Prime reciprocal product sets

Independently reproduce or precisely reject the Bonfioli v1.0.0 universal barrier, then harden certificate verification.

First action: Reuse /root/.cache/proof-factory/erdos307-v1.0.0 and invoke scripts/check_bonfioli_barrier.sh once through run_experiment.py with memory above 4096 MB, or first document a supported single-thread setting; do not reclone.

Stop or redirect when: Stop after one successful lake build plus Check.lean axiom audit, or the first new precisely diagnosed failure. If successful, proceed only to one proof-carrying primality control.

• Complete the pinned Lean barrier and axiom audit under adequate or serialized memory.
  Run scripts/check_bonfioli_barrier.sh once through the experiment recorder with memory above 4096 MB or a documented supported single-thread cache configuration.
• Add proof-carrying primality to the exact certificate format.
  Define a Pratt or Pocklington JSON schema, accept one known certified prime under two implementations, and reject every one-field mutation.
• Test construction families solely through squarefree arithmetic-derivative closure.
  Specify a family reaching a greater than 2e56, evaluate exact squarefreeness and closure on a fixed small parameter sample, and stop if it cannot reach the scale or only optimizes decimal proximity.

• counterexample and witness search
  Replace trial-division primality with certificate-carrying Pratt or Pocklington verification while retaining exact forced-partner closure.
• formal proof with premise retrieval
  Finish the pinned kernel build and inspect the exact axiom footprints of card_ge_59 and erdos307_barrier_closed.
• counterexample and witness search with formal barrier audit
  Reduce every candidate to exact integer forcing equations, validate a fixed finite shell under independent encodings, provide a fail-closed verifier, and audit the pinned formal lower-bound source before permitting search
  Reopen only if: For the formal audit, provide memory above 4096 MB or a documented supported serialization setting. For frontier search, provide a barrier-scale exact construction family with measurable squarefree forced-partner output.

• Another arbitrary bounded search beneath the reported 2e56 side-product scale.
  It would repeat a range already excluded by the closest source and would not improve its result.
  Reopen only if: The formal barrier fails an independent kernel audit, or the computation targets a separately motivated structural statistic with an explicit acceptance path.
• Reporting 60 as the sharp consequence of the elementary reciprocal-sum argument.
  The exact crossing occurs at the 59th prime.
  Reopen only if: None; the argument yields 59.
• Independent two-sided subset search or floating-point genetic optimization as a proof-bearing route.
  Q is forced by A(P), while decimal proximity does not certify either integer forcing equality.
  Reopen only if: An exact theorem or construction showing that the proxy preserves or efficiently predicts forced-partner closure.