e_332013
plain-language theorem explainer
For the six-index tuple (3,3,2,0,1,3) 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 256 kernel point checks. The proof is a single decide tactic evaluating both integer sides.
Claim. For indices $(a,b,c,d,i,j)=(3,3,2,0,1,3)$ 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 sits in the 4D Regge gravity analysis that certifies an exact midpoint identity for the M2TT coupling. The local setting is chunk 15 of a 256-point kernel: each point is one six-tuple of Fin 4 indices, and the claim is that the folded numerator equals eight times an explicit table.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a contribution function over a fixed coupling list: start at 0 and add contrib t a b c d i j for each term t. The explicit kernel $Z$ is a total function Fin 4^6 → ℤ given by a large pattern-match table (sample entries include 4 on diagonal-like pairs and −2 on mixed pairs).
Upstream, both definitions live in the KernelCert module. The present lemma is one concrete evaluation of that table against the fold, not a structural rewrite of either definition.
proof idea
One-line computational proof: by decide. Lean reduces both sides of the integer equality for the concrete Fin 4 literals 3,3,2,0,1,3. The left side evaluates the fold that defines m2Num; the right side multiplies the matched explicitZ clause by 8. No lemmas are invoked beyond the decidable equality instance on ℤ.
why it matters
The parent theorem is m2Num_eq_eight_explicitZ, which asserts the identity for every six-tuple in Fin 4. Its proof introduces the six indices and runs nested fin_cases, producing 4^6 = 256 goals; this declaration discharges the single goal at (3,3,2,0,1,3).
In the Recognition gravity stack, the exact midpoint M2TT identity is part of the discrete curvature bookkeeping that must match continuum limits on the phi-ladder side. Closing the kernel pointwise (rather than by a single symbolic argument) keeps the certificate fully computational and free of hidden analytic hypotheses. Chunk 15 is one of the parallel decide batches that together assemble the universal statement.
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