e_313101
plain-language theorem explainer
Case identity: the Regge midpoint m2 numerator at multi-index (3,1,3,1,0,1) equals eight times the explicit Z kernel at that same index. Gravity analysts cite it only as one cell of the 4^6 case split that builds the global m2Num = 8·explicitZ theorem. The proof is a single kernel decide on a concrete integer equality.
Claim. For indices $(a,b,c,d,i,j)=(3,1,3,1,0,1)$ in $\{0,1,2,3\}^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer kernel table.
background
This module is chunk 13 of a brute-force certification that the four-dimensional Regge midpoint $m_2$ numerator coincides with eight times an explicit integer kernel on every multi-index in $(\mathrm{Fin},4)^6$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The comparison target $\mathrm{explicitZ}$ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by a finite pattern table (e.g. $(0,0,1,1,2,2)\mapsto 4$, $(0,0,1,2,1,2)\mapsto -2$, and so on).
The local claim is only the single cell with indices $3,1,3,1,0,1$. Sibling theorems cover the other cells in the same chunk.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers (the fold that defines $m_2^{\mathrm{num}}$ at this fixed six-tuple, and eight times the table lookup for $\mathrm{explicitZ}$) and closes the equality in the kernel. No lemmas are invoked beyond the definitions of $m_2^{\mathrm{num}}$ and $\mathrm{explicitZ}$.
why it matters
Parent theorem $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ (forall on $(\mathrm{Fin},4)^6$) is assembled by exhaustive fin_cases on all six indices; each leaf is one of these chunk theorems. Without the cell-wise identities the global midpoint Regge $m_2$ numerator certification does not close.
In the Recognition gravity stack this is pure analysis scaffolding: it underwrites the exact discrete curvature/numerator identity used downstream in Regge-style continuum matching, not a forcing-chain (T0–T8) step. It touches no open physics hypothesis; it only discharges one of the 4^6 kernel obligations.
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