2025 C.L. Smith. All rights reserved.
We propose a theory of spacetime fundamentally grounded in logarithmic scaling, referenced to Schwarzschild radii. By leveraging the log-log structure of factor symmetries and geometric means, we develop a rigorous geometric framework based on multiplicative duality, discrete tiling, and scale self-similarity. This structure suggests novel interpretations of discrete curvature, symmetry, and field coherence, with implications for black hole physics, conformal invariance, and scale-relative observables.
This work introduces a novel geometric reformulation of spacetime in which observable space and time (i.e., linear spacetime) arise as exponential projections from a deeper logarithmic spacetime manifold. By redefining coordinates in terms of logarithmic variables, the model reveals an invariant causal depth and reinterprets the observed speed of light $c$ as a projection of a deeper constant $c' = c^2$. Within this framework, simultaneity is defined geometrically, the lightcone emerges as a scale-invariant surface, and gravitational and quantum phenomena are unified through scale-dependent operators and causal structure. The reformulation resolves classical singularities, regularizes field-theoretic divergences, and predicts a generalized uncertainty principle tied to causal scale. Gravitational geometry is recast without divergences, while the Klein–Gordon equation and standard model fields are embedded in scale-invariant Lagrangians. Observable cosmological redshifts, supernova dimming, and cosmic acceleration are explained as projection effects without invoking dark energy. A causal interpretation of inflation, entropy, and time asymmetry arises naturally from logarithmic time. The model offers experimentally testable predictions, including modifications to redshift–distance relations, particle scattering, atomic transitions, and the running of the fine-structure constant. It proposes specific adaptations for detectors and explores implications for propulsion, communication, and perception. A scale-resolved standard model is constructed, introducing new predicted particles and clarifying coupling unification, with implications for GUT and SUSY. This framework resolves long-standing open problems including the horizon problem, black hole information paradox, and the hierarchy problem, offering a unified, causal, and scale-aware foundation for modern physics.
We present a complete axiomatic formulation and analytic resolution of Hilbert’s Sixth Problem using a logarithmic reformulation of spacetime and phase space. By defining all kinetic observables in log-coordinates \( (\tau, \xi, \eta) = (\log t, \log x, \log p) \), we derive and analyze a unified log-kinetic theory encompassing classical, quantum, relativistic, and thermodynamic dynamics. We prove existence, uniqueness, entropy production, and macroscopic limits for both interacting and mean-field systems, including a complete quantum-classical transition and relativistic coupling. This framework yields a logically complete, physically predictive theory that fulfills Hilbert’s original program.
We present a complete mathematical proof of the global existence and uniqueness of smooth solutions to the three-dimensional incompressible Navier--Stokes equations on \( \mathbb{R}^3 \), using a novel formulation in logarithmic spacetime coordinates. By transforming the classical velocity and pressure fields into a logarithmic coordinate system \( x^\mu = e^{\chi^\mu} \), we obtain a modified geometric structure that regularizes short-scale behavior and enhances dissipative control. We define weighted Sobolev spaces \( H^k_{\log} \) tailored to the log-coordinates and establish energy inequalities, coercivity, and higher-order enstrophy bounds. A Galerkin approximation in \( H^1_{\log} \), together with log-weighted Aubin--Lions compactness, yields global smooth solutions. Mapping back to physical space, we demonstrate that the transformed solution defines a unique smooth solution \( u(x,t) \in C^\infty(\mathbb{R}^3 \times [0,\infty)) \) with finite energy, thus resolving the Clay Millennium Problem. This geometric approach introduces a new class of scale-resolved fluid models with intrinsic regularization, offering a pathway to analytic and numerical advances in turbulence, compressible flow, and quantum fluid analogues.
We present a mathematically rigorous resolution of the Yang--Mills existence and mass gap problem by reformulating non-Abelian gauge theory in logarithmic spacetime coordinates. This framework introduces a scale-dependent geometric weighting via the Jacobian $J(\chi) = e^{\sum \chi^\mu}$, which regularizes ultraviolet divergences, induces confinement, and dynamically generates a spectral mass gap. We construct the log-Yang--Mills Hamiltonian in canonical form, prove its self-adjointness, and show that the energy spectrum is discrete and bounded below by a strictly positive value $\Delta > 0$. A constructive Euclidean formulation is provided via a log-lattice discretization and verified to satisfy Osterwalder--Schrader axioms, leading to a unitary quantum theory upon reconstruction. The resulting quantum gauge theory is ultraviolet-finite, nonperturbative, and exhibits confinement by construction. We compare this formulation to standard lattice QCD and renormalization group approaches, and discuss generalizations to log-QCD, gravitational coupling, and early-universe cosmology. This work completes a constructive, geometrically grounded solution to the Clay Millennium Yang--Mills mass gap problem.
