e_311323
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (3,1,1,3,2,3) equals eight times the explicit integer kernel entry. Gravity analysts cite it as one cell of the 256-case kernel that underwrites the exact midpoint M2 TT identity in 4D Regge calculus. Proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,1,3,2,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 13 of a 256-cell decide kernel establishing $m_2^{\mathrm{num}}=8\cdot Z$ on all of $(\mathbb{F}_4)^6$. The setting is the exact midpoint M2 TT identity for 4D Regge calculus: a discrete curvature/mass-squared numerator assembled from coupling contributions must match a closed-form integer table.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing each term's contribution at those six indices. The companion table $Z$ is an explicit pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (sample clauses return $4$, $-2$, etc.). The claim is the numerical equality of those two integers at one fixed multi-index.
proof idea
One-line kernel proof: decide evaluates both sides as concrete Int values (the fold that defines the numerator versus the pattern match that defines the explicit table) and discharges equality by computation. No lemmas beyond the two definitions are invoked.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles all 256 cells by exhaustive fin_cases on the six indices; this declaration is the cell for $(3,1,1,3,2,3)$. That global identity is the algebraic certificate that the folded Regge midpoint numerator is exactly eight times the explicit kernel, a required step in the 4D TT/M2 analysis chain inside the Gravity domain. It does not itself touch the T0–T8 forcing landmarks; it is pure discrete-kernel bookkeeping supporting the continuum-limit gravity identities built on top.
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