e_020213
plain-language theorem explainer
For the fixed index sextuple (0,2,0,2,1,3), the folded coupling numerator equals eight times the tabulated explicit kernel entry. Gravity analysts certifying the 4D Regge midpoint M2TT identity cite this as one concrete point in a 256-way kernel split. The proof is a single kernel decide on closed integer arithmetic.
Claim. With six arguments in $\mathrm{Fin}\,4$, the folded numerator $m_2^{\mathrm{num}}(0,2,0,2,1,3)$ equals $8$ times the explicit kernel value $Z(0,2,0,2,1,3)$.
background
In the 4D Regge exact-midpoint analysis, two integer kernels on six $\mathrm{Fin},4$ indices are compared pointwise. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a contribution at each index sextuple. The companion table $Z$ is a sparse pattern-matched map into small integers (entries such as $4$, $-2$, and so on).
This module is chunk 2 of a 256-way split of the identity $m_2^{\mathrm{num}}=8\cdot Z$. Each chunk discharges a block of concrete index combinations by decision procedure, so the full universal statement can later be assembled by exhaustive $\mathrm{Fin}$ case splits.
proof idea
One-line wrapper: decide. With all six arguments fixed to concrete $\mathrm{Fin},4$ values, both sides reduce to closed integer expressions, and the kernel equality checker closes the goal. No named lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
The assemble theorem m2Num_eq_eight_explicitZ proves the identity for every sextuple by nested fin_cases and consumes this chunk fact (and its siblings) as the leaf evidence. That universal equality is part of the Regge midpoint M2TT certification stack in the gravity analysis layer. It is computational scaffolding for an exact discrete-gravity kernel identity, not a step on the T0–T8 forcing chain, the RCL, or the $\varphi$-ladder mass formula.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.