Searcharxiv⌕ Search

arXiv subjects

Asvin G

Publications and source records attributed to Asvin G.

9 recordsLinked to original sources
arXiv2026

The Assistant as a Privileged Persona: A canonical reference in cross-persona self-recognition

Post-trained language models can recognize their own outputs from a sentence or two out of context. In a companion paper \citep{jack2026twomodes} we showed they can also recognize when they are currently acting on-policy, through the sharp entropy drop of assistant-mode generation. Both signals are tied to the Assistant persona that post-training mainly shapes. This paper widens the frame to cross-persona authorship judgement on Llama-3.1-70B-Instruct. We measure a matrix of authorship claim rates over a panel of evaluator and generator personas spanning librarian to dragon to Shakespeare, and make two claims. \emph{First}, on the Assistant's own row of the matrix, the Assistant's claim rate, the persona-vector distance from the Assistant in activation space, and the entropy gap between the Assistant's surprise on a persona's text and the persona's surprise on its own text are all tightly coupled. This extends the entropy signature of \emph{acting} from the companion paper to a retrospective signature of \emph{having acted}. \emph{Second}, this coupling fails off the Assistant's row: the natural symmetric extension of the entropy gap does not predict authorship for distinctive evaluators (pirate, dragon, Shakespeare); what does is asymmetric -- the evaluator's surprise compared to the Assistant's surprise on the same text, not to the generator's. We rule out the alternative that any persona could play this reference role by trying many candidate substitutes; none does. We interpret the asymmetry as the model performing an implicit Bayesian likelihood-ratio test against the Assistant as the canonical alternative hypothesis, with the persona-vector geometry of \citet{chen2025persona} (every persona a delta off the Assistant) ensuring that the Assistant is the only persona universally accessible to that test.

cs.LG↗
arXiv2025

Matrix Points on Varieties

We study the cohomology of $C_n(X)$, the moduli space of commuting $n$-by-$n$ matrices satisfying the equations defining a quasi-projective scheme $X$. This space can be viewed as a non-commutative Weil restriction from the algebra of $n$-by-$n$ matrices to the ground field. We introduce a semi-simple counterpart $S_n(X)$, defined as the quotient of $X^n \times \mathrm{GL}_n/\mathrm{T}_n$ by the diagonal $S_n$ action. We show that there exists a natural map $\sigma \colon S_n(X) \to C_n(X)$ inducing isomorphism on $\ell$-adic cohomology under mild restrictions on $X$ or the characteristic of the field. This confirms a heuristic derived from Weil restrictions. Furthermore, we provide explicit combinatorial formulae for the Betti numbers of $C_n(X)$ and prove a Macdonald-type generating series. A version for Hermitian matrix point is also proved in the last section.

math.AG↗
arXiv2025

On the mechanical creation of mathematical concepts

Any act of problem-solving combines prior knowledge, local search, and a third element that is less often discussed: the extraction of information from search to update understanding. I propose a model of mathematical problem-solving as a belief-update loop in which the mathematician generates auxiliary questions, resolves them through computation, and uses the outcomes to shift confidence in conjectures. The information yield of this loop depends on the vocabulary available to the solver, and I distinguish two forms of concept that reshape this vocabulary: implicit concepts, which improve pruning within a fixed language of moves, and explicit concepts, which introduce new moves that were previously inexpressible. I argue that explicit concept creation is the characteristic step of mathematical discovery, driven by necessity when no computation in the existing vocabulary can resolve the problem, and yielding shareability and composability as byproducts. Current AI systems, including those that achieve superhuman performance in games and formal theorem proving, operate exclusively through implicit concept formation. I discuss what it would take for machines to create explicit concepts, and consider how differing computational tradeoffs between humans and machines may lead to fundamentally different styles of mathematics.

math.HO↗
arXiv2025

Computational Platonism

We offer a view of mathematics as an experimental science where axioms play the role of foundational theories like general relativity and quantum mechanics in physics. Under this view, axioms are provisional and inferred from experience with the experiental substrate of mathematics which we locate within computation rather than encoding intuitive and absolute truths. This offers a reframing of Godel's theorem, placing its impact sharply upon the incompleteness rather than the potentially contradictory nature of any computational set of axioms. The essay originated in an attempt to make precise the nature of mathematics in order to estimate how AI might impact it. This exploration is continued in the paired essay "The mechanical creation of mathematical concepts."

math.HO↗
arXiv2022

A Chebotarev Density Theorem over Local Fields

We compute the $p$-adic densities of points with a given splitting type along a (generically) finite map, analogous to the classical Chebotarev theorem over number fields and function fields. Under some mild hypotheses, we prove that these densities satisfy a functional equation in the size of the residue field. This functional equation is a direct reflection of Poincar\'e duality in \'etale cohomology. As a consequence, we prove a conjecture of Bhargava, Cremona, Fisher, and Gajovi\'c on factorization densities of p-adic polynomials. The key tool is the notion of admissible pairs associated to a group, which we use as an invariant of the inertia and decomposition action of a local field on the fibers of the finite map. We compute the splitting densities by M\"obius inverting certain p-adic integrals along the poset of admissible pairs. The conjecture on factorization densities follows immediately for tamely ramified primes from our general results. We reduce the complete conjecture (including the wild primes) to the existence of an explicit "Tate-type" resolution of the "resultant locus" over the integers and complete the proof of the conjecture by constructing this resolution.

math.NT↗
arXiv2022

On the variation of the Frobenius in a non abelian Iwasawa tower

For varieties over a finite field $\mathbb F_q$ with "many" automorphisms, we study the $\ell$-adic properties of the eigenvalues of the Frobenius operator on their cohomology. The main goal of this paper is to consider towers such as $y^2 = f(x^{\ell^n})$ and prove that the characteristic polynomials of the Frobenius on the \'etale cohomology show a surprising $\ell$-adic convergence. We prove this by proving a more general statement about the convergence of certain invariants related to a skew-abelian cohomology group. Along the way, we will prove that many natural sequences $(x_n)_{n\geq 1} \in \mathbb Z_\ell^{\mathbb N}$ converge $\ell$-adically and give explicit rates of convergence. In a different direction, we provide a precise criterion for curves with many automorphisms to be supersingular, generalizing and unifying many old results.

math.NT↗
arXiv2022

Unlikely and just likely intersections for high dimensional families of elliptic curves

Given two varieties V,W in the n-fold product of modular curves, we answer affirmatively a question (formulated by Shou-Wu Zhang's AIM group) on whether the set of points in V that are Hecke translations of some point on W is dense in V. We need to make some (necessary) assumptions on the dimensions of V,W but for instance, when V is a divisor and W is a curve, no further assumptions are needed. We also examine the necessity of our assumptions in the case of unlikely intersections and show that, contrary to exceptions, two curves in a high dimensional space over a finite field can intersect infinitely often up to Hecke translations.

math.AG↗