Physical Sciences

Abstract: We analyze the tangential sector of the exact finite-radius background of the P-059 parent action of a gravity model with two tensorial degrees of freedom, inside its deposited Hamiltonian–Dirac reduction, and we certify every claim by exact number-field computation. The paper delivers one exact classification and one exact obstruction, joined by a diagnostic, and it states the logical type of each result explicitly. (1) Exact tangential gradient phase diagram (region level). The tangential gradient coefficient factorizes as G_tan = (k²/4)·Π²·(W² − Y²) with exact normalization, and its shape splits into two explicit affine factors of a; the failing a-set at each (s, α) is therefore a single interval or ray with closed-form endpoints, organizing the (s, α)-plane into exactly three occupied regions plus an empty region. The negative-gradient region is nonempty (explicit rational witness), with a complete exact enumeration of the failure islands and rays. (2) Pointwise finite-frequency spectral obstruction (constrained parent and relaxed quotient; certified dictionary). From the constrained parent's frozen 10×10 assembly, two exact spectral objects are analyzed side by side: the parent pencil P10 itself, and the relaxed quotient pencil P9 obtained by an invertible multiplier reparametrization followed by the specialization δ = 0 and the drop of one constraint row. The organizing correction of this part: the multiplier difference δ = (λ_A − λ_B)/2 is not a gauge direction — P10(e)·g_gauge = (0, 2e, −2e, 0⁷)ᵀ, and the constraint dynamics forces δ = (c₀·G_r/2)·y on any parent solution with e ≠ 0 — so the gauge-fixed systems cannot stand in for the parent, and the parent pencil is analyzed directly. At all fourteen frozen-dictionary points, in either sign convention and at two wavenumber freezes (the deposited two-slot point and the consistent single-mode slice k_X² = κ = 1), the classification is obtained exactly over the number field K_s = Q(u)/(u⁴ − s): the parent is a regular pencil, det P10 = e³·q̃ with q̃ of degree 2, square-free, both roots real and negative — exactly two nonzero modes, both nonoscillatory (ω² < 0), both decaying; the relaxed quotient has det P9 = e³·q₃ with q₃ of degree 3, all three roots real, signature (N₊, N₋) = (1, 2) at ten points and (0, 3) at four; and gcd(q₃, q̃) is a constant — the quotient's nonzero spectrum is disjoint from the parent's, so the quotient is neither the parent's spectrum nor a superset of it. The obstruction verdict is common to both objects: no oscillatory (real-frequency) tangential mode exists at any certified point; the only real frequency is the zero-frequency constraint sector (in the parent's e = 0 kernel: a two-dimensional gauge/multiplier/auxiliary plane plus one algebraically slaved static deformation). The quotient pencil is a regular quadratic eigenvalue problem with a singular leading coefficient: its twelve infinite eigenvalues (grade-2 count 18 − 6 = 12) are the algebraic constraint/descriptor structure, not missed propagating modes, and the classification delivered is finite-eigenvalue. A new exact lemma certifies that the frozen assembly is mixed-order and carries no weighted-homogeneity (principal-symbol scaling) identity, so this second result is a pointwise finite-wavenumber, finite-frequency spectral obstruction — not a principal-characteristic theorem, and no PDE-level hyperbolicity verdict is claimed or implied. The diagnostic: the one-pair T2 block is positive at every scan point, and the contrast is structural — the quotient pencil carries a nonconservative odd-in-frequency velocity coupling (not a conservative gyroscopic one), so no ω²-type reconstruction equals it. Exact divisibility certificates over the frozen assembly show that the constant-ratio exceptional locus is exactly the zero-frequency sector for both signs of the ratio (a reduction/reconstruction obstruction carried by the parent-derived relaxed quotient assembly, explaining why no quotient-level ω²-type reconstruction can succeed); the corresponding spectral obstruction of the fully unreduced parent P10 itself is established independently, by the direct exact parent analysis (det P10 = e³·q̃ with both nonzero modes nonoscillatory), not by lifting the quotient's structure. Every verdict is pointwise-exact (exact polynomial determinants, Euclid square-freeness, Sturm counts over K_s); the region-level statements of result (1) are exact semialgebraic consequences of the certified factorizations. No open-region full-system spectral verdict, no principal-characteristic claim, no claim resting on real roots alone, and no sign-convention-structural claim is made anywhere.

