Aperiodic Pseudocoherent Phase

• Aperiodic pseudocoherent phase is a state where engineered coherence persists in systems lacking strict periodic order, characterized by partial phase memory and irregular interference features.
• Research employs atomistic wave-packet simulations, DFT, and phase-space diagnostics to reveal multi-interface interference, localized steady states, and mode conversion in various systems.
• The concept has broad implications across phonon transport, grain-boundary reconstructions, quantum localization, and photonic applications, serving as a diagnostic for coherence without conventional symmetry.

to=arxiv_search.^^^^^^^^10query10^^^^^^^^ 天天彩票中大奖_json_schema={"^^^^^^^^10query10 Pseudocoherent Phase^^^^^^^^10\10 OR pseudocoherence aperiodic10^^^^^^^^", "^^^^^^^^10max_results10 ^^^^10\10query10^^^^} code to=arxiv_search.^^^^^^^^10query10^^^^^^^^ 天天彩票是_json_schema={"^^^^^^^^10query10 OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^^^^^", "^^^^^^^^10max_results10 ^^^^10\10query10^^^^} code to=arxiv_search.^^^^^^^^10query10^^^^^^^^ 北京赛车计划_json_schema={"^^^^^^^^10query10 of Spatial Coherence on Phonon Transmission across Aperiodically Arranged Interfaces^^^^^^^^10\10 OR ^^^^^^^^10\10 Dissipation as a Mechanism for Steady-State Localization^^^^^^^^10\10 "^^^^^^^^10max_results10 ^^^^10\10query10^^^^} code Aperiodic pseudocoherent phase is a context-dependent designation for regimes in which coherence-like organization persists, or is operationally engineered, in systems that lack strict periodic order. Across the literature, the term does not denote a single universal order parameter. Instead, it refers to partial phase memory across non-equidistant interfaces in coherent phonon transport, locally near-coherent but quasi-aperiodic interface reconstructions in grain boundaries, weakly interfering localized phase-space states in aperiodic electronic chains, mixed yet coherent localized steady states in dissipative lattices, intermittent synchronization-like temporal organization in non-normal stochastic dynamics, pseudospectrum-diagnosed coherent edge behavior in aperiodic topological superconductors, and spectrally uniform phase-gradient control in aperiodic metalenses (&&&^^^^10query10^^^^&&&, &&&^^^^10\10^^^^&&&, &&&^^^^10 OR pseudocoherence aperiodic10^^^^&&&, &&&^^^^10max_results10^^^^&&&, &&&^^^^10query10^^^^&&&, &&&^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^&&&, &&&^^^^10max_results10^^^^&&&).

^^^^10\10^^^^. Conceptual scope and domain-specific definitions

The shared feature across these uses is not periodic Bloch order, but a residual or engineered coherence that survives aperiodicity. In the phonon-transport setting, aperiodic pseudocoherence means that the phonon maintains phase correlations over multiple, non-equidistant interfaces, while the lack of periodicity prevents regular Bragg bandgaps and instead produces broadened, irregular interference features in transmission spectra (&&&^^^^10query10^^^^&&&). In the grain-boundary setting, “pseudocoherent” refers to segments that locally maintain near-registry with adjoining lattices while overall misfit is accommodated by localized distortions or disconnections (&&&^^^^10\10^^^^&&&). In open quantum systems, the term is defined operationally as a localized steady state with finite eigenbasis coherence, finite off-diagonal correlations, and mixedness rather than purity (&&&^^^^10max_results10^^^^&&&). In non-normal stochastic dynamics, pseudo-coherence denotes intermittent, synchronization-like temporal organization without intrinsic oscillators, Hopf bifurcation, or fixed-frequency resonance (&&&^^^^10query10^^^^&&&).

One paper explicitly notes that “pseudocoherent” is not standard nomenclature in topological superconductivity; there it is used only in an operational sense, through the Clifford pseudospectrum and coherent, protected edge phenomena in aperiodic media (&&&^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^&&&). In photonics, the phrase denotes coherent spectral-spatial phase behavior across a globally non-periodic aperture, enabled by local periodicity modulation of identical nanostructures (&&&^^^^10max_results10^^^^&&&). Collectively, these definitions suggest an umbrella concept whose precise meaning is observable-dependent and model-dependent rather than universal.

