e_320102
plain-language theorem explainer
At multi-index (3,2,0,1,0,2) the folded Regge midpoint numerator equals eight times the explicit integer Z-table entry. Gravity analysts cite it as one of the 256 kernel point checks that assemble the global m2Num = 8·explicitZ identity. The proof is a single kernel decision on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,2,0,1,0,2)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 14 of a 256-way split of the 4D Regge exact-midpoint identity for the $m_2$ numerator. The ambient setting is discrete gravity on a 4-simplex skeleton: one checks that a folded sum of coupling contributions matches a closed-form integer table, up to the universal factor 8.
Upstream, $m_2^{\mathrm{num}}$ is defined by folding couplingZList and accumulating each term's contribution at a six-index in $\mathbb{F}_4$. The companion table $Z$ is an explicit pattern-matched integer function on the same six indices (sample clauses return $4$, $-2$, etc.). The claim is the pointwise equality of those two integers at one fixed multi-index.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 tuple $(3,2,0,1,0,2)$ and checks integer equality in the kernel. No algebraic rewriting or named lemmas are invoked beyond the definitions of the numerator fold and the explicit $Z$ table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-index in $(\mathbb{F}_4)^6$ by exhausting cases. That global equality is the certified kernel step for the 4D Regge exact-midpoint $m_2$ TT identity used in the gravity analysis stack. Within Recognition Science this is pure discrete-geometry bookkeeping supporting the continuum limit of the recognition-derived gravitational sector; it does not itself invoke the forcing chain (T5–T8) or the RCL, but it hardens the discrete curvature side those continuum claims rest on.
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