The Hypergeometrical Universe: Cosmology and Standard Model

Scientific Paper
Title	The Hypergeometrical Universe: Cosmology and Standard Model
Read in full	Link to paper
Author(s)	Marco Pereira
Keywords	Cosmology, universe, Quantum Gravity, Unification, Topology, dilator, dilaton, Fabric of Space, hyperspherical expansion
Published	2012
No. of pages	27

Read the full paper here

Abstract

This paper presents a simple and purely geometrical Grand Unification Theory. Quantum Gravity, Electrostatic and Magnetic interactions are shown in a unified framework. Newton's, Gauss' and Biot-Savart's Laws are derived from first principles. Unification symmetry is defined for all the existing forces. This alternative model does not require Strong and Electroweak forces. A 4D Shock-Wave Hyperspherical topology is proposed for the Universe which together with a Quantum Lagrangian Principle and a Dilator based model for matter result in a quantized stepwise expansion for the whole Universe along a radial direction within a 4D spatial manifold. The Hypergeometrical Standard Model for matter, Universe Topology and a new Law of Gravitation are presented. Superluminal neutrinos are explained.

Overview

Marco A. Pereira's paper sets out the Hypergeometrical Universe (HU), a unification programme built not on new formalism but on a replacement of the underlying picture. He states the intention plainly: most research "is concerned on how to express some view of Nature in a mathematically elegant formalism while keeping what we already know," whereas "this work concentrates on what to say, the conceptual framework of Nature instead." Accordingly the familiar primitives — mass, charge, colour, hypercharge — are all discarded and replaced by two: dilators, which are metric modulators, and dilatons, the travelling metric modulations they emit.

The cosmological setting is a 4D spatial manifold in which the Big Bang was a four-dimensional explosion. Our 3D universe is the shockwave hypersurface of that explosion, the Fabric of Space (FS), expanding radially at light speed in quantised de Broglie steps whose length is the Compton wavelength of a hydrogen atom. Cosmological time is a genuine fifth dimension and an absolute time frame "as envisioned by Newton and Mach"; proper time is its projection onto a given frame. The departure from the mainstream account is therefore total: there is no strong force and no electroweak force, no quark composition, no Dark Energy and no inflation, and general relativity's curvature of spacetime is replaced by torsion of a 3D hypersurface embedded in four spatial dimensions.

The argument

Stress and strain across two cross-sections

The formal core is a geometric tautology. Newton's second law is rewritten as a rate of torsion of the local Fabric of Space, F = m₀c² d[tanh(α_τ)]/dτ. Adding the fourth spatial dimension gives a second expression for the same stress on the other cross-section, involving tan(α_r) and dr. Setting the two equal — the stress is the same on both cross-sections while the strain is expressed differently in each — is what permits, in Pereira's account, the derivation of Newton's, Gauss's and Biot–Savart's laws "from first principles." Mass appears as the area over which the strain acts, so that Newton's equation "can be thought as a Stress-Strain description."

The Fundamental Dilator

Matter is built from a single object. The Fundamental Dilator is a coherence between two 4D metric deformation states in a rotating four-dimensional double-well potential — the electron-proton coherence. Electron, positron, proton and antiproton are "just four faces of the same coin," distinguished by phase and orientation: p = (2/3, 2/3, −1/3), p* = (2/3, −1/3, 2/3), e = (0, −2/3, −1/3), e* = (0, −1/3, −2/3), these triplets being the semi-axes of a 4D ellipsoid of revolution. Pereira calls the four states of a coherence "dimensional notes," pressing the analogy between dilators and musical instruments.

Two kinds of mass follow. 4D mass is the capacity to generate a dilaton field and is the same for electron and proton patterns, which is why the derivations use a "fat electron" of 1.007825 a.m.u. 3D or inertial mass is the overlap of the dilator with the FS at the phases 0, π, 2π where interaction occurs. Because the overlap is non-zero only at particular spinning angles, "interaction is quantized and 'existence' is quantized" — Pereira's paradigm being "I interact, thus I exist." Since the overlap vanishes between those phases, the neutron-scale world is a Stroboscopic Universe in which interaction is intermittent.

Composite particles are polymers of this one unit. A neutron is a dimer: an electron coherence, a 90-degree rotation within the FS, a proton coherence, another rotation, the rotations being "transmutation notes," two of which make an antineutrino. The electron neutrino is the subcoherence between (2/3, 2/3, −1/3) and (2/3, −1/3, 2/3) — a shrinking of the metric along Y with simultaneous expansion along Z, hence an asymmetric dilaton field. A pion minus is a trimer of two electron and one positron subcoherences, rotating 120 degrees per de Broglie step; a coherence of n dilators pseudo-rotates by 2π/n per step. Neutron decay is given a specific number: the electron/proton FS twisting is 43.90266/−0.07294 degrees for a neutron at rest, and the stored twist is the nuclear energy released.

