Physical Sciences

Abstract: Physical interactions are instantiated on finite spacetime domains. Windowed quantum field theory encodes this with a fixed gauge-singlet, BRST-inert profile ⋄(x) that restricts the action while leaving the interior dynamics unchanged. The present paper asks which gauge structures admit the corresponding finite-domain gauge-fixed construction. Three necessary conditions follow. Under a regulated local group-averaging prescription with normalized invariant measure, the internal gauge group must be compact, because a normalized invariant probability measure on each local group factor exists only for compact G. The perturbative and global gauge-anomaly obstructions of the chiral matter must vanish, because the window leaves the ordinary interior BRST cohomology and the single-valuedness of the fermion determinant untouched. The global form of the gauge group, together with the allowed bundle, line-operator, boundary, and Gribov sectors, must be specified on the bounded domain. These conditions are necessary, not a construction of a nonperturbative continuum theory. The scalar window supplies no representation-selection functional, so it does not fix the generation number or exclude sterile and vectorlike matter. The result is not a derivation of the matter spectrum, generation number, or minimality criterion from the scalar window alone. It is a finite-domain admissibility derivation: given the observed chiral multiplet pattern, a nonzero charge normalization, the observed up- and down-type singlet identification, and the stated minimality criterion, the windowed Ward/BRST consistency conditions recover the Standard Model gauge algebra as the minimal admissible solution.

Abstract: An exchangeable pair-indexed Hawkes model for the collision statistics of a dilute hard-sphere gas is introduced. The event types are given by unordered particle pairs. It is proved that the integrated kernel belongs to the Johnson association scheme and that it has exactly three symmetry modes: a global mode, a tagged-particle mode and a cycle mode. The scalar aggregation of the total collision count is obtained, and the clock scaling induced by the kinetic collision frequency is identified. A cumulant obstruction is then proved: for every stationary positive multitype Hawkes process with a Poisson-cluster representation, each projected count has third long-time cumulant at least as large as its second long-time cumulant. The Sonine benchmark of Visco, van Wijland and Trizac for tagged hard-sphere collisions in dimension three has second-to-first cumulant ratio 363/320 and third-to-first cumulant ratio 1129729/1024000, so the third cumulant rate is smaller than the second. This comparison indicates that the positive Hawkes model yields a reference model, whereas the hard-sphere collision statistics require inhibition or latent kinetic states.

Abstract: We perturbatively study kinks in the discrete Josephson transmission line and non-linear Klein-Gordon $(\phi^4$ and sine-Gordon) chains. The expansion parameter of the perturbation theory is the ratio of the system period to the kink width. The next to leading order approximation changes the shape of the kinks calculated previously in the leading order approximation.

Abstract: A filtered local reconstruction scheme is formulated for codimension-three Codazzi defects in four-dimensional Lorentzian branches. The closure defect of the self-reconstruction loop is organized as a lexicographic residual whose entries fix, in order, the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion. For a worldline defect with resolved link $\mathbb{CP}^1_\Gamma$, the scalar two-jet leaves two principal non-scalar types, $V_1$ and $V_2$. Faithful reconstruction of these Gauss-local charges, together with $\mathbb{CP}^1$ Toeplitz visibility, selects the separated support $E_3\oplus E_2$; reduced finite visibility fixes the degree-one line. After this carrier has been selected, the split top-form condition gives the familiar $S(U(3)\times U(2))/\mathbb Z_6$ global form and the standard one-generation exterior package, with the usual hypercharge normalization and anomaly checks. This determinant package is used as the structural comparison layer for the reconstructed carrier. The remaining construction keeps the full $\mathbb Z_6$ finite shadow, reads its projective-color projection as a quantized boundary shadow of the Alena-Codazzi vorticity sector, realizes it as a projective-color finite Hodge-spectral module, and organizes the locked low sector by a $B-L$-filtered Schur-Kuranishi completion. Yukawa, neutrino, mixing, running, and contact coefficients are thereby treated as completed-branch data rather than as inputs to the carrier selection. A scale-free charged-lepton balance residual is recorded as a Schur-layer diagnostic; its zero-correction form gives the Koide-type Hodge-balance on the reconstructed torsor carrier, while the observed deviation is left as a finite Schur-tensor datum.