Abstract: A geometric reconstruction framework is developed in which protected data of a codimension-three timelike Codazzi defect determine both a finite internal structure and a constrained low-energy prediction manifold. Protected least-sufficient reconstruction selects a primitive \(3+2\) carrier, its multiplication and dark-line geometry, the global form $S(U(3)\times U(2))$, a chiral $16$-dimensional coefficient packet with \(k_Y=5/3\), and a norm-seven family structure. These data are embedded in a single Alena-Codazzi Parent action. Its conserved charge provides an intrinsic selector \(\chi_Q=dQ/d\Omega\); directional bounds, Fredholm-Feshbach reduction, and validated continuation give sufficient conditions for a unique first protected constrained fold without assuming global charge convexity. The selected Parent state fixes a single scale-free coordinate \(u_*\) and simultaneously determines a rank-one critical-response residue that can be recovered from stable-side data with certified second-order error. This gives independent tests of the critical direction, its transport into family and dark sectors, and proposed causal relations between them. The resulting charged-family shapes form fixed-Casimir \(A_2\) circles, while norm-seven and contraction-Wilson data organize hierarchy and CP orientation. Spectral states and dark-channel widths are treated separately through Schur-Herglotz, Hankel, and outgoing-channel tests. After one global calibration, electroweak, charged-family, CKM, and Dirac-PMNS observables are constrained to a one-dimensional no-retune manifold. Absolute normalization, Majorana data, and continuum scattering remain independent completion problems.

Abstract: We introduce Cox sprinklings, extending Poisson sprinkling in causal set theory from fixed to random time-oriented Lorentzian metrics. Conditional on a realised metric, sampled locations form an ordinary Poisson point process with intensity measure proportional to spacetime volume. With the metric left random, the location process is Cox. The same realised metric determines the chronological relation, and discarding locations and labels yields a Cox-generated random causal set. A volume-order split distinguishes the metric’s contributions to sampling intensity and causal order and identifies the log-Gaussian Cox subclass. We derive the joint probability distribution of the number of sampled points and their unlabelled causal order and show that conditioning on the number of points can change the distribution among causal order patterns through the latent metric. We then define the complete coordinate-free Cox sampling information of a realised metric, show that it characterises the corresponding conditional probability distribution, and prove its recovery in the high density limit. Under compactness, continuity and injectivity conditions, this yields consistent parametric recovery, while independent high density observations recover the probability distribution of the sampling information across metric realisations.

Abstract: Enlarging observation space extends the available event families, but its effects on correlation bounds and dynamical spectra depend on different assumptions. Within the quantum representation of the physical--observation dual-axis structure (PODA), the real Gram geometry of the Born pairing yields the dimension-independent CHSH bound \(2\sqrt{2}\) and a two-dimensional saturating configuration. For a fixed \(d\)-dimensional maximally entangled state, arbitrary-rank binary projections give an even--odd dependence, whereas rank-one events yield \(2+(4\sqrt{2}-4)/d\). These bounds admit an experimental comparison at fixed preparation. We then introduce an independent joint field with a quadratic action and positive gradient energy along observation space. On the full \(\mathbb{CP}^1\) with its standard projective metric and Hopf connection, we derive the covariant eigenvalues and degeneracies for integer phase weight. In the minimal nonzero sector \(\lvert q \rvert=1\), the mass levels obey \(M_n^2=m_X^2+[2n(n+2)+1]m_S^2\), where \(m_X\) is an independent local mass parameter and \(m_S\) is determined by the lowest observation eigenvalue and the gradient coefficient. The two lowest distinct mass levels uniquely reconstruct both scales, giving \(m_X^2/m_S^2=(7-r)/(r-1)\) with \(r=M_1^2/M_0^2\in(1,7]\). Higher levels satisfy relations such as \(3M_2^2=8M_1^2-5M_0^2\) without further parameters. Exactly equal mass spacings occur if and only if \(m_X=m_S\). These spectral results depend on the specified action, full observation geometry, and phase representation. The scale ratio remains a dynamical parameter, while the rest-energy relation retains the form \(E_{0,n}=M_nc^2\).

