e_032123
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (0,3,2,1,2,3) equals eight times the explicit integer table at those same indices. Gravity analysts assembling the 4D midpoint M2 TT identity cite it as one decided kernel cell. The proof is a single native decide on a closed integer equality.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,1,2,3)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $N(0,3,2,1,2,3)=8\,Z(0,3,2,1,2,3)$.
background
In the 4D Regge midpoint analysis, two integer kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $N=\mathrm{m2Num}$ is the fold of a fixed coupling list: each term contributes an integer via a local contribution function, summed from zero. The table $Z=\mathrm{explicitZ}$ is a sparse pattern-matched closed form on the same six indices (typical nonzero values $\pm 2,,4$).
The module is chunk 3 of a 256-cell kernel certification: each cell asserts $N=8Z$ at one concrete multi-index. The ambient goal is the exact midpoint M2 TT identity in four dimensions, reduced to these finite integer checks.
Upstream, $N$ and $Z$ are defined in the kernel certificate module; this declaration only evaluates one ordered sextuple.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers (the fold for $N$, the match for $Z$), and Lean’s decision procedure discharges the equality. No lemmas beyond the definitions of $\mathrm{m2Num}$ and $\mathrm{explicitZ}$ are invoked.
why it matters
Feeds the universal assembly theorem $\mathrm{m2Num_eq_eight_explicitZ}$, which states $\forall a,b,c,d,i,j,,N=8Z$ by exhausting all $\mathrm{Fin},4$ cases. That global identity is the algebraic core of the Regge exact-midpoint M2 TT certification in 4D gravity analysis inside the monolith.
Within Recognition Science gravity work, such kernel equalities underwrite discrete curvature bookkeeping on the eight-tick / $D=3$ side of the forcing chain; they are bookkeeping lemmas, not new physics postulates. Closing every chunk removes scaffolding from the midpoint identity path.
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