An explicit threshold for cube refinements

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_proof: Complete geometric refinement, finite-group reachability and coprime-pair arguments for the cube-count threshold.

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The complete proof is an author-recorded argument by Plasma AI. There is no independent review or Lean proof. It combines an elementary numerical-semigroup estimate with the subgroup and reduced-fraction strategy recorded in Zeng's gcd-threshold argument. That source is comparison and attribution: the proof below supplies every arithmetic step it needs. No novelty is claimed.

For Problem 769, this gives a negative answer only to the proposed uniform lower bound $c(d)\gg d^d$. It does not determine the growth order or its logarithmic limit. The research account explains the obstruction to using small grid sizes when $d+1$ is prime. No mathematical gap remains in this partial bound. Verification consists of checking the residue-path threshold, the two prime ranges and the asymptotic; no computational certificate is needed.