Research proof index

The witness assembler is correct , the hmk input to the generic assembly coordinateWiseSpecialSound_of_mkWitness (SingleRound.lean), and the mathematical content of Hachi Lemma 8's three-case split: at every star-shaped family of relOut-accepting branches, buildWitness lands in relIn.

Eval-consistency of the extracted opening (Hachi Lemma 8, case (C), part 2 , Eq. (15)): the shared-ŵ c3 row plus the coordinate-isolated, unit-divided c4 rows discharge the w/c3/c4 hypotheses of evalConsistency_of_star at the shared recomposed carrier w := G_{2^r} *ᵥ ŵ.

The c5-side unit-cancellation of the subtract-and-divide extraction (Hachi Lemma 8, case (C)): from the c5-subtract chain c̄ᵢ •ᵥ (G_{n_A} t̂ᵢ) = A *ᵥ Δz and IsUnit c̄ᵢ, the extracted message block sᵢ := Ring.inverse c̄ᵢ •ᵥ Δz satisfies the weak-opening inner gadget relation (VerifiedBlock.inner_eq).

The main lemma: Evaluating a residual derived from partially evaluating a well-typed expression is equivalent to that of evaluating the original expression, provided that requests and entities are consistent. The equivalency is defined using Except.toOption.