e_313211
plain-language theorem explainer
At multi-index (3,1,3,2,1,1), the folded Regge numerator equals eight times the explicit integer table value. Gravity analysts building the 4D midpoint M2TT identity cite this as one kernel case in chunk 13 of the exhaustive check. Proof is a single kernel `decide` on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,3,2,1,1)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table entry: $m_2^{\mathrm{num}}(3,1,3,2,1,1)=8\,Z_{\mathrm{expl}}(3,1,3,2,1,1)$.
background
In the 4D Regge exact-midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a contribution map over a fixed coupling list, for six indices each ranging in $\mathrm{Fin},4$. The companion $Z_{\mathrm{expl}}$ is a piecewise integer function on the same index space, serving as the closed-form target (so the desired identity is $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ pointwise).
This module is chunk 13 of that pointwise verification: a battery of kernel decides covering one block of the $4^6$ index tuples. The local setting is pure integer equality after unfolding the fold and the table; no continuum geometry is invoked at this layer.
proof idea
One-line wrapper: both sides reduce to concrete Int values once the six indices are fixed, and decide discharges the equality in the kernel. No lemmas beyond the definitions of the numerator fold and the explicit table are required.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full $\forall$-statement over $(\mathrm{Fin},4)^6$ by exhaustive fin_cases on the six indices; each concrete case lands on a chunk lemma of this form. That global identity is the algebraic backbone of the Regge midpoint M2TT analysis in 4D gravity inside the monolith. It does not itself touch the RS forcing chain (T0–T8), but it is infrastructure for the discrete gravity side that must eventually match continuum limits consistent with $D=3$ spatial dimensions and the eight-tick structure.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.