e_032002
plain-language theorem explainer
For the six-index slot (0,3,2,0,0,2) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the global identity m2Num = 8·explicitZ over the 4D Regge midpoint kernel. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,0,0,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ 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 exact-midpoint numerator equals eight times a closed-form integer table. The ambient setting is discrete gravity analysis: couplings on six Fin-4 indices are summed into a numerator, then matched against an explicit pattern-matched integer function.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and accumulating each term's contribution at the six indices. The comparison target $Z$ (explicitZ) is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by a finite pattern match (sample values include $4$, $-2$, and other small integers on distinguished index patterns).
Chunk 3 isolates a block of those index sextuples so each equality can be discharged by native decision rather than a single huge proof term.
proof idea
One-line decide proof. Both sides reduce to concrete integers: the left by evaluating the fold of contributions at the fixed indices $(0,3,2,0,0,2)$, the right by multiplying the pattern-matched $Z$ value by eight. Lean’s kernel decision procedure checks integer equality; no lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $\mathrm{Fin},4^6$ by exhaustive fin_cases and invokes each chunk equality such as this one. That global identity is the certified algebraic core of the Regge exact-midpoint $M_2$ TT analysis in the Gravity domain: it replaces a summed coupling expression by an eightfold explicit kernel, enabling downstream curvature and continuum-limit arguments without re-expanding the fold.
Within Recognition Science gravity work this is bookkeeping infrastructure rather than a forcing-chain landmark (T5–T8, RCL, $\varphi$). It closes one concrete cell of the 256-case decide grid so the assembled forall is sorry-free.
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