e_021122
plain-language theorem explainer
For the Fin-4 index tuple (0,2,1,1,2,2), the folded coupling numerator m2Num equals eight times the explicit integer kernel explicitZ. Gravity analysts certifying the 4D Regge midpoint M2TT numerator identity cite this as one atomic kernel case among 256. The proof is a single decide on concrete integers.
Claim. For indices $a=0$, $b=2$, $c=1$, $d=1$, $i=2$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, m2Num is the integer obtained by folding a fixed coupling list: each term contributes an integer contrib at a six-index pattern drawn from Fin 4, and the fold starts at 0. The companion explicitZ is a sparse pattern-match table returning small integers (4, -2, and similar) on selected sextuples and (implicitly) 0 elsewhere.
This module is chunk 2 of a 256-case kernel certification that m2Num = 8 · explicitZ holds pointwise. The local setting is pure finite enumeration: every index lives in Fin 4, so the universal claim is a finite conjunction of concrete integer equalities.
proof idea
One-line computational proof. Both sides are closed Int expressions at fixed indices, so decide evaluates the fold that defines the numerator and the pattern match that defines the explicit kernel, then checks equality. No algebraic lemmas or rewrites are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in Fin 4 and discharges it by nested fin_cases, routing each concrete case to a sibling decide lemma such as this one. That pointwise match is the numerator half of the 4D Regge exact-midpoint M2TT kernel certification in the Gravity analysis layer. It does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law; it is infrastructure for the discrete curvature side of RS gravity.
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