e_200333
plain-language theorem explainer
For the six-index tuple (2,0,0,3,3,3) on Fin 4, the folded numerator coupling m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,0,3,3,3)$ in $\mathrm{Fin}\,4$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification: the claim that the folded numerator coupling equals eight times an explicit sparse kernel on every six-tuple of Fin 4 indices. The module doc states the local goal as "m2Num = 8·explicitZ, chunk 8 (256 kernel decides)."
Upstream, m2Num is defined by folding contrib over couplingZList, accumulating an Int for fixed indices a b c d i j. explicitZ is a pattern-matched Int table on the same six Fin 4 arguments (nonzero only on a short list of index patterns such as (0,0,1,1,2,2) ↦ 4 and several −2 entries). The present declaration fixes one concrete cell of that 4^6 grid.
proof idea
One-line closed proof: by decide. Both sides reduce to concrete integers once the six Fin 4 arguments are literals, so the kernel compares the folded sum against 8 times the matched explicitZ entry and accepts equality. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full universal identity ∀ a b c d i j, m2Num = 8 * explicitZ by nested fin_cases on all six indices; each leaf is one of these chunk theorems (here e_200333 for the cell 2,0,0,3,3,3). Without the cell-wise equalities the assemble proof has nothing to discharge. In the gravity stack this identity is part of the exact midpoint M2TT kernel certification for 4D Regge analysis, a discrete curvature bookkeeping step rather than a continuum Einstein equation. It does not itself invoke the RS forcing chain (T5–T8) or the J-cost; it is pure finite combinatorial support for the discrete gravity layer.
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