e_201212
plain-language theorem explainer
For the Fin-4 index tuple (2,0,1,2,1,2), the folded M2 numerator coupling equals eight times the explicit integer Z-table entry. Gravity analysts certifying the 4D Regge midpoint M2TT identity cite it as one cell of the 256-case kernel. Proof is a single kernel decide on integer arithmetic.
Claim. For indices $a=2$, $b=0$, $c=1$, $d=2$, $i=1$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge-exact midpoint M2TT analysis, the numerator coupling is the integer
$$m_2^{\mathrm{num}}(a,b,c,d,i,j)=\sum_{t\in\mathrm{couplingZList}}\mathrm{contrib}(t;a,b,c,d,i,j),$$
obtained by folding a fixed contribution map over a static coupling list. The companion table $Z$ is an explicit six-argument function $\mathrm{Fin},4^6\to\mathbb{Z}$ with closed integer values on selected index patterns (e.g. $\pm 2,\pm 4$) and zero elsewhere.
The local module is chunk 8 of the 256 kernel decides that discharge $m_2^{\mathrm{num}}=8\cdot Z$ pointwise on every six-tuple. Upstream definitions supply both the fold and the table; no analytic closed form is assumed beyond those defs.
proof idea
One-line proof by decide. Both sides are closed integer terms: the left reduces by evaluating the fold that defines the numerator at $(2,0,1,2,1,2)$; the right reduces by pattern-matching the explicit $Z$ table at the same indices and multiplying by 8. The kernel checks the resulting $\mathbb{Z}$ equality.
why it matters
Parent theorem m2Num_eq_eight_explicitZ states the universal identity for all $a,b,c,d,i,j:\mathrm{Fin},4$ and proves it by six nested fin_cases, invoking one kernel cell per tuple. This declaration is the cell for $(2,0,1,2,1,2)$.
That universal identity is the algebraic core of the Regge-exact midpoint M2TT certification in 4D gravity analysis. It converts a summed coupling expression into an eightfold multiple of a sparse explicit table, which is the form needed for later assembly and continuum comparison steps in the Recognition gravity stack.
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