e_011310
plain-language theorem explainer
For the six-index slot (0,1,1,3,1,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}1$, $c{=}1$, $d{=}3$, $i{=}1$, $j{=}0$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(0,1,1,3,1,0)=8\,Z(0,1,1,3,1,0)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2 TT identity certification. The local goal, stated in the module header, is to prove $m2Num=8\cdot explicitZ$ by discharging 256 concrete kernel cases.
The numerator $m2Num(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start from 0 and add each contribution $contrib,t,a,b,c,d,i,j$. The closed form $explicitZ$ is an integer-valued pattern match on the six Fin-4 indices (sample clauses return $4$, $-2$, etc.).
Both definitions live in the KernelCert module. The present declaration fixes one ordered sextuple and asserts numerical equality after the fold.
proof idea
One-line proof by decide. Lean evaluates both sides as concrete Int values: the left-hand fold of m2Num on $(0,1,1,3,1,0)$ and the right-hand $8\cdot explicitZ$ on the same indices. No lemmas are invoked; the kernel simply checks integer equality.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m2Num=8\cdot explicitZ$ by exhaustive fin_cases over all six indices. Each chunk such as this one supplies one (or a block of) decided equalities that the assembler consumes.
In the Recognition gravity stack this identity is part of the exact midpoint treatment of the M2 TT sector in 4D Regge calculus: the folded coupling numerator is replaced by an explicit integer kernel, clearing a computational bottleneck before continuum or continuum-limit arguments. It does not itself touch the T0–T8 forcing chain, but it is infrastructure for the discrete gravity side of the framework.
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