Domain	Operational meaning	Principal observables
Coherent phonon transport	Partial phase memory across aperiodically spaced interfaces	PRESERVED_PLACEHOLDER_^^^^10query10^^^^, standing waves, mode conversion
Grain boundaries	Near-coherent local registry with quasi-aperiodic reconstructions	GB atomic density PRESERVED_PLACEHOLDER_^^^^10\10^^^^, PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10^^^^, HRTEM contrast
Aperiodic electronic chains	Weakly interfering localized phase-space lobes between anticrossings	Wigner negativity, IPR, PRESERVED_PLACEHOLDER_^^^^10max_results10^^^^, PRESERVED_PLACEHOLDER_^^^^10query10^^^^
Open quantum systems	Localized mixed steady state with finite coherence	PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^, PRESERVED_PLACEHOLDER_^^^^10max_results10^^^^, PRESERVED_PLACEHOLDER_^^^^10query10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10^^^^, PRESERVED_PLACEHOLDER_^^^^10 OR \10^^^^
Non-normal stochastic systems	Intermittent, drifting coherent bands without oscillators	PRESERVED_PLACEHOLDER_^^^^10\10query10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10\10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10^^^^, spectral peaks
Aperiodic topological superconductors	Real-space pseudospectrum diagnosis of coherent edge modes	PRESERVED_PLACEHOLDER_^^^^10\10max_results10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10query10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10max_results10^^^^
Aperiodic metalenses	Uniform phase-gradient scaling across a non-periodic aperture	focal-shift RMSE, FWHM, efficiency RMSE

A common misconception is to treat the phrase as if it referred to a single equilibrium phase in the condensed-matter sense. The literature instead uses it as an operational descriptor for coherence-like behavior under aperiodic structure, dissipation, or non-normal amplification, with distinct diagnostics in each setting (&&&^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^&&&, &&&^^^^10max_results10^^^^&&&, &&&^^^^10query10^^^^&&&).

^^^^10 OR pseudocoherence aperiodic10^^^^. Coherent transport across aperiodically arranged interfaces

In coherent phonon transport, the basic quantities are the phase PRESERVED_PLACEHOLDER_^^^^10\10query10^^^^, amplitude PRESERVED_PLACEHOLDER_^^^^10\10\10^^^^, and the first-order spatial coherence function

PRESERVED_PLACEHOLDER_^^^^10\10 OR \10^^^^

The spatial coherence length PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10^^^^ is the characteristic length over which PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10\10^^^^ decays, and in the wave-packet formulation it is represented in practice by the full-width at half-maximum PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10 OR pseudocoherence aperiodic10^^^^ of the Gaussian envelope. For a narrowband packet, PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10max_results10^^^^, so larger PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10^^^^ corresponds to smaller PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ and PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10max_results10^^^^ (&&&^^^^10query10^^^^&&&).

The phonon-transport study models coherent LA-mode packets in an aperiodic superlattice using the atomistic wave-packet technique adapted from Schelling et al. in a Lennard-Jones solid argon-based model tuned to emulate stronger bonded materials. The relaxed lattice constant is PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10^^^^, the device length is PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10\10^^^^, the cross-section is PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10 OR \10^^^^ unit cells, and the aperiodic device is placed between two semi-infinite periodic superlattice contacts of approximately PRESERVED_PLACEHOLDER_^^^^10max_results10query10^^^^ each. Transmission is computed by energy accounting,

PRESERVED_PLACEHOLDER_^^^^10max_results10\10^^^^

and time-frequency wavelet analysis is used to resolve mode conversion during scattering (&&&^^^^10query10^^^^&&&).

Within this framework, the aperiodic pseudocoherent regime occurs when PRESERVED_PLACEHOLDER_^^^^10max_results10 OR pseudocoherence aperiodic10^^^^ decays over several interface spacings, with PRESERVED_PLACEHOLDER_^^^^10max_results10max_results10^^^^ of order a few times the average interface spacing PRESERVED_PLACEHOLDER_^^^^10max_results10query10^^^^ and potentially smaller than the full device length. Phase memory then persists long enough to support multi-interface interference, but the lack of periodicity prevents regular Bragg cancellation. The phase accumulated across layers can be approximated by

PRESERVED_PLACEHOLDER_^^^^10max_results10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^

so the net phase PRESERVED_PLACEHOLDER_^^^^10max_results10max_results10^^^^ accumulates irregularly rather than periodically (&&&^^^^10query10^^^^&&&).

