MATH, 10 PRIMO, 10 - Vol. 16: pi Meets the Primes - The Digit-Offset Exploration (the Complete Step-by-Step Documentation of the Ten-Stage Dialogue; the Corrected Digit-Restriction Theorem with the Three Exceptional Triplets (3,5,7), (3,7,11), (3,11,19); the Exact Census of 55 Coincidences to 10^4 Digits, All with d_n = 6; the Hardy-Littlewood Reconciliation with the Correct Singular Series S(0,6,12) = 5.7165 = 2 x S(0,2,6), Expected 56.2 vs Observed 55; Twelve Documented Corrections Including C

Abstract

Volume 16 of the series documents, step by step and in full, the author's exploration of the relation between the digits of pi and the primes - conducted as a ten-stage dialogue with an AI interlocutor - and then submits every quantitative claim of that dialogue to an independent 59-check audit with its own sieve and 10200-digit digit generation. The census survives exactly: precisely 55 positions n <= 10^4 have p_n - d_n, p_n, p_n + d_n prime and pairwise distinct, all with d_n = 6, and the observed count matches the Hardy-Littlewood independence prediction to within one unit (56.2 expected from 1021 sixes and exactly T = 550 triplet centers among the first 10^4 primes; HL itself predicts 576, a 4.8 percent error, with the CORRECT singular series S(0,6,12) = 5.7165 = 2 x S(0,2,6) - the dialogue had used the (0,2,6) constant 2.858 for the wrong pattern). The audit also corrects twelve errors, three of them load-bearing: the digit restriction theorem needs three exceptional triplets - (3,5,7), (3,7,11), (3,11,19) - beyond d = 6 (certified in Lean 4.33.1, core only, axiom profile propext/Quot.sound, no Classical.choice, no sorryAx); three of the six pi phase-table rows rest on composites claimed prime (3149 = 47*67, 31411 = 101*311, 31421 = 13*2417; true pairs (3137,3163) and (31397,31469)); and Pi_6 = 314159 is itself prime, so the perfect zero returns and the claimed stabilization law E_k^- -> -8 and the no-zeros-after-k = 2 claim are void. The phi row k = 5 likewise claims 16177 = 7*2311 prime. The novelty ledger is unchanged: the sum sequence is OEIS A109635, the triplet centers are OEIS A006489, the decimal rulers are OEIS A011545, and the density theory is Hardy-Littlewood (1923) - the volume's contributions are the corrected theorem, the exact census, the corrected densities, twelve documented corrections, and a case study in verification debt. Sources, verification scripts, JSON data, Lean file with axiom certificates and SHA256 manifest included. License CC BY 4.0.