e_211010
plain-language theorem explainer
For multi-index (2,1,1,0,1,0), the midpoint Regge mass-squared numerator equals eight times the explicit Z-kernel entry. Gravity analysts assembling the 4D TT identity cite this as one kernel decision inside chunk 9. The proof is a single computational `decide` on the closed integer fold that defines the numerator.
Claim. At indices $(a,b,c,d,i,j)=(2,1,1,0,1,0)$ in $(\mathrm{Fin}\,4)^6$, the integer mass-squared numerator equals $8$ times the explicit $Z$-kernel value: $N_2(2,1,1,0,1,0)=8\,Z(2,1,1,0,1,0)$.
background
This module sits in the 4D Regge midpoint analysis of the transverse-traceless mass-squared identity. The local goal, stated in the module header, is to certify $N_2=8Z$ on a 256-point chunk of the six-index kernel by pure decision procedure.
The numerator $N_2(a,b,c,d,i,j)$ is the integer fold of coupling contributions over a fixed list: start at $0$ and add each contrib term for the six Fin-4 indices. The comparison target $Z$ is an explicit piecewise table $\mathrm{Fin},4^6\to\mathbb{Z}$ (sample clauses include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).
Chunk theorems such as this one pin one concrete six-tuple so the universal assembly can recombine them without re-expanding the fold.
proof idea
One-line computational proof: decide. Both sides are closed integer expressions once the six concrete Fin-4 values are substituted into the fold definition of the numerator and the piecewise table for $Z$. Lean’s decision procedure evaluates the equality in $\mathbb{Z}$ and closes the goal. No lemmas beyond the two defining defs are invoked.
why it matters
Feeds the universal identity m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,,N_2=8Z$ and discharges the six Fin-4 quantifiers by fin_cases. That universal fact is the algebraic backbone of the exact midpoint M2 TT identity in 4D Regge calculus inside the Gravity analysis stack.
In the broader Recognition framework this is bookkeeping, not a forcing-chain step: it certifies a numerical kernel relation needed for the discrete gravity side, rather than T5–T8 (J-uniqueness, $\phi$, eight-tick, $D=3$). Its value is that the 4D TT midpoint identity can quote a fully decided integer table instead of an unexpanded fold.
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