e_201320
plain-language theorem explainer
For the six-index combination (2,0,1,3,2,0), the midpoint mass-squared numerator equals eight times the explicit kernel integer. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of 256 certified kernel cells. Equality is discharged by a single kernel decision on concrete integers.
Claim. At indices $(2,0,1,3,2,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit integer kernel entry: $m_2^{\mathrm{num}}(2,0,1,3,2,0)=8\,Z_{\mathrm{expl}}(2,0,1,3,2,0)$.
background
In the 4D Regge midpoint analysis, the mass-squared numerator is the integer obtained by folding a fixed coupling list: starting from zero, each list entry adds a contribution at a sextuple of indices in $\mathrm{Fin},4$. The companion object is an explicit sparse integer kernel on the same six indices, with nonzero values (for example $4$ or $-2$) on selected matched and crossed patterns, and zero elsewhere.
This module is chunk 8 of a 256-cell kernel certification whose local slogan is that the folded numerator equals eight times the explicit kernel, cell by cell. The surrounding development targets an exact midpoint identity in the M2TT (mass-squared / transverse-traceless) sector of discrete gravity.
Both the fold and the explicit kernel are defined upstream in the KernelCert module. The present declaration only evaluates one concrete sextuple.
proof idea
One-line wrapper: by decide. Both sides are closed integer terms at fixed indices. The left-hand side evaluates the coupling fold (sum of contributions at $(2,0,1,3,2,0)$); the right-hand side multiplies the matching explicit-kernel clause by eight. The kernel checks the resulting $\mathrm{Int}$ equality.
why it matters
This cell is one of the 256 kernel decisions that underwrite the global statement: for every sextuple in $(\mathrm{Fin},4)^6$, the folded numerator equals eight times the explicit kernel. The parent theorem assembles those cells by exhaustive case splits on all six arguments and quotes each chunk equality in turn.
In the Recognition Science gravity stack, the Regge midpoint M2TT identity is part of the discrete bridge for the mass-squared sector in four dimensions. Closing the numerator identity at the exact midpoint removes a certification gap in that analysis. The constant factor eight is the combinatorial multiplicity being checked, not a dynamical claim.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.