e_013130
plain-language theorem explainer
Pointwise identity: the midpoint M2 numerator at multi-index (0,1,3,1,3,0) equals eight times the explicit Z-coupling at the same indices. Gravity analysts cite it as one cell of the 256-entry kernel table that underwrites the exact midpoint TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,1,3,1,3,0)$ on $\mathrm{Fin}\,4$, the folded midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer coupling $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer tables are compared indexwise. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at a sextuple of $\mathrm{Fin},4$ indices, and the fold starts from zero. The comparison target explicitZ is a sparse pattern-matched table on the same sextuples, returning small integers such as $4$, $-2$, and so on.
The local module is chunk 1 of the 256 kernel decides that assert $m_2^{\mathrm{num}}=8\cdot Z$ at every concrete multi-index. The ambient claim is the exact midpoint TT identity for the 4D Regge kernel: once every cell matches, the closed-form $Z$ table may replace the folded numerator in downstream curvature and mass-window arguments.
proof idea
Both sides are closed integer expressions at fixed $\mathrm{Fin},4$ values, so the equality is a decidable proposition on $\mathbb{Z}$. The proof is the single tactic decide, which evaluates the fold defining $m_2^{\mathrm{num}}$ and the pattern match defining $Z$ and checks $8Z$ against the result. No lemmas beyond the two definitions are invoked.
why it matters
This cell feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8Z$ and discharges the universal claim by exhausting all sextuples. That universal equality is the algebraic backbone of the exact midpoint M2 TT identity in 4D Regge gravity inside the Recognition stack: it lets the sparse $Z$ table stand in for the folded numerator when matching continuum limits and discrete curvature weights. Within the broader RS gravity line it is bookkeeping rather than a forcing-chain step (T0–T8), but without the full 256-cell cover the midpoint identity stays conditional on an unchecked kernel.
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