e_010222
plain-language theorem explainer
For the Fin-4 index sextuple (0,1,0,2,2,2), the folded coupling numerator equals eight times the explicit integer kernel entry. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one cell of the 256-point kernel. The proof is a single computational decide on concrete integer arithmetic.
Claim. On indices $(0,1,0,2,2,2)\in(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit kernel value: $N(0,1,0,2,2,2)=8\,Z(0,1,0,2,2,2)$, where $N$ is the fold of coupling contributions and $Z$ is the explicit integer table.
background
In the 4D Regge exact-midpoint M2TT analysis, two integer-valued maps on sextuples of $\mathrm{Fin},4$ indices are compared. The numerator folds a fixed coupling list: each term contributes an integer via a local contribution function, and the fold starts at zero. The explicit kernel is a finite case table returning small integers (examples in the table include $4$ and $-2$ on selected patterns).
This module is chunk 1 of the 256 kernel decides that discharge the pointwise claim numerator $= 8\cdot$ explicit kernel. The local setting is pure discrete arithmetic on $\mathrm{Fin},4$; no continuum geometry is invoked at this layer. Upstream, both maps are defined in the kernel-certificate module that this chunk imports.
proof idea
One-line wrapper: by decide. Lean reduces both sides on the six concrete $\mathrm{Fin},4$ literals, evaluates the fold that defines the numerator and the case table that defines the explicit kernel, and checks integer equality. No named lemmas are applied beyond those definitions.
why it matters
This cell is consumed by the assembly theorem that states the identity for every sextuple in $(\mathrm{Fin},4)^6$. That assembly runs nested fin_cases on all six indices and lands each branch on a chunk decide of this form. The global equality is part of the Regge exact-midpoint M2TT certificate chain in the gravity analysis layer, tying the folded coupling numerator to a closed-form integer kernel. Within Recognition Science it supports the discrete geometric bookkeeping behind 4D Regge structure used in the gravitational side of the framework.
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