e_000132
plain-language theorem explainer
For the six-index slot (0,0,0,1,3,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these micro-identities when assembling the exact midpoint M2 TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. With indices $a=b=c=0$, $d=1$, $i=3$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and the fold starts from zero. The companion map $Z$ is an explicit piecewise-constant integer table on the same six indices (typical values $\pm 2,,4$, and zero off the listed patterns).
The module is one of the 256-case decide chunks that certify the pointwise identity $m_2^{\mathrm{num}}=8Z$ on a single index sextuple. Local setting: chunk 0 of the kernel decides that assemble the global equality used in the exact midpoint M2 TT identity.
proof idea
One-line decide on the fully concrete integer equality after substituting the six literal $\mathrm{Fin},4$ values. Both sides reduce by unfolding the fold definition of $m_2^{\mathrm{num}}$ and the pattern-match table for $Z$; the kernel checks the resulting $\mathbb{Z}$ equality.
why it matters
Feeds the assembler theorem $m_2^{\mathrm{num}}=8Z$ for all six indices, which is proved by exhaustive fin_cases and dispatches each sextuple to a chunk identity of this form. That global equality is the algebraic backbone of the exact midpoint M2 TT identity in the 4D Regge gravity analysis. Within Recognition Science gravity work it is pure discrete kernel bookkeeping: no continuum limit, no forcing-chain step (T0–T8), just certified integer arithmetic supporting the curvature/mass side of the framework.
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