e_021223
plain-language theorem explainer
For the six-index slot (0,2,1,2,2,3) on Fin 4, the folded M2 numerator equals eight times the explicit Z kernel value. Gravity analysts cite it when discharging one cell of the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the M2 numerator at $(0,2,1,2,2,3)$ equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(0,2,1,2,2,3)=8\,Z_{\mathrm{expl}}(0,2,1,2,2,3)$.
background
This module is chunk 2 of a 256-way kernel certification that the folded M2 numerator agrees with eight times an explicit integer table on every 6-tuple in $(\mathrm{Fin},4)^6$. The setting is 4D Regge calculus at the exact midpoint: one checks a discrete M2–TT identity by comparing two integer-valued kernels rather than symbolic curvature expressions.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, accumulating each contribution at those six indices. The comparison target $Z_{\mathrm{expl}}$ is a pattern-matched integer function on the same six indices (sample clauses send $(0,0,1,1,2,2)$ to $4$ and $(0,0,1,2,1,2)$ to $-2$). Both live in the kernel-cert module imported here.
The full pointwise identity is assembled downstream by exhausting all Fin-4 cases; each chunk theorem such as this one pins a single concrete tuple.
proof idea
One-line proof by decide. Both sides reduce to closed integers once the six Fin-4 arguments are literals: the fold that defines the numerator runs over a finite coupling list, and the explicit kernel is a finite pattern match. Lean’s kernel decision procedure checks integer equality; no lemmas are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by nested fin_cases over all six indices. That global identity is the certified algebraic core of the 4D Regge exact-midpoint M2–TT check in the Gravity analysis stack.
Within Recognition Science this is infrastructure for discrete gravity bookkeeping (Regge edge/hinge combinatorics in $D=3$ spatial dimensions, T8), not a direct citation of J-uniqueness or the RCL. It closes one of 256 decide-cells so the assembly has no sorry and no residual case split.
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