e_033202
plain-language theorem explainer
For the six-index slot (0,3,3,2,0,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,3,2,0,2)$ 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 midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times a sparse explicit integer table $Z$. Both maps take six arguments in $\mathrm{Fin},4$ and return an Int.
Upstream, $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing local contributions contrib at the six indices. The table $Z$ is a pattern-matched closed form that is nonzero only on a thin set of index patterns (e.g. diagonal blocks returning $4$, off-diagonal swaps returning $-2$).
The local setting is pure finite enumeration: no continuum limit, no curvature hypothesis, only exact integer identity on the discrete index cube.
proof idea
One-line decide proof. Lean evaluates both sides at the concrete six-tuple $(0,3,3,2,0,2)$: the fold that defines $m_2^{\mathrm{num}}$ and the pattern match that defines $Z$, then checks the integer equality $m_2^{\mathrm{num}}=8Z$ by kernel computation. No lemmas are invoked beyond decidable equality on Int.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for all six indices by fin_cases and dispatches each cell to a chunk theorem of this form. That global identity is the algebraic core of the exact midpoint M2 TT certificate in the 4D Regge analysis stack.
In the Recognition gravity line this closes a discrete bookkeeping step: the folded coupling numerator collapses to an eightfold multiple of a sparse kernel, so later continuum or continuum-limit arguments can quote a single closed table rather than a fold. It does not itself touch T0–T8 or the J-cost; it is infrastructure under the Regge side of the gravity module.
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