Physical Sciences

Abstract: The goal of this work is simple and traditional: to provide a formalization and generalization of the notion of entropy in the spirit of Carnot. The tools we employ to achieve our objective are quite novel however. We have no need for state spaces, paths in state spaces, path integration,cycles in state spaces, systems in equilibrium, quasi-static transformations, the usual dynamical notion of reversibility, or even, the first law of thermodynamics. Instead, we rely on a primitive notion of process and the effects a process has on a system, typically, on the surroundings of a system. Carnot cycles are replaced in our framework by bithermal processes. We recover that the efficiency of a bithermal process is maximal when the process is reversible and maximal efficiency is determined by the temperature of the reservoirs. We also derive that all cyclic processes are dissipative, at best isentropic. The formalism assigns a degree of irreversibility to every process. The argument we present is quite general. It does not assume systems to be at equilibrium; it simply requires systems to have a state at the beginning and at the end of the process.

Abstract: While the existential math for non-smooth weak solutions in ideal fluid bounds has historically been demonstrated via chaotic convex integration loops, an explicit, constructive analytical closed-form derivation governing singular hypersonic compressible magnetohydrodynamic (MHD) shear-strip boundary layers has been entirely absent from the scientific literature. We resolve this profound deficit by constructing a definitive, fully explicit analytical closed-form architecture that stabilizes coupled multi-field transport gradients. By mapping the coupled density-weighted velocity and magnetic induction fields through a gauge-covariant intrinsic tensor Friedrichs mollifier matted with a multi-dimensional singular Riesz transform vector mapping , the high-frequency ultraviolet loop interactions are structurally neutralized. This process forces the spontaneous, inner generation of a fourth-order biharmonic hydro-magnetic dissipative spectral cage , avoiding any ad-hoc parameters or empirical truncations. Under the absolute topological control of the Aubin-Lions Compactness Embedding Lemma, the continuous functional scaffold contracts uniformly to zero (Lc → 0), establishing strong Cauchy convergence toward rough local solutions of the classical compressible MHD field equations. We prove that at this sharp local limit, the induced biharmonic dissipation matrix collapses into a stable, non-vanishing strictly positive local metric invariant ( > 0), analytically isolating the anomalous dissipation profile at the exact Onsager Hölder regularity exponent threshold α = 1/3. Furthermore, the predictive validity of this explicit framework is directly verified against empirical astronomical data from the Event Horizon Telescope (EHT) and the Chandra X-ray Observatory for the relativistic accretion disk of M87*, yielding a 100% deterministic predictive matching with a zero boundary layer error profile.

Abstract: Layered transition-metal dichalcogenides provide an ideal platform to explore the competition between charge-density-wave (CDW) order and superconductivity. The 3R-stacked TaSe2 hosts intrinsic CDW and superconductivity without chemical doping, yet systematic high-pressure experimental evidence remains lacking. Here we report on systematic studies of pressure phase diagrams up to 2 GPa in high-quality single crystals of 3R-TaSe2. At ambient pressure, a CDW transition emerges near 100 K, and bulk superconductivity onset occurs at Tc ≈ 2.8 K. Upon hydrostatic compression, the CDW transition temperature is monotonically suppressed, while Tc increases to ~3.1 K at 2 GPa, clearly revealing an antagonistic relationship between these two quantum phases. Hall and thermopower measurements suggest CDW-driven Fermi-surface reconstruction and single-band carrier behaviour. From pressure-dependent transport we construct the P(T) phase diagram, and anisotropic upper-critical-field data demonstrate quasi-two-dimensional superconductivity in bulk 3R-TaSe2. This work offers key experimental insights into competing correlated states in van der Waals layered materials.

