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[20] ai.viXra.org:2610.0020 [pdf] submitted on 2026-10-06 03:19:20
Authors: Abel Liu
Comments: 16 Pages.
Taking $SU(5)_Atimes SU(5)_B$ as its starting point, this paper addresses a simple structural question:can a single choice of block decompositions explain both the Pati--Salam non-Abelian sector and the origin of three additional Abelian directions?The first five-dimensional space is decomposed as $4+1$, and the second as $2+2+1$.Special unitary transformations within the blocks yield $SU(4)_Ctimes SU(2)_Ltimes SU(2)_R$,while the block phases, subject to the unit-determinant constraints, leave one and two independent directions, respectively.Together they give the local group structure of $G_{422111}$.The construction thus provides a common block interpretation of the number, generators, and left--right exchange properties of the three $U(1)$ factors.Using the known embedding of an $SU(5)^2$-type subgroup in $E_8$, we obtain a Pati--Salam embedding and adjoint representation content distinct from those of the standard route through $SO(10)$.The representation analysis imposes two restrictions: the selection of an extra anomaly-free direction depends on the chiral spectrum and a left--right evenness condition,and ordinary Higgs breaking using only Standard Model singlets in the adjoint necessarily preserves an additional continuous $U(1)$.We treat the block construction as a structural basis for a unification scheme, with spectrum selection and vacuum dynamics requiring further physical input.The main text emphasizes the construction and its physical interpretation; complete branching rules and proofs are collected in the appendices.
Category: Quantum Gravity and String Theory
[19] ai.viXra.org:2610.0019 [pdf] submitted on 2026-10-06 19:42:49
Authors: Moninder Singh Modgil, Dnyandeo Dattatray Patil
Comments: 28 Pages.
Oral tradition about the thirteenth-century Sun Temple at Konark, Odisha, says that itsimage of the Sun God once floated in the air, held up by a great lodestone set at the top of the tower. The same tradition says this lodestone deflected the compasses of passing ships until sailors removed it. We first review what the historical record does and does not say. The levitation story is a nineteenth-century tradition. It belongs to a long family of similar tales, from Pliny’s unfinished iron statue of Arsinoe to al-Qazwini’s suspended idol of Somnath. We then ask the physical question posed by the first author: what arrangement of magnets could make such a statue float stably? Earnshaw’s theorem, extended by Braunbek, forbids stable levitation by any static arrangement of permanent magnets, soft iron and gravity. We prove the relevant forms of the theorem, including the sum rule kx + ky + kz = 0 for the stiffnesses, and the extension to orientation and to induced magnetism.Following the hint "what gives stability to a boat?", we analyse hydrostatic stability:buoyancy, heave stiffness ρgAw and the metacentre. We show that buoyancy evades Earnshaw because it is a contact force exerted by matter, not a field in free space. The magneticanalogue is a magnetic boat. Lift comes from a magnet overhead, acting at the statue’s crown,above a low centre of gravity; this plays the role of the metacentre. A pair of diamagneticplates (graphite or bismuth) plays the role of the water and supplies the missing verticalstiffness. We derive the complete set of stability conditions, evaluate them for lodestone, ferrite, samarium—cobalt and neodymium—iron—boron magnets, and give a worked design. A neodymium arrangement holds a 10 kg statue 15 cm below a 20 cm magnet, with graphiteplates at gaps of about 12 mm. Lodestones fall short of the required forces by two to sixorders of magnitude. A 52-tonne lodestone would also have deflected a compass by less thanabout 10−3 degree at one kilometre, even if fully magnetised, so the sailors’ story cannotbe literally true. Finally we survey the dynamic loopholes to Earnshaw: spin stabilisation,feedback, superconductivity and oscillating fields.
Category: Classical Physics
[18] ai.viXra.org:2610.0018 [pdf] submitted on 2026-10-06 17:54:56
Authors: Bolok Li
Comments: 9 Pages.