The paper isolates this regime by comparing two coherence lengths, PRESERVED_PLACEHOLDER_^^^^10max_results10query10^^^^ and PRESERVED_PLACEHOLDER_^^^^10max_results10\10^^^^, for four representative wavelengths: PRESERVED_PLACEHOLDER_^^^^10max_results10 OR \10^^^^, PRESERVED_PLACEHOLDER_^^^^10query10query10^^^^, PRESERVED_PLACEHOLDER_^^^^10query10\10^^^^, and PRESERVED_PLACEHOLDER_^^^^10query10 OR pseudocoherence aperiodic10^^^^. Longer-wavelength packets, especially PRESERVED_PLACEHOLDER_^^^^10query10max_results10^^^^, show significantly lower transmission for large PRESERVED_PLACEHOLDER_^^^^10query10query10^^^^ than for small PRESERVED_PLACEHOLDER_^^^^10query10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^, whereas the curves converge as PRESERVED_PLACEHOLDER_^^^^10query10max_results10^^^^ decreases. Real-space snapshots show strong amplitude modulation, multi-reflection splitting, phase lags between sub-packets, and standing-wave formation. Wavelet transforms further show that for large PRESERVED_PLACEHOLDER_^^^^10query10query10^^^^ at long PRESERVED_PLACEHOLDER_^^^^10query10\10^^^^, scattering excites modes away from PRESERVED_PLACEHOLDER_^^^^10query10 OR \10^^^^, indicating enhanced mode conversion, while short-PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10query10^^^^ packets show weaker coherence-length sensitivity (&&&^^^^10query10^^^^&&&).

The conditions for strong interference suppression are stated explicitly as PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10\10^^^^, PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10 OR pseudocoherence aperiodic10^^^^, and a spectrally narrow source. Under those conditions, reduced average transmission, broadened irregular interference features, standing-wave structure, and enhanced mode conversion constitute the hallmarks of an aperiodic pseudocoherent phase. The same study also suggests a temperature connection: if PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10max_results10^^^^ decreases with increasing temperature, then reduced pseudocoherent destructive interference could contribute to the increase of aperiodic superlattice thermal conductivity PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10query10^^^^ with temperature, complementing previously reported inelastic-interface effects (&&&^^^^10query10^^^^&&&).

^^^^10max_results10^^^^. Quasi-aperiodic and pseudocoherent grain-boundary phases

In refractory body-centered cubic metals, the term denotes an interfacial structural state rather than a transport regime. The relevant systems are PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ and PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10max_results10^^^^ symmetric tilt grain boundaries in Nb, Ta, Mo, and W, with misorientations of approximately PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10query10^^^^ and PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10\10^^^^, respectively. The canonical periodic structure is the kite-unit array, obtained with grain-boundary atomic density PRESERVED_PLACEHOLDER_^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10 OR \10^^^^ and a small repeat cell defined by the coincidence site lattice. The alternative phase is a quasi-aperiodic split-kite structure obtained when the boundary core can exchange atoms with reservoirs under grand-canonical modeling (&&&^^^^10\10^^^^&&&).

The split-kite phase contains additional atoms in the boundary core, and the kite apex columns split into distinct positions. Along the tilt axis, these tip columns form wave-like arrangements whose period depends on reconstruction length, so there is no single periodic unit. The interface is termed pseudocoherent because portions remain close to coherent registry with one or both adjoining lattices, while the split tips and intermediate core positions generate locally incommensurate modulations that accommodate misfit (&&&^^^^10\10^^^^&&&).

