e_110202
plain-language theorem explainer
For the multi-index (1,1,0,2,0,2) on Fin 4, the folded numerator m2Num equals eight times the tabulated kernel value explicitZ. Gravity analysts cite this as one cell of the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. Let $a,b,c,d,i,j\in\{0,1,2,3\}$ with $(a,b,c,d,i,j)=(1,1,0,2,0,2)$. Then the integer numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ obtained by folding coupling contributions equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit six-index kernel table.
background
This module is chunk 5 of a 256-way case split proving $m_2^{\mathrm{num}}=8\cdot Z$ on all Fin-4 sextuples for the exact midpoint M2–TT identity in 4D Regge analysis.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at those six indices. The explicit kernel $Z$ is a pattern-matched integer table on Fin 4^6 (sample entries include 4 on diagonal-type pairs and -2 on mixed pairs).
The local goal is purely arithmetic: check one concrete sextuple so the assembler can recombine all cells into a universal identity.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values (the fold for $m_2^{\mathrm{num}}$ and the table lookup for $Z$), and the kernel checks equality to $8Z$ with no lemmas or rewriting.
why it matters
Feeds the parent assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhausting Fin 4 via fin_cases and dispatching each cell (including this one). That universal identity is the certified numerator half of the 4D Regge exact-midpoint M2–TT kernel comparison used in the gravity analysis stack. It is bookkeeping infrastructure, not a new physical law: it closes one of 256 decide cells so the closed-form kernel table can replace the folded sum everywhere downstream.
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