Works – Andrea Brussi

A pure mathematical version of the Clay Mathematics Institute’s Navier-Stokes Millennium Problem

Abstract
This report provides a rigorous transition from the physical formulation of the incompressible Navier-Stokes equations to their pure mathematical representation as defined by the Clay Mathematics Institute for the Millennium Prize Problems. We detail the physical meaning of each term in the classical equations and subsequently reformulate the problem strictly in the language of partial differential equations (PDEs) and operator theory, establishing the axiomatic constraints that govern the variables and parameters of the system.

Brussi 2026_Navier-Stokes Millennium Problem

Version uploaded to Zenodo: DOI – zenodo.21852870

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Mathematical and historical foundations of Extended Theories of Gravity (ETG)

A short paper, in english, on a fascinating development of General Relativity.

Abstract
Extended Theories of Gravity (ETG) represent a prominent framework in modern theoretical physics and cosmology, aimed at modifying or extending Albert Einstein’s General Relativity to address open phenomenological questions at both ultraviolet (early universe) and infrared (galactic and cosmological) scales.

Brussi 2026 Mathematical and historical foundations of ETG (305 KB)

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Galactic Energy Profiles II

Galactic Energy Profiles: GEP II. In-System and Background Energy Densities

Update: I decided to publish this work, part of the GEP series, on Zenodo without peer review and to use it as a basic tool for building a subsequent dataset (GEP III). Note: This work is version 2.0 of the previous NAGE II work, which has been superseded.

Abstract
Standard galactic energy inventories typically focus on virialized components, namely kinetic and gravitational potential energy derived from baryonic mass distributions, together with energy inferred from bolometric luminosities. In the first paper of the Galactic Energy Profiles (GEP) series, we introduced a reproducible geometric framework for the radiant and relativistic fluxes of galactic origin, accounting for their finite escape time across the Halo. In this second work, we extend the framework toward a volumetric energy inventory, where bolometric luminosity is complemented by energy densities persistently present within the galactic volume. We consider the thermal and kinetic energy of baryonic matter—including bulk motions, turbulence, and rotational degrees of freedom from large-scale dynamics down to the intrinsic angular momentum of bound systems—together with cosmic rays confined by magnetic fields and extragalactic backgrounds (photons, neutrinos, and diffuse fields) permeating the Halo. This approach distinguishes between flux-based contributions and volume-based reservoirs, providing a complementary description of the total galactic energy budget. Crucially, by resolving these volumetric reservoirs into local radial profiles, we characterize the multi-component pressure support (including thermal, magnetic, cosmic ray, and radiant pressures) available to sustain the circumgalactic medium (CGM) in hydrostatic equilibrium. Although these components vary with morphology and evolutionary state, this study is purely theoretical and simplifies the extreme variability of galactic systems to derive reference estimates and ‘standard cases’. The resulting parametrized framework, supported by publicly available Python scripts, is intended as a further methodological baseline for the sample-wide application planned in later work.

Keywords
galactic energy inventory; galactic energy density; galactic Halo energy; energy distribution based on galactic morphology

Here is the Zenodo version of the paper (1.1 MB):

Brussi 2026_GEP II

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Galactic Energy Profiles I

Galactic Energy Profiles: GEP I. Radiant and Relativistic Fluxes

Abstract
Standard galactic energy inventories typically focus on the virialized components of the system, namely the kinetic and gravitational potential energy, derived from baryonic mass distributions and stellar luminosity. This paper, the first in the Galactic Energy Profiles series, introduces a reproducible geometric framework for accounting radiant and relativistic fluxes as a standing energy reservoir. By considering the photon and neutrino escape time τ across the galactic Halo, we quantify the energy density of radiation currently in transit as a component of the total galactic budget, complementary to the standard flux-based description. The resulting closed-form expressions, together with the accompanying open-source Python implementation, are intended as a calibrated methodological baseline for the subsequent application of this framework to observed galactic samples.

Keywords
galactic energy inventory; energy residence time; galactic radiant fluxes; galactic Halo energy; galaxy morphology

Here is the Zenodo version of the paper (1.1 MB):

Brussi 2026 GEP I

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The URD Framework – under development

The URD Framework: Cosmological Distances from Redshift in a Non-Standard Context

While I am still developing the project —including through parallel studies on the Tolman test and the quality of indirect data obtained from recent galaxy surveys— I am making my work public as a preprint. I always hope that someone might be interested in my conjectures or perhaps even willing to collaborate with me to address the critical issues of the URD Framework.