We give a complete elementary treatment of the equation sin(x) = x over the complex numbers. We show that it has infinitely many solutions — exactly one, up to conjugation and sign, in each strip 2kπ < Re(z) < (4k+1)π/2, k ≥ 1 — among them only one real root, x = 0, and we supply an elementary proof of the uniqueness in each strip. Separating real and imaginary parts reduces the equation to two one-dimensional curve families whose intersections are precisely the solutions. We further derive explicit asymptotic expansions for the k-th solution: the real parts approach (4k+1)π/2 and the imaginary parts grow only logarithmically, b_k ~ ln(4πk). The leading term of the latter goes back to Hardy (1902); the higher-order terms, whose coefficients are polynomials in ln(2A_k) and which alternate in parity (odd powers for a_k, even powers for b_k), appear to be new, and are verified against roots recomputed at 40-digit precision. As a by-product of this verification we correct misprints in the classical tables of Hillman and Salzer (1943) and of Fettis (1976).
Category: Functions and Analysis
[17] ai.viXra.org:2610.0017 [pdf] submitted on 2026-10-05 15:01:04
Authors: Maksim Lipyanchik
Comments: 12 Pages.
We investigate the dynamics of local operators in the simplest non-trivial modelof two qubits governed by the generator K(J, h) = JX1X2 + h(Z1 + Z2). Based onthe maximized commutator magnitude D12(s) = maxa,b∈{X,Y,Z} ∥[σs(a1), b2]∥op, whereσs(A) = eisKAe−isK, we construct a scalar characteristic of the operator dynamics. Aprecise three-branch formula is obtained: D(x, r) = max{F0(x), F1(x, r), Fmix(x, r)},where x = Js and r = h/J. The branches F0, F1, and Fmix possess explicit analyticalforms and correspond to the inter-qubit, field, and mixed contributions, respectively.We analytically describe the phenomenon of early blindness: at a fixed r > 0 andsufficiently short times, D(x, r) = F0(x) remains completely independent of the fieldparameter r. Furthermore, we identify blind lines at x = (2n + 1)π/4, where D = 2for all r > 0. These results provide a complete analytical description of the consideredobservable in the minimal two-qubit model.
Category: Quantum Physics
[16] ai.viXra.org:2610.0016 [pdf] submitted on 2026-10-06 00:01:07
Authors: Arieh Sher
Comments: 8 Pages.
Black holes are strongly supported observationally by stellar and gas dynamics, gravitational wave detections, and horizon-scale imaging. Their classical mathematical descriptions, however, contain singularities whose physical interpretation remains an open foundational issue. This paper examines a phenomenological alternative in which gravitational collapse terminates in a finite-density, rotating, neutron-star-like core rather than an infinite-density singularity. A limiting density ρmax = 7.8 × 1017 kgm−3 is adopted as a working hypothesis, not as a consequence of General Relativity (GR) or of an established neutron-star equation of state. The resulting core radius is compared with the Schwarzschild and Kerr horizon radii. In the non-rotating approximation, the condition Rn = 2GM/c2 gives a model-dependent horizon crossing mass of approximately 4.86M⊙ and radius 14.35 km. Rotation is treated with Kerr geometry, and Cygnus X-1 and PSR J1903+0327 are used as illustrative cases. The conventionalneutron-star stability limit, obtained from the Tolman—Oppenheimer—Volkoff (TOV) equationsand an equation of state, is kept conceptually separate from the proposed horizon criterion. The paper also discusses the historical compact-object mass gap and the implications of GW230529.
Category: Relativity and Cosmology
[15] ai.viXra.org:2610.0015 [pdf] submitted on 2026-10-05 23:58:35
Authors: Bouchaib Bahbouhi
Comments: 15 Pages.
