e_030212

In the 4D Regge exact-midpoint analysis, two integer-valued maps on sextuples of $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer via a local weight, and the fold starts at zero. The explicit kernel $Z$ is a closed case table of small integers (entries such as $4$, $-2$, and so on) indexed by the same six $\mathrm{Fin},4$ arguments.

The local module is chunk 3 of a 256-cell decide grid that checks $m_2^{\mathrm{num}}=8Z$ pointwise. The parent assembly theorem will range over all sextuples by nested fin_cases; each chunk theorem discharges one concrete cell so the kernel need not re-derive the whole table at once.

One-line computational proof: decide evaluates both sides at the fixed indices $(0,3,0,2,1,2)$. The left side runs the fold that defines the numerator; the right side looks up the explicit kernel entry and multiplies by eight. Equality of the resulting integers is decided in the kernel with no lemmas or rewriting.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhausting indices and invoking the chunk cells. That global equality is the algebraic backbone of the 4D Regge midpoint M2TT identity certification in the Gravity analysis stack: it converts a folded coupling sum into an eightfold multiple of a sparse explicit kernel, enabling later exact midpoint identities without residual summation. Within Recognition Science gravity work this is bookkeeping infrastructure rather than a forcing-chain landmark (T0–T8), but it is required scaffolding for certified discrete curvature identities that sit downstream of the continuum limit story.