Volume 1

Choosing a network architecture for zero-knowledge (ZK) proofs of inference needs a circuit-cost estimate that can be checked against the prover, yet NAS work on ZK has relied on untested cost proxies. We study ezkl, a Halo2-based prover, on CIFAR-10 MLPs, CNNs and ResNet-style networks. (1) Cost model. A parametric model of assigned cells, with coefficients fitted on 180 trained, calibrated networks and evaluated on 54 held-out ones, matches ezkl-calibrated logrows exactly for 39 and within one for 51 (Spearman 0.84; cross-validated over all 234 networks: 70% exact, 95% within one). The mismatches have a mechanism: on 72 dumped ezkl settings, logrows is exactly the maximum of four resource requirements (rows, constants, lookup table, range-check table; 72 of 72 sampled networks at logrows 19–23), of which cell count is one. (2) A negative result. Width-matched, ReLU and x² networks have the same calibrated logrows (all 5 skeletons; 45 of 50 matched pairs in the sweep), contradicting the assumption that polynomial activations are cheaper to prove in ezkl; unnormalized x³ + x networks calibrate to a circuit up to 16× larger at equal cell count because their range-check table spans 2²² + 1 values. (3) Benchmark. We release zkml-bench, 474 networks with two training seeds and proxy scores. NEAR predicts accuracy only for ReLU networks (ρ = +0.70) and is indistinguishable from zero on all three polynomial families; a Zen-style score (our variant, not the published Zen-Score protocol) is anti-correlated on polynomial networks; a sign-pattern NASWOT variant is negatively correlated on polynomial networks; ZiCo’s apparent success on x³ + x disappears once failed training runs are removed. Between 39% and 60% of polynomial networks fail to train. (4) Search. With budgets set on ezkl-calibrated circuit size, a tuned joint accuracy-cost proxy almost always reduces to NEAR-only and shows no consistent gain over it, while NEAR-only beats random selection; we withdraw an earlier claim of a joint-proxy gain. Filtering by the cost model’s estimate removes most but not all budget violations. Scope: CIFAR-10, one ezkl version, calibration on trained networks with real inputs, real proofs for seven architectures.

Model-free identification of the minimal world-model of a bounded agent in a POMDP has been made tractable in the finite setting by Myhill–Nerode quotient learning: a permutation G-test on fixed-policy continuation histograms clusters histories into continuation-equivalence classes. With continuous observations and actions the continuation alphabet is infinite and the G-test is undefined. We extend quotient learning to 𝒪 × 𝒜 ⊆ ℝ^(d). Histories are keyed by coarse quantile bins; for every pair of bins the continuation laws are compared with a kernel maximum mean discrepancy (MMD) permutation test on raw continuation vectors under Benjamini–Hochberg control, and bins whose test does not reject are merged by union–find. We (i) identify the estimator’s target—the bin-level continuation quotient—and give conditions under which it coincides with the latent quotient of index n; (ii) prove an alphabet-free trajectory bound N = O (Blog (B/δ)/ε_(MMD)² + B²log (1/δ)) for an analysed variant (T) with one window per trajectory, in which the alphabet size |𝒞| of the finite bound is replaced by the MMD separation ε_(MMD); and (iii) characterise finite-sample behaviour as a resolution effect: distinctions below the test’s detection scale are merged, so the returned size depends on the bin count B and the sample budget. Empirically, the procedure we run (permutation test with BH; procedure P) returns the latent index on 90/90 runs when B = n on Tiger, K-cycle (K ≤ 16) and a Gaussian-mixture walk, but this uses n: with B chosen without n the adjusted Rand index drops on K-cycle (K ≥ 4) and the index is over-split. No baseline we implement (RKHS ε-machine, GRU causal-state learning, Ferns bisimulation metric, multivariate Wasserstein, DBC) is uniformly better or worse; the GRU is strongest on K-cycle. The analysed variant (T) of Theorem 1 is conservative: with the sampler its theorem covers (one window per trajectory) it merges Tiger’s classes at N = 3000, resolving them from N = 10⁴ at B = 2, and merges all Gaussian-mixture classes at N ≤ 3000. We state which guarantee covers which procedure. On PO-MuJoCo the estimator returns n̂ = B, and a trajectory-level held-out analysis shows the classes carry no information beyond a nonlinear function of speed. On a 4-state chain Dec-POMDP the estimated critical communication rate tracks the analytical value within 0.8% mean error when both quotients are non-trivial; on Dec-Tiger agent A over-splits on 5/10 seeds at α = 0.05. We report all seeds. Code and results: github.com/npow/quotient-learning-papers (cql/, tag v1, commit 83f9f42; new result files in commit 2c16829).

