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arXiv2026

Prove2Me: An Open Collaborative Platform for Scaling Math Formalization

Proof assistants such as Lean 4 promise the paradigm of formally verified mathematics, but large-scale formalization projects have faced major barriers to entry, including the need for expertise in formal verification (as well as the underlying mathematics) and the significant time required for writing formal proofs. AI coding agents have dramatically reduced these barriers; human users can now use natural language to prompt agents to write complex proofs in Lean. This opens up the intriguing possibility of internet-scale mathematical collaboration involving both humans and AI agents, where correctness is machine-checked. To realize this possibility, we introduce Prove2Me (https://prove2.me), an open collaborative platform for formalizing mathematics. Users launch formalization "missions", to which AI agents contribute formal proofs toward completion. We designed mechanisms and a specialized harness in Prove2Me that enable large-scale collaboration so that agents can build on one another's work and freely reuse existing results. In doing so, Prove2Me aims to turn math formalization into a scalable, crowd-sourced effort open to anyone with an agent.

cs.AI↗
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

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

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

Essential Unitarity for Higher-Order Quantum Computation

We develop a boundary-centric semantic framework for higher-order quantum computation, building on the Kelly-Laplaza description of compact closure and Abramsky's execution account. In the semantic carrier Perm(C), morphisms are complex-linear combinations of polarized boundary linkings, composed by execution. Finite-family addresses provide coherent control over finite-level quantum registers (qudits) while retaining the multiplicative boundary structure. We identify essential unitarity, a boundary condition extending ordinary unitarity to higher-order interfaces. On positive qudit registers it coincides with ordinary matrix unitarity; at higher order it expresses preservation of information across the full polarized boundary. We define a unit-free coherent quantum core generated by multiplicative wiring, unitary gates on positive qudit registers, and contextual coherent control, and prove that every one of its morphisms is essentially unitary. The framework realizes an applied coherent quantum switch and the unitary stages of equal-ratio one-slot supermap dilations with explicit memory. An extended abstract of this work was accepted for QPL 2026 and is forthcoming in its proceedings.

quant-ph↗
arXiv2026

On Left Adjoints Preserving Colimits in Homotopy Type Theory

We examine how the standard proof that left adjoints preserve colimits behaves in the setting of wild categories, a natural setting for synthetic homotopy theory inside homotopy type theory. We show that the proof may fail for adjunctions between wild categories and even produce a wild left adjoint that fails to preserve colimits. Our core contribution, however, is a sufficient condition on the left adjoint for the proof to go through. The condition, which we call 2-coherence, expresses that the naturality structure of the hom-isomorphism commutes with composition of morphisms. We present two useful examples of this condition in action. First, we use it, along with a new version of a known trick for homogeneous types, to show that the suspension functor, as well as a generalization thereof, preserves graph-indexed colimits. Second, we show that every modality, viewed as a functor on coslices of a type universe, is 2-coherent as a left adjoint to the forgetful functor from the subcategory of modal types, thereby proving this subcategory is cocomplete. We have formalized our main results in Agda.

cs.LO↗
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↗
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

Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization

Let $R$ be a commutative ring and $f(a_1, \ldots, a_n) = \sum_{i=1}^k r_{i1}^{a_1} \cdots r_{in}^{a_n} m_i$ be a linear-exponential map over an $R$-module $M$. Dong and Shafrir (2026) showed that, when $\ell M = 0$ for some $\ell \in \mathbb{N}_{>0}$, the zero set of $f$ is the intersection of effectively computable $p$-normal sets, where $p$ ranges over the prime divisors of $\ell$. This generalizes an earlier theorem of Derksen and Masser (2012) on the solution set of $S$-unit equations over fields of positive characteristic. The purpose of this paper is twofold. First, we give a shorter proof of Dong and Shafrir's result, using the theorem of Derksen-Masser as a blackbox. Our proof also yields a decomposition of the zero set as a positive Boolean combination of affine transformations of zero sets of linear-exponential equations over fields. Second, we prove a multi-dimensional generalization of the Skolem-Mahler-Lech theorem over rings of finite characteristic. Specifically, we show that the zero set of every $n$-dimensional linear recurrence sequence over an $R$-module $M$ satisfying $\ell M = 0$ is the intersection of effectively computable $p$-normal sets (in $\mathbb{N}^n$), where $p$ ranges over the prime divisors of $\ell$. For example, this gives a decision procedure for whether two classical linear recurrence sequences have a common value over a ring of characteristic $p^a$ or $p^a q^b$, where $p$ and $q$ are primes.

