e_020033
plain-language theorem explainer
For the six-index tuple (0,2,0,0,3,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts cite it inside the exhaustive 4^6 case split that proves the Regge midpoint M2 TT numerator identity in 4D. The proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,0,0,3,3)$ with each index in $\{0,1,2,3\}$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,2,0,0,3,3)=8\,Z_{\mathrm{explicit}}(0,2,0,0,3,3)$.
background
This module is one chunk of the 4D Regge-exact midpoint M2 TT identity certification. The local goal, stated in the module header, is to check $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{explicit}}$ on a block of the $4^6$ multi-indices, each discharged by a kernel decide.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding couplingZList and summing the local contribution of each coupling term at those six Fin-4 indices. $Z_{\mathrm{explicit}}$ is the matching closed-form integer table on the same six indices (sample values include $4$, $-2$, and other small integers on the listed patterns).
The identity equates the summed numerator to eight times that table entry, index by index. Chunk 2 holds the present tuple among its siblings.
proof idea
One-line proof: by decide. Both sides are closed integer expressions once the six concrete Fin-4 indices are substituted into m2Num and explicitZ, so the kernel reduces the equality to true with no further lemmas or rewriting.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every Fin-4 six-tuple into the global statement $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$, via exhaustive fin_cases. This chunk supplies the (0,2,0,0,3,3) cell of that grid.
In the Recognition gravity stack the identity is bookkeeping for the Regge midpoint discretization of the M2 TT sector in 4D: the folded coupling numerator must match the explicit integer kernel before continuum or continuum-limit claims can be attached. It does not itself invoke the forcing chain (T5–T8) or the J-cost; it is pure discrete tensor arithmetic supporting the gravity analysis layer.
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