The paper introduces persistent magnitude homology as a functorial invariant for quantitative equational theories, providing a barcode that captures the metric structure of the free algebra generated by a metric space of generators. It shows how this invariant combines graded magnitude homology with persistence, yielding stability estimates and a method to compare barcodes when theories are extended. Four concrete examples illustrate the theory in each homological degree.
arXiv:2303. 04203v4 Announce Type: replace Abstract: The theory of computation was built to answer Turing's question: what is effectively calculable by an unbounded, immortal, disembodied agent following rules?
arXiv:2507. 03065v2 Announce Type: replace Abstract: Why do some macroscopic structures remain identifiable even though their microscopic constituents continually change?
arXiv:2609. 11326v2 Announce Type: replace-cross Abstract: We ask whether it can be certified algorithmically that a self-modifying computational system preserves a safety property at its next step (preservation) and along its whole evolution (persistence).
The paper investigates integer‑sequence benchmarks from the OEIS by applying a two‑part minimum description length (MDL) learner that searches for P‑recursive recurrences. It finds that MDL difficulty correlates with a combinatorial parameter count, that most sequences fit a recurrence on a prefix but not at full length (the “wilderness” regime), and that language models do not hallucinate in the wilderness but instead hedge, showing that memorisation dominates perceived competence. The study provides a cheap, contamination‑free difficulty signal for OEIS‑derived benchmarks.
The paper addresses the limitations of spectral dimension reduction for dynamical systems composed of weakly interacting components, where standard rank‑based methods either require exponentially many modes or omit entire components (a phenomenon termed linear masking). It proposes scoring the σ‑algebra generated by coordinates instead of individual modes, using a χ²‑divergence criterion that guarantees an embedding with twice the intrinsic dimension captures the full operator spectrum. Experiments on benchmark systems show that this algebraic approach recovers masked components and enables accurate prediction from few labels, outperforming traditional rank‑based and VAMP methods.
arXiv:2608. 10986v1 Announce Type: cross Abstract: A growing class of methods probes a language model by feeding it its own output: self-consistency, iterated refinement, agentic loops.
arXiv:2606. 07627v1 Announce Type: new Abstract: Transfer learning presumes that a representation learned on source tasks carries structure that remains usable on related target tasks.
arXiv:2603. 25414v4 Announce Type: replace-cross Abstract: A prevailing assumption in machine learning is that model correctness must be enforced after the fact.
arXiv:2606. 16541v1 Announce Type: new Abstract: Autoformalization, translating natural-language mathematics into formal proof assistants, is bottlenecked not by translation fluency but by \emph{faithfulness}: a formal statement can typecheck and be provable, yet still encode a different theorem than the source intended.
arXiv:2606. 14688v1 Announce Type: cross Abstract: AI systems coupled to proof assistants now generate formal mathematics at scale, and the gap between what a checker can verify and what a mathematician would value has become the binding constraint.
The paper proposes a new dimension‑reduction strategy for transfer‑operator models of dynamical systems that focuses on scoring the σ‑algebra generated by coordinates rather than the operator’s spectral span. By using a χ²‑divergence criterion between embedded present and future states, the method guarantees that twice the intrinsic system dimension suffices to capture the full operator spectrum, even for systems with weakly interacting components that would otherwise require exponentially many modes. Experiments on benchmark systems show that this algebraic approach recovers masked components missed by rank‑based methods and enables accurate prediction of those components from few labels.
arXiv:2609.36279v1 Announce Type: cross Abstract: The shallow embedding of higher-order modal logic in classical higher-order logic, used in Benzm\"uller and Scott's Notes on G\"odel's and Scott's va...