e_201230
plain-language theorem explainer
For the concrete multi-index (2,0,1,2,3,0) on Fin 4, the folded Regge midpoint mass-squared numerator equals eight times the explicit integer table Z. Gravity analysts building the 4D TT midpoint identity cite this as one of 256 kernel cells. The proof is a pure `decide` on fixed integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,2,3,0)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer lookup table on six $\mathrm{Fin}\,4$ arguments.
background
This module is chunk 8 of a 256-cell kernel certifying that the midpoint Regge mass-squared numerator coincides with eight times an explicit closed-form integer table on all six-tuples in $(\mathrm{Fin},4)^6$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each local contribution contrib t a b c d i j. The table $Z$ is a pattern-matched Int-valued function on six Fin 4 indices (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
The ambient setting is 4D Regge calculus at the exact midpoint, aimed at a transverse-traceless (TT) identity for the discrete graviton mass-squared operator.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed, so the kernel decision procedure discharges the equality with no lemmas or algebraic rewriting.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple by exhaustive fin_cases on $a,b,c,d,i,j$. That global equality is the certified bridge between the folded coupling definition and the explicit $Z$ table used in the 4D midpoint TT analysis.
Within Recognition Science gravity work, this is pure discrete-geometry bookkeeping: it locks one cell of the kernel so the midpoint mass-squared numerator is interchangeable with $8Z$ everywhere the TT identity is invoked. It does not itself touch the forcing chain (T0–T8), $\varphi$, or continuum limits; it only hardens the combinatorial substrate those arguments sit on.
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