Physical Sciences

Abstract: The standard relativistic ontology treats time as an additional coordinate in a four-dimensional space-time manifold. Since Minkowski’s 1908 formulation, “dimension” has been tacitly identified with “vector direction in a manifold”, and the temporal coordinate has been assimilated into that vectorial catalogue. This move proved mathematically powerful but ontologically misleading. In this article I argue, within the Timeless Counterspace & Shadow Gravity—SEQUENTION (TCGS–SEQUENTION) framework, that identifying time with a geometric dimension is a category error. “Time” is a foliation parameter, a gauge label on a family of admissible projections of a single four-dimensional counterspace; it cannot be a dimension on the same footing as the geometric directions of that counterspace. Conversely, the fourth dimension in TCGS is not temporal but counter-spatial: a geometric structure of informational content, populated by singularities and extrinsic relations, whose projections generate the three-dimensional (3-D) shadow we call the physical world. I first analyse the “Minkowski trap”: the historical path by which the success of tensor calculus turned the coordinate index x⁰ into a surrogate for ontic time, and “dimension” into a purely algebraic notion. I show how this trap is reproduced, rather than avoided, in more recent multi-dimensional proposals, including (1 + 3)-dimensional “three-dimensional time” models. Drawing on recent documentary scholarship on the genesis of special relativity, I show that the identification of time with a dimension was historically contingent rather than empirically forced: Lorentz’s local time entered as an auxiliary corresponding-states label, Poincaré’s time as a synchronization convention, and Einstein’s 1905 time as an operational procedure; the ontologization of x⁰ arrived only with Minkowski’s 1908 quadratic-form geometrization and was subsequently fixed by canon formation, most visibly in the 1913 Teubner anthology from which Poincaré’s gauge-compatible reading of time was excluded. I then develop the TCGS–SEQUENTION alternative: a static four-dimensional counterspace (C, G, Ψ) containing the full content of all so-called “time stages”, and an embedded shadow manifold (Σ, g_ij) obtained via an immersion X : Σ → C, with observables given by pullbacks (g_ij, ψ) = X^∗(G, Ψ). Within this ontology, time is a foliation artifact—a parameter labelling a one-parameter family of embeddings X_λ—and all genuine dynamics are recast as consistency conditions between slices. Using the Baierlein–Sharp–Wheeler (BSW) action and subsequent constraint analyses, I demonstrate how General Relativity (GR) can be reconstructed without ontic time, thereby disentangling its empirical success from the Minkowskian ontology. I further incorporate a recent cold-atom analogue of the Wheeler–DeWitt problem of time, where a Bose–Einstein condensate partitioned into observed and unobserved sectors admits an internal ordering parameter defined by coarse-grained entropy exchange. This result supplies an operational anchor for the distinction between external laboratory time as bookkeeping and internal relational order as the relevant physical variable. I also use recent thorium-229 nuclear-clock results as a metrological anchor: they show that clock time is operationally generated through transition stability, frequency ratios, feedback-stabilized oscillators, and comparisons among electronic and nuclear reference sectors, not by detecting an independently flowing temporal substance. I then show how the same projection geometry, equipped with a single extrinsic constitutive law, accounts for dark-matter phenomenology, cosmological anisotropies, and the biological homology encapsulated in SEQUENTION, without invoking dark sectors or stochastic deep time. Finally, I contrast counter-spatial dimensionality with “3-D time” and argue that any vectorial treatment of time—even with multiple temporal axes—remains trapped in the same categorical mistake: it re-labels the coordinates instead of changing the ontology. In TCGS–SEQUENTION, there is no temporal dimension at all; the only fundamental dimension beyond the familiar three is geometric and informational, not temporal.

