e_033233
plain-language theorem explainer
Pointwise identity: the folded Regge numerator at multi-index (0,3,3,2,3,3) equals eight times the explicit integer table at those same indices. Gravity analysts cite it as one of 256 kernel cells feeding the global m2Num = 8·explicitZ assembly. The proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(0,3,3,2,3,3)\in(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(0,3,3,2,3,3)=8\,Z_{\mathrm{expl}}(0,3,3,2,3,3)$.
background
This module is chunk 3 of a 256-cell kernel certifying that the folded Regge midpoint numerator agrees with an explicit integer table, scaled by eight. The ambient setting is 4D discrete gravity analysis (Regge-type midpoint identities) inside the Recognition Science gravity stack.
The numerator m2Num is defined by folding a fixed coupling list: sum the local contributions of each coupling term at a six-tuple of Fin 4 indices. The comparison object explicitZ is a closed-form integer table on the same six-index domain (sample values include 4, -2, and other small integers on distinguished patterns).
The global claim is that these two agree up to the universal factor 8 on every cell of (Fin 4)^6. Each chunk theorem discharges one concrete cell so the assembler can finish by exhaustive fin_cases.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete Int values once the six indices are fixed literals, so Lean's decision procedure evaluates the fold defining the numerator and the matching clause of the explicit table and checks equality with the factor 8. No lemmas are invoked beyond kernel reduction of the two defs.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six-tuples in (Fin 4)^6 by nested fin_cases and appeals to the 256 pointwise cells. Without each cell (including this one at (0,3,3,2,3,3)), the assembler cannot close.
In the gravity analysis layer this certifies that the folded coupling numerator is exactly eight times a sparse explicit integer kernel. That algebraic simplification is the bookkeeping step needed before midpoint/Regge identities are used as exact discrete curvature input. It is pure integer kernel work, not a continuum limit claim, and sits downstream of the coupling-list definition rather than of the T0–T8 forcing chain.
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