e_030222
plain-language theorem explainer
Pointwise kernel identity: the Regge midpoint numerator at multi-index (0,3,0,2,2,2) equals eight times the explicit integer table value at that same index. Gravity analysts cite it only as one cell of the 4D Fin-4 kernel certification. The proof is a pure `decide` on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,0,2,2,2)$ in $(\mathrm{Fin}\,4)^6$, 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 3 of a 256-cell certification that the 4D midpoint Regge numerator coincides with eight times a closed-form integer table. The ambient setting is discrete gravity analysis: couplings on a 4-index lattice are summed into a numerator, then matched against an explicit sparse kernel.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and accumulating a local contribution at each sextuple of $\mathrm{Fin},4$ indices. The comparison table $Z$ is an explicit pattern-matched integer function on the same domain (nonzero only on a short list of index patterns such as $(0,0,1,1,2,2)\mapsto 4$).
The present declaration fixes one concrete sextuple and asserts numerical equality after the factor of eight.
proof idea
Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted. The tactic decide runs the kernel decision procedure on that closed integer equality; no algebraic rewriting or intermediate lemmas are required.
why it matters
The parent assembly theorem m2Num_eq_eight_explicitZ states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases. Each chunk lemma such as this one discharges one residual goal after that case split, so the full midpoint numerator identity is only as strong as the conjunction of these 256 decides.
In the Recognition gravity stack this closes the algebraic certificate that the discrete Regge midpoint second-moment numerator is exactly eight times the explicit kernel $Z$, a prerequisite for later continuum or continuum-limit comparisons. It does not itself invoke the forcing chain (T0–T8) or the Recognition Composition Law; it is pure kernel arithmetic inside the gravity analysis layer.
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