Physical Sciences

Abstract: We study integrable variable-mass sine-Gordon model (vmSG) with point defects. Using the Lax and B\"acklund-gauge formulations, we construct type-I and type-II defect matrices, derive the corresponding sewing conditions, and generate the bulk and defect contributions to an infinite hierarchy of conserved charges. The lowest members reduce to the standard sine-Gordon energy and momentum in the homogeneous limit, while for inhomogeneous backgrounds they define integrability-generated energy- and momentum-type quantities. Defect compatibility imposes matching conditions on the variable-mass functions across the defect. An analytical transmission factor \( z \)characterizes soliton transmission, topological conversion, and absorption/emission processes. We then deform the type-I sewing conditions by parameters\( \alpha \) and \( \beta \)and derive the associated defect anomalies. Consistent one-soliton transmission is recovered for \( \alpha \beta =1 \), whereas for \( \alpha \beta \neq 1 \) parity-centered kink–kink and kink–antikink transmissions display vanishing lowest order integrated anomalies but no generic vanishing at higher order. Numerical simulations reproduce the integrable transmission/conversion regimes and show that non-integrable defects generate weak radiative tails and lowest order quasi-conserved charges, while quasi-conservation of the higher order charges remains unestablished. These results elucidate how exact defect integrability deforms into charge-dependent quasi-conservation and establish a framework for defect-controlled soliton transport in inhomogeneous media, with potential applications to nonuniform Josephson junctions, magnetic and nonlinear-optical systems, and effective molecular and DNA models.

Abstract: We study \( 2\to 2 \) scattering of a stable, \( \mathbb{Z}_2 \)-odd real scalar \( \chi \) in four-dimensional Minkowski space, assuming analyticity, crossing, locality in the Jin--Martin sense, and partial-wave unitarity. Galactic-structure inferences of a self-interaction cross section per mass in the window \( \sigma/m\in[0.1,10]\,\mathrm{cm}^2/\mathrm{g} \) are used only as a phenomenological input, not as a discovery claim. Within the weakly coupled class defined by a crossing-symmetric subtraction constant \( |g_0|\le(4\pi)^2 \)and a twice-subtracted dispersive representation whose unresolved absorptive support starts at a scale \( \Lambda \), an independently replayable dual functional yields \( |M_{\mathrm{thr}}|\le B \) with \( B\simeq 158 \) at \( \Lambda/m=10 \). Consequently \( m_\chi\le 0.238\,\mathrm{GeV}\left(\frac{1\,\mathrm{cm}^2/\mathrm{g}}{\sigma/m}\right)^{1/3} \) in that class. An explicit contact countermodel (\( m_\chi=10\,\mathrm{MeV} \), \( \lambda=1 \)) lies inside the SIDM window with no extra poles, so the unqualified conjecture that a new state at \( M_X\le C m_\chi \) is always required is false. For \( m_\chi \) above the certified ceiling, SIDM-sized threshold scattering is incompatible with weak coupling and a gap: a new singularity or the failure of weak coupling is required. No universal \( C \) exists. In the short-range subclass the effective-range pole sits at \( M_X\le 2m_\chi \); a tree-level \( 0^{++} \) mediator at \( m_\chi=1\,\mathrm{GeV} \) instead allows \( M_X/m_\chi\lesssim 0.47 \)for \( |g|\le 4\pi \). Nature-level existence of SIDM or of any new dark-sector state is not established. No proof assistant was used.

