TFNR - Perturbation - Gradient

Perturbation, or Perturbative Action, is the instantiation through which "Variation" manifests at the most elementary level of Physical Reality.

It is the most fundamental Mode of manifestation of Elementary Action, its first Component. The first of the Modes of Physical Action.

From a causal perspective, we can represent it as the Operator Perturbation, the Operational Agent responsible for the emergence of inhomogeneities in the pre-temporal distributions (from now on "tau-distributions", or more briefly "τ-dist") of the sub-elementary pre-spatial fluctuations ("sigma-fluctuations", or "σ-fluct") of the Elementary Field (resulting from an expansion/contraction of infinitesimal volumes). Inhomogeneities that we can represent as gradients.

At the same time, from a variational point of view, Perturbation appears as inhomogeneities (gradients) in the tau-distributions of the Elementary Events, namely the tiny sigma-fluctuations of the Elementary Field resonating around the Planck scale (discrepancies in the amplitudes of the sub-elementary sigma-fluctuations), emerging under the incessant Action of the Fundamental Force in its functional / operational form of the Operator Perturbation.

These gradients of Elementary Events, precisely because they occur on extremely small scales and at the level of individual event-points of the Field, can be called Micro-gradients.

Cause (expression of force)	Sub-elementary dynamism	Dynamic effect	Observed variation of the micro-dynamism of event-points and its geometric implications
Internal interaction between the Force and the Field	Local inhomogeneities in the density of τ-distributions of σ-fluctuations -> amplitude	Perturbation / Gradient formation	Variation of the infinitesimal event-point volume (local dilation or contraction) within the micro-dynamism of event-points
Field reaction (counter-force) produced by the manifestation of gradients	Tendency to restoration of homogeneity of the density of τ-distributions of σ-fluctuations -> orientation	Translation / Divergence formation	Stretching of σ-fluctuations along the direction of divergence within the micro-dynamism of event-points
Combined field reactions (composed counter-forces) produced by the manifested divergences	Interaction of divergences, tendency to conservation of homogeneities in the density of τ-distributions of σ-fluctuations -> vorticity -> chirality and axis orientation	Rotation / Curl formation	Vorticity of σ-fluctuations within the micro-dynamism of event-points

Perturbation is the way in which the Elementary Field deforms under the Action of the Fundamental Force, the way in which the Field receive the expression of the Fundamental Force, the product of the expression of Causality at the most elementary level of Reality. Perturbation is the root of the emergence of the Dimensionalities of Reality that I call Spatiality and Temporality, the fundaments of Space and Time.

The Modes / Components of Action are intimately connected with the Dimensionalities of Physical Reality: Physical Causality and Physical Variationality, from which derive the Dimensions of Physical Reality: Cause, Space and Time.

The causal expression and the variational manifestation—two sides of the same coin—of this Mode of Elementary Action are governed by the Principle of Variation, which, at the Level of Elementary Action, takes the form of the Principle of Uncertainty, which describes how the relation of indetermination between pre-spatial and pre-temporal aspects and between the emerging spatial and temporal aspects allows for the emergence of existence through differentiation and inhomogeneity.

Perturbation is the root of Metric / Mass, encompassing both the curvature of Spacetime, gravity and gravitational attraction / repulsion, and thus the foundation of metric and gravitational Phenomena.

Gravity represents the point of conjunction between Perturbation and Translation Elementary Action Modes. While Perturbation constitutes the origin of the gravitational potential (the metric deformation), Translation represents the variational effect, manifesting as potential motion (acceleration) that, through interaction, produces the effective motion toward or away from the massive Structures/Forms (negative or positive Mass, corresponding to negative or positive Perturbation).

From an epistemic point of view, Perturbation is not a "state" but a dynamic Process of incessant mutation, the manifestation of a specific modality of the dynamism of Elementary Field. It provides the unstructured, unmanifest support for the structuring of the Field, acting as the "elementary grammar" that precedes the "emerging syntax" of observable physical laws.