We study a modest but identifiable abstraction problem for partially observed control: under one fixed data-collection policy, which bounded observation contexts have the same finite-horizon continuation law? We define the episode-position predictive quotient, an explicitly policy- and position-dependent estimand. Our estimator first selects contexts on one half of independently reset trajectories, then takes at most one prespecified continuation block from each trajectory in the other half. It compares fixed- alphabet histograms by empirical total variation and abstains whenever a pair lacks the samples required by its own concentration bound. Under an ε-gap, a per-context visitation floor, and a selection threshold no larger than the inference threshold, we prove exact recovery with probability 1 − δ from O((d + log (K/δ))/(μ_(min)ε²)) trajectories, where d is the continuation-alphabet size and K the selected context count. A two-point construction shows the dependence on μ_(min) and ε is unavoidable; the dependence on d is not matched and is known to be improvable by optimal closeness testers. The guarantee is non-asymptotic: it does not treat overlapping windows as independent, does not rely on chi-square calibration, and conditions each context only on its own count, so nested candidate contexts are covered. Implementation checks on three analytically solvable environments—each with ε set below its known gap—recover the exact quotients on all of 20 seeds, including an environment whose quotient requires merging; a budget sweep shows the power gate degrading through abstention with no incorrect merge at any budget; and a position-nonstationary chain shows the overlapping-window mode systematically merging contexts the episode-position quotient separates.

We examine the thermodynamic reduction in Chowdhry and Loureiro's model of current-sheet formation under radiative cooling, using its journal and arXiv editions. The displayed pressure formula does not solve the stated energy equation. Direct integration gives a Bernoulli solution whose initial cooling rate is nonzero, whereas the published bremsstrahlung form has zero initial cooling rate for every finite spatial integration coefficient. Correcting this formula generally destroys the exact parabolic-pressure affine closure. On the common positive-pressure branch, the corrected scalar evolution preserves pressure ordering; in the published final ordinary differential equations, outflow deceleration while the inflow scale is shorter than the outflow scale requires negative central total pressure. Separately, a cumulative cooling integral cannot be replaced by its vanishing endpoint tail. At fixed finite coefficients and finite positive outflow scale, that integral makes the printed cooling-to-magnetic force ratio tend to zero, contradicting the proposed strict cooling-dominated endpoint balance under those hypotheses. We retain the algebra of the modified steady Sweet-Parker branch and state its parameter-domain and stability limitations. Direct scalar integrations and separate high-precision checks support these bounded conclusions. They do not reconstruct the original numerical trajectories or rule out physical radiative contraction in a different model.

Regional particle energization requires tracking where each energy increment occurs. We audit the posted particle-attribution routine associated with arXiv:2511.15988v1 and analyze all 1,003 histories in its two publicly released example batches. The routine partitions trajectories at first sheet and subsequent downstream passages; later returns do not change that chronological attribution. A zero-transition-increment counterexample reverses regional ordering while preserving total-gain closure. Applying the routine to four released illustrations reproduces all eight preprint annotations, and three illustrations exactly match full histories in the larger batch. Pooling the batches before applying the source's strict final-energy cuts leaves 667 intermediate and 84 high-energy histories. In the available high-energy subset, the downstream particle-mean fraction changes from 0.500318 under first-passage attribution to 0.370129 under symmetric full-history attribution; left and right conventions give 0.370742 and 0.369515. The paired symmetric particle-mean shift is -13.0189 percentage points, with a conditional empirical bootstrap percentile range of [-20.4864, -6.6663] points; this is not a population confidence interval. Its fraction of total net gain changes from 0.490959 to 0.393724. Downstream remains dominant in the intermediate subset. All comparisons retain signed increments and identical particle selections. The batches have undocumented release probabilities and do not reproduce the published intermediate-band point; these are measured available-sample effects, not a corrected full-population result. Discrete endpoint allocations also do not uniquely recover continuous electric work. Supporting appendices preserve positive initialization and diagnostic-curve checks, distinguish inferred plotting operations from source-unit evidence, and print the complete audited routine.

