e_213221
plain-language theorem explainer
Pointwise identity: the Regge midpoint mass-squared numerator at multi-index (2,1,3,2,2,1) equals eight times the explicit integer table at those same indices. Gravity analysts assembling the 4D exact midpoint M₂TT identity cite this as one of 256 kernel cells. The proof is a single kernel decision on concrete integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(2,1,3,2,2,1)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit table value: $N_2(2,1,3,2,2,1)=8\,Z(2,1,3,2,2,1)$.
background
This module is chunk 9 of a 256-cell kernel certifying $N_2=8Z$ on all of $(\mathbb{F}_4)^6$ for the 4D Regge exact-midpoint mass-squared identity.
The numerator $N_2=\mathrm{m2Num}$ is the fold of a fixed coupling list: it sums contribution terms over that list at a six-index slot. The closed form $Z=\mathrm{explicitZ}$ is an explicit integer-valued table on the same six $\mathbb{F}_4$ arguments (sample entries include $4$, $-2$, and so on by pattern match).
The local goal is purely arithmetic: evaluate both sides at one concrete multi-index and check equality in $\mathbb{Z}$.
proof idea
One-line kernel proof: decide evaluates both the fold defining the numerator and the pattern-match table for $Z$ at the concrete indices $(2,1,3,2,2,1)$, then checks the integer equality $N_2=8Z$. No lemmas beyond the two definitions are required.
why it matters
Feeds the assembly theorem $\mathrm{m2Num_eq_eight_explicitZ}$, which states $\forall a,b,c,d,i,j,, N_2(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ by exhaustive fin_cases on all six $\mathbb{F}_4$ arguments. Each chunk such as this one discharges one cell of that $4^6=4096$-free but 256-pattern kernel partition used in the Regge midpoint M₂TT identity work in the Gravity analysis layer. It is bookkeeping infrastructure for the discrete curvature/mass-squared side, not a new physical law; it closes a decidable arithmetic obligation so the global identity can be cited cleanly.
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