e_312330
plain-language theorem explainer
For the six Fin-4 indices (3,1,2,3,3,0), the integer numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For the index tuple $(a,b,c,d,i,j)=(3,1,2,3,3,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint M2TT analysis, two integer-valued maps on six Fin-4 indices appear. The numerator m2Num folds a fixed coupling list, summing a local contribution at each tuple. The companion explicitZ is a sparse case table of small integers (entries such as 4, -2, and so on) meant to be the closed form of that fold.
The module is chunk 13 of a 256-cell kernel certification that m2Num = 8 · explicitZ pointwise. Each cell fixes one concrete six-tuple and discharges the equality by computation. Upstream, m2Num and explicitZ are pure definitions in the KernelCert module; no analytic lemma is required beyond evaluating those defs at the given indices.
proof idea
One-line proof by decide. Lean reduces both sides at the concrete Fin-4 values 3,1,2,3,3,0 to closed integers (via the fold definition of m2Num and the match table of explicitZ) and checks equality in the kernel. No lemmas are invoked.
why it matters
The parent theorem m2Num_eq_eight_explicitZ assembles all 256 cells by nested fin_cases on a,b,c,d,i,j; this declaration is the cell for (3,1,2,3,3,0). That universal identity is the algebraic backbone of the exact midpoint M2TT relation in the 4D Regge gravity analysis. Within Recognition Science it supports the discrete gravity side of the forcing chain (spatial D = 3, eight-tick structure) by certifying the numerator identity that feeds continuum-limit and coupling checks. It closes no open scaffold by itself; it is pure kernel bookkeeping inside an already-proved assembly.
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