e_130332
plain-language theorem explainer
For the single index sextuple (1,3,0,3,3,2) on Fin 4, the discrete Regge midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it only as one cell of the 4^6 case split that builds the global identity. The proof is a pure kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,3,0,3,3,2)$ in $\{0,1,2,3\}^6$, the folded midpoint 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
This module is chunk 7 of a 256-way kernel certification that the discrete midpoint numerator on a 4D Regge-type index set agrees with an explicit closed form, up to the universal factor 8. Indices run over $\mathrm{Fin},4$, i.e. ${0,1,2,3}$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term evaluated at the six indices. The comparison target $Z$ is an explicit pattern-matched integer table on the same six indices (sample clauses include values $4$, $-2$, and so on).
The local claim is one concrete cell of that table equality. Upstream, both sides are pure defs with no hypotheses; downstream assembly will quantify over all $4^6$ cells.
proof idea
One-line kernel proof: decide. Both sides reduce to closed integer expressions for the fixed sextuple $(1,3,0,3,3,2)$, so Lean’s decision procedure checks numeral equality with no lemmas, rewrites, or case splits inside this declaration.
why it matters
Parent theorem m2Num_eq_eight_explicitZ states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases, each leaf a chunk theorem of this form. This declaration is the leaf for $(1,3,0,3,3,2)$.
In the gravity analysis stack it certifies that the folded coupling numerator matches the hand-written kernel on that cell, so the assembled identity can be used without residual case obligations. It is bookkeeping inside the Regge exact-midpoint 4D identity, not a new physical law; it does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law, but it hardens the discrete geometric substrate those layers sit on when gravity is derived.
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