The Quantum Lagrangian Principle

The single dynamical rule is: dilators always dilate in phase with the surrounding dilaton field. A dilator therefore "surfs" the dilaton field, its position coinciding with the local field maximum at each de Broglie step. Because the interference pattern around a proton is azimuthally uncertain, the electron's trajectory is uncertain — this, with the pseudo-time quantisation, is offered as the basis of quantum mechanics. Since dilators are never dephased by interaction, all dilators in the universe stay in phase, giving a Cosmological Coherence that Pereira says is why the field concept works at all.

Deriving the force laws

A dilaton field is written as a decaying cosine, cos(k₁r)/[P(1 + f(k₁,r))], with the phase volume P = 3. Superposing the fields of a probe dilator and a 1 kg_4D body and locating the shifted maximum gives the displacement per de Broglie step, hence the acceleration, hence the force. Two results are claimed. Setting the anisotropy coefficient α = 1 recovers the electrostatic coupling G_{Electrostatic} = 8.23558 × 10²⁵ without ever using charge or the permittivity of vacuum; setting α = 1/R₀ and fitting to Newton's G = 6.6720 × 10⁻¹¹ gives the "elasticity of spacetime." Running the identification backwards yields ε₀ = 8.85418782 × 10⁻¹², which Pereira presents as a perfect match derived from only the electron charge, c, Avogadro's number and Planck's constant.

Setting the derivative of the summed waveform to zero gives sin(k₁r₀) = cos(k₁r₀), i.e. a limiting angle of 45 degrees — the speed of light as a limit "without any postulate as in the Theory of Relativity." The same construction extended to two current elements, with the 5D rotation matrix expanded to first order in v/c, reproduces the Biot–Savart law and recovers the relation between μ₀ and ε₀. The velocity-dependent form is then carried over to gravitation to give the Hypergeometrical Universe Law of Gravitation, a single expression that reduces to Gauss's law at rest and to Newton's law at zero relative velocity.

Cosmological consequences

Several distinctive claims follow from the topology.

The true speed of light is √2 c. Light travels at 45 degrees to the radial direction; the observed c is the projection.
The Pioneer anomaly is geometric. Working the light path back from a receding spacecraft on the expanding hypersphere and taking the second derivative gives a deceleration 2c²/R₀ − 2v²/R₀. Fitting the observed 8.75 × 10⁻¹⁰ m/s² yields R₀ = 2.05 × 10²⁶ m and an age of 21.72 billion years rather than 15.36. The Hubble constant is recovered only in the small-angle limit; at large distances H₀ is replaced by an expression in cos(α).
Redshift is geometric. Looking further out changes the observation angle from 45 degrees to zero, which changes the light's wavelength by geometry alone.
G weakens with time, being inversely proportional to R₀. Gravity was stronger in the early universe, so stellar candles were less massive and less luminous in the past — which, Pereira argues, corrupts intensity-based distance measurements and removes the need for dark energy or inflation.
Grand unification is a moment, not a group. Since gravitational and electrostatic strengths differ by the factor 14.57 × 10⁴/R₀, they were equal at R₀ = 14.57 × 10⁴λ₁, i.e. 6.40 × 10⁻¹⁹ s into the universe's life.
Antimatter falls up. The unification equation "shows that anti-matter will have gravitational repulsion or anti-gravity with respect to normal matter."
A natural frequency for gravitational waves of 37.6 kHz (given as 37.02 kHz in the conclusions), independent of any mass.
Black hole jets. Because the velocity-dependent gravitational force is asymmetric — inward pull stronger than the retarding pull — a black hole can propel itself, giving single or, for symmetric matter distributions, double jets.
Superluminal neutrinos. Light is a symmetric dilaton mode; the electron neutrino is an asymmetric one of higher frequency, and "the asymmetric dilaton mode with higher frequency has a higher speed than the symmetric dilaton mode."

Classical tests

Two standard tests are addressed. Setting V₁ = 0 in the velocity-dependent potential yields, Pereira states, exactly Gerber's 1898 potential, which reproduces the 42.3 arcseconds per century precession of Mercury's perihelion. For lensing, photons are dilaton fields with m = ħk/c, and the phase-matching condition at closest approach gives a deflection Δk/k = GM/c²R₀.

