e_021032
plain-language theorem explainer
For the discrete index sextuple (0,2,1,0,3,2), the folded 4D midpoint Regge numerator coupling equals eight times the tabulated explicit integer kernel entry. Gravity analysts cite it as one cell of the exhaustive Fin-4 kernel identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,1,0,3,2)$ in $\mathrm{Fin}\,4$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D midpoint Regge identity that asserts the folded numerator coupling equals eight times an explicit integer table. The ambient setting is discrete gravity analysis: six indices in $\mathrm{Fin},4$ label edge and face data in a 4-simplex midpoint scheme.
The numerator side $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each term. The right-hand side is the lookup table explicitZ, an integer-valued function on six $\mathrm{Fin},4$ arguments with finitely many nonzero patterns (for example $4$ on matched diagonal pairs and $-2$ on certain crossed pairs).
The module documentation states the goal bluntly: $m_2^{\mathrm{num}}=8\cdot Z$, proved by 256 kernel decides split across chunks. This declaration is the cell for indices $(0,2,1,0,3,2)$.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six indices are fixed: the left-hand fold over the coupling list evaluates to a single Int, and the right-hand side is eight times the matching table entry of the explicit kernel. No lemmas are invoked beyond kernel evaluation of closed integer arithmetic.
why it matters
The parent theorem is the assembled identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$, proved by exhaustive fin_cases on all six $\mathrm{Fin},4$ arguments. Each chunk cell such as this one discharges one concrete sextuple so the universal statement is a pure case split rather than a symbolic simplification.
In the Recognition gravity stack this identity is bookkeeping for the discrete midpoint Regge kernel in 4D: it certifies that the folded numerator matches the closed-form integer table used downstream in curvature and mass-squared couplings. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is infrastructure inside the gravity analysis layer that those continuum limits eventually rest on.
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