Riemann Zeta Function

We study the ordered Bernoulli-word kernel f(p,n,k)=p^k(1-p)^(n-k) and the geometry generated by its inverse-integer level sets. The binary level 2^(-n) selects p=1/2 as the unique real split-independent anchor. Under complement-preserving complex continuation, the pair becomes z=1/2+iu and 1-z=1/2-iu, producing a conjugation-symmetric vertical geometry before any zeta-function input is introduced. The quadratic coordinate Q(z)=z(1-z)=1/4+u^2 has a sharp minimum at the central point and admits an exact integer quantization. For critical-line zero ordinates gamma_k, the induced levels L_k=1/4+gamma_k^2 are decomposed exactly as L_k=N_k+delta_k, where N_k is the nearest integer and delta_k is a periodic first-Bernoulli residual. Circularization gives Z_k=exp(2 pi i delta_k), isolating gamma_k^2 mod 1 as the residual phase variable. Unique factorization resolves the integer shells into prime-generator coordinates, while a distinct complex exponent s lifts the same construction to the Dirichlet atoms m^(-s), linking the Dirichlet-series and Euler-product assemblies. Exact identities, classical zeta connections, numerical controls, and open conditional Weyl tests are kept explicitly separate. No proof of the Riemann Hypothesis is claimed.

The Zarankiewicz number Z(m,n,s,t) is the maximum number of edges in a bipartite graph with parts of orders m and n containing no copy of Ks,t. We give one combined, certificate-based computer-assisted proof for two finite slices and a corrected neighboring frontier: Z(12,n,3,3) = 6n (18 <= n <= 22), Z(13,22,3,3) = 137, Z(13, 18, 3, 3) = 116, Z(14, 18, 3, 3) = 124, Z(15,18,3,3) = 132, Z(14, 17, 3, 3) = 118, Z(15, 17, 3, 3) = 126, 132 <= Z(16,17,3,3) <= 133. The load-bearing new upper bounds are the exact 12 x 18 and 13 x 18 certificate packages. Their orbit certificates exclude every hypothetical matrix at the next edge count. Deletion lemmas and explicit witnesses close four neighboring cells, while the 16 x 17 entry is deliberately reported as an interval because only its 132-edge lower witness and the published 133 upper bound are certified here. Separately, the 13 x 22 proof excludes 138 ones by reducing to 83 degree profiles, rationally separating 77 of them, and eliminating the remaining six by marked-row congruences, leave enumeration, modular Gram tests, and exact Farkas certificates. All accepted claims are replayed by standard-library Python and exact integer/rational arithmetic; floating-point optimization is used only to discover certificates.

In Question~3.1 of his 1995 paper on depth and transfer, Carlson asked whether the depth of a finite-group cohomology ring is always realized by the dimension of one of its associated primes. We give a negative answer. Let $$G=\SG{128}{859},\qquad k=\kbar.$$ An exact presentation certificate proves that $\depth H^*(G;k)=2$. Okuyama's associated-prime theorem would convert an associated prime of dimension two into a rank-two elementary abelian subgroup $E\leq G$ satisfying $\depth H^*(C_G(E);k)=2$. We enumerate all $75$ rank-two elementary abelian subgroups of $G$ and obtain six centralizer types. Duflot's theorem gives depth at least three for four types, while exact ideal-quotient certificates exhibit regular sequences of length three for the remaining two. Hence every rank-two centralizer has cohomological depth at least three, so $H^*(G;k)$ has no associated prime of dimension two. The finite group presentation, the three cohomology-ring presentations, the enumeration summary, and the exact algebraic certificates are included for independent verification.

