Physical Sciences

Abstract: This paper presents PySkyLumos, an open-source Python framework for simulating skylight polarization measurements and metadata. It implements the Rayleigh, Berry, and Pan models, capturing both classical scattering and elevation-expressed singularity dynamics. Efficiency benchmarks show a fivefold overall speedup and up to two orders of magnitude faster Stokes-parameter image formation compared to existing tools. Validation against real polarization camera data confirms its accuracy and suitability for bio-inspired computer vision, navigation, robotics, and sensing applications.

Abstract: Investment decisions during an energy crisis are usually driven by capital cost, levelized cost, carbon intensity and nominal conversion efficiency. These indicators are necessary but incomplete, because every energy technology also consumes energy for extraction, manufacture, conversion, storage, control and replacement. This review uses energy returned on energy invested (EROEI) as a common physical basis and follows the energy chain from conventional oil and gas to photovoltaics, wind, hydropower, hydrogen and electrical storage. It separates four quantities that are often conflated: source EROEI, round-trip efficiency, manufacturing energy per unit of storage capacity, and lifetime energy stored on energy invested (ESOI). For lithium-ion batteries, a widely cited 2017 review estimated 350 to 650 MJ of manufacturing energy per kWh of capacity, about 97 to 181 kWh, before the energy embodied in upstream materials is added; newer gigafactory data show large reductions but do not yet support a single universal value. A simple combination rule shows that the complete-chain return of stored electricity is always lower than both the operational return and the ESOI of the storage device, so that cycle life can matter as much as chemistry. At small energy-harvesting scales, sensing and control can consume a material fraction of the captured energy; Energy Returned on Invested Energy for Embodied Intelligence (EROIE) extends the same accounting to Reflex Policy control architectures. The paper concludes with a minimum energetic due-diligence protocol to be applied before major energy investments.

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: We develop an operational framework for testing small frequency-dependent departures from standard electromagnetic vacuum propagation. The vacuum is treated phenomenologically as a causal and passive linear-response system whose electromagnetic properties may exhibit weak spectral structure while preserving the reference vacuum impedance and recovering standard Maxwell behaviour at high frequency. A central feature of the framework is the separation between the hypothesized vacuum response and the spectral sensitivity of the measuring apparatus. Each experiment is represented by a calibrated sensitivity function, while its response is summarized by a corresponding band-averaged observable. This construction provides a common metrological interface through which different precision experiments can constrain the same underlying response model while retaining their instrument-specific calibration and uncertainty information. The proposed reporting scheme is intended to support reproducible cross-platform comparison and subsequent re-analysis for alternative admissible response models. First-order mappings are developed for resonant cavities, interferometric phase measurements, group delay, radiometry, and Casimir-type observables. For Casimir measurements, the analytically continued response and the experiment-dependent sensitivity are retained explicitly. A published optical-resonator constraint on the isotropic photon sector of the Standard-Model Extension is conditionally recast within the present framework, yielding an experimentally anchored sensitivity at approximately the ten-to-the-minus-eight level. The result is statistically consistent with zero and does not constitute evidence for a nontrivial vacuum response or a model-independent constraint on arbitrary dispersion. The framework is deliberately phenomenological rather than microscopic. It does not derive the vacuum degrees of freedom, the electromagnetic constants, the numerical value of the speed of light, or Lorentz symmetry. Its contribution is a causal, reproducible, and falsifiable interface between hypothetical vacuum-response models and precision photonic experiments, together with a candidate standardized metrological reporting structure for future cross-platform tests.

Abstract: The parameters that determine practical applications oxide ferrimagnets are the values of saturation magnetizations and the magnitudes and signs of the magnetocrystalline anisotropy fields. Measuring magnetocrystalline anisotropy fields using ferromagnetic resonance (FMR) on single-crystal or textured polycrystalline hexaferrite samples is straightforward. However, these methods are not applicable for hexaferrite which typically synthesized as macroscopically magnetically isotropic polycrystalline or powder samples. This paper describes two methods to calculating the components of the permeability tensor of polycrystalline (or powder) hexaferrites in the independent grain approach. The proposed method is illustrated results of a FMR study of Y-type polycrystalline Ba₂Ni₂–_xCu_xFe₁₂O₂₂ (0.0 ≤ x ≤ 2.0) system, synthesized using a standard ceramic method. The measurements were performed on spherical samples in the frequency range of 37–50 GHz. The metod allows one to evaluate the magnitudes of the magnetocrystalline anisotropy fields and magnetomechanical ratios of both the target Y-phase and impurity phases. The phase composition estimates obtained from FMR data were compared with the results of X-ray phase analysis. It has been shown that up to a composition with x = 1.8, these materials are the ferroxplana-type hexaferrites. Hexaferrite Ba₂Cu₂Fe₁₂O₂₂ exhibits easy-axis type of the magnetocrystalline anisotropy.

