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arXiv2026

Proof verification by polynomial Fingerprinting

To cater to the needs of fast verification for mathematical proofs, we describe a method to encode formal sentences in $2 \times 2$ - matrices over multivariate polynomials with integer coefficients. This correspondence is homomorphic: usual proof-steps like modus-ponens or variable substitution in terms and formulae become operations with matrices. By evaluating the polynomial variables in random elements of a suitably chosen finite field, the proof is replaced by a numeric sequence. Only the values corresponding to axioms and tautologies have to be computed from scratch. The values corresponding to derived formulas are computed from the values corresponding to their ancestors by applying the homomorphic properties. The polynomial matrix corresponding to the conclusion of the proof is also evaluated in the chosen random values. If the last term of the numeric sequence equals the evaluation of the conclusion, by the Schwartz-Zippel Lemma, the proof is with high probability correct.

math.LO↗
arXiv2026

Examples and counterexamples of injective types

It is known that, in univalent mathematics, type universes, the type of $n$-types in a universe, reflective subuniverses, and the underlying type of any algebra of the lifting monad are all (algebraically) injective. Here, we further show that the type of ordinals, the type of iterative (multi)sets, the underlying type of any pointed directed complete poset, as well as the types of (small) $\infty$-magmas, monoids, and groups are all injective, among other examples. Not all types of mathematical structures are injective in general. For example, the type of inhabited types is injective if and only if all propositions are projective. In contrast, the type of pointed types and the type of non-empty types are always injective. The injectivity of the type of two-element types implies Fourman and Ščedrov's world's simplest axiom of choice. We also show that there are no nontrivial small injective types unless a weak propositional resizing principle holds. Other counterexamples include the type of booleans, the simple types, the type of Dedekind reals, and the type of conatural numbers, whose injectivity implies weak excluded middle. More generally, any type with an apartness relation and two points apart cannot be injective unless weak excluded middle holds. Finally, we show that injective types have no non-trivial decidable properties, unless weak excluded middle holds, which amounts to a Rice-like theorem for injective types.

math.LO↗
arXiv2026

Coexact completion of profinite Heyting algebras and uniform interpolation

This paper shows that the sheaf representation of finitely generated free Heyting algebras constructed by Ghilardi and Zawadowski can be factored as the profinite completion of Heyting algebras, followed by identifying the dual category of profinite Heyting algebras as a full subcategory of a sheaf topos. We show that the dual category of profinite Heyting algebras is an infinitary extensive regular category, and its ex/reg-completion is exactly the aforementioned sheaf topos, which we refer to as the K-topos. We show how certain properties of uniform interpolation can be generalised to the context of arbitrary profinite Heyting algebras, and that they are consequences of the internal logic of the K-topos. Along the way we also establish various topos-theoretic properties of the K-topos.

math.LO↗
arXiv2026

Trace-Tree Magmas: Proof-Producing Infinite Countermodels and 28 New Order-Five Austin Classifications

Finite model finders cannot witness an Austin law: an identity whose finite models are all trivial but which has a nontrivial infinite model. We introduce rank-decreasing sparse trace-tree magmas, finitely presented total operations on a countably infinite constructor-tree carrier. The default product pairs its arguments; finitely many positive Horn clauses define exceptions. Our main procedure derives clauses from symbolic evaluation traces. For every model found, it proves functionality of the exceptional relation by descent on constructor size, proves the identity by exhaustive symbolic case analysis, and emits a self-contained Lean 4 certificate. A least simultaneous fixed point gives an implementation-independent semantics, so bounded search may miss models but cannot invalidate certified results. On ETP's 96 order-five Austin candidates, we discover and Lean-verify infinite countermodels for 28 identities with no prior public classification in our audit. They form 14 duality classes and establish 28 new Austin classifications. Four ALPS-known cases bring the total to 32 certified candidates. On Canonical-4187, the deduplicated union of Order5-130 and the 4,141-row ALPS pool, a fresh trace run produces 636 certificates, all accepted by Judge v3. At equal resource limits, Vampire 5.0.1, E 3.5.1, and complete Twee 2.6.1 jointly prove implications in 94 canonical classes. Only Twee returns trusted counter-satisfiable outcomes, for 18 classes; independent finite-side certificates force 16 to be infinite. None of these ATPs emits an explicit model or Lean certificate, and none decides the 28 new classifications. To the best of our audit, this is the first automated system to synthesize this trace-tree model family, generate well-founded inversion proofs, and emit self-contained Lean 4 certificates.