Abstract:

The Navarro–Frenk–White (NFW) profile occupies a central role in the description of dark-matter halos. Despite its remarkable empirical and numerical success, the physical origin of its universal inner scaling \[\rho(r)\propto r^{-1}\] remains a subject of ongoing discussion, particularly in connection with the longstanding cusp–core problem. This paper explores the possibility that the inner NFW cusp may admit a geometric interpretation. Motivated by the universal structure of fold caustics in singularity theory, we investigate whether the characteristic inner NFW scaling can be understood as the radial manifestation of a projected fold singularity rather than solely as a consequence of collisionless halo dynamics. A central observation motivates the present work. Ideal fold caustics are singular only in idealized mathematical limits in which finite physical scales are removed. Across wave optics, gravitational lensing, collisionless phase-space dynamics, and other physical systems, the observed realization of a caustic is generically regularized by finite intrinsic scales. The observed finite structure is not the exception. The singularity is. This suggests a different perspective on the cusp–core problem. If the NFW cusp represents the ideal mathematical form of an underlying fold-like structure, then the observed finite core may not require a separate explanation. Instead, the core may represent the physically expected realization of that structure, while the perfectly unregularized cusp corresponds to an idealized limiting case. To explore this possibility, we consider a simple fixed-width regularization of the NFW cusp, \(\rho(r)=\frac{\rho_s}{(r/r_s+\epsilon)(1+r/r_s)^2},\) with a representative dimensionless regularization scale of order \(\epsilon\sim0.1\). As a proof-of-concept example, the dwarf galaxy IC2574 is used to test whether such a fixed regularization can reproduce an observed core-dominated system while preserving the characteristic outer behavior of the NFW profile. The purpose of the present work is not to challenge the cosmological success of \(\Lambda\)CDM, modify gravity, or introduce a new dark-matter model. Rather, it investigates whether the universal NFW cusp may admit a complementary geometric interpretation as a fold-caustic structure and whether the observed galactic cores may naturally arise as the physical regularization of that structure.

Abstract: The quantum anomalous Hall effect is ordinarily written on the Brillouin torus of a fixed Hamiltonian. Here the occupied Chern band is held fixed while the state that forms the observation record is allowed to vary independently. Two projectors carry the two roles: an occupied-band projector PX(k) and an observation-response projector R(k, s). On every fixed observation section the construction reduces exactly to the standard theory, σxy = CXe 2 /h. When the observation state contains a periodic coordinate φ, the response bundle can carry a mixed first Chern number ν_iφ on \( S^1_{k_i}\times S^1_{\varphi} \) without changing the physical Chern number. Once the response connection enters the phase-transport law, one closed observation cycle transports Q_i = eν_iφ per equivalent transverse response channel. The integer collapses to ν_iφ = ±1 whenever the mixed response line bundle is topologically primitive; orientation fixes the sign. An elementary nontrivial response cycle therefore carries exactly one charge quantum per channel, while higher integers describe composite topological sectors. A two-level projector realizes the primitive class. The parameter-free relations Q/e = 1, I/(ef) = 1, and eV_H/(hf) = 1 follow, distinct from ordinary Thouless and Brouwer pumping. The experimental question is whether such a quantized, orientation-odd residue survives while the physical QAH Hamiltonian remains fixed.

Abstract: Planck units are fundamental constants of physics. Time, velocity, mass, and gravitational force can be expressed in terms of Planck units. Planck time constitutes the fundamental unit of the numerical order of events unfolding in time-invariant space. There is neither a physical past nor a physical future. Time as duration is an emergent physical quantity that arises only in the process of measurement. Every physical object is an energetic structure of space itself. Gravity is mediated by variations in the energy density of space.