Abstract: Complex one-particle Hilbert spaces are often obtained from real phase or solution spaces by choosing a compatible imaginary unit. We treat this choice as a representation-theoretic invariant. For an orthogonal representation $U$ on a real Hilbert space $H_{\R}$, with complexification $V$ on $H_{\C}$ and canonical conjugation $C$, the Real commutant \[\AR(U)=\{T\in B(H_{\C}):TV(g)=V(g)T\text{ for all }g,\ TC=CT\}\] is the intrinsic real $C^*$-algebra controlling this choice. We prove that symmetry-compatible orthogonal complex structures are exactly the skew-adjoint square roots of $-\Id$ in $\AR(U)$. In fixed central/disjoint type~I direct-integral realizations, canonical conjugation determines a multiplicity-side antiunitary transport gauge class, yielding a decomposable formula for $\AR(U)$. Abelian spectral orientations and compact Frobenius--Schur decompositions appear as scalar and atomic cases. For strongly continuous orthogonal flows, the sign of the Stone generator selects a canonical complex structure on the nonzero-energy sector and factors the real generator as $A=JB=BJ$ with $B\ge0$ on its natural domain; this positive-energy factorization is unique. With a compatible invariant symplectic form, the selected structure yields the complex one-particle space, a pure gauge-invariant quasifree Weyl state, and, under a norm-equivalence hypothesis, the ground-state Fock representation. The stationary Klein--Gordon formula $J(q,p)=(\omega^{-1}p,-\omega q)$ is obtained as a direct example.

Abstract: Traditional elastic shell theory relies heavily on cumbersome tensor calculus in curvilinear coordinates and is usually confined to a small-deformation linear framework. This not only obscures intrinsic physical and geometric meanings but also makes theoretical derivations under large deformations extremely arduous. This paper systematically constructs a geometrically exact (finite deformation) Kirchhoff-Love shell dynamic theory based on modern differential geometry—specifically fiber bundle theory and differential forms—while retaining linear constitutive relations. We model the elastic shell as a fiber bundle with the mid-surface as the base manifold and the thickness interval as the fiber. By employing Cartan’s method of moving frames, we replace traditional Christoffel symbols with sparse antisymmetric connection 1-forms, thereby completely avoiding the algebraic disaster caused by coordinates. In kinematics, by introducing a nonlinear deformation mapping and a pull-back metric, we define the Green-Lagrange strain tensor, which naturally decomposes into geometrically exact membrane and bending strains, ensuring no spurious strains arise under large rotations. In dynamics, based on the Second Piola-Kirchhoff (PK2) stress, we define a vector-valued stress 2-form and adopt the principle of virtual work as the core statement of global equilibrium. Through integration along the normal fiber, the three-dimensional principle of virtual work is naturally and coordinate-independently reduced to a two-dimensional one on the mid-surface, where complex geometric nonlinear coupling terms automatically emerge through the exterior derivative structure. Finally, the Saint-Venant-Kirchhoff linear constitutive model is introduced to close the system of equations. This theory explicitly targets ”large rotation, small strain” conditions, revealing the profound meaning of geometric nonlinearity and laying a rigorous theoretical foundation for engineering structures such as flexible electronics and post-buckling of thin plates.

