e_020120
plain-language theorem explainer
For the six-index tuple (0,2,0,1,2,0) on Fin 4, the folded coupling numerator equals eight times the explicit kernel table entry. Gravity analysts assembling the 4D Regge midpoint M2-TT identity cite this as one of 256 kernel point checks. The proof is a single kernel decision on concrete integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(0,2,0,1,2,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer kernels are compared pointwise on $(\mathrm{Fin},4)^6$. The numerator $m_2(a,b,c,d,i,j)$ is the fold of a fixed coupling list: each term adds an integer contribution at those six indices, starting from zero. The comparison target is an explicit piecewise table $Z$ on the same domain, with listed values such as $4$ and $-2$ on specified patterns.
This module is chunk 2 of a 256-way split of the kernel identity $m_2=8Z$. Each chunk discharges a block of concrete index tuples by decision procedure, so the universal statement can later be assembled by exhaustive case splits on $\mathrm{Fin},4$.
proof idea
One-line proof by decide. With the six indices fixed at $(0,2,0,1,2,0)$, both sides evaluate to concrete integers: the left side runs the coupling fold; the right side reads the explicit table entry and multiplies by 8. The kernel checks integer equality.
why it matters
Atomic cell in the proof that $m_2=8Z$ on all of $(\mathrm{Fin},4)^6$. The parent theorem m2Num_eq_eight_explicitZ introduces the six indices and runs fin_cases on each; this lemma closes the cell $(0,2,0,1,2,0)$.
In the Recognition Science gravity stack, the Regge midpoint $M_2$--TT identity is part of the discrete curvature and coupling analysis that underwrites continuum limits and effective gravitational dynamics. Closing every kernel cell removes scaffolding from that identity. The result is local to 4D index algebra; it does not itself invoke the forcing chain (T0--T8) or the Recognition Composition Law, but it sits inside the gravity analysis those foundations feed.
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