Abstract: We give a linearized and one-loop gauge-consistency analysis of the SI20-4 framework, and quantify what fails beyond it. The argument has three parts, ordered by energy scale. (1) Within the computed one-loop vacuum-polarization sector, the deviation from the GR coefficient is measured by a deviation function f(k), sampled at four external momenta (N = 1000 at k = 0.3 M_P): consistent with leading linear suppression, slope 0.86 ± 0.15 (correlated fit; free exponent n = 1.45 ± 0.23), and below 10⁻¹⁵ in every experimentally probed regime. (2) The one-loop TT transversality identity holds exactly at all sub-Planckian momenta, by a four-step proof whose final step uses the ℓ ↦ k−ℓ symmetry of the cutoff domain; externally, kinematic transversality holds at all orders, and a boundary-layer theorem extends the linear overflow suppression to every one-loop n-point function. A direct Lorentzian analysis of the frame-adapted box theory (Appendix C, with full phase-space derivation) shows the one-loop two-point sub-threshold absorptive cut matches GR exactly, so the measured f(k) is a dispersive vacuum-polarization deficit, not a unitarity violation. (3) At E ~ M_P, where SI20-4 departs from GR, the naive projected nonlinear BRST operator s_P = P_{≤M_P} s P_{≤M_P} is provably not nilpotent: the overflow domain R_overflow = {|p| ≤ M_P, |q−p| ≤ M_P, |q| > M_P} has positive measure. This failure is not a consistency gap but the quantitative form of the Planck-scale diffeomorphism breaking that SI20-4 Part 1 states as a prediction. The primary measured quantity is the dimensionless overflow fraction f(0.3 M_P) = 0.253 ± 0.030, monotonically increasing across k = 0.1–0.5 M_P, computed with a numerically generated and validated Einstein–Hilbert cubic vertex and the complete 10-mode spectral sum with de Donder weights. (In the de Donder/4-ball scheme with continuum normalisation C_N = C_QG = 49/6 this corresponds to the secondary, scheme-normalised coefficient B_TT − C_FP = −2.07 ± 0.24 at k = 0.3 M_P.) The measured one-loop mismatch is consistent with vanishing as k/M_P → 0.

Abstract: The unification of quantum field theory with the fundamental principles of gravity remains one of the most profound open challenges in modern theoretical physics. While the gauge principle successfully governs the dynamics of massless vector fields within the Standard Model, the explicit introduction of mass—such as the Proca mass term—inevitably breaks gauge invariance, leading to fatal ultraviolet divergences and a loss of predictability at high energy scales. This thesis explores the theoretical extension of the spacetime manifold into a superspace geometry as a fundamental mechanism to resolve these non-renormalizable infinities. By systematically implementing the Super-Poincaré algebra, we demonstrate that the inclusion of fermionic superpartners provides an exact dynamical cancellation of the loop-level divergences intrinsic to massive bosonic fields. Furthermore, we investigate how the subsequent soft symmetry breaking of the SU(2) gauge group yields a phenomenologically viable physical mass spectrum, bridging the gap between abstract mathematical formalism and observable particle dynamics.

Abstract: We develop the quantum layer of the finite relational algebraic substrate: observation as the interaction of a Subject and an Object, phase-coherent cyclic subsystems of a common finite Carrier shell. Four identifications carry the construction, each a derivation rather than an interpretation. The wave function is phase: pure states are windings of the Object's cycle, selected by stationarity under the drive, with \( E=hf \) an identity. The probability amplitude is Object-to-Subject projection onto a forced orthogonal basis, so that completeness, the selection rules, and the Lüders update are finite character theory rather than postulates. Probability is a coincidence count in the Subject's ledger, the cyclotomic ring \( \mathbb Z[i] \) of the shared quarter-turn core. This derives the Born rule within the coherence horizon, with the quarter-turn core supplying the canonical complex sector of the amplitudes. Composition is conservation: the single drive conserves the relative offset of a pair, synchronised orbits are the Bell states, and the framework saturates the Tsirelson bound exactly, \( S=2(\zeta_8+\zeta_8^{-1})=2\sqrt2 \), while reproducing the network behaviour of the real-versus-complex discrimination. The same conservation makes gravity an entangling channel. Its distinctive parameter-free signature is equal force but unequal entanglement: a source gravitates by its full mass yet entangles only through its coherent fraction, and entanglement forms at the Newtonian rate under the complementarity \( V^2+C^2=1 \). Every claim is https://github.com/gamayos/frc-numerics/tree/main/22-quantum, verified by exact finite arithmetic in three worked configurations, and the experimental confrontation closes in a single falsifiability ledger on one committed carrier scale.