Abstract: Fundamental physics faces three profound difficulties: ontological incompatibility ($G,c,\hbar$ presuppose different ontological categories), the chasm between continuity and discreteness (classical continuum vs. quantum discreteness), and divergent mathematical languages (differential geometry, Hilbert spaces, renormalization groups). This paper proposes a first-principles axiomatic system for the Cosmic Continuum. A0 asserts: All state changes in the universe arise from energy redistribution; there is no state change without energy change. This axiom replaces Newton's "inertia" as the most fundamental physical assumption—inertia is merely the special case $\Delta E = 0$. Built upon this first principle, the axiomatic system is structured through ontological axioms (A1–A3) as the cornerstone, structural axioms (A4–A7) as the skeleton, dynamical axioms (A8–A12) as the realization, and meta-axioms (A13–A14) as verification. The framework clearly distinguishes mass, energy, and dark mass beings and their corresponding space, time, and dark space dimensions, unified via the New Equivalence Principle (A3). The component (A4) serves as the skeleton, unifying particles and gauge fields into an inseparable tensor product. The scale topos and the $U_{q}(\mathfrak{e}_{8})$ modular tensor category provide the mathematical realization, rigorously capturing the relative continuum (A6) and embedding the Standard Model gauge group. From these axioms we derive: the mirror 2-morphism $M$ is equivalent to CPT; the singularity is a phase boundary from ordinary spacetime to dark space; the singularity flux is quantized as $dN/dt = \mathrm{sgn}(-t)/t_{P}$; dark space entropy $S_{\mathrm{dark}} = k_{B}\ln 2\cdot N$ resolves the black hole information paradox; the mirror cyclic universe predicts $w_{a} > 0$ (dark energy weakens over time), consistent with current DESI/Planck data at $1.3\sigma$; wavefunction collapse is interpreted as a natural projection in the fibred category, with coherent information entering dark space. The framework is fundamentally deterministic (causal): the Born rule probabilities arise from limited access to information stored in dark space, not from intrinsic randomness. Testable predictions include Planck-scale CMB oscillations ($\alpha \approx 0.032$), a gravitational wave background peak at $f_{\mathrm{peak}}\approx 0.2\,\mathrm{Hz}$, and quantum corrections to black hole shadows ($\gamma \approx 0.3$). The core dynamical part is falsifiable: exclusion of $w_{a} > 0$ at $>5\sigma$ etc. would falsify the framework.

Abstract: A physical claim rests on three separable supports: internal formal control, operational experimental contact, and reproducible computation. At laboratory scales these three move together. At extreme scales (Planck-regime gravity, cosmic horizons, high-dimensional simulations) they decouple. nWe develop the Clampdown Effect as a precise mathematical theory of that decoupling. A validation score $V_Q\in[0,1]$, formal pressure $W_Q$, operational pressure $S_Q$, and a six-channel error budget $E_Q$ are defined. A Pareto obstruction theorem is proved: under generic inter-channel coupling no admissible theory-instrument-computation triple simultaneously minimises all six error channels. The empirical equivalence classes of $T_Q$ under a fixed corridor threshold carry a natural metric $d_\theta$, and $S_Q$ descends to a Lipschitz-$1$ function on the quotient. Seven foundational axioms are shown to be mutually independent by explicit countermodels. The dynamic corridor criterion, the controlled avoidance theorem, the channel separation bound, and the stochastic Chebyshev corridor complete the calculus.

Abstract: The cosmological horizon is not a smooth surface but a mosaic of Planck‑sized quantum pixels – spin‑network punctures. In the early Universe, a far‑from‑equilibrium Coherent–Decoherent Spacetime Transition (CDST) scrambled these pixels, leaving behind a decohered gas of gravitational quanta: a virtual foam. This work shows that the entropy of this foam is the dark energy. Using the boundary‑state counting of Group Field Theory (GFT), we reproduce the Bekenstein–Hawking horizon entropy S = A/(4G). We then define the active foam fraction α_foam, the share of horizon microstates actually occupied by the foam. In the simplest picture, α_foam is simply the fraction of horizon punctures that are decohered and contribute to the cosmic acceleration. Horizon thermodynamics yields the scaling \( \rho_{\mathrm{DE}} \simα_{foam}^{\mathrm{eff}} M_{\mathrm{P}}^{2} H^{2} \), where \( α_{foam}^{\mathrm{eff}} \) absorbs the dynamical temperature correction. The present‑day dark‑energy density parameter is thus \( {\Omega_{\mathrm{DE},0} = \alpha_{\mathrm{foam},0}^{\mathrm{eff}}} \). The observed Ω_DE,0 ≈ 0.69 fixes \( \alpha_{\mathrm{foam},0}^{\mathrm{eff}} \) ≈ 0.69; for the benchmark late‑time background adopted here, the underlying fraction of activated horizon punctures is α_foam,0 ≈ 0.95, a perfectly natural order‑one efficiency. To obtain a consistent expansion history, we embed this holographic scale into a thawing quintessence model with an exponential potential \( V(\bar\phi)=V_{0}\,e^{-\lambda_{\mathrm{DE}}\bar\phi/M_{\mathrm{P}}} \). The slope λ_DE ≃ 0.65 is semi‑analytically estimated from the scaling dimension of the dominant foam operator at the GFT fixed point. Numerical integration yields the equation of state w₀ ≃ -0.86, w_a ≃ -0.15 and a mild suppression of structure growth relative to ΛCDM – predictions that can be stringently constrained by Stage‑IV surveys. A Gaussian entropy formula for α_foam and an illustrative horizon‑cell constraint \( f_{\mathrm{act}}\,\bar{s}\simeq 5.97 \) provide concrete targets for future GFT decoherence calculations. The model also offers a microscopic sequestering argument for the absence of leading fifth forces. The work unifies the microscopic origin of black‑hole entropy with the late‑time cosmic acceleration, turning the dark‑energy puzzle into a quantitatively well‑posed target for future GFT decoherence calculations.

