e_202030
plain-language theorem explainer
For the six-index tuple (2,0,2,0,3,0) on Fin 4, the folded numerator coupling m2Num equals eight times the explicit integer kernel explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on two closed integer expressions.
Claim. For indices $(a,b,c,d,i,j)=(2,0,2,0,3,0)$ in $\mathrm{Fin}\,4$, the folded numerator coupling equals eight times the explicit kernel value: $m_2^{\mathrm{num}}(2,0,2,0,3,0)=8\,Z(2,0,2,0,3,0)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator coupling is defined by folding a fixed coupling list: each term contributes an integer via a local contrib function, and the fold starts at 0. The explicit kernel is a piecewise integer table on the same six indices (sample values include 4, -2, and other small constants on distinguished patterns).
The module is chunk 8 of a 256-way case split that certifies the pointwise identity numerator = 8 · explicit kernel. The local setting is pure finite enumeration: every index runs in Fin 4, so the full identity is a finite check once the two maps are fully defined.
Upstream, both maps live in the kernel certificate module; this chunk only evaluates them at one concrete six-tuple.
proof idea
One-line proof by decide. Lean reduces both sides at the concrete indices (2,0,2,0,3,0) to closed integers (the fold of contrib over the coupling list on the left; the matching table entry of the explicit kernel on the right) and checks equality in Int. No lemmas beyond the two definitions are invoked.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the universal statement for all six Fin-4 indices by exhaustive fin_cases. Each leaf of that case tree is discharged by a chunk identity of this form; the present declaration is the leaf for (2,0,2,0,3,0).
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint M2TT kernel in 4D Regge calculus: once numerator = 8 · Z holds pointwise, downstream curvature and mass-ladder comparisons can quote a single closed table instead of a fold. It does not itself touch T5–T8 or the RCL; it is infrastructure inside the gravity analysis layer that keeps the 4D kernel certificate fully computational.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.