Abstract
A phenomenological framework for determining cosmological distances is proposed, based on a non-expanding spacetime interpretation consistent with current observational datasets. The model introduces a redshift mechanism associated with cumulative photon energy attenuation along the line of sight, formulated in a way that is not equivalent to standard tired light scenarios. The approach is constructed to remain compatible with established relativistic principles whereas adopting boundary conditions distinct from the metric expansion paradigm. Distance estimation is achieved through the URD equation (U energy density, Redshift, Distance), which incorporates two phenomenological parameters that describe conformal geometric dissipation in extended gravity and an effective relativistic path-length correction. Within this formulation, the model addresses the primary shortcomings traditionally associated with energy dissipation effects, specifically reconciling time dilation in supernova light curves and the Tolman surface brightness relation. The model is calibrated using the Pantheon+ supernova sample by comparing and cross-validating distances inferred from the proposed redshift mapping with redshift-independent indicators, with additional comparison performed against baryon acoustic oscillations, radio galaxy samples, and high-energy quasars. Across these datasets, the framework reproduces observational constraints over a wide redshift range within a phenomenological, non-standard cosmological setting. The results indicate that alternative, non-expansion-based mappings between redshift and distance can be constructed that remain consistent within the observational range considered, suggesting a phenomenological alternative worth further study.
Keywords
cosmological distance ladder; redshift-distance relation; non-expanding spacetime; cosmic microwave background; extended gravity; Pantheon+ sample; baryon acoustic oscillations; Tolman test; time dilation; geodetic correction

Here is the last version of the preprint (5.1 MB):

Brussi 2026_URD Framework Preprint

Version uploaded to Zenodo:

DOI – zenodo.20763682

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Cosmology for a static universe

Abstract

This brief work, then became an Appendix to my URD Framework study, proposes a cosmological model characterized by a globally flat, spatially infinite, and non-time-limited spacetime. It challenges the standard application of the Friedmann-Lemaître-Robertson-Walker metric by reinterpreting general relativity as a strictly local phenomenon within a non-continuum matter distribution. In this framework, the observed flatness of the universe is a fundamental geometric property rather than a dynamical result of inflation, and cosmological redshift is modeled as a cumulative energy dissipation process rather than metric expansion.

Brussi 2026_Cosmology for a static universe

Ai generated image from free Adobe stock

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The virial theorem and stellar equilibrium

The virial theorem is a fundamental analytical result in the mechanics of particle systems, as it establishes a rigorous connection between the time averages of kinetic and potential energies. Its validity extends to systems in dynamic equilibrium, where internal forces are governed by potentials that depend on distance according to a power law, as in the case of universal gravitation or electrostatics.
The work is written in italian.

Brussi 2026_Il teorema del viriale e l’equilibrio stellare

Teorema del viriale

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The error function as an integration trick

The error function, erf(x), represents a statistical function, but it can also be used as an integration trick in the case of complex exponentials, which occur fairly frequently in the study of physics, for example, in differential equations describing heat propagation. It is also used in quantum mechanics to describe particles represented by a wave function, and in astrophysics for the spectroscopic analysis of spectral lines, as in the examples discussed in detail (in english and italian translation).

Brussi 2026_The error function as an integration trick

Brussi 2026_La funzione di errore come trucco di integrazione

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Dedekind’s theorem: a reconsideration of the demonstration

I had second thoughts about the draft proof already posted here.
Using a couple of theorems (called 1.1.3 and 1.1.8, attached), the proof becomes much simpler and more straightforward.
But I think I am justified in my oversight, as I studied these theorems so long ago that Tim Berners Lee had yet to invent the www ; )

Brussi 2026_Theorem_1.1.3 and 1.1.8

Brussi 2026_Dedekind theorem 2

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Dedekind’s theorem: a draft proof

A draft written in Italian proves the existence and uniqueness of an element that separates two non-empty sets representing the field of real numbers.
This allows us to prove that the real numbers are a complete space (i.e., that every Cauchy sequence is convergent).

Note: in the case of irrational numbers instead of rational numbers, it does not work because they are not complete, i.e., it is not possible to define an element that is an extreme.
In layman’s terms, one could say that if rational numbers are removed from the real numbers to obtain irrational numbers, then ‘gaps remain’ and completeness no longer exists.
upgrade: a reconsideration of the demonstration

Brussi 2026_Dedekind theorem

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