Abstract: We present a predictive dark‑energy scenario rooted in the Group Field Theory (GFT) condensate cosmology framework. Matter particles are localised coherent excitations — private spacetimes — that perturb the global FLRW metric. The homogeneous condensate gives a negligible vacuum energy, while the incoherent virtual foam of Planck‑scale 4‑simplices provides dark energy. Its collective effect is modelled by a minimally coupled scalar field \( \bar\phi \) with a natural initial amplitude ~ M_P and an exponential potential \( V(\bar\phi)=V_0 e^{-\lambda_{\rm DE}\bar\phi/MP} \), adopted as a well‑motivated effective ansatz. The minimal coupling is supported by an explicit projection calculation showing that the foam collective mode is orthogonal to the original condensate modulus. Working in the physical frame where the Planck mass is constant and matter is minimally coupled, we avoid fifth‑force issues. Using the covariant entropy bound on the apparent horizon, the energy scale of the foam is shown to be parametrically of order M_P²H₀², naturally explaining the meV scale. The slope λ_DE is estimated semi‑analytically from the scaling dimension of the leading foam operator, giving λ_DE ~ 0.65, and further constrained by cosmological data. After fixing the potential normalisation V₀ to the observed dark‑energy density, the dynamics is controlled by this single parameter. A numerical integration of the thawing scalar yields an equation of state with w₀ ≃ -0.86, w_a < 0, and a suppression of linear matter growth relative to ΛCDM, in broad agreement with current observations. A Fisher forecast indicates that Stage‑IV surveys (DESI, Euclid) will decisively distinguish this model from a cosmological constant. In contrast to recent GFT‑based phantom dark‑energy models, this model predicts a thawing evolution with a clear observational signature. Detailed derivations and all numerical checks are provided in the accompanying Supplementary Material.

Abstract: We give a comprehensive finite-field realisation of the Riemann Hypothesis within the Finite Ring Cosmology framework, where arithmetic is carried by a local observer shell operating on a large but ultimately finite relational algebraic substrate. The Riemann zero heights are realised as the finite spectrum of the scale-evolution operator \(\hat{H}=-i(x\partial_x+1/2)\) and are computed from the primes alone: the von Mangoldt comb drives a secular condition and an explicit matrix whose eigenvalues recover the heights to \(4\times10^{-5}\), with \(\zeta\) never evaluated. On the shell the half-turn 1/2 is three things at once: the self-dual locus \(\mathrm{Tr}=1\), the symmetrizer that makes the operator self-adjoint, and the critical real part. The real part of a zero is therefore fixed at \(1/2\) by construction, for the same reason that primes are natural numbers fixed to the real line: both live where the observer and Carrier frames agree. The Riemann Hypothesis reduces to a single clause, the completeness of this spectrum for the prime vector. Completeness is a \(\Pi^0_1\) statement, a single unbounded quantifier over heights: \emph{no zero off the line, ever}. The finite totality closes the domain of that quantifier. Below the observer horizon \(\Omega^{1/4}\sim10^{30}\) the clause is decidable height by height, and is verified to \(3\times10^{12}\). Between \(\Omega^{1/4}\) and the coherence horizon \(\sqrt\Omega\sim10^{61}\) it is bounded but unresolvable in any finite observer frame. Beyond \(\sqrt\Omega\) the height chart wraps onto the sub-horizon cycle, so the trans-horizon region is identified, not populated. Two premises carry this closure: the finitude of the totality, concluded by a formal reductio ad absurdum argument against completed infinity, currently under review, and the frame-covariance of the height chart. The entire content of the hypothesis is then the bounded completeness clause below the horizon: relocated, not removed. The generalized hypothesis reduces on the same footing through the \(\chi\)-twist. The Davenport-Heilbronn function, which lacks an Euler product, carries off-line zeros exactly as the construction predicts; we demonstrate this with a positive control. The paper closes with the physical reading of the construction, from the cosmological to the quantum end of the observable range.

