e_212300
plain-language theorem explainer
For the Fin-4 multi-index (2,1,2,3,0,0), the folded Regge numerator m2Num equals eight times the tabulated kernel explicitZ. Analysts assembling the 4D Regge midpoint M2TT identity cite this cell among the 256 decided kernel equalities. The proof is a single decide on concrete integers.
Claim. For indices $a=2$, $b=1$, $c=2$, $d=3$, $i=0$, $j=0$ in $\{0,1,2,3\}$, the integer $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ obtained by folding coupling contributions equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit kernel table.
background
This module sits in the 4D Regge midpoint analysis for the M2TT identity. The local goal, stated in the module header, is to verify $m_2^{\mathrm{num}}=8\cdot Z$ cell by cell across a 256-cell decide kernel (chunk 9).
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 the six Fin-4 indices. The comparison value $Z(a,b,c,d,i,j)$ is an explicit integer table on the same six indices (sample entries include $4$, $-2$, and so on for distinguished patterns).
Both sides are pure integers once the six indices are fixed, so each cell is a closed arithmetic identity rather than a symbolic rewrite.
proof idea
One-line wrapper: decide evaluates both sides of the equality at the concrete indices $(2,1,2,3,0,0)$. The left side runs the fold that defines the numerator; the right side looks up the explicit kernel entry and multiplies by 8. No lemmas are invoked beyond the computational kernel for Int equality.
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 six-tuple in $\mathrm{Fin},4$. That assembler introduces the six indices and discharges them by nested fin_cases; each concrete residual goal is one of these named cell theorems.
In the Recognition gravity stack this identity is bookkeeping for the Regge midpoint M2TT kernel in four dimensions: once every cell matches, the folded coupling form may be replaced by the closed explicit table. It does not itself invoke the forcing chain (T0–T8), the Recognition Composition Law, or the $\varphi$-ladder mass formula; it is pure discrete kernel certification supporting the continuum gravity side.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.