e_010331
plain-language theorem explainer
For the six-index slot (0,1,0,3,3,1) 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 M2TT identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}1$, $c{=}0$, $d{=}3$, $i{=}3$, $j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is one chunk of a 256-way case split establishing the algebraic identity m2Num = 8·explicitZ on all six-tuples of Fin 4 indices. The setting is the 4D Regge midpoint analysis of the M2TT gravity kernel: discrete curvature couplings are reduced to integer arithmetic on a finite index set.
The numerator m2Num(a,b,c,d,i,j) is defined by folding a fixed coupling list, accumulating a contribution term at each entry. The comparison target explicitZ is a piecewise integer table on the same six Fin 4 arguments (typical nonzero values are ±2, ±4 on selected diagonal and off-diagonal patterns).
Chunk theorems such as this one pin a single concrete six-tuple so the assembler can recombine them by exhaustive fin_cases.
proof idea
One-line proof by decide. Both sides evaluate to concrete integers once the six Fin 4 arguments are fixed: m2Num reduces by folding the coupling list, explicitZ matches its table clause (or default), and Lean’s kernel decision procedure checks the equality left = 8 · right.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple of Fin 4 indices and is proved by nested fin_cases dispatching to these chunk lemmas. That global equality is the certified numerator form of the Regge exact midpoint M2TT identity in 4D, a discrete gravity consistency check inside the Recognition Science gravity stack.
Within RS, such kernel identities support the discrete geometric side of the forcing chain (spatial dimension D = 3 and the eight-tick octave structure appear upstream in the continuum limit story). This declaration itself is pure finite arithmetic; it closes one of 256 decide cells rather than an open physical conjecture.
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