e_112332
plain-language theorem explainer
For the index sextuple (1,1,2,3,3,2) on Fin 4, the midpoint Regge numerator equals eight times the explicit kernel table entry. Gravity analysts proving the 4D TT mass-matrix identity cite this as one closed cell of the 4^6 case split. The proof is a single kernel decide on the integer fold versus the table.
Claim. For $a=1$, $b=1$, $c=2$, $d=3$, $i=3$, $j=2$ in $\{0,1,2,3\}$, the summed coupling numerator at those indices equals $8$ times the explicit integer kernel value at the same indices.
background
This module is chunk 5 of the finite case analysis that the 4D midpoint Regge numerator equals eight times a closed-form kernel table. Indices run over Fin 4 (four discrete directions in the TT sector).
The numerator m2Num is the fold of all coupling contributions from couplingZList at a fixed sextuple $(a,b,c,d,i,j)$. The table explicitZ is a piecewise integer function on the same sextuples (sample values include $4$, $-2$, and other small integers on the diagonal and off-diagonal patterns).
The local claim is one cell of the identity m2Num = 8 · explicitZ that the assemble theorem will quantify over all $4^6$ sextuples.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers (the fold defining the numerator versus eight times the matching table clause) and closes the equality in the kernel. No lemmas beyond the definitions of the numerator and the explicit table are required.
why it matters
Feeds the parent assemble theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in Fin 4 by exhaustive fin_cases. That assemble step is the bridge from the certified kernel table to the exact midpoint M2 TT identity in 4D Regge analysis.
In the Recognition gravity stack this is bookkeeping, not a new physical law: it discharges one of the 256-style decide cells so the global numerator-table match is machine-checked rather than hand-waved. It does not itself invoke the forcing chain (T0–T8), RCL, or the phi ladder; those enter only if the TT identity is later tied to continuum limits or continuum constants.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.