The causal perspective

The causal perspective focuses on the active aspect of the Agency of the Source, on its causal function, on the Force and its expression in the incessant construction of Reality, from the most elementary levels to the most complex.

Technically, the Perturbation Operator should be defined as a sub-Entity, a functional / operational derivation of the Primary Agent—the active aspect of the Primary Source of Reality, the Fundamental Force. In a broader sense, Perturbation is the substrate of Existence in its most elementary form.

Like all other Operators of Reality, Perturbation should not be regarded as an Entity in the common sense of the term. It is therefore not something that exists in and of itself. Rather, it must be conceived as a Phenomenon—something that occurs, something that exists in the very act of generating Existence: a portion of the immense Phenomenon—the Universe in its incessant formation—produced by the dynamics of the internal interaction between the two aspects of the Source of Reality.

Therefore, from the causal perspective, Operator Perturbation acts as the functional derivation of the Fundamental Force, transmuting its active pressure into an incessant Process of inhomogeneities (micro-gradients) within the Field's fundamental dynamism. This Mode does not act in isolation or temporal succession but operates in parallel with Operators Translation and Rotation in the incessant construction of the elementary dynamic basis of Reality.

The variational perspective

The variational perspective focuses on the passive aspect of the Agency of the Source, on its variational function, on the Field and its manifestation in the incessant construction of Reality, from its most elementary Levels to the most complex.

Here we must focus on what happens to the Field under the incessant Action of the Fundamental Force in the form of the Perturbation Operator. On the Variations produced by the internal interaction between Force and Field, between Agent and Acted, on what occurs in this interaction—on the Events I call Elementary precisely because they represent the most elementary form of Existence, the simplest occurrences that can manifest within Reality and that lie at the foundation of its very Existence. This is the essence of the sub‑Process of Creation, as part of the Process of Formation of Reality.

As already stated countless times in this work, by Elementary Events I mean the Events that characterize the elementary dynamism of the Elementary Field near the Planck scale. The tiny sub-elementary pre-spatial sigma-fluctuations of the Field in their pre-temporal tau-distributions. Sometimes, for brevity, in this work I use the term "spatial", instead of "pre-spatial", even though Space, the spatial dimensions, technically emerge at an evolutionary stage of Reality that is temporally concurrent but causally subsequent to the expression of the Perturbation Operator. The same consideration applies to the use of the term ‘temporal’ instead of "pre-temporal".

It is precisely this expression that produces the premises—the causal and variational substrate—for the emergence of the spatial dimension, of Space itself. We could say that pre-spatial Dimensionality, inherent in the variational aspect of the Agency of the Source, through the expression of the Perturbation Operator, is the causal seed that enables the subsequent manifestation of Space and of the spatial dimension of Reality (as well as of Time and of the temporal dimension).

From the variational perspective, Perturbation manifests as the re-Action of the Field to the incessant Action of the Fundamental Force, reaction that deforms the originally uniform basic sub-elementary dynamism of the Field in the form of Elementary Events. This reaction produces gradients (Micro-gradients) in the temporal distributions of the minute elementary spatial fluctuations resonating near the Planck scale.

In other words, Perturbation manifests as discrepancies in the tau-distributions of Elementary Events. Crucially, Perturbation focuses exclusively on the pre-spatial amplitude of the sub-elementary fluctuations, the basic dynamism of the Field, distinguishing it from the directional aspects of other Modes of Elementary Action. This focus on the pre-temporal distribution of the amplitudes of pre-spatial fluctuations underlies the nature and structure of Space and Time, and therefore of Metric and Mass — respectively the potential and the actual manifestations of Spacetime curvature, in particular of Time, given the hypothesis (adopted in this research) that Space is flat at scales far larger than the Planck scale.

Micro-gradients define the primitive dynamic elements of the informational scaffold from which temporal Dimensionality arises. As said, in this model, while Space remains flat at macroscopic scales, Time is conceived as "curvable"—undergoing contraction or dilation according to the local value of Perturbation. Thus, the "curvature of Spacetime" is more accurately described as the curvature of Time.

