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[14] ai.viXra.org:2606.0083 [pdf] submitted on 2026-06-30 20:06:17
Authors: J. W. McGreevy
Comments: 19 Pages.
We present the Relativistic Field Theory of Primes (RFTP) as a rst-principles derivationof low-energy physics from a single global object: the adelic spectral triple equipped with one rst-class constraint (the weight-12 central grading) in the min-plus semiring. Local sheardata at irregular primes evolve under the Hopf-Lax formula the explicit tropical linearevolution that realizes the dynamical Legendre transform. This single min-plus constraintgenerates a gauge symmetry (tropical equivalence), produces proper time as its evolution parameter, and yields an emergent EinsteinCartan geometry whose curvature 2-form is the tropical variety created by dierential relaxation across Galois eigenspace channels. Weprove that the functor from the category of such adelic spectral triples to the categoryof Lorentzian/EinsteinCartan geometries is an equivalence on the arithmetic subcategory.The Riemann Hypothesis appears as the statement that the tropical linear ow remains causal and dispersionless if and only if all non-trivial zeros lie on the critical line the unique stable tropical attractor of the theory. Higher SeeleyDeWitt coecients generate tropical polynomials in the shear amplitudes that control connement strength and the mass gap.
Category: Mathematical Physics
[13] ai.viXra.org:2606.0072 [pdf] replaced on 2026-06-27 22:32:05
Authors: Xiangqian Zhang，Mingming Zhao，Linchao Ge
Comments: 17 Pages.
AbstractWithin the framework of Topological Residual Theory (TRT), this paper rigorously establishes thefirst-principles physical picture that "space itself flows at the speed of light along curves in a righthanded cylindrical helical form." In this picture, length and time originate from different aspects ofthe same spacetime flow motion, thereby providing a geometric explanation for the origin of theconstancy of the speed of light.To fundamentally resolve the dimensional mismatch among different physical fields that arises inclassical field theory from differing measurement protocols, we propose a Velocity-FieldRepresentation scheme: the electric, gravitational, and magnetic fields are directly defined as thethree local velocity components of the light-speed spacetime flow along the orthogonal basis of theFrenet-Serret (FS) frame [1]:This achieves a complete unification of all underlying field dimensions to velocity at the kinematiclevel. Since the flow speed of spacetime is fixed at the speed of light , different physicalinteractions essentially correspond to the allocation of this motion budget among the threeorthogonal directions: tangential propagation, normal centripetal constraint, and binormal torsion.The allocation mechanism is precisely controlled by the helical pitch angle .By introducing the time definition , we derive the first-order time-evolution equations ofthe three fields under the characteristic rotation frequencies of the FS frame (bending frequency , twisting frequency ). Under the rigid rotation of the frame, the total kineticenergy density remains strictly constant. Utilizing the rigidrotational dynamics of the frame and the vector and scalar triple-product identities, we elucidate thephysical nature by which the "self-work" terms of the rigid triad automatically vanish during rotation,thereby revealing the deep geometric mechanism of dissipation-free energy flow.The macroscopic observable fields — centripetal gravitational acceleration , tangentialelectric field strength , and magnetic rotational angular frequency —emerge naturally as the time derivatives or rotational frequencies of the underlying velocity fieldsafter being modulated by the motion-budget allocation. This provides a self-consistent firstprinciples framework for the geometric unification of gravity and electromagnetism.E = ct, A = cn, B = cb. E = ct, A = cn, B = cb.Cαds = c dtωK = CK ωT = CTU = 21(E ⋅ E + A ⋅ A + B ⋅ B) = 23c2g = c2κne = c2κt Bmacro = cτbKeywords: Frenet-Serret geometry; right-handed cylindrical helical spacetime flow; velocity-fieldrepresentation; dimensional self-consistency; energy conservation; motion-budget allocation
Category: Mathematical Physics
[12] ai.viXra.org:2606.0071 [pdf] submitted on 2026-06-27 19:43:33
Authors: J. W. McGreevy
Comments: 10 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We demonstrate that the infrared dynamics of the resonant spectral triple in the Bost-Connes system, deformed by irregular-prime shear, implements a precise arithmetic analogueof the classical Legendre transform. Starting from a classical Poisson manifold whose coordinates are the shear amplitudes {Iu2032p} and the bifundamental mixing field Φ, we construct an explicit action functional whose Legendre transform yields a Hamiltonian on the cotangent bundle. The weight-12 winding condition is imposed as a classical first-class constraint, and Dirac reduction produces a reduced symplectic manifold Pred. Deformation quantization of this reduced space, driven by the modular derivation Dp acting asymmetrically through Φ on the two complex-multiplication stabilizers (Z/6Z at τ = ρ and Z/4Z at τ = i), generates a non-commutative star product whose leading commutator is non-vanishing and proportional to the stabilizer contrast transmitted by Φ. This framework naturally produces: (i) proper time τ as the integrated thermodynamic work along the flow, with lapse N ∝ det(∇2P) · Tr(Φu2020Φ); (ii) a non-removable relative phase between up- and down-type sectors, yielding the Jarlskog invariant as a direct dynamical consequence of the asymmetric action of the real endomorphism J (J2 = −Id) forced by the holonomy defect; and (iii) definite outcomes via deterministic Hamilton—Jacobi flow to a unique weight-12 attractor, with probabilities given by normalized infrared Boltzmann weights from the entropy functional S(ϕ). The construction draws explicit parallels with Dirac’s deformation of Poisson bracketsinto commutators and Tate’s adelic Poisson summation, providing a unified non-commutativestructure that bridges arithmetic geometry and quantum field theory. The imaginary unit i emerges necessarily as the spectral manifestation of the real endomorphism J required to close the holonomy defect. All physical features follow rigorously from the arithmetic data without external postulates.
