e_203003
plain-language theorem explainer
For the fixed index sextuple (2,0,3,0,0,3) on Fin 4, the folded coupling numerator equals eight times the tabulated explicit kernel integer. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single `decide` on concrete integers.
Claim. Let $a,b,c,d,i,j\in\{0,1,2,3\}$. Writing $N$ for the folded coupling numerator and $Z$ for the explicit integer kernel, one has $N(2,0,3,0,0,3)=8\,Z(2,0,3,0,0,3)$.
background
This module is one chunk of the 4D Regge-exact midpoint analysis: it certifies the cellwise identity that the folded numerator equals eight times an explicit integer table. Indices run over Fin 4, i.e. the four discrete directions of the 4-simplex edge labeling.
The numerator m2Num a b c d i j is defined by folding a fixed coupling list and summing a local contribution at each term. The companion table explicitZ is a pattern-matched integer function on six Fin 4 arguments; only a sparse set of patterns is nonzero (typical values $\pm 2,\pm 4$).
The local setting is purely algebraic bookkeeping inside the gravity analysis stack: no continuum limit or variational principle is invoked here, only equality of two integer-valued maps on a finite domain of size $4^6=4096$, discharged in 256-sized decide chunks.
proof idea
One-line computational proof: by decide. Lean reduces both sides on the concrete sextuple $(2,0,3,0,0,3)$ to closed integers (via the fold definition of the numerator and the pattern match of the explicit table) and checks equality by the kernel decision procedure. No lemmas are invoked beyond the two definitions.
why it matters
This cell is one of the 256 decide fragments that the assembler m2Num_eq_eight_explicitZ case-splits over with nested fin_cases. That parent theorem states the identity for every sextuple and is the certified bridge from the folded coupling definition to the closed-form explicit kernel used downstream in the Regge midpoint M2TT analysis.
In the broader Recognition gravity stack, such kernel identities keep the discrete curvature bookkeeping exact before continuum or phenomenological layers are attached. The chunk does not itself touch forcing-chain landmarks (T5–T8) or the RCL; it is infrastructure for the 4D Regge identity layer.
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