e_022230
plain-language theorem explainer
For the six-index slot (0,2,2,2,3,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,2,2,2,3,0)=8\,Z_{\mathrm{explicit}}(0,2,2,2,3,0)$.
background
This module is chunk 2 of an exhaustive verification that the 4D Regge midpoint M2 numerator equals eight times a sparse explicit integer table. The ambient setting is discrete gravity analysis: couplings on Fin-4 multi-indices are summed, then matched to a closed form.
Upstream, m2Num a b c d i j is the fold of contrib over couplingZList, starting from 0. explicitZ is the matching lookup table on six Fin-4 arguments, returning small integers such as 4, -2, and so on for the nonzero patterns. The identity claimed here is one concrete six-tuple from that table.
The parent assembly theorem quantifies over all $4^6$ index tuples and discharges them by fin_cases; each chunk module supplies a block of the atomic equalities the kernel can decide.
proof idea
One-line proof: by decide. Both sides reduce to concrete Int values once the six Fin-4 indices are fixed, so the kernel compares the folded sum against 8 * explicitZ with no further lemmas.
why it matters
Feeds the universal statement m2Num_eq_eight_explicitZ in the assemble module, which asserts $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$. That identity is the certified algebraic core of the Regge exact-midpoint M2–TT relation in 4D used by the gravity analysis stack.
Within Recognition Science gravity work, such kernel-certified numerator identities keep the discrete curvature bookkeeping exact before continuum or phenomenological limits are taken. This declaration is pure casework scaffolding: it closes one cell of the $4^6$ grid so the quantified theorem can finish by fin_cases without sorry.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.