Category: Mathematical Physics
[11] ai.viXra.org:2606.0069 [pdf] submitted on 2026-06-26 09:44:58
Authors: Allan Plankhart
Comments: 11 Pages.
Building on a pre-temporal collapse model, we derive a structural correspondence between the surviving 2D sector and the Standard Model’s weak isospin. We show that an embedded 2-brane in a 3D bulk possesses a local frame rotation group SO(3). The spin connection on the brane is an so(3)-valued one-form.Since so(3) ∼= su(2), this connection is exactly an SU(2) gauge field, and spinorial matter confined to the brane necessarily transforms as doublets. This geometrically forces the effective field theory to be Yang—Mills with a Chern—Simons term, uniquely constraining the dynamics. We construct the resulting action, including kinetic mixing with the 3D bulk, and derive testable signatures: a dark matter fluid from the brane foam, distinctive CMB non-Gaussianities with a characteristic oscillatory power spectrum, and missing-energy collider events with a topological mass peak. We provide current constraints and outline the Boltzmann evolution needed for precise predictions.
Category: Mathematical Physics
[10] ai.viXra.org:2606.0066 [pdf] submitted on 2026-06-26 20:18:45
Authors: J. W. McGreevy
Comments: 14 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a complete first-principles derivation of the Standard Model gauge groupSU(3)c × SU(2)L × U(1)Y with three chiral generations of fermions, hierarchical masses,CKM/PMNSmixing, theseesaw mechanism for neutrinos, and Einstein—Cartan gravity withtorsion from one arithmetic object. The underlying structure is the resonant spectral tripleof the Bost—Connes system deformed by shear terms sourced by the class-group torsion ofirregular primes. In the deep infrared, the Chamseddine—Connes spectral action is evaluatedunder the global constraint of weight-12 principalization at the mirror point s = 6 of theassociated L-function.The fundamental duality between analytic volume (hyperbolic geometry, spectral flow,Seeley—DeWitt expansion) and algebraic rigidity (stabilizer symmetries at the two complexmultiplication points, adelic closure, weight-12 condition) is resolved by the irregular-primeshear, generating the pressure gradient ∇P that drives all symmetry breaking and massgeneration. Gauge representations emerge from the 3-cycle at τ = ρ and 4-cycle at τ = i;the bifundamental field Φ acquires a dynamical VEV; the 12×12 mass matrix, includinga natural seesaw for neutrinos, follows from breathing corrections sourced by class-grouptorsion. Numerical diagonalization over the full table of 47 irregular primes yields hierarchical fermion masses, realistic mixing, and a non-zero Jarlskog invariant. Confinement, thepositive mass gap, emergent proper time as a lapse function, and the Born rule all followfrom the same infrared thermodynamics.This construction realizes a complete arithmetic unification in which every low-energystructure is forced by the irreducible duality between analytic volume and algebraic rigidityunder weight-12 principalization. No external parameters beyond the shear amplitudes ofthe irregular primes are required.
