e_130100
plain-language theorem explainer
For the multi-index (1,3,0,1,0,0) on Fin 4, the summed coupling numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one cell in the exhaustive 4^6 case split that proves the Regge midpoint M2–TT identity. The proof is a single `decide` on the concrete integer equality.
Claim. For indices $a=1$, $b=3$, $c=0$, $d=1$, $i=0$, $j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(1,3,0,1,0,0)=8\,Z_{\mathrm{explicit}}(1,3,0,1,0,0)$.
background
This module is chunk 7 of a 256-way kernel certification that $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{explicit}}$ pointwise on $(\mathrm{Fin},4)^6$. The setting is the exact midpoint form of the 4D Regge M2–TT identity used in the gravity analysis stack.
$m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding contrib over couplingZList, yielding an integer. $Z_{\mathrm{explicit}}$ is the closed-form integer kernel on the same six indices (pattern-matched cases such as $(0,0,1,1,2,2)\mapsto 4$ and mixed pairs $\mapsto -2$).
The parent assembly theorem quantifies over all six indices and discharges each cell by fin_cases; each chunk theorem such as this one supplies one concrete cell.
proof idea
One-line computational proof: by decide. Lean reduces both sides of the integer equality for the fixed indices $(1,3,0,1,0,0)$ and checks they match. No lemmas are invoked beyond the evaluable definitions of m2Num and explicitZ.
why it matters
Feeds directly into m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ and proves it by nested fin_cases over all six $\mathrm{Fin},4$ arguments. That universal identity is the certified algebraic core of the Regge exact-midpoint M2–TT relation in 4D.
Within Recognition Science gravity work, the identity constrains the discrete curvature/coupling bookkeeping that sits behind continuum limits and effective Newtonian structure. This declaration is pure scaffolding closure: one of 4096 index cells, grouped into decide-chunks so the kernel stays machine-checkable.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.