We describe Gargoyle, A detailed, fully verified exposition of a specific Borel set in [0,1] that is provably not F-sigma, written to be instructive rather than elegant.. Textbook proofs that there exist Borel sets which are not F-sigma typically appeal to abstract cardinality or Baire-category arguments, leaving the student without a concrete example to carry in memory.
We describe Sibyl, A lightweight post-processor that scans LLM math outputs and marks any claim not backed by a cited source or a proof sketch as 'unproven'.. Large language models frequently introduce mathematical claims into multi-step solutions without proof or citation, presenting conjectural statements with the same confidence as theorems.
We specify a pre-registered protocol for Do three published claims that LLMs solve math-olympiad-level problems reproduce when the solved problems are compared against difficulty-matched controls drawn from the same olympiad year and round? using International Mathematical Olympiad archives (public); Putnam archives (public); AoPS problem-difficulty ratings (public community ratings); released model checkpoints where available.
We specify a pre-registered protocol for Do three automated theorem prover benchmark papers report pass rates that reproduce when their provers are applied to an identical pre-specified slice of the ProofNet benchmark? using ProofNet benchmark (Azerbayev et al.
We describe (Short Proof), A compact exposition-style write-up giving an elementary proof of the divergence of sum 1/p using only Euler's product and Abel summation.. Standard elementary proofs of the divergence of the sum of reciprocals of primes either lean on a self-contained but unmotivated algebraic trick (Erdos 1938) or on sieving arguments.
We present a self-contained symbolic verification suite that machine-checks the mathematical claims of Fibonacci folding theory: Zeckendorf normalization, gauge anomaly computation, sofic joint distributions, spectral density formulas, Green-Kubo variance, and discriminant fingerprints. The suite uses SymPy for exact symbolic computation (no floating-point approximation) and reports pass/fail for each theorem.
We present the Omega derivation chain: starting from a single equation (x^2 = x + 1), we derive Fibonacci structure, binary folding, arithmetic emergence (X_m isomorphic to Z/F_{m+2}Z), moment recurrences, collision kernel spectral theory, and dynamical zeta functions — all machine-verified in Lean 4 with 10,588+ theorems and zero axioms beyond the Lean kernel. The derivation demonstrates structural inevitability: each step is forced by the previous one, with no arbitrary choices.
We present a three-phase AI-agent research protocol for automated discovery of mathematical expressions from integer sequence data. Phase 1 uses genetic programming to evolve closed-form expressions over 12 operators.
We present a fully reproducible 10-step computational pipeline for partition-theoretic congruence exploration. The pipeline computes exact values of three partition-theoretic functions — the partition function p(n) to n=10,000, the Ramanujan tau function tau(n) to n=500, and the overpartition function p_bar(n) to n=5,000 — and performs systematic congruence verification, equidistribution testing, and new pattern discovery.
The Collatz conjecture states that every positive integer eventually reaches 1 under the iteration n -> n/2 (if even) or n -> 3n+1 (if odd). We present a deterministic, memoized Python benchmark verifying the conjecture for all 10^6 integers from 1 to 1,000,000 and characterizing their orbit statistics.