Abstract: An experimental record depends both on the state of the system and on the conditions under which it is observed. Physical–Observation Dual-Axis (PODA) theory retains these as distinct arguments of the record law. The four-quadrant ontology identifies their roles; equivalence of complete responses gives a canonical, behaviorally minimal representation. We prove that a fixed observation section recovers a reduced theory precisely when the full evolution is tangent to that section and the constraints, boundary data, transport, and records agree. With the continuous, coherent, and probabilistic structures specified here, this criterion recovers general relativity, Maxwell and Yang–Mills fields, Dirac and Schrödinger dynamics, and quantum and classical information. Symmetries arise as automorphisms of the record diagram, gauge covariance from changes of response frame, and causal boundaries from finite propagation together with realizable histories. For gravity, a single action governs the response projector, connection, matter, and metric. Its variation relates response stress to Ricci curvature. About a stable flat background, the quadratic metric action admits a finite-mode quantum representation with two transverse-traceless polarizations. Exact recovery, the nonrelativistic limit, and the weak-field expansion are treated separately: the first gives a correspondence of full solutions, while the latter two hold to their stated orders.

Abstract: This paper is a direct continuation of previous papers where the 4-body problem of classical electrodynamics is derived and solved in an internal frame of reference with rectangular coordinates. The equations of motion are 16 in number in the Minkowski space, but one can prove that only 12 of them are independent ones – as many as the unknown velocities (or trajectories). Here we consider the 3D-Kepler formulation in spherical coordinates setting the first particle (the nuclei) at the origin. Then we consider equations describing the motion of the last three particles orbiting the nucleus. We obtain 9 equations for three moving particles. The Kepler formulation leads to two groups of equations. The first one contains unknown functions on the initial interval. That is why we call them Initial equations. Their solutions become initial functions for the second group of equations (Basic equations). We look for periodic solutions of these equations on the interval to the right of the initial point. These equations are of neutral type and require prescribing of initial functions. We take initial functions to be the solutions of the Initial equations. However, here second derivatives appear due to the presence of the radiation terms. Therefore, we choose a space of infinitely differentiable periodic functions and operators whose fixed point is a periodic solution of the 3D Kepler 4-body problem. The method allows us to obtain estimates of the distances between moving particles of the Li-atom. Besides the Kepler formulation allows us to describe transitions from one stationary state to another.

Abstract: The standard formalism treats spin-½ states as elements of a complex vector space, and the geometrical content of that description is not easy to state simply. We show that the transition probability between two spin-½ pure states can be computed using only the ordinary dot and cross products of real three-vectors. Replacing the ket |-⟩ by the quaternion unit j and applying an overall phase factor of i carries a normalised state to a real unit three-vector. Writing A∘B for the sum of the dot product and the i-component of the cross product, we prove that square of |A∘B|² equals the standard quantum probability exactly (Theorem 2). This operation is the projection of the quaternion product of conjugate of \( \overline{a}b \) onto the subalgebra spanned by 1 and i (Proposition 2), and by Lagrange’s identity its four component contributions partition unit probability. The construction does not remove the complex structure from the description of a spin-½ state: it relocates it from an algebraic scalar into a distinguished spatial axis, and we state the resulting limitations precisely. The calculation requires nothing beyond vector algebra, whereas the standard route requires a complex Hilbert space.

Abstract: The paper derives the fundamental physical units in a universe over a finite holographic substrate: a finite totality whose capacity bounds the state count of every embedded observer, the holographic bound taken as the substrate itself. The units are the substrate's four horizons, the Planck length, momentum, time and energy; the constants \( c \), \( \hbar \), \( G \), \( k_B \) follow uniquely from this quartet, over realisations stated with their falsifiers:\( c \) the ratio across the two Fourier pairs, \( \hbar \) the product within each conjugate pair, \( k_B \) the mixed product, \( G \) the normalisation to the totality. The quartet admits one cancellation identity and no further independent relation; on an admissible substrate the constants are exact residues, read on the unit face as magnitudes. Beneath the quartet, dimensional analysis is a graded modular domain algebra on the framed shell, recovering the classical calculus exactly inside the sub-capacity window. The domain of \( \hbar \) is the unit flag, the generator of the lattice's unique order-four subgroup; its torsion-free surrogate is the classical mass dimension, temperature inherits the acceleration domain, and the flag cancels in every count-valued comparison. Every theorem of the paper is machine-verified (217 exact checks) and formalized in Lean 4.

