e_012132
plain-language theorem explainer
Pointwise identity: the Regge midpoint mass-squared numerator at index sextuple (0,1,2,1,3,2) equals eight times the explicit Z-table entry. Gravity analysts cite it as one of 256 kernel cells in the m2Num = 8·explicitZ certification. Proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,2,1,3,2)$ in $(\mathbb{F}_4)^6$, 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 coupling table.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer contrib and the accumulator starts at 0. The companion map $Z$ is an explicit pattern-matched table of small integers (entries such as $4$, $-2$, and so on).
The local module is chunk 1 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ holds at every index sextuple. The full quantified statement is assembled downstream by exhausting all $\mathrm{Fin},4$ cases. This declaration fixes one concrete cell: $(0,1,2,1,3,2)$.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices. The left side reduces the fold that defines the numerator; the right side looks up the explicit table and multiplies by 8. Equality of the resulting integers is discharged by the kernel with no lemmas and no case split beyond what decide performs internally.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhausting all six $\mathrm{Fin},4$ indices. That global identity is the algebraic backbone of the exact midpoint mass-squared / TT identity in the 4D Regge sector of the Gravity analysis.
Within Recognition Science this is bookkeeping infrastructure, not a forcing-chain step: it certifies a discrete curvature/coupling identity used when matching continuum limits and effective gravitational couplings. Closing all 256 cells removes a scaffolding gap between the explicit $Z$ table and the folded numerator.
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