Abstract:

A complete observation may change because the prepared-object condition changes or because the physical process forming its record changes. We ask when the latter dependence becomes an irreducible state responsibility. The primitive input is an operationally role-faithful closed-world table μ0 : P × A → Y_⊥. Here the object and record-forming roles are certified by experimental protocol or antecedent physics, rather than inferred from the recorded values, and ⊥ denotes certified nonimplementability. If two operations give different entries at one fixed object label, no exact encoding that preserves the certified first-role map can identify the witnessed pair. Quotienting complete row and column behaviour removes precisely duplicate labels and yields reduced behavioural objects X and S, a descended law eF : X × S → Y⊥, and an admissible joint domain B. The second quotient is canonically bijective with the implemented family H_F ⊆ Map(X , Y⊥), thereby acquiring a behaviour-minimal observer-state identity. Record-forming non-triviality makes S non-singleton, while injectivity of the exponential transpose forbids any further factorwise lossless quotient. The reduced realization is terminal among active, product-separable, pointwise-faithful realizations and is invariant under admissible role-preserving empirical covers, including redundant refinements and relabellings. These results do not assert that every arbitrary joint encoding admits a global S-decoder; such a decoder is an additional compatibility condition. For a fixed reduced interface, whether the observer state is fixed, recorded, hidden or actively controlled changes later inference, not the complete conditional law. We state the resulting interface retrospectively as the Dual-Axis Axiom of Observation. Within the operationally role-faithful domain, it expresses a foundational structural law: witnessed same-first-role dependence cannot be merged or made to factor through the first role alone without changing the complete observational content or abandoning the certified first-role semantics.

Abstract: We introduce the **Information-Geometric Grammar Principle (IGGP)**, a mathematical framework that unifies concepts from formal language theory, information geometry, Bayesian inference, and complex adaptive systems. In conventional grammar-based descriptions of adaptive systems, individual agents constitute the alphabet, interaction rules define the grammar, and collective behaviors emerge as the language generated by repeated applications of these rules. We generalize this paradigm by replacing symbolic representations with probabilistic ones. Each agent is identified with a local statistical manifold endowed with a Fisher information metric, while communication acts as a geometric production operator that transforms probability distributions through Bayesian updating and information exchange. Repeated communication generates an evolving collective statistical manifold whose geometry encodes the emergent organization of the system. Within this formulation, communication assumes the role of a probabilistic grammar: the elementary statistical states of individual agents constitute the alphabet, communication operators define the production rules, and the family of admissible collective probability distributions forms the language generated by the adaptive dynamics. The resulting Fisher geometry provides a quantitative description of syntactic organization, with off-diagonal metric components measuring communication-induced statistical dependencies among agents. Emergent collective structures are therefore interpreted as geometric properties of a statistical language rather than as purely symbolic constructions. The Information-Geometric Grammar Principle naturally extends traditional grammar-based models by introducing intrinsic geometric observables, including curvature, geometric entropy, and communication-induced coupling measures, thereby linking symbolic organization with differential geometry. The framework is sufficiently general to encompass biological populations, social and communication networks, distributed robotic systems, scientific communities, and human–artificial intelligence ecosystems. More broadly, it suggests that the emergence of collective organization can be understood as the generation of an increasingly structured statistical language whose syntax is encoded by an evolving Fisher geometry. This perspective establishes a unified mathematical language for studying adaptive organization across disciplines and opens new directions for the geometric analysis of learning, communication, and emergence in complex systems.

Abstract: This paper explores the geometrical connection between the electromagnetic gauge group and the rotational invariance of a massive fermion's spin axis, building upon a half-century of research. A brief history is given, highlighting what is and is not understood within the modern literature. Using the Algebra of Physical Space, the manifestly gauge-covariant Pauli-Schrödinger equation is derived from first principles, and it is clear that the rotational symmetry of a local spin axis generates the electromagnetic gauge-covariance. The equivalency with conventional theory is demonstrated. This forces a clear meaning onto the geometrical connection, and has philosophical implications that are discussed: Pauli spinors/wavefunctions and the electromagnetic gauge-potential exist within the same ontological category. The results of this paper do not imply views, like Hestenes' and Baylis', that interpret the geometrical connection as describing physical rotations. Lastly, there is now an apparent tension between traditional Gauge Theory and the geometrical understanding of electromagnetism and spin—this will be the subject of future work.

