e_032131
plain-language theorem explainer
For the six-index slot (0,3,2,1,3,1) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact Regge midpoint M2 identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,1,3,1)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint certification: it checks the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on a block of the $4^6$ index space (here chunk 3), each case discharged by a kernel decide.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing local contributions at those six $\mathrm{Fin},4$ slots. The comparison value $Z$ is an explicit integer table on the same six indices (sample entries include $4$, $-2$, and other small integers on distinguished patterns).
The local goal is purely algebraic bookkeeping inside the gravity analysis stack: no continuum limit or variational argument is invoked at this layer.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values for the fixed tuple $(0,3,2,1,3,1)$: the left via the fold definition of $m_2^{\mathrm{num}}$, the right via the pattern-match table for $Z$, and the kernel checks equality to $8Z$.
why it matters
Feeds the assembler theorem $m_2^{\mathrm{num}}=8Z$ for all six $\mathrm{Fin},4$ indices, which fin-cases over the full grid and invokes these chunk lemmas. That global identity is the algebraic core of the Regge exact-midpoint M2/TT certification in 4D gravity analysis.
Within Recognition Science this sits in the discrete gravity layer that supports continuum recovery after the forcing chain has fixed $D=3$ spatial dimensions (T8) and the eight-tick structure (T7). It does not itself touch J-uniqueness, $\phi$, or the mass ladder; it is infrastructure for the Regge curvature bookkeeping.
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