e_033300
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at multi-index (0,3,3,3,0,0) equals eight times the explicit Z-table entry. Gravity analysts cite it when assembling the full 4D Regge midpoint M2–TT kernel identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,3,3,0,0)$ in $(\mathrm{Fin}\,4)^6$, 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 3 of a 256-way split certifying $m_2^{\mathrm{num}}=8\cdot Z$ on the 4D Regge midpoint kernel. Indices run over $\mathrm{Fin},4$ (four discrete directions). The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each local contribution at the six indices. The table $Z$ is an explicit pattern-matched integer function on the same six indices (sample clauses give values such as $4$, $-2$, and so on).
The local claim is one concrete sextuple in that table. Upstream, both $m_2^{\mathrm{num}}$ and $Z$ live in the kernel certificate module; this chunk only evaluates them at $(0,3,3,3,0,0)$.
proof idea
One-line computational proof: decide reduces both sides to concrete integers (the fold of contributions versus eight times the matched $Z$ clause) and checks equality in the kernel. No lemmas are invoked beyond the definitions of the numerator and the explicit table.
why it matters
Feeds the assembly theorem that states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ by exhausting all $\mathrm{Fin},4$ sextuples. That universal identity is the algebraic core of the 4D Regge exact midpoint M2–TT certificate in the Gravity analysis stack. Within Recognition Science it supports the discrete curvature/mass-squared bookkeeping used in the gravity sector (companion to the forcing chain’s geometric constraints, not a T0–T8 step itself). Closing every chunk removes scaffolding from the midpoint identity.
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