We present a novel argument for proving \( P \ne NP \) grounded in the causal and geometric properties of log-scaled spacetime. By embedding computation within a causal metric where time and space scale logarithmically, we define a new framework of causal depth constraints. Within this architecture, we show that polynomial-time verification tasks in \( \text{NP} \), such as Boolean satisfiability, exceed log-causal propagation limits under any deterministic model respecting local causality. This approach bypasses the relativization, natural proofs, and algebrization barriers. We formulate the argument in terms of bounded causal volume and information flux, demonstrating that for every polynomial-time verifier of an \( \text{NP} \)-complete language, a contradiction emerges in resource scaling. These findings constitute a direct and physically interpretable proof that \( P \ne NP \), with implications for both computational theory and foundational physics.
We present a mathematically rigorous spectral framework for proving the Riemann Hypothesis (RH). Building on the Hilbert--Pólya philosophy, we construct a real, self-adjoint Schrödinger-type operator $\widehat{H}_{\log} := -\frac{d^2}{d\chi^2} + V_{\log}(\chi)$ on $L^2(\mathbb{R})$, where $\chi = \log x$ is a logarithmic coordinate. The potential $V_{\log}(\chi)$ incorporates a harmonic term and oscillatory components encoding prime arithmetic. We prove that $\widehat{H}_{\log}$ has pure point spectrum and establish that its eigenvalues $\lambda_n = \gamma_n^2$ match the squares of the imaginary parts $\gamma_n$ of the nontrivial zeros $\rho_n = \frac{1}{2} + i \gamma_n$ of the Riemann zeta function $\zeta(s)$. Through spectral zeta functions and zeta-regularized determinants, we express $\zeta(s)$ in the form \[ \zeta(s) = \Phi(s) \cdot \det(s - \widehat{H}_{\log}^{1/2})^{-1}, \] where $\Phi(s)$ is entire and chosen to match the Hadamard product. The trace of the heat kernel $\mathrm{Tr}(e^{-t \widehat{H}_{\log}})$ is shown to reproduce the prime number theorem and the Riemann explicit formula via its Mellin transform. We demonstrate that assuming any zero $\rho$ off the critical line would imply a complex eigenvalue of a real self-adjoint operator---a contradiction. This work provides a fully analytic construction of an operator whose spectral and functional properties precisely encode the nontrivial zeros of $\zeta(s)$. Numerical simulations confirm agreement with the first several hundred Riemann zeros, and our operator formalism naturally extends to Dirichlet $L$-functions and other automorphic cases. Our findings confirm that the nontrivial zeros of $\zeta(s)$ lie on the critical line $\Re(s) = \frac{1}{2}$.
We present a spectral and operator-theoretic framework for the Generalized Riemann Hypothesis (GRH), formulating Dirichlet \( L \)-functions \( L(s, \chi) \) as spectral zeta functions of self-adjoint Schrödinger-type operators \(\widehat{H}_{\log, \chi}\) defined on logarithmic coordinate space. The operator \(\widehat{H}_{\log, \chi} = -\frac{d^2}{d\chi^2} + V_{\log,\chi}(\chi)\) is constructed with a potential \( V_{\log,\chi} \) that encodes arithmetic structure via Dirichlet characters and primes. We rigorously establish the essential self-adjointness and discreteness of the spectrum, prove analyticity and regularization properties of the associated spectral zeta function, and derive the determinant identity \[ L(s, \chi) = \Phi_{\chi}(s) \cdot \det(s - \widehat{H}_{\log,\chi}^{1/2})^{-1}, \] where \(\Phi_{\chi}(s)\) is an explicit, entire factor. Through inverse Mellin transforms of the trace of the heat kernel, we recover the explicit formula for the weighted prime counting function \(\psi_{\chi}(x)\), thereby demonstrating an arithmetic-spectral correspondence. Numerical simulations of the spectrum of \(\widehat{H}_{\log, \chi}\) show strong agreement with the imaginary parts of known zeros of \(L(s, \chi)\), and we prove that any hypothetical off-line zero would yield a complex eigenvalue of a real self-adjoint operator, contradicting spectral theory. These results constitute a step-by-step resolution of GRH under this framework, connecting classical number theory, spectral analysis, and quantum chaos.
We investigate the transcendental nature of the Euler-Mascheroni constant \( \gamma \) and its complementary constant \( \delta \), using a decomposition of their definitions into infinite series of geometric area differences between harmonic sums and natural logarithms. Each term in these decompositions is shown to involve a transcendental number due to logarithmic components of rational arguments. We further show that the termwise sum of the corresponding components of \( \gamma \) and \( \delta \) is rational and converges to 1. This leads to a contradiction under the assumption that either \( \gamma \) or \( \delta \) is algebraic, providing strong evidence for the transcendence of both constants.
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