MATH, 10 PRIMO, 10 - Vol. 1-S (v1.1): Methods and From-Zero Reconstruction Guide for the Shifted Euler Champion n^2-61n+971 (71 Consecutive Primes)

Abstract

Methods, environment specification, and a step-by-step from-zero reconstruction guide for the PRIMES-LOOP discovery that n^2-61n+971 is prime for all 71 consecutive integers n=0..70 (Euler-Rabinowitsch m^2+m+41 shifted by t=31, Heegner number 163). Includes the completed exhaustive census of all nine Heegner families (72,000,016 shifts): vertex optimality in every family, the rigid staircase below run 80, and elementary proofs of the small-family ceilings (D=-3 run<=3, D=-4 run<=2), each confirmed computationally. All claims re-verified by the self-contained verify_chain.py (three independent primality pathways; about 1 s). The record-gap chain 132-320 is literature-complete: all 14 first occurrences machine-verified (v1.1 incorporates an external audit that restored three omitted records, corrected the next-milestone value to 336 at 3,842,610,773, and fixed the MR-12 bound citation). The corrected next-milestone was subsequently certified live: the engine's deep scan crossed 3,842,610,773 and machine-verified the record gap 336 (merit 15.2247) with three-path confirmation. Complete source code and LaTeX sources included per the series reconstruction contract.