Abstract: Energy research and automotive engineering often optimize individual components while leaving fixed system topology unchanged. This Communication argues that protected, real-time topology change within a Reflex-Policy architecture can act as a system-integration multiplier across energy harvesting, storage and use. Historical transitions from carburetion to electronic injection, turbocharging and regenerative braking show how sensing, actuation and control create value at the system level. Two illustrative cases quantify the opportunity. In a 3 x 3 photovoltaic proof-of-concept under partial and moving shadow, protected substring bypass followed by MPPT increased aggregate static power by 73.2% and representative dynamic energy by 31.5% in behavioural simulation. In a Model 3-class LFP battery extrapolation, selective non-dissipative charge injection could reduce correction time by 4.2-8.3 times and intentional balancing heat by approximately 89% relative to 240 mA passive balancing. These figures are simulations and engineering extrapolations, not final hardware measurements. The same local-evidence, bounded-action and supervisory-policy contract can scale from cells and micro-harvesters to vehicles, buildings, microgrids and utility assets.

Abstract: Artificial intelligence (AI) is increasingly applied to laboratory medicine as health systems confront infectious diseases, antimicrobial resistance, chronic disease burdens, and increasingly complex diagnostic requirements. This critical narrative review examines AI-enabled laboratory technologies for precision diagnostics and global health surveillance. A structured literature search was conducted across major biomedical, multidisciplinary, engineering, and health-informatics databases, supported by supplementary citation searching. Included evidence addressed diagnostic accuracy, laboratory workflow, digital pathology, microscopy, molecular diagnostics, biosensors, surveillance, implementation, governance, and equity. Findings indicate that AI can improve pattern recognition, automate selected laboratory processes, support molecular interpretation, strengthen digital pathology, and enhance outbreak detection through integrated clinical, genomic, and epidemiological data. However, strong technical performance does not consistently translate into clinical validity, external generalisability, improved patient outcomes, or equitable implementation. Major limitations include retrospective study designs, weak external validation, algorithmic bias, cybersecurity risks, fragmented interoperability, workforce shortages, infrastructure constraints, and underrepresentation of low-resource settings. AI should therefore augment, rather than replace, qualified laboratory professionals. Sustainable implementation requires representative datasets, rigorous validation, interoperable information systems, cybersecurity safeguards, skilled personnel, regulatory oversight, and context-sensitive infrastructure. When these conditions are met, AI-enabled laboratories can strengthen precision diagnostics, surveillance capacity, and global health decision-making across diverse health system contexts.

Abstract: The unsteady compressible laminar boundary layer is the natural setting in which the earliest, pre-transitional history of a high-speed viscous flow is written, yet almost no exact solutions of the two-dimensional unsteady compressible boundary-layer equations are known. We extend a recently proposed incompressible similarity transformation [B. H. Sun, Phys. Fluids 36, 083616 (2024)] to compressible flow. Building on the unsteady Howarth--Dorodnitsyn reduction introduced by Stewartson [Q. J. Mech. Appl. Math. 4, 182 (1951), Section 7], and combining it with an undetermined boundary-layer thickness \(\delta(x)\) and a diffusion time \(\tau=C\nu_r t/\delta^{2}\), we show that the compressible mass, momentum and energy equations collapse onto a pair of partial differential equations in two variables (\(\eta\),\(\tau\)) whose coefficients are independent of the streamwise coordinate. Remarkably, the compressible momentum equation differs from its incompressible counterpart only through the replacement 1\(\rightarrow\Theta\)g, where \(g\) is the reduced total enthalpy and \(\Theta=1+\tfrac{1}{2}(\gamma-1)M_e^{2}\). Three applications are developed. (i) The steady limit of the system is shown to be exactly the compressible Falkner--Skan--Dorodnitsyn system, which we solve numerically as a validation. (ii) For the impulsively started semi-infinite plate we obtain two exact solutions of the momentum equation of distinct similarity type---one in Kummer functions, one diffusive---and, at Pr=1, a closed-form Crocco integral for the temperature; for arbitrary Prandtl number the energy equation is integrated in quadrature and yields the closed-form adiabatic recovery factor \(r=(4/\pi)\sqrt{Pr/(2-Pr)}\arctan\sqrt{(2-Pr)/Pr}\), which differs from the classical steady value \(\sqrt{Pr}\). (iii) For m ≠ 0 we solve the unsteady system globally in a Williams--Rhyne chart, in which both ends are characteristic, and show that the transformed wall shear is Mach independent for all time when \(m=0\) but not otherwise, and that the recovery factor migrates continuously from the unsteady closed form to \(\sqrt{Pr}\). Throughout, the sense in which the solutions are exact---exact under the Chapman--Rubesin and constant-Prandtl-number assumptions, in the same sense as the classical steady results---is stated explicitly, and the transformation is validated against the Blasius, Falkner--Skan, Crocco and compressible-Rayleigh results of the classical literature, including the induced outflow \(v_\infty=(\gamma-1)M_w^{2}\sqrt{C\nu_r/2\pi t}\).