We introduce an extremal invariant associated with Chowla-type order conditions in finite groups. A nonempty subset $S$ of a finite group $G$ is called a Chowla set if every element of $S$ has order greater than $|S|$, and we write $C(G)$ for the maximum cardinality of such a set. We first show that $C(G)$ is determined by the distribution of element orders in $G$. For cyclic groups, we derive an exact divisor formula and characterize the integers $n$ for which $C(\mathbb{Z}/n\mathbb{Z})=\varphi(n)$. We prove that $\liminf_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=1$, whereas $\limsup_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=\infty$, and we determine the corresponding lower and upper limits under normalization by $n$. For finite abelian groups, we obtain an explicit formula in terms of the invariant-factor decomposition, together with a closed formula for finite abelian $p$-groups. We then develop a linear analogue for finite field extensions. A nonzero $K$-subspace $A$ of an extension $L/K$ is called a Chowla subspace if $[K(a):K]>\dim_K A$ for every nonzero $a\in A$. Since this condition depends on $\dim_K A$, it does not generally require every nonzero element of $A$ to generate $L$ over $K$. Nevertheless, when $L/K$ is finite and separable, we prove the exact formula $C(L/K)=[L:K]-d_{\max}(L/K)$, where $d_{\max}(L/K)$ is the largest degree over $K$ of a proper intermediate field. For finite fields, we give a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human-AI collaboration. A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies. The authors formulated the problem, independently verified and completed all arguments, and wrote the final proofs.

I propose a new methodology to attack the fascinating Gilbreath's conjecture about prime numbers, first posted in 1878 and unsolved to this day. The problem statement is rudimentary: kids can understand it. However, despite decades of research, almost no progress has been made. This paper changes the game by presenting a new approach based on sieving, a number of new results with proof, a precise path to the solution, and solid references. It also introduces the concept of reverse sieving, along with applications to testing randomness, pattern and fraud detection, cybersecurity, synthetic data, sequence categorization and normalization, or to detect and quantify a new type of chaos in time series including Brownian motions. Magic primes, forbidden prime number constellations, cellular automata, and reduction via classes of equivalent sequences, are some of the innovative and promising topics discussed in the paper.

To help evaluate the mathematical skills of current AI systems, we present a set of formulas for fundamental mathematical constants. These problems are attractive for AI evaluation because they are concrete and can be checked numerically to arbitrary precision, yet proving them may require non-obvious mathematics. Mathematical constants such as $π$, $e$, Catalan's constant, and special values of the Riemann zeta function have fascinated mathematicians for centuries. The search for formulas evaluating mathematical constants has produced some of the most beautiful mathematics in the field, especially in cases that yield irrationality proofs or fast convergence rates. Ramanujan's legacy is emblematic of this tradition. The list we provide contains two types of problems: formulas whose proofs are known to the authors but will remain encrypted for a short initial period; and formulas that are not yet proven. We are curious to see the achievements of AI in both cases.

Euler's Gradus Suavitatis (1739) assigns a dissonance value to a musical interval p/q by the formula G(p/q) = 1 + Ω^(p) + Ω^(q), where Ω^(n) = \sum_i e_i(p_i - 1) sums the weighted prime exponents of n. We propose the simpler asymmetric formula f(p/q) = p + Ω^(q), which treats numerator and denominator differently and performs comparably on standard consonance data. We also show that, under a model in which harmonics are integer-indexed and counted uniformly up to a fixed truncation level, Gradus is equivalent to a weighted harmonic coincidence count with weights w(n) = Ω^(n), connecting it to Galileo's earlier pulse-coincidence model (1638). The formula naturally generates a coprime integer triangle T(n,k) = n + Ω^(k), whose rightmost diagonal gives the two-stage dissonance of the superparticular (consecutive-harmonic) intervals. The formula f admits a simple two-stage interpretation in terms of harmonic context and partial recognition, which we offer as a speculative perceptual hypothesis.

