Daily Papers

We study Krasner quotient hyperfields arising from finite fields, F_q/G_r, where G_rle F_q^times has index r. Building on the structure theorem of Baker--Jin, we determine the characteristic and C-characteristic of all sufficiently large such quotients: they depend only on the parity of r and, when r is even, on the residue class of q modulo 2r. In particular, the two Baker--Jin stable classes for even r are separated by characteristic 2 versus 3, while the C-characteristic is always 1. We complement this structural result with a computational laboratory: sharp Weil thresholds for Baker--Jin large-q isomorphism, empirical minimal stabilization bounds N_r^{emp}, complete finite-field quotient atlases for hyperfield orders nle 7, and comparisons with the enumerations of Ameri--Eyvazi--Hošková-Mayerová (orders le 6) and Massouros--Massouros (order 7). Among other findings, exactly 15 isomorphism types of order 7 arise as finite-field quotients, out of 277 hyperfields of that order -- a concrete data point toward the Baker--Jin rarity conjecture for quotients. All algorithms and tables are available in an open-source package suitable for independent verification and arXiv ancillary material.

This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert--Pólya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law A_k(a) = log(1/a) + log(k-2) + B_0 + O(a), with B_0 = log q - 2log 2, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient B(ν) and shows B_0 depends only on the conductor q, independent of the Archimedean parameter. (R2b) The degree d of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, S(a) sim (2a)^3/6. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line σ^*(a) descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue λ_1(a) is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function W(a,0;z) is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.

RED-2400 is a public benchmark of algorithmically-rejected trading events from a live Solana decentralized-exchange filter stack. I logged the data continuously between 2026-04-10 and 2026-05-02. The benchmark contains 6,659 rejection events linked to 169,122 post-rejection price and liquidity observations and 1,836 graveyard-tracker snapshots. Outcome labels follow the five-tier classification of Kamat (2026c): saved (windowed), saved (early-death), missed, flat, and unclassifiable. Thresholds use the trough-to-reference and peak-to-reference price ratios within a 24-hour window. Most filter-design datasets cover the accept side only. That gap leaves reject-side outcomes unmeasured and biases filter validation. RED-2400 lets researchers replicate filter-precision claims directly. RED-2400 is the first window in a planned dataset series; subsequent windows will extend the time horizon and enable regime-stratified analysis.

We apply the Weibull distribution -- a two-parameter family from extreme-value theory -- as a diagnostic framework for element-wise weight magnitude distributions in transformers. At initialization, i.i.d. Gaussian weights give |w| ~ HalfNormal, yielding k ~ 1.20 via middle-80% probability-plot fit (the protocol used throughout this work). This anchor makes k a principled, architecture-independent measuring stick for training dynamics; fitting each weight matrix independently at every layer at every checkpoint enables per-component, per-layer, and per-step diagnostics that aggregate statistics cannot resolve. Applying this framework to 12 model entries spanning 7 architectural families (Pythia, OLMo-1/2, LLaMA-3, Mistral, Qwen2.5/3) reveals three findings. First, FFN modules and the attention output projection W_o -- the Transmission Class -- fall in a narrow k band: median terminal k in [1.186, 1.204] across 12 entries (cross-family CV = 0.51%), shared across SwiGLU/GeLU activations, Pre-LN/QK-Norm placements, and 70M-14B sizes. Second, the attention input projections W_q, W_k -- the Selection Class -- depart from the Weibull family, with severity shaped by storage: separately-stored Q/K (OLMo-1, OLMo-2) yields k in [0.76, 0.99] (deep); GQA models yield k in [1.10, 1.16] (mild); Pythia's merged W_qkv occupies a transitional zone tracking training budget T/tau monotonically. Third, lambda grows substantially during training and scales with sqrt(eta/lambda_wd) within the Pythia family (Pearson r = 0.94, three Transmission kinds), directionally consistent with Fan et al. (2025). The two parameters carry independent information: k labels the functional class, lambda labels training progress. We release npm-weibull-py v0.4 (Python library) and DATABASE_v9_1 at https://github.com/tiexinding/NPM-Weibull-public .

Clinical decision-support systems are expert systems whose recommendations clinicians act on directly, yet they are usually cleared on one aggregate accuracy number from a held-out test set. That number says nothing about input reliability under encoding shifts, subgroup gaps, threshold sensitivity, or operational feasibility. We present RISED, a pre-deployment evaluation framework operationalising five dimensions (Reliability, Inclusivity, Sensitivity, Equity, Deployability) through BCa bootstrap 95% confidence intervals, literature-grounded thresholds, and Holm-Bonferroni-corrected PASS / FAIL / INCONCLUSIVE verdicts; Equity is a proxy-dependence diagnostic rather than a gating test. Applied to seven cohorts spanning 35 years (n from 303 to 99,492), RISED surfaces failures invisible to AUROC: on Diabetes 130, Reliability passes by three orders of magnitude (PSS = 0.0004) while Inclusivity (AUC parity gap = 0.262) and Sensitivity (max threshold-flip rate 49.1%) fail decisively; both NHIS cohorts reproduce this. NHANES 2021-2023, with a complete feature profile, achieves INCONCLUSIVE verdicts; BRFSS 2024 produces the suite's most severe Sensitivity failure (max threshold-flip rate 64.2%) after instrument rotation removed hypertension and cholesterol. The pattern recurs on credit- and income-prediction cohorts, confirming domain-agnosticity; a multi-model check shows the failures are data-driven, not model-specific. RISED ships as an open-source Python package complementing TRIPOD+AI, FUTURE-AI, and Fairlearn with the structured numerical evidence those standards require but do not prescribe.

We consider the convergence of the ESD for non-Hermitian random band matrices with independent entries to the circular law, which is the uniform measure on the unit disk in the center of the complex plane. We assume that the bandwidth of the matrix scales like n^γ for some γin(0,1], where n is the matrix size, and the variance profile of the matrix is only assumed to be doubly stochastic with no additional assumption on its specific mixing properties. We prove that the circular law limit holds either (1) when γ>5{6} and the entries are independent Gaussians, (2) or when γ>8{9} and the entries are independent subgaussian random variables. This new threshold improves the previous threshold γ>32{33} which was only proven for block band matrices and periodic band matrices. After the initial version of this paper, the author further extended the range of circular law for much smaller values of γ in 2508.18143 and 2511.01744 when the variance profile has specific mixing properties, but not for an arbitrary doubly stochastic variance profile. Thus the main contribution of this paper is the circular law for a genuine power law bandwidth for any doubly stochastic variance profile. We also prove an extended form of product circular law with a growing number of matrices. Weak delocalization estimates on eigenvectors are also derived. The new technical input is new polynomial lower bounds on some intermediate small singular values, and this estimate does not depend on the specific structure of the variance profile beyond the fact that it is doubly stochastic.

We study the problem of detecting the correlation between two Gaussian databases XinR^{ntimes d} and Y^{ntimes d}, each composed of n users with d features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation sigma over the set of n users (or, row permutation), such that X is rho-correlated with Y^sigma, a permuted version of Y. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of n and d. Specifically, we prove that if rho^2dto0, as dtoinfty, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of n. This compliments the performance of a simple test that thresholds the sum all entries of X^TY. Furthermore, when d is fixed, we prove that strong detection (vanishing error probability) is impossible for any rho<rho^star, where rho^star is an explicit function of d, while weak detection is again impossible as long as rho^2dto0. These results close significant gaps in current recent related studies.