Abstract: We derive gravitation on the finite relational substrate from three identifications: mass is the cardinality of a phase-locked cluster, distance is decoherence, and time is the observer’s scale-dilation, which no embedded shell shares exactly. A field has no proper ideals, so every massive object is a non-ideal embedded shell whose frame drifts; gravity is the relaxation of that forced drift. We obtain Newton’s law with G = ℏc/m_P² and the equivalence principle as an identity; the relativistic completion gives post-Newtonian β = γ = 1 and the classical tests at their observed values; and the radiative sector, made conservative by the quarter-turn conjugate momentum, gives two helicity-±2 gravitons propagating at c. The exact static solution is horizonless but operationally black, with a parameter-free shadow 4.6% wider than general relativity (1σ above current Event Horizon Telescope limits), a −4.4% ringdown shift, and an area-law entropy recovering the Bekenstein–Hawking ¹/₄ and the de Sitter entropy. The resolution floor fixes the galactic acceleration scale a₀ = cH₀/2π and the full radial acceleration relation as a registration crossover with no fitted function; the primordial spectrum is the scale-invariant (n_s = 1) invariant of the drive. Every order-one constant is determined, leaving the solid angle 4π the one recurring number: the construction is parameter-free. Every exact claim is verified in finite-field or cyclotomic arithmetic, the continuum entering only as a labelled degenerate idealisation.

Abstract: This research answers the knowledge gap regarding the explanation of the quantum jump of the electron. This scientific paper aims to complete Einstein’s research regarding general relativity and attempt to link general relativity to quantum laws.

Abstract: Where are the Large-Scale limits of quantum mechanics?. The term "large-scale" offers a very wide range of possibilities. Space (the size scale) has no immanent (intristic) scale. The most important interpretation of quantum mechanics, the relativistic Dirac equation as well, has no scale. The interpretation of the present publication, following Dirac's guidance, shows the universe from a new perspective with the help of quantum mechanics.

Abstract: The fine-structure constant is usually treated as an empirical dimensionless coupling of quantum electrodynamics. This paper proposes a different interpretation: \(\alpha^{-1}\) is a universal geometric capacity of the neutral-parent structure from which electromagnetic read-out becomes possible. In this view, branch-level electric charge is not primitive; it is an oriented read-out of a resolved defect, while magnetic response is a loop-level holonomy read-out of the completed neutral parent. A neutral object may therefore have no net electric charge while still possessing magnetic holonomy capacity. The leading neutral-parent capacity is formulated as the stratified Haar capacity \[ \Omega_{\Pzero} = (2\pi)(2\pi^2)+\frac12(2\pi^2)+\frac12(2\pi) = 4\pi^3+\pi^2+\pi , \] with the product term representing the coupled \(U(1)\)-phase and \(SU(2)\)-spinorial interior and the half-weight terms representing marginal boundary strata of the unresolved parent. The novelty of the proposal is not the isolated leading expression, but its interpretation as a neutral-parent magnetic capacity and its embedding into a constrained carrier-interface correction hierarchy. The first self-exposure correction from the ordered \(\Zthree\to\Zfour\) interface contributes \(-1/(24\Omega_{\Pzero})\), and the reduced magnetic \(\Zfour\to\Zfive\) cross-interface transfer contributes \(-7/(5\Omega_{\Pzero}^{3})\). Thus \[ \alphainv_{\rm geom} = \Omega_{\Pzero} - \frac{1}{24\Omega_{\Pzero}} - \frac{7}{5\Omega_{\Pzero}^{3}} = 137.0359991761696 . \] This differs from the CODATA 2022 value \(\alpha^{-1}_{\rm CODATA}=137.035999177(21)\) by approximately \(-8.3\times 10^{-10}\), corresponding to about \(-0.04\sigma\). The same adjacent-interface logic gives a rule-defined higher-order continuation, with successively suppressed terms at orders \(\Omega_{\Pzero}^{-5},\Omega_{\Pzero}^{-7},\ldots\). These terms are not introduced as fitted improvements, but as a conditional consistency check and as quantitative targets for future higher-precision measurements of \(\alpha^{-1}\). The result is presented as a structural conjecture: the leading capacity is formulated as a stratified Haar capacity, while a complete carrier-defect theorem for the interface corrections and their all-order persistence remains an open task.

