e_012332
plain-language theorem explainer
For the Fin-4 index sextuple (0,1,2,3,3,2), the folded Regge midpoint mass-squared numerator equals eight times the corresponding explicit Z-table entry. Gravity analysts cite it as one atomic cell of the 4D midpoint M2TT identity kernel. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,2,3,3,2)$ in $(\mathbb{F}_4)^6$, 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
This module is chunk 1 of a 256-cell kernel that certifies $m_2^{\mathrm{num}}=8\cdot Z$ pointwise on $(\mathbb{F}_4)^6$. The setting is the exact midpoint identity for the 4D Regge mass-squared (TT) sector in the Gravity analysis stack.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the six indices. The comparison table $Z$ is an explicit pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and so on).
The parent assembly theorem states the universal equality for every sextuple and discharges it by exhaustive fin_cases, so each concrete cell such as this one is a pure computational obligation.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,1,2,3,3,2)$. The left side runs the fold that defines $m_2^{\mathrm{num}}$; the right side looks up $8\cdot Z$ at those indices. No lemmas are invoked beyond kernel reduction of the two definitions.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by six nested fin_cases over $\mathrm{Fin},4$. Each chunk theorem closes one concrete branch of that case tree.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge M2TT kernel in 4D: once every cell matches, the closed-form $Z$ table may replace the folded numerator in downstream curvature and mass-squared identities. It is scaffolding closure for the kernel certificate, not a new physical law; the physics content sits in the parent identity that this cell helps discharge.
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