Everything occurs in parallel from the variational point of view (pre-spatial and pre-temporal), but from the causal point of view the order of expression of the Operators at the level of Elementary Action is crucial and fundamental for the Creation of Physical Reality as we know it, in its causal, spatial, and temporal Dimensions.

We will then see how Translation and Rotation represent, from the causal point of view, the subsequent steps in defining the dimensional framework of Physical Reality, the first being understood as Propagation and Conservation in Space and the second as Propagation and Conservation in Time. Taken together with Perturbation, all three constitute the elementary dynamic basis (the dynamism of the Elementary Field under the incessant Action of the Fundamental Force, the expression/manifestation of the Agency of the Primary Source) upon which the Fundamental Physical Quantities are founded: Metric/Mass, Motion, Charge, Spin, and the related phenomenological domains that combine in different forms within gravity, electromagnetism, and the more complex phenomenologies of the weak and strong nuclear interactions, as well as all the natural phenomena that contribute to the incessant Formation of Reality and to the complex life of the Universe.

From the point of view of the Elementary Events and the Dynamics of Elementary Action

The Operator Perturbation, as a derived causal expression of the Force, transmutes the active pressure of the Fundamental Force on the Elementary Field into an incessant Process of Variation of the dynamism of the same Field at the most elementary level of Complexity. At the same time, Perturbation, as a variational manifestation, is the result of that Process of Variation. Causality produces Variationality (Spatiality and Temporality).

The key question is: what really varies? What varies are the pre-temporal distributions of the amplitude of Elementary Events, the tiny sub-elementary sigma-fluctuations of the Field. Or, more precisely, what varies is the pre-temporal aspect of the amplitudes of Elementary Events.

Elementary Events, and precisely their pre-temporal distributions of the pre-spatial fluctuations, must be regarded as the pre-physical substance Reality is based on, the direct manifestation of the expression of the Action of the Force upon the Field.

All this lies beyond molecules and atoms, beyond Particles; it dwells in the dimensional abysses, at infinitesimal scales where indetermination sketches a blurred boundary between here and there, between before and after, where Space and Time are but a faint hint, indistinguishable from the creative chaos out of which Physical Reality incessantly emerges and evolves.

We are dealing with a pre‑physical condition, grounded in the aforementioned abstract, conceptual, and purely descriptive hypothesis of independence in the dynamism of the individual event‑points we can imagine within the Field. This pre‑physical condition becomes physical—we might even say real—the moment we begin to consider these event‑points as observational localizations, within a collective and continuous dynamism, as spatial and temporal localizations of the incessant dynamism that sustains Physical Reality.

And it is precisely from this perspective—moving from the Level of Elementary Events to that of Elementary Action, when we begin to evaluate the independent local dynamism of the single event‑point in the light of the extended dynamism of broader regions of the Field—that the relevance of the inhomogeneities of local dynamism, relative to the universal dynamism or to a significant neighborhood of the event‑point under consideration, emerges.

These inhomogeneities constitute the Elementary Action, which manifests in Modes / Components of which Perturbation is the most elementary. These inhomogeneities are the true support, the true ‘physical substance’ from whose organizational correlations Information/Energy emerges—the Structures of Information that compose the Material Forms.

Here we are at an extremely elementary level. I call it sub-elementary precisely to highlight that we are at the most elementary level of Reality, and that we are examining its most basic aspects (from a speculative and entirely hypothetical conceptual perspective). Here we must consider the dynamism of the Field not in its entirety, nor over spatially extended portions or significant durations, but independently for each event-point of the Field, with its own independent, specific and peculiar dynamics.

A way of looking (so to speak) at the most elementary aspects of Physical Reality that is entirely unrealistic and purely theoretical. The Elementary Field is not a collection of independent event‑points. It is an interconnected continuum, exhibiting features both of the classical continuous fields found in General Relativity and of the quantum fields found in Quantum Field Theory and in the Standard Model of Particles.

