e_201213
plain-language theorem explainer
For the single 4D multi-index (2,0,1,2,1,3), the Regge midpoint mass-squared numerator equals eight times the explicit Z coupling value. Gravity analysts cite it as one atomic case in the 4^6 case-split that identifies m2Num with 8·explicitZ. The proof is a pure kernel decide on the two closed integer expressions.
Claim. For indices $a=2$, $b=0$, $c=1$, $d=2$, $i=1$, $j=3$ in $\mathrm{Fin}\,4$, the midpoint mass-squared numerator equals eight times the explicit integer coupling: $m_2^{\mathrm{num}}(2,0,1,2,1,3)=8\,Z_{\mathrm{explicit}}(2,0,1,2,1,3)$.
background
This module is one chunk of the 4D Regge exact-midpoint $M_2$ TT-identity certification. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each triple contributes an integer term depending on six $\mathrm{Fin},4$ indices, and the fold starts at zero. The comparison target $Z_{\mathrm{explicit}}$ is a pattern-matched integer table on the same six indices (typical values $\pm 2,,4$, and zero off the listed patterns).
The local claim is the equality $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ at one concrete multi-index. The module header records the strategy: 256 kernel decides, split across chunks, later reassembled by exhaustive fin_cases.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers (the fold that defines the numerator, and the pattern match that defines explicit $Z$) and checks equality in the kernel. No lemmas are invoked beyond the two definitions.
why it matters
The parent theorem m2Num_eq_eight_explicitZ states the identity for every six-tuple in $(\mathrm{Fin},4)^6$ by nested fin_cases; each leaf is one of these chunk theorems. Establishing the factor-of-eight match converts the folded coupling sum into the closed explicit $Z$ table, which is the algebraic core of the 4D Regge midpoint $M_2$ TT identity used on the gravity side of the monolith. It does not itself touch the T0–T8 forcing chain, but it is infrastructure for the discrete curvature/mass-squared identities that sit under RS gravity analysis.
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