e_012231
plain-language theorem explainer
Pointwise identity: the folded 4D Regge midpoint numerator m2Num at multi-index (0,1,2,2,3,1) equals eight times the explicit kernel entry at that same index. Gravity analysts cite it as one cell of the 256-case kernel that underwrites the global m2Num = 8·explicitZ identity. Proof is a single kernel decide on two concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,1,2,2,3,1)$ with each coordinate in $\mathbb{F}_4$, the folded coupling numerator equals eight times the explicit integer kernel value: $N(0,1,2,2,3,1)=8\,Z(0,1,2,2,3,1)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add each list contribution at the given multi-index. The comparison target $Z=\mathrm{explicitZ}$ is a closed-form pattern-matched table of small integers (entries such as $4$, $-2$, and so on).
The module is chunk 1 of a 256-cell decide grid that checks $N=8Z$ at every multi-index. The local setting is purely combinatorial: no continuum limit or curvature hypothesis is in play, only equality of two explicitly computable $\mathbb{Z}$-valued functions on a finite domain of size $4^6$.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete integers (the fold of couplingZList versus eight times the matched explicitZ clause), and Lean’s decision procedure discharges the equality. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and is proved by exhaustive fin_cases over the six indices. That global identity is the certified algebraic core of the 4D Regge midpoint $M_2$ TT analysis in the Gravity domain: it replaces an opaque fold by an explicit sparse integer table, making later curvature and mass-ladder comparisons inspectable. Within Recognition Science this sits on the gravity side of the forcing chain (after T8 fixes $D=3$ spatial dimensions), supplying a machine-checked numerator identity rather than a continuum postulate. It closes one cell of the 256-decide scaffold; the open obligation is only that every sibling cell and the assembler stay sorry-free.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.