e_022110
plain-language theorem explainer
Pointwise identity: the folded m2 numerator at multi-index (0,2,2,1,1,0) equals eight times the explicit integer table Z at that same index. Gravity analysts cite it when assembling the full 4^6 kernel identity m2Num = 8·explicitZ. Proof is a single kernel decide on two concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,2,2,1,1,0)$ with each coordinate in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit closed-form integer $Z(a,b,c,d,i,j)$.
background
This module is chunk 2 of a 256-way kernel split verifying $m_2^{\mathrm{num}}=8\cdot Z$ on all of $(\mathbb{F}_4)^6$. The setting is the 4D Regge exact-midpoint TT identity analysis in the Gravity domain.
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 table on six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and other small integers on the listed patterns).
Upstream, both $m_2^{\mathrm{num}}$ and $Z$ live in the KernelCert module; this chunk only evaluates one concrete multi-index against that shared infrastructure.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete Int values (the fold for $m_2^{\mathrm{num}}$ at $(0,2,2,1,1,0)$ versus $8$ times the matching explicitZ clause), and the decidable equality checker discharges the goal with no lemmas or case splits inside this declaration.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and is proved by exhaustive fin_cases over the six $\mathrm{Fin},4$ indices. Each chunk theorem such as this one covers one residual concrete cell so the global identity is a finite conjunction of kernel facts rather than a symbolic expansion.
In the broader Recognition gravity stack, the identity certifies that the midpoint Regge TT numerator matches the closed integer table used downstream in 4D coupling bookkeeping. It is pure finite combinatorics: no continuum limit, no forcing-chain step (T0–T8), and no appeal to $J$, $\varphi$, or the eight-tick octave.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.