e_220121
plain-language theorem explainer
Kernel equality: the Regge midpoint mass-squared numerator at multi-index (2,2,0,1,2,1) equals eight times the explicit Z-table entry. Gravity analysts cite it as one of 256 finite-case checks assembling the full 4D identity. Proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,2,0,1,2,1)$ in $(\mathbb{F}_4)^6$, 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 lookup table on six $\mathbb{F}_4$ arguments.
background
In the 4D Regge midpoint analysis, the mass-squared numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on six indices in $\mathbb{F}_4$, and the fold starts from zero. The companion table $Z$ is an explicit piecewise integer function on the same six indices (sample clauses return $4$, $-2$, etc.).
The local module is chunk 10 of a 256-way case split whose sole job is to certify $m_2^{\mathrm{num}}=8Z$ pointwise. The ambient setting is the exact midpoint M2 TT identity kernel for discrete gravity, imported from the KernelCert definitions of both $m_2^{\mathrm{num}}$ and $Z$.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(2,2,0,1,2,1)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the fold $m_2^{\mathrm{num}}$ and the lookup $Z$.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhausting all $\mathbb{F}_4$ cases. That global identity is the algebraic backbone of the Regge exact midpoint M2 TT certificate in 4D gravity analysis. Each chunk (here case 10) closes one cell of the 256-kernel decide grid so the assembler can finish by fin_cases alone. No T0–T8 forcing step is touched; the link is pure discrete-gravity bookkeeping inside the RS gravity stack.
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