The atomistic energetics are expressed through the grain-boundary energy

PRESERVED_PLACEHOLDER_^^^^10max_results10query10^^^^

with an analogous DFT expression for slab calculations that corrects for atom-count differences in split-kite structures. The optimized grain-boundary atomic densities reported for split kites are PRESERVED_PLACEHOLDER_^^^^10max_results10\10^^^^ for Nb PRESERVED_PLACEHOLDER_^^^^10max_results10 OR pseudocoherence aperiodic10^^^^, corresponding to insertion of PRESERVED_PLACEHOLDER_^^^^10max_results10max_results10^^^^ of the PRESERVED_PLACEHOLDER_^^^^10max_results10query10^^^^ plane; PRESERVED_PLACEHOLDER_^^^^10max_results10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ for Ta PRESERVED_PLACEHOLDER_^^^^10max_results10max_results10^^^^, corresponding to insertion of PRESERVED_PLACEHOLDER_^^^^10max_results10query10^^^^ of the PRESERVED_PLACEHOLDER_^^^^10max_results10\10^^^^ plane; PRESERVED_PLACEHOLDER_^^^^10max_results10 OR \10^^^^ for metastable Mo PRESERVED_PLACEHOLDER_^^^^10query10query10^^^^; and PRESERVED_PLACEHOLDER_^^^^10query10\10^^^^ for W PRESERVED_PLACEHOLDER_^^^^10query10 OR pseudocoherence aperiodic10^^^^ with empirical potentials (&&&^^^^10\10^^^^&&&).

Large-area reconstructions with PRESERVED_PLACEHOLDER_^^^^10query10max_results10^^^^, PRESERVED_PLACEHOLDER_^^^^10query10query10^^^^, and PRESERVED_PLACEHOLDER_^^^^10query10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ repeats show that the split-kite tip columns form wave-like patterns whose energies oscillate and converge only for large sizes greater than PRESERVED_PLACEHOLDER_^^^^10query10max_results10^^^^ repeats. This near-degeneracy across different reconstruction lengths is a central reason the phase is described as quasi-aperiodic rather than simply defective. DFT grain-boundary energies confirm that split kites are the ground state for PRESERVED_PLACEHOLDER_^^^^10query10query10^^^^ in Ta, Mo, and W, while kites remain favored in Nb at the accessible reconstruction sizes. For example, the DFT values for Ta PRESERVED_PLACEHOLDER_^^^^10query10\10^^^^ are PRESERVED_PLACEHOLDER_^^^^10query10 OR \10^^^^ for kites and PRESERVED_PLACEHOLDER_^^^^10\10query10^^^^ for split kites, whereas for W PRESERVED_PLACEHOLDER_^^^^10\10\10^^^^ they are PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10max_results10^^^^, respectively (&&&^^^^10\10^^^^&&&).

The structure search uses the Grand-canonical Interface Predictor (GRIP), combined with dynamic sampling by MD in a PRESERVED_PLACEHOLDER_^^^^10\10query10^^^^-thick grain-boundary region at PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^–PRESERVED_PLACEHOLDER_^^^^10\10max_results10^^^^ for up to PRESERVED_PLACEHOLDER_^^^^10\10query10^^^^, followed by PRESERVED_PLACEHOLDER_^^^^10\10\10^^^^ conjugate-gradient relaxation. First-order transitions between kite and split-kite phases are observed with open surfaces that act as sources and sinks of atoms. Phase-contrast HRTEM simulations for Nb PRESERVED_PLACEHOLDER_^^^^10\10 OR \10^^^^ show broadened intensity at the kite tips for split-kite structures and lack the rigid-body translation along [^^^^10query10query10\10^^^^] that regular kites exhibit, providing experimentally accessible markers for the quasi-aperiodic pseudocoherent phase (&&&^^^^10\10^^^^&&&).

^^^^10query10^^^^. Quantum and open-system realizations

In one-dimensional aperiodic electronic chains generated by Fibonacci and Thue–Morse sequences, the relevant framework is the Wigner phase-space description under a homogeneous electric field. The one-particle Hamiltonian is

PRESERVED_PLACEHOLDER_^^^^10 OR \10query10^^^^

and the state-resolved Wigner distribution is

PRESERVED_PLACEHOLDER_^^^^10 OR \10\10^^^^

The study defines nonclassicality through the Wigner negativity measure

PRESERVED_PLACEHOLDER_^^^^10 OR \10 OR pseudocoherence aperiodic10^^^^

and transition probability through the Wigner-overlap integral

PRESERVED_PLACEHOLDER_^^^^10 OR \10max_results10^^^^

Localization is quantified via the Husimi-based inverse participation ratio (&&&^^^^10 OR pseudocoherence aperiodic10^^^^&&&).

