e_132011
plain-language theorem explainer
Single kernel point check: the summed coupling numerator at indices (1,3,2,0,1,1) equals eight times the explicit integer table entry. Gravity analysts cite it only as one cell of the 4D Regge midpoint M2 identity. Proof is a pure `decide` on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,3,2,0,1,1)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(1,3,2,0,1,1)=8\,Z_{\mathrm{expl}}(1,3,2,0,1,1)$.
background
This module is chunk 7 of a 256-point kernel certification that the 4D Regge exact-midpoint M2 numerator matches a closed integer table. The ambient setting is discrete gravity analysis: index sextuples in $\mathrm{Fin},4$ label edge/face pairings in a 4-simplex style coupling sum.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding couplingZList and accumulating each term's contribution contrib. The comparison target explicitZ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample clauses return $4$, $-2$, etc.). The claim is the pointwise identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at one fixed sextuple.
proof idea
One-line kernel decision: both sides reduce to concrete Int values once the six indices are literals, so decide closes the equality with no lemmas, rewrites, or case splits inside this declaration.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):(\mathrm{Fin},4)^6$ and discharges the universal quantifier by exhaustive fin_cases. That global equality is the certified algebraic core of the 4D Regge exact-midpoint M2/TT numerator identity in the Gravity analysis stack. Within Recognition Science this is bookkeeping infrastructure for discrete curvature/coupling numerics, not a forcing-chain (T0–T8) step; it locks one cell of the 256-decide kernel so the assembled identity can be quoted without residual sorry.
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