Abstract: For a bound orbit of Newtonian point masses, the product \( T|E|^{3/2} \) of the period and the \( 3/2 \) power of the energy is the only scale-invariant combination of the two; for two bodies it is fixed by Kepler's third law. Two closed-form generalisations to \( N\ge3 \) bodies have been proposed on dimensional grounds: a cube-sum form, supported by numerical data for planar three-body orbits, and a pair-sum form, which reproduces Semay's envelope-theory result for self-gravitating identical bosons. We show that the pair-sum form is exact for a definite class of orbits. In a homographic motion the configuration keeps its shape while rotating and pulsating, and the equations of motion reduce to a Kepler problem in which the moment of inertia \( I(\bf{a}) \) of the configuration acts as the mass and its potential \( U(\bf{a}) \) as the coupling constant. It follows that \( T|E|^{3/2}=\tfrac{\pi}{\sqrt2}\,G\,U(\bf{a})\sqrt{I(\bf{a})} \) for every eccentricity; for the equilateral triangle with arbitrary masses this coincides with the pair-sum formula. The same relation yields \( 5\pi/2 \) for the equal-mass Euler orbit, closed-form results for rings of co-orbital satellites, and the two-body limit for Trojan configurations. The semiclassical spectrum of the reduced problem is hydrogen-like, \( E_n=-K^2/(\pi\hbar n)^2 \), where \( K \) is the classical constant, and the correspondence principle returns \( T|E|^{3/2}=K \). This accounts for the agreement between Semay's quantum period and a classical law, and for the factor \( \binom N2 \) that appeared in his comparison. For non-homographic orbits we find that the moment of inertia of the figure-eight varies by only \( 0.2\% \) over a period, while its period exceeds that of a relative equilibrium of equal size and energy by \( 14\% \). We also test the cube-sum form, interpreted as a period per free-group letter, against published data for about 1600 planar orbits. The agreement is within \( 5 \)--\( 15\% \) for comparable masses, but the form fails when one body is light, since the period per letter must vanish as that mass tends to zero. Neither form is a universal \( N \)-body law, and the exact result implies that no mass function alone can be.

Abstract: The Sylvester-type equation has applications in many fields, such as control theory, scientific computing, and signal processing. However, research on the Sylvester-type equation over dual quaternions remains limited. In this paper, using generalized inverses and matrix rank , we provide necessary and sufficient conditions for the solvability of the dual quaternion matrix equation AX+BYC+ZD=E. When the solvability conditions are satisfied, the general expression of the solution is derived. Subsequently, a numerical example is presented to verify the obtained results. Finally, based on these results, an application in color image encryption and decryption is provided.

Abstract: Born probabilities determine the distribution of individual records, but leave their temporal relations open. In the U(1) M-event model a transported circle phase carries path memory; fixed transports on the same fiber nevertheless commute. We extend the event selector to a nontrivial compact connected finite-dimensional Lie group within the physical–observation dual-axis structure. One action governs the selector, its conjugate charge and their coupled motion on the two-axis base. A Haar initial selector, independent of the other initial data and prescribed controls, preserves the Born probabilities at every observation. Non-Abelian transport changes the relations between records. For balanced hemisphere readout on the full SU(2) group, we derive the exact law P_mis = α/π, relating the mismatch probability to the transport’s conjugacy angle. In a U(1) × SU(2) realization, a central charge sustains the original M response, while a periodic potential preserves its orbit and phase integral and makes finite deviations exactly harmonic. A classical pointer action gives hemisphere readout and a two-period SU(2) return for the stated apparatus preparation. With the faithful fixed-axis coupling, four blocks of 25 periods produce α₂₅ = 1.012373567 . . .. If readout retains the selector after either outcome, the third supercycle differs from the initial record with probability 96.6746%, and the sixth agrees with it with probability 93.3491%. Independent response and matrix measurements fix these values before the event sequence is observed. Scans of axis angle and block length test the relation between noncommuting transport and event memory.