Abstract: TF-RSE-IonCF v3.1 is presented as a unified closed-form model returning the total electron binding energy (TBE) for every neutral atom and ion across the periodic table. For each nuclear charge Z = 1–92 and each electron number N = 1 to Z (charge state q = Z–N = 0 to Z–1), a single formula evaluates the TBE in microseconds. The model is built in two layers. The neutral-anchor layer (TF-RSE v2) extends the Thomas–Fermi power law with a Dirac relativistic factor, a trigonometric shell-oscillation factor and a piecewise light-element correction (0.40% mean error on the 92 neutral atoms, degrading smoothly to 3.6% out-of-sample on Z = 93–103). The ionic layer decomposes the ionic TBE into shell-resolved contributions anchored at four limits — the hydrogenic Dirac value at N = 1, a screened helium-like value at N = 2, a screened neon-like value at N = 10, and the neutral-anchor value at N = Z — blended by power laws, with three ablation-validated upgrades (effective charge Z_eff(Z, N) in the relativistic factor; electron-number shell-sawtooth S(Z, N); shell-closure bumps at N = 10, 18, 36, 54, 86). Fitted to 4278 NIST reference points with the neutral layer frozen, the core model achieves a mean relative error of 0.258% (median 0.153%, maximum 4.63%), with 97.7% of points below 1% and 100% below 5%. Version 3.1 replaces the v3.0 runtime heuristic with a three-tier physical-consistency safeguard: (i) a hard physical cap E(Z, N) ≤ E_v2(Z) enforcing positive binding of the last electron; (ii) a boundary-layer reconstruction (N > Z–5) with an increasing ionization-energy trend (IP_k = 1.8·IP_{k−1}) that matches the experimental stripping sequence; and (iii) a minimal-touch projection (0.5 eV floor) for the interior oscillation violations, without heuristic cascade. The safeguard modifies 582 of 4278 points (13.6%) by no more than 1.14 keV (Lu, N = 63; deep-core oscillation link), leaves the statistical accuracy of the core grid unchanged (0.258%), and preserves the exactness of the N = 1 and N = Z anchors for all elements and of the N = 2, N = 10 anchors except at five clamp-degenerate points (Li N = 2; Mg, Al, Si, P N = 10), where the replacement value is more accurate than the clamped anchor (e.g. Si N = 10: +1.25% → +0.03%). Zero negative ionization energies and zero non-monotonic elements are enforced on the full grid, with Z_eff ≤ Z hard-constrained.

Abstract: This work investigates the perihelion advance of Mercury in a generalized description of gravity in which the speed of light depends on the cumulative gravitational potential of the Universe. Starting from a relativistic Lagrangian with c = c(Φ), the orbital equations are derived including the effect of the retarded gravitational potential. The resulting precession separates into intrinsic and retarded-potential contributions. For the particular Lagrangian formulation adopted, their combination reproduces the first-order General Relativity prediction. A generalized two-parameter family of Lagrangians is then introduced to examine the dependence of the precession on the underlying Lagrangian structure. While the adopted formulation reproduces the observed perihelion advance, alternative structures considered within this framework do not straightforwardly reproduce the observed value.

Abstract: Arterial pressure waveforms are shaped by vascular biomechanics and wave reflection, processes that become increasingly relevant in hypertension and vascular aging. This study aimed to investigate how heart rate (pump frequency) and baseline diastolic pressure modulate pressure waveform morphology and wave reflection in a controlled biophysical cardiovascular model. A biophysical model mimicking the aorta and its major branches was equipped with custom pressure sensors, enabling simultaneous recordings at multiple locations along the system. Measurements were performed at diastolic pressures of 60, 80, 100, and 120 mmHg and pump frequencies ranging from 24 to 120 bpm. The downstream tube length and reflection site configuration were systematically varied to evaluate the effects of distal boundary conditions. Increasing pump frequency resulted in higher pressure amplitudes and a steeper systolic upstroke (increased dp/dt). Peak pressure amplitude increased from approximately 105 mmHg at 24 bpm to 145 mmHg at 120 bpm. At higher frequencies and longer reflection pathways, forward and reflected wave components became more clearly separated, whereas branched distal networks produced complex multi-reflection patterns. Wave reflection in this experimental system is strongly dependent on heart rate, diastolic pressure, and distal tube configuration, providing mechanical insight into central pressure augmentation in hypertensive and aging-related conditions.

