Endpoint Tangents and a New Universal Lower Bound for the Ropelength of Nontrivial Knots

Preprint; not peer reviewed. For two decades, the universal ropelength bound L_DDS = 15.6606092679785... of Denne, Diao and Sullivan (DDS) has remained the strongest published lower bound for nontrivial knots, reflecting the depth and durability of their work. Lower bounds for the length of a knotted arc are typically obtained by calibration: one exhibits a scalar field X ≤ 1 along the arc, writes ℓ = ∫X + ∫(1−X), and discards the second, non-negative term. We show that this discarded term can be recovered quantitatively from tangent data at a single endpoint. Suppose the terminal tangent makes spherical angle at least β with the calibrating direction. The curvature bound and the drift of the direction field limit how quickly that angle can close along the curve. Integrating this constraint gives an explicit excess J(β,μ). For the radial calibration about a point of a unit-thickness knot the Lipschitz constant is 3, and J(β,μ) exceeds the naive quadratic defect estimate by a factor of about 2.58 at the configuration that governs the application. We apply this estimate to the essential-arc argument of Denne, Diao and Sullivan. Near an essential alternating quadrisecant, the obstacle avoided by an adjacent arc forces a one-sided bound on an endpoint tangent of the essential middle arc. Equality in the spherical-projection estimate would require a tangent direction incompatible with that bound. Retaining the projection deficit makes the resulting bound independent of any branch threshold and reduces the defect angle to the closed form β = arccos(p) − arcsin(s/2). An explicit finite interval certificate then gives, for every nontrivial knot K, Rop(K) ≥ L_DDS + 0.0298 > 15.69040926, where 0.0298 = 149/5000 exactly. The value 0.0298 is a convenient rounded-down target below the narrowest non-corner grouped certificate enclosure 0.029834205355; the fraction has no intrinsic geometric significance. The archived release includes the PDF, exact LaTeX and text source, certificate and tests, machine-readable outputs, reproduction instructions, licenses, citation metadata, SHA-256 checksums, and a self-contained release ZIP.