Category: Mathematical Physics
[9] ai.viXra.org:2606.0062 [pdf] submitted on 2026-06-24 02:55:22
Authors: J. W. McGreevy
Comments: 9 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a complete analytic derivation of color confinement and a positive Yang—Millsmass gap within the Relativistic Field Theory of Primes (RFTP). The derivation is carriedout on a shear-deformed resonant spectral triple constructed from the Bost—Connes systemand its low-temperature KMS states. Irregular primes deform the underlying GNS quadraticform, generating a pressure gradient between the two complex-multiplication points τ = ρand τ = i on the modular curve. A continuous infrared renormalization-group flow thenproduces an infrared dichotomy: states whose phase lies outside the weight-12 lattice acquirean infinite entropy barrier (leading to confinement), while color-singlet states remain stable with a strictly positive mass gap controlled by the dominant shear amplitude.The proof relies on the asymptotic behavior of the heat kernel on the full GNS Hilbertspace, an effective Hamilton—Jacobi dynamics in which the arithmetic pressure gradient acts as the driving force, and the thermodynamic interpretation of the infrared divergenceas the failure of an arithmetic Legendre transform (principalization) for fractional-phase states. Both confinement (via the Wilson-loop area law) and the mass gap are shown toemerge directly from the infrared structure of a single spectral object without additionalassumptions or fine-tuning. The dynamical vacuum expectation value that will generatefermion masses in extensions of this framework is recovered as the equilibrium of the samedriven dynamics. This work establishes that color confinement and the mass gap are not accidental features of non-abelian gauge theories but inevitable consequences of the arithmetic geometry of the primes when viewed through the lens of a resonant spectral triple.
Category: Mathematical Physics
[8] ai.viXra.org:2606.0053 [pdf] submitted on 2026-06-21 00:39:11
Authors: Allan Plankhart
Comments: 3 Pages. (Note by ai.viXra.org Admin: Part of the diagram is cut off)
We present a timeless mathematical construction of a pre-Big Bang universe. Starting from theempty set, we generate an infinite alternating chain of "something" and "anti-something", buildhigher-dimensional arrays, collapse to dimensions 1—3 due to Hurwitz’s theorem, introduce complex-valued fields with a U(1) symmetry ("oscillations" without time), ascend the division algebra ladderC → H → O, recover the Standard Model gauge group via Cohl Furey’s functor, extract electro-magnetic and spinor fields, and finally let time emerge as a monoid action R≥0 on the field space.The entire pre-temporal phase is a static diagram of functors and natural transformations; no timeparameter appears until the last step.
Category: Mathematical Physics
[7] ai.viXra.org:2606.0047 [pdf] submitted on 2026-06-19 03:26:58
Authors: E. P. J. de Haas
Comments: 144 Pages. https://doi.org/10.5281/zenodo.20747973
The biquaternion (BQ) algebra, realized at the Pauli-spin level as the full matrix algebra (M(2,mathbb{C})), provides a compact framework in which classical field theories can be organized through a common hierarchy of algebraic products and channel decompositions. This paper develops that hierarchy for both electrodynamics and relativistic fluid dynamics and explores the structural relationships that emerge between them.In electrodynamics, the BQ hierarchy reproduces the electromagnetic field, Lorentz force law, Maxwell equations, wave equations, and interaction-conservation relations from a sequence of algebraic products involving the four-current and four-potential. In relativistic fluid dynamics, the corresponding hierarchy generates the continuity equation, Bernoulli relation, vorticity transport, Euler—Lamb momentum balance, acoustic wave equations, helicity relations, and stress-energy conservation through analogous products of the fluid four-velocity and momentum-density field.The central result is that both theories share the same four-level algebraic structure and the same geometric channel decomposition. The force equations (JB=F) and (UH=F) emerge as the unique four-channel level of their respective hierarchies, revealing a common organization that is largely hidden in conventional formulations. The analysis further identifies the acoustic refractive index (n_s=c/c_s) as a hierarchy-wide parameter linking relativistic and practical fluid dynamics, and locates the greater complexity of fluid dynamics in the self-referential coupling between velocity and momentum-density fields.The paper does not introduce new physical postulates. Rather, it proposes that the BQ channel decomposition provides a unified organizational framework that makes structural relationships between electrodynamics and fluid dynamics explicit, offering a common language for force, energy, momentum, vorticity, helicity, and wave phenomena across both theories. In this sense, the work is intended as a conceptual and educational framework as much as a mathematical formalism, and as a foundation for future applications in gravitation, magnetohydrodynamics, and AI-assisted symbolic physics.