Abstract: The quantum–classical transition is one of the most frequently invoked concepts in modern physics. Yet the notion of classicality itself is far from unique. Depending on the physical context, classical behavior may be associated with decoherence, the semiclassical limit, thermodynamic averaging, suppression of correlations, emergence of collective order, or geometric simplification of statistical state space. These viewpoints are often presented as if they described a single phenomenon, although they emphasize different physical mechanisms and different operational criteria. In this article we examine the principal notions of classicality that appear across quantum theory, statistical physics, condensed matter physics, and information geometry. We compare the corresponding mechanisms of classical emergence and analyze the physical quantities commonly used to characterize them, including coherence, entanglement, fluctuations, correlation length, Fisher information, statistical complexity, and information-geometric curvature. We argue that many apparently distinct routes toward classical behavior share a common structural feature: a reduction of effective fluctuation freedom. From this perspective, classicality may be interpreted as an emergent regime in which the accessible fluctuation manifold becomes progressively constrained, stabilized, or geometrically simplified. This viewpoint naturally unifies decoherence, semiclassical localization, thermodynamic averaging, decorrelation, and collective organization within a common conceptual framework. Rather than representing a unique physical process, the quantum–classical transition appears as a family of related mechanisms through which complex quantum fluctuation structure gives rise to effective macroscopic classical behavior.

Abstract:

We provide a coordinate-free characterisation of phase boundaries in field theory on globally hyperbolic spacetimes with boundary. For a complex scalar field, we prove that a diffeomorphism-invariant local scalar functional $P[\Phi,g]$, partitioning spacetime into strata across a level set $\mathcal{H} = \{P=P_\star\}$, induces a stratified covariant phase space in the sense of Sjamaar--Lerman, in which the admissible variation class jumps discontinuously across $\mathcal{H}$. Concretely, on the dense stratum $M_{dens} = \{P\geq P_\star\}$ a diverging phase-stiffness functional $\kappa(P)$ enforces, by a finite-action selection rule, the vanishing of phase variations $\delta\theta = 0$, restricting the tangent space to amplitude fluctuations alone. The principal result, which we call the Phase Boundary Characterisation Theorem, states that this single energetic condition produces two algebraically equivalent effects on the augmented covariant phase space: it enlarges the presymplectic kernel of the augmented form $\Omega^{\mathrm{aug}}_\Sigma$ in the phase sector, and it suppresses the explicitly represented boundary 2-cocycle of the boundary charge algebra, $K_dens = 0$. The phase boundary $\mathcal{H}$ is identified intrinsically as the unique locus of this stiffness-induced phase-sector degeneracy, with no reference to the trigger functional once the construction is complete. Along the way, we exhibit a concrete mechanism by which the Iyer--Wald--Zoupas freedom in a phase-dependent boundary density is removed on the constrained stratum by explicit mixed boundary conditions; this is a model calculation within the IWZ setting, not a general resolution of the ambiguity. We show that the algebraic structure on each stratum is compatible with reduced phase-space quantization carried out independently on each stratum, and verify that the unstratified limit $P_\star \to \infty$ recovers the standard Lee--Wald / Iyer--Wald formalism identically. Throughout the paper the metric is treated as fixed Lorentzian background data, and the trigger functional $P[\Phi,g]$ is prescribed for purposes of the variational problem; dynamical metric variation and fully dynamical trigger functionals are identified as natural extensions rather than assumptions of the present theorem.