cs.LO↗
arXiv2026

Univalence without function extensionality

It is a well-known theorem of homotopy type theory, originally due to Voevodsky, that function extensionality holds inside any univalent universe. We consider a weaker variant of the univalence axiom, asserting that the wild category formed by the universe is univalent, which we call categorical univalence. We show that categorical univalence does not imply function extensionality by an analysis of Von Glehn's polynomial model construction, which produces models of Martin-Löf type theory that always refute function extensionality. We find in particular that when the base model has a univalent universe, its polynomial model has a universe that is categorically univalent but lacks function extensionality.

cs.LO↗
arXiv2026

A SAT Attack on Tarski's High School Algebra Problem

Tarski's high school algebra problem asks whether every true identity concerning addition, multiplication, and exponentiation of positive integers follows from a list of 11 elementary identities. Surprisingly, Wilkie showed that the following identity is valid over the positive integers and yet does not follow from Tarski's axioms: \begin{align*} &\left((1+x)^y + (1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x + (1+x^2+x^4)^x\right)^y = \\ &\left((1+x)^x + (1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y + (1+x^2+x^4)^y\right)^x. \end{align*} Gurevič gave an algebra on 59 elements that satisfies Tarski's axioms but not Wilkie's identity, and over the years several authors whittled down the size of such a countermodel, culminating in a countermodel of size 12 due to Burris and Yeats. On the other hand, Zhang proved that there is no countermodel with fewer than 11 elements. Using SAT, we prove that the smallest countermodels are of size 12, as conjectured by Burris and Yeats. Moreover, we show that there are exactly 8,957,952 countermodels on 12 elements up to isomorphism and provide a simple classification of them. Our SAT approach outperforms dedicated tools for finding countermodels in equational theories, namely Mace4 and SEM. Furthermore, using autoformalization, we prove the correctness of our main result in Lean.

math.LO↗
arXiv2026

Quantified propositional calculi and narrow implicit proofs

In the implicit version of a propositional proof system Q, we work with Q-proofs that are not written down directly, but are succinctly encoded by circuits. Thus implicit Q-proofs are potentially exponentially shorter than usual Q-proofs. We study narrow implicit proofs, a restricted version of this notion, in which lines in the encoded proof can only have polynomial size. We use a cut-elimination construction to show that G_{i+1} is equivalent to narrow implicit G_i, for i >= 1, where G_i is the extension of Frege allowing reasoning with Sigma^q_i quantified propositional formulas. We show that G_1 is equivalent to implicit resolution.

cs.LO↗
arXiv2026

Modalities in non-classical variations of $\mathsf{S4}$

A classical result in modal logic states that $\mathsf{S4}$ has $14$ modalities, that is, every sequence of negations, boxes, and diamonds is equivalent to one in a set of $14$ such sequences. We study analogous results for the non-classical analogues $\mathsf{CS4}$, $\mathsf{IS4}$, $\mathsf{GS4}$, and $\mathsf{GS4^c}$ of $\mathsf{S4}$. First, we show that, while all these logics have finitely many $\{\Box,\Diamond\}$- and $\{\neg,\Box\}$-modalities, the logic $\mathsf{CS4}$ has infinitely many $\{\neg,\Diamond\}$-modalities. Second, we show that $\mathsf{IS4}$ and $\mathsf{GS4}$ have finitely many $\{\neg,\Diamond\}$-modalities, but they have infinitely many $\{\neg,\Box,\Diamond\}$-modalities. At last, we show that $\mathsf{GS4^c}$ has finitely many $\{\neg,\Box,\Diamond\}$-modalities.

