When is anti-holomorphic dependence forced? Exchange non-Hermiticity of one-way transfer kernels and a proposed time-domain test in a recirculating fiber loop

Anthony Monnerot

Published Jul 4, 2026• 0 citations

Abstract

A theoretical and numerical study of when the anti-holomorphic (conjugate-variable) dependence of a two-point transfer kernel is forced by the underlying open-system dynamics rather than removable by gauge, basis, or preparation. For two counter-winding mode ladders coupled unidirectionally (reservoir-engineered, Lindblad-type, per-pair Jordan double-pole structure), the transfer kernel sums to a closed form involving the Lerch transcendent in the product of conjugated coordinates. Reciprocity is shown to be equivalent to the existence of an exchange-Hermitian mirror partner of the kernel; unidirectionality removes that partner. All symbolic claims are certified by exact Wirtinger-derivative computation. A recirculating-fiber-loop implementation and a time-domain protocol are proposed, with noise/imperfection analysis and explicit falsifiability criteria. Version 3, additive over v2 — the boundary arc (Sections 3–5): the kernel's intrinsic boundary at |x| = 1 with its generic-existence statement (the wall belongs to the class; its location is set by the tail of the weights); the chiral phase law of the branch-cut jump (machine-exact ratio, linear drift at rate kappa/2delta, family-independent); the log-periodic interference calendar on the log-scale coordinate, with prior-art positioning against discrete scale invariance (Sornette; Derrida–Itzykson–Luck); the Fourier floor theorem with full proof (the residual relief of large kernel sums equals the window-contrast of the Fourier transform of the weighted winding measure; finite atomic ladders keep the calendar forever by Bohr almost-periodicity, continua forget by Riemann–Lebesgue); closed-form floors (Ci/Si integrals, cardinal sine); and an inverse spectroscopy reading the hidden weighted distribution of winding speeds from the measured relief, with proven limits (exact mirror blindness of modulus-only data; window = aperture). Non-Hermitian point-gap topology references added (Wanjura et al. 2020; Zirnstein et al. 2021). Title-date correction (July 10, 2026). No physical claim is added; all results are machine-verified structural characterizations. Note added at deposit time: subsequent to the sealing of this version, the mirror-blindness limit of the inverse spectroscopy has been lifted by a reference-needle heterodyne protocol (demodulation law, asymmetry law, quadrature recovery of the full complex transform), machine-verified and judge-certified; traces and harnesses are in the public repository below. This upgrade will be the subject of the next linked deposit. All FINDINGS and machine harnesses backing every number in Sections 3–5, plus an independent external from-scratch reproduction audit, are public: https://github.com/antparis/oneway-kernel (MIT).

DOI: 10.5281/zenodo.21195310

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