e_231301
plain-language theorem explainer
At the six Fin-4 indices (2,3,1,3,0,1), the folded midpoint mass-squared numerator equals eight times the explicit integer Z-coupling. Gravity analysts certifying the 4D Regge midpoint M2TT identity cite it as one discharged kernel cell. Proof is a single decide on concrete integer arithmetic.
Claim. For indices $a=2$, $b=3$, $c=1$, $d=3$, $i=0$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis the mass-squared numerator is the fold of a fixed coupling list: $m_2^{\mathrm{num}}(a,b,c,d,i,j):=\sum_{t}\mathrm{contrib}(t;a,b,c,d,i,j)$ over six indices in $\mathrm{Fin},4$. The comparison object is an explicit piecewise integer map $Z$ on the same six indices (values such as $\pm 2$, $4$, and so on).
This module is chunk 11 of a 256-way kernel-decide partition of the pointwise claim $m_2^{\mathrm{num}}=8\cdot Z$. The ambient setting is the exact midpoint M2TT identity certificate for discrete 4D gravity in the Recognition Science stack.
proof idea
One-line wrapper: decide. Both sides reduce to concrete Int values (left via the fold definition of the numerator, right via the pattern-match clauses of the explicit $Z$ table), and the kernel checks equality.
why it matters
Feeds the universal assembly theorem m2Num_eq_eight_explicitZ, which states $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ for every six-tuple in $(\mathrm{Fin},4)^6$. That identity is the algebraic core of the exact midpoint M2TT certificate in 4D Regge calculus; each chunk theorem discharges one concrete cell so the global statement carries no sorry.
Within Recognition Science this lives in the gravity-analysis layer that underwrites discrete-to-continuum checks for the effective Newtonian sector. The 4D index set is consistent with the forced spatial dimension $D=3$ (T8) plus time, though this lemma itself is pure finite integer arithmetic.
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