Abstract: Filtered minimal reconstruction is applied to a codimension-three timelike Codazzi defect with primitive \(\mathbb{CP}^1\) link and nonzero first- and second-order normal source grades. Marked Borel-Weil descent selects the multiplicity-free carrier \(E_3\oplus E_2\), whose determinant-preserving automorphisms have faithful global form \((\mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y) / \mathbb{Z}_6 \). Clifford completion gives the even exterior package of one chiral generation with a neutral singlet, on which the local anomaly and mod-two parity coefficients vanish as consequences rather than conditions. The same centered integral degree fixes hypercharge and generates a \(\mu_6\) finite shadow whose parity-selected corner is three-dimensional. Under the Alena-Codazzi realization hypotheses, the isolated low sector is obtained through a microscopic Gram-Riesz splitting and carries the gauge connection as projected transport, not as added structure. One determinant-tangent normalization fixes both the neutral cell and the central mixing seed, giving \(\lambda_{\rm cen}=0.22501\) and the parameter-eliminated relation \(15m_h^2=266m_Z^2-306m_W^2\). The first Hodge incidence step gives the Eisenstein packet $(1,4,5;7)$; the Hodge-Eisenstein jet completion fixes the CKM magnitudes, while the commensurate torsor branch gives \(\delta_{\rm CKM}=65.360^\circ\) and \(\sin2\beta=0.71030\). A normalized neutral completion gives the three PMNS angles within $0.1\sigma$ of the NuFIT 6.1 central values, \(\delta_{\rm PMNS}=207.13^\circ\), and \(\sum m_\nu=0.0599\,{\rm eV}\). Independently, parity-paired adjoint thresholds select \(M_H^{\rm RG}\simeq6.32\times10^{14}\,{\rm GeV}\), coincident at the quoted accuracy with the neutral half-flux scale \(M_R\). Structural, conditional, and normalized results are kept distinct throughout.

Abstract: The objectless, phase sector of quantum dynamics is exhibited complete on the smallest non-trivial finite holographic substrate determined by a pair of finite fields: the Carrier \(\mathbb F_{233}\) and the Subject \(\mathbb F_{13}\). The total state of physical space-time is the wave function, realised as the register on the holographic encoding of the observer's sky. Schrödinger unitary evolution is the rigid rotation of that encoding. The Dirac flow is a gauge deformation forced by the finiteness itself: the Carrier is a torsor, a field only once the frame of reference is selected by the Subject, and no bounded observer embeds without deficit. The deficit carries a finite rate, one Carrier tick per chronon, which no infinite substrate can host; its two faces, apsidal advance and clock dilation, support the gravitational realisation: gravity as the finite-substrate correction to the rotation, standing with its strong-field falsifier, the double cover supplying exactly the factor that relates the faces. The observable sky is the past light cone, realized event by event as a section of a finite Hopf fibration. Every structural theorem is proven at general cardinality p, drawn at the instance in a live interactive laboratory, and machine-verified in integer arithmetic end to end; the continuum enters only as labelled charts. The paper is the laboratory's certificate.

Abstract: The finite ring cosmology framework consumes one quantitative input, the de Sitter entropy S; every capacity-layer statement inherits its standing from how S is measured. We fix the estimation methodology and price its evidence on two declared registers: metrology, governed by a circularity criterion (a confrontation measures S only if channel-disjoint from the fitting channel), and explanation, priced by zero-freedom derivation on disclosed premises. Four instrument classes read the totality on its two faces, returning S to ±17% half-range on the count face S (conventional channels alone: ±14%). The realisation-conditional channels derive the two standing late-universe coincidences: the octant lemma fixes the temporal reading at (π/4)rH/c=13.8 Gyr, making the fitted dark-energy fraction an output (ΩΛ=tanh2⁡(3π/8)=0.684 against 0.685±0.007, 0.19σ --- same-channel, a derivation, not a measurement), and the floor a0=cH/2π closes with zero remaining freedom. Three falsifiable claims stand under live confrontation: the floor runs as a0(z)∝H(z); the (Λ,H0) locus gives H0=67.4±0.7 km/s/Mpc, siding with CMB/TRGB in the Hubble tension; and w=−1 exactly, against the DESI-preferred drift. A registrable triangle between the Compton and Schwarzschild walls closes the account at object level. The numeral of S is the prototypical Ω-hard residue --- import is its permanent status, this metrology the complete epistemic access to it - and every numeral reproduces by the accompanying scripts.