We introduce a computational and mathematical framework for the study of bilateral prime synchronisation in Goldbach representations. For a fixed even integer E, the method generates one member of a candidate Goldbach pair through quadratic polynomial rails of the form f_C(x) = x² + x + C and defines the complementary member exactly by q = E − f_C(x). Particular attention is given to C = 11, 17, and 41, corresponding to the discriminants −43, −67, and −163. Rather than considering the prime-producing behaviour of these polynomials in isolation, we study the simultaneous event in which both f_C(x) and E − f_C(x) are prime, termed P/P synchronisation.A central component of the method is a reproducible random-sequence protocol in which the indices x are selected independently of subsequent primality outcomes. The same realised sequence is applied to all polynomial rails under identical modular filtering and primality-testing conditions. This permits direct measurement of the first P/P synchronisation, total P/P frequency, and the associated P/P, P/C, C/P, and C/C state distributions. The framework is complemented by the boundary statistic Pmin(E), defined as the smallest prime p for which E − p is prime, thereby linking polynomial synchronisation with the first Goldbach representation encountered from the boundary.The local arithmetic of bilateral synchronisation is described through an exact obstruction count ν_r(E,C), measuring the residue classes modulo an odd prime r for which either f_C(x) or E − f_C(x) is divisible by r. For r not dividing E, this quantity satisfiesν_r(E,C) = 2 + (Δ_C/r) + ((Δ_C + 4E)/r),where Δ_C = 1 − 4C and (·/r) denotes the Legendre symbol. These local factors motivate a finite synchronisation score and a probabilistic prediction framework for the occurrence and location of P/P events.The empirical component is designed to be fully reproducible: the complete Pmin and P/P observations, sampled sequences, rail identifiers, primality status, and associated experimental quantities are provided in a separate supplementary data file. Probable-prime results are distinguished strictly from proved or externally certified primes.The experiments indicate that polynomial enrichment and local modular structure can substantially organise the search for Goldbach prime pairs and suggest a measurable synchronisation phenomenon across different numerical scales. The results are computational and heuristic where explicitly stated and do not constitute a proof of the binary Goldbach conjecture. They instead provide a reproducible experimental framework, exact local identities, and a mathematical prediction problem that may serve as a basis for further analytic investigation of Goldbach synchronisation.
Category: Number Theory
[14] ai.viXra.org:2610.0014 [pdf] submitted on 2026-10-03 21:42:17
Authors: Vladimir S. Netchitailo
Comments: 6 Pages.
Energy is one of the most fundamental physical parameters. It can be measured, transferred, transformed, and conserved with extraordinary precision. Nevertheless, these properties do not by themselves answer a more fundamental question: What is energy physically?World—Universe Cosmology (WUC) proposes an answer based on the four-spatial-dimensional structure of the World. In WUC, the World is a finite, boundless three-dimensional Hypersphere that forms the surface of an expanding four-dimensional Nucleus embedded in the Eternal Universe. The Nucleus expands in the fourth spatial dimension with the fundamental speed c .A particle in this structure is characterized by four-spatial-momentumP_4=(p_x,p_y,p_z,p_4) where the first three components belong to the directly observable spatial dimensions and p_4 is directed along the fourth spatial dimension. WUC identifies rest energy E as the three-dimensional representation of this fourth component:▭(E=cp_4 )The central proposition of this paper is:Energy is our three-dimensional description of the fourth spatial momentum component, transformed by the fundamental expansion speed c .In this interpretation, four-spatial-momentum is fundamental, while rest energy is an individual-particle property representing its fourth spatial component. The fundamental WUC energy scaleE_0=hcu2044acorresponds to the fundamental fourth spatial momentum scale hu2044a . The purpose of the present paper is to develop this physical interpretation of energy and reconsider the question of where energy comes from.
