arXiv subjects

Ibrahim Mian

Publications and source records attributed to Ibrahim Mian.

3 recordsLinked to original sources

arXiv2026

A Kernel-Checked Exclusion Certificate for Erd\H{o}s Problem 647

Erd\H{o}s problem 647 asks whether any $n > 24$ satisfies $\max_{m<n}(m + \tau(m)) \le n + 2$, where $\tau$ is the divisor-count function. Computational searches have excluded solutions up to $10^{12}$ by direct sieve and up to roughly $9.17 \times 10^{18}$ within a modular reduction whose Lean component relies on native_decide; those computations sit outside any proof kernel. We give the first exclusion checked end to end by one: no solution exists with $24 < n \le 10^9$, proved in Lean 4 with axiom closure exactly {propext, Classical.choice, Quot.sound} -- no sorry, no native_decide, no problem-specific axiom. The proof replays a chain of 6,685,922 factorization witnesses whose excluded intervals concatenate across $(24, 10^9]$; it needs no primality facts beyond primes below 1024, and it is the finite, fully proved form of a domination-interval argument whose asymptotic step was the identified gap in a withdrawn January 2026 claim on this problem. The generation pipeline is cross-checked by two further independent implementations, the compiled development replays through the standalone lean4checker, and two from-source verification legs -- Lean toolchains compiled from source by gcc and by clang, mathlib rebuilt with no cache -- reproduce the committed certificates byte for byte, with olean digests identical across three builds on two architectures. Our range is three to ten orders of magnitude below the computational frontiers we cite; the contribution is the trust base, not the range.

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arXiv2026

Machine-Checked Certificates for the Geometric Half of the Minimum Kochen-Specker Bound

The best known lower bound for the minimum Kochen-Specker vector system in $\mathbb{R}^3$ -- 24 vectors -- rests on a computational proof whose combinatorial half emits DRAT proofs but whose geometric half does not: the non-embeddability of thousands of candidate graphs is established by Z3's nonlinear real arithmetic, which produces no checkable proof objects. We close this gap for the proof's blocking database. We introduce exact rational case-tree certificates of real non-embeddability, whose splits are polynomial factorizations and rational sum-of-squares decompositions and whose leaves are discharged by injectivity, ideal-membership, or Positivstellensatz-shaped positivity arguments, and we certify all 291 source lines (180 distinct graphs) of the published pipeline's order-10 to order-13 blocking lists. Certificates are replayed by two independent checkers that share no code with the generator: a pure-Python replay over exact fractions, and a total checker implemented and proved sound in Lean 4. The soundness theorem -- acceptance implies that no injective-on-rays, orthogonality-respecting assignment of nonzero real vectors realizes the graph -- is kernel-checked with axiom closure {propext, Classical.choice, Quot.sound}, and a gcd-free rational arithmetic layer makes the entire verdict computation kernel-reducible, so each per-graph non-embeddability result is a closed kernel theorem proved by decide. The formalization surfaced findings about the published pipeline, including a load-bearing injectivity side condition in its embeddability notion, hidden WLOG case obligations invisible to Z3-based workflows, and an unreproducible candidate count that we resolve against the published artifacts. All certificates, checkers, and proofs are available and replayable from a single build.

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arXiv2026

Kernel-Checked Exclusions for the Erd\H{o}s-Selfridge Odd Covering Problem: Any Odd Covering of $\mathbb{Z}$ Has lcm Exceeding 10000

The Erd\H{o}s-Selfridge odd covering problem (Erd\H{o}s problem #7) asks whether a covering system of $\mathbb{Z}$ exists whose moduli are all odd, distinct, and greater than 1. The problem is open. We present a Lean 4 formalization, checked end to end by the proof kernel, of the exclusion: any covering of $\mathbb{Z}$ by finitely many congruence classes with distinct odd moduli > 1 has lcm of the moduli exceeding 10000. The proof composes a formalized density argument (a covering by divisors of $N$ exceeding 1 forces $2N \le \sigma_1(N)$, so the lcm is abundant or perfect), a kernel-checked abundancy floor (no odd $N < 945$ qualifies), a family of Chinese-Remainder capacity certificates -- decidable per-$N$ arithmetic inequalities each refuting every covering with distinct moduli > 1 dividing that $N$ -- for all 23 odd abundant numbers below $10^4$, and a kernel-checked enumeration establishing that those 23 are the only odd non-deficient candidates. The result is transported to the official StrictCoveringSystem $\mathbb{Z}$ formulation of Erd\H{o}s #7 in google-deepmind/formal-conjectures, with a bidirectional periodicity bridge between coverings of $\mathbb{Z}$ and finite checks over $\mathbb{Z}/N\mathbb{Z}$ suitable for consuming future SAT-style search output. All 63 published theorems depend on exactly propext, Classical.choice, and Quot.sound: no sorry, no native_decide, no solver in the trusted base. The mathematical content is known -- the density argument is folklore, and far larger uncertified classifications of covering numbers exist -- so the contribution is epistemic rather than mathematical: these exclusions are theorems of the Lean kernel, with an axiom gate enforced mechanically in continuous integration.

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