The Riemann Hypothesis remains one of the central unsolved problems in mathematics. Rather than claiming proof, we investigate whether a verifiable AI-assisted reasoning system can produce reliable, formally checked partial progress while explicitly identifying the remaining mathematical obstructions. We apply the Verifiable Growing Physical Transformer with Recursive Self-Improvement (VGPT-RSI) to two RH-adjacent certification tasks. First, we construct and verify a finite RH-boundary certificate for inequality on a parameterized safe lower curve over a region. The numerical boundary curve is converted into a certificate-backed lower curve, audited using outward-rounded interval arithmetic and Arb/FLINT ball arithmetic, and then checked in Rocq/CoqInterval for the parameterized theorem. Second, we initiate a formal Lagarias-route certificate. Lagarias criterion states that RH is equivalent to the global inequality. We formalize the finite quantity and produce a Coq-checked finite certificate. The final system identifies the exact unresolved mathematical bottlenecks: formalizing the Lagarias equivalence, proving the global tail theorem beyond any finite cutoff, and potentially reducing counterexamples to colossally abundant or related extremal integers. These results demonstrate that VGPT-RSI can produce certified RH-adjacent formal progress, organize proof dependencies, and avoid overclaiming when the remaining obstruction is genuinely mathematical.

Synthetic healthcare data are widely proposed as privacy-preserving substitutes for real patient data, yet their evaluation remains dominated by statistical similarity and predictive performance that do not reflect clinical validity. We introduce a multi-dimensional evaluation framework grounded in epidemiology, assessing descriptive fidelity, clinical utility, and structural validity, corresponding to descriptive, predictive, and causal questions. We evaluate four representative generative paradigms - GAN-based, VAE-boosted, diffusion-based, and masked modelling - using PRIME-CVD, a 50,000-person cohort with known ground-truth structure. While all models reproduce marginal distributions, none simultaneously preserve subgroup structure, effect estimates, and dependency structure. Notably, models with strong distributional fidelity can exhibit poor calibration and distorted relationships, leading to unreliable inference. These results show that current evaluation practices can overestimate synthetic data quality and motivate domain-informed assessment based on the ability to support valid clinical and scientific conclusions.

The 2026 disproof of Erdős's unit-distance conjecture and Sawin's quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $δ=0.014114\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $δ=0.015263\ldots$ and supporting the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. For extended ramified prime ranges, the Emmerich--Cordella certificate obtained with the same framework reports $u(n)>n^{1.031}$ for $\#T=67$, illustrating the importance of enlarging $T$. Very recent MathOverflow discussions, brought to the author's attention as of version~4, report further improvements, including certificates above $δ>0.035$ and beyond $δ>0.036$. Some of these improvements may rely not only on larger prime ranges but also on modified constraint systems and additional degrees of freedom that deviate from Sawin's original formulation. Beyond this application, the work illustrates how randomized optimization heuristics can improve, verify, and refine explicit certificates for combinatorial geometry through nonlinear integer optimization.

Fact-checking articles encode rich supporting evidence and reasoning, yet this evidence remains largely inaccessible to automated verification systems due to unstructured presentation. We introduce PrimeFacts, a methodology and resource for extracting fine-grained evidence from full fact-checking articles. We compile 13,106 PolitiFact articles with claims, verdicts, and all referenced sources, and we identify 49,718 in-article hyperlinks as natural anchors to pinpoint key evidence. Our framework leverages large language models (LLMs) to rewrite these anchor sentences into stand-alone, context-independent premises and investigates the extraction of additional implicit evidence. In evaluations on cross-article evidence retrieval and claim verification, the extracted premises substantially improve performance. Decontextualized evidence yields higher retrievability, achieving up to a 30 percent relative gain in Mean Reciprocal Rank over verbatim sentences, and using the evidence for verdict prediction raises Macro-F1 by 10-20 points over the baseline. These gains are consistent across different verdict granularities (2-class vs. 5-class) and model architectures. A qualitative analysis indicates that the decontextualized premises remain faithful to the original sources. Our work highlights the promise of reusing fact-checkers' evidence for automation and provides a large-scale resource of structured evidence from real-world fact-checks.

We prove lower bounds on learning the Möbius or Liouville function with a variety of standard learning techniques, including kernel methods, noisy gradient methods, and correlational statistical query algorithms. These results follow from quantitative bounds on the correlation of Möbius with digital characters of various finite abelian groups, where the group is dictated by the type of input data the algorithm is given. Using residues mod $p$ for many different primes corresponds to a cyclic group, and using the base $p$ expansion for a fixed prime corresponds to an elementary abelian $p$-group. We also note that lower bounds of this form are closely related to certain types of digital prime number theorems.

