e_313220
plain-language theorem explainer
Concrete identity: the Regge midpoint m2 numerator at index sextuple (3,1,3,2,2,0) equals eight times the explicit kernel integer Z at those same indices. Gravity analysts cite it as one closed case in the 4D midpoint TT identity. The proof is a single kernel decide on two fully evaluated integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,3,2,2,0)$ in $\{0,1,2,3\}^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer kernel table.
background
In the 4D Regge midpoint analysis, two integer-valued maps on sextuples of $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums a local contribution at each coupling term for the given indices. The companion map $Z$ is an explicit piecewise table of small integers (entries such as $4$, $-2$, and so on) on those same sextuples.
The module is chunk 13 of a 256-case kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. Each chunk discharges a block of concrete index tuples so the global assembler need only case-split on $\mathrm{Fin},4$.
proof idea
Both sides are closed integer expressions once the six indices are fixed. The tactic decide evaluates $m_2^{\mathrm{num}},3,1,3,2,2,0$ via the fold definition and $8\cdot Z,3,1,3,2,2,0$ via the explicit table, then checks equality in $\mathbb{Z}$. No lemmas beyond the two definitions are required.
why it matters
The parent theorem is the universal statement that $m_2^{\mathrm{num}},a,b,c,d,i,j=8,Z,a,b,c,d,i,j$ for every sextuple in $(\mathrm{Fin},4)^6$. That assembler introduces the six indices and runs fin_cases on each; the present lemma is the discharged branch for $(3,1,3,2,2,0)$.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint TT kernel in 4D Regge calculus: it certifies that the summed coupling numerator is exactly eight copies of the closed-form kernel table, with no residual. It does not itself invoke the forcing chain (T5–T8) or the RCL; it is pure discrete linear-algebra support under the gravity analysis layer.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.