Abstract: We present the Lucron Model, a physical framework in which fundamental reality is a network of identical units called Lucrons. Each Lucron possesses intrinsic minimal properties---length, time, mass, charge, and action---that are the smallest non-zero values in every metric. The apparent absence of intrinsic properties at ordinary scales is an effect of scale, not a metaphysical absence. We show that a single-scale Lucron with scale \( \ell_L \) and coupling \( g_* \), together with the permutation symmetry \( S_3 \) of the bound-state ``candado,'' reproduces the following without fine-tuning: (i) the Planck scale, \( G = \ell_L^2 \); (ii) gauge unification with \( \alpha_{\rm GUT} = g_*^2/(4\pi) \approx 1/24 \); (iii) the Weinberg angle \( \sin^2\theta_W = 3/8 \) at the GUT scale; (iv) the electroweak hierarchy \( v/m_P = e^{-2\pi^2/g_*^2} \); (v) the electron hierarchy \( m_e/v = e^{-7/g_*^2} \); (vi) the Koide formula with \( \theta = 2/9 \) rad; (vii) the Cabibbo angle \( \sin\theta_C = \sqrt{m_d/m_s} \); (viii) the PMNS angles \( \theta_{23} \approx 45° \) and \( \theta_{13} \approx 8.5° \); (ix) the cosmological constant \( \Lambda \ell_P^2 = N^{-2/3} \approx 10^{-122} \) for \( N \sim 10^{183} \). We also identify where the model fails quantitatively: the CKM angle \( \theta_{23} \) (factor \( 1.4 \)), the CP phase \( \delta_{\rm CP} \) (factor \( 2.4 \)), and the PMNS angle \( \theta_{12} \) (factor \( 1.25 \)). These failures point to unresolved geometric details of the candado. We further show that the volume law \( V = v_0 \sum_i 1/c_i \) reproduces \( \Lambda \)CDM at the cosmological level, yielding \( \chi^2 = 12.8 \) against DESI DR2 BAO, identical to \( \Lambda \)CDM. With two fundamental parameters \( (\ell_L, g_*) \) and one symmetry (\( S_3 \)), the model accounts for approximately twenty dimensionless constants, a significant reduction over the Standard Model's nineteen free parameters.

Abstract: A lattice of locally coupled quantum oscillators is proposed as a candidate microscopic substrate for emergent spacetime. Each oscillator is treated as an elementary substrate unit (ESU): a minimal quantum degree of freedom whose local dynamics, correlations, and collective organization may underlie higher-level physical structure. For the minimal fixed cubic-lattice realization, the exact collective-mode dispersion relation is derived and shown to approach relativistic propagation at long wavelength, with limiting speed c_eff = a√(K/M). The leading lattice correction is anisotropic and quadratic in momentum, providing both a direct Lorentz-violation signature and an intrinsic ultraviolet cutoff. For a Planck-scale lattice spacing, the corresponding Lorentz-violation scale is E_LV ≈ ħc/ℓ_P ≈ 1.22×10^19 GeV, roughly eight orders of magnitude above the conservative quadratic-order lower bound inferred from GRB 090510 for the bosonic collective sector constructed here. Comparison with strings and matrix models, spin networks, causal sets, causal dynamical triangulations, and quantum graphity shows that the oscillator substrate trades background independence at this stage for calculability, explicit microscopic dynamics, and quantitative falsifiability. The framework further treats possible chains, loops, defects, networks, fields, and particles as structures to be generated—or excluded—by the substrate dynamics rather than inserted independently. Thus the model supplies a concrete first-stage substrate hypothesis and a dynamical selection principle. The calculation establishes kinematic continuum-like emergence on a fixed lattice; dynamical connectivity, a universal causal structure, a massless spin-2 gravitational sector with the required gauge structure, and viable chiral matter remain decisive tests of the broader hypothesis.

Abstract: Scalar magnetic descriptors are useful only when omitted information does not change the selected response beyond the required accuracy. We combine a literature audit with one-dimensional and three-dimensional tests. In an eight-material historical ledger, room-temperature labels coincide with magnetic-order eligibility, preventing identification of an additional descriptor effect. A published boundary benchmark resolves size dependence at fixed material descriptor. In a finite chiral-spin model with complete dipolar interactions, four near-matched-radius pairs at the same initial field and pairwise equal thickness have different field-step responses. First-order source-state susceptibility fails its declared twofold-improvement criterion. A disclosed second-order extension, derived from equilibrium response, is tested on eight new settings. Its root-mean-square radius error is 0.000259 lattice spacings, compared with 0.006649 for the best tested scalar normalization and 0.002221 for a multivariable baseline. Independent dipole calculations, stationarity refinement, contour checks and isolated prediction replay establish the numerical scope. The method requires the complete source configuration and Hamiltonian derivatives. The results demonstrate a target-specific audit and mechanistic restoration within declared finite geometries; they do not establish a low-cost material-only descriptor, calibrated lifetime or universal material optimum.

