e_032121
plain-language theorem explainer
Concrete kernel check: the folded m2 numerator at multi-index (0,3,2,1,2,1) equals eight times the explicit integer table entry. Gravity analysts cite it only as one cell of the 4D Regge midpoint identity. The proof is a pure `decide` on fixed Fin-4 indices.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,1,2,1)$ in $(\mathrm{Fin}\,4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator side $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib and the fold starts at 0. The comparison target is explicitZ, a closed-form case table on the same six indices (typical nonzero values are $\pm 2,,4$).
This module is chunk 3 of the 256-point kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The local setting is pure finite enumeration: every index lives in $\mathrm{Fin},4$, so each sextuple is a decidable integer identity.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are substituted, and Lean’s decision procedure checks equality. No algebraic lemmas are invoked beyond the definitions of the fold and the explicit table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by fin_cases on all six indices and discharge of each cell. That global equality is the algebraic backbone of the Regge exact-midpoint $M_2$–TT identity in 4D gravity analysis inside the monolith. The chunk exists only to keep kernel certification modular and machine-checkable; it does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law, but it hardens the discrete curvature side used downstream in RS gravity.
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