Discrete Knot Theory via Lattice-Filtered Move Graphs

We introduce lattice-filtered move graphs as finite-state experimental models for knot types. At level $N$, vertices are lattice-polygon representatives of a fixed knot type with lattice length at most $N$, modulo orientation-preserving lattice isometries, and edges are prescribed local moves. Connected components are discrete analogues of admissible components in ropelength-filtered knot spaces, and the first level at which two initial components become connected defines a discrete merge scale; after subtracting the birth level this gives an ultrapseudometric whenever the relevant components eventually merge. The theoretical part is move-system independent. We then specialize to the simple cubic lattice and to BFACF-type moves, treated as a chosen local move system rather than a complete lattice-isotopy calculus. A separate conditional theorem shows that, for PL-realizable local move systems, discrete paths project to finite Reidemeister certificates. The experimental part reports reproducible seed-generated computations from a prototype Python implementation and a deterministic C++ accelerator using the same BFACF rules and canonicalization. For a 30-edge figure-eight seed $\omega$ and its reflected mirror $\omega !$, the seed-generated components are separated at $N=30$ and merge at $N=32$. For a 40-edge minimal simple cubic seed $\sigma$ of the other amphichiral six-crossing knot $6_3$, they are separated at $N=40$ and $N=42$, and merge at $N=44$. Explicit, independently verified move paths give $m_{\mathrm{s}\mathrm{e}\mathrm{e}\mathrm{d}}^{\mathrm{B}\mathrm{F}\mathrm{A}\mathrm{C}\mathrm{F}}(\omega ,\omega !)=32$ and $m_{\mathrm{s}\mathrm{e}\mathrm{e}\mathrm{d}}^{\mathrm{B}\mathrm{F}\mathrm{A}\mathrm{C}\mathrm{F}}(\sigma ,\sigma !)=44$, and the $6_3$ result is confirmed by a second, independently constructed minimal seed. These are seed-specific and BFACF-specific length-barrier computations, not claims about the global merge matrices of the full minimal layers.