e_020211
plain-language theorem explainer
For the six-index slot (0,2,0,2,1,1) on Fin 4, the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite these micro-identities when assembling the global Regge midpoint M2TT numerator identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(0,2,0,2,1,1)$ equals eight times the tabulated explicit kernel entry at the same indices: $N(0,2,0,2,1,1)=8\,Z(0,2,0,2,1,1)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list and summing a local contribution at each tuple. The comparison target $Z=\mathrm{explicitZ}$ is a sparse case table of small integers (entries such as $4$, $-2$, and defaults).
The module is chunk 2 of a 256-way kernel split: each chunk theorem pins one concrete six-tuple equality $N=8Z$. The ambient claim is that this holds for every $(a,b,c,d,i,j)\in(\mathrm{Fin},4)^6$, which is the algebraic content needed for the exact midpoint M2TT identity certification.
proof idea
Both sides evaluate to concrete integers once the six indices are fixed. Lean closes the equality by decide on that ground arithmetic; no algebraic rewriting or induction is required. The chunk is independent of neighboring index tuples.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which introduces all six Fin-4 variables and discharges the universal statement by fin_cases on each coordinate, invoking one chunk theorem per cell. That global identity is the numerator half of the Regge exact-midpoint M2TT certification in four dimensions, inside the Gravity analysis stack of Recognition Science. It is pure discrete kernel bookkeeping rather than a forcing-chain (T0–T8) step, but it is load-bearing for the certified 4D gravity identities that sit above it.
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