Abstract: This paper develops a local complement-topological formulation of codimension selection for primary homotopy linking obstructions. Let D ⊂ Md be a closed embedded submanifold of codimension c. Rather than attempting to classify all homotopy classes of M \ D, the paper focuses on primary local defect-generated classes arising from the punctured normal complement of D. By the tubular neighborhood model, a punctured normal fiber has the homotopy type of Sc−1. Hence the primary local linking sphere associated with a codimension-c defect has dimension c −1, and primary k-sphere detection selects the relation c = k + 1. The paper also introduces an explicit tubular neighborhood model of admissible modification. In this model, when modification is supported near the portion of D inside a region of scale R, there exists a tubular modification whose support is bounded above by O(Rd−c), yielding a subextensive upper bound for the minimal admissible modification cost relative to ambient volume Rd. Worked examples for a codimension-two line defect, a codimension-three defect, and the general product model verify the hypotheses explicitly. Finally, the paper recalls the standard abelianization constraint on abelian loop-valued invariants: in the loop-detectable case, globally composable abelian holonomy-type invariants factor through first homology. The result is not a classification of all homotopies of complements, but a precise local formulation of primary defect-generated linking obstructions and their subextensive tubular admissible-modification behavior.

Abstract: We analyse the unique positive, spherically symmetric ground state of the stationary Schrödinger–Newton system. Using the regularity and symmetry of solutions, we construct the full even-power Taylor expansions of the wave function and the Newtonian potential at the origin, and we express all coefficients through explicit recurrence relations depending polynomially on the initial data (a₀,b₀)=(y(0),V(0)) with a₀>0 and b₀>1. In the far field we derive the complete Coulomb-corrected asymptotic expansion of the ground state, y(r)=Ce^−rr−^1/2(1+O(r⁻¹) and V(r)=1/r+O(e^−2r), and we show that the Newtonian potential admits no algebraic corrections beyond the leading Coulomb term. The mass constraint and the virial identity provide global relations that uniquely determine the admissible pair (a₀,b₀) and fix the asymptotic amplitude C, thereby linking the origin expansion to the far-field behaviour. These results complement the classical analysis of Moroz–Penrose–Tod and the asymptotic theory of Moroz–Van Schaftingen for the Choquard equation. In the Schrödinger–Newton setting, the Coulomb tail and the mass-preserving scaling interact in a subtle way, and the present work clarifies this interaction by giving explicit expansions, precise recurrence relations, and a unified analytic description of the ground state across all spatial scales.

Abstract: In this article, the authors present a methodological approach for solving the Klein-Gordon-Fock equation by means of the separation of variables method, with emphasis on its formulation in the light-front coordinate system. The justification for the study arises from the fact that, although plane-wave solutions are frequently presented in the literature within the context of relativistic quantum mechanics, the solving process leading to their derivation is not always developed explicitly. Initially, they revisit the Klein-Gordon-Fock equation for a free particle in Minkowski spacetime, showing how the separation between the temporal and spatial parts leads to the general solution and allows the interpretation of the components associated with particle and antiparticle propagation. Next, they rewrite the equation in the light-front coordinate system, adopting α_LF=2, and again apply the separation of variables method to the coordinates x⁺, x⁻, and x^⊥. The results show that the solution obtained in this frame preserves the plane-wave structure and recovers, under suitable choices of the superposition coefficients, the wave function expected from the covariant transformation of the scalar product pμxμ. In this way, they demonstrate the coherence between the direct approach through coordinate transformation and the explicit solution of the differential equation. Moreover, the development presented reinforces the didactic potential of separation of variables in the introductory study of relativistic quantum mechanics and indicates the usefulness of the light-front formalism in future treatments involving external potentials, interaction fields, and applications in quantum field theory.

