Hey Codex,

This folder claims a verified solution of Erdős Problem #320 (see problem.md). Please verify it end to end — trust nothing you don't check:

1. Read problem.md (the exact Erdős problem). Open verified_math/F-038_two-sided-full-product-estimate/lean/ResearchPNT/FinalEstimate.lean and judge for yourself that the formal statement faithfully encodes the problem — quantifiers, ranges, and edge cases. check_answer/README.md documents the encoding choices; be skeptical and check them.
2. Reproduce the verification from a cold start by running, from this folder:
   bash check_answer/verify-lean-theorem.sh verified_math/F-038_two-sided-full-product-estimate/lean ResearchPNT.FinalEstimate ResearchPNT.exists_two_sided_full_product_estimate
3. Scan the proof sources yourself for escape hatches (sorry, admit, added axioms, native_decide) — scoped to the proof project, excluding .lake build dirs.
4. The axiom audit must list only [propext, Classical.choice, Quot.sound]. Confirm the build/verifier output shows it.

Then give your verdict: does this bundle contain a complete, faithful, kernel-checked solution of the stated problem?
