e_203310
plain-language theorem explainer
For the six-index slot (2,0,3,3,1,0) on Fin 4, the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite this as one atomic decide-cell in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator at $(2,0,3,3,1,0)$ equals eight times the tabulated explicit kernel entry at the same indices: $N(2,0,3,3,1,0)=8\,Z(2,0,3,3,1,0)$.
background
This module is chunk 8 of a 256-cell decide grid that certifies the algebraic identity $N=8Z$ pointwise on $(\mathrm{Fin},4)^6$. The setting is the exact midpoint analysis of the 4D Regge M2TT kernel used in the Recognition gravity stack.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start from 0 and add each contribution $\mathrm{contrib},t,a,b,c,d,i,j$. The right-hand side $Z=\mathrm{explicitZ}$ is a sparse integer table on six $\mathrm{Fin},4$ indices (nonzero only on a short list of patterns such as $(0,0,1,1,2,2)\mapsto 4$ and several $-2$ off-diagonal slots).
The full quantified statement is assembled downstream by exhaustive fin_cases; each chunk theorem such as this one discharges one concrete sextuple.
proof idea
One-line decide proof. Both sides reduce to concrete Int values for the fixed indices $(2,0,3,3,1,0)$: the left by evaluating the fold of contrib over couplingZList, the right by looking up explicitZ (or the default 0 branch). Lean’s kernel decision procedure checks integer equality; no lemmas are invoked.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,,N=8Z$ by six nested fin_cases over $\mathrm{Fin},4$. That global identity is the certified numerator form of the Regge exact-midpoint M2TT relation in 4D, a computational pillar of the Recognition gravity analysis.
Within the framework this sits in the gravity domain rather than the T0–T8 forcing chain; it is infrastructure that keeps the discrete curvature/kernel bookkeeping exact before continuum or phenomenological limits are taken. Closing all 256 decide cells removes scaffolding risk from the M2 numerator identity.
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