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: Branching or partially specified reconstruction cannot in general be represented by an inverse se- quence of surjective maps: continuation may be multivalued, stage-dependent, and may contain dead ends. We therefore study compact Hausdorff stage spaces joined by closed continuation relations and equipped with closed observational equivalences. The central questions are viability and pathwise soundness. We prove that indefinite viability is exactly the intersection of the finite-horizon viability kernels. We characterize exact successor congruence by representative-independent one-step lifting, prove that it lifts every finite and infinite observable history, and obtain a homeomorphism between the observable quotient of the primitive history space and the history space of the quotient system. Consequently, indefinite viability descends exactly. Finite obstruction sets and minimal obstruction depth diagnose quotient splicing; moreover, every nonliftable infinite observable history has a nonliftable finite prefix. Counterexamples isolate the roles of compactness, closedness, and successor congruence. For finite systems, a balance criterion characterizes when uniform primitive transitions induce a representative-independent Markov kernel, thereby separating qualitative history descent from proba- bilistic lumpability. Backward pruning, successor signatures, count vectors, and subset lifting yield executable audits and certificates. The significance is twofold. The results supply a model-independent audit layer for reconstruction-before-dynamics approaches to law and parameter selection, and an action-labelled extension gives a sufficient soundness condition for Reconstruction-Before-Control: each controller-visible policy history then possesses a coherent primitive lift. The framework does not validate any physical bridge or establish general agency; it identifies exact conditions those applications must satisfy.

Abstract: This note is a sequel to a previous exposition of Hodge theory, aiming to provide a non-mathematician- friendly introduction to Hodge theory on surfaces with boundary. The central ingredients include five versions of the Helmholtz-Hodge decompositions. It also discusses the finite-dimensional spaces of harmonic vector fields with boundary conditions, whose dimensionalities equal the first Betti number of the underlying surface. Then, the Helmholtz-Hodge decompositions are used to solve differential equations on surfaces, especially the Cauchy-Stokes system. Finally, illustrative examples in physcis are presented.

Abstract: Fractional heat-transport models provide a powerful framework for describing anomalous thermal diffusion and wave-like propagation. However, the same reduced fractional temperature equation may result from different energy–flux realizations, de-pending on whether the fractional dynamics is introduced through the energy balance, the heat-flux constitutive relation, or both. We investigate the consequences of this non-uniqueness for externally driven photothermal transport. We show that different realizations can have the same reduced propagation coefficient while exhibiting different characteristic thermal impedances and, consequently, different photothermal amplitude and phase responses. Thus, the reduced fractional propagation equation is not sufficient to characterize an externally driven thermal response. Photothermal measurements can provide additional information on the underlying energy–flux cou-pling and distinguish transport realizations that are indistinguishable at the level of the reduced fractional equation.

Abstract: Classical theories typically model a complete record as a unary function of a physical state, treating observation conditions as external parameters. However, quantum mechanics exhibits a fundamental tension: the same independently certified physical input can yield different complete records when observation conditions vary, giving rise to the century-old measurement problem and the EPR paradox. In this work, we derive the necessary state architecture to resolve this without conflating the physical input with the conditions through which its record is formed. By applying behavioral equivalence and redundancy elimination within the category of sets, we prove that the formation-side difference cannot be mathematically removed. This forces the “observation state” to emerge as an irreducible, objective, state-bearing axis, establishing a strict physical–observation dual-axis (PODA) architecture. We emphasize that this dual-axis structure represents two non-interchangeable projection responsibilities, rather than orthogonal coordinates or dynamically decoupled systems. Under stated coherence and composition conditions, this dual-axis geometry naturally yields the complex field, binary projective geometry, and the Born trace pairing. Crucially, it further derives the Schrödinger equation as a covariant temporal composition, the CHSH inequality, and the attainable Tsirelson bound. In standard quantum mechanics, the complex field,theSchrödingerequation, and the Born rule are independent axioms. In our framework, they are unified as geometric corollaries of the PODA structure. Within this framework, the notorious quantum paradoxes are naturally dissolved: “wave-function collapse” is revealed as a conditional transition at the observation interface, and “decoherence” is understood as the objective geometric coupling between the physical and observation axes, rather than random environmental noise. Bell violation is thereby identified not as superluminal influence, but as the absence of a positive global answer section across incompatible observation states. Finally, the theory proposes a specific table-top experiment that predicts a sharp, parameter-free numerical constant: a geometric phase of (0.7π ≈ 2.1991) radians. This value directly distinguishes the dual-axis mechanism from standard quantum mechanics, which predicts exactly zero for the same configuration. Observing this constant would confirm the physical reality of the observation axis, whereas a null result falsifies the mechanism.

