e_021222
plain-language theorem explainer
For the six-index slot (0,2,1,2,2,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one atomic decide-cell in the 4D Regge midpoint M2 identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(a,b,c,d,i,j)=(0,2,1,2,2,2)$ equals $8$ times the explicit integer kernel $Z$ at those same indices: $N(0,2,1,2,2,2)=8\,Z(0,2,1,2,2,2)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $N=\mathrm{m2Num}$ is obtained by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The comparison target $Z=\mathrm{explicitZ}$ is a sparse closed-form table of small integers (entries such as $4$, $-2$, and defaults).
The module is chunk 2 of a 256-cell decide grid that checks $N=8Z$ pointwise. Each cell fixes one concrete six-tuple; the present cell is $(0,2,1,2,2,2)$. Upstream, $N$ and $Z$ are pure definitions with no analytic hypotheses, so equality is a finite integer identity.
proof idea
One-line proof by decide. Lean evaluates both sides as concrete Int values (the fold that defines the numerator, and the pattern match that defines the explicit kernel) and checks numerical equality. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over all six indices. That global identity is the algebraic backbone of the Regge-exact midpoint $M_2$ TT certificate in 4D gravity analysis inside the monolith.
Within Recognition Science gravity work, such kernel identities underwrite discrete curvature bookkeeping on the eight-tick / $D=3$ side of the forcing chain; they are computational certificates rather than continuum GR derivations. This cell closes one of the 256 decide obligations in chunk 2.
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