e_310100
plain-language theorem explainer
Pointwise identity: the summed Regge midpoint coupling numerator at multi-index (3,1,0,1,0,0) equals eight times the explicit integer kernel Z at that same index. Gravity analysts cite it as one of the 256 kernel cells feeding the global m2Num = 8·Z assembly. Proof is a single kernel decide on concrete Fin-4 integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(3,1,0,1,0,0)$ with each coordinate 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 $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator mass-squared coupling is defined by folding a fixed list of elementary contributions:
$$m_2^{\mathrm{num}}(a,b,c,d,i,j)=\sum_{t\in\mathrm{couplingZList}}\mathrm{contrib}(t;a,b,c,d,i,j)\in\mathbb{Z}.$$
The companion map $Z$ is an explicit, pattern-matched integer kernel on $(\mathbb{F}_4)^6$ (nonzero only on a short list of index patterns such as $(0,0,1,1,2,2)\mapsto 4$ and several $-2$ off-diagonal slots).
This module is chunk 13 of the 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ holds at every multi-index. The local setting is pure finite enumeration: every argument is a concrete Fin 4 value, so both sides reduce to integers.
proof idea
One-line computational proof: decide evaluates both sides at the fixed six-tuple $(3,1,0,1,0,0)$. The left side runs the fold that defines the numerator; the right side multiplies the pattern-matched kernel value by eight. Equality of the resulting integers is discharged by the kernel decision procedure. No algebraic lemmas are invoked.
why it matters
The parent assembly theorem m2Num_eq_eight_explicitZ states the universal claim
$$\forall a,b,c,d,i,j:\mathbb{F}_4,\quad m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$$
and proves it by exhaustive fin_cases on all six indices, each leaf citing one chunk identity of this form. Without the pointwise cells, the assembly cannot close.
In the broader Recognition gravity stack this identity is bookkeeping for the exact midpoint TT sector in four dimensions: it certifies that the summed coupling numerator is exactly eight copies of the closed-form kernel, so later curvature and mass-ladder arguments may substitute the cheap explicit $Z$ for the expensive fold. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete linear-algebra support for the 4D Regge analysis.
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