensor Networks,
arXiv:2609.30148v1, September 24, 2026; current record only v1.
https://arxiv.org/html/2609.30148v1
Theorems 2.1--2.3 and explicit-comparator hypotheses were read. Arbitrary mixed
inputs are allowed, but proper output belongs to a bounded-bond MPS/TTN class.
The general reduction requires suitable improper-learning and explicit-target
optimization data. Those hypotheses are not supplied here for the full pure
stabilizer class. General stabilizers need not have small bond on a given
geometry; optimizing in a containing tensor class also does not enforce a
stabilizer output. No same-resource subsumption follows.

10. Letian Tang, Thomas Steckmann,
Learning Noise-Robust Stabilizer Structure via Bell Sampling,
arXiv:2610.03307v2, October 5, 2026; current version v2.
https://arxiv.org/html/2610.03307v2
Abstract, noise model and qubit formulation were read. It reconstructs low-nullity
ideal-qubit structure under promised Pauli noise. Its useful polynomial/noise
regimes do not solve arbitrary-mixed, pure-comparator optimization for odd primes.

11. Xingya Fan, Yongzhi Luan, Clifford Invariants and Three-Copy Quantum Tests,
arXiv:2610.14758v1, October 10, 2026; current record only v1.
https://arxiv.org/html/2610.14758v1
Density-invariant scope and Section 5.1 were read. Its qutrit cubic-invariant
and testing setting is related algebra, but the stabilizer-fidelity bound there
is introduced for pure inputs. No equivalent proper mixed all-prime learner was
identified. Removing it from the consolidated short bibliography is not a
material attribution defect because no cubic invariant is claimed as invented.

12. Srinivasan Arunachalam, Arkopal Dutt, Sabee Grewal, Aparna Gupte,
Marton's conjecture in polynomial time, arXiv:2609.20771v1, September 17, 2026;
current record only v1.
https://arxiv.org/html/2609.20771
Found via Chen's date-sorted forward citations. Section 1.2.2 and Section 8
give polynomial-time IMPROPER learning for arbitrary PURE QUBIT inputs, via
algorithmic PFR and boosting. Both comparator/output and input/local-dimension
guarantees differ. This is important complexity context, not a proper mixed
odd-prime theorem. No polynomial-time properness conclusion transfers silently.

ADDITIONAL PRIMARY LEADS SCREENED FOR MODEL MISMATCH

13. Rebecca Chang, Matthias C. Caro, Martin Larocca, Maxwell West,
Sample-optimal learning of stabilizer states, arXiv:2609.10974v1, September 10,
2026. Abstract/current record and Theorem 1 read: exact PURE QUBIT identification,
not arbitrary mixed agnostic optimization.
https://arxiv.org/html/2609.10974

14. Gerard Aguilar, Sofiene Jerbi, Jens Eisert, Lennart Bittel,
Efficient learning of Clifford disentanglers and typical t-doped unitaries with
exponentially more T gates, arXiv:2609.27565v1, September 23, 2026.
Current abstract read: hidden product structure in promised PURE states and
typical doped unitaries, not arbitrary density inputs/proper stabilizer optimum.
https://arxiv.org/abs/2609.27565

15. Tobias Haug, Efficient Learning of Clifford-Scrambled Product States,
arXiv:2609.27128v1, September 22, 2026. Current abstract read: structured
qubit product ansatz behind a Clifford; no same mixed agnostic guarantee.
https://arxiv.org/abs/2609.27128

16. Jicun Li, Wei Xie, Jun Wu, Honglin Chen, Xiang-Yang Li,
Asymptotic Entanglement Hiding under Stabilizer Restrictions,
arXiv:2608.18440v2, September 30, 2026. Read introduction/main task and
Theorem 1: restricted-measurement visible entanglement and distillation, not
pure-comparator learning. This new Google lead is a task mismatch, not a
contradiction with the present four/six-copy sampling bounds.
https://arxiv.org/html/2608.18440v2

SEARCH AND CITATION COVERAGE / ACCESS LIMITS

Actual Google, through this reviewer's background IAB browser:
https://www.google.com/search?q=%22mixed%22+%22odd-prime%22+%22agnostic%22+stabilizer
The visible result list was read; Qi was opened directly. Related code algebra,
resource witnesses and entanglement-hiding leads were screened by task. An
initial alias-based IAB opening timed out; a numeric browser binding succeeded.
No Chrome window, other chat's browser tab, account control or setting was used.

