e_213200
plain-language theorem explainer
At multi-index (2,1,3,2,0,0) on Fin 4, the folded Regge midpoint numerator equals eight times the explicit kernel table entry. Gravity analysts proving the global 4D M2TT numerator identity cite this as one atomic kernel check among the chunked decide suite. The proof is a single kernel decide on integer equality.
Claim. For $a=2$, $b=1$, $c=3$, $d=2$, $i=0$, $j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel value: $N(2,1,3,2,0,0)=8\,Z(2,1,3,2,0,0)$.
background
This module sits in the 4D Regge exact-midpoint M2TT analysis. The numerator $N=\mathrm{m2Num}$ is defined by folding a contribution map over a fixed coupling list: start at 0 and add each term's contribution at the six $\mathrm{Fin},4$ indices. The comparison table $Z=\mathrm{explicitZ}$ is a closed-form integer function on the same six indices, with sparse nonzero cases (e.g. $4$, $-2$) and default zero elsewhere.
The local goal, stated in the module header, is the pointwise identity $N=8Z$ discharged by 256 kernel decides in this chunk. Upstream, both $N$ and $Z$ are pure definitions in the kernel-cert module; no analytic lemma is required beyond evaluating those definitions at a concrete tuple.
proof idea
One-line proof: decide. Lean reduces both sides at the concrete indices $(2,1,3,2,0,0)$ to closed integers (the fold for $N$, the match for $Z$) and checks equality in the kernel. No lemmas are invoked beyond definitional unfolding of the two upstream defs.
why it matters
The parent theorem m2Num_eq_eight_explicitZ asserts $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six indices. Each concrete case lands on a chunk theorem of this form; this declaration covers the single point $(2,1,3,2,0,0)$.
In the broader gravity stack, the identity certifies that the folded midpoint numerator matches the explicit kernel used in the 4D Regge M2TT analysis, so later curvature and continuum-limit arguments can quote the closed table $Z$ instead of the fold. It is bookkeeping inside the gravity domain, not a forcing-chain (T0–T8) step, but it is load-bearing for any claim that the midpoint discrete action reproduces the certified kernel.
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