Causality in linear response is conventionally treated as a binary property: a response function is either analytic in the upper half-plane or it is not. We show that in a PT-symmetric open dimer it instead carries a topological charge. As the gain-loss parameter crosses the exceptional point, a single pole of the reflection coefficient migrates into the upper half-plane, the Blaschke winding number jumps from 0 to 1, and standard Kramers-Kronig (KK) reconstruction acquires a Lorentzian residual fixed by the pole residue. The transition is sharp, protected by the codimension-one structure of the exceptional point, and directly measurable in a one-port reflection experiment. Most strikingly, the violation magnitude scales as Delta_KK ~ |gamma - gamma_c|^nu with nu ~ -1.08 in the single-port geometry: the breakdown of standard KK is strongest at threshold and weakens deeper in the broken phase. We derive the exact reflection coefficient, verify the residue-corrected dispersion relation, and propose a THz time-domain spectroscopy protocol that detects the topological charge through the residual itself.
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4 pages, 4 figures. Supplemental Material included (2 pages)
Causality in linear response is conventionally treated as a binary property: a response function is either analytic in the upper half-plane or it is not. We show that in a PT-symmetric open dimer it instead carries a topological charge. As the gain-loss parameter crosses the exceptional point, a single pole of the reflection coefficient migrates into the upper half-plane, the Blaschke winding number jumps from 0 to 1, and standard Kramers-Kronig (KK) reconstruction acquires a Lorentzian residual fixed by the pole residue. The transition is sharp, protected by the codimension-one structure of the exceptional point, and directly measurable in a one-port reflection experiment. Most strikingly, the violation magnitude scales as Delta_KK ~ |gamma - gamma_c|^nu with nu ~ -1.08 in the single-port geometry: the breakdown of standard KK is strongest at threshold and weakens deeper in the broken phase. We derive the exact reflection coefficient, verify the residue-corrected dispersion relation, and propose a THz time-domain spectroscopy protocol that detects the topological charge through the residual itself.
Retarded support, upper-half-plane holomorphy, and Hardy boundary control are distinct properties of a memory kernel. We give sufficient conditions linking them for the Nakajima--Zwanzig kernel of an open system with finite-dimensional system Liouville space. If the projected kernel has an absolutely continuous real-axis representation with density w in L1 intersect Lp0, no singular part, and p0 > 1, its transform lies in the operator-valued Hardy class Hp(B) for 1 < p <= p0. A finite-dimensional componentwise argument yields the corresponding principal-value and once-subtracted Kramers--Kronig (KK) boundary formulas; the H1 endpoint is separate. Microscopic unitary evolution gives holomorphy and a trace-norm bound for reduced-state transforms for arbitrary trace-class initial states, without that spectral hypothesis. On a common analytic domain, a perturbative force-fit equation has a first-order quotient pole at a simple baseline state-transform zero zeta only when the first-order inhomogeneous numerator I_tilde^(1)(zeta) is nonzero; at finite perturbation the actual zero and numerator must be checked, while matrix reconstructions additionally require adjugate non-cancellation. For rational reconstructions, the printed local or global clearing and no-cancellation hypotheses convert a rational kernel pole into a genuine upper-half-plane propagator pole. Such a pole is incompatible with a uniformly bounded CPTP family when the transforms agree on an open set, with Vieta's formula giving the imaginary-part budget. We formalize these paper-specific implications in Lean 4 from named classical inputs. A finite Jaynes--Cummings calculation provides a shifted, distributional KK check with a 0.024 percent fixed-truncation residual.
