e_030131
plain-language theorem explainer
For the six-index slot (0,3,0,1,3,1) on Fin 4, the folded numerator coupling m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one atomic kernel check. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,0,1,3,1)$ in $\mathrm{Fin}\,4$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 3 of a 256-way case split certifying $m_2^{\mathrm{num}}=8\cdot Z$ on all six-tuples in $(\mathrm{Fin},4)^6$. The ambient setting is the 4D Regge exact-midpoint M2TT identity kernel: discrete curvature couplings assembled from a fixed contribution list.
Upstream, $m_2^{\mathrm{num}}$ is defined by folding couplingZList and summing a local contribution at each tuple. The comparison target explicitZ is a closed-form integer table on the same six indices (nonzero only on a sparse pattern of matched pairs, with values such as $\pm 2,,4$).
Chunk theorems pin one concrete multi-index at a time so the global identity can be reassembled by exhaustive fin_cases without re-deciding the whole kernel in one go.
proof idea
One-line proof: decide. Both sides reduce to concrete Int values for the fixed indices $(0,3,0,1,3,1)$; the kernel checks equality. No lemmas beyond the definitions of m2Num and explicitZ are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ and discharges the claim by six nested fin_cases over Fin 4. Each chunk such as this one is a named witness for one of the $4^6$ slots, keeping the decide obligations small and auditable.
In the Recognition gravity stack this identity is infrastructure for the exact midpoint Regge M2TT analysis in 4D: it locks the numerator coupling against the explicit kernel before continuum or continuum-limit arguments. It does not itself invoke the forcing chain (T5–T8) or the RCL; it is pure discrete-kernel bookkeeping inside the gravity analysis layer.
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