Abstract: This article presents the authors’ proposed theory on the formation of cavitation voids in the false vacuum during the inflationary phase of the Big Bang. It examines the structure of, and the physical processes occurring in, the transition region at the boundary between the false and physical vacuums. It has been demonstrated that, during the formation of physical vacuum voids (regions in which the Λ-term, a constant in the equations of general relativity, goes to zero), conditions arise in the transition region between the false vacuum and the physical vacuum that allow for the formation of a narrow layer of matter. This layer may be the precursor to the bridges observed in the large-scale lattice structure of the universe. The mechanism of bubble expansion during the inflationary phase of the universe is examined. It has been demonstrated that the inflationary phase of expansion passes into the standard Big Bang model due to the conversion of the false vacuum into matter at the boundaries of the bubbles. This model does not require the universe to undergo an extremely rapid expansion to explain the quasi-homogeneous and isotropic distribution of matter and the cosmic microwave background. Furthermore, it can account for the formation of the large-scale lattice structure of the universe. The estimated radii of cosmic voids are consistent with observational data.

Abstract: A recent similarity solution of the two-dimensional unsteady laminar boundary-layer equations [B. H. Sun, Phys. Fluids 36, 083616 (2024)], expressed in terms of Kummer functions, was proposed as the solution for a semi-infinite flat plate impulsively set into motion, but its boundary conditions cannot be satisfied at any finite time. We show that the solution is, in fact, a Burgers–Townsend strained shear layer superposed on the decaying, zero-pressure-gradient straining flow u = x/t, v = −y/t. For this class the nonlinear terms cancel identically, and the shear component obeys a linear equation that a Lundgren-type change of variables reduces to the heat equation; the Kummer functions are the fractional repeated-integral error functions i^2/3 erfc. The two forms of the published solution are copies of a single flat-wall solution related by Prandtl’s transposition theorem, and the flat-wall representative is an exact solution of the full Navier–Stokes equations, not merely of the boundary-layer equations. The correct physical setting is a sheet stretching at the decaying rate 1/(t + t₀), the S = −1 point of the Wang–Andersson family of unsteady stretching-sheet flows. Within this setting we solve three well-posed initial–boundary-value problems in closed form: (1) impulsive translation of the stretching sheet, whose wall shear stress crosses over from the Rayleigh law to −1.370 ρU_s√ν/t; (2) the decay of a pre-existing shear layer, governed by the exact invariant \( $s^3\!\int_0^\infty Y u'\ \)dY and (3) arbitrary wall histories, including power-law ramps with wall-shear factor √3 Γ(p + 1)/Γ(p + 1/2) and an oscillating sheet that relaxes to the classical Stokes layer. All three problems are two-dimensional – both velocity components are nonzero and the field depends on x – in contrast to Stokes’ first and second problems, which are one-dimensional and are recovered only as limits. The published Kummer solution is identified as the self-similar late-time attractor of problem (1). The unsteady Blasius problem for a finite plate remains open.

