e_220102
plain-language theorem explainer
Pointwise identity: the folded coupling numerator equals eight times the explicit integer kernel at multi-index (2,2,0,1,0,2). One of 256 kernel cases in chunk 10 of the 4D Regge midpoint M2TT certification. Assembly cites it when discharging the universal equality by exhaustive Fin 4 case split. Proof is a single kernel decide.
Claim. For indices $(a,b,c,d,i,j)=(2,2,0,1,0,2)$ 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
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six $\mathbb{F}_4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at zero. The comparison target is an explicit piecewise integer function $Z$ on the same index domain (tabulated by pattern-matching on the six coordinates).
The local module is chunk 10 of a 256-way partition of that domain. The claim under audit is the single-point equality $m_2^{\mathrm{num}}=8Z$ at one concrete multi-index. Upstream, both kernels live in the kernel-certificate module that supplies the fold definition and the closed-form table.
proof idea
One-line proof by decide. Lean reduces both sides at the concrete Fin 4 sextuple: the left-hand side evaluates the fold of coupling contributions; the right-hand side multiplies the tabulated explicit integer by eight. No lemmas are invoked beyond kernel computation of the two definitions.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the universal identity $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases over $\mathbb{F}_4$. Each leaf is one of these pointwise decides; this declaration is the leaf for $(2,2,0,1,0,2)$.
In the gravity analysis stack, that universal equality is the certified bridge between the folded coupling numerator and the closed-form kernel used in the 4D Regge midpoint M2TT identity. It is bookkeeping infrastructure inside the discrete gravity certification, not a forcing-chain landmark (T0–T8), but it is required so the midpoint identity can quote a single clean algebraic relation rather than a fold.
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