e_130313
plain-language theorem explainer
For the six-index tuple (1,3,0,3,1,3) on Fin 4, the discrete midpoint numerator equals eight times the explicit kernel integer. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one kernel cell. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the midpoint numerator $m_2^{\mathrm{num}}(1,3,0,3,1,3)$ equals $8\,Z(1,3,0,3,1,3)$, where $Z$ is the explicit integer kernel and $m_2^{\mathrm{num}}$ is the folded coupling sum.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each multi-index. The comparison target is an explicit piecewise integer kernel $Z$ given by pattern match on the six indices (typical nonzero values $\pm 2,4$).
The module is chunk 7 of a 256-cell decide grid that exhausts the identity $m_2^{\mathrm{num}}=8Z$ pointwise. Local setting: certify one concrete cell so the assembler can reassemble the universal statement by fin_cases.
proof idea
One-line kernel proof: decide evaluates both sides at the concrete indices $(1,3,0,3,1,3)$ and checks integer equality. No lemmas beyond the definitions of the folded numerator and the explicit kernel are required; both reduce to closed integers at this point.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple and is proved by nested fin_cases that discharge each cell (including this one). That universal equality is the algebraic core of the Regge exact midpoint M2–TT identity in 4D, tying the discrete coupling sum to a sparse explicit kernel. Within Recognition gravity analysis it is bookkeeping infrastructure rather than a new physical law: it closes one decide cell so the assembled identity can be cited downstream without residual case splits.
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