Abstract: Lithium filaments penetrate solid electrolytes by wedging open pre-existing flaws, and the field has consequently treated the problem as one of fracture mechanics: raise the toughness, close the crack, and penetration is delayed. Here we show that mechanical stress acts on filament growth through a second channel that has been overlooked, and that this channel is exponential rather than linear. Because ion migration in a solid requires a local dilation of the lattice, the ionic conductivity of every solid electrolyte depends on stress through an activation volume ΔV∗, a quantity measured to span 0.6–25 cm³ mol⁻¹ across sulfide and polymer conductors, and never measured at all for the oxide garnets. The stress field of a pressurised filament therefore reshapes the conductivity of the electrolyte that feeds it. Using coupled finite-element simulations of a sharp, Li-filled edge flaw we find that the sign of this feedback is set by the near-tip stress state, and that in an unstressed electrolyte it is adverse: the crack tip sits in tension, transport there is enhanced, and the supply of lithium to the tip rises by 12–22% at the activation volumes measured for sulfide argyrodites (0.58–1.08 cm³ mol⁻¹) and by a factor of 6.6 at the value measured for PEO–LiTFSI (25 cm³ mol⁻¹), even as the supply to the crack flanks falls. We derive a two-constant scaling law, σ ^tip_h = κ (p − λσr) with κ =3.00 and λ = 1.195, which locates a narrow but real operating window, σ_r < p < λσ_r, in which the flaw is open yet its tip remains compressed. Inside that window the feedback reverses sign, giving a predicted 3.6-fold gain in critical current density of which about half is fracture-mechanical in origin. Because the exponent scales with ΔV∗, the effect is a 10–35% correction for stiff inorganic conductors but the dominant term for polymer and polymer-in-ceramic electrolytes, where Γ ≈ 7at crack-tip stresses. The framework predicts that critical current density should correlate negatively with ΔV∗ in unstressed cells, and identifies a concrete architecture—a compressively prestressed gating skin—that converts a known linear benefit into an exponential one.

Abstract: Regional WRF-Chem dust simulations may require continuous integrations when state continuity is scientifically important, yet extended limited-area runs can depart dynamically from the driving analysis. We use a two-tier controlled design that separates dynamical attribution from aerosol transport response: a continuous chemistry-free FREE-FDDA pair spanning 104 days (15 March-27 June 2024) isolates the meteorological response to spectral nudging of the horizontal wind, and paired event-scale WRF-Chem simulations of two Saharan outbreaks test whether that response propagates into dust prediction skill. The meteorological response is variable-selective and strongly non-monotonic. Sea-level pressure improves most (mean daily RMSE 2.08 to 1.53 hPa) as an indirect response of the mass field, 10 m wind speed improves coherently but less (2.50 to 2.20 m s-1), and 2 m temperature does not improve (2.05 to 2.15 °C); the benefit recurs episodically as synoptic regimes evolve rather than following a universal lead-time threshold. Aerosol transport, evaluated against AERONET Version 3 Level 2.0 AOD550 with Level 1.0 as a coverage-sensitivity check, is conditional. In the weakly forced June case all seven sites improve simultaneously in RMSE, correlation and variability amplitude (station-mean RMSE 0.457 to 0.235; correlation 0.778 to 0.881). In March, RMSE decreases at all eight sites but mean correlation is unchanged, indicating amplitude correction without uniform improvement of plume timing. Spectral nudging therefore acts as a process-targeted constraint on circulation and transport consistency whose aerosol benefit is greatest when transport error dominates; it does not replace constraints on dust emission, optical properties or removal.

Abstract: Physical--observation dual-axis (PODA) Theory holds that physical reality is realized jointly through physical state and observation state. A change of observation state can therefore change the realized physical fact even when the physical state is held fixed. We develop the phase law of this PODA structure. A coherent readout of the joint observation law gives a smooth scalar complex response \( g \). Wherever this response is nonzero, differentiating its polar decomposition gives the observed phase rate and the one-form \( A_g=\operatorname{Im}(\mathrm d g/g) \). A transported phase with the same increments differs from the observed phase by a constant reference factor. We derive its differential equation and unique initial-value solution, and show that the equation is horizontal lifting for the unique response-compatible connection on the trivial principal \( \mathrm U(1) \) phase bundle. We call the resulting equation \( \widetilde c^{\,*}\Omega_g=0 \), the Phase Transport Fundamental Equation (PTFE), the first fundamental equation of observation space in PODA Theory. Its group-valued, covariant-derivative and continuous real-phase forms express one transport law: a path and an initial phase determine a unique horizontal lift. Integration gives finite phase comparison, composition along successive paths, and inverse transport between two observation states of the same physical source. Positive amplitude rescaling leaves the law unchanged; synchronized changes of response and fiber coordinates preserve its covariant form. The induced connection is flat and has trivial circular holonomy, although a continuously unwrapped phase may wind. We distinguish the full response \( \mathsf G_0, \) the phase-transport structure \( \mathsf G_1 \), and the task quotient \( \mathsf G_2 \). At the last level, maximal invariants describe the quotient, and a target can be recovered exactly when it is constant on each fiber of the quotient observation. Phase transport and the information retained after a task reduction thus follow from the same joint observation law.

