e_211323

In the 4D Regge exact-midpoint analysis, the M2 numerator is defined by folding a fixed coupling list: each term contributes an integer depending on six Fin-4 indices, and the fold sums those contributions from zero. The companion table explicitZ is a sparse pattern-matched integer function on the same six indices (nonzero only on a short list of index patterns such as $(0,0,1,1,2,2)\mapsto 4$ and several $-2$ off-diagonal patterns).

The local module is chunk 9 of a 256-way kernel split: each chunk discharges a block of concrete index tuples so that a later assembler can recombine them. The theoretical claim being certified is the exact numerical identity $m_2^{\mathrm{num}}=8\cdot Z$ at every multi-index, which underwrites the midpoint M2 TT identity used in the discrete gravity kernel.

One-line computational proof: decide evaluates both sides at the concrete six-tuple $(2,1,1,3,2,3)$. The left side runs the fold that defines the numerator; the right side looks up (or defaults) the explicit Z entry and multiplies by eight. Equality of the resulting integers is discharged by the kernel decision procedure; no algebraic rewriting is required.

This lemma is one atomic tile in the proof of the universal statement that for all $a,b,c,d,i,j:\mathrm{Fin},4$, the folded numerator equals eight times explicit Z. The parent assembler m2Num_eq_eight_explicitZ introduces the six indices and runs nested fin_cases, each leaf landing on a chunk theorem of this form.

In the Recognition gravity stack, that global identity is the certified algebraic core of the Regge exact-midpoint M2 TT relation in four dimensions. Closing every chunk removes scaffolding from the discrete curvature/mass-side bookkeeping that feeds higher gravity claims. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure integer kernel support for the 4D midpoint calculus.