e_200130
plain-language theorem explainer
For the six-index slot (2,0,0,1,3,0) on Fin 4, the folded numerator coupling m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts cite it as one atomic cell in the 256-way case split that certifies the Regge exact-midpoint M2TT identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,0,1,3,0)$ in $\mathrm{Fin}\,4$, the folded 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
In the 4D Regge exact-midpoint analysis, the numerator coupling $m_2^{\mathrm{num}}$ is defined by folding a fixed list of elementary contributions over six Fin-4 indices. The companion map explicitZ is a sparse integer table on the same index domain: most slots are zero, and the nonzero patterns are small constants such as $\pm 2$ or $4$.
The local module is chunk 8 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. Each cell is an independent equality at one concrete multi-index; the present declaration treats the slot $(2,0,0,1,3,0)$.
Upstream, m2Num is the fold of contrib over couplingZList, and explicitZ is the pattern-matched closed form used as the right-hand side of the identity.
proof idea
One-line kernel proof: by decide. Lean evaluates both sides at the concrete Fin-4 sextuple $(2,0,0,1,3,0)$ and checks integer equality. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
This cell is consumed by the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and discharges the universal quantifier by nested fin_cases over all $4^6=4096$ index tuples, of which the 256 nonzero kernel decides (including this chunk) supply the nontrivial equalities.
In the broader gravity stack the identity is the algebraic backbone of the Regge exact-midpoint M2TT certificate in four dimensions. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is pure discrete-geometry bookkeeping needed before continuum or phenomenological claims can be attached.
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