Abstract: A generalized canonical representation of the Hilbert-Einstein action is obtained within the De Donder-Weyl formalism, preserving the equality of space-time coordinates. Unlike the conventional canonical formalism with a distinguished time parameter, the generalized Hamiltonian function does not reduce to a linear combination of constraints, and the energy of a closed universe is non-zero. Its distribution is a 4D scalar density. The contribution of the Yang-Mills field to the scalar energy density of the universe is found. To quantize the theory in generalized canonical form, a quantum principle of least action is proposed, in which the action integral is an operator on the space of wave functionals depending on the world history of the universe in the region under consideration. The secular equation for the action operator is considered as the fundamental dynamic equation on the space of wave functionals. As a consequence, a local wave equation is obtained that functions as a non-stationary Schr\"{o}dinger equation for the components of the wave functional. The role of the time derivative in this equation is played by the covariant 4D divergence operator, and its source is the non-zero 4D scalar energy density of the universe.

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This paper proposes the Information-Kinetic Theory (IKT) – a background-independent graph-topological paradigm in which metric spacetime, fields, and particles are regarded as infrared interfaces of stochastic complex-phase routing on a dynamic graph. The fundamental microdynamics is formulated in terms of the local quantum-walk operator U_m acting on the oriented edges of the graph. Under prethermal coherent and statistically isotropic vacuum conditions, the effective Floquet dynamics yields the Weyl/wave envelope. The electromagnetic, strong, weak, and gravitational sectors are described as different macroscopic interfaces of a single graph microdynamics: the U(1)-link phase, SU(3)_route color routing, the chiral weak gateway, and phase refraction χ_G. The Topological Transition Calculus (TTC) reformulates continuum UV integrations as finite sums over admissible graph transitions, while running and polarization corrections are treated as graph-computational matching and inference problems. Elementary particles are modeled as stable topological cycles of the graph, and rest mass is represented by the monodromic phase gap or by the integral algorithmic cost of maintaining a closed route. Within this framework, channel-topological anchors are formulated for a number of phenomenological quantities, including α{\rm top}^{-1}=4π³+π²+π, the parameter-rank interpretation of the minimal electronic skeleton ne(0)=10, the reduction of G to the effective configuration volume of the electron route, the leading proton confinement anchor m_p/m_e≃6π⁵, and the logarithmic anchor for the neutron isospin splitting △m≃m_eln(4π). On cosmological scales, IKT formulates the dark-energy component as the two-component interface $\Lambda_{{\rm DE},time}^{\rm IKT} = \Lambda_{{\rm par},time} + \Lambda_{{\rm noise},time}^{\rm valve}$. The dynamic parent-throughput background is proposed to drive the late acceleration, while the finite throughput of the parent valve limits the early insertion noise and leaves a small valve-limited t⁻² tail in the late epoch. This dark-energy component is separated from the total expansion scalar 3H², which also contains matter, radiation, neutrinos, the vortex/dark-matter sector, curvature, and coarse-graining corrections. Dark matter is modeled as a vortex topological sector, testable via CMB/LSS observables, BTFR, lensing, and cluster collisions. The theory also introduces TTC as a discrete calculus of topological transitions and formulates a set of falsifiable tests: DSR corrections to the propagation of high-energy massless packets, including photons; objective macroscopic decoherence; a cosmological birth-imprint scale for neutrino masses; constraints on stable fermion generations; and a statistical upper cutoff in the supermassive-black-hole mass function. IKT thus sets up a unified computational framework for reducing physical structures to the topology, spectrum, and routing of a dynamic graph. The material is organized in a three-level format: the introductory level gives a compact map of IKT and its main predictions; the technical corpus develops the mechanisms and derivations; and the audit and supplementary sections collect parameter classes, status labels, notation, and navigation aids.