Abstract: Understanding turbulent channel flow remains a central problem in fluid dynamics, particularly in relation to near-wall structures and scaling behavior. In this work, we investigate analytical solutions of turbulent channel flow using both the classical Navier-Stokes (NS) equations and the Alexeev hydrodynamic equations (AHE), derived from the Generalized Boltzmann Equation, which accounts for non-local effects and finite collision-time dynamics. Analytical solutions are obtained under the assumption that stationary states correspond to mean flow quantities. The AHE solutions are validated against numerical results and compared with NS predictions and experimental data from multiple studies spanning Reynolds numbers from 3×103 to 3.5×107. The AHE-based solutions show consistently improved agreement with experimental velocity profiles across this wide range, while the NS-based solutions exhibit systematic deviations. The analysis reveals a new similarity parameter associated with the boundary-layer thickness, providing additional structure to turbulent scaling. A reduced formulation for the transverse velocity component yields analytical solutions that are consistent with experimental observations of secondary flows. Furthermore, coupled numerical solutions for the streamwise and transverse velocities exhibit a family of kink-type structures, which are interpreted as representations of streamwise streaks. This provides a mechanistic link between transverse velocity dynamics and the formation of coherent near-wall structures. Overall, the results offer a unified framework for mean velocity profiles, secondary flows, and streak formation in turbulent channel flow, and suggest that non-local hydrodynamic effects play an important role in wall-bounded turbulence.

Abstract: The paper explores the provenance of the fundamental physical units in a physical universe defined over a finite relational arithmetic substrate. The units are identified as the four horizons of the substrate: the Planck length, momentum, time, and energy, one per representation domain of the framed shell. The four physical constants \( c \), \( \hbar \), \( G \), \( k_B \) follow uniquely from this quartet, over declared identifications whose epistemic status is tagged throughout. The speed of light is the ratio across the two Fourier pairs of the domain square. The Planck constant is the product within each conjugate pair. The Boltzmann constant is the mixed product. The gravitational constant is the normalisation of the quartet to the totality. The quartet admits exactly one internal cancellation identity. No further algebraically independent relation exists within its lattice. On an admissible substrate the four constants instantiate as exact residues; their unit-face magnitudes are their observer-facing manifestation. Beneath the quartet, dimensional analysis is developed as a graded modular domain algebra over the framed shell. Classical dimensional analysis is recovered exactly inside the sub-capacity window. The domain of the Planck constant is the unit flag: the generator of the unique order-four torsion subgroup that the domain lattice itself carries. The flag marks the crossing between the observer's counts and the substrate-anchored measures. Its torsion-free surrogate is the classical mass dimension. Temperature inherits the acceleration domain. The flag cancels in every count-valued comparison.