In this setting, an aperiodic pseudocoherent phase is a field interval between anticrossings where electronic states form localized, quasi-Gaussian Husimi lobes occupying well-defined, non-overlapping phase-space regions, with saturated IPR, relatively small and stable uncertainty products, diminished Wigner negativity, and negligible overlap with neighboring states. For Fibonacci bottom states, the ground-state IPR saturates at approximately PRESERVED_PLACEHOLDER_^^^^10 OR \10query10^^^^ for PRESERVED_PLACEHOLDER_^^^^10 OR \10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ a.u., and the first excited-state IPR saturates at approximately PRESERVED_PLACEHOLDER_^^^^10 OR \10max_results10^^^^ for PRESERVED_PLACEHOLDER_^^^^10 OR \10query10^^^^ a.u. Near the anticrossing at PRESERVED_PLACEHOLDER_^^^^10 OR \10\10^^^^ a.u., nonclassicality increases, the uncertainty product changes rapidly, and PRESERVED_PLACEHOLDER_^^^^10 OR \10 OR \10^^^^ peaks, signaling exit from the pseudocoherent regime (&&&^^^^10 OR pseudocoherence aperiodic10^^^^&&&).

A distinct open-system realization appears in a clean one-dimensional tight-binding lattice with aperiodically phase-modulated Lindblad jump operators,

PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^

PRESERVED_PLACEHOLDER_^^^^10\10query10\10^^^^

PRESERVED_PLACEHOLDER_^^^^10\10query10 OR pseudocoherence aperiodic10^^^^

Here the Hamiltonian eigenstates are extended plane waves, so any localization is driven by engineered dissipation rather than by disorder or a quasiperiodic Hamiltonian potential (&&&^^^^10max_results10^^^^&&&).

The steady state is characterized by the relative entropy of coherence,

PRESERVED_PLACEHOLDER_^^^^10\10query10max_results10^^^^

the purity

PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^

the participation ratio

PRESERVED_PLACEHOLDER_^^^^10\10query10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^

and the two-point coherence

PRESERVED_PLACEHOLDER_^^^^10\10query10max_results10^^^^

The operational definition of the aperiodic pseudocoherent phase is a localized steady state with small PR and finite localization length PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^, together with large PRESERVED_PLACEHOLDER_^^^^10\10query10\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10query10 OR \10^^^^ (&&&^^^^10max_results10^^^^&&&).

The paper gives an explicit parameter regime: for the incommensurate case PRESERVED_PLACEHOLDER_^^^^10\10\10query10^^^^, slow modulation PRESERVED_PLACEHOLDER_^^^^10\10\10\10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10\10 OR pseudocoherence aperiodic10^^^^, and PRESERVED_PLACEHOLDER_^^^^10\10\10max_results10^^^^, one finds PRESERVED_PLACEHOLDER_^^^^10\10\10query10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^, and PRESERVED_PLACEHOLDER_^^^^10\10\10max_results10^^^^. As PRESERVED_PLACEHOLDER_^^^^10\10\10query10^^^^ increases from PRESERVED_PLACEHOLDER_^^^^10\10\10\10^^^^ to PRESERVED_PLACEHOLDER_^^^^10\10\10 OR \10^^^^, coherence and localization collapse, with PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10query10^^^^ changing from PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10\10^^^^ to PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10 OR pseudocoherence aperiodic10^^^^, PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10max_results10^^^^ from PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10query10^^^^ to PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^, and PR from PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10max_results10^^^^ to PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10query10^^^^. A practical criterion stated in the paper is that, at fixed PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10 OR pseudocoherence aperiodic10 OR \10^^^^, the pseudocoherent phase persists for PRESERVED_PLACEHOLDER_^^^^10\10max_results10query10^^^^, whereas for PRESERVED_PLACEHOLDER_^^^^10\10max_results10\10^^^^ the state becomes delocalized and incoherent. In the commensurate case PRESERVED_PLACEHOLDER_^^^^10\10max_results10 OR pseudocoherence aperiodic10^^^^, localization is weaker, with PRESERVED_PLACEHOLDER_^^^^10\10max_results10max_results10^^^^ and lower coherence PRESERVED_PLACEHOLDER_^^^^10\10max_results10query10^^^^–PRESERVED_PLACEHOLDER_^^^^10\10max_results10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ (&&&^^^^10max_results10^^^^&&&).