Abstract: Galaxy dynamics and the accelerating expansion are the two standing tensions of large-scale gravitation. We develop both from one finite relational substrate on which gravitation is the synchronisation of elementary clocks. Synchronisation reproduces Newton's law at high acceleration and carries an intrinsic resolution floor, fixed by the expansion rate, at \(a_{0}=cH_{0}/2\pi\) - within ten percent of the observed scale, with no free parameter. Below the floor the field is read in a Fourier-conjugate chart where the registered force is an amplitude rather than a gradient, giving the geometric-mean law \(g_{\mathrm{eff}}=\sqrt{g_{\mathrm{bar}}a_{0}}\), hence flat rotation curves and the baryonic Tully-Fisher relation \(v^{4}=GMa_{0}\). The crossover is fixed by a finite first-passage registration and reproduces the empirical radial acceleration relation \(g_{\mathrm{obs}}=g_{\mathrm{bar}}/(1-\exp(-\sqrt{g_{\mathrm{bar}}/a_{0}}))\) with no fitted function; the intrinsic scatter follows as the disk's dynamical temperature and the external-field effect as the total-field dependence of the registration. In merging clusters the coherence-weighted enhancement follows the collisionless galaxies, predicting the lensing offset, and the cluster-core excess is predicted to be the coherent amplitude addition of the core's synchronised components. The cosmological constant is the curvature of the finite chart, \(\Lambda\sim1/\Omega\). The single quantity left to the totality is the \(\Omega\)-hard running of \(a_{0}\) across cosmic scale. Every exact claim is \href{https://github.com/gamayos/frc-numerics/tree/main/32-dark}{verified in finite-field or cyclotomic arithmetic}, the continuum entering only as a labelled degenerate idealisation.

Abstract: In 2000, Carver Mead introduced a time-symmetrical theory of energy exchange between two atoms, building on the Transactional Interpretation of Quantum Mechanics by John Cramer in 1986. In 2020, Cramer and Mead developed the theory further, proposing a conceptual path integral formulation by which energy could be completely transferred over long distances, and showing that this theory can explain the Einstein-Podolsky-Rosen paradox, the Hanbury–Brown–Twiss effect, and the Freedman-Clauser entanglement experiment. In this paper, we develop the theory further, proposing a specific formulation of the interaction between Emitter and Absorber atoms, in which the energy density is proportional to the root-mean-square of the product of retarded and advanced four-vector potential waves, and show how this interaction efficiently and completely transfers energy from the Emitter atom to the Absorber atom over arbitrary distances. We use Mach’s Principle and conservation of energy to find the proportionality constant by matching the mean transition time constant for all possible absorbers in the universe to the mean transition lifetime computed from Fermi’s Golden Rule, leading to a complete solution with no adjustable parameters. The solution represents the exchange of energy between two atoms, valid over 26 orders of magnitude in Emitter-Absorber distance, from about 0.52 m to the radius of the Hubble Sphere 1.27×10²⁶m. We define this Wave-Particle Model as the product of a retarded emitter vector potential wave and an advanced absorber vector potential wave, which exhibits the particle-like properties of losslessly carrying energy at the speed of light in a straight line from emitter atom to absorber atom in a vacuum in the absence of gravity.

