e_201222
plain-language theorem explainer
At multi-index (2,0,1,2,2,2) the Regge midpoint mass-squared numerator equals eight times the explicit integer Z-kernel. Gravity analysts cite it as one of 256 concrete kernel checks that assemble the global identity. The proof is a single kernel decide on fixed Fin-4 indices.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,2,2,2)$ in $(\mathbb{F}_4)^6$, 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 8 of a 256-way split of the 4D Regge exact-midpoint identity $m_2^{\mathrm{num}}=8\cdot Z$. Each chunk discharges a block of concrete multi-indices by kernel decision rather than symbolic algebra.
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 Fin-4 slots. The explicit kernel $Z$ is a closed integer table on $(\mathbb{F}_4)^6$, with nonzero pattern values such as $\pm 2,\pm 4$ on selected index patterns and (implicitly) $0$ elsewhere.
The local goal is only the single point $(2,0,1,2,2,2)$. Upstream definitions supply both sides as pure integer data, so equality is a finite check.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple. The left side runs the fold that defines the numerator; the right side looks up (or defaults) the explicit $Z$ table and multiplies by $8$. No lemmas beyond the two definitions are required.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full $\forall$-statement by fin_cases on all six indices; each leaf is one of these chunk equalities. Without the pointwise facts the global midpoint identity does not close in Lean.
In the gravity analysis stack this certifies that the discrete Regge mass-squared numerator is exactly eight copies of the explicit $Z$-coupling kernel in 4D, a structural step toward exact midpoint identities used in the RS gravity sector. It is bookkeeping, not a new physical law, but it is load-bearing for the certified identity.
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