e_220201
plain-language theorem explainer
At multi-index (2,2,0,2,0,1) on Fin 4, the Regge midpoint M2 numerator equals eight times the explicit kernel integer Z. Gravity analysts cite it as one atomic cell in the 4D TT-identity kernel table. The proof is a single kernel `decide` on closed integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(2,2,0,2,0,1)$ in $(\mathrm{Fin}\,4)^6$, 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.
background
This module sits in the 4D Regge exact-midpoint analysis for the M2 TT identity. The local goal, stated in the module header, is to certify $m_2^{\mathrm{num}}=8\cdot Z$ by chunked kernel decisions (here chunk 10 of the 256-cell partition).
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: sum the local contributions of each coupling term at a six-index in $(\mathrm{Fin},4)^6$. The comparison target $Z$ is an explicit integer-valued kernel on the same index space, given by a finite pattern-match table (sample entries include $4$ on diagonal-type slots and $-2$ on selected off-diagonal slots).
Upstream, both $m_2^{\mathrm{num}}$ and $Z$ are pure definitions in the kernel-certificate module; no analytic hypotheses remain once the indices are ground.
proof idea
One-line kernel proof: decide. With all six indices concrete in Fin 4, both sides reduce to closed Int expressions (a finite fold versus a table lookup), and the decidable equality checker discharges the goal with no lemmas or case splits in this file.
why it matters
Feeds the universal assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhausting $(\mathrm{Fin},4)^6$. This cell is one of the sibling ground instances (the $e_{220\ldots}$ family) that make the assembly's fin_cases tree close.
In the broader gravity stack, the identity is the algebraic core of the Regge exact-midpoint M2 TT certificate in 4D: once every kernel cell matches, the continuum TT structure is under exact discrete control. It does not itself invoke the RS forcing chain (T5–T8) or the J-cost; it is pure index arithmetic supporting the discrete gravity side.
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