e_222000
plain-language theorem explainer
For the six Fin-4 indices (2,2,2,0,0,0), the folded coupling numerator equals eight times the explicit integer kernel. Gravity analysts certifying the 4D Regge midpoint M2TT identity cite this as one finite case among the 256 kernel checks. The proof is a single decide on concrete integers.
Claim. For $a=b=c=2$ and $d=i=j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator at $(2,2,2,0,0,0)$ equals $8$ times the explicit integer kernel value at the same multi-index.
background
In the Regge-exact midpoint analysis for the 4D M2TT identity, two integer kernels on six indices in $\mathrm{Fin},4$ are compared pointwise. The numerator is defined by folding a fixed coupling list and summing a contribution at each multi-index. Its companion is a sparse explicit table of small integers (entries such as $4$, $-2$, and so on) that encodes the closed-form kernel.
This module is chunk 10 of the 256 kernel decides that discharge the identity numerator $= 8\cdot$ explicit table. The local setting is pure finite enumeration: both sides are total functions $\mathrm{Fin},4^6\to\mathbb{Z}$, so each fixed tuple is a ground integer equality.
proof idea
One-line proof by decide. With all six arguments fixed to concrete Fin 4 values, both the fold that defines the numerator and the pattern-match that defines the explicit kernel reduce to numerals; the kernel decision procedure closes the equality with no further lemmas.
why it matters
The parent theorem m2Num_eq_eight_explicitZ states the identity for every six-tuple in $\mathrm{Fin},4$ and proves it by nested fin_cases. Each generated goal is discharged by one of these chunk theorems (or an analogous decide). The global certificate that the folded numerator is everywhere eight times the explicit kernel is infrastructure inside the Gravity.Analysis pipeline for the Regge midpoint M2TT identity in four dimensions. It does not invoke the Recognition forcing chain (T0–T8), the J-cost, or the phi ladder; it is a discrete computational step on the gravity side of the mirror.
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