But in order to understand how everything we observe in the Universe emerges from the ceaseless internal interaction between the Fundamental Force and the Elementary Field, we must exercise abstraction to imagine the unimaginable.

To do this, we must attempt to picture how the dynamism of the Field manifests at the level of a single event‑point of infinitesimal extension (both spatial and temporal—or rather, pre‑spatial and pre‑temporal) at the aforementioned sub‑elementary level, a conceptual level that is, as noted, purely descriptive.

The next step is to imagine the dynamics of the pre-temporal distributions of the pre-spatial fluctuations that I call Elementary Events—the units of Existence—and then to contextualize this dynamics within the continuous dynamism of larger regions of the Field, with the corresponding correlations that lead us from the plane of Action to that of Information, Energy, and the patterns that emerge and evolve within the Field, the patterns I call InfoStructures: Waves, Particles, Dark Halos, and so on, which compose the Forms—the objects we observe in the Universe.

Returning to the Process that unfolds at the sub-elementary level within each event-point we can imagine in the Elementary Field, I remind the reader that this is a dynamic condition (what in conventional physics is, in my view improperly, referred to as a "state" of the system) that is unstructured (an unstructured Elementary Field), characterized by separation and independence, and describable through an independent stochastic process.

This process could be assimilated to that of a "continuous random harmonic oscillator" (similar, for example, to a Damped Harmonic Oscillator with White Noise Forcing, as in the standard model for Brownian Motion with elastic return—confining potential), where what oscillates are the pre-spatial distances, area and volume, according to a pre-temporal distribution around a mean value and with a given variance (confined random fluctuations). The amplitude of the fluctuations would depend on the intensity of the Fundamental Force, while the damping and rigidity would depend on the properties of the Elementary Field.

In this context, by inhomogeneity of the tau-distributions of Elementary Events I mean any deviation from the mean value of this dynamics.

It is evident that these tau-distributions define not only the rhythm of Time at the particular event-point under consideration, but also the velocity of the sub-elementary sigma-fluctuations, and the variation of that velocity over Time, which we may call "acceleration". We shall later see how the relationship between the interactions among the dynamic conditions of event-points in connected portions of the Elementary Field—describable as networks of interactions (correlation schemes) among the punctual stochastic processes mentioned above—will lead us to the concepts of Metric (Space‑Time) and Mass, and to the related metric phenomena (curvature, dilation‑contraction of Times, etc.) and gravitational phenomena (attraction‑repulsion of Structures endowed with Mass—ordinary Matter and Dark Matter/Energy).

And we shall see how the equivalence between gravitational attraction and acceleration (the indistinguishability between an external gravitational field and the acceleration of one's own reference system) is grounded in the Properties and Behaviors of the elementary dynamism of the Field, as outlined above and further integrated by what we shall discuss shortly concerning the Operator and the Mode/Component that I call Translation.

From the point of view of the Fundamental Principles of Evolutionary Dynamics

Perturbation is an instantiation of the Operator of Reality Variation that expresses the Fundamental Principle I call Principle of Variation: everything that exists is in incessant mutation, variation—or better, everything that exists "is" incessant variation.

The way Perturbation expresses and manifests itself in the construction of Reality is described by the Principle of Uncertainty, which is precisely an instantiation of the Principle of Variation.

This principle, in particular, describes how elementary variations—small pre-spatial fluctuations—are pre-temporally distributed with a certain element of uncertainty. It is precisely in the relation of indetermination that binds these two aspects of Variationality, which underlies all that exists, in the interplay between Spatiality and Temporality (pre-spatiality and pre-temporality at the sub-elementary pre-physical Level that describes the Elementary Events), that lies the foundation of the incessant Creation of Elementary Action in its primary form, which I here call Perturbation. If, in response to the expression of the Fundamental Force, the Spatiality and Temporality manifested by the Field in the Creation of Elementary Action were bound by a fixed, constant, determined relation, nothing could exist. It is only through differentiation, uncertainty and inhomogeneity, that Existence and the Essence/Form of Reality in Formation can emerge.

