e_222100
plain-language theorem explainer
For the six-index slot (2,2,2,1,0,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact midpoint M2–TT identity in 4D cite this as one of the 256 kernel cases. The proof is a single decide on concrete integer arithmetic.
Claim. With indices in $\{0,1,2,3\}$, the six-argument numerator $m_2^{\mathrm{num}}(2,2,2,1,0,0)$ obtained by folding the coupling list equals $8$ times the explicit kernel value $Z(2,2,2,1,0,0)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six Fin-4 indices appear. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution function, and the fold starts at 0. The companion map explicitZ is a piecewise closed form that returns small integers (typically $\pm 2,\pm 4$, or 0) on the same six-index domain.
The module is chunk 10 of a 256-case kernel certification whose sole goal is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$. Indices run over spatial/temporal edge labels in the discrete 4-geometry; the factor 8 is the combinatorial multiplicity that converts the folded sum into the explicit table.
Upstream, only the two definitions are needed: the fold that builds $m_2^{\mathrm{num}}$ and the case table for $Z$. No analytic continuum limit is invoked at this layer.
proof idea
One-line kernel decide. Both sides evaluate to concrete integers once the six Fin-4 arguments are fixed at $(2,2,2,1,0,0)$: the left side by folding the coupling list, the right side by looking up (or defaulting) explicitZ and multiplying by 8. Lean’s decide closes the equality by normalized integer computation; no lemmas beyond the two definitions are applied.
why it matters
This case is one brick in the exhaustive proof of $m_2^{\mathrm{num}}=8\cdot Z$ for every six-tuple in $(\mathrm{Fin},4)^6$. The parent theorem $m_2^{\mathrm{num}}=8\cdot Z$ (forall) proceeds by nested fin_cases and dispatches each concrete slot to a chunk lemma of this form.
In the broader Recognition gravity stack, the identity certifies that the midpoint Regge M2–TT numerator matches the explicit kernel used downstream in curvature and mass-ladder bookkeeping. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete 4D kernel algebra supporting those later geometric claims.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.