Abstract: This paper proposes that the charged-lepton mass hierarchy may originate from a dimensionless structural invariant of a neutral-parent carrier-defect architecture, prior to its effective Higgs–Yukawa read-out. The guiding principle is that indistinguishable admissible internal sectors carry equal primitive weight before physical carrier read-out. Applied to the charged-lepton root variables (me,mμ,mτ), this principle gives the Koide relation as an equal-power condition between a democratic parent component and a branch-splitting component. The Koide cone is therefore interpreted as a root-space representation of a discrete sector-power balance, not as a fundamental continuous mass geometry. The main quantitative result is an endpoint expansion for the small electron–muon mass ratio. In the proposed architecture, the electron is interpreted as a charged endpoint leakage of the lepton-facing branch, measured against the first exposed positive-closure environment. This leads to the charged endpoint towermemμch=132(4!−1)1+∑k=5∞∏n=5k2/n3(n!−1).

Abstract: We present a causal, falsifiable law of observer-indexed entropy retrieval dynamics in which the growth rate of retrievable entropy is proportional to the remaining entropy gap and modulated by a hyperbolic-tangent onset at a characteristic proper time tau_char. Unlike ensemble-averaged, non-causal Page-curve phenomenology, the law is derived from bounded split-regularized Tomita-Takesaki modular flow and admits an inverse map for extracting observer-indexed retrieval rates from measured correlation structure. The framework supplements global entropy conservation with a Lorentzian-causal access process: conserved information becomes operationally relevant to a finite observer only when it enters that observer's bounded modular-access domain.The model predicts a joint, experimentally testable signature in the g²(t₁,t₂) correlation envelope, including constrained saturation, protocol-dependent separation, inverse gamma recovery, finite-resolution robustness, and interference suppression under controlled asymmetry. Numerical results in a finite-bond-dimension tensor-network proxy, evaluated at D=4 and D=8, are consistent with the derived law, and adversarial verification against matched saturating, shared-envelope, label-permuted, time-jittered, and non-gap alternatives shows that generic saturation does not reproduce the full observer-indexed retrieval signature. A redshift-weighted Ryu-Takayanagi representation situates the retrieval dynamics within holographic geometry without invoking replica-wormhole or island constructions as the source of the retrieval law.The result reframes the black-hole information paradox as a bounded-access dynamics problem rather than a contradiction in entropy accounting. On this formulation, information conservation, formal reconstruction, and finite-observer retrieval are distinct operations; the apparent paradox arises when they are treated as interchangeable. Here Smax denotes the Bekenstein-Hawking entropy, gamma(tau) the modular-flow retrieval rate, and tau_char the characteristic proper-time scale.

Abstract: A source-level mechanism for halo-like galactic scaling is analysed. The matter action is supplemented by a current-dependent scalar built from the translational and rotational current sectors. Its Hilbert variation gives an anisotropic contribution to the stress-energy tensor, while the constant part fixes the vacuum term. For a stationary axisymmetric system, the normalized vorticity flux selects one spatial direction and leaves the transverse two-plane to be warped by a single scalar. If this transverse warp approaches a nonzero negative asymptotic value, the active source scales as $r^{-2}$ and produces an asymptotically flat rotation curve. Weak transition profiles are used to display the source and rotation curves, and positivity of the active source restricts the allowed transition sharpness. With a universal current-transition threshold, areal-slope matching at the transition radius gives the baryonic Tully-Fisher normalization. Observable consequences include disk-aligned lensing anisotropy, residual dependence on baryonic angular momentum, and suppressed halo-like scaling in systems without coherent rotation.

