e_310001
plain-language theorem explainer
For the six-index tuple (3,1,0,0,0,1), the discrete midpoint mass numerator equals eight times the explicit kernel value. Gravity analysts cite this as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single `decide` on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,0,0,0,1)$ in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the Regge exact-midpoint 4D analysis, the mass-side numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at six Fin 4 indices, and the fold starts from zero. The companion kernel explicitZ is a closed-form integer table on the same six indices (nonzero only on a sparse set of patterns such as $(0,0,1,1,2,2)\mapsto 4$ and sign-flipped off-diagonal pairs $\mapsto -2$).
The local module is chunk 13 of a 256-way case split: the claim is that $m_2^{\mathrm{num}}=8\cdot Z$ holds pointwise on every six-tuple in $(\mathrm{Fin},4)^6$. This file discharges one concrete tuple by kernel decision; the sibling chunks cover the rest of the grid.
proof idea
One-line computational discharge: both sides are closed integer expressions at fixed indices, so decide evaluates $m_2^{\mathrm{num}}(3,1,0,0,0,1)$ via the fold definition and compares it to $8\cdot\mathrm{explicitZ}(3,1,0,0,0,1)$. No algebraic rewriting or intermediate lemmas are required beyond the two definitions.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ by exhaustive fin_cases over the six indices and invocation of the chunk decides. That global identity is the algebraic bridge between the folded coupling definition and the sparse explicit kernel used in the Regge midpoint $M_2$ TT analysis in 4D gravity.
Within Recognition Science gravity work, such kernel identities underwrite exact discrete curvature bookkeeping on the eight-tick / $D=3$ side of the forcing chain; they are infrastructure rather than a named T0–T8 step, but they keep the midpoint mass formulae free of residual summation error.
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