e_312331
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at index sextuple (3,1,2,3,3,1) equals eight times the explicit integer kernel at those indices. One of 256 kernel cases assembled into the universal m2Num = 8·explicitZ statement for the 4D Regge midpoint M2TT identity. Proof is a single decide on concrete integers.
Claim. For the index sextuple $(a,b,c,d,i,j)=(3,1,2,3,3,1)$ with each index in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 13 of a 256-case kernel certification for the exact midpoint M2TT identity in 4D Regge analysis. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add a local contribution at each list entry for the six $\mathrm{Fin},4$ indices. The comparison target is an explicit integer table $Z$ on $(\mathrm{Fin},4)^6$, given by pattern-matched constants (e.g. $\pm 2,,4$ on selected index patterns).
The local claim is one concrete sextuple evaluation of the global relation $m_2^{\mathrm{num}}=8Z$. Downstream assembly exhausts all $4^6$ index tuples by finitary case split and invokes these chunk lemmas.
proof idea
One-line computational proof: decide evaluates both sides as concrete Int values (the fold for the numerator versus eight times the table lookup) and closes the equality by kernel decision. No algebraic rewriting or named lemmas beyond the two definitions.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j\in\mathrm{Fin},4$, $m_2^{\mathrm{num}}=8Z$, proved by six nested fin_cases over the indices. That universal identity is the certified numerator half of the 4D Regge exact-midpoint M2TT kernel. In the gravity analysis stack it underwrites exact discrete curvature bookkeeping at the midpoint, rather than a floating-point or asymptotic check. Landmark contact is indirect: it is infrastructure for the discrete geometric side of Recognition gravity, not a T0–T8 forcing step.
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