e_313033
plain-language theorem explainer
At multi-index (3,1,3,0,3,3) on (F_4)^6, the folded Regge midpoint mass-squared numerator equals eight times the explicit integer Z-kernel. Gravity analysts cite it as one of the 256 kernel point checks that assemble the global identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,3,0,3,3)$ with each coordinate in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer kernel on six $\mathbb{F}_4$ arguments.
background
This module is chunk 13 of a 256-way case split proving $m_2^{\mathrm{num}}=8\cdot Z$ on the 4D Regge midpoint mass-squared identity. Indices run in $\mathrm{Fin},4\cong\mathbb{F}_4$, labeling discrete edge/face slots in the midpoint stencil.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the six indices. The comparison target $Z$ is an explicit integer-valued pattern match on those same six indices (typical values $\pm 2,4$, and so on).
The local claim is only the single tuple $(3,1,3,0,3,3)$. Sibling chunks cover the other tuples; the assemble theorem glues them by exhaustive fin_cases.
proof idea
One-line computational proof: decide. Lean reduces both sides at the concrete $\mathrm{Fin},4$ sextuple to closed integers (the fold for the numerator versus the pattern match for $Z$) and checks equality in $\mathbb{Z}$. No lemmas beyond the two kernel definitions are invoked.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by six nested fin_cases over $\mathrm{Fin},4$. Each chunk such as this one discharges one leaf of that case tree.
In the gravity analysis stack this identity is the algebraic certificate that the midpoint Regge mass-squared numerator is exactly eight times a sparse explicit kernel, so later continuum or continuum-limit arguments can quote a closed form rather than a fold. It is bookkeeping inside the 4D Regge sector, not a forcing-chain (T0–T8) step, but it hardens the discrete gravity side of the monolith.
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