e_031102
plain-language theorem explainer
For the six Fin-4 indices (0,3,1,1,0,2), the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one cell of the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,1,1,0,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(0,3,1,1,0,2)=8\,Z_{\mathrm{ex}}(0,3,1,1,0,2)$.
background
This module is chunk 3 of a 256-cell kernel certification that the 4D Regge exact-midpoint numerator matches an explicit closed form. The ambient setting is discrete gravity analysis: edge and face couplings on a 4-simplex lattice with indices in $\mathrm{Fin},4$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each term's contribution at the six indices. The comparison target $Z_{\mathrm{ex}}$ is an explicit integer-valued pattern on those same six indices (sample clauses include values $4$, $-2$, and so on).
The local claim is one concrete sextuple equality inside that certification grid.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values (the fold for the numerator versus the pattern match for the explicit kernel), and the kernel checks equality. No lemmas are invoked beyond the two definitions.
why it matters
Parent consumer is m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and dispatches each cell to a chunk theorem of this form. Without the cell equalities, the assemble step cannot close.
In the gravity stack this is bookkeeping for the exact midpoint M2–TT identity in 4D Regge analysis: it certifies that the folded coupling numerator is uniformly eight times the explicit kernel, a normalization step before continuum or continuum-limit comparisons. It does not itself touch the T0–T8 forcing chain; it is infrastructure under the discrete gravity side of the mirror.
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