e_320222
plain-language theorem explainer
Pointwise identity: the midpoint Regge numerator coupling at multi-index (3,2,0,2,2,2) equals eight times the explicit kernel integer Z at that same index. Gravity analysts cite it as one cell of the 256-case kernel table that assembles the global m2Num = 8·explicitZ identity. The proof is a single kernel decide on concrete Fin-4 indices.
Claim. For indices $(a,b,c,d,i,j)=(3,2,0,2,2,2)$ in $(\mathbb{F}_4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 14 of a 256-cell case split proving that the midpoint Regge numerator coupling equals eight times an explicit integer kernel on every 4D multi-index. The ambient setting is exact algebraic identities for the 4D Regge midpoint TT sector used in the gravity analysis stack.
The numerator m2Num is defined by folding a fixed coupling list: it sums contribution terms contrib t a b c d i j over couplingZList, yielding an integer for each six-tuple of Fin 4 indices. The comparison target explicitZ is a closed-form integer table on the same domain (pattern-matched constants such as $4$, $-2$, and so on for distinguished index patterns).
The local claim is one concrete cell of that table: indices $(3,2,0,2,2,2)$. Sibling theorems in the chunk cover the neighboring cells needed before the assembler quantifies over all of $(\mathbb{F}_4)^6$.
proof idea
One-line computational proof: by decide. Both sides are closed integer expressions once the six Fin 4 arguments are fixed, so the kernel evaluates m2Num 3 2 0 2 2 2 and 8 * explicitZ 3 2 0 2 2 2 and checks equality. No lemmas are invoked beyond the definitions of the fold and the explicit table.
why it matters
The parent theorem m2Num_eq_eight_explicitZ states the universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$, proved by exhaustive fin_cases on the six indices. Each chunk cell such as this one discharges one branch of that case tree, so the assembler can conclude the full TT midpoint numerator identity in 4D.
Within Recognition Science gravity analysis, this is bookkeeping infrastructure for the exact Regge midpoint sector rather than a forcing-chain landmark (T5–T8). It locks the algebraic factor of eight between the folded coupling numerator and the explicit kernel, which downstream curvature and mass-ladder arguments treat as settled input. No open scaffold remains on this cell: the decide closes it.
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