e_201013
plain-language theorem explainer
Pointwise kernel identity: the midpoint Regge m₂ numerator at multi-index (2,0,1,0,1,3) equals eight times the explicit Z-table entry. Gravity analysts cite it as one of 256 concrete decides that assemble into the universal m₂Num = 8·explicitZ identity. Proof is a single native decide on fixed Fin-4 integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,0,1,0,1,3)$ with each coordinate in $\mathbb{F}_4$, the summed coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on $(\mathbb{F}_4)^6$ are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each triple contributes an integer contrib, and the fold accumulates the total. The companion table explicitZ is a closed-form case-split on the six Fin-4 indices, returning small integers such as $4$, $-2$, and so on.
The module packages one 256-cell chunk of the pointwise claim $m_2^{\mathrm{num}}=8\cdot Z$. The present declaration is the cell at indices $(2,0,1,0,1,3)$. Upstream, both kernels live in the KernelCert module; downstream they are reassembled by exhaustive Fin-case analysis into a single universal equality.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the case table for $Z$. Lean’s native decision procedure checks the resulting integer equality; no algebraic rewriting or intermediate lemmas are required.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathbb{F}_4,; m_2^{\mathrm{num}}=8\cdot Z$ and is proved by six nested fin_cases that discharge every cell, including this one. That universal identity is the algebraic backbone of the exact midpoint $M_2$ TT identity in 4D Regge calculus inside the Gravity analysis stack. Within Recognition Science it sits in the discrete-geometry layer that supports continuum limits and curvature bookkeeping; it does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law, but it certifies a kernel step those continuum arguments rely on.
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