math.LO↗
arXiv2026

A categorical formulation of Kraus' paradox

We give a categorical formulation of Kraus' "magic trick" for recovering information from truncated types. Rather than type theory, we work in Van den Berg-Moerdijk path categories with a univalent universe, and rather than propositional truncation we work with arbitrary cofibrations, which includes truncation as a special case. We show, using Kraus' argument that any cofibration with homogeneous domain is a monomorphism. We give some simple concrete examples in groupoids to illustrate the interaction between homogeneous types, cofibrations and univalent fibrations.

math.CT↗
arXiv2026

Proof-Carrying Analytic Approximation: Local-to-Global Evidence Transport at Encoding Cost

Under quasi-uniform refinement, bounded local-encoding hypotheses, and local $W^{r,2}$ approximation of order $r\ge 2$ in a rational piecewise-polynomial presentation of $W^{1,2}(0,1)$, carrying the complete proof genealogy up to the level required by an accuracy $\varepsilon$ costs the same asymptotic bit order as the finest-level conventional coefficient encoding. If $B_n=Θ(M_nβ_n)$ denotes that level-$n$ encoding size, our compiler transports supplied local approximation and overlap witnesses through exact partition-of-unity synthesis and geometric refinement to a represented limit with total certificate size $O(B_{m(\varepsilon)})$, where $m(\varepsilon)=O(\log(1/\varepsilon)/(r-1))$. The construction makes no oracle query to an independently supplied semantic target name ($Q_{\rm target}=0$). When $β_n=O(n+1)$, this becomes $O(\varepsilon^{-1/(r-1)}(1+\log(1/\varepsilon)))$. The surrounding framework is intentionally separated from this resource theorem. Every real computable Banach presentation admits a uniformly computable linear isometric embedding into standard computable $C([0,1])$, with computable inverse on its represented range. Complete metric evidence with rational strict slack collapses extensionally to the represented analytic metric once effective names are available, while chosen evidence transformations retain construction history and resource information. For the Lipschitz grammar used here, qualitative evidence-local lifting is canonical; the nontrivial question is therefore which evidence is retained and at what cost.

math.FA↗
arXiv2026

A Borel Concept Class of VC Dimension One with a Non-PAC Consistent Learner in ZFC

The fundamental theorem of statistical learning states that, under suitable measurability assumptions, finite Vapnik--Chervonenkis (VC) dimension guarantees that every proper consistent learning rule is probably approximately correct (PAC). Blumer, Ehrenfeucht, Haussler, and Warmuth showed, assuming the Continuum Hypothesis, that the "well-behavedness" condition of the concept class cannot be omitted: they constructed a concept class of Borel sets of VC dimension one admitting a consistent learning rule that is not PAC. We show that the Continuum Hypothesis is unnecessary. Working in Zermelo--Fraenkel set theory with the Axiom of Choice (ZFC) alone, we construct a concept class of Borel sets on $[0,1]$ of VC dimension one and a proper consistent learning rule that is not PAC. More precisely, for a suitable Borel probability measure and target concept, the rule has true risk one at every sample size on a set of samples of outer probability one. Consequently, finite VC dimension and Borel measurability of the individual concepts do not suffice to guarantee that every proper consistent learning rule is PAC. The result shows, with no need of extra set-theoretical assumptions, that the additional regularity assumption in the fundamental theorem cannot in general be omitted.

math.LO↗
arXiv2026

Exponential Gaps Between Intuitionistic Linear Extended Frege Systems

In this paper, we establish exponential separations between Extended Frege systems for a range of intuitionistic substructural and linear logics. More precisely, for any logic $L$ below the intuitionistic logic obtained by extending $\mathbf{ILL}$ with structural rules, and any logic $M$ not contained in $L$, we construct a family of $\mathsf{FL_e}$-provable formulas that have short proofs in $M$-Frege but require proofs of exponential size in $L$-Extended Frege. The same result holds in the $!$-free settings, using $\mathbf{IMALL}$ and $\mathbf{FL_e}$ in place of $\mathbf{ILL}$. The key ingredient in proving these separations is a variant of the feasible disjunction property for $L$-Frege, which may be of independent interest.