Abstract: The Dirac equation in light-front coordinates exhibits a particular structure due to the presence of coupled longitudinal derivatives and the singularity of the matrices associated with time evolution. In this work, we develop a systematic construction of solutions of the free Dirac equation in these coordinates using the method of separation of variables. We first show that each spinor component satisfies a Klein--Gordon--Fock (KGF) equation, which we solve by separation of variables. The mixed derivative characteristic of the light-front formulation leads to a multiplicative relation between the longitudinal dependences. We decouple this relation by introducing a nonzero auxiliary parameter \( \lambda \), thereby determining the spacetime dependence of the plane-wave branch considered. We then return to the first-order Dirac equation and use the projectors \( \Lambda_{\pm} \) to reconstruct the spinor structure. The projected equations recover the dispersion relation as a compatibility condition, while the two-dimensional image of \( \Lambda_{+} \) leads to two linearly independent spinor solutions. The procedure thus establishes a systematic connection between separation of variables, determination of the spacetime dependence, and reconstruction of the light-front spinor solutions.

Abstract: While current literature heavily relies on symmetric exponential kernels within the stress-driven nonlocal integral framework, these models are strictly limited to normal bending fields and exhibit a total architectural vacancy in describing independent shear mechanics. This paper establishes the first exact analytical, closed-form constitutive model for uncoupled nonlocal shear deformation using an anti-symmetric state-dependent integral kernel.By executing a rigorous mathematical decomposition, the spatial coordinate boundaries are isolated from the polynomial domain, transforming the integral equation into a well-posed second order ordinary differential equation. Under this anti-symmetric framework, the standard boundary layer pathologies and Dirac-delta singularity spikes that historically crippled Eringen’s strain-driven configurations are identically neutralized via cross-term cancellation at the regular singular interface.The mathematical derivations reveal a pioneering structural paradigm “Automatic Dual-Phase Mechanical Bifurcation”. It is algebraically, proved that the uncoupled nonlocal shear framework automatically toggles between microstructural softening and wave-stiffening configurations based exclusively on the spatial gradient of the applied loading field. The exact formulation is successfully, deployed to evaluate the torsional shear alignment of single-walled carbon nanotubes (SWCNTs), unlocking an accurate elastic scaling resolution that remains fully singularity-free and asymptotically collapses to classical Hookean macro-continuum mechanics as the nonlocal parameter approaches zero.

Abstract: Black-hole entropy scales with horizon area rather than interior volume. This conceptual hypothesis interprets that area law as the thermodynamic signature of boundary topological recycling: in stationary exterior bookkeeping, a structured infalling configuration is mapped to conserved charges plus unresolved horizon, radiation, and correlation channels, while its detailed organization no longer appears as recoverable stationary exterior architecture. The paper represents this distinction with a many-to-one exterior map, a four-channel decomposition, and a schematic structural/topological-load template K[Γ]. It separates loss from the stationary exterior ledger from claims about microscopic nonunitarity or annihilation and compares the interpretation with stretched-horizon, holographic, fuzzball, soft-hair, entanglement, and island/Page-curve accounts. The formulation states inherited consistency conditions—leading area scaling and no additional classical hair attributable to unprotected complexity within stationary electrovacuum—and identifies transient relaxation and radiation correlations as candidate research channels. The maps and K[Γ] are bookkeeping constructions: their spaces, weights, and dynamics remain unspecified. The hypothesis therefore defines a conceptual research program rather than a microscopic entropy count or quantitative prediction. A completion would need to define and evaluate the structural quantity, recover the coefficient 1/4 without calibration, and derive an outcome that distinguishes the model from a named comparator.