How much data is enough to make a scientific discovery? As biomedical datasets scale to millions of samples and AI models grow in capacity, progress increasingly depends on predicting when additional data will substantially improve performance. In practice, model development often relies on empirical scaling curves measured across architectures, modalities, and dataset sizes, with limited theoretical guidance on when performance should improve, saturate, or exhibit cross-over behavior. We propose a scaling-law framework for cross-modal discoverability based on spectral structure of data covariance operators, task-aligned signal projections, and learned representations. Many performance metrics, including AUC, can be expressed in terms of cumulative signal-to-noise energy accumulated across identifiable spectral modes of an encoder and cross-modal operator. Under mild assumptions, this accumulation follows a zeta-like scaling law governed by power-law decay of covariance spectra and aligned signal energy, leading naturally to the appearance of the Riemann zeta function. Representation learning methods such as sparse models, low-rank embeddings, and multimodal contrastive objectives improve sample efficiency by concentrating useful signal into earlier stable modes, effectively steepening spectral decay and shifting scaling curves. The framework predicts cross-over regimes in which simpler models perform best at small sample sizes, while higher-capacity or multimodal encoders outperform them once sufficient data stabilizes additional degrees of freedom. Applications include multimodal disease classification, imaging genetics, functional MRI, and topological data analysis. The resulting zeta law provides a principled way to anticipate when scaling data, improving representations, or adding modalities is most likely to accelerate discovery.

Machin-like arctangent relations are classical tools for computing $π$, with efficiency quantified by the Lehmer measure ($λ$). We present a framework for discovering low-measure relations by coupling the PSLQ integer-relation algorithm with number-theoretic filters derived from the algebraic structure of Gaussian integers, making large scale search tractable. Our search yields new 5 and 6 term relations with record-low Lehmer measures ($λ=1.4572, λ=1.3291$). We also demonstrate how discovered relations can serve as a basis for generating new, longer formulae through algorithmic extensions. This combined approach of a constrained PSLQ search and algorithmic extension provides a robust method for future explorations.

Slice-to-volume reconstruction (SVR) is the standard method for obtaining high-resolution (HR) 3D fetal brain volumes from motion-corrupted 2D MRI slice stacks acquired in multiple orientations. Existing SVR methods are optimized and validated only for clinical-range echo times (TEs), limiting their use at non-clinical TEs and making them incompatible with quantitative T2 mapping, a protocol- and center-independent biomarker of fetal brain maturation requiring HR reconstructions across multiple TEs. We present PRIME-SVR, the first implicit neural representation (INR) framework for joint HR reconstruction from multi-echo MRI. A single fully connected network models a continuous function from spatial coordinates to signal intensities across TEs, while a second network estimates slice-specific acquisition degradations. Cross-TE coherence is enforced via a Bloch equation-derived regularization penalizing deviations from expected T2 decay, with adaptive weighting that strengthens coupling for degraded stacks. The method is fully self-supervised. We validate PRIME-SVR on 39 in vivo fetal acquisitions (13 subjects x 3 TEs) from two centers, two vendors, and two field strengths (1.5 T and 0.55 T). Compared to state-of-the-art SVR, PRIME-SVR improves reconstruction sharpness by 47%, anatomical accuracy by 30%, and cross-TE structural consistency by 14%. It enables reconstruction at late TEs previously inaccessible to SVR, yielding the first 0.8 mm isotropic T2 maps at 0.55 T and the first T2 maps derived from INR-based SVR. PRIME-SVR also accelerates quantitative imaging by reducing the data needed for multi-TE reconstruction, cutting acquisition from 15 to 10 minutes while keeping T2 accuracy within 1.7% in white and deep gray matter, or to 5 minutes with a mean T2 error of 2.3% for high-quality acquisitions.