e_010232
plain-language theorem explainer
Pointwise check that the Regge midpoint numerator at multi-index (0,1,0,2,3,2) equals eight times the explicit integer table. Gravity analysts cite it when assembling the full 4D m2Num = 8·explicitZ identity. The proof is a single kernel decide on fixed Fin 4 indices.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,2,3,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer value $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint identity certification: verify $m_2^{\mathrm{num}}=8\cdot Z$ at every 6-tuple of $\mathrm{Fin},4$ indices by kernel decision (256 cases per chunk family).
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of a fixed coupling list: sum of local contributions contrib over couplingZList. The comparison target explicitZ is a closed integer table on six $\mathrm{Fin},4$ arguments (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
The local setting is pure finite enumeration in the Gravity analysis stack: no continuum limit, only exact integer equality of the discrete midpoint kernel against its tabulated form.
proof idea
One-line computational proof: decide evaluates both sides at the concrete indices $0,1,0,2,3,2$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the folded numerator and the explicit table.
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 exhausting all $\mathrm{Fin},4$ cases. That global identity is the algebraic certificate that the 4D Regge midpoint $M_2$ numerator matches eight times the closed kernel table, a discrete gravity bookkeeping step in the RS gravity analysis path.
It does not itself touch the forcing chain (T0–T8), RCL, or continuum constants; it is infrastructure that keeps the exact discrete identity machine-checked before any continuum or phenomenological claim.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.