Abstract: The article examines the origins of modern mathematical physics in Kyiv and provides an overview of M. Bogolyubov’s manuscripts produced at the Institute of Mathematics.

Abstract: The present paper explores the inherent algebraic and geometric relation between the operations of scale-shift, i.e.~zoom, and rotation in representation space composed of the space and spectrum cardinal dimensions. Specifically, the fractional Fourier transform and a reversible meridian-step scale-shift are two facets---sharing the cyclic meridian index \( \Phi_t \) and its four cardinal interpretations---of one finite cyclic rotation in representation space, exact over the symmetry-complete prime shells \( F_p,\ p=4t+1 \) of the Finite Ring Continuum framework. Three theorems carry the meridian cycle of length \( p-1 \): (i) an FRC-native FrFT family, additive in the meridian index, reaching Fourier quarter-turn, parity, and inverse quarter-turn before returning to identity at the four cardinal meridians; (ii) a finite-field Weil realization in \( \mathrm{SO}(2,F_p) \) matching it on the cardinal Fourier sub-skeleton; (iii) a reversible scale-shift \( S_r:x\mapsto e_t^r x \) on \( F_p \)that recasts framed-rational refinement as a coordinate-side rotation. The same algebraic structure implies physical interpretation as chronon-indexed finite Schrödinger evolution, as well as Hubble-like cosmic dilation at the two opposed ends of the epistemic horizon. The construction is finite and algebraic; no continuum angle, trigonometric kernel, or limit is invoked.

Abstract: In this work, truth is modeled as a continuous field variable rather than as a set of discrete symbolic values. The interactions between propositions are described by saturating nonlinear dynamics, and it is shown that, under sufficiently strong collective coupling, the system separates into bipolar stable truth phases. After this separation, the application of a threshold projection leads to the emergent appearance of classical Boolean logic. Within this framework, Boolean separation is not interpreted as an axiomatic structure imposed from the outset, but as the stabilized macroscopic limit of an underlying continuous truth field. Pitchfork bifurcation analysis and local Lyapunov stability are investigated through numerical simulations, showing that Boolean-like separation arises as a stable phase organization generated by collective nonlinear interactions among continuous truth variables. Numerical analyses are also performed for finite N-dimensional systems, demonstrating that Boolean-like separation persists beyond the one-dimensional reduction and appears in higher-dimensional collective regimes. The study further proposes that arithmetic and Gödel incompleteness should not be regarded as direct properties of continuous truth dynamics, but as conse quences of discretized recursive symbolic regimes. Finally, logic gates and digital computational structures are reinterpreted as stable threshold phases emerg ing from continuous truth dynamics, suggesting a conceptual framework for analog computation, neuromorphic systems, and emergent artificial intelligence architectures.

Abstract: Symmetry operations are usually studied within the frameworks of group theory, geometry, and operator algebra. In the present work, a Ramsey-theoretic approach to symmetry is developed. Symmetry operations are treated as operators serving as vertices of complete bi-colored graphs, called symmetry graphs (SGs). Two symmetry operators are connected by a maroon edge when they commute and by a teal edge when they do not commute. Thus, the commutation structure of a symmetry group is transformed into a combinatorial object suitable for Ramsey-theoretic analysis. The introduced coloring is generally non-transitive, leading naturally to nontrivial complete bi-colored graphs constrained simultaneously by group-theoretical and combinatorial principles. It is shown that every symmetry graph containing six vertices necessarily contains either a monochromatic commuting triangle or a monochromatic non-commuting triangle as a direct consequence of the classical Ramsey theorem R(3,3)=6. The framework is illustrated for the symmetry groups of the equilateral triangle, regular tetrahedron, crystallographic point groups, infinite Cairo pentagonal tilings, and the triangular Ising ferromagnet. Higher-order structures, including teal quadrangles, second-order graph symmetries, infinite monochromatic cliques, and Lie-algebraic constraints arising from the Jacobi identity, are discussed. The proposed framework establishes a new connection between symmetry theory, Ramsey theory, graph theory, crystallography, and operator algebra.