e_311210
plain-language theorem explainer
Pointwise kernel identity: for multi-index (3,1,1,2,1,0) the midpoint M2 numerator equals eight times the explicit Z coupling. Gravity analysts cite it when assembling the full 4D Regge midpoint TT identity over all Fin-4 sextuples. The proof is a single kernel decide on two closed integer expressions.
Claim. For the multi-index $(a,b,c,d,i,j)=(3,1,1,2,1,0)$ with each entry in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer coupling $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contrib kernel, and the fold starts at zero. The comparison target explicitZ is a sparse case table of small integers (entries such as $4$, $-2$, and so on) that encodes the closed-form Z coupling for the midpoint TT identity.
This module is chunk 13 of a 256-way kernel split: each chunk discharges a block of concrete index sextuples of the claim $m_2^{\mathrm{num}}=8\cdot Z$. The local setting is pure integer arithmetic on a finite index set; no continuum limit or metric signature is invoked here.
proof idea
One-line kernel proof: decide evaluates both sides as concrete Int values for the fixed sextuple $(3,1,1,2,1,0)$ and confirms equality. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ (the fold over couplingZList) and explicitZ (the case table) are required.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8\cdot Z$ by exhausting all sextuples. That global identity is the algebraic core of the 4D Regge exact-midpoint M2 TT certificate in the Gravity analysis stack. Within Recognition Science this sits in the discrete geometric layer that underwrites continuum gravity limits; it does not itself touch the T0–T8 forcing chain, but it is part of the certified scaffolding that keeps the Regge side exact before continuum identification.
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