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The Collatz hypothesis is a theorem of the algorithmic theory of natural numbers. We prove the (algorithmic) formula that expresses the halting property of Collatz algorithm. The observation that Collatz's theorem cannot be proved in any elementary number theory completes the main result.
We introduce the Graph Machine (GM), an architecture that maintains an $O(n)$-sized state and accesses it through sparse, dynamic routing. Unlike methods with fixed-size states or sparse but static routing, GM preserves $O(n)$ complexity in its sparse layers without restricting the potentially accessible state size to $O(1)$. Instead, GM uses edges - pointer-like objects updated differentiably by a referral mechanism resembling pointer chasing. We replace 75% of the dense Transformer layers in Qwen3-0.6B with GM sparse layers and pretrain from scratch on 15.7B tokens. With only 2 of 4,096 tokens retrieved per KV head in each sparse layer, loss degrades only slightly; with 4, the best model marginally improves loss.
We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.
BioMedRAG introduced retrieval-augmented generation with a learned chunk scorer for biomedical information extraction. However, it relies on fixed-size chunking which can fragment semantic evidence. We propose a configurable semantic chunking framework that addresses this limitation by combining entity-preserving windows, trigger-centered chunking, proposition-first extraction, tiered trigger prioritization, and hierarchical relation resolution. The framework integrates with BioMedRAG by replacing only the chunk construction stage while preserving the embedding model, learned chunk scorer, generator, and evaluation protocol. We evaluate the framework on biomedical relation extraction benchmarks (GM-CIHT, DDI, ChemProt) and adverse event classification (ADE). On GM-CIHT, the full hybrid configuration achieves 82.6% F1, improving over the fixed-size baseline (74.2% F1) by 8.4 points under our experimental setup. Cross-dataset analysis shows that semantic chunking improves extraction datasets with explicit relation cues, such as GM-CIHT and DDI, while fixed chunking remains competitive or stronger for dense biochemical extraction and binary classification settings such as ChemProt and ADE. By externalizing chunking logic into configuration files, the framework provides an interpretable and adaptable alternative to rigid fixed-size chunking for biomedical RAG pipelines.
Background and Objectives: Full-image objectives in brain magnetic resonance imaging (MRI) super-resolution (SR) can underweight tissue-transition regions affected by the partial-volume effect (PVE), as these regions occupy a small fraction of the image. Binary boundaries further provide only a discrete approximation of continuous tissue mixtures within a voxel. Methods: We propose Anatomy-Guided Gaussian-Parameter Warping with PVE-Balanced Reconstruction (AGW-PBR), combining a low-resolution (LR)-only reconstruction backbone with a PVE-aware training objective. The backbone uses LR-derived anatomical guidance, soft latent assignment, and bounded residual warping. Quality-controlled tissue fractions are converted into tissue-mixture entropy to spatially weight reconstruction within validated PVE support. PVE sidecars are used only during training, while inference requires only the LR image. Downstream utility is further evaluated through zero-shot transfer to whole-tumor segmentation on BraTS2023. Results: AGW-PBR improves reconstruction across 2x and 4x SR on IXI and achieves the lowest normalized gradient-vector reconstruction error at both CSF--GM and GM--WM interfaces at 4x. Ablation studies verify the contributions of PVE-aware weighting and soft latent assignment. The PVE-free AGW backbone also maintains strong performance on fastMRI. On BraTS2023, AGW-PBR achieves competitive whole-tumor Dice and the lowest HD95 under direct zero-shot transfer. Conclusions:AGW-PBR improves brain MRI SR while preserving tissue-transition information relevant to downstream analysis. The results support tissue-mixture entropy as an effective supervision signal for partial-volume-aware MRI reconstruction.
Deep Imbalanced Regression (DIR) is pervasive in continuous prediction tasks across diverse modalities, such as age estimation, depth prediction, and protein mutation activity prediction, where label-scarce tail samples often carry higher practical value. However, most existing methods still learn deterministic point mappings under mean squared error or its simple variants, implicitly assuming a uniform uncertainty level across all samples and thereby overlooking the instance-wise heteroscedasticity that is widespread in long-tailed data. We further point out that even heteroscedastic negative log-likelihood suffers from a gradient coupling issue, which, under DIR scenarios, weakens the learning signal of hard tail samples and leads to optimization inertia as well as tail underfitting. To address this, we propose DUO, an uncertainty-aware long-tailed regression framework. Specifically, the proposed method models the regression target as a conditional Gaussian distribution to explicitly characterize instance-level predictive uncertainty, and transforms uncertainty into a dynamic enhancement signal for tail samples through decoupled mean-variance optimization. Furthermore, we design a distribution-guided contrastive learning mechanism that adaptively constructs positive and negative pairs based on the overlap between sample distributions, thereby alleviating feature looseness and cross-label semantic entanglement. Across visual and biological DIR benchmarks, DUO achieves the best few-shot bMAE and GM on IMDB-WIKI-DIR, AgeDB-DIR, and AAV2-DIR while remaining competitive on few-shot MAE.