The root of everything lies in this creative chaos and randomness, the primary form of Existence, the raw material of the evolutionary organization that gradually emerges and shapes the Form of Reality.

Chaos, but not uncontrolled chaos. Chaos that follows a law, a principle—this very Principle. Where the relation of indetermination binds Causality and Variationality, the expression of the Force and the manifestation of the Field, within the Action that is continuously expressed and manifested. The Relation between Force, Space, and Time dance incessantly around a mean value, and this dance incessantly generates the dynamism upon which all of Reality is founded. If this dynamism—let me say "perfectly" indeterminate—were suddenly to cease, Reality would immediately vanish, and nothing would exist, as if Reality had never existed at all!

The Information / Energy perspective

Here we are speaking of Action (Force, Space, and Time). To move from the level of Action (the level of Existence, of that which exists insofar as it occurs) to the level of Information / Energy (the level of Essence, the Form of what occurs), it is necessary to ideally stop Time flow, focusing on Force and Space.

Now, if we observe the stochastic processes that describe the physical Processes of Elementary Action as mentioned above, we may imagine that the sub-elementary sigma-fluctuations at a given instant characterize a local pre-spatial deformation of each event-point of the Elementary Field. A kind of curvature at the infinitesimal level, which compensates itself when we observe portions of the Elementary Field at a scale a few orders of magnitude larger than the Planck scale, around which the elementary fluctuations resonate. The Field, therefore, while showing incessantly variable pre-spatial curvature at such an infinitesimal level, appears absolutely flat at higher scales.

If we freeze the flow of Time and take a snapshot of the Field at this scale, we will observe an infinity of tiny inhomogeneities, micro‑gradients in pre-spatial dimensionality. At each event-point in the instant considered, we may associate to such pre-spatial deformation a value of proto-Energy, which, when summing all these small punctual contributions, will display an immense, infinite value—just as infinite is the Elementary Field and the Fundamental Force that animates it.

If we consider the Field as a whole, this overall Elementary Energy will be null, since the total (infinite) spatial extension of the Field does not vary, as all punctual contributions compensate one another. The total Elementary Energy of the Field (considering only the dynamics of pre-spatial dimensionality) will therefore be null.

If we also consider the pre-temporal dimensionality, the situation changes.

The tau-distributions—the pre-temporal intervals at which such local sub-elementary sigma-fluctuations occur (the greater the amplitude of the pre-spatial fluctuation, the shorter the interval in which it may occur)—oscillate around a universal mean value (empty, homogeneous, unstructured Field). Yet they may exhibit local inhomogeneities of varying extent (turbulent unstructured Field and structured Field when those inhomogeneities are organized in the patterns that characterize the InfoStructures like Waves, Halos/Particles, and Interactions).

The total Energy of the Field will no longer be null, but will increase due to the incessant Action of the Force, which progressively exerts pressure toward an ever greater structuring of the Field, ever greater compactness, ever more Energy and Matter, thereby increasing that Energy balance of the Universe, which is so studied and so crucial for understanding its origin and evolution.

The correlation between the Perturbation values in a given portion of the Elementary Field (Relations between Elementary Action of a contiguous set of event-points) defines the Elementary Energy of that portion of the Field in the Component Perturbation, which we can call "deformation of the Metric" or "Mass" (two equivalent concepts, the first seen from the point of view "of" the Field, the second from that of the Structures "in" the Field).

At the level of Elementary Action, Perturbation is associated with Elementary Pre-Spatial and Pre-Temporal unmanifested Proto-Energy (E_PST_PE). This is a non-relational and unmanifested form of Energy that represents the minimal intrinsic dynamical discrepancy of an event-point compared to the Field's average dynamics.

