Uniform cubic lower bounds for strict three-unit-fraction representations at primes

Version v0.8 extends the v0.7 fixed-numerator result to a uniform growing-numerator prime-block theorem. Let R_II,m(p) count distinct positive integer triples x < y < z satisfying m/p = 1/x + 1/y + 1/z, with p dividing y and z and p not dividing x. There exist absolute constants c,C,X0 > 0 such that, for every real X >= X0 and each integer 4 <= m <= (log X)^3, at most C m X/(log X)^4 primes p in (X,2X] have R_II,m(p) < c(log X)^3/m. The numerator is fixed within each prime block; the result does not assert that one common good-prime set works simultaneously for every admissible numerator.

The fixed-4 v0.7 theorem and certificate are retained. The v0.8 extension has a completed additive Lean certificate: two authenticated foundation modules produce 55 standard-axiom reports, and a fresh 31-module closure produces 164 reports with zero compiler warnings or errors. The final export is UniformPrime.full_uniform_strict_typeII. The source/review archive includes the unchanged portable v0.7 certificate, every v0.8 Lean source, raw logs, frozen source locks, accepted build results, proof map, and internal statement/source/artifact audits. The accepted environment used Lean 4.32.0 and credited FormalPantheon for the pinned Bombieri-Vinogradov development.

This does not prove the universal Erdos-Straus or Schinzel conjecture. Numerical constants and onset, a common good-prime set for all numerators, normal-order statements, and the deferred wider-envelope extensions are not claimed. The research is AI-assisted. Independent human specialist review and a literature-wide originality assessment remain outstanding; internal AI-assisted reviews are not human peer review.

The manuscript and original contributions are licensed CC BY 4.0. Third-party sources retain their own licenses.