e_200100
plain-language theorem explainer
For the six-index slot (2,0,0,1,0,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single decide on concrete integers.
Claim. For indices $a{=}2,\,b{=}0,\,c{=}0,\,d{=}1,\,i{=}0,\,j{=}0$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(2,0,0,1,0,0)=8\,Z(2,0,0,1,0,0)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification. The local goal, stated in the module header, is to prove the pointwise relation numerator = 8 · explicit kernel on a block of the 4^6 index space by kernel decides.
The numerator m2Num is defined by folding a fixed coupling list: it sums contribution terms contrib t a b c d i j over that list, yielding an integer for each six-tuple in Fin 4. The comparison target explicitZ is a closed-form integer table on the same six Fin-4 arguments (sample values include 4, −2, and so on for distinguished index patterns).
The present declaration fixes one concrete six-tuple, (2,0,0,1,0,0), inside chunk 8 of that enumeration.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete integers once the six Fin-4 indices are substituted into the fold definition of the numerator and the pattern-match table for the explicit kernel, so the equality is discharged by decidable integer arithmetic with no further lemmas.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for all six Fin-4 indices and proves it by exhaustive fin_cases on each coordinate, invoking the chunk decides (including this one) as the leaves. That global identity is the certified algebraic core of the Regge exact-midpoint M2TT analysis in the Gravity domain: it replaces a folded coupling sum by an eightfold multiple of a sparse explicit kernel, which is the form needed for downstream curvature and mass-ladder bookkeeping in Recognition Science gravity. It does not itself touch T0–T8 or the RCL; it is infrastructure under the discrete gravity side.
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