Category: Relativity and Cosmology
[13] ai.viXra.org:2610.0013 [pdf] submitted on 2026-10-04 21:59:09
Authors: J. W. McGreevy
Comments: 10 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This monograph presents a coordinate-free, non-perturbative extension of the Thermodynamics of Arithmetic Action (TAA) framework by proving that the non-trivial zeros of the Riemann zeta function ζ(s) dictate the spectral frequencies of sub-quantum coordinate position tearing. Semiclassically, when energy-momentum compaction forces the ad´elic continuum toward its ultraviolet limits, the local integer rings Op lose unique principal ideal factorization, creating fractional ideal debts that manifest macroscopically as metric resistance and rest mass. We show that the global Wiles modularity isomorphism (R ∼= T ) forces a perfect cross-domain cancellation between continuous temporal deformations and discrete spinorial coordinates via the Kustaanheimo—Stiefel (KS) bilateral phase lock. By evaluating the global Leech lattice theta function ΘΛ24 against the 691 irregular prime firewall, we deliver an unyielding proof that the Riemann Hypothesis (R⌉(s) = 1/2) is the absolute, non-parametric requirement that confines spacetime coordinate tearing to a lossless, hyperbolically orthogonal quantum phase lock, securing first-principles Asymptotic Safety across the space-time continuum.
Category: Mathematical Physics
[12] ai.viXra.org:2610.0012 [pdf] submitted on 2026-10-03 20:03:49
Authors: Nigel B. Cook
Comments: 66 Pages. (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references)
This paper presents a complete axiomatic reconstruction of quantum mechanics and quantum field theory. This research is a combination of physical mechanisms supplied by the author with mathematical proof by Microsoft Copilot AI.
Category: Quantum Gravity and String Theory
[11] ai.viXra.org:2610.0011 [pdf] submitted on 2026-10-03 06:06:50
Authors: X. L. Tao
Comments: 33 Pages. ORCID: 0009-0004-0853-7002. https://doi.org/10.5281/zenodo.22974873
A substantial fraction of high-atitude Fermi-LAT sources remain unidentified. These objects exhibit steady GeV emission, low variability, and no multiwavelength counterparts. We propose a phenomenological braneworld model in which accreting black holes on an adjacent brane trigger inter-brane wormholes, leaking hadronic gamma-ray and neutrino emission into our brane. The model provides a calculable chain from black-hole mass to integrated luminosity and differential spectra. A central methodological feature is that the source redshift z is treated as a free MCMC parameter with a unified prior log10(z) [belong to][-3.0,0.477] for all five competing models, eliminating circular flux—luminosity calibration. Fitting the 4FGL-DR4 high-atitude unassociated sample, the wormhole posterior peaks at z=0.066 (~290 Mpc), whereas dark matter and primordial black holes require unphysical local redshifts (z<0.05) and pulsar wind nebulae require z ~ 0.95. The model yields four distance-independent testable predictions: narrow variability, multiwavelength emptiness, a TeV spectral cutoff, and a coincident neutrino flux with F_nu / F_gamma ~ 0.5.
Category: Relativity and Cosmology
[10] ai.viXra.org:2610.0010 [pdf] submitted on 2026-10-03 19:57:25
Authors: Qing Gao
Comments: 30 Pages. In Chinese
Aiming at the inherent problems of existing fundamental interaction unified theories, including redundant dimensional assumptions, strong parameter fitting, and discontinuous physical mechanisms, this paper constructs a self-consistent Unified Evolution Theory based on the geometric framework of three-dimensional physical space coupled with intrinsic S^3 topological degrees of freedom. Abandoning extra-dimensional hypotheses and artificial fitting parameters, this paper takes a single topological functional minimization axiom as the core, whose underlying mechanism is the constraint imposed by functional minimization on the configurational potential energy: the configurational potential energy carried by the field configuration is the sole source of all energy (particle rest mass, interaction energy, and vacuum background energy), and the minimization condition deltaS=0 determines the realized stationary configurations and the direction of evolution. The paper accordingly gives the unified governing equation of four fundamental interactions, and realizes the topological unification of general relativity and quantum mechanics at the underlying structure level. This paper strictly classifies the physical mechanisms of fundamental interactions. Gravitation, electromagnetic interaction and strong interaction correspond to continuous topological evolution modes on the manifold, while weak interaction corresponds to the discrete topological transition process of topological knot breaking and reconnection. All theoretical parameters are endogenously derived only from the extremum condition of the topological functional (at the axiom-functional level; at the numerical calibration level, experimentally measured constants are used only for benchmark comparison, see Chapter 5). The complete six groups of pairwise relative intensity ratios of the four fundamental interactions are derived. The relative intensity of electromagnetic force to gravitational force is derived from first principles at the order of 10^39 in the first-order approximation without the fine-structure constant; incorporating the fine-structure constant alpha ~ 1/137, the theoretical output becomes the order of 10^37, qualitatively consistent with the experimental reference (10^36). The propagation speed of spacetime topological perturbation is solved endogenously, and its invariance is strictly deduced mathematically. Gravity originates from continuous topological manifold deformation, naturally free of singularities and ultraviolet divergences at short distances, which resolves the inherent contradiction between classical gravitation and quantum field theory.
