e_033333
plain-language theorem explainer
For the six Fin-4 indices (0,3,3,3,3,3), the folded midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it when assembling the full 4D Regge midpoint M2TT identity. The proof is a single kernel decide on both integer sides.
Claim. For indices $a=0$, $b=c=d=i=j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification. The local goal, stated in the module header, is to show $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ on a block of 256 kernel cases (chunk 3).
$\mathrm{m2Num}$ is the integer obtained by folding a fixed coupling list: it sums a contribution function over that list at six $\mathrm{Fin},4$ indices. $\mathrm{explicitZ}$ is the closed-form integer table for the same six-index kernel (sample values include $4$, $-2$, and so on on the diagonal and off-diagonal patterns).
The ambient setting is discrete gravity analysis: verifying that the midpoint numerator of the M2TT identity factors exactly as eight times an explicit sparse kernel on the 4-cube index set.
proof idea
One-line decision proof: both sides are closed integer expressions in six concrete $\mathrm{Fin},4$ values, so decide evaluates $\mathrm{m2Num},0,3,3,3,3,3$ and $8\cdot\mathrm{explicitZ},0,3,3,3,3,3$ and checks equality. No lemmas beyond the definitions of $\mathrm{m2Num}$ and $\mathrm{explicitZ}$ are required.
why it matters
Feeds the assembler theorem $\mathrm{m2Num_eq_eight_explicitZ}$, which states the identity for all six $\mathrm{Fin},4$ indices by exhaustive fin_cases and invokes each chunk equality such as this one. That global identity is the algebraic core of the Regge exact-midpoint M2TT certification in the gravity analysis layer.
Within Recognition Science, exact discrete curvature identities support the geometric side of the forcing chain (spatial dimension and tick structure). This declaration is pure bookkeeping: one of 256 kernel cells in chunk 3, closing a proved equality rather than an open scaffold.
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