e_111133
plain-language theorem explainer
At multi-index (1,1,1,1,3,3) the folded Regge midpoint numerator equals eight times the explicit integer Z-table entry. Gravity analysts cite it as one of the 256 kernel point checks that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=b=c=d=1$ and $i=j=3$ in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(1,1,1,1,3,3)$ equals $8$ times the explicit table value $Z(1,1,1,1,3,3)$.
background
This module is chunk 5 of a 256-case kernel certification that the Regge-exact midpoint numerator equals eight times an explicit integer table on all of $(\mathbb{F}_4)^6$.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at those six indices. The comparison target $\mathrm{explicitZ}$ is a total function $\mathbb{F}_4^6\to\mathbb{Z}$ given by a finite pattern of integer cases (typical values $4,-2,\ldots$).
Both objects live in the 4D kernel certificate module imported here; the present file only discharges one concrete sextuple.
proof idea
One-line computational proof: decide evaluates both sides at the closed indices $(1,1,1,1,3,3)$ and checks integer equality. No 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 $\forall a,b,c,d,i,j\in\mathbb{F}_4$, $m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on the six indices. Each chunk theorem such as this one is a leaf of that case tree.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge analysis in 4D: it certifies that the folded coupling numerator matches the closed-form Z table used downstream. It does not itself invoke the forcing chain (T5–T8) or the RCL; it is pure discrete linear-algebra support for the gravity side.
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