e_332000
plain-language theorem explainer
At multi-index (3,3,2,0,0,0) the folded numerator coupling equals eight times the explicit kernel integer. Gravity analysts certifying the 4D Regge midpoint M2TT identity cite this as one of 256 pointwise kernel checks. The proof is a single decide on concrete integer arithmetic.
Claim. At indices $(3,3,2,0,0,0)\in(\mathrm{Fin}\,4)^{6}$, the folded numerator coupling equals eight times the explicit integer kernel: $N(3,3,2,0,0,0)=8\,Z(3,3,2,0,0,0)$.
background
In the 4D Regge midpoint M2TT analysis two integer maps on six Fin 4 indices appear. The numerator coupling is a fold over a fixed coupling list, accumulating a contribution at each entry for the given indices. The explicit kernel is a piecewise table of small integers (entries such as $4$, $-2$, and so on) on selected index patterns.
This module is chunk 15 of a 256-way split of kernel point checks. The module header states the local goal: confirm numerator $=8\cdot$ explicit kernel at each multi-index by a decide tactic. The present declaration treats the single point $(3,3,2,0,0,0)$.
Both maps are defined in the upstream kernel-certificate module. The global identity is later assembled by exhaustive case analysis on all six coordinates.
proof idea
One-line wrapper: by decide. Both sides evaluate to concrete integers. The left-hand side runs the coupling-list fold at the fixed indices; the right-hand side looks up the piecewise explicit kernel and multiplies by 8. Lean’s decision procedure checks the resulting integer equality with no named lemmas.
why it matters
This cell is one point in the exhaustive verification that the numerator coupling equals eight times the explicit kernel on all of $(\mathrm{Fin},4)^{6}$. The parent theorem m2Num_eq_eight_explicitZ packages every cell into a single universal statement, proved by nested fin_cases on the six indices.
That universal identity sits inside the exact midpoint M2TT analysis for 4D Regge calculus in the Recognition Science gravity stack. It certifies a discrete coupling identity used when matching the midpoint scheme to the continuum TT sector. Chunking into 256 decide lemmas keeps each certificate small and kernel-checkable. No open scaffolding remains on this cell.
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