e_213112
plain-language theorem explainer
Pointwise identity: the midpoint Regge mass-squared numerator at multi-index (2,1,3,1,1,2) equals eight times the explicit integer table entry. Gravity analysts cite it as one of 256 kernel cells that assemble the global m2Num = 8·explicitZ theorem. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,3,1,1,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit closed-form integer: $N(2,1,3,1,1,2)=8\,Z(2,1,3,1,1,2)$.
background
In the 4D Regge midpoint analysis, the mass-squared numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each local contribution at the six $\mathrm{Fin},4$ indices. The companion table $Z$ is an explicit integer-valued function on the same six indices (sample values include $4$, $-2$, and so on for listed patterns).
This module is chunk 9 of a 256-cell kernel certification that $N=8Z$ holds at every multi-index. The local setting is pure finite enumeration over $\mathrm{Fin},4$, not continuum GR: each cell is an independent integer equality.
Upstream, $N$ and $Z$ are the kernel certificate definitions; the present theorem instantiates them at one concrete sextuple.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(2,1,3,1,1,2)$ and checks integer equality. No lemmas beyond the definitions of the folded numerator and the explicit table; the kernel reduces the equality to a closed arithmetic fact.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six indices and discharge of each cell. Without the full 256-cell cover, the global midpoint TT identity for the 4D Regge mass-squared numerator stays uncertified.
In the Recognition gravity stack this is bookkeeping infrastructure: it locks the discrete curvature/mass-squared algebra before continuum or phenomenological claims. It does not itself invoke the forcing chain (T5–T8), RCL, or $\varphi$-ladder masses; it is a finite kernel step those layers can trust once assembled.
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