Exact even-odd phase offsets in the geometric threshold spectrum of a Lorentzian well

The attractive one-dimensional Lorentzian well belongs to the class of Schrödinger potentials solvable in terms of confluent Heun functions and has a marginal inverse-square tail. Above the critical coupling it supports infinitely many negative bound states accumulating at the continuum threshold. General inverse-square spectral theory determines the geometric scaling of this accumulation but not the parity-resolved placement of the even and odd sequences. Here the threshold problem is exploited in its simpler zero-energy Gauss hypergeometric form. Exact large-distance connection coefficients provide closed gamma-function phases encoding propagation through the nonsingular Lorentzian core. Matching these inner solutions to the recessive inverse-square tail yields separate asymptotic quantization laws for the even and odd threshold sequences. The common geometric ratio is universal, while the leading parity-dependent asymptotic constants and the even-odd displacement are obtained explicitly as gamma-function phases, with the physical branch fixed by Sturm ordering. Inward integration of the full Lorentzian Schrödinger equation, independent of the zero-energy hypergeometric core matching but initialized from the common inverse-square Bessel tail, verifies both the universal slope and the parity-dependent intercepts without fitting spectral data. The result gives an analytically explicit example of how a long-range critical tail fixes spectral scaling while a smooth core fixes the channel-dependent threshold phase.