From the Qi arXiv abstract, followed the displayed Google Scholar link:
https://scholar.google.com/scholar_lookup?arxiv_id=2607.15559
Its one exact record had no visible cited-by link. The related-articles first
page showed about 100 results; ten visible entries were screened. This supplied
Bittel and Chang. From that page, followed Chen's Cited by 47 link. The first
ten relevance-ranked entries and first ten date-sorted entries were screened.
The latter exposed recent symmetry, tensor-network, noise, product-structure
and algorithmic-PFR work. This is bounded citation coverage, not all 47 full
texts or a complete forward-citation graph. No missing cited-by control was
treated as evidence of zero citations.

Additional generic web discovery (engine not identified as Google):
- mixed/qudit/agnostic/stabilizer combinations and proper mixed learners;
- mixed-state escape, anticoncentration and bootstrap with prime dimensions;
- Qi identifier and exact-title forward disclosures;
- MUB purity / stabilizer-fidelity inequalities;
- Holt/Subramanian agnostic terminology and mixed-qudit learners.
Relevant primary texts were read for every consequential comparison above.
AI literature summaries, search snippets and discussion-board entries were
discovery leads only. No paywall, CAPTCHA or browser security warning was
bypassed; no author was contacted; no manuscript uploaded. Unindexed or private
work and complete global-priority exclusion remain beyond this assessment.

VALUE, ORIGINALITY AND PRESENTATION JUDGMENTS

Validity: Separate full proof review is necessary and is not replaced by these
literature findings. The full manuscript was read for model matching; the
local-to-global argument selects a mixed conditioned block at the global
optimizer, retains its purity in a complete-MUB inequality, and transports a
positive gap through a physical convolution. These are the claims requiring
fresh analytic verification, not saved internal PASS labels.

Originality: Supported provisionally. Pure-to-mixed extension is not automatic
by linearity: the target optimizer can change across a mixture and escape is
nonlinear in rho. Pure Fourier self-duality fails. The new argument addresses
that issue and completes an operational guarantee. Kernel positivity itself,
uncertainty/projector bounds and capped rejection are standard mathematics.

Novelty: Supported with moderate, bounded-search confidence, conditional on
validity. No inspected primary statement subsumes the complete arbitrary-mixed
odd-prime proper learner. Do not use first-ever, globally unique or proved
priority language. A stated predecessor research direction does not by itself
constitute a completed theorem; it still deserves attribution.

Uniqueness: The distinguishing feature set is every odd prime, arbitrary density
input, exact globally maximizing stabilizer, every proper linear subspace,
positive mixed local gap, constant-size multi-copy draws, and a fully capped
proper additive learner. This is a useful characterization, not proof that
no alternative construction or stronger theorem can exist.

Mathematical contribution/value: A worthwhile focused extension in theoretical
quantum learning, stronger than a constant-only improvement or reformatting.
It resolves the mixed-input restriction within its exact operation model and
shows an actual downstream learning consequence. It has significant known-method
composition and conservative budgets; practical cost and wider impact remain
unproved. A specialist paper/preprint is proportionate. There is no basis to
advertise a broad complexity breakthrough, optimized hardware algorithm,
sharp constants or a new sampling/bootstrapping framework.

Attribution/readiness: Main attribution is accurate and unusually explicit
about known architecture and arithmetic. I recommend optional concise additions
for Iyer--Liang (mixed positive Fourier law), Holt--Subramanian (recent mixed
qudit exact-group/Fourier difference learning), and Bittel et al. (single-copy
high-fidelity alternative plus exact prime realizable learning). These are
editorial improvements; the current manuscript does not falsely claim their
specific primitives/results. Independent proof/code verdicts control mathematical
readiness. This report alone is not release clearance or external refereeing.

FAMILY / PUBLICATION RELATIONSHIP

The all-prime paper contains the private qutrit theorem at p=3, with exactly
F^6/18. The older qutrit package and its completed Publisher review remain
preserved evidence. The consolidated manuscript is the current strongest
assessment/publication candidate in the SAME escape/learner family. It should
supersede the qutrit-specific candidate editorially, not be counted as a second
independent overlapping paper. Subtree, theta and self-transport remain separate.
The qutrit p=3 result and all-prime statement have distinct source checkpoints
but one coherent paper is the appropriate publication form.

RECOMMENDED ACCURATE CONTRIBUTION LANGUAGE

We establish purity-sensitive mixed-state escape estimates in every odd-prime
local dimension and combine them with existing weighted Weyl measurements and
stabilizer bootstrapping to obtain a proper additive agnostic learner from
ordinary copies, with explicit bounded preparation costs.

Preserve the existing limits: no new measurement primitive/arithmetic split,
no optimal-copy claim, no composite dimensions, no approximate-local list
guarantee, no practical efficiency claim, no finite universal gate compilation
theorem, no worldwide priority and no external human peer-review claim.
