e_232001
plain-language theorem explainer
For the six Fin-4 indices (2,3,2,0,0,1), the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one of the 256 exhaustive kernel checks that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2$, $b=3$, $c=2$, $d=0$, $i=0$, $j=1$ in $\mathrm{Fin}\,4$, the midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-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 integer kernel on the same six indices (sample values include $4$, $-2$, and so on for the listed patterns).
The local module is chunk 11 of the 256 kernel decides that establish $m_2^{\mathrm{num}}=8Z$ pointwise. The theoretical setting is pure integer arithmetic on a finite index set: no continuum limit or curvature hypothesis enters these checks.
Upstream, $m_2^{\mathrm{num}}$ and $Z$ are the two definitions being compared; the present lemma fixes one concrete sextuple.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six indices are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the pattern-match definition of $Z$; the kernel then checks equality.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8Z$ for all six $\mathrm{Fin},4$ indices, which is proved by exhaustive fin_cases and therefore depends on each pointwise chunk (including this one). That global identity is the algebraic backbone of the Regge exact-midpoint $M_2$ TT identity in 4D gravity analysis inside the monolith.
Within Recognition Science this sits in the gravity layer that supports continuum and discrete curvature bookkeeping; it does not itself invoke the forcing chain (T5–T8), the Recognition Composition Law, or the $\varphi$-ladder mass formula. It closes one of the 256 finite cases needed so the assembled identity can be cited without sorry.
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