e_310133
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at Fin-4 indices (3,1,0,1,3,3) equals eight times the explicit integer table entry. Gravity analysts assembling the 4D Regge midpoint M2TT kernel cite it as one of 256 kernel cells. Proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,0,1,3,3)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(3,1,0,1,3,3)=8\,Z_{\mathrm{expl}}(3,1,0,1,3,3)$.
background
In the 4D Regge midpoint M2TT identity certification, two integer-valued maps on sextuples of Fin 4 indices are compared. The numerator m2Num folds a fixed coupling list, summing a local contribution at each term. The companion explicitZ is a closed pattern-match table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the simplified form of that fold.
This module is chunk 13 of a 256-cell kernel partition: each cell fixes one concrete sextuple and asserts the scalar identity m2Num = 8 · explicitZ at that point. The factor eight is the global normalization relating the folded sum to the table. The local setting is pure finite enumeration over a discrete index cube, not continuum GR.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are literals, so the kernel decision procedure discharges equality with no lemmas or case splits inside this declaration.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and invokes each chunk cell. Closing all 256 cells certifies that the folded Regge coupling numerator is identically eight times the explicit table, a discrete algebraic step in the Gravity analysis path toward the exact midpoint M2TT identity. It is bookkeeping infrastructure, not a continuum curvature claim.
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