A finite free-entropy gap for two self-adjoint variables

Starting from the stipulated finite-entropy separation in the pinned OpenAI Math preprint, we construct exactly two bounded self-adjoint elements with −∞ < χ < χ* < ∞. Microstates entropy uses the original operator-norm cutoff and ordinary limsup over all matrix sizes. For the source tuple of length 2^64+1 and k = 2^32, a normal-slice encoding preserves at least k^(−2) times the source gap in the small-noise limit. An unconditional Fisher-information estimate is coupled with a microstates-volume bound that recovers matrix units from arbitrary pair microstates after removing a vanishing dimensional fraction. Source separation and named classical interfaces are inputs. No ambient von Neumann algebra generation assertion or effective noise parameter is claimed.

AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review or worldwide-priority certification. Identified OpenAI source inputs at revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb are stipulated true. Lean verifies scoped supporting results only, not the full theorem. Original prose: CC BY 4.0; original code and Lean: MIT; third-party rights preserved. See the supporting archive for precise assumptions, coverage, provenance and corrections.