e_203113
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (2,0,3,1,1,3) equals eight times the explicit integer table at that point. One of 256 kernel decides in chunk 8 of the 4D midpoint M2TT certification. Downstream assembly cites it to finish the universal equality. Proof is a single decide on concrete Int values.
Claim. For indices $(a,b,c,d,i,j)=(2,0,3,1,1,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit closed-form integer: $N(2,0,3,1,1,3)=8\,Z(2,0,3,1,1,3)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathbb{F}_4$ indices are compared. The numerator $N=\mathrm{m2Num}$ is obtained by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib},t$ at the six indices. The comparison target $Z=\mathrm{explicitZ}$ is a sparse explicit table $\mathbb{F}_4^6\to\mathbb{Z}$ (nonzero only on a short list of patterns such as $(0,0,1,1,2,2)\mapsto 4$ and sign-flipped off-diagonal pairs $\mapsto -2$).
The module certifies the scalar relation $N=8Z$ by partitioning the $4^6=4096$ index tuples into decide-chunks. This file is chunk 8 (256 kernel decides). The local claim is the single tuple $(2,0,3,1,1,3)$.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six Fin 4 arguments are substituted into the fold definition of the numerator and the pattern-match table for the explicit kernel, so the equality is a closed Int comparison discharged by the kernel decision procedure.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and proves it by exhaustive fin_cases on all six indices, invoking one pointwise lemma per cell. That universal identity is the algebraic core of the Regge exact midpoint M2TT certification in four dimensions: it replaces the folded coupling sum by an eightfold multiple of a sparse closed form, enabling later gravity-analysis steps that need a compact integer kernel rather than a list fold.
Within Recognition gravity, this is bookkeeping infrastructure for the discrete curvature/edge-length calculus, not a forcing-chain landmark (T5–T8). It closes one cell of the 256-decide chunk so the assemble layer can finish without sorry.
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