e_110330
plain-language theorem explainer
Pointwise check that the Regge midpoint coupling numerator at multi-index (1,1,0,3,3,0) equals eight times the explicit integer table value there. Gravity analysts cite it as one of the 256 kernel cells assembling m2Num = 8·explicitZ in 4D. The proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,1,0,3,3,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table entry: $m_2^{\mathrm{num}}(1,1,0,3,3,0)=8\,Z_{\mathrm{expl}}(1,1,0,3,3,0)$.
background
This module is chunk 5 of a 256-cell kernel certification that the 4D Regge exact-midpoint TT numerator agrees with a closed integer table. The ambient setting is discrete gravity analysis: curvature/coupling contributions on a 4-index simplex skeleton are reduced to integer arithmetic on $\mathrm{Fin},4$ labels.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, summing each term's contribution at those six indices. The comparison object $Z_{\mathrm{expl}}$ is an explicit pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (nonzero only on a sparse set of index patterns, with values such as $\pm 2,\pm 4$).
The global claim targeted by the chunks is $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at every multi-index. Each chunk theorem fixes one concrete sextuple and discharges the integer equality.
proof idea
Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed. The proof is the one-tactic kernel decision decide, which reduces the equality $m_2^{\mathrm{num}}(1,1,0,3,3,0)=8,Z_{\mathrm{expl}}(1,1,0,3,3,0)$ by computation of the fold defining the numerator and the pattern match defining the explicit table. No lemmas beyond those definitions are required.
why it matters
The parent assembly theorem m2Num_eq_eight_explicitZ states the identity for all $a,b,c,d,i,j:\mathrm{Fin},4$ and discharges it by exhaustive fin_cases; this declaration is one named cell in that 256-way split (chunk 5). Establishing $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ certifies that the TT midpoint coupling numerator collapses to a sparse closed form, which is the algebraic backbone of the 4D Regge exact-midpoint identity used in the gravity analysis stack.
In the broader Recognition framework this sits on the gravity side of the discrete geometric ledger (eight-tick / $D=3$ forcing is upstream ontology; here one is checking a concrete 4D curvature-coupling identity). It does not itself touch $\phi$-ladder masses or $\alpha$, but it hardens the Regge-side exactness claim those continuum limits rely on.
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