e_120232
plain-language theorem explainer
For the multi-index (1,2,0,2,3,2) on Fin 4, the folded numerator coupling equals eight times the tabulated explicit kernel entry. Gravity analysts assembling the exact midpoint M2 TT identity in 4D cite this as one of the 256 kernel cases. The proof is a single decide on the concrete integer equality.
Claim. With indices $a=1$, $b=2$, $c=0$, $d=2$, $i=3$, $j=2$ in $\mathrm{Fin}\,4$, the folded numerator coupling satisfies $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8\,Z_{\mathrm{explicit}}(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six Fin 4 indices are compared. The numerator is defined by folding a fixed coupling list and summing each term's contribution at the given indices. The comparison target is a sparse explicit table of small integers (typical nonzero entries are $4$, $-2$, and similar constants).
This module is chunk 6 of a 256-way kernel decision: each chunk discharges a block of concrete index tuples toward the global identity that the numerator equals eight times the explicit table. The setting is pure finite enumeration over $\mathrm{Fin},4$; no continuum limit is taken inside the chunk.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(1,2,0,2,3,2)$ and checks integer equality. The left-hand side runs the fold that defines the numerator; the right-hand side multiplies the explicit table entry by eight. No intermediate lemmas are invoked beyond those two definitions.
why it matters
This case is one brick in the assembly theorem that asserts the numerator equals eight times the explicit kernel for every six-tuple in $\mathrm{Fin},4$, proved there by exhaustive fin_cases. That global equality is certified kernel infrastructure for the Regge exact-midpoint M2 TT identity in the 4D gravity analysis stack. It does not itself invoke the forcing chain (T0–T8), the Recognition Composition Law, or the $\varphi$-ladder; it only closes one discrete coupling identity needed downstream.
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