The memory kernel of an open quantum system obeys Kramers--Kronig (KK) relations if and only if its Laplace transform is analytic in the upper half-plane -- a property known as Hardy-space analyticity. Here we show that non-unitary exchange statistics, the defining property of paraparticles, intrinsically breaks Hardy-space analyticity. The metric η that guarantees a real closed-system spectrum for these particles necessarily differs from the physical Born inner product (∥η−I∥F/∥I∥F=0.51) -- a mathematical consequence of the R-matrix's non-unitarity, not a parameter choice. This metric is a "shadow metric": Schur's lemma forces it to commute with every bilinear observable, making the distortion physically invisible in the closed system. But when the paraparticle is coupled to a bath, any coupling operator that lies outside the symmetry algebra -- that is, any interaction that sees the internal flavour structure -- exposes the distortion. The memory kernel then develops upper-half-plane poles at coupling gc≈0.1, breaking standard dispersion relations before the closed-system spectrum complexifies. Fermions and bosons, whose exchange is unitary (η=I as an analytic fact of the canonical anticommutation algebra), are immune at any coupling, because there is no distortion to expose. The violation is intrinsic: it distinguishes non-unitary exchange statistics from ordinary particle statistics at the level of the memory kernel's analytic structure.
The Jaynes-Cummings (JC) model, introduced in 1963 and central to cavity quantum electrodynamics, describes a two-level system coupled to a single bosonic mode under the rotating-wave approximation. When the mode is projected out via the Nakajima-Zwanzig (NZ) formalism, the memory-kernel generator QLQ is manifestly non-Hermitian -- yet we prove analytically that its spectrum is strictly real at every coupling and every finite truncation, for both vacuum and thermal baths. For the vacuum bath the characteristic polynomial factorises completely; the nonzero eigenvalues reproduce the JC dressed-state ladder for n >= 2, while the lowest mode is suppressed by exactly sqrt(2) relative to the bare-Hamiltonian prediction. For any thermal Gibbs state, the squared generator reduces to an asymmetric rank-one perturbation symmetrised by a closed-form diagonal metric, with eigenvalues guaranteed real by Cauchy interlacing. We construct a positive-definite metric eta_osc intertwining QLQ with its adjoint on the nonzero spectral subspace, proving hidden pseudo-Hermiticity. A classification theorem extends these results to the full U(1)-conserving single-photon-exchange class with arbitrary complex couplings. The phase boundary under counter-rotating deformation is mapped analytically at resonance and numerically across the coupling-truncation plane, revealing a weak-coupling protected wedge, re-entrant real-spectrum bubbles, and N-universal plateaus organised by a three-family band catalog with closed-form level spacings. The full phase structure is proved well-defined in the N -> infinity thermodynamic limit. The sqrt(2) suppression provides a platform-independent experimental falsification target spanning seven orders of magnitude in coupling strength.
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).
Companion to "The Windowed Reading", sharpening its regrade of the two-modes proposal at the one point where that note stopped: the Wick identity - one spectral number appearing as Euclidean decay rate and as unitary oscillation frequency - was displayed there on the free spectrum only. This note displays it with interaction, exactly, and adds an equivalence. The positive-type property of the joint kernel - the engine of the link-reflection-positivity theorem - makes the transfer operator of the interacting quartic chain positive semidefinite; hence H = -ln T exists on the support of T as a real self-adjoint Hamiltonian, and the interacting excitation energies exist as theorems of the Euclidean data. They are then extracted twice, independently: as Euclidean decay rates (agreement with the spectrum to 9e-16 - machine precision) and as unitary oscillation frequencies (FFT, agreement at the 1e-3 resolution limit); a matrix-pencil analysis of the Euclidean correlator alone recovers two interacting levels with the correct parity selection (Delta E_1 to 1e-12, Delta E_3 to 5e-7); the identity holds across couplings g = 0, 0.5, 2.0, with the free control an exact benchmark (Delta E_1 = ln 2, derived, passed at 1e-6); the gap is grid-converged to nine digits (Delta E_1 = 1.303200265 at g = 0.5); the Euclidean reading passes into the unitary one as the boundary value of the tube (linear convergence); and the interacting poles sit at s = +-i Delta E_n, on-axis and gapped - the mechanism premise verified at the interacting level in this truncation. The control is the sharpest statement: with the antiferromagnetic joint, the transfer operator acquires a macroscopic negative eigenvalue (-0.391 against 1e-16 noise), -ln T turns complex, and no real Hamiltonian generating these data - no unitary reading - exists in this class. In this class, reflection positivity is thereby displayed as equivalent to the existence of the indefinite face: the second mode stands and falls, exactly and computably, with RP. No number is derived; every claim carries a kill criterion; all computations are supplied as self-contained scripts.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.