e_010323
plain-language theorem explainer
For the six-index tuple (0,1,0,3,2,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these pointwise identities when assembling the global Regge midpoint M2–TT identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}1$, $c{=}0$, $d{=}3$, $i{=}2$, $j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is one chunk of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times an explicit integer table. Indices run over $\mathrm{Fin},4$, matching the four spacetime directions in the discrete gravity setup.
The numerator $N$ is defined by folding a fixed coupling list: start at $0$ and add a contribution for each coupling term at the six indices. The explicit kernel $Z$ is a pattern-matched integer table on those same six indices (typical values $\pm 2,,4$, and defaults elsewhere).
The local claim is the single instance $N(0,1,0,3,2,3)=8Z(0,1,0,3,2,3)$. Sibling theorems cover the other tuples in the same chunk; the assembler later quantifies over all of $\mathrm{Fin},4^6$.
proof idea
One-line decide proof. Both sides reduce to concrete integers: the left by evaluating the fold of contributions at the fixed indices, the right by looking up (or defaulting) the explicit table and multiplying by 8. Lean’s kernel decision procedure checks integer equality; no lemmas are invoked beyond definitional unfolding.
why it matters
Feeds the assembler 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 branch by a chunk identity of this form.
In the Recognition gravity stack this closes the algebraic identity between the midpoint Regge M2 numerator and the explicit TT kernel in 4D, a computational step toward matching discrete curvature couplings to continuum limits. It is pure finite-case bookkeeping, not a dynamical law, but without the pointwise certificates the universal equality does not typecheck.
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