Abstract: Artificial intelligence is becoming denser: equivalent capability can increasingly be achieved with fewer parameters, less memory and lower computational cost. For Physical AI, however, this is only the first of three efficiencies. A system may possess very large computational capacity, extract only a fraction of its possible useful intelligence from that capacity, and still devote more computation than the physical action actually requires. This Perspective argues that the first optimization should therefore occur before processor or model selection: reformulate the task so that only the information required for the next useful physical action must be computed. Nature repeatedly follows this strategy, from optic-flow control in insect flight to the vestibulo-ocular reflex and retinal preprocessing. Recent honeybee-inspired robot navigation provides an engineering example in which changing the representation of the navigation problem enables extremely compact computation. We propose embodied intelligence density as a family of system-level measures and position Reflex-Policy as one architecture for problem reduction, lowest-sufficient computation and selective escalation.

Abstract: We show that Bažant’s Type-2 size effect law follows exactly from a single premise---that the cross-scale energy flux is constant within a bounded range of scales---once the fracture process zone is allowed to saturate at a material length rather than remaining geometrically similar. The resulting effective energy scaling exponent is \(\alpha_{\mathrm{eff}}(l)=2+\left(1+l/\ell_p\right)^{-1}\), decreasing from 3 to 2 with increasing size, and the transitional size is not a fitting parameter: it is Irwin's characteristic length \(\ell_p=E'G_c/f_t^{2}\), fixed by independently measurable properties. We note that the alternative assumption of a geometrically similar process zone, which appears in parts of the multiscale-damage literature, yields an exponent that increases with size and a size effect law inverted relative to observation. The same constant-flux premise is then used to organise two extensions. For fractal crack surfaces we prove that the energy scaling exponent equals the surface fractal dimension, \(\alpha=D_f\), within the surface-controlled regime only; we set out explicitly why this weaker statement escapes Bažant’s critique of the fractal size-effect hypothesis, which we accept in its original target. For anisotropic media we develop a structural-tensor perturbation of the exponent and organise it by an angular decomposition on the orientation sphere, which yields a sharp prediction: the \(\ell=2\) component must vanish identically for cubic symmetry, so that a cubic polycrystal must show a four-lobed rather than a two-lobed angular pattern. We also show that the universal anisotropy of fracture surfaces reported by Ponson and co-workers implies a direction-dependent excess-area exponent even in nominally isotropic materials, which bounds the accuracy of any single-\(D_f\) description. Throughout, the constant-flux premise is treated as a hypothesis rather than a result. Section~7 states four falsifiable predictions and a validation protocol using existing published size-effect data. The connection to Kolmogorov's inertial-range argument is used as a method of organisation, not as a physical claim about any equivalence between fluids and solids.