Abstract: Canonical decompositions are among the most powerful organizing principles in mathematics. Prime factorization, Jordan canonical form, spectral decomposition, and Hodge decomposition demonstrate that complicated mathematical objects often admit unique descriptions in terms of elementary structural components. Surprisingly, despite the ubiquity of finite relational systems throughout mathematics, no general decomposition theory appears to exist for finite architectures themselves. Existing approaches typically begin with graphs, operators, Green functions, networks, or application-specific models rather than with architecture as the primitive mathematical object.In this paper we develop a decomposition framework for finite architectures generated by four primitive structural processes: transport, interaction, closure, and hierarchy. Within the free architectural algebra introduced here, every architecture possesses a unique canonical decomposition into these primitive generators. This decomposition induces canonical architectural coordinates and leads to a representation theorem showing that every positive multiplicative structural invariant is uniquely determined by its values on the primitive generators.The framework is formulated independently of any particular realization. Graph theory, Green-function methods, operator theory, and Green-Function Architecture Theory arise naturally as representations of the same underlying structural decomposition. The resulting theory separates structural decomposition from mathematical representation and provides a foundation for studying architectural invariants independently of their realization. The extension of the decomposition theory beyond the free architectural algebra and the determination of distinguished architectural basis constants are formulated as central problems for future research.

Abstract: This paper establishes a formal, information-theoretical derivation of gravitational acceleration as an emergent phenomenon driven by the spatial gradient of the Semantic Geometric Entropy. Operating within an n-dimensional Euclidean space, we evaluate the system at the exact conformal scaling limit where the smooth boundaries of the functional domain transition through a critical embedding threshold. At this specific boundary, the fractional Sobolev space marks the analytical phase transition where localized, heavy-tailed geometric fluctuations stabilize into continuous, bounded spatial distributions. We demonstrate that a naive application of standard field gradients under the fractional scaling exponent yields an incorrect, hyper-localized force drop-off due to strict chain-rule differentiation. To resolve this dimensional conflict, I define a specialized radially symmetric gradient profile that accounts for the internal geometry of the fractional exponent, which uniquely recovers the classical macroscopic inverse-square law. The topological integrity and stability of this coordinate manifold are verified via numerical optimization paths executed along curved matrix search spaces under strict orthogonality constraints. The resulting trajectories demonstrate that while unconstrained updates induce absolute structural collapse into a geometric singularity at a topological saturation limit, implementing a dimensionally mandated reset parameter successfully eliminates parallel transport truncation errors. This analytical and numerical alignment proves that pseudo-Riemannian spacetime is an autonomous, self-stabilizing metric configuration required to maintain global tracking coherence across the informational domain.

Abstract: The holographic principle suggests that the physically accessible information contained inside a causal region may ultimately be constrained by the entropy associated with its boundary. Motivated by this idea, we investigate a phenomenological framework in which causal horizons possess a finite effective information capacity that limits the local semiclassical representability of bulk degrees of freedom while preserving global unitary evolution. The proposed framework does not modify microscopic quantum dynamics. Instead, it assumes that as a causal horizon approaches an effective saturation regime, the reconstruction of bulk physics by a local semiclassical observer gradually becomes incomplete, whereas the complete physical state remains encoded globally through nonlocal quantum correlations. An effective entropy--area response density is introduced as a phenomenological quantity describing the local informational cost associated with representing bulk excitations on a causal boundary. A complementary geometric--informational partition is then postulated as a normalization hypothesis motivated by spherical geometry. The framework is further compared with the observational discrepancy between early- and late-time determinations of the Hubble parameter, expressed as an effective density excess of approximately 17\%. This observational quantity is not identified with the geometric normalization itself, but rather provides a phenomenological scale against which the proposed mechanism may eventually be tested. The present work therefore introduces a falsifiable theoretical program rather than a complete cosmological model. Its principal objective is to establish a mathematically consistent framework from which microscopic horizon dynamics may later derive quantitative corrections to cosmological evolution.

