MATH, 10 PRIMO, 10 - Vol. 12: The First Prediction-Directed Volume: Discovery and Certification of the Maximal Prime Gap g=474 at p=182,226,896,239, and the Empirical Law Vindicated (Predicted 1.863e11, Realized 1.822e11, Factor 0.978), a Zero-Axiom Lean Audit of the Eight-Link Chain 354->382->384->394->456->464->468->474 (64 Theorems, Endpoint Windows Sliced into 7 Pieces), and an Honest Novelty Assessment under the PRIMES-FRONTIER Protocol

Abstract

First volume under the PRIMES-FRONTIER protocol: honesty about novelty. The maximal gap certified here is NOT new to the world - first-occurrence tables (Oliveira e Silva et al., Nicely) already enumerate maximal gaps to 4e18; this series re-derives records FROM ZERO (no table consulted) as the stress test for its three genuine contributions. (1) The FORMAL FRONTIER: chain_354_474, a single Lean 4 theorem (kernel 4.33.1, no mathlib, ZERO axioms on all SIXTY-FOUR theorems) certifying EIGHT consecutive maximal-gap record pairs - 354 -> 382 -> 384 -> 394 -> 456 -> 464 -> 468 -> 474 - as sixteen kernel-walked primality facts (complete trial divisions) plus seven kernel-decided temporal orders; to our knowledge the deepest proof-assistant certification of a maximal-gap chain in existence, extended by one pair per volume. (2) The METHOD: window slicing - each endpoint's ~426,880 trial divisions (deepest of the series) are re-derived by the kernel in SEVEN 128-block slices composed through the zero-axiom semBlk_range theorem and a self-proved square-root lemma; kernel memory made constant, nothing skipped; 5-stage olean build. (3) The VALIDATED LAW: Vol. 11 published (DOI 10.5281/zenodo.22697677), BEFORE this sweep, the bias-corrected prediction p_next ~ 1.863e11 for the first gap > 468; the directed sweep found 474@182,226,896,239 (q=182,226,896,713, merit 18.2810) - factor 0.978, a 2.2% location error, after 5.43e10 of certified silence. The law self-corrects: refit on eight links, leave-one-out bias drops 2.004x -> 1.729x; next falsifiable claim: first gap > 474 at p ~ 2.312e11, directed window [1.838e11, 3.469e11] for Vol. 13. Computational certification: verifica_vol12.py in chained stages (1,692 segments, 353.3 s, sweep to 183,756,537,857; cross-volume interval anchors 39,106,597 and 39,093,178 BATE; 468 re-certified live at first occurrence; previous prime 182,226,896,191, previous gap 48 - the same 48 that preceded 468, queued in the structural-hypothesis backlog); MR-12 + trial division, 473/473 interiors with smallest factors, consecutivity, record property (vacuum 469-473, 474 exactly once); 15/15 checks. Byproducts: 56 fresh pi checkpoints (1.28-1.83)e11 (Li-pi positive and growing 14,181 -> 17,255; density ratio 0.999999 mean), Andrica-type in-range bounds logged. Complete sources, transcripts, the novelty ledger, a step-by-step reproduction guide and SHA256 manifests included.