e_013030
plain-language theorem explainer
For the fixed index sextuple (0,1,3,0,3,0) on Fin 4, the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one cell of the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,3,0,3,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(0,1,3,0,3,0)=8\,Z_{\mathrm{explicit}}(0,1,3,0,3,0)$.
background
This module is chunk 1 of a 256-cell kernel certification that the 4D Regge midpoint numerator equals eight times a closed-form integer table. The ambient setting is exact midpoint analysis of the M2TT identity in discrete gravity.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each triple's contribution at those six Fin-4 indices. The comparison table $Z_{\mathrm{explicit}}$ is a pattern-matched integer function on the same six indices (sample values include 4 on diagonal-like pairs and $-2$ on crossed pairs).
The local claim is only the single sextuple $(0,1,3,0,3,0)$. Sibling theorems cover the other cells; the assembly theorem quantifies over all of $(\mathrm{Fin},4)^6$.
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 definitions of the two functions.
why it matters
Feeds the universal identity m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ and discharges the claim by exhaustive fin_cases on all six indices, invoking one cell theorem per case.
In the Recognition gravity stack this cell is bookkeeping for the exact midpoint form of the 4D Regge M2TT identity: once every cell matches, the folded coupling numerator is interchangeable with the closed kernel, simplifying later curvature and mass-ladder comparisons. It does not itself touch T0–T8 or the RCL; it is infrastructure under the discrete gravity analysis.
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