Abstract: The fractional Fourier transform and the reversible scale-shift, or zoom, are exhibited as two facets of one finite cyclic rotation in representation space, exact over the arithmetic symmetry shells defined over finite fields. Three results share the shell's meridian cycle. A native fractional Fourier family is additive in the meridian index, takes the Fourier quarter-turn, parity, and inverse quarter-turn as its cardinal values, and is faithful on the full cycle. A finite-field Weil realization matches this family on the cardinal Fourier skeleton. A reversible meridian-step scale-shift recasts framed-rational refinement as a coordinate-side rotation. Under a declared cyclotomic observer readout the spatial and spectral bases form a mutually unbiased pair, the finite entropic uncertainty relation is saturated by basis-localized states, and the Shannon-entropy interpolation along the cycle takes an exact closed form. The shell rotation conserves entropy; entropy is produced only by the observer readout, locating irreversibility at coarse-graining. The transform construction is finite and algebraic, with no continuum angle, trigonometric kernel, or limit invoked; the continuum enters only through the declared observer readout.

Abstract: Abstract: Numerous studies on non-Hermitian physics have remained confined to isolated exceptional points (EPs) and relied on complex parameters to analyze their physical characteristics [1–20]. Because manipulating such complex variables poses significant engineering constraints in real-world applications, recent efforts have focused on shifting the operational framework towards pure real-parameter spaces or continuous regimes [21–23]. To overcome these limitations, a recent study investigated a purely real-parameter passive system, successfully elucidating an L-surface where innumerable loci of EPs cluster together [24]. To demonstrate that this phenomenon is not an accidental occurrence but a generalized physical trait, we applied this framework to a three-degree-of-freedom semi-definite damped system, thereby establishing its systemic universality. Importantly, we discovered that the inertia of this bare primary mass acts as a critical topological switch: varying its scale governs whether the system coalesces into the L-surface or branches into alternative physical regimes. By treating this ungrounded mass as a key scaling parameter, we define the exact boundaries that map these diverse topological states. Given its high configurability, this framework opens up new avenues for multi-functional wave guiding, adaptive energy harvesting, and highly sensitive topological sensors, expanding the practical utility of non-Hermitian systems previously explored [24].

Abstract: We study the finite-volume nearest-neighbor energy generated by the symmetric reciprocal-ratio penalty, or equivalently the gradient potential \( V(t)=\cosh t-1 \) in logarithmic variables. Uniform convexity places the model in the standard class of noncompact height theories and yields weighted-Laplacian Hessians, strict convexity on fixed-mean slices, ground-state characterization, and coercive finite-graph bounds. The specific \( \cosh \) form gives several closed-form results. In each cohomology class of oriented edge fields there is a unique minimizing representative, characterized by the nonlinear coclosed equation \( \delta\sinh\omega=0 \), and the sector energy admits a quadratic gap. On twisted discrete tori this representative is explicit, so affine fields are the unique minimizers modulo constants and the finite-volume energy density equals \( \sum_{i=1}^d(\cosh a_i-1) \), independent of torus size. For boxes and discrete tori in \( \mathbb{Z}^d \), the spectral gaps are explicit, giving in \( d=3 \)an \( o(L) \) sufficient condition under which the normalized logarithmic field vanishes in averaged \( L^2 \).

Abstract: Similar to electroweak interaction, strong force and electromagnetism can have similar Higgs mechanism mediated interaction. Thus, gluons can acquire mass. And, neural colored gluons have larger mass than colored gluons. Total, we can have eight gluons without red-anti-red gluon. The puzzle of proton or neutron mass can be solved. We can also derive a new SU(5) model to include all the above eight gluons, three W/Z bosons, photon, Higgs boson, three generations of leptons and quarks to make a new 5x5 SU(5) model. Wightman axioms can be fulfilled in this final SU(5) without causing proton decay crisis. 16f matter field construction is used to put all the particles. We can also add the 4x4 four dimensional spacetime tensor integrating mass-energy density, light pressure, electric fields, and magnetic fields as well as four gradients to make a new contravariant SO(10) model. Weyl tensor and Ricci tensor related to the final SO(10) model are also given. 16f matter fields and 10 dimensional Higgs field with X charge conservation are used in constructing this final SO(10) model. Thus, grand unified theory or theory of everything can be obtained, that is compatible with four dimensional spacetime without extra-dimension needed in string theories.