Sharing analog integrated circuit designs remains difficult: foundry non-disclosure agreements restrict the process details a design depends on, and the testbenches behind published results are rarely released. We present analog-db, an open-source, versioned database built on a shareable design representation. A domain-specific language captures each design as a process-neutral topology, reusable testbenches, and a machine-readable datasheet under one schema, so a design is shared in full and re-simulates on the process kits it is bound to. A parameterization scheme exposes functional sub-blocks and device sizes as named parameters that carry their matching constraints, making circuits composable and retargetable; a schema-governed contract and queryable catalog let AI design agents discover and reuse them directly. Across the regulator corpus, all 23 circuit-kit bindings on three open kits meet their own recorded specification bands (typical corner, matched devices, no layout) and 10 of 23 meet a common class band. Seventeen of the 23 imported sizings failed their testbenches and closed under a gm/ID sizing loop driven by the annotated sub-block roles, typically within one to three iterations. In a supervised case study, a coding agent working from the released artifacts sized the op-amp cores of a chopper instrumentation amplifier on an open 130nm kit, locating four hand-entry defects and a missing common-mode feedback loop that the sizing-only baseline did not repair. The database holds 68 circuits across sixteen classes, verifiable at schematic level under a tiered harness and tracked on a power/performance scoreboard, released at https://github.com/MacAnalog/spicexplorer-release.
Artificial intelligence models are promising for medical diagnosis, but they require large numbers of unbiased data, which in medicine are distributed across hospitals and cannot be centralized to protect patient privacy. Federated Learning (FL) addresses this, since hospitals train one shared diagnostic model while patient data remain local. Training proceeds in communication rounds, in which each hospital trains the shared model locally and returns it to the server for merging by weighted average. This aggregation weight determines whose institutional knowledge shapes the result. Federated averaging (FedAvg) sets it in proportion to local sample count, so a small but informative hospital is permanently assigned a small influence, andl argest clients could dominate the global model even when they are less informative. We propose Federated Dual Reputation Annealing Weighting (FedDRAW), a server-side aggregation method that combines a data-size prior with the cosine similarity between client and global parameters under two coupled annealing schedules. An inner schedule shifts client reputation from the size prior towards similarity. An outer, deferred annealing schedule on the softmax inverse temperature keeps the weighting selective in the early and middle rounds and relaxes it to uniformity at convergence. We evaluate FedDRAW on 12 simulated client-partition scenarios of two chest radiograph datasets (CheXpert and ChestMNIST), against seven federated baselines under identical local training settings. FedDRAW achieved the highest average rank among all eight methods under both AUC and the geometric mean (GM) of sensitivity and specificity, which a Friedman test with Nemenyi post-hoc analysis confirmed to be a statistically significant difference between the methods. Scheduling two signals, rather than fixing the weights by sample count alone, could enable less biased diagnostic models.
We give an algorithm for calculating the maximum entropy state as the least fixed point of a Scott continuous mapping on the domain of classical states in their Bayesian order.
In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.
Let $C$ be an $[n,k]_q$ maximum distance separable (MDS) code with $4\le k\le q-3$, and suppose that it has a $2$-transitive permutation automorphism group. In this paper we show that $n\le q+1$, so the MDS conjecture holds for this class of codes.
The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.
We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.
Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single set theory formula with parameters unrelated to formulas. Exotic expressions involving sets related to formulas of unbounded quantifier depth appear mostly in esoteric or foundational studies. Recognizing the internal to math (formula-specified) and external (parameter-based) aspects of math objects greatly simplifies foundations. I postulate that external sets (not internally specified, constituting the domain of quantifiable variables) are hereditarily countable and independent of purely formula-defined classes, i.e. with finite algorithmic information about them. Variables for classes are not explicitly quantified. This opens a way to eliminate all non-integer quantifiers in set theory sentences. The restrictions seem to require almost no changes in math papers, only reinterpreting some formalities.
We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.
Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.
A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.
We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2 57$. This is the first equilateral set of more than $n+1$ points in $\ell_p^n$ for any finite $p\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\le p\le4$, the infimum of exponents at which it fails lies in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.