e_100122
plain-language theorem explainer
Pointwise identity: the folded M2 numerator coupling at multi-index (1,0,0,1,2,2) equals eight times the explicit Z-table entry. Gravity analysts cite it when assembling the full 4D Regge midpoint M2–TT numerator identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,0,0,1,2,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer table value: $m_2^{\mathrm{num}}(1,0,0,1,2,2)=8\,Z(1,0,0,1,2,2)$.
background
This module is chunk 4 of a 256-case kernel certifying $m_2^{\mathrm{num}}=8\cdot Z$ on all $\mathrm{Fin},4$ sextuples. The setting is the exact midpoint Regge analysis of the 4D M2–TT identity in the gravity stack.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the integer obtained by folding a fixed coupling list: each term contributes via a local contrib and the accumulator starts at 0. The comparison table $Z$ is an explicit six-index function $\mathrm{Fin},4^6\to\mathbb{Z}$ with sparse nonzero cases (e.g. $Z(0,0,1,1,2,2)=4$, $Z(0,0,1,2,1,2)=-2$).
Both objects live in the kernel certificate module; this chunk only evaluates one concrete sextuple.
proof idea
One-line computational proof: decide evaluates both sides at the closed indices $(1,0,0,1,2,2)$. The left side reduces by unfolding the fold over the coupling list; the right side reduces by the matching clause of the explicit $Z$ table (or default 0). Equality of the resulting integers is discharged by the kernel decision procedure. No lemmas beyond the two definitions are required.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $\mathrm{Fin},4$ by exhausting cases. That universal equality is the numerator half of the exact midpoint M2–TT certificate used in the 4D Regge gravity analysis.
In the broader Recognition stack this sits inside the gravity domain that supports continuum limits and effective Newtonian structure built on the forced $D=3$ and eight-tick scaffolding (T7–T8). The chunking (256 decides) keeps the kernel certificates small and machine-checkable rather than a single giant proof term.
No open scaffold remains on this point: the equality is closed by decide for this index.
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