Abstract: Exceptional points (EPs) offer new paradigms in non-Hermitian physics, but their realization has long relied on active gain-and-loss mechanisms. In Parts I and II, we established that clusters of innumerable EP loci can be achieved within purely real-parameter passive systems under single-fixed and semi-infinite boundaries. However, extending this framework to double-ended boundaries introduces highly coupled transcendental boundary conditions, thereby increasing the complexity of the conventional exact tracking. This study addresses this challenge by considering a both-fixed two-degree-of-freedom (2DOF) passive system. Through rigorous algebraic proofs, we achieve an analytical mapping of the global coalescing trajectories, specifically encompassing the complex degenerate roots (loci of EPs) and real double roots (loci of critical damping points) within a purely real state-space via discrete boundary points. We demonstrate that the dual-boundary constraints fundamentally morph the topologies of the EP clusters, giving rise to novel localized phase transitions and critical damping loci. These exact analytical boundaries reveal that dual confinement acts not as a restriction but as an unprecedented design flexibility to precisely manipulate non-Hermitian singularities, even under purely real-parameter passive conditions. Our exact formulation provides a foundational theoretical framework that is potentially applicable to the environmentally robust passive tuning of next-generation device architectures, ranging from micro-scale electromechanical resonators to high-frequency communication components, offering a passive alternative to mitigate the instabilities inherent in conventional active non-Hermitian systems.

Abstract: We give the operator core of Schrödinger and Dirac dynamics on a framed prime shell \(p=4\kappa+1\), whose capacity \(\kappa\) counts its quarter-turn packets: the shell's information budget. The Lorentzian form depends only on the square class of its temporal coefficient; we prove the drive multiplier canonical and identify this class with the chronon-parity grading. Over the quadratic extension, finite Hamiltonians are self-adjoint and their Cayley steps exactly unitary; the Cayley map identifies the self-adjoint projective line with the norm-one torus, also the boost group transporting the finite Dirac operator (the boost content is deliberately 1+1). The Dirac operator factors exactly into a spinorial Klein-Gordon operator, and a spinor twist forced by the arithmetic makes its evolution unitary, with massless periods a power of \(p\) and mass phases on the same torus. Free evolution is the drive itself; wave transports are isentropic, orbit periods are classified by the shell's two tori, and power maps lose distinguishability by an exact Hartley count. The two wave equations emerge as the two Fourier pairs of one finite shell. The identities are proved in finite-field arithmetic, every quoted example is reproduced by exact scripts, and the continuum enters only as a labelled degenerate idealisation.

Abstract: Research on exceptional points (EPs) as singularities, where eigenvalues and eigenvectors coalesce, has sparked a revolution in non-Hermitian physics [1–7], offering unprecedented sensitivity and wave manipulation. Previous studies have predominantly focused on isolated points in the complex plane [8–12], often relying on nonphysical complex parameters or active gain–loss modulation. However, such approaches introduce significant system complexity and hinder scalability, leaving the realization of continuous EP structures in purely passive, real-world systems an open challenge [13–18]. To address this challenge, this study reports the discovery of an EP surface within a purely passive, real-parameter, two-degree-of-freedom (2DOF) damped system. A hidden physical landscape was unveiled, termed the L-surface, representing clusters of loci of innumerable EPs. Leveraging the L-surface derived from exact analytical solutions, this study identified and validated fundamental topological phenomena, including topological jump, imprint, and nucleation, all of which were previously obscured by numerical noise [19–23]. The comprehensive analysis and precise identification of this manifold required a physical parameter precision of 100 decimal places. This regime has been conventionally dismissed as mere numerical noise in standard 64-bit double-precision floating-point formats. The unveiled L-surface provides a robust foundation for next-generation ultrasensitive sensing and perfect energy absorption across frontiers, ranging from quantum computing [24–26] to advanced biosensing [27, 28], extending its transformative impact to the broader realms of electronics [29, 30] and optics [31–33].