e_202013
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at index sextuple (2,0,2,0,1,3) equals eight times the explicit integer table at those indices. Gravity analysts cite it as one of 256 kernel cells assembling the global m2Num = 8·explicitZ theorem. Proof is a single decidability check on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,2,0,1,3)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit integer table: $N(2,0,2,0,1,3)=8\,Z(2,0,2,0,1,3)$.
background
This module is chunk 8 of a 256-cell kernel certifying that the folded numerator of a 4D Regge midpoint mass-squared identity equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$, so the full statement is a finite case split.
The numerator $N=m2Num$ is defined by folding a contribution function over a fixed coupling list: start at 0 and add each term's contribution at the six indices. The table $Z=explicitZ$ is an explicit pattern-matched $\mathrm{Int}$-valued function on six $\mathrm{Fin},4$ arguments (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and several $-2$ off-diagonal patterns).
The local claim is only the single cell $(2,0,2,0,1,3)$. Sibling theorems cover the other cells in the same chunk.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six indices are plugged into the fold definition of the numerator and the pattern match of the explicit table; Lean’s decision procedure discharges the resulting numeral equality. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and proves it by exhaustive fin_cases on all six indices. Each cell theorem such as this one is the leaf that a corresponding case of that split lands on.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint form of a 4D Regge $M^2$ / TT kernel relation: the folded coupling sum is replaced by a sparse explicit integer table scaled by 8. It does not itself invoke the forcing chain (T5–T8) or the J-cost; it is pure finite combinatorial certification inside the gravity analysis layer.
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