^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^. Non-normal stochastic pseudo-coherence without oscillators

A further generalization appears in linearly stable overdamped stochastic systems with no intrinsic oscillators. The underlying dynamics is a multivariate Ornstein–Uhlenbeck process,

PRESERVED_PLACEHOLDER_^^^^10\10max_results10max_results10^^^^

with Gaussian white noise covariance PRESERVED_PLACEHOLDER_^^^^10\10max_results10query10^^^^. The defining assumptions are linear stability, PRESERVED_PLACEHOLDER_^^^^10\10max_results10\10^^^^, real eigenvalues only, and non-normality, PRESERVED_PLACEHOLDER_^^^^10\10max_results10 OR \10^^^^. In the reduced two-dimensional reaction subspace, the drift matrix is

PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^

with PRESERVED_PLACEHOLDER_^^^^10\10query10\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10query10 OR pseudocoherence aperiodic10^^^^ controlling non-normality (&&&^^^^10query10^^^^&&&).

Pseudo-coherence here is not a periodic rhythm. It is an intermittent, aperiodic form of collective organization characterized by drifting spectral peaks, finite-time coherent bands, transient growth of Kuramoto-like cluster order parameters, broken time-reversal symmetry, circulating probability currents, and positive entropy production, all without eigenvalue crossings or Hopf bifurcation (&&&^^^^10query10^^^^&&&). The relevant geometric control parameter is

PRESERVED_PLACEHOLDER_^^^^10\10query10max_results10^^^^

with threshold

PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^

Crossing PRESERVED_PLACEHOLDER_^^^^10\10query10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ does not alter eigenvalues; it changes transient amplification geometry (&&&^^^^10query10^^^^&&&).

Irreversibility is quantified through the lagged-covariance imbalance

PRESERVED_PLACEHOLDER_^^^^10\10query10max_results10^^^^

which in the reduced model becomes

PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^

with PRESERVED_PLACEHOLDER_^^^^10\10query10\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10query10 OR \10^^^^. Entropy production is

PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10query10^^^^

so PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10\10^^^^ whenever effective non-normality is nonzero (&&&^^^^10query10^^^^&&&).

The spectral signature is equally distinctive. The matrix power spectral density is

PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10 OR pseudocoherence aperiodic10^^^^

and the reaction-mode spectrum develops a PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10max_results10^^^^ tail in an intermediate asymptotic regime once the system is sufficiently non-normal. Windowed Fourier spectra then show interior peaks that drift over time, while Morlet wavelet scalograms show intermittent bursts concentrated within evolving frequency bands. The literature therefore uses “aperiodic pseudocoherent phase” to identify collective temporal organization generated by non-normal pseudospectral amplification rather than by oscillators, resonance, or classical synchronization (&&&^^^^10query10^^^^&&&).

^^^^10max_results10^^^^. Topological and photonic realizations without periodicity

In aperiodic weak topological superconductors, the key issue is how to diagnose weak topology without a Brillouin torus. The system considered is a two-dimensional class D BdG Hamiltonian realized on an Ammann–Beenker tiling, with on-site and nearest-neighbor terms

PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10query10^^^^

PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^

and real-space particle-hole symmetry. After a basis change that makes the Hamiltonian purely imaginary, the strong Clifford-pseudospectrum invariant is

PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10max_results10^^^^

while the weak real-space invariant along PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10query10^^^^ is

PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10\10^^^^

The associated scattering invariant is

PRESERVED_PLACEHOLDER_^^^^10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10 OR \10^^^^

The paper states that “pseudocoherent” is not standard nomenclature here; the term is only an operational interpretation of coherent-like protected boundary behavior in aperiodic media (&&&^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^&&&).

