e_213231
plain-language theorem explainer
One multiplet entry in the 4D Regge midpoint kernel: the numerator functional m2Num at index (2,1,3,2,3,1) equals eight times the explicit closed form explicitZ at the same index. Gravity analysts checking the M2 TT identity cite these chunk lemmas as the exhaustive case split. The proof is a pure kernel decision (`decide`) with no algebraic rewriting.
Claim. For the multi-index $(2,1,3,2,3,1)$, the midpoint numerator $m_2^{\mathrm{Num}}(2,1,3,2,3,1)$ equals $8$ times the explicit closed-form value $Z^{\mathrm{expl}}(2,1,3,2,3,1)$.
background
The ambient module is a chunk of the 4D Regge exact-midpoint analysis for the M2 TT identity. The global claim of the chunk family is the pointwise identity $m_2^{\mathrm{Num}}=8,Z^{\mathrm{expl}}$ on a finite discrete index set; the module doc states this is chunk 9 of a 256-entry kernel decision table.
Here $m_2^{\mathrm{Num}}$ is the numerator side of the midpoint mass-squared (or curvature) expression arising in the Regge discretization, and $Z^{\mathrm{expl}}$ is the fully expanded closed form against which that numerator is checked. Indices are small natural numbers labeling discrete edge/face multiplet components in the 4D complex.
No external lemmas are imported beyond Mathlib and the sibling kernel-certificate module; each entry is self-contained and decided independently.
proof idea
One-line kernel proof: by decide. Lean reduces both sides of the equality on concrete natural-number arguments to ground numerals and checks definitional equality. No rewriting, induction, or named upstream lemmas are used.
why it matters
The parent goal is the exact midpoint M2 TT identity in 4D Regge gravity analysis: every multiplet component must satisfy $m_2^{\mathrm{Num}}=8,Z^{\mathrm{expl}}$. Chunk 9 contributes one of the 256 kernel cases that together close that identity.
In the Recognition Science gravity stack this sits under the discrete geometric side of the forcing chain (spatial dimension and tick structure already fixed at T7–T8), supplying a machine-checked algebraic identity rather than a continuum approximation. Downstream consumers are the aggregated kernel certificate and any theorem that quantifies over the full multiplet table; this single entry has no further dependents listed.
No open scaffold remains for this line: the equality is fully decided.
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