e_200211
plain-language theorem explainer
For the six-index slot (2,0,0,2,1,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the global identity m2Num = 8·explicitZ over the 4D Regge midpoint kernel. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2,b{=}0,c{=}0,d{=}2,i{=}1,j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(2,0,0,2,1,1)=8\,Z(2,0,0,2,1,1)$.
background
In the 4D Regge midpoint analysis the numerator m2Num is defined by folding a fixed coupling list: start at 0 and add each local contribution contrib t a b c d i j. The companion explicitZ is a total function on six Fin 4 indices that returns a small integer by pattern match (typical values $\pm 2,\pm 4$, and zero off the listed patterns).
This module is chunk 8 of a 256-way split of the kernel identity $N=8Z$. Each chunk theorem pins one concrete six-tuple so the assembler can discharge the universal statement by exhaustive fin_cases without re-running the full fold in one place.
The surrounding Gravity.Analysis development certifies exact midpoint identities for the discrete curvature/mass kernel used in the Recognition gravity sector.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left-hand side evaluates the fold of couplingZList at the fixed indices $(2,0,0,2,1,1)$; the right-hand side is $8$ times the pattern-matched explicitZ entry. Lean’s kernel decision procedure checks integer equality and closes the goal.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; N=8Z$ by nested fin_cases over all six indices. Each chunk such as this one supplies one of the 256 ground instances the assembler invokes.
In the Recognition gravity stack these exact kernel identities underwrite the discrete midpoint mass/curvature bookkeeping that sits behind continuum limits and phenomenological mass ladders. The module doc frames the whole family as “m2Num = 8·explicitZ, chunk 8 (256 kernel decides),” so this declaration is pure certification scaffolding rather than a new physical law; it closes one cell of the exhaustive check.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.