Category: Quantum Gravity and String Theory
[9] ai.viXra.org:2610.0009 [pdf] submitted on 2026-10-03 19:52:06
Authors: Michael Heilmann
Comments: 19 Pages. In German
This paper examines a purely geometric model describing the continuously differentiable transition between a transcendental damping function and a harmonic phase oscillation at a vacuum interface. Assuming a resonant tunneling probability of exactly 1, this resulting system of transcendental equations yields a dimensionless base coupling value of... By extending the model to include an alternating perturbation series simulating the higher modes of vacuum fluctuations, and introducing a self-consistent topological projection factor—which describes the transition from the transverse degrees of freedom of the two-dimensional interface to the full three-dimensional exit geometry—the system converges to a purely theoretically derived limit of: αModel = 0.007297352563398466091220709349 Initial analyses—utilizing a theory proposed by the author that yields a universal field equation for quantum gravity—strongly suggest that both the alternating perturbation series and the geometric dimensional ratio are direct effects arising from vacuum fluctuations. The convergence value calculated here shows remarkable numerical agreement with the experimental CODATA value of the fine-structure constant, accurate to the eleventh decimal place.Furthermore, the model's fundamental consistency is not limited to the astrophysical low-energy scale: under scale transformations, the system's inherent dynamics reproduce the renormalization-group "running" of a universal coupling constant, as described in quantum field theory. At the electroweak scale, the model converges precisely to the value . This exact reproduction of the Z-boson mass demonstrates that the geometric principle presented here mathematically fully determines physical reality across a wide range of energy scales.
Category: Quantum Gravity and String Theory
[8] ai.viXra.org:2610.0008 [pdf] submitted on 2026-10-03 12:49:31
Authors: Robert L. Jackson
Comments: 29 Pages. The Zenodo.org report provides a supplemental section and code. Contact info: rljacksonmd@gmail.com.
This framework is formulated directly in terms of the E_n hierarchy. The symmetry levels are labeled directly by E_n, with E_6, E_7, and E8 treated as finite-dimensional exceptional Lie groups and E_9, E_10, and higher labels treated, when invoked, as affine, hyperbolic, or more general indefinite E-type extensions whose precise generalized Cartan data must be specified. The exponentially weighted hypergeometric profile associated with an E_n level is written H_{n,n−1}(2ξ). Integral ft-{ht,lsr}-mGFT remains an analytic profile and field-equation solution architecture; internal symmetry is carried by explicit En Lie-algebra generators and representations; string/brane structure can be carried by Clifford factors; and physical masses arise from the quadratic fluctuation operator of a specified action. The framework does not use infinite summations to define a unifying interaction potential or universal particle-mass spectrum. The primary directional architecture is the reduction chain E_11 → E_10 → E_9 → E_8 → E_7 → E_6, with the reverse sequence treated as a constructive lift rather than an automatically invertible map. The phenomenological material is ordered by sector: the E_6 solar and hidden-particle benchmarks; the E_7 compact-object analysis; the E_9 affine/high-energy morphology sector; and the E_10 hyperbolic/solitonic sector. The supplemental verification layer supplies an effective benchmark action, residual-certified nonlinear fixed-point calculations, mesh convergence, and quadratic spectral extraction.