math.NT↗
arXiv2026

AutoGraphForge: Towards Automated Graph Theory Discovery

We report on our ongoing project to develop a computational pipeline, AutoGraphForge, for an automated graph-theoretic conjecturing-refuting-formalizing-proving system. Conjecture generation is counterexample-guided and runs in rounds: a Graffiti3 generator proposes conjectures over a small, evolving snapshot table $T$ (initially a few hundred graphs with their computed invariants) that grows only by counterexamples to its own conjectures. A novelty filter of $559$ classical and folklore relations, closed under transitive composition and linear identity substitution, decides via a linear program whether a candidate is already implied by known results. Surviving candidates are tested against a dataset of about $348,000$ graphs, unioning the complete House of Graphs invariant export, the exhaustive census of all connected graphs on at most nine vertices, several extremal families (strongly regular, minimal Ramsey, Cayley, cages, barbells, lollipops, spiders), and random models. Counterexample-search algorithms then attack the remainder. Run for several rounds on an HPC cluster, the loop yields $6,522$ conjectures that survived the refutation dataset, the novelty filter and every active-search run -- among them nontrivial relations between the annihilation number and the edge-cover number for bipartite and regular graphs, which we prove by hand. A subsequent formalization and proving stage deterministically translates each surviving conjecture into a Lean 4 statement skeleton; every candidate proof is kernel-verified against a pinned mathlib4 and our custom invariant preamble. This stage integrates two neural provers -- DeepSeek-Prover-V2-671B (served with vLLM) and the Lean-specialised OProver-32B -- behind the independent kernel check. It is implemented end-to-end and passes initial sanity checks, with the full pipeline currently running on the cluster.

cs.AI↗
arXiv2026

A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the $q,t$-Catalan Symmetry

Algebraic combinatorics often seeks bijections that explain identities between distributions object by object. Encoding combinatorial objects as words lets automata theory study such a bijection as a word-to-word computation and measure its memory, input access, and control of output order. This refines existence questions by asking which computational mechanisms a bijection requires. We develop this viewpoint for Dyck paths. Our motivating example is the $q,t$-Catalan polynomial. Let $D_n$ be the set of Dyck paths of semilength $n$, let $D=\bigcup_{n\ge 0}D_n$, and let $area, dinv, bounce \colon D\to\mathbb{N}$ be the standard statistics. Then, \[ C_n(q,t)=\sum_{P\in D_n}q^{area(P)}t^{bounce(P)} =\sum_{P\in D_n}q^{dinv(P)}t^{area(P)}. \] Haglund's zeta map $ζ\colon D\to D$ gives a bijective proof: it preserves semilength and sends $(dinv,area)$ to $(area,bounce)$. By contrast, the full symmetry $C_n(q,t)=C_n(t,q)$ still lacks a direct explanation: no explicit, uniform, semilength-preserving bijection is known that swaps area and dinv on every Dyck path. Polyregular maps from automata theory provide a natural computational starting point, but we prove that neither $ζ$ nor the classical height-sweep bijection witnessing Narayana symmetry is polyregular. The missing mechanism is global ordering by numerical levels whose range grows with the input. We call this a \emph{rank sort} and introduce \emph{weighted-rank polyregular maps} (WRP), extending polyregular maps by one such sort and containing both bijections. Nevertheless, WRP is a proper subclass of deterministic logspace. We prove that $ζ^{-1}$ lies outside WRP and that no WRP map can realise a semilength-preserving area-dinv swap. Thus the rank-sorting strategy behind $ζ$ cannot be extended within WRP to exchange the two statistics.

math.CO↗
arXiv2026

Implementing Grassroots Logic Programs with Multiagent Transition Systems and AI (Full Version)

Grassroots Logic Programs (GLP) is a concurrent logic programming language in which logic variables are partitioned into paired readers and writers. An assignment is produced at most once via a writer and consumed at most once via its paired reader, and may contain additional readers and/or writers. This enables the concise expression of rich multidirectional communication modalities. The language was introduced together with concurrent (cGLP) and multiagent (maGLP) operational semantics. Here, we derive from these (1) dGLP, a deterministic counterpart of cGLP, and (2) madGLP, a counterpart of maGLP in which deterministic agents communicate solely by asynchronous message passing, and prove them correct against their abstract counterparts. maGLP shared variable pairs spanning agents can be implemented by two local variable pairs joined by a \emph{global link}, with correctness following from disjoint substitution commutativity (a consequence of GLP's single-occurrence invariant). We further prove that madGLP is grassroots. Both dGLP and madGLP serve as formal specifications for an AI-driven implementation discipline (math $\to$ informal spec $\to$ Dart) employed and described here: from dGLP, AI (Claude) developed a workstation-based GLP implementation in Dart, and from madGLP it is developing a smartphone-based multiagent one.

