e_020133
plain-language theorem explainer
For the fixed multi-index (0,2,0,1,3,3) on Fin 4, the summed Regge coupling numerator equals eight times the tabulated explicit kernel value. Gravity analysts cite it as one cell of the 4D midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. Let $a,b,c,d,i,j\in\{0,1,2,3\}$. Writing $N$ for the folded coupling numerator and $Z$ for the explicit integer kernel table, one has $N(0,2,0,1,3,3)=8\,Z(0,2,0,1,3,3)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on sextuples of Fin 4 indices appear. The numerator $N=\mathrm{m2Num}$ is the fold of a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at zero. The companion map $Z=\mathrm{explicitZ}$ is a sparse case table of small integers (entries such as $4$, $-2$, and defaults).
The local module is chunk 2 of a 256-cell decide grid whose sole job is to check $N=8Z$ pointwise. Upstream, both maps are defined in the kernel certificate module; the assembly theorem will later quantify over all Fin 4 sextuples by fin_cases.
proof idea
One-line proof: decide. Both sides reduce to concrete Int values for the fixed indices $(0,2,0,1,3,3)$, so the kernel closes the equality with no lemmas and no case split inside this declaration.
why it matters
This cell is consumed by m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,,N=8Z$ and discharges the universal claim by fin_cases on every coordinate. That identity is the algebraic backbone of the Regge exact-midpoint M2–TT certificate in 4D gravity analysis: once numerator and explicit kernel agree up to the constant 8, downstream curvature and mass-side identities can quote a single closed form rather than a fold. It sits in the Gravity domain of the Recognition mirror; it does not itself touch T5–T8 or the J-cost, but it hardens the discrete geometric layer those forcing steps eventually feed.
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