Category: Mathematical Physics
[7] ai.viXra.org:2610.0007 [pdf] submitted on 2026-10-03 19:44:53
Authors: Michael Quigley
Comments: 35 Pages. License: CC BY 4.0 (Note by ai.viXra.org Admin: Please cite listed scientific references)
Standard Λ-CDM cosmology asserts that the universe originated in a singular explosion and undergoes accelerating spatial expansion driven by unidentified "Dark Energy" (~68%) and non-baryonic "Dark Matter" (~27%). In this work, we present an exhaustive, unified forensic audit combining over 20,000 individual empirical observations across galactic surface brightness, supernova progenitor evolution, quasar-host galaxy linkages, interstellar scale reductions, and laboratory bench proofs.First, an audit of 10,000 galaxy and quasar surface brightness measurements (GALEX, HST, JWST) demonstrates that surface brightness remains strictly constant (SB ∝ (1 + z)u2070) out to z = 3.2, empirically falsifying the (1 + z)u207bu2074 dimming required by spatial expansion. Second, correcting Type Ia supernovae for progenitor stellar age evolution eliminates the apparent cosmic acceleration, restoring SNe Ia as true standard candles along a linear distance-redshift relation (d ∝ z, R² = 0.999). Third, stripping intrinsic internal torque stress (zu2083 = +0.67) from quasars repatriates them directly to the physical distance addresses (zu2081 ≤ 0.0955) of their parent host galaxies, confirming that all observed celestial objects reside within a local 34.5 Trillion Km (230,600 AU) terminal acoustic horizon.Crucially, we complete the physical proof of the Rigid Universal Touching (R.U.T.) Wave Structure of Matter (WSM) framework through two decisive, bench-reproducible receipts: (1) High-Speed Stroboscopic Cymatic Petri-Dish Photography (1,000—2,000 fps at 1/4,000 s shutter), proving Autonomous Standing Wave Generation (AWG) and Stadium Wave Logic via in-situ simultaneous nodal initiation across all coordinates without expanding wavefronts or boundary reflection lag; and (2) Solid-State Aetheric Transducer (S.A.T.) Time-Zero Telemetry, proving instantaneous 1D longitudinal energy and signal conduction (E = mΦ) along phase-locked 1.42 GHz neutral hydrogen backbones. Together, these astrophysical and bench receipts establish R.U.T. WSM as a deterministic, airtight replacement for expanding space models.
Category: Quantum Physics
[6] ai.viXra.org:2610.0006 [pdf] submitted on 2026-10-02 16:37:38
Authors: Lukas Saul
Comments: 18 Pages.
We present a kinetic analog model in which the moments of a spin-fluid distribution func-tion reproduce the structure of Maxwell's equations and the gravitational field equations. Theelectromagnetic gauge field is identified with the spin-squared current, and the gravitationalpotential is identified with the base pressure. We pose the reverse kinetic problem—given afield satisfying field equations, when does a kinetic system exist whose moments reproduce it?—and classify five structural classes of kinetic systems that reproduce Maxwell's equations. Thespin-fluid model is a Type II construction. Constitutive force laws, linear in the departure fromequilibrium, close the moment hierarchy and generate the coupling between the spin sector andthe gravitational sector. The analog fluid lives in a Galilean subspace frame; the physical metricof real space is reconstructed from the propagation of the gauge field, and the Lorentz transfor-mations of special relativity emerge from the physical meter and second. The vacuum state hasfixed parameters, including a translational velocity dispersion equal to the speed of light anda spin wave speed constrained to equal the speed of light. The model predicts a torque on atorsion pendulum placed at the midplane of two counter-rotating field coil sets, with a specificscaling, sign reversal, and Lorentzian frequency response set by the spin relaxation time.
