MATH, 10 PRIMO, 10 - Vol. 7: The Chain Extends to 384: Discovery and Independent End-to-End Certification of the Maximal Prime Gap g=384 at p=20,678,048,297 (Merit 16.1668), Twelve Fresh pi(1.1e10..2.2e10) incl. pi(2e10)=883,554,787, and a Zero-Axiom Lean 4 Audit of the Chain 354->382->384

Abstract

Volume 6 of the "MATH, 10 PRIMO, 10" series certified the seventeenth element of the maximal-gap chain, 382@10,726,904,659. This volume discovers and certifies the eighteenth: the first prime gap greater than 382 above 10^10, found by an exact segmented sweep of the odd integers - not quoted from any table - at p = 20,678,048,297, with p+384 = 20,678,048,681 and merit 384/ln p = 16.1668. Every claim is established twice, on disjoint paths that share no data. (i) Computationally: verifica_384.py, re-executed live at build time, discovers the record by sweeping a 1.2e10-wide window (363 segments, about one minute), then certifies both endpoints by two disjoint primality paths (Miller-Rabin deterministic 12 bases + complete trial division to sqrt(p)), all 383 interior numbers composite with their smallest factors, consecutivity, and - by composition with Vol. 6's integral certificate of [2, 10^10) - the record property itself: within the window the largest gap with left endpoint below p is exactly 382@10,726,904,659 (the previous record re-certified live), no gap of length 383 occurs before p, 384 occurs exactly once up to p, and exactly two gaps of size >=382 exist in the whole window. The sweep also yields twelve fresh primary-source prime-counting certifications pi(1.1e10) through pi(2.2e10), including pi(2e10) = 883,554,787, composed on the Vol. 6 anchor pi(10^10) = 455,052,511. (ii) Formally: a Lean 4 file (kernel 4.33.1, no mathlib) built as a verbatim superset of the Vol. 6 audit proves, with ZERO axioms on all TWENTY-FOUR theorems (audited by #print axioms; no propext, no Quot.sound, no Classical.choice): the primality of both new endpoints by complete kernel-executed trial division (sqrt(p) ~ 143,798; the Vol. 6 envelope primoSq_ok_big, n < 2099202^2, already covered it - no new foundation needed), the compositeness of all 383 interiors by witness certificates (one kernel-checked division per interior), the full record gap384, the link chain_382_384, and - new for this volume - the entire three-link chain 354 -> 382 -> 384 as a single theorem (chain_354_384). The volume also corrects a prose error of Vol. 6, which quoted the "next literature record" from memory as 486@171,231,342,420 - an even number, hence not prime and impossible as a gap endpoint; the true successor is certified here. All transcripts are injected verbatim; the paper cannot diverge from the mathematics it reports. Complete source code, witness tables, Lean sources, a step-by-step reproduction guide and SHA256 manifests included per the series reconstruction contract.