e_010220
plain-language theorem explainer
For the six-index slot (0,1,0,2,2,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,2,2,0)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint $M_2$ TT-identity certification. The local goal, stated in the module header, is to show $m_2^{\mathrm{num}}=8\cdot Z$ on a block of the $4^6$ index space by kernel decides.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of a fixed coupling list: each term contributes an integer weight at the six Fin-4 indices, and the fold starts from 0. The comparison target $Z$ is an explicit piecewise integer table on those same indices (e.g. $Z(0,0,1,1,2,2)=4$, $Z(0,0,1,2,1,2)=-2$, and so on).
The identity $m_2^{\mathrm{num}}=8Z$ is the algebraic content needed before assembling the full midpoint TT relation for the discrete curvature kernel.
proof idea
One-line proof by decide. Both sides are closed integer expressions once the six Fin-4 arguments are fixed to $(0,1,0,2,2,0)$: the left side evaluates the fold that defines $m_2^{\mathrm{num}}$, the right side multiplies the matching clause of the explicit $Z$ table by 8. Lean’s kernel decision procedure checks the resulting integer equality.
why it matters
Feeds the universal statement m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on Fin 4. Each chunk theorem such as this one discharges one concrete cell so the assembler can finish without a monolithic decide.
In the gravity stack this is bookkeeping for the exact midpoint $M_2$ TT identity in 4D Regge calculus: the factor 8 relates the summed coupling numerator to the closed kernel $Z$. It does not itself invoke the Recognition forcing chain (T5–T8) or the RCL; it is a discrete-geometry certificate used downstream in the RS gravity analysis.
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