e_020331
plain-language theorem explainer
For the six Fin-4 indices (0,2,0,3,3,1), the folded midpoint numerator equals eight times the explicit integer table entry. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one atomic kernel case among 256. The proof is a single kernel decide on concrete integers.
Claim. At indices $(a,b,c,d,i,j)=(0,2,0,3,3,1)$, the folded coupling numerator equals eight times the explicit closed-form integer: the left-hand sum over the coupling list matches $8$ times the table value at those indices.
background
In the 4D Regge exact-midpoint stack, two integer kernels on six Fin 4 indices are compared pointwise. The numerator folds a fixed coupling list, accumulating a contribution at each tuple; the companion is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on).
This module is chunk 2 of a 256-case kernel certification that the folded numerator equals eight times the explicit table everywhere. The local setting is pure finite enumeration: each declaration pins one concrete six-tuple. Upstream, the two kernels are defined in the shared KernelCert module; downstream assembly recombines the chunks into the universal statement.
proof idea
One-line wrapper: decide. Both sides evaluate to concrete Int values at the fixed indices $(0,2,0,3,3,1)$, and the kernel checks equality with no lemmas or rewriting.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for every six-tuple of Fin 4 indices by nested fin_cases. That universal equality is the algebraic core of the Regge exact-midpoint M2TT identity certification in the gravity analysis layer: it converts a folded coupling sum into a sparse closed-form table, enabling later continuum or discrete comparisons. The constant factor $8$ is the structural bridge between the two presentations.
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