For a weak phase constructed from PRESERVED_PLACEHOLDER_^^^^10\10max_results10query10^^^^ weakly coupled quasi-one-dimensional strips with PRESERVED_PLACEHOLDER_^^^^10\10max_results10\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10max_results10 OR pseudocoherence aperiodic10^^^^, the strong invariant vanishes, PRESERVED_PLACEHOLDER_^^^^10\10max_results10max_results10^^^^, while the weak invariant is nontrivial, PRESERVED_PLACEHOLDER_^^^^10\10max_results10query10^^^^, with PRESERVED_PLACEHOLDER_^^^^10\10max_results10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ as well. Only the top and bottom boundaries are gapless, and the total conductance is PRESERVED_PLACEHOLDER_^^^^10\10max_results10max_results10^^^^. With disorder PRESERVED_PLACEHOLDER_^^^^10\10max_results10query10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10max_results10\10^^^^, together with random inter-chain couplings in PRESERVED_PLACEHOLDER_^^^^10\10max_results10 OR \10^^^^, the single-edge conductance distribution matches the universal bimodal form reported in the paper, establishing that weak topology and localized zero-energy edge modes persist without translational symmetry (&&&^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^&&&).

A photonic realization appears in aperiodic metalenses made from structurally identical silicon nanocylinders of diameter PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^ and height PRESERVED_PLACEHOLDER_^^^^10\10query10\10^^^^ on fused silica. The phase is controlled solely through local period PRESERVED_PLACEHOLDER_^^^^10\10query10 OR pseudocoherence aperiodic10^^^^, with

PRESERVED_PLACEHOLDER_^^^^10\10query10max_results10^^^^

At PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^, FDTD shows a smooth monotonic PRESERVED_PLACEHOLDER_^^^^10\10query10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ with full PRESERVED_PLACEHOLDER_^^^^10\10query10max_results10^^^^ coverage as PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^ varies from PRESERVED_PLACEHOLDER_^^^^10\10query10\10^^^^ to PRESERVED_PLACEHOLDER_^^^^10\10query10 OR \10^^^^. The target focusing phase is

PRESERVED_PLACEHOLDER_^^^^10\10\10query10^^^^

and the achromatic condition requires the phase gradient to scale linearly with frequency at every radius (&&&^^^^10max_results10^^^^&&&).

The decisive mechanism is a linear effective-index scaling with fill factor,

PRESERVED_PLACEHOLDER_^^^^10\10\10\10^^^^

with PRESERVED_PLACEHOLDER_^^^^10\10\10 OR pseudocoherence aperiodic10^^^^. Because PRESERVED_PLACEHOLDER_^^^^10\10\10max_results10^^^^ is effectively invariant over PRESERVED_PLACEHOLDER_^^^^10\10\10query10^^^^–PRESERVED_PLACEHOLDER_^^^^10\10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^, the phase-gradient scaling remains nearly ideal across the aperture. Supplement S^^^^10\10^^^^ reports deviation from the ideal achromatic condition of approximately PRESERVED_PLACEHOLDER_^^^^10\10\10max_results10^^^^ across PRESERVED_PLACEHOLDER_^^^^10\10\10query10^^^^–PRESERVED_PLACEHOLDER_^^^^10\10\10\10 specifically PRESERVED_PLACEHOLDER_^^^^10\10\10 OR \10^^^^ at PRESERVED_PLACEHOLDER_^^^^10\10 OR \10query10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10 OR \10\10^^^^ at PRESERVED_PLACEHOLDER_^^^^10\10 OR \10 OR pseudocoherence aperiodic10^^^^ relative to PRESERVED_PLACEHOLDER_^^^^10\10 OR \10max_results10^^^^. This coherent gradient scaling in a globally non-periodic structure is what the paper calls an aperiodic pseudocoherent phase (&&&^^^^10max_results10^^^^&&&).