cs.LO↗
arXiv2026

Subgroup Accessibility in Group Order Logic

We investigate the expressive power of fixed-point logics (FP) and their extensions in defining generating sets for accessible subgroups of definable permutation groups. This operation, computable in polynomial time via the Schreier-Sims algorithm, plays a central role in the group-theoretic approach to Graph Isomorphism and Graph Canonisation. In particular, it underpins polynomial-time canonisation for bounded colour-class graphs--a class for which no natural logic capturing P is currently known. We first show that this operation cannot, in general, be expressed in any logic for P. This limitation arises from the fact that accessible subgroups need not admit symmetric generating sets of polynomial size. However, we prove that when the base group admits a definable ordered generating set, the accessible subgroup operation becomes definable in fixed-point logic with the group order operator (FP + ord). This is achieved by partially simulating the Schreier-Sims algorithm within FP + ord. As a corollary, we show that fixed-point logic with counting (FPC) can also define the operation when the base group is abelian. In particular, FPC can define the automorphism group of any graph with abelian colours--despite being unable to canonise such graphs.

cs.LO↗
arXiv2026

Bootstrapping Mutual Attestation with Kleene's Second Recursion Theorem

Mutual attestation among nodes with no central trusted operator requires each node to hold reference values (expected code measurements) for its peers. The naïve approach of mutually embedding these reference values in the nodes' code leads to an infinite regress. We call the problem of resolving this infinite regress the reference-value bootstrapping problem for mutual attestation. Existing solutions avoid this regress by relying on a trusted third party (TTP), externally supplied reference values, or architecture-specific measurement mechanisms. We instead express the bootstrapping problem as a system of mutual fixed-point equations and solve it by Kleene's second recursion theorem. The construction produces nodes that mutually reference one another's code and reconstruct every peer's exact source from built-in data alone. When a deployed source file is measured directly, as with a Python script, a node obtains the peer's reference value by applying the measurement function directly to the reconstructed source. When a built image is measured, as with AWS Nitro Enclaves, a node instead reproducibly rebuilds the peer's image from the reconstructed source and derives its reference measurement. For the first case, we develop PyReflect, a Python transpiler, and use it to implement a TPM mutual-attestation PoC. For the second, we develop NixReflect, a Nix transpiler, and use it in a PoC in which two Nitro Enclaves reproduce each other's reference PCRs from built-in data alone. Our solution is architecture-independent, requires neither a TTP nor externally supplied reference values, and works with existing attestation stacks unchanged.

cs.CR↗
arXiv2026

Unconditional $V^0_1$-independence of a certified hitting-set principle

We show that a certified formalization of the hitting-set-existence axiom of Atserias and Tzameret, instantiated on the parity-based Nisan-Wigderson compression class of Khaniki, is independent of the two-sorted theory $V^0_1$ of $\mathrm{AC}^0$-reasoning, unconditionally: $V^0_1$ proves neither it nor its negation. The same holds for the corresponding certified dual weak pigeonhole principle, whose refutation is witnessed by a single seed that certified-computes every string of the model simultaneously. The mechanism is a bounded-arithmetic transfer of Atserias-Tzameret's reduction from hitting sets to the dual weak pigeonhole principle: the amplification half of that reduction, the sole source of its NP-oracle, is unnecessary at the native stretch of the Nisan-Wigderson map, and the compression half becomes a $V^0_1$-provable implication once circuit evaluation is replaced by its certified $Σ^B_0$ unfolding. This is, to our knowledge, the first independence result for a derandomization-flavoured existence principle at the $\mathrm{AC}^0$-reasoning level, and it makes explicit the bridge between the Khaniki Nisan-Wigderson line and the Atserias-Tzameret reverse mathematics of hitting sets.

cs.CC↗
arXiv2025

Security Science (SecSci), Basic Concepts and Mathematical Foundations

This textbook compiles the lecture notes from security courses taught at Oxford in the 2000s, at Royal Holloway in the 2010s, and currently in Hawaii. The early chapters are suitable for a first course in security. The middle chapters have been used in advanced courses. Towards the end there are also some research problems.

cs.CR↗