We shall later see how these elementary proto-energetic and energetic aspects, linked to Perturbation, will translate into the Physical Quantities of Metric and Mass—of which they are the root—and into the related metric and gravitational Phenomena, as well as into the existence, form, and dynamics of the Structures that compose the Energy, Radiation, and both Dark and Ordinary Matter that we observe in the Universe.

In other words, the transition to manifest Physical Energy and the fundamental quantities of Metric and Mass occurs only at a higher level of Reality through spatial correlations (corr(PER)):

• Metric deformation (Time deformation): emerges from the correlation of Perturbation as seen from the Field's perspective,
• Mass: emerges as the "effective" manifestation of these same correlations, where negative Perturbation profiles (below average density) relate to attractive Mass (Ordinary/Dark Matter) and positive Perturbation profiles relate to repulsive effects (Dark Energy).

We shall also see, as described by the Principle of General Equivalence, how Perturbation constitutes the causal and variational substrate of many physical quantities and Phenomena which, in conventional physics, do not appear to share origin and nature.

I refer in particular to:

• the deformation of Metric, the so called Space-time curvature (or rather the curvature of Time, in this model)
• Mass density (positive, producing attraction, or negative, producing repulsion)
• the limit of the velocity of propagation of Causality / Action / Information in the Elementary Field (the speed of light in the "vacuum", a limit but not a constant in this model)
• the temperature of the Elementary Field

At any level of Complexity, from the most elementary to the Universe as a Whole, Reality can be described as a network of more or less independent / correlated stochastic / deterministic processes. Metric, Space, Time, Mass, Motion, Charge, Spin and all the more complex Entities (Agents / Forces and Acted / Fields) and Phenomena (Processes of Events organized by Relations) that we can see in Nature, at their foundation, are made of Events, of Action, which can be described be sets, or better networks, of Relations among an infinite number of Processes. Infinite are the ways in which the Relations among those Processes organize, produce, the wonderful Complexity of the Universe in Evolution.

Let's try to summarize.

From the Level of Action to the Level of Information/Energy

• Action (Force, Space, Time) describes what happens.
• To access the level of Information/Energy (Essence, Form of what happens), one must ideally stop the flow of Time.

Sub‑elementary Fluctuations and Proto‑Energy

• Pre-spatiality
  • Sub‑elementary sigma‑fluctuations generate micro‑pre‑spatial deformations at each event‑point.
  • At infinitesimal scales: the Field is "curved".
  • At scales larger than the Planck scale: these deformations compensate -> Field appears flat.
• Snapshot of the Field
  • Freezing Time reveals infinitely many micro‑inhomogeneities.
  • Each deformation corresponds to a value of proto‑Energy.
  • Summed over the entire Field, this proto‑Energy is infinite, as is the Field itself.
• Total Pre-spatial Energy
  • Considering the Field as a whole, all contributions compensate/ cancel -> total Energy is null (pre‑spatial dynamics only).

Introducing Pre-temporality

• Tau-distributions
  • Fluctuations occur in pre‑temporal intervals (τ) oscillating around a universal mean.
  • Local inhomogeneities may arise:
    • turbulent unstructured Field
    • structured Field (Waves, Halos/Particles, Interactions)
• Total Energy of the Field
  • When pre‑temporality is included, total Energy is no longer null.
  • The Force drives:
    • increasing structuring
    • increasing compactness
    • more Energy and more Matter -> raising the Universe’s energetic balance.

Perturbation and Elementary Energy

• Correlations of Perturbation
  • Correlations among Perturbation values in a region define its Elementary Energy.
  • This corresponds to:
    • Metric deformation (Field perspective)
    • Mass (perspective of Structures within the Field)
• Pre-spatial / Pre-temporal Proto‑Energy
  • At the level of Elementary Action, Perturbation corresponds to unmanifested proto‑Energy, the intrinsic dynamical discrepancy of an event‑point relative to the Field's average.