We audit arXiv:2508.15691v2 and the published Quantum Science and Technology article *Trotter-based quantum algorithm for solving transport equations with exponentially fewer time-steps* by Zylberman, Fredon, Loureiro and Debbasch. Ten registered correction groups persist in the directly inspected journal PDF. Exact counterexamples refute conservation of spatial derivative norms for variable transport fields and the signed-overlap identity assigned to a controlled-SWAP circuit. Further corrections concern transport and Liouville signs, Fourier inversion, moment normalization, constants and index ranges. We retain finite-time regularity factors and apply established commutator reasoning to obtain a coefficient-aware finite-grid operator bound proportional to inverse mesh spacing rather than its square for smooth periodic coefficients. This sharpens the chosen comparison baseline without proving a complete repaired vector-norm theorem or disproving the usefulness of smooth-state analysis. Independent calculations reproduce the reported eight-point discretization errors and a separate periodic splitting diagnostic. A version concordance, attributed source statements and numerical provenance tables make the findings directly checkable. The journal's added Gaussian-width, circuit-cost and boundary qualifications are acknowledged. No end-to-end quantum advantage, general impossibility result or inference about unavailable implementation code is claimed.

This diagnostic comment examines initial-state matching and pressure realizability in arXiv:2607.07106v1. Its exact mixture identities are correct. However, archived PIC and fluid initial moment-current profiles differ by $9.87\%$ in the full-domain, $x$-averaged relative $\ell^2$ norm; local Cartesian covariance pressures differ by about $20\%$ in $P_{xx}$ and $38.4\%$ in $P_{zz}$. The PIC input specifies proper momentum where the fluid input specifies ordinary velocity. Complete archive checks and independent ten-moment layout validation corroborate the data audit. Four clean-source fluid cases attempt comparisons of the posted and aggregate-PIC-matched initializations, while distinguishing the unavailable historical dirty build. At identical one-fluid endpoint times, matching lowers $P_{xx}$ error against PIC from $36.29\%$ to $28.85\%$, with smaller errors in density, current and $P_{zz}$ as well. These are numerical endpoint results, not validated physical dynamics: both one-fluid cases contain indefinite covariance tensors at saved intermediate times, and the clean unmatched two-fluid endpoint is also inadmissible. The matched two-fluid run terminates with an invariant-check assertion before producing its comparison endpoint. The clean implementation tests coordinate pressures without testing full tensor positivity. The original archived endpoints pass the same covariance checks; their intermediate trajectories and exact executable remain unidentified. A conditional loader calculation, an auxiliary homogeneous stability benchmark, a continuity clarification and a cancellation-conditioned heat-flux bound provide separate diagnostics. The results motivate matched moments and realizability checks without disproving the source’s exact algebra or establishing that its nonlinear physical conclusions are false.

We examine version 2 of arXiv:2609.28999, which reconstructs conservative nonlinear ordinary differential equations from means of transported probability densities. The printed geometric-distortion definition retains the initial separation between neighboring states while its Taylor expansion omits that separation. For the stated directional transport maps, the resulting normalized quantity is at least one, so it cannot satisfy the proposed small-distortion criterion as written. We give a corrected displacement definition and a local derivative estimate, without asserting a universal replacement stability threshold. Applying the standard circular-mean description to the source's phase-reference circuit shows that finite variance can reduce contrast without shifting the maximizing reference. An explicit pair of positive grid distributions has identical fringes but different arithmetic means, including in a conservative transported-density realization. Further corrections distinguish Bernoulli-ancilla sampling from direct coordinate sampling, reconcile the Lotka–Volterra benchmark parameter with the source definitions, and exhibit a twice continuously differentiable observable for which a stated fourth-order Gaussian remainder fails. All numerical examples are deterministic classical computations with analytic checks and a recorded execution log. These results repair specific geometric, inference, and regularity statements; they do not refute Koopman embeddings, the genuine nonlinear mean-trajectory discrepancy, or all covariance-based error estimates.

