e_033132
plain-language theorem explainer
For the six Fin-4 indices (0,3,3,1,3,2), the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the full Regge midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on integer equality.
Claim. With indices $(a,b,c,d,i,j)=(0,3,3,1,3,2)$ in $\mathrm{Fin}\,4$, 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
This module is chunk 3 of a 256-cell kernel certification that the Regge exact-midpoint numerator equals eight times an explicit integer table on $(\mathrm{Fin},4)^6$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at the six indices. The comparison target $\mathrm{explicitZ}$ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by a finite case table (sample entries include $4$ on diagonal-type pairs and $-2$ on mixed pairs).
Local setting (module doc): prove $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ cellwise by kernel decide, then reassemble.
proof idea
One-line computational proof: by decide. Lean reduces both sides at the concrete indices $(0,3,3,1,3,2)$ to closed integers (the fold for $m_2^{\mathrm{num}}$ versus the matching explicitZ clause times $8$) and discharges equality in the kernel. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ and proves it by exhaustive fin_cases on all six indices; each cell is one of these chunk theorems.
In the Gravity analysis stack this is bookkeeping for the Regge exact-midpoint M2TT identity in 4D: once every cell matches, the folded coupling form may be replaced by the closed kernel $Z$ (scaled by $8$) in downstream curvature or mass-response arguments. It is not itself a forcing-chain step (T0–T8); it is certified arithmetic support for the discrete gravity side of the RS mirror.
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