e_111102
plain-language theorem explainer
For the concrete multi-index (1,1,1,1,0,2) on Fin 4, the folded Regge midpoint coupling numerator equals eight times the explicit integer table entry. Gravity analysts cite it as one cell of the 4D kernel identity m2Num = 8·explicitZ. The proof is a single kernel decide on the evaluated integers.
Claim. Let $a=b=c=d=1$, $i=0$, $j=2$ in $\mathbb{F}_4$. Then the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 5 of a 256-way kernel split proving the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on all sextuples in $(\mathbb{F}_4)^6$. The setting is the exact midpoint analysis of the 4D Regge TT sector in the Gravity.Analysis 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 table $Z$ is an explicit case-split function $\mathbb{F}_4^6\to\mathbb{Z}$ hard-coding the closed-form integer values (examples: $Z(0,0,1,1,2,2)=4$, $Z(0,0,1,2,1,2)=-2$).
Both definitions live in the KernelCert module. The present lemma fixes one concrete sextuple and checks the scalar equality at that point.
proof idea
One-line computational proof: by decide. After substituting the six concrete Fin 4 arguments, both sides reduce to closed integers (the fold over the coupling list on the left, the matching table clause on the right), and the kernel decision procedure discharges the equality.
why it matters
Parent theorem is m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by exhaustive fin_cases on all six indices, invoking one chunk lemma per cell. This declaration is the cell for $(1,1,1,1,0,2)$.
In the broader Recognition gravity stack, the identity certifies that the midpoint-evaluated TT mass-squared numerator matches the explicit combinatorial table used downstream in the Regge exact analysis. It is pure integer bookkeeping, not a dynamical claim, but without the full 256-cell cover the assembly theorem cannot close.
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