e_211220
plain-language theorem explainer
Pointwise identity: the Regge midpoint M2TT numerator at multi-index (2,1,1,2,2,0) equals eight times the explicit integer kernel Z there. Gravity analysts cite it when assembling the global m2Num = 8·explicitZ certificate over Fin 4^6. Proof is a single decide on the concrete six-tuple.
Claim. For indices $(a,b,c,d,i,j)=(2,1,1,2,2,0)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each local contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The comparison target $\mathrm{explicitZ}$ is a sparse case table of small integers (entries such as $4$, $-2$, and so on) encoding the closed-form Z coupling.
This module is chunk 9 of a 256-way partition of the $\mathrm{Fin},4^6$ domain. Each chunk discharges a block of concrete equalities $m_2^{\mathrm{num}}=8\cdot Z$ by kernel decision, so the global identity can be assembled by exhaustive case split rather than a symbolic closed form.
proof idea
Both sides evaluate to concrete integers once the six indices are fixed. The proof is the single tactic decide, which reduces the equality of those integers in the kernel. No named lemmas are invoked beyond the definitions of $m_2^{\mathrm{num}}$ (the fold over the coupling list) and $\mathrm{explicitZ}$ (the case table).
why it matters
The parent theorem is $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ for all six $\mathrm{Fin},4$ indices, proved by nested fin_cases that land on these pointwise certificates. This chunk entry supplies the $(2,1,1,2,2,0)$ cell of that cover. In the Recognition gravity stack the identity certifies that the midpoint Regge M2TT numerator matches the explicit Z kernel up to the universal factor eight, a numerical checkpoint inside the discrete curvature analysis rather than a forcing-chain landmark (T5–T8).
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