e_212023
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at index sextuple (2,1,2,0,2,3) equals eight times the explicit integer table at those indices. Gravity analysts cite it as one of 256 kernel cells in the Regge midpoint M2TT 4D certificate. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,2,0,2,3)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit integer table entry: $N(2,1,2,0,2,3)=8\,Z(2,1,2,0,2,3)$.
background
This module is chunk 9 of a 256-cell kernel that certifies the algebraic identity between two integer-valued tensors on $(\mathrm{Fin},4)^6$ arising in the 4D Regge exact-midpoint M2TT analysis.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at 0 and add each list term's contribution at the six indices. The comparison object $Z=\mathrm{explicitZ}$ is an explicit case table $\mathrm{Fin},4^6\to\mathbb{Z}$ (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
The local claim is the single cell of $N=8Z$ at $(2,1,2,0,2,3)$. Sibling theorems cover the other cells in the same chunk pattern.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple and checks integer equality. No lemmas are invoked beyond the reducible definitions of the folded numerator and the explicit table; the kernel closes the ground term.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six indices and discharge of each cell. That global identity is the algebraic core of the Regge exact-midpoint M2TT 4D kernel certificate in the Gravity analysis stack.
Within Recognition Science gravity work, such exact integer identities underwrite discrete curvature bookkeeping before continuum or phenomenological limits. This cell is pure scaffolding glue: it does not itself touch T0–T8, the RCL, or the phi ladder, but it is required for the certified 4D midpoint identity those layers may later cite.
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