e_103230
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at index sextuple (1,0,3,2,3,0) equals eight times the explicit Z-coupling there. Gravity analysts assembling the 4D Regge midpoint M2–TT kernel cite it as one of the 256 concrete cells. The proof is a single kernel `decide` on the evaluated integers.
Claim. For indices $a{=}1$, $b{=}0$, $c{=}3$, $d{=}2$, $i{=}3$, $j{=}0$ in $\{0,1,2,3\}$, the folded midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer coupling $Z(a,b,c,d,i,j)$.
background
This module is chunk 4 of a 256-cell kernel certification that the numerical midpoint mass-squared numerator equals eight times an explicit integer coupling table on every sextuple of $\mathrm{Fin},4$ indices. The local slogan is $m_2^{\mathrm{num}}=8\cdot Z$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at the six indices. The explicit coupling $Z$ is a closed pattern-match table $\mathrm{Fin},4^6\to\mathbb{Z}$ (sample cells: $(0,0,1,1,2,2)\mapsto 4$, $(0,0,1,2,1,2)\mapsto -2$). Both live in the kernel certificate module imported here.
The setting is 4D Regge calculus analysis of an exact midpoint M2–TT identity: one checks, index by index, that the assembled numerator matches the closed-form coupling scaled by eight.
proof idea
One-line kernel proof: decide. Both sides are concrete integers once the six Fin 4 arguments are plugged into the fold definition of the numerator and the pattern-match definition of explicit $Z$; Lean’s decision procedure discharges the equality with no further lemmas.
why it matters
Feeds the universal assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8Z$ and proves it by exhaustive fin_cases on the six indices. Each chunk cell such as this one supplies (or is the decide target for) one branch of that case split.
In the Recognition gravity stack this closes a purely algebraic identity inside the 4D Regge midpoint analysis: the numerical coupling fold agrees with the explicit table everywhere, so downstream curvature or mass-squared identities may quote the closed form $8Z$ instead of the fold. It is scaffolding for the exact midpoint M2–TT identity, not a dynamical field equation.
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