Abstract: We present a self-contained formulation of finite-deformation Kirchhoff–Love shell theory built directly on the differential-geometric objects that the problem itself supplies: the shell body is a fibre bundle π : S → M over the mid-surface, kinematics are expressed through pull-backs of the ambient metric, stresses are bundle-valued differential forms, and dimensional reduction is fibre integration. Working in an orthonormal moving coframe and using Cartan’s structure equations [14,24], all metric information is carried by the connection 1-forms, so that the Gauss and Codazzi–Mainardi conditions appear as the vanishing of a curvature 2-form rather than as separately imposed cons traints. The resulting membrane and bending strain measures are Green–Lagrange tensors, hence exactly invariant under the full Euclidean group; no small-rotation, moderate-rotation or shallowness hypothesis is used anywhere. Relative to the usual presentation of this material we (i) give the exact through-thickness expansion of the pulled-back metric, including the O(z²) remainder that is normally discarded silently, and the exact shifter determinant μ₀(z) = 1 + 2Hz + Kz²; (ii) derive the constitutive resultants by honest fibre integration, which produces curvature-induced membrane–bending coupling of relative size O(h²K) absent from the classical uncoupled form; (iii) derive the strong form of the balance laws from the variational statement and exhibit the shear–curvature coupling and the effective membrane resultant \(\hat N^{ab}=N^{ab}+M^{ac}\tilde b_{c}{}^{b}\) that are frequently omitted; (iv) supply the Kirchhoff boundary conditions with corner forces, which are unavoidable for a C¹ shell theory; and (v) show explicitly how Koiter’s linear theory, the von Kármán plate equations, the Föppl membrane and inextensional bending are recovered as limits. The theory is then exercised numerically. Finiterotation invariance of the strain measures is confirmed to machine precision (∼ 10⁻¹⁷) on a generic doubly curved patch, against which the corresponding linearised measures accumulate spurious strain growing like 1 − cos θ—already 0.3% at 5◦, equivalent to 207MPa of fictitious stress in aluminium. A C¹ Hermite discretisation of the axisymmetric reduction reproduces the classical clamped-circular-plate solutions in both the linear and large-deflection regimes, converges at the expected rate, and traces thecomplete snap-through equilibrium paths of clamped spherical caps, including the vanishing of the fundamental vibration frequency exactly at the limit points.

Abstract: Magnetic reconnection describes the rapid conversion of stored magnetic energy during solar flares, yet the physical mechanism initiating the earliest release remains incompletely resolved. This article introduces magnetic breakdown as a proposed threshold-driven phenomenon in which a pre-existing, current-supported magnetic structure loses local stability and releases energy at the onset of magnetic reassignment and reconnection. The interpretation is developed through Photony theory, in which internal free electrons associated with current transduce dynamic elemental charge photons into linked magnetic chains that provide a proposed physical organization underlying magnetic fields and magnetically confined plasma structures. These chains assemble into magnetic fibrils whose magnetic structure embodies the substantial energy required for their formation, organization, confinement, and continued maintenance within the solar environment. Additional loading develops through chain density, curvature, compression, interaction, twist, and confinement. When this loading exceeds the capacity for stable accommodation or continuous reassignment, localized chain fragmentation is proposed to initiate radiation, particle acceleration, plasma heating and motion, current redistribution, and subsequent magnetic reconfiguration. Solar-flare observations, including precursor electromagnetic emission, rapid nonthermal electron acceleration, magnetic shear, fibril interaction, and the timing of energy release relative to macroscopic reconnection, are examined as evidence relevant to this proposed sequence. Electric-field and voltage-breakdown phenomena are considered only as needed to distinguish them from magnetic breakdown and will be developed separately in a companion article.

Abstract: Portable, noninvasive analysis of biofluids demands methods that couple chemical specificity with low cost and minimal instrumentation. We report a self-powered triboelectric spectrometry platform based on an aqueous-analyte triboelectric nanogenerator (LS-TENG) that converts the motion of a single droplet into an electrode-ordered sequence of transferred charge, forming a “triboelectric spectrum.” A 16-electrode copper array on a PMMA substrate, overlaid with FEP, encodes the droplet’s passage and enables spatially resolved signal acquisition. Across eight metabolite-relevant solutions (NaCl, KCl, CaCl₂, NH₄Cl, glucose, urea, lactic acid, uric acid), three-dimensional Q–position–concentration maps (5–30 mM) reveal concentration-dependent modulation of the response: ionic solutions show attenuation at higher concentration consistent with interfacial site saturation and electrostatic screening, whereas polar non-ionic solutes exhibit divergent trends (growth for glucose/urea; suppression for lactic/uric acid) attributable to interfacial orientation, double-layer formation, and hydronium-mediated screening. To decouple classification from amplitude–concentration collinearity, we train exclusively on 5 mM data. After standardized preprocessing and pulse segmentation, a three-convolution CNN operating on time-by-electrode tensors achieves 95.3% five-fold cross-validated accuracy with diagonal-dominant confusion matrices and stable learning curves. The LS-TENG platform is self-powered, structurally simple, and supports sub-second, single-droplet measurements, yielding generalizable composition fingerprints suitable for rapid home- or field-based identification and offering a practical route toward portable, point-of-care chemical sensing.

