e_023312
plain-language theorem explainer
For the six-index slot (0,2,3,3,1,2) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact 4D Regge midpoint M2TT identity cite this as one of the 256 kernel decides. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}2$, $c{=}3$, $d{=}3$, $i{=}1$, $j{=}2$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $\mathrm{m2Num}(0,2,3,3,1,2)=8\cdot\mathrm{explicitZ}(0,2,3,3,1,2)$.
background
The module is chunk 2 of a 256-case kernel certification that the midpoint numerator of the 4D Regge M2TT identity equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$, matching the four spacetime directions of the discrete simplex.
The numerator m2Num is the fold of contrib over couplingZList at a fixed six-tuple of indices. The table explicitZ is a pattern-matched integer function on the same six-tuple (sample values include $4$, $-2$, and other small integers on the diagonal and off-diagonal patterns).
Local goal: discharge one concrete equality so the assembler can recombine all chunks into a universal statement.
proof idea
One-line kernel proof: by decide. Lean evaluates both sides as concrete Int values (the fold defining the numerator versus eight times the pattern match) and accepts the equality by computation. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the universal assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,;\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ by exhaustive fin_cases on all six indices. Each chunk theorem such as this one is a leaf of that case split.
In the gravity stack this closes the exact midpoint M2TT numerator identity in 4D Regge calculus, a discrete precursor to continuum curvature identities used downstream in the Recognition gravity analysis. It is pure integer certification, not a continuum limit claim.
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