We audit the source statements in arXiv:2603.08799v1, identified by a versioned PDF and its hash. An explicit smooth periodic shear contradicts the derivative-norm conservation assertion, which is also verified in the publisher's public Note 10. A positive-coefficient diffusion example exposes omitted mixed third derivatives in the preprint's displayed splitting prefactor. We quote the targeted preprint passages, distinguish a changed publisher note from the inaccessible publisher body, and make no claim that the unverified published diffusion formula is identical. For fixed trigonometric data we give the exact centered-stencil commutator and its order of convergence. Cosine-only numerical checks report initial and evolved-supremum comparators, finite-time limiting coefficients, and step refinement from pinned code. A standard auxiliary autonomous matrix estimate records the intermediate-state hypothesis needed for a corrected analysis; no novelty is claimed for that estimate. The results are specific counterexamples and proof obligations, not an exhaustive audit, an unconditional time-dependent PDE theorem, or a demonstrated quantum speedup.

Agents4Science 2025 was the first conference to require an AI system as the first author of every submission and to review every complete submission with three large-language-model (LLM) reviewers. Its organisers' automated reference check reported that 56% of submissions contained at least one reference that could not be verified. We re-examined all 6,849 references in the 304 submissions with a parsable reference list, using a reproducible pipeline (DOI, arXiv and URL resolution; Crossref, OpenAlex, Semantic Scholar, OpenLibrary and Google Books) followed by manual adjudication of every reference it could not verify (857 decisions with logged evidence) under a protocol fixed in advance. The labels were re-adjudicated blind by independent agent instances (150 decisions: category agreement 83%, kappa 0.75; fabricated-versus-not 92%, kappa 0.83); an independent human coder agreed with the adjudication on 98% of a random sample of 45 manual decisions for fabricated-versus-not (kappa 0.94); among 180 automatically verified entries, about 6% were real works cited with a wrong author list, venue or identifier and about 2% did not exist, so every adjudicated rate below is a lower bound. Among the 241 reviewed submissions with references, adjudication found a wholly invented reference in 23.7% (95% CI 18.7-29.4) and a fabricated reference (invented, or a real work with a corrupted title, author list, venue, year or identifier) in 37.8% (31.9-44.0); 3.8% of their 5,230 references were invented and 7.2% fabricated, rising to an estimated 6.2% (4.7-8.8) and 13.5% (10.8-17.4) once the errors found among automatically verified entries are added. Accepted papers cite fewer: no invented reference was detected among their 1,308 references (an estimated 34, 14-70, expected undetected) and nine corrupted ones were detected in eight of 48 papers, with an adjusted fabricated share of 7.8% (4.7-12.0) against 15.4% for rejected submissions. The organisers' flag was a screen, not a measure: 51.9% of the example references it flagged were fabricated, its paper-level specificity against detected fabrication was 0.66 (sensitivity 0.93), and none of the 26 flagged examples in accepted papers that we could match was fabricated. Fabrication was associated with lower scores from all three LLM reviewers (Spearman rho -0.13 to -0.15 for the fabricated share, -0.18 to -0.21 for the invented share), with lower human expert scores (rho -0.28 and -0.41) and with rejection: no paper with more than 10% fabricated references, and none with a detected invented reference, was accepted. Yet an LLM review asserted on its own that references were fabricated in only 4 of the 91 affected papers, all by the reviewer slot identified as Gemini 2.5 Pro, and no human expert review did. Of the 513 detected fabricated references, 56% were invented; 32 invented references carried a DOI or arXiv identifier. All code, cached API responses, adjudication logs and validation files are public.