e_003322
plain-language theorem explainer
For the Fin-4 index sextuple (0,0,3,3,2,2), the folded numerator m2Num equals eight times the tabulated kernel value explicitZ. Gravity analysts certifying the 4D Regge midpoint M2 TT identity cite this as one atomic case. The proof is a single decide on concrete integer equality.
Claim. For indices $(a,b,c,d,i,j)=(0,0,3,3,2,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,0,3,3,2,2)=8\,Z_{\mathrm{explicit}}(0,0,3,3,2,2)$.
background
This module is one chunk of a 256-way kernel certification that the 4D Regge midpoint numerator equals eight times an explicit integer table. The ambient setting is discrete gravity analysis: exact midpoint identities for the M2 TT sector on a 4-index lattice.
The numerator m2Num is defined by folding a fixed coupling list: sum the contribution of each coupling term at the six Fin-4 indices. The comparison table explicitZ is a total function Fin 4^6 → ℤ with finitely many nonzero pattern clauses (e.g. (0,0,1,1,2,2) ↦ 4, mixed permutations ↦ −2).
Chunk 0 discharges a block of those sextuples by kernel decide; the present declaration is the single cell (0,0,3,3,2,2).
proof idea
One-line wrapper: by decide. Both sides reduce to concrete integers (left via the fold definition of m2Num, right via the pattern match on explicitZ), and the kernel checks equality. No lemmas are invoked beyond evaluation of the two upstream defs.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states ∀ a b c d i j, m2Num a b c d i j = 8 * explicitZ a b c d i j and proves it by exhaustive fin_cases on all six Fin-4 arguments. Each atomic equality such as this one is a leaf of that case tree.
In the Recognition gravity stack this closes a numerical identity needed for the exact midpoint M2 TT certification in 4D Regge analysis. It is pure discrete linear algebra over the coupling kernel, not a continuum GR claim, and does not itself touch the forcing chain (T0–T8) or the J-cost uniqueness step.
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