e_200030
plain-language theorem explainer
One kernel instance of the 4D Regge midpoint identity: the folded numerator coupling at multi-index (2,0,0,0,3,0) equals eight times the explicit integer kernel Z at those indices. Gravity analysts assembling the universal m2Num = 8·explicitZ statement cite this cell. The proof is a single decide on concrete Fin-4 data.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,0,0,0,3,0)$ on $\mathrm{Fin}\,4$, the folded numerator coupling $m_2(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator coupling $m_2$ is the fold of a fixed coupling list: each term contributes an integer via a local contribution function, and the fold starts at $0$. The explicit kernel $Z$ is a sparse case table on the same six indices, returning small integers such as $4$, $-2$, or $0$ off the listed patterns.
The module is chunk 8 of a 256-cell decide grid that checks $m_2=8Z$ pointwise. The ambient goal is the TT (transverse-traceless) midpoint identity for the Regge curvature coupling in four dimensions, reduced to finite arithmetic on index tuples.
proof idea
One-line kernel proof: decide evaluates both sides at the concrete tuple $(2,0,0,0,3,0)$. The left side runs the fold that defines $m_2$; the right side looks up $8\cdot Z$ at those indices. No lemmas are invoked beyond decidable equality on $\mathbb{Z}$.
why it matters
This cell is one of the 256 pointwise facts consumed by the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2=8Z$ by exhausting $\mathrm{Fin},4$ in each slot. That universal identity is the certified numerator half of the Regge exact-midpoint M2/TT relation in 4D gravity analysis inside the monolith. It does not itself touch the Recognition forcing chain (T5–T8) or the J-cost; it is infrastructure for the discrete gravity side of the framework.
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