e_022021
plain-language theorem explainer
Pointwise kernel identity: at multi-index $(0,2,2,0,2,1)$ the summed coupling numerator equals eight times the explicit integer table. Gravity analysts cite it when assembling the full $m_2$ numerator identity over $(\mathbb{F}_4)^6$. The proof is a single kernel decision on two concrete integers.
Claim. For indices $a{=}0,b{=}2,c{=}2,d{=}0,i{=}2,j{=}1$ in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit closed-form integer $Z(a,b,c,d,i,j)$ at that sextuple.
background
This module is chunk 2 of a 256-case kernel certification that the Regge midpoint $m_2$ numerator in 4D equals eight times an explicit integer table. Indices run over $\mathbb{F}_4$ (four discrete directions).
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the given sextuple. The explicit table $Z$ is a pattern-matched integer function on six $\mathbb{F}_4$ arguments (sample values include $4$, $-2$, and other small integers on the listed patterns).
The local goal is purely algebraic bookkeeping: check the scalar identity $m_2^{\mathrm{num}}=8Z$ at one concrete multi-index so a later exhaustive fin_cases assembly can quote every cell.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed: the left side evaluates the fold of contrib over couplingZList; the right side is eight times the matching clause of the explicit table (or its default). The kernel discharges equality of those two integers with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall(a,b,c,d,i,j)\in(\mathbb{F}_4)^6$, $m_2^{\mathrm{num}}=8Z$, by nested fin_cases on all six indices. Each chunk theorem such as this one supplies one cell of that case tree.
In the Recognition gravity stack this identity is infrastructure for the exact midpoint Regge $m_2$ TT analysis in 4D: it replaces a summed coupling expression by a closed integer table, enabling later exact (not approximate) curvature and mass-side identities. It does not itself touch the forcing chain T0–T8, $\phi$, or $\alpha$; it is pure discrete-tensor bookkeeping inside the gravity analysis layer.
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