e_013331
plain-language theorem explainer
For the six-index tuple (0,1,3,3,3,1) on Fin 4, the integer numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts cite it as one cell in the exhaustive 4^6 case split that certifies the Regge midpoint M2TT identity. The proof is a single kernel decide on two fully evaluated integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,3,3,3,1)$ in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,1,3,3,3,1)=8\,Z_{\mathrm{explicit}}(0,1,3,3,3,1)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer via a local kernel, and the fold starts at zero. The comparison target $Z_{\mathrm{explicit}}$ is a sparse closed-form table on the same six indices (nonzero only on a short list of patterns such as $(0,0,1,1,2,2)\mapsto 4$ and several $-2$ off-diagonal patterns).
The module is chunk 1 of a 256-way decide partition of that table identity. The local claim is only the single cell $(0,1,3,3,3,1)$; neighboring cells live in sibling theorems of the same shape.
proof idea
One-line kernel proof: decide evaluates both sides as concrete integers (the fold for $m_2^{\mathrm{num}}$ and the pattern match for $Z_{\mathrm{explicit}}$) and checks equality. No lemmas beyond the two definitions are invoked.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every cell by nested fin_cases on the six indices; this declaration discharges the $(0,1,3,3,3,1)$ branch. That global identity is the algebraic backbone of the Regge-exact midpoint M2TT certification in the gravity analysis stack: it replaces a summed coupling expression by an eightfold multiple of a sparse explicit table, which is what later curvature and continuum-limit arguments consume. It is bookkeeping inside the gravity domain, not a forcing-chain (T0–T8) step, but it is required for the discrete-to-continuum bridge those landmarks assume.
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