e_211012
plain-language theorem explainer
For the six-index tuple (2,1,1,0,1,2) on Fin 4, the folded midpoint numerator equals eight times the explicit integer kernel. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of 256 concrete kernel cases. The proof is a single decide on fully evaluated integers.
Claim. For indices $a=2$, $b=1$, $c=1$, $d=0$, $i=1$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel integer: $m_2^{\mathrm{num}}(2,1,1,0,1,2)=8\,Z_{\mathrm{expl}}(2,1,1,0,1,2)$.
background
This module is chunk 9 of a 256-case kernel certification that the midpoint numerator equals eight times an explicit integer table on all six Fin-4 indices. The local goal is the pointwise identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ needed for the 4D Regge exact-midpoint M2TT analysis.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at 0. The explicit kernel $Z_{\mathrm{expl}}$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by a finite pattern of integer values (for example 4 on diagonal-like pairs and $-2$ on selected off-diagonal patterns).
Upstream, both maps live in the kernel-cert module imported here. Downstream assembly will range over all six indices by fin_cases and invoke one such equality per cell.
proof idea
One-line computational proof: decide. Both sides are closed integer terms once the six Fin-4 arguments are substituted, so the kernel reduces $m_2^{\mathrm{num}}(2,1,1,0,1,2)$ and $8\cdot Z_{\mathrm{expl}}(2,1,1,0,1,2)$ to concrete integers and checks equality by evaluation. No lemmas beyond the definitions of the numerator fold and the explicit table are required.
why it matters
Parent theorem m2Num_eq_eight_explicitZ states the universal identity for all $a,b,c,d,i,j:\mathrm{Fin},4$ and discharges it by nested fin_cases; each leaf is one of these chunk equalities. This declaration is the leaf for indices (2,1,1,0,1,2).
In the gravity stack, the identity converts a folded coupling expression into an explicit sparse integer kernel, which is the algebraic content needed to certify the 4D Regge exact-midpoint M2TT relation. It is pure bookkeeping relative to the Recognition forcing chain (T0–T8), but it is load-bearing for the discrete gravity side of the monolith: without the 256-case cover, the midpoint numerator cannot be replaced by the closed kernel table.
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