e_202021
plain-language theorem explainer
Pointwise identity: the folded midpoint numerator at multi-index (2,0,2,0,2,1) equals eight times the explicit integer kernel value there. Gravity analysts cite it when assembling the full 4D Regge midpoint M2TT numerator identity over (F_4)^6. Proof is a single kernel decide on closed integer arithmetic.
Claim. For the multi-index $(2,0,2,0,2,1)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit integer table entry: $N(2,0,2,0,2,1)=8\,Z(2,0,2,0,2,1)$.
background
This module is chunk 8 of a 256-way split of the 4D Regge exact-midpoint M2TT numerator identity. The claim is the pointwise relation $N=8Z$ on one sextuple of indices in $\mathbb{F}_4$.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: sum the local contributions of each coupling term at the six indices. The closed form $Z=\mathrm{explicitZ}$ is an explicit integer-valued table on $(\mathbb{F}_4)^6$ (sample entries include $4$, $-2$, and so on for distinguished index patterns).
The ambient goal is the exact algebraic identity between that fold and eight times the table, used in the gravity analysis of the midpoint M2TT kernel in four dimensions.
proof idea
One-line computational proof: decide. Both sides are closed integer expressions once the six Fin 4 arguments are concrete literals, so the kernel reduces the equality of two Int values with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall(a,b,c,d,i,j)\in(\mathbb{F}_4)^6,,N=8Z$ by exhausting all sextuples (fin_cases on each coordinate). This chunk is one of the 256 concrete cells in that cover (module doc: "m2Num = 8·explicitZ, chunk 8").
In the Recognition gravity stack, the identity certifies that the folded midpoint numerator matches the explicit kernel table up to the universal factor 8, a bookkeeping step toward exact Regge/M2TT control in $D=3+1$. It does not itself invoke the forcing chain (T0–T8) or the J-cost; it is pure discrete kernel arithmetic downstream of those structural choices.
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