An almost-entropic Ingleton score below -0.09738: Conditional hashing of a binary parity structure

We prove an improved lower bound of 0.097387436922 on the largest normalised Ingleton violation among four-variable almost-entropic vectors. Previously reported constructions reach approximately 0.0925, used as a rounded benchmark; the best published universal upper bound is 19/124. Conditional parity hashing combines local binary parity labels, one common linear hash, simultaneous conditional privacy amplification and joint conditioning. Applied to a 24-atom source of Matúš and Csirmaz, it subtracts a multiple of the uniform-matroid rank function U(3,4) from all fifteen entropy coordinates. The limiting signed score lies in [-0.097387436923, -0.097387436922], and finite-alphabet distributions with score below -0.09738 exist. This record contains the preprint PDF, manuscript sources, Lean proof sources, exact numerical checkers, build instructions and verification evidence. The main theorem, including the closed rate interval, score enclosure and finite-alphabet existence statement, was rebuilt and kernel-checked in Lean 4.35.0-rc2 with pinned Mathlib dependencies. The finite sampler and fixed-path optimality argument are written proofs rather than separate Lean declarations. Søren Riis is the creator and coordinator and directed the research. Astra/Codex and Claude contributed mathematical development, computation and verification; Claude contributed the Lean formalisation and adversarial review. The source construction is due to Matúš and Csirmaz; the paper gives full references to the earlier entropy-region and privacy-amplification work. AI checks do not constitute independent human peer review. Licences by component: the paper PDF, LaTeX sources and original figures are CC BY 4.0; Lean sources, Python checkers and build scripts are Apache-2.0. Third-party dependencies retain their own licences. Repository: https://github.com/SR123/conditional-parity-hashing