The resulting performance is quantified for two numerical apertures. For moderate NA PRESERVED_PLACEHOLDER_^^^^10\10 OR \10query10^^^^, the longitudinal chromatic focal-shift RMSE over PRESERVED_PLACEHOLDER_^^^^10\10 OR \10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^–PRESERVED_PLACEHOLDER_^^^^10\10 OR \10max_results10^^^^ is reduced from PRESERVED_PLACEHOLDER_^^^^10\10 OR \10query10^^^^ in a conventional geometry-variant design to PRESERVED_PLACEHOLDER_^^^^10\10 OR \10\10^^^^ in the aperiodic identical-rod design, a reduction of nearly PRESERVED_PLACEHOLDER_^^^^10\10 OR \10 OR \10^^^^. The efficiency variation RMSE is PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10query10^^^^ versus PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10\10^^^^, and the focal-spot FWHM at PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10 OR pseudocoherence aperiodic10^^^^ is PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10max_results10^^^^, only PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10query10^^^^ above the diffraction limit of PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^. At NA PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10max_results10^^^^, focal-length RMSE is comparable to the conventional case, but the average FWHM remains smaller, PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10query10^^^^ versus PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10\10^^^^, with smoother spectral efficiency (&&&^^^^10max_results10^^^^&&&).

^^^^10query10^^^^. Limits, ambiguities, and open questions

The literature treats aperiodic pseudocoherence as a useful but non-universal descriptor, and that fact defines several open issues. One ambiguity concerns terminology itself. In the topological-superconductor context, the term is explicitly nonstandard, whereas in phonon transport, open-system localization, stochastic dynamics, and photonics it is introduced as an operational regime with directly specified diagnostics (&&&^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^&&&, &&&^^^^10query10^^^^&&&, &&&^^^^10max_results10^^^^&&&, &&&^^^^10query10^^^^&&&, &&&^^^^10max_results10^^^^&&&). This suggests that comparisons across fields must be made through observables rather than through a presumed shared microscopic definition.

Model limitations are equally prominent. The phonon study uses a scaled Lennard-Jones solid argon model at nominal PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10query10 OR \10^^^^, injects only coherent LA phonons along [^^^^10\10query10query10^^^^], and does not compute PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10\10query10^^^^ directly, so finite-temperature anharmonic decoherence, TA modes, oblique incidence, and explicit coherence-function measurement remain open problems (&&&^^^^10query10^^^^&&&). The grain-boundary study is limited by DFT-accessible reconstruction sizes, potential sensitivity, and focus on PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10\10\10^^^^ symmetric tilt boundaries; the true DFT ground state may require larger repeats, and the role of alloying, interstitials, and finite-temperature free energies remains unresolved (&&&^^^^10\10^^^^&&&).

In the phase-space electronic problem, the pseudocoherent regime is empirical, defined through Wigner negativity, IPR, uncertainty products, and anticrossings in finite chains with PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10\10 OR pseudocoherence aperiodic10^^^^, PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10\10max_results10^^^^, and PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10\10query10^^^^ wells, rather than by a closed-form phase boundary (&&&^^^^10 OR pseudocoherence aperiodic10^^^^&&&). In dissipative localization, analytical control of the Liouvillian spectrum and exact localization length PRESERVED_PLACEHOLDER_^^^^10 OR pseudocoherence aperiodic10\10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^ remains challenging, and extension beyond the single-particle setting to interacting many-body systems is open (&&&^^^^10max_results10^^^^&&&). In the non-normal stochastic theory, the pseudo-critical transition is sharply characterized in the reduced model, but broader questions remain about rigorous bounds on observables in high-dimensional applications and the relation to empirical phenomena in neural, ecological, climate, and fluid systems (&&&^^^^10query10^^^^&&&).

The photonic and topological examples introduce further practical limits. In metalenses, high-NA performance is constrained by nonlocal coupling, angular acceptance, and standard off-axis aberrations, while in aperiodic topological superconductors the real-space invariants require a well-conditioned localizer and preserved class D particle-hole symmetry (&&&^^^^10max_results10^^^^&&&, &&&^^^^10(Maranets et al., 2024) OR (Chen et al., 27 Jan 2025) OR (Spisak et al., 2012) OR (Roy et al., 18 Jun 2025) OR (Troude et al., 7 Mar 2026) OR (Fulga et al., 2015) OR (Moreno et al., 7 Apr 2026)10^^^^&&&). A plausible implication is that future work will continue to differentiate cases where aperiodicity merely broadens or perturbs coherent phenomena from cases where aperiodicity itself is the enabling resource.

Taken together, the literature presents the aperiodic pseudocoherent phase not as a single canonical phase of matter, but as a recurrent structural and dynamical motif: coherence-like behavior that remains diagnostically meaningful even when strict periodicity, translation symmetry, commensurability, or oscillator-based synchronization is absent.