e_211122
plain-language theorem explainer
For the six-index slot (2,1,1,1,2,2) on Fin 4, the folded Regge numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one atomic case in the 256-way kernel certificate that m2Num = 8·explicitZ. The proof is a pure kernel decide on concrete integers.
Claim. For indices $a{=}2,\,b{=}1,\,c{=}1,\,d{=}1,\,i{=}2,\,j{=}2$ in $\mathbb{F}_4$, the folded coupling numerator $m_2(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 9 of the 256 kernel decides that certify the four-dimensional Regge exact-midpoint identity $m_2 = 8\cdot Z$ pointwise on $(\mathbb{F}_4)^6$. The ambient setting is discrete gravity analysis: edge and face couplings on a 4D simplicial complex, reduced to integer arithmetic on Fin 4 labels.
Upstream, $m_2$ is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. The comparison target explicitZ is a closed pattern-match table on the six Fin 4 arguments, returning small integers such as $4$, $-2$, and so on. The identity asserts that the fold reproduces eight times that table entry at every multi-index.
Chunk theorems isolate one multi-index each so the kernel can discharge them by pure evaluation, without case-splitting tactics at this layer.
proof idea
One-line computational proof: by decide. Both sides are closed integer terms once the six Fin 4 literals are substituted into m2Num (a fold of concrete contributions) and explicitZ (a pattern match). Lean’s decision procedure evaluates the equality in $\mathbb{Z}$ and closes the goal. No lemmas beyond the two definitions are invoked.
why it matters
Parent consumer is m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2=8\cdot Z$ and proves it by exhaustive fin_cases on all six indices, routing each leaf to a chunk theorem such as this one. Without the pointwise certificates the assembly cannot finish.
In the Recognition gravity stack this identity is bookkeeping infrastructure for the Regge exact-midpoint $M_2$ TT analysis in 4D: it replaces a summed coupling expression by an explicit sparse kernel, enabling later curvature and continuum-limit arguments. It does not itself invoke the forcing chain (T0–T8), RCL, or $\phi$-ladder mass formulae; it is pure discrete-kernel algebra supporting those layers.
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