e_010001
plain-language theorem explainer
For the six-index slot (0,1,0,0,0,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(0,1,0,0,0,1)$ equals eight times the explicit integer kernel at the same indices: $N(0,1,0,0,0,1)=8\,Z(0,1,0,0,0,1)$.
background
This module is one chunk of the exhaustive check that the 4D Regge midpoint M2TT numerator equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$ (four discrete directions). The numerator $N=m2Num$ is defined by folding a fixed coupling list and summing local contributions at six indices $(a,b,c,d,i,j)$. The table $Z=explicitZ$ is a pattern-matched integer kernel: only a handful of index patterns are nonzero (e.g. $4$ or $-2$), and all other sextuples default to zero.
The local setting is purely algebraic bookkeeping inside the gravity analysis stack: no continuum limit is taken here. The parent assembly theorem will range over all $4^6$ index tuples by fin_cases; each chunk such as this one discharges a single concrete equality so the kernel can decide it.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left-hand side by evaluating the fold that defines the numerator on the fixed indices $(0,1,0,0,0,1)$, the right-hand side by evaluating the pattern match for the explicit kernel (or its default zero) and multiplying by eight. Lean’s decision procedure closes the integer equality with no lemmas beyond the two definitions.
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 over $\mathrm{Fin},4$. That identity is the certified algebraic core of the Regge exact midpoint M2TT analysis in four dimensions: it replaces a folded coupling sum by a sparse closed form, which is what downstream curvature and gravity identities actually consume.
Within Recognition Science gravity work this is scaffolding for exact discrete identities rather than a forcing-chain landmark (T5–T8). It does not itself invoke $J$, $\phi$, or the eight-tick octave; it is infrastructure so those layers can quote a machine-checked 4D kernel equality without re-expanding the fold.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.