e_210223
plain-language theorem explainer
For the six-index slot (2,1,0,2,2,3) 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. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(a,b,c,d,i,j)=(2,1,0,2,2,3)$ equals $8$ times the explicit integer kernel $Z$ at those same indices: $N(2,1,0,2,2,3)=8\,Z(2,1,0,2,2,3)$.
background
This module is one chunk of a 256-case kernel certification that the 4D Regge midpoint mass-squared numerator equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$, i.e. coordinate labels $0,1,2,3$.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The table $Z=\mathrm{explicitZ}$ is a pattern-matched integer function on six $\mathrm{Fin},4$ arguments (typical nonzero values $\pm 2,\pm 4$).
The local claim is the single sextuple $(2,1,0,2,2,3)$ inside chunk 9 of that decide grid. Sibling lemmas cover the neighboring index tuples in the same chunk.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left-hand fold over the coupling list and the right-hand pattern match on explicitZ are fully computational at this fixed sextuple, so the kernel closes the equality with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over all six $\mathrm{Fin},4$ arguments. Each chunk lemma such as this one discharges one concrete cell so the global identity is a pure case split rather than an analytic argument.
In the gravity stack this identity is the algebraic core of the exact midpoint M2 TT certificate for 4D Regge calculus: once $N=8Z$ holds pointwise, downstream curvature and mass-squared identities can quote a closed integer kernel instead of the folded sum. It is bookkeeping inside the Recognition gravity analysis, not a forcing-chain (T0–T8) step, but it hardens the discrete geometric side that later couples to continuum limits.
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