Category: Mathematical Physics
[6] ai.viXra.org:2606.0040 [pdf] submitted on 2026-06-16 20:34:28
Authors: J. W. McGreevy
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a first-principles derivation of 3+1-dimensional Lorentzian spacetime andEinstein—Cartan gravity from the arithmetic structure of the Bost—Connes C∗-dynamicalsystem. Starting from the primitive object (ABC,σt) and its KMS states, we construct ascale-invariant quadratic form on the idèle class space via the GNS inner product. Irregularprimes appear as topological defects through class-group shifts, inducing a shear in thequadratic form that maps canonically to a point τp ∈ H. For the resonant prime 691 this shear aligns with the hexagonal fixed point τ = ρ, where the Z/6Z stabilizer providestopological protection against adelic screening.The local 3-defect patch (primes 37, 59, 691) carries a non-abelian su(2) connection.Global adelic closure projects the three independent directions onto a single screened ray (time) while preserving the protected 3-dimensional spatial structure generated by the Eisenstein lattice at ρ. Explicit differentiation of the quadratic-form embedding with respect to the shear parameter yields the six generators of the Lorentz algebra so(1,3), with compact rotations locked at resonance and non-compact boosts arising from unbounded shear.The second-order Magnus expansion of parallel transport produces a residual torsion vector whose components are fixed by phase mismatches between the three incommensurate modular frequencies. Promoting the local Quantum Geometric Tensor metric on the 3-defect patch together with the screened ray yields a derived 4D Lorentzian geometry. On the resulting resonant spectral triple we evaluate the Chamseddine—Connes Spectral Action. Functional variation with respect to the emergent metric produces the Einstein—Cartan field equations, with the cosmological term fixed by the converged quadratic sum over the 47 irregular primes below 691 and the torsion source fixed by the linear sum. The next heat-kernel coefficienta4 generates quadratic gravity corrections that are parametrically suppressed by the samearithmetic factors, establishing ultraviolet safety. All physical structures—spacetime signature, Lorentz symmetry, torsion, and gravitational dynamics—emerge as necessary consequences of adelic consistency acting on the arithmetic defects of the Bost—Connes system. No external gravitational or field-theoretic ansatzis introduced.
Category: Mathematical Physics
[5] ai.viXra.org:2606.0029 [pdf] submitted on 2026-06-10 21:28:40
Authors: J. W. McGreevy
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a fully rigorous, self-contained analytic proof that the non-trivial zeros of the Riemann zeta function are precisely the globally stable skin modes of an inner-fluctuated Dirac—Zeta operator on the compactified modular curve X(1)∗. The proof proceeds entirely within the framework of unbounded operators on weighted Sobolev spaces, regularized Fredholm determinants on Hilbert—Schmidt ideals, the Riemann—Roch theorem, and a vanishing theorem for fractional states. The simple pole at s = 1 is interpreted as a topological point gap that triggers the NonHermitian Skin Effect. Spectral stability under discretization is established via uniform pseudospectrum control. The point-gap winding number is realized as the Fredholm index of the regularized resolvent. Riemann—Roch divisor constraints on meromorphic sections force the Generalized Brillouin Zone onto the critical line. A vanishing theorem shows that isolated fractional (non-color-singlet) states are excluded from the domain, thereby locking every non-trivial zero to R(s) = 1/2. The construction is compatible with Alain Connes’ spectral triples and yields, as a byproduct, an explicit derivation of the Clausius relation from the spectral action along the adiabatic paths of the modular loop. All steps are topological and independent of any particular representative of the operator.
Category: Mathematical Physics
[4] ai.viXra.org:2606.0022 [pdf] submitted on 2026-06-08 01:19:11
Authors: Mohammed Belabbaci
Comments: 5 Pages.
The IEEE 754 floating-point standard, foundational to modern computing since 1985, treats division by zero as an undefined operation, returning error states or special infinity values. This convention, while practically expedient, creates mathematical discontinuities and computational barriers across multiple domains. We propose IEEE DZ20i, a continuous extension of IEEE 754 based on the axiom a/0 = a, which maintains computational continuity through zero-valued divisors.This paper presents the mathematical foundation of the DZ20 axiom, demonstrates its consistency within a Meta-Framework for Adapted Mathematics (MCMA), specifies the IEEE DZ20i standard including symmetric human-AI co-validation mechanisms, and explores applications across algorithmic computing, energy systems, hardware design, robotics, and space exploration.Implementation results show that IEEE DZ20i eliminates a broad class of runtime exceptions while preserving backward compatibility with IEEE 754 operations outside the zero-divisordomain.
