arXiv2026
We develop a filtered geometric framework for essential surfaces in knot exteriors. Let $γ$ be a unit-thickness representative of a knot type $K$ with $\mathrm{Len}(γ)\leΛ$, and let $F$ be a properly embedded essential surface in the exterior of a fixed-radius tube about $γ$, with $\mathrm{Area}(F)\leΔ$ and relative thickness at least $τ$, formulated through Federer reach and controlled boundary collars. We prove that this bounded-geometry pair space contains only finitely many pair-isotopy classes, and that equality of explicitly bounded canonical layered codes at resolution $\varepsilon\le c\min\{1,τ\}$ implies ambient pair-isotopy. In a fixed exterior $E$, the surface systems visible in a geometric window form finite subcomplexes ${ES}_{Δ,τ}(E)$ that exhaust the essential-surface complex; isometries act levelwise and $C^{1,1}$ self-diffeomorphisms act with controlled reindexing. For a fixed two-sided surface, compressing disks are filtered in the same way, giving finite geometric witnesses for compressibility and weak reducibility and recovering the index-one characterization in Bachman's topological index theory. On the peripheral torus, every visible numerical boundary slope lies in an explicit writhe window, so that $|r|\le C_{\mathrm{BS}}Λ^{4/3}+w(Δ,τ)$. Together with finite Reidemeister certificates, the faithful codes give two independent finite recognition mechanisms, one for the knot and one for the carried essential-surface type. The framework is a smooth, triangulation-free analogue of the finiteness philosophy of normal surface theory; it does not assert that a knot exterior has only finitely many essential surfaces without geometric bounds.