Assessment

There is real ambition and real internal discipline here. The programme is genuinely reductive: one object (the fundamental dilator), one rule (the Quantum Lagrangian Principle) and one topology are asked to deliver electrostatics, magnetism, gravitation, inertia, the particle spectrum and cosmology, and Pereira does not help himself to extra machinery when a case gets hard. The stress–strain reading of Newton's second law is an elegant way to make force geometric without a metric tensor, and the observation that a single velocity-dependent expression can reduce to Gauss's law at rest and Newton's law at zero relative velocity is the kind of structural unification the paper advertises. The identification of ε₀ from e, c, ħ and Avogadro's number would be striking if it held. The theory is also unusually willing to be shot at: 21.72 billion years, 37 kHz, √2 c, antimatter antigravity, G ∝ 1/R₀ and superluminal neutrinos are all numbers or signs, not hedges.

Most of them have since gone the wrong way, and several were already in trouble. The superluminal neutrino result the paper builds on — OPERA's 2011 announcement — was withdrawn in 2012 after a loose fibre-optic connector and a clock oscillator were identified as the cause; SN 1987A neutrinos, arriving within hours of the light after 168,000 years, already bounded any excess to about one part in 10⁹, and the paper does not address that constraint. The Pioneer anomaly, the sole quantity from which R₀ and hence the age of the universe are fixed, was resolved in 2012 by Turyshev and colleagues as anisotropic thermal re-radiation from the spacecraft's own RTGs and electronics; that removes not merely a supporting datum but the paper's only numerical anchor, and with it the 21.72-billion-year age and the elasticity coefficient. Antimatter antigravity has since been excluded: ALPHA-g's 2023 measurement finds antihydrogen falls down with roughly Earth's gravitational acceleration. The predicted 37 kHz natural frequency of gravitational waves is in the band LIGO is most sensitive to, and the events actually detected are chirps sweeping tens to hundreds of hertz from binary inspirals, matching general-relativistic waveforms.

There are also difficulties internal to the argument. The number-theoretic assignments of the standard model — the triplets (2/3, 2/3, −1/3) and the rest — are presented as displacement-volume semi-axes, but no calculation is offered connecting them to any measured mass; Pereira concedes as much, writing that "further research should follow to pinpoint the exact angles and their application to the mass calculation of the subatomic particles and isotopes." Until that is done, the claim to replace quark composition is a relabelling rather than a derivation, and it inherits no account of the hadron spectrum, of deep inelastic scattering structure functions, of jets, or of the running of the strong coupling. Similarly, dispensing with the electroweak force leaves parity violation in beta decay, the measured W and Z masses and the weak mixing angle unaddressed. The 43.90266/−0.07294 degree figures for neutron decay are quoted to seven figures without a shown derivation.

The two classical tests are the weakest passages, and for a specific reason. Reproducing Gerber's potential is not a success but a warning: Gerber's 1898 expression gives the correct Mercury precession yet is known to give the wrong light deflection — half the observed value — and was criticised on those grounds by Seeliger and by von Laue. Pereira's own lensing result, Δk/k = GM/c²R₀, contains R₀, the radius of the universe, where the observed deflection depends on the impact parameter at the deflecting mass; as written it does not reproduce the measured 1.75 arcseconds at the solar limb, and the assertion that it is "the observed Gravitational Lensing" is not demonstrated. More generally the theory is nowhere tested against the post-Newtonian parameters, the Shapiro delay, or binary pulsar orbital decay.

Finally, the claim that G varies as 1/R₀ is directly measurable and directly excluded. Lunar laser ranging and binary pulsar timing bound the fractional rate of change of G to below about 10⁻¹³ per year, whereas 1/R₀ scaling implies roughly 10⁻¹⁰ per year. Big Bang nucleosynthesis abundances and the acoustic peak positions in the Cosmic Microwave Background add independent constraints. The proposal that this variation explains away Dark Energy by corrupting standard candles also runs into the fact that the Type Ia supernova result is corroborated by the (1+z) time dilation of the light curves themselves, by baryon acoustic oscillations and by CMB geometry, none of which depends on candle luminosity.

Read as a conceptual experiment — what would physics look like if matter were metric deformation and force were torsion of an expanding shockwave? — the paper is coherent and worth the reading. Read as a competing theory, it needs its numerical anchor replaced, its lensing calculation redone, and its particle assignments turned into computed masses before the comparison with experiment can begin.