e_200212
plain-language theorem explainer
For the six-index tuple (2,0,0,2,1,2) on Fin 4, the midpoint M2 numerator equals eight times the explicit Z kernel entry. Gravity analysts cite it as one cell of the 256-case kernel identity that underwrites the Regge exact midpoint M2–TT relation in 4D. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2,\,b{=}0,\,c{=}0,\,d{=}2,\,i{=}1,\,j{=}2$ in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 8 of a 256-cell case split proving $m_2^{\mathrm{num}}=8\cdot Z$ on all of $(\mathbb{F}_4)^6$. The setting is the Regge exact-midpoint analysis of the 4D M2–TT identity in the Gravity.Analysis stack.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: sum the local contribution of each coupling term at the six Fin-4 indices. The comparison target $Z$ is an explicit integer-valued kernel on the same six indices, given by a finite pattern-match table (typical nonzero values $\pm 2,,4$).
Both definitions live in the KernelCert module. The present declaration fixes one concrete multi-index and asserts the scalar identity at that point.
proof idea
One-line computational proof: decide. Lean reduces both sides at the concrete Fin-4 sextuple $(2,0,0,2,1,2)$ to closed integers (the fold for $m_2^{\mathrm{num}}$ and the pattern match for $Z$) and checks equality. No lemmas beyond the two kernel definitions are invoked.
why it matters
Parent theorem is m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):\mathbb{F}_4$ and discharges the universal claim by exhaustive fin_cases. Each chunk cell such as this one supplies one decided equality in that cover.
In the Recognition gravity stack this kernel identity is bookkeeping for the Regge exact-midpoint M2–TT relation in four dimensions: the folded coupling numerator must match eight times the tabulated $Z$ so later curvature and mass-ladder arguments can treat $Z$ as the closed form. It does not itself invoke T5–T8 or the RCL; it is pure discrete kernel certification upstream of those physics claims.
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