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Preprint; not peer reviewed.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. Lower bounds for knotted arcs are often obtained by calibration: one writes ell = integral X + integral (1-X) for a scalar field X <= 1, then omits the nonnegative second term. We prove that tangent data at one endpoint recovers part of this term. Curvature and calibrating-field drift limit how quickly the terminal tangent angle can close, giving an explicit excess J(beta,mu) that improves the naive quadratic estimate by a factor of about 2.58 at the governing configuration. Applied to the DDS essential-arc argument, an obstacle avoided by an adjacent arc imposes a one-sided endpoint-tangent bound incompatible with equality in the spherical-projection estimate. Retaining the projection deficit removes the branch threshold and gives the closed-form angle beta = arccos(p) - arcsin(s/2). A finite rational interval certificate then proves Rop(K) >= L_DDS + 0.0298 > 15.69040926 for every nontrivial knot K. Version 3.0.4 strengthens the exposition, DDS dependency audit, and reproducibility package without changing the theorem's numerical value.