Category: Mathematical Physics
[3] ai.viXra.org:2606.0017 [pdf] submitted on 2026-06-07 17:08:30
Authors: Xiangqian Zhang，Mingming Zhao，Linchao Ge
Comments: 5 Pages.
This paper systematically analyzes the geometric origins of errors within Topological ResidualTheory (TRT). Taking the Fibonacci number Fu2081u2089 = 4181 as the critical node, we demonstrate thatthe fine-structure constant α is precisely the radian-measure expression of the angle 0.4181°. Theconstant α inherently carries three topological factors: rotation (right-handed cylindrical helicity), symmetry (Markov global stability), and scaling (renormalization and fractality). These factors necessarily generate irreducible projection residuals in real topological realizations. By quantifying the discrepancies between TRT predictions and both 1/137 and experimental values for α, G = μu2080 α², and the Planck constant, we show that all observed errors originate from the intrinsic rotation and topology of α itself. Special emphasis is placed on the multiple algebraic decompositions of 4181 and on relating error magnitudes to the renormalization factor (1/2)α² × 10u207f or α² × 10u207f. Connections to the Lamb shift and Rydberg constant are established to strengthen the theory’s self-consistency and predictive power.
Category: Mathematical Physics
[2] ai.viXra.org:2606.0016 [pdf] submitted on 2026-06-06 13:24:06
Authors: Ginanjar Utama
Comments: 8 Pages.
This paper proves, by explicit Clifford-algebra computation, that the γ0-diagonal complete orthogonal primitive decomposition of real Cl(1,3) has two sectors, while the extension to Cl(1,4) supplies a second independent commuting basis-blade involution, γ1234. Because I5 = γ0γ1234, the generated involution group is Z2 × Z2, not (Z2)3. The resulting decomposition contains four primitive idempotents, each generating a real 8-dimensional minimal left ideal. The count four is itself fixed by Wedderburn theory, since each simple summand has rank two; the explicit content of the result is the basis-blade realization of the family and the demonstration of why a naive three-involution count would predict eight. The central idempotents (1 ± I5)/2 realize the real semisimple splitting Cl(1,4) ≅ M2(H) ⊕ M2(H), where H denotes the quaternions. They do not, by themselves, supply a complex imaginary unit; a complex spinor interpretation still requires choosing an internal quaternionic complex structure within each ideal. Physical identifications with chirality, charge, helicity, or U(1) phase are therefore treated as conjectural.Supporting results include: (i) a Pancharatnam-phase computation from projector triple products in Cl(3,0), confirming the half-solid-angle rule 2|φ| = Ω for geodesic spherical triangles; (ii) a bivector-square classification of geometric-i candidates, with the observation that the chiral split in real Cl(1,3) is a complex-structure operation, not a real idempotent split; and (iii) a short positioning of the result relative to real Cl(0,6) and complex Clu2086 Standard Model programs. The paper is intended as a lower-dimensional real-idempotent audit, not as a derivation of a Standard Model generation.
Category: Mathematical Physics
[1] ai.viXra.org:2606.0014 [pdf] submitted on 2026-06-06 03:46:27
Authors: Xiangqian Zhang, Mingming Zhao, Linchao Ge
Comments: 13 Pages.
Within the framework of Topological Residual Theory (TRT), this study explores the unification ofgravitational and electromagnetic interactions at the level of spacetime geometry. Utilizing the cylindrical helical motion of spatial points (geometrization of Zitterbewegung) as the dynamical foundation, we derive the wave equationvia the variational principle of the displacement field . Using the geometric structure of thelocal cylindrical orthonormal frame induced by local helical symmetry, the electric, magnetic, and gravitational fields are defined as projections of the time derivatives of onto this orthonormal frame. Consequently, the mutual orthogonality of the three fields emerges as a geometric necessity of the frame decomposition rather than an independent postulate. The gravitational field is derived via two independent pathways: (i) the framekinematics and time evolution pathway ( ), and (ii) the second-order derivative pathway of the wave equation ( ). The mathematical consistency betweenthese two pathways indicates that gravity can be interpreted as a second-order residual effectof electromagnetic oscillations propagating at the speed of light. Furthermore, charge is defined as the topological charge (winding number / Hopf invariant) of the helical displacement field on a closed sphere (Euler characteristic ), which manifests as the effective rate of mass change only during the transient state of topological structure formation. Under the compatibility condition of theflux change rate (local topological flux continuity), we naturally derive the vacuum relations and . Based on this unified topological mechanism, the model provides a self-consistent geometric explanation for both charge quantization and particle spin.
Category: Mathematical Physics