MATH, 10 PRIMO, 10 - Vol. 10: The Chain Extends to 464: Discovery and Independent End-to-End Certification of the Maximal Prime Gap g=464 at p=42,652,618,343 (Merit 18.9571), the Largest Inter-Record Span of the Series (1.76e10), Three Interval Anchors, an Erratum to Vol. 9 Fresh pi (Double Count +41,951,193, Corrected by Three Independent Paths), and a Zero-Axiom Lean 4 Audit of the Six-Link Chain 354->382->384->394->456->464

Abstract

Volume 9 of the "MATH, 10 PRIMO, 10" series discovered and certified the twentieth element of the maximal-gap chain, 456@25,056,082,087. This volume discovers and certifies the twenty-first: the first prime gap greater than 456 above 2.2e10, found by an exact segmented sweep of the odd integers - not quoted from any table - at p = 42,652,618,343, with p+464 = 42,652,618,807 and merit 464/ln p = 18.9571, reached after the longest silent stretch the series has certified (inter-record span 17,596,536,256 = 1.76e10, six times the previous maximum); no gap of length 457 through 463 occurs before p, and both previous records (394@22,367,084,959 and 456@25,056,082,087) are re-certified live as first occurrences inside the same window. Every claim is established twice, on disjoint paths that share no data. (i) Computationally: verifica_vol10.py, re-executed live at build time (661 segments, 111 s, sweep to 44,179,479,553), certifies both endpoints by two disjoint primality paths (Miller-Rabin deterministic 12 bases + complete trial division to sqrt(p) ~ 206,525), all 463 interiors composite with smallest factors, consecutivity (previous prime 42,652,618,301, gap 42), and the record property; 19/19 checks. (ii) Formally: a Lean 4 file (kernel 4.33.1, no mathlib), a verbatim superset of the Vol. 9 audit, proves with ZERO axioms on all FORTY-EIGHT theorems (#print axioms; no propext, no Quot.sound, no Classical.choice): the endpoints (kernel-walked trial division, the series' deepest), the 463 witness certificates, the full record gap464, the link chain_456_464, and the entire SIX-link chain 354 -> 382 -> 384 -> 394 -> 456 -> 464 as a single theorem (chain_354_464: twelve kernel-walked primality facts, five kernel-decided order facts). The volume's second subject is the erratum it carries: the three fresh pi values printed in Vol. 9 v1 carried a double count of +41,951,193 (the sweep window was summed on top of its own base point). Three disjoint paths now agree exactly - an independent single-interval arbitration, a corrected re-run (verifica_vol9.py v1.1), and this volume's from-scratch sweep - establishing pi(2.4e10) = 1,050,186,367, pi(2.5e10) = 1,091,987,405, pi(2.6e10) = 1,133,717,820; Vol. 9's interval anchors were never at risk (a constant error cancels in differences) and the record 456 never depended on pi; Vol. 9 was corrected in place (v2) with fixed metadata. Eighteen fresh certifications pi(2.7-4.4)e10 extend the ladder (next anchor pi(4.4e10) = 1,875,482,847). The Lean build in this memory-limited sandbox is documented in two stages via a kernel-checked olean (stage A 5m49s + stage B 1m53s), with the maxRecDepth/maxHeartbeats pitfalls recorded; the canonical single-shot file ships for normal environments. All transcripts are injected verbatim; the paper cannot diverge from the mathematics it reports. Complete source code, witness tables, Lean sources, the arbitration tooling, a step-by-step reproduction guide and SHA256 manifests included per the series reconstruction contract.