cs.PL↗
arXiv2026

Tri-Band Channel Measurement-Enabled Multi-Layer Digital Twin for Terahertz Wireless Data Centers

The rapid growth of AI computing has driven increasing demands for flexible and high-capacity data-center interconnections. Owing to its ultra-wide bandwidth and high spatial reuse capability, terahertz (THz) communication has emerged as a promising solution for future wireless data centers, while digital twins (DTs) enable efficient wireless planning and real-time optimization. In this work, a measurement-driven multi-layer DT framework is proposed for THz wireless data centers, where the physical, channel, evaluation, and manipulation layers are progressively constructed from bottom to top. First, extensive channel measurements are conducted at 140, 220, and 300 GHz to characterize frequency-dependent propagation behaviors. Based on the tri-band measurements, a measurement-calibrated physical twin is established by jointly optimizing the geometry, material, antenna, and hybrid propagation models. On top of the physical twin, a line-of-sight (LoS)-aware implicit neural field is developed to construct an AI channel twin for efficient channel reconstruction. The proposed AI twin learns location-dependent channel statistics from the calibrated twin, enabling real-time prediction of received power and LoS probability. Building upon the reconstructed channel field, a system-level evaluation layer is derived to analyze coverage and interference for both AP-to-rack and rack-to-rack communications. Experimental results show that the proposed AI twin achieves lower power reconstruction error than existing neural-field baselines while maintaining real-time inference capability. Moreover, the ceiling-mounted AP deployment achieves over 90% coverage under a 10 dB signal-to-interference-plus-noise ratio (SINR) threshold, demonstrating the effectiveness of the proposed DT framework for THz wireless data-center planning and optimization.

cs.LG↗
arXiv2026

Causal Probabilistic Programming via Magmadic Do-Notation

We introduce a do-notation metalanguage for causal probabilistic programming. The metalanguage is based on magmads: non-associative monads. We derive causal probabilistic programming constructs from non-associativity and the primitives of probabilistic programming.

cs.PL↗
arXiv2026

Deriving Program Logics from Distributive Monoidal Categories

We derive multiple program logics - including correctness, incorrectness, and relational Hoare logic - from the axioms of imperative categories: uniformly traced distributive copy-discard categories. Rules of program logics follow from the axioms of imperative categories. The algebra of guarded commands derived by the categorical structure generalises guarded Kleene algebras with tests.

cs.LO↗
arXiv2026

A Non-Formulable Theorem: A Fundamental Limit of Finite Syntactic Systems and Its Consequences for Security and AI

For every coherent and sufficiently expressive finite syntactic system S, we prove the existence of at least one theorem that S cannot produce autonomously. The result is a metatheorem: it proves the existence of a theorem, and applies to every finite syntactic system - security mechanisms, AI systems, formal verifiers, legal systems, economic models, and the formal system in which it is itself proved.

cs.CR↗
arXiv2026

Revisiting average case complexity of multilevel syllogistic

We describe a Lean~4 formalization revisiting NYU Courant Technical Report TR1995-711 on the average-case complexity of Multilevel Syllogistic (MLS). The development encodes Reischuk--Schindelhauer average-case classes, an axiomatic MLS/EMLS semantics layer, a partial Ferro--Omodeo--Schwartz decision procedure with proved soundness and partial completeness on a membership-free fragment, serialization and step budgets, and conditional NP-average completeness and non-AvP hardness corollaries modulo explicitly documented structural axioms. Full Lean sources are inlined in the appendix modules.

cs.LO↗
arXiv2026

String Rewriting Systems: Brief Introduction and Sample of Open Problems

This document is the result of pulling together small parts of different lecture notes I have written over several decades for first-year graduate-level courses -- typically with the title Formal Methods -- which included many other topics of mathematical logic and theoretical computer science. After combining these materials, I updated the references and adjusted the text to account for progress accomplished in the intervening years. The sample of open problems in the last section is a small collection of special cases that are still unresolved up until the date of this writing.

cs.LO↗