Discrete p-Density and Compression Radii of Lattice Knots

We introduce a finite-state framework for discrete $p$-density and compression-radius profiles of lattice knots. At a lattice length level $N$, we consider representatives of a fixed knot type modulo orientation-preserving lattice isometries, and optionally restrict to subgraphs generated from specified seeds by a local move system such as BFACF moves. Raw density and compression functionals are attached directly to lattice polygons; a smoothed version permits controlled perturbation, corner rounding, thickness certification, and normalization. For a fixed explored graph and finite smoothing scheme, we prove finite computability, monotonicity under increasing length caps, and componentwise inequalities under graph mergers. We also introduce a two-parameter filtration by lattice length and by a density or compression threshold, giving a computable discrete counterpart of continuous filtered knot spaces. The analytic lattice-to-continuous approximation problem remains open. As preliminary computations, we give raw-lattice data for the square unknot and a $24$-edge cubic-lattice trefoil, together with a restricted trefoil profile for $N=24,26,28,30,32$. We further evaluate density and compression along verified seed-generated BFACF mirror-merge certificates for the amphichiral knots $4_1$ and $6_3$. The certificates have, respectively, $21$ states and $20$ moves under the cap $N=32$, and $153$ states and $152$ moves under the cap $N=44$. Exact separation at the preceding admissible levels yields seed-specific mirror barriers $2$ and $4$. Along both paths, density and compression behave as competing rather than redundant objectives. All numerical values are explicitly raw-lattice or seed-generated and are not claimed to be exhaustive smoothed knot-type invariants.