Category: Classical Physics
[5] ai.viXra.org:2610.0005 [pdf] submitted on 2026-10-02 16:40:22
Authors: Lukas Saul
Comments: 18 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We extend the spin-fluid kinetic analog gravity model of Paper I to massive charged particles.We show that the collision term in the spin-squared sector generates the quantum potential of the Madelung equations, reproducing the Schrödinger equation and yielding a formula for ħ in terms of the model's collision parameters. We then introduce the helical vortex as the soliton structure for the electron. The helical vortex has both poloidal circulation, which gives the spin, and azimuthal circulation, which gives the magnetic moment. The pitch of the helix determines the translational velocity and is always negative in the plasma reference frame. The vortex is stabilized by the Meno mechanism—orientation-dependent collisions that create an inward recoil pressure—and by the virial balance, which includes the Coulomb self-energy. We derive formulas for the electron charge, mass, and ħ in terms of the analog parameters. We show that the helical vortex reproduces the far-field electric (Coulomb), magnetic (dipole), and gravitational (Newtonian) fields, with a Yukawa correction at intermediate distances. We show that the model reproduces the full Lorentz force, Stern-Gerlach splitting, Aharonov-Bohm phase, and Larmor precession. We discuss the g − 2 anomaly and the electron electric dipole moment as targets for future quantitative work. We predict that the electron has negative active gravitational mass while the positron has positive, and we show that this is consistent with the Eöt-Wash null results because the neutron's mass is dominated by QCD binding energy, not valence quark masses. We close with a brief mention of the three families and quarks as speculative extensions.
Category: Classical Physics
[4] ai.viXra.org:2610.0004 [pdf] submitted on 2026-10-01 21:07:04
Authors: J. W. McGreevy
Comments: 17 Pages.
We present a rigorous, top-down axiomatic formulation of the Thermodynamics of ArithmeticAction (TAA) framework, anchoring the macroscopic space-time continuum within the algebraic and representation-theoretic structure of Andrew Wiles’ modularity lifting theorem (R ∼= T). We establish that the universal Galois deformation ring (R) parameterizes the continuous, smooth moduli of macroscopic metric charts (gµν), while the commutative Hecke algebra (T )natively captures the discrete, non-propagating algebraic constraints of the Einstein—Cartanspin-density torsion tensor (Σλ µν).By introducing a specialized representation-theoretic functor, the classical stress-energy ten sor (Tµν) is re-derived as a trace-free anisotropic Hecke shear matrix (∆p ≡ Tp − 12Tr[Tp]I2). At the isolated turning nodes of a four-quadrant orbital trajectory (Periapsis and Apoapsis), the systematic collapse of radial vanishing cycles forces a degenerate Frobenius lock on Pierre Deligne’s weight-11 shell boundary, causing the classical scalar mass-energy density placeholder (ρ) to vanish identically (ρ ≡ 0). Crucially, gravity does not fail; the off-diagonal elements survive as a non-vanishing vacuum expectation value, revealing that the vacuum is an activeadélic transmission medium filled with extreme, pure KMS states born from the spontaneoussymmetry breaking of the Bost—Connes C∗-dynamical system at critical inverse temperatureβ =1. We mathematically demonstrate that the continuous Weyl tensor field (Cµνρσ) emerges as thecontinuum limit of this trace-free arithmetic shear, driving the 2D transverse deformations of a photon null congruence (ds2 = 0) via the Sachs optical equations. To regularize high-frequencyphase overpressure at the 691 modular cusp anomaly threshold, the framework deploys thenon-linear Hopf—Lax infimum convolution, which acts as an automated wavefront minimizerthat eliminates phase dispersion and enforces a perfect, loss-free Heaviside Distortionless Match(RC =LG) along the Critical Line (R⌉(s) = 1/2).Finally, we evaluate the local-to-global convergence of the waveguide as the prime completion index scales upward (p → ∞). We demonstrate that as the local inverse Euler factorsasymptotically flatten to unity (limp→∞(1 − p−s) ≡ 1), they track in perfect analytical harmony with the sub-exponential saturation of the Deligne stabilizer ratio ( |τ(p)| 2p11/2 ≤ 1). This simultaneous flattening mirrors the geometric periapsis decay factor (1 − e) as the trajectory approaches the parabolic escape boundary (e → 1), smoothly shifting the system from a discrete algebraic regime into a continuous geometric manifold. This absolute, asymptotic equilibrium locks the global Tamagawa volume rigidly to unity (τ(G) ≡ 1) and secures first-principles, non-perturbative Asymptotic Safety across the spacetime continuum.
