e_310123
plain-language theorem explainer
For Fin-4 indices (3,1,0,1,2,3), the midpoint M2 numerator equals eight times the explicit Z kernel entry. Gravity analysts building the 4D Regge exact-midpoint identity cite it as one of 256 pointwise kernel cases. Proof is a single kernel `decide` on two closed integer terms.
Claim. For indices $a=3$, $b=1$, $c=0$, $d=1$, $i=2$, $j=3$ 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 4D Regge exact-midpoint stack, the M2 numerator is an integer obtained by folding a fixed contribution map over a coupling list, one value per sextuple of Fin 4 indices. The companion object is an explicit integer table $Z$ on the same domain, with sparse nonzero pattern (e.g. $4$ on matched pairs, $-2$ on selected off-diagonal patterns).
This module is chunk 13 of the 256 kernel decides that discharge the pointwise claim numerator $= 8\cdot Z$. The setting is pure finite-index integer arithmetic; continuum geometry has already been reduced away in the certificate layer.
proof idea
One-line kernel decision. Both sides are closed Int terms: the left is a fold of contributions at the fixed indices $(3,1,0,1,2,3)$, the right is eight times a pattern-match lookup in the explicit $Z$ table. decide evaluates both and checks equality; no lemmas are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:, m_2^{\mathrm{num}}=8Z$ and discharges it by nested fin_cases over all six Fin 4 arguments, landing on each chunk equality such as this one. That universal identity is the algebraic backbone of the 4D Regge midpoint M2-TT certificate in the Gravity analysis path. Within Recognition Science gravity work, these exact discrete kernel identities underwrite later continuum and curvature-matching steps; this declaration is one brick in the 256-case wall, not a physics claim on its own.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.