Abstract: Background: Infrared thermography (IRT) has become a valuable non-invasive tool for monitoring exercise-induced physiological responses in soccer players. However, evidence regarding the influence of sex on skin temperature responses following soccer-specific exercise remains limited. This study compared changes in skin temperature (ΔTsk) and thermal asymmetry between women and men soccer players following a standardised small-sided game (SSG). Methods: Forty-six university soccer players (22 women and 24 men) completed a standardised 4 vs. 4 SSG. Thermographic images were acquired before (Pre), immediately after (Post), and 10 min after exercise (Post_10). Skin temperature changes (ΔPost–Pre, ΔPost_10–Post, and ΔPost_10–Pre) and bilateral thermal asymmetries were calculated for the anterior and posterior thigh and lower leg. Between-sex comparisons were performed using independent-samples Student's t-tests, and effect sizes were estimated using Hedges' g. Results: Women players exhibited significantly greater increases in ΔTsk than men players across most anatomical regions. The largest differences were observed during recovery (ΔPost_10–Pre), when all regions demonstrated greater thermal responses in women (p < 0.001), with large to very large effect sizes (g = 1.42–1.89). In contrast, thermal asymmetry did not differ significantly between sexes at any anatomical region or assessment time point (p > 0.05). Conclusions: Women soccer players demonstrated a greater acute thermoregulatory response than men players following an SSG, particularly during the early recovery period, whereas thermal asymmetry remained stable regardless of sex. These findings indicate that sex should be considered when interpreting exercise-induced skin temperature responses and suggest that assessment performed 10 min after exercise represents the most sensitive time point for detecting physiological differences using infrared thermography.

Abstract: The Gassmann–Nur critical-porosity model, as used for amplitude and pore-pressure analysis, assigns a single critical porosity to a mineral — equivalently, it holds the dry-frame Poisson’s ratio constant with porosity. This paper generalizes the model to distinct bulk and shear critical porosities, ϕs ≤ ϕc. Three results follow. First, the Vp–Vs hyperbola which follows when Gassmann is combined with Nur is exact for any constant pair of critical porosities; the equal-porosity condition fixes only the anchor velocity (Vp as Vs approaches zero), not the shape of the curve. Second, a granular frame is physically expected to lose its shear-supporting backbone before its bulk-supporting one, placing ϕs at or below ϕc. Third, the dry sandstone frames of Han (1986) give ϕc = 0.402 and ϕs = 0.350, with the dry-frame Poisson’s ratio rising across the measured porosity range at better than four sigma, rejecting the constant-Poisson idealization within the data. The generalization preserves the hyperbolic form while changing its details — the anchor and the dry-frame Poisson’s ratio — and leaves the single-porosity numerics a close approximation, the separation in critical porosities being small and constant and its effects pronounced only as porosity approaches critical; it identifies where single-porosity pore-pressure prediction carries a systematic bias. It also resolves a standing feature of the published model — the dry- rock residual of its fluid parameter d, which the single-porosity form must carry as an unexplained correction — as the same shear–bulk separation, corroborated independently in dry shale.

Abstract: Most AI-hardware research focuses on reducing the cost of inference: moving memory closer to compute, reducing data movement, and performing operations inside or near memory. Physical Spatial AI raises an additional question: once spatial or physical evidence exists, must a machine always build a richer model before acting, or can it sometimes produce a safe first action through a simpler local rule? The Pallottoliere/Abacus Paradox states that, for some embodied problems, a low-bit local rule may outperform an advanced AI accelerator in the time and energy needed to deliver the first useful physical action. The reason is not that simple logic is more intelligent than neural inference, but that the immediate physical answer is often lower-dimensional than the perceptual model. This Short Note frames the paradox as a complementarity principle, not as a replacement of inference.

Abstract: Spatial AI is moving from representation to physical action. Single-camera streaming 3D reconstruction illustrates the shift: it can provide pose and geometry on small machines, but a reconstructed model is not yet a safe actuator command. A robot, vehicle, drone, wearable or industrial machine must still decide which spatial events require immediate local action, which can be summarized for a higher layer, and which require richer world-model reasoning. LingBot-Map is used only as a representative case of this class of pipelines; no experimental benchmark claim is made. This paper proposes a Reflex-Policy approach for Spatial AI. Local reflex layers convert urgent evidence from cameras, event sensors, MEMS, tactile, acoustic or other physical sources into low-bit events; execute bounded first actions such as stop, slow, inhibit, freeze, keyframe request, fallback or quarantine; and report compact event-action traces. Policy and world-model layers remain outside the urgent path, updating thresholds, permissions, context and rules. The contribution is an event-contract level of analysis: a spatial reflex event dictionary, structured physical observability with cardinality control, and an EROIE measurement discipline for the energy and value of event-level control. The framework does not claim that layered control is new. It makes the sensor-to-actuator contract, permission envelope, measured feedback, containment state and audit trace explicit, so that Spatial AI can remain energy-proportional, actuation-aware and diagnosable.

