e_320200
plain-language theorem explainer
At multi-index (3,2,0,2,0,0) the folded Regge midpoint numerator equals eight times the explicit integer kernel Z. Gravity analysts cite it as one of the 256 pointwise kernel checks that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete Fin-4 data.
Claim. For indices $(a,b,c,d,i,j)=(3,2,0,2,0,0)$ in $(\mathbb{F}_4)^6$, 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 14 of the 256-point kernel certification that the Regge exact-midpoint mass-squared numerator in 4D equals eight times a closed-form integer table. Indices run over $\mathbb{F}_4$ (four values), matching the discrete 4D simplex/edge labeling used in the TT-sector identity.
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 case-table $\mathbb{F}_4^6\to\mathbb{Z}$ (sample entries include $4$ on diagonal-type patterns and $-2$ on crossed pairs).
Local setting (module doc): prove $m_2^{\mathrm{num}}=8\cdot Z$ pointwise by kernel decides, then assemble.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(3,2,0,2,0,0)$. The left side runs the fold that defines the numerator; the right side looks up the explicit kernel and multiplies by 8. No lemmas are invoked beyond decidable equality on Int.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on all six $\mathbb{F}_4$ arguments. Each chunk such as this one discharges one concrete cell so the global identity is a pure case split over certified points rather than a symbolic fold argument.
In the gravity analysis stack this identity is the algebraic backbone of the Regge exact-midpoint M2 TT relation in 4D: once numerator and explicit kernel match up to the universal factor 8, downstream curvature/mass-squared comparisons can quote the closed form instead of the coupling list. It is bookkeeping inside the discrete gravity sector, not a forcing-chain (T0–T8) step, but it hardens the 4D Regge calculus layer that sits under continuum limits.
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