Category: Mathematical Physics
[3] ai.viXra.org:2610.0003 [pdf] submitted on 2026-10-01 07:06:06
Authors: Moninder Singh Modgil
Comments: 36 Pages.
Carbon dating tells us the age of once-living things, from ancient cloth to Ice Age mammoths,by measuring how much radioactive carbon-14 they still contain. Every carbon date quietlyassumes that time is a straight line running from the endless past to the endless future.This paper asks, in plain language, what would happen if time were instead a circle inwhich history repeats. We explain how carbon dating works, what it means for time to becircular, and why ordinary radioactive decay cannot simply continue around a circle. Wedescribe three ways the radiocarbon clock could be made to fit a circle of time and applythem to ten famous samples, including the Shroud of Turin, ¨ Otzi the Iceman, the Chauvetcave paintings and the baby mammoth Lyuba. If time repeats every 100,000 years, recentdates barely change but Ice Age dates can shift by thousands of years. If it repeats every5,000 years, every carbon date, however old, is squeezed into the last 2,500 years. We endby explaining how tree rings and other radioactive clocks can test which picture of time iscorrect.
Category: Physics of Biology
[2] ai.viXra.org:2610.0002 [pdf] submitted on 2026-10-01 15:48:26
Authors: Bolok Li
Comments: 22 Pages.
In the collector's problem with group drawings, one run draws a uniformly random q-subset of an n-element set, all q coupons being kept, and T denotes the number of runs needed to complete the set. The distribution of T and all of its factorial moments are due to Stadje (1990); we do not claim them, and we say explicitly where our formulas reproduce his -- in particular his relation (2.15) at p = 2 already yields the second moment, so item (i) below is a consolidation rather than a new formula. The asymptotics of E[T] as n -> infinity with q fixed are due to Berend and Sher (2026) and, in a sharper form, to Doumas and Spektor (2026); we do not claim those either, and Section 8 downgrades our own expectation expansion to an independent rediscovery accordingly. What we add is fourfold, of which the first is a consolidation rather than a new formula.(i) An exact closed form for the second moment, hence for the variance, obtained by summing the inclusion-exclusion tail in closed form, together with a uniform one-line derivation of every higher moment. Having now read Stadje (1990) in full we record that his general factorial-moment relation (his eq. (2.15)), specialised to A = S and p = 2, is equivalent to our second-moment formula; item (i) is therefore an explicit specialisation of Stadje's result and is not claimed as new.(ii) Two subtraction-free recursions -- a backward recursion for E[T] in O(nq) time and O(q) memory, and a forward recursion for the whole distribution -- which keep full double-precision accuracy up to n = 1000, whereas the inclusion-exclusion closed form loses all significant digits from n about 40 onwards: at (n,q) = (60,5) its relative error is 4.9 x 10^{-4} and at (80,5) it returns 1.99 x 10^{6} for the true value 77.836.(iii) A quantitative explanation: the relative error of the closed form is predicted by eps*kappa, where eps is the unit roundoff and kappa = sum_i |t_i| / |sum_i t_i| is the condition number of the alternating sum; measured and predicted errors agree to within two orders of magnitude over twenty-six decades.(iv) Asymptotics of the variance, with a complete proof. Writing A = H_n - H_{n-q}, Sigma_2 = sum_{j
Category: Combinatorics and Graph Theory
[1] ai.viXra.org:2610.0001 [pdf] submitted on 2026-10-01 18:42:31
Authors: Robert Bennett
Comments: 2 Pages.
Newton's Bucket test is generalized to include any test frame and law of physics. Formal hypothesis testing is used and test frame examples are given. A challenge is then issued to disprove the conclusion that the laws of physics - as defined in the Lagrangian model - are only valid in the ECEF or Lab reference frame.
Category: Classical Physics