Abstract: The inconsistency between the classical formula and the conventional relativistic formula for the Doppler effect in electromagnetic waves is revisited. The classical formula can be derived based on the hypothesis of phase invariance of a plane wave under the Galilean transformations, while the conventional relativistic formula can be derived based on the same hypothesis under the Lorentz transformations. In this paper, we propose to derive a Lorentz type relativistic formula for the Doppler effect by strictly solving the radiation fields of a moving Hertzian dipole with the Lorentz transformations and inverse Lorentz transformations. The resultant formula is exactly of the same form as the classical one instead of the conventional relativistic formula. Our analysis shows that the inconsistency is due to the fact that the angular frequencies defined in different frames have different bases because of time dilation. It may be more natural to evaluate the Doppler effect between the angular frequency of the source and that of the fields received by the observer in the same time base. Moreover, we have derived a general expression for the Doppler effect from the far field of a moving Hertzian dipole. The result clearly shows that the classical formula and the Lorentz type relativistic formula are approximate forms of the general formula by representing the far field with a plane wave.

Abstract: Fracture and crack propagation in flexible shells under extreme loading represent a fundamental challenge in continuum mechanics. Traditional shell fracture theories rely heavily on local coordinate systems and asymptotic expansions, often entangled in the contradiction between three-dimensional solid fracture and two-dimensional shell theory. Taking the geometrically exact Kirchhoff-Love shell theory based on fiber bundles and differential forms previously established by the author as a starting point, this paper strictly generalizes it to a two-dimensional mid-surface manifold topology containing evolving cracks. We model through-cracks as evolving internal boundaries and one-dimensional submanifolds on the two-dimensional mid-surface manifold, introducing a rigorous kinematic mapping for crack propagation. In terms of dynamics, based on the elastic strain energy on the two-dimensional mid-surface, we derive a geometrically exact two-dimensional Eshelby configuration stress tensor and express it as a vector-valued configuration stress 1-form. Through the generalized virtual work principle applied to the variation of the crack front, a coordinate-independent J-integral (energy release rate) is naturally defined. Based on the Griffith criterion and the maximum energy release rate principle, this paper strictly derives the control equations for the crack propagation direction vector and propagation velocity on the tangent space of the manifold. This theory implicitly contains the complex curvature-fracture coupling within the structures of exterior differentiation and pullback metrics. Furthermore, we present a complete Discrete Exterior Calculus (DEC) discretization framework for the theory.

Abstract: Remote swell generated far from land can deliver hazardous wave energy to island coasts many tens of hours after the responsible wind event has ceased locally. Forecasting this hazard typically relies on full spectral wave models forced by global winds, which require specialized input data, computational infrastructure and expert interpretation. Here we present a reduced physical framework that constrains the amplitude envelope of long-range swell along a single dominant ray, using deep-water dispersion, steepness-limited saturation and post-source attenuation. The reduced model describes three successive regimes. First, remote wind input increases the significant wave height H_s from approximately 3 m to a physically derived saturation level of about 5.8 m. Second, H_s reaches a quasi-steady plateau when the bulk steepness ε=kH_s/2 approaches a threshold ε_th≈0.06, so that whitecapping dissipation balances further wind-driven growth. Third, after leaving the forcing region, the swell decays slowly during propagation and can still arrive offshore with H_s of order 4 m after approximately 72 h. For a representative 14 s swell packet, this corresponds to a deep-water group velocity of about 10.9 m s^-1, a travel distance of about 2830 km, and an effective attenuation length of about 7500 km. The analytical envelope is tested against quality-controlled measurements from the Graciosa buoy, Azores, in the North Atlantic. In the analysed record, 117 observations satisfied H_s≥4 m and 12≤T_p≤16 s. The strongest selected long-period candidate reached H_s=5.98 m with T_p=15.4 s, remaining below the corresponding steepness-limited ceiling for ε_th=0.06. The buoy data support the proposed envelope as a physically grounded reference for energetic long-period swell near the Azores. The framework provides an interpretable first-alert diagnostic for assessing whether a remote swell packet is capable of producing hazardous offshore conditions at Azorean island coasts, while remaining complementary to full spectral wave forecasting.