e_030200
plain-language theorem explainer
For the six-index slot (0,3,0,2,0,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,3,0,2,0,0)=8\,Z_{\mathrm{explicit}}(0,3,0,2,0,0)$.
background
This module is chunk 3 of a 256-cell kernel certification that the Regge exact-midpoint numerator equals eight times an explicit integer table on $(\mathrm{Fin},4)^6$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of couplingZList under a local contribution map, so it is a pure integer sum. The companion table $Z_{\mathrm{explicit}}$ is a pattern-matched closed form on the same six indices (sample clauses give values such as $4$, $-2$, etc.).
The local claim is the equality at one fixed multi-index inside that finite grid; sibling theorems cover the other cells of the same chunk.
proof idea
One-line computational proof: by decide. Lean reduces both sides at the concrete indices $(0,3,0,2,0,0)$ to integers (folding the contribution list on the left, pattern-matching the explicit table on the right) and checks equality in the kernel. No lemmas beyond the two definitions are invoked.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every cell by nested fin_cases on the six Fin-4 indices; this declaration is the witness for the slot $(0,3,0,2,0,0)$ inside that case tree. The identity is the algebraic backbone of the 4D Regge exact-midpoint M2TT certification in the Gravity analysis stack. It does not itself touch the T0–T8 forcing chain, but it hardens the discrete curvature bookkeeping used when RS gravity is reduced to certified integer kernels.
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