e_300031
plain-language theorem explainer
Pointwise identity: the midpoint Regge M2TT numerator at multi-index (3,0,0,0,3,1) equals eight times the explicit Z-table entry. Gravity analysts cite it as one cell of the 4D kernel certification. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,0,0,0,3,1)$ with each index in $\{0,1,2,3\}$, the summed 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
This module is chunk 12 of a finite case-split certifying $m_2^{\mathrm{num}}=8\cdot Z$ on the full $(\mathrm{Fin},4)^6$ index set for the 4D midpoint Regge M2TT kernel.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it accumulates integer contributions contrib at the six indices. The comparison table explicitZ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by a finite pattern of nonzero entries (e.g. $4$ on matched pairs such as $(0,0,1,1,2,2)$, $-2$ on crossed pairs) and zero elsewhere.
The local claim is the single cell at $(3,0,0,0,3,1)$. Upstream, both sides are pure definitions in the kernel certificate module; no analytic continuum limit is invoked here.
proof idea
One-line computational proof: decide. After reducing the six Fin 4 arguments to the concrete naturals $3,0,0,0,3,1$, both sides evaluate to closed integers (the fold for $m_2^{\mathrm{num}}$ and the pattern match for $Z$), and the kernel checks equality to $8Z$. No lemmas beyond the definitions of m2Num and explicitZ are required.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every cell by nested fin_cases on $a,b,c,d,i,j$; this declaration discharges the branch $(3,0,0,0,3,1)$. That global identity is the algebraic backbone of the exact midpoint M2TT kernel certificate in the RS gravity analysis stack: it replaces a symbolic sum by an explicit sparse integer table scaled by eight.
In the broader Recognition framework this sits in the discrete gravity / Regge sector rather than the T0–T8 forcing chain, but it is the kind of fully decided finite identity the monolith uses to keep curvature-coupling bookkeeping free of sorry. Open analytic questions (continuum limit, physical units) are out of scope for this cell.
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