e_123322
plain-language theorem explainer
Kernel identity: the folded midpoint numerator at multi-index (1,2,3,3,2,2) equals eight times the closed-form table entry. Gravity analysts cite it when assembling the full 4D Regge midpoint m2-numerator identity. The proof is a single computational decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,2,3,3,2,2)$ in $(\mathbb{F}_4)^6$, 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 6 of a 256-case kernel that certifies $m_2^{\mathrm{num}}=8\cdot Z$ pointwise on $(\mathbb{F}_4)^6$. The setting is the exact midpoint analysis of the 4D Regge/TT identity in the Gravity.Analysis stack.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The closed form $Z$ is an explicit six-argument integer table on $\mathrm{Fin},4$, with sample values such as $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$.
The present declaration fixes one concrete multi-index and asserts numerical equality of the fold against eight times the table entry.
proof idea
One-line computational proof: decide evaluates both sides as concrete Int expressions (the fold of contrib over couplingZList versus 8 * explicitZ 1 2 3 3 2 2) and discharges the equality by kernel decision. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the assembler theorem 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 $\mathrm{Fin},4$ arguments. Each chunk theorem such as this one supplies one of the $4^6$ leaves of that case tree.
In the Recognition gravity stack, the identity is part of the exact midpoint/TT certification for the 4D Regge analysis: once the numerator matches the explicit table everywhere, downstream curvature and mass-ladder arguments can treat $Z$ as the closed form. It does not itself touch T0–T8 or the RCL; it is pure discrete kernel bookkeeping inside the gravity analysis layer.
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