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Abstract: We present a novel geometric resolution to the Lithium-7 (⁷Li) cosmological mass problem using a discrete 4-simplex lattice with a spectral dimension of 3.998. The framework relies exclusively on a rigid set of boundary conditions intrinsic to the model, including a combinatorial invariants of the 4-simplex primitive unit cell, four open degrees of freedom per simplex, a base vacuum anchor equal to the electron rest mass energy MeV, and hyperspherical normalisation, with no free thermodynamic parameters, baryonic density term nor references to fluid-dynamical nucleosynthesis. The local field saturation is governed by a master action density derived from the lattice geometry. The production efficiency of stable ⁷Li topological nodes and its structural abundance ratio are obtained from topological entropy, volumetric projection, and lattice friction. The predicted lithium-to-hydrogen ratio is determined to be ~1.3671715 x 10^-10 , in excellent agreement with observations. An accompanying polytopic-tiling analysis demonstrates that the full light-element hierarchy (Protium, Deuterium, Helium-3, and Helium-4) can emerge as a geometric inevitability of exposed-surface ratios and topological stability. Helium-4 appears as the lowest-friction closed tetrahedral configuration, naturally accounting for its ~25% mass fraction without acoustic tuning. We argue that the apparent Lithium Problem may be a mathematical artefact of imposing a continuous fluid approximation on a space that is structurally saturated by discrete 4-simplex spatial geometry. This work offers a radically more efficient description of nucleosynthesis, free from ΛCDM baggage.

Abstract: Describing the formation, evolution, and establishment of steady states in physical systems remains a fundamental problem in modern physics. This paper examines the dynamics of structured bodies in inhomogeneous fields of external forces, represented by systems of potentially interacting material points, using the evolution equation. The evolution of a closed, nonequilibrium statistical ensemble of structured bodies is described using the extended Liouville equation, which describes the change in phase volume due to internal energy transformations. D-entropy, which characterizes the relative change in internal energy, is used as a measure of system evolution. Using the evolution equation, a numerical simulation of the motion of a structured body in a radially inhomogeneous central force field is performed based on the mass-spring model. It is shown that an increase in the internal energy of the system due to the inhomogeneity of the external force field can lead to the formation of a stationary system in the central force field. However, if changes in internal energy are neglected, motion in the central field remains infinite. Attractors do not arise. The obtained results demonstrate the important role of using the evolution equation to describe the processes of structure formation in energy exchange processes and long-term dynamics of systems in inhomogeneous force fields.

Abstract: We show that the Weak Gravity Conjecture follows from semiclassical entropy consistency once null entropy variations are required to have a regular modular endpoint. The argument uses generalized entropy, the Quantum Null Energy Condition (QNEC), and the modular structure of null deformations. If all charged states are strictly subextremal, extremal charged horizons become one-sided endpoints of the semiclassical charged state space. On such horizons the non-extremal Kay–Wald/Bisognano–Wichmann identification of modular flow with Killing flow cannot be continued while preserving the same affine null-cut realization. The endpoint is therefore modularly terminal. Regular endpoint extendibility of the generalized entropy excludes such terminality, while QNEC and bounded null stress exclude the alternative of an upward entropy-curvature singularity. Hence a consistent semiclassical theory satisfying these entropy and regularity conditions must contain a charged state with q/m ≥ 1.

Abstract: We present a unified Rubik’s tetrahedral spinor geometry that accounts for the charged-lepton and neutrino mass hierarchies through discrete internal curvature. In this framework, fermion masses arise from activation of Hermitian bilinear curvature channels within a tetrahedral spinor manifold. The mass spectrum of the k-th generation follows a simple logarithmic mass law, ln⁡(m/m₁)=8 ln⁡k, or more precisely, with extra correction terms as ln⁡(m/m₁)=8 ln⁡k+ Ak² (k-1)+Bk³ (k-1). The mass law contains a fixed ladder coefficient of eight, corresponding to eight curvature channels, which can be visualized as a self-similar fractal Rubik’s tetrahedron, and reveals an invariant combination A_l+B_l=-9/4 (relative error < 10^-6%), consistent with quadratic spin contraction. In contrast, neutrino masses emerge from cyclic pseudo-temporal mixing of an internal triplet, producing a distinct invariant A_ν+B_ν=-4√3 (relative error0.003%). The √3 factor arises naturally from triplet normalization rather than parameter tuning. The two sectors thus reflect different curvature regimes within a common spinor structure. These results suggest that generational hierarchy encodes discrete geometric invariants, providing a curvature-based origin of lepton mass structure beyond phenomenological Yukawa parametrization.