e_012000
plain-language theorem explainer
For the index sextuple (0,1,2,0,0,0), the midpoint mass-squared numerator equals eight times the explicit Z-coupling table entry. Gravity analysts assembling the 4D Regge exact-midpoint identity cite this atomic check. The proof is a single kernel decision on concrete Fin-4 values.
Claim. The midpoint numerator at indices $(0,1,2,0,0,0)$ equals eight times the explicit coupling value at those same indices: $m_2^{\mathrm{num}}(0,1,2,0,0,0)=8\,Z(0,1,2,0,0,0)$.
background
In the 4D Regge exact-midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is an integer obtained by folding a fixed coupling list and summing a contribution function over six labels in $\mathrm{Fin},4$ (simplex and edge indices). The companion table $Z$ is an explicit integer-valued function of the same six indices, with finitely many nonzero pattern cases.
The local module certifies the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ by chunked kernel decisions (256 decides per chunk). This declaration is one cell of chunk 1: the concrete sextuple $(0,1,2,0,0,0)$.
proof idea
One-line wrapper: both sides are closed integer expressions at fixed $\mathrm{Fin},4$ arguments, so decide evaluates them and discharges equality. No lemmas beyond the definitions of the numerator fold and the explicit $Z$ table are required.
why it matters
Parent assembly theorem: the universal statement that $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ for all six $\mathrm{Fin},4$ indices, proved by exhaustive case splits that land on these chunk cells. That identity is computational scaffolding inside the Regge exact-midpoint M2/TT analysis in the gravity sector. It does not itself invoke the forcing chain (T5–T8), RCL, or the phi-ladder; it underwrites a discrete curvature/mass-squared bookkeeping step used downstream in the continuum match.
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