e_320133
plain-language theorem explainer
Pointwise kernel identity: the midpoint mass-squared numerator at multi-index (3,2,0,1,3,3) equals eight times the explicit Z-coupling there. Gravity analysts assembling the 4D Regge exact midpoint M2TT identity cite this cell among the 256 decides. Proof is a single computational `decide` on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,2,0,1,3,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer coupling table.
background
This module is chunk 14 of the 256-cell kernel that certifies $m_2^{\mathrm{num}}=8\cdot Z$ on every 6-tuple of $\mathrm{Fin},4$ indices. The setting is the 4D Regge exact midpoint analysis for the M2TT identity in the gravity stack.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at those six indices. The comparison target $Z$ is an explicit integer table on $(\mathrm{Fin},4)^6$, with sample values such as $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$.
Both definitions live in the kernel certificate module; this chunk only evaluates one concrete cell of the equality.
proof idea
One-line computational proof: decide. After substituting the six concrete Fin 4 values, both sides reduce to closed integer expressions (a fold of contributions versus a table lookup times 8), and the kernel decides equality of those integers. No lemmas are invoked beyond the definitions of the numerator fold and the explicit $Z$ table.
why it matters
Parent theorem is the assembled identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$, proved by exhaustive fin_cases on all six indices. Each cell such as this one discharges one branch of that case split.
In the Recognition gravity analysis, the factor-of-eight match between the folded midpoint numerator and the explicit coupling table is the algebraic core of the Regge exact midpoint M2TT identity in 4D. Closing all 256 cells removes a scaffolding gap in that identity and lets downstream curvature and mass-ladder arguments treat the equality as proved rather than assumed.
No T0–T8 forcing step is restated here; the result is local linear-algebraic bookkeeping inside the gravity kernel.
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