e_010002
plain-language theorem explainer
One kernel case of the 4D Regge midpoint identity: the integer numerator at index sextuple (0,1,0,0,0,2) equals eight times the explicit closed-form kernel entry. Gravity analysts cite it only as a brick inside the full Fin-4 assembly. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,0,0,2)$ in $(\mathrm{Fin}\,4)^6$, the folded midpoint 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
This module is chunk 1 of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times an explicit integer table. The ambient setting is discrete gravity analysis: couplings on a 4-index lattice are folded into an integer numerator, then matched against a hand-written closed form.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list, accumulating a contribution at each sextuple of $\mathrm{Fin},4$ indices. The explicit kernel $Z$ is a pattern-matched integer table on the same domain (sample entries include $4$, $-2$, and so on for distinguished index patterns).
The local claim is only the single sextuple $(0,1,0,0,0,2)$. Sibling theorems cover the other concrete points in the same chunk.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 indices and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit kernel table; the kernel reduces the equality to true.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases. That global equality is the certified midpoint numerator identity used in the 4D Regge analysis stack.
Within Recognition Science gravity work, such kernel certificates pin discrete curvature bookkeeping before continuum or phenomenological limits are taken. This declaration closes one of the 256 decide cells; it does not itself touch the forcing chain (T0–T8), RCL, or the $\varphi$-ladder, but it is infrastructure those continuum claims rely on once the discrete identity is assembled.
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