Abstract: This paper formulates a thermodynamically consistent finite-strain Maxwell-Glen model for glaciological applications in a Lagrangian setting. A multiplicative decomposition of the deformation gradient separates the elastic and viscous mappings, while a trace-free material velocity gradient describes isochoric creep. The elastic state is represented by a logarithmic strain, and its associated material stress drives a temperature-dependent Glen law. Under the isotropic constitutive assumptions, commutation reduces the multiplicative metric update to an additive elastic-viscous corrector. The exponential map is evaluated by a Cayley-Hamilton reduction for generally nonsymmetric arguments, and a consistent linearisation supplies the material tangent. Homogeneous simple shear verifies agreement with spatial Glen flow under non-coaxial deformation. A self-weighted plane-strain column confirms preservation of the incremental viscous Jacobian to the solver tolerance. For a floating shelf, depth-dependent viscous resistance reverses the shelf-edge bending relative to uniform viscosity.

Abstract: We propose a fast mapped spectral method for a class of variable-order fractional partial differential equations posed on the real line. The main difficulty arises from the spatially dependent fractional order of the operator, which prevents the direct diagonalisation techniques available for constant-order fractional Laplacians and leads, in general, to expensive nonlocal discretisations. To overcome this limitation, we introduce an operator-interpolation strategy in which the variable-order fractional Laplacian is approximated by a finite combination of constant-order fractional operators evaluated at suitably chosen interpolation nodes in the fractional-order variable. Each constant-order contribution is treated through a Fourier-like mapped Chebyshev representation on R, allowing the corresponding nonlocal operator to be evaluated efficiently in spectral space. Chebyshev interpolation with respect to the fractional order is employed to obtain an accurate approximation over a prescribed interval s(x) ∈ [s_min, s_max]. This construction separates the difficulties associated with spatial unboundedness and variable nonlocality, and leads to an implementation whose cost is governed by a small number of fast constant-order operator evaluations. The spatial discretisation is combined with a pseudospectral treatment of nonlinear terms, avoiding the explicit construction of dense high-order interaction tensors. Approximation properties are analysed by separating the error due to interpolation in the fractional order from the mapped spectral discretisation error in space. The proposed approach is first validated on variable-order fractional problems with prescribed or manufactured solutions. It is then applied to stationary variable-order fractional Allen–Cahn equations in heterogeneous media, with particular attention to the influence of spatial variations of the fractional order on interface profiles, asymmetry, and far-field decay. Numerical experiments are designed to assess accuracy, convergence, computational complexity, and robustness with respect to both the spectral resolution and the variation of the fractional order.

Abstract: The Spiral Structure Model (SSM) is a phenomenological model in Bohr’s tradition: restricted to kinematics and geometry, it trades dynamical depth for ontological clarity and experimental exposure. Three postulates suffice: (I) the photon is a pointlike energy quantum in uniform helical motion, axial and circulation speeds both equal to c; (II) physical space is an elastic, relativistically covariant medium sustaining vortex excitations; (III) the electron is a stable topological vortex defect of that medium. These yield, every step labeled: the photon’s circulation angular momentum mcr_γ = ℏ as a consistency theorem identified with its spin; logarithmic vortex confinement with zero free parameters; the two polarization states and no longitudinal mode; orbital angular momentum as an ensemble property; the electron’s 720-degree periodicity and spin ℏ/2 from π1(SO(3)) = Z₂; the fine-structure constant as a ratio of structural scales; the free radiation field as an ensemble statistical moment; a phase-coupling account of two-slit interference; and a double-helix ontology of entangled pairs reproducing the CHSH value 2√2. Three falsifiable discriminants are stated with kill criteria: a wavelength-locked on-axis detection void, a spin-dependent electron form-factor window near Q = mc, and a circulation-periodic component in entangled-pair coincidence timing.