Emergence of Metric and Mass The transition to manifest physical Energy arises through spatial correlations:

• Metric deformation (Time deformation): correlation of Perturbation from the Field's viewpoint -> Space‑time curvature
• Mass: correlation of Perturbation from the viewpoint of InfoStructures. Negative Perturbation -> attraction (Ordinary/Dark Matter) - Positive Perturbation -> repulsion (Dark Energy)

Principle of General Equivalence Perturbation is the common causal substrate underlying phenomena that conventional physics treats as unrelated:

• Metric deformation (Space‑time curvature)
• Mass density (positive/attractive or negative/repulsive)
• limit of the velocity of propagation of Causality/Action/Information (speed of light)
• temperature of the Elementary Field

Reality as a Network of Processes At every level of complexity:

• Reality is a network of mixed stochastic/deterministic Processes with varying degrees of correlation that organizes in patterns that we observe as waves, particles, dark halos, etc.
• All fundamental physical quantities (Metric (Space, Time, Space-time)/ Mass, Motion, Charge, Spin) and all Structures and Phenomena emerge from:
  • Events -> Action
  • Relations (which organize Events)-> Information
  • Processes (Events organized by Relations) -> InfoStructures and Forms
• The infinite ways these Relations organize give rise to the evolving complexity of the Universe.

Formulation in vector calculus

Perturbation can be understood as inhomogeneity in the tau-distributions of sub-elementary sigma-fluctuations. In vector calculus, such inhomogeneities in a scalar field are represented by gradients—vector quantities that express the rate and direction of variation of scalar magnitudes in Space.

We speak of a scalar field, which might at first suggest something static. Yet, as with the scalar field representing temperature—for example, in a gas—even if the gas appears motionless, we know its molecules are in constant motion. It is precisely the average intensity of this molecular motion that defines what we call temperature, reflecting the distribution of thermal energy (effective Energy, in this model) within the volume of gas considered. Yet, what often matters physically is not only the temperature itself, but its spatial variation. This variation is expressed by the thermal gradient, a vector quantity that indicates the rate and direction of change of temperature within the gas, and which governs processes such as heat conduction. Thus, here we are referring to a gradient in the dynamics underlying the observed phenomenon: the molecular motion that explains the spatial distribution of temperature in the gas.

The same reasoning applies to the Elementary Field and its micro-dynamism. When we speak of Structures of Information within the Field, we recognize them as manifestations of the organization of aspects (amplitude, orientation, chirality and axis orientation of vorticity) of the Field's dynamism. These Structures embody the Information that provides form and coherence to what becomes manifest—what we perceive as observable reality.

Such InfoStructures—interpreted and named as Waves, Vortices/Halos/Particles, and Interactions—constitute both Dark Matter and Ordinary Matter. Gradients of complex and changing forms, interactions among gradients, and so on.

As mentioned, gradients are representations in vector calculus of the most elementary form of inhomogeneity: variations of the dynamics of Elementary Field along a chosen direction.

At the fundamental level, they are produced by the Fundamental Force, which disturbs and deforms the Elementary Field.

At higher levels, they arise from the Derived Action of Derived Forces, acting on the Derived Fields—organized modes of the Elementary Field—whose inhomogeneities manifest as Variation.

In this model, a gradient is any variation in the dynamism of the Elementary Field along a direction. It represents discrepancy, inhomogeneity, anisotropy, deformation, and the primary manifestation of Action.

The phenomenological perspective

Perturbation manifests phenomenologically through Mass, Mass density, Spacetime curvature, gravity, gravitational attraction/repulsion, etc.. These phenomena are different expressions of how local inhomogeneities and variations in the Elementary Field organize themselves into observable manifestations.

Why do we observe in nature Phenomena that we trace back to inhomogeneous distribution of Mass and/or inhomogeneities in Metric?

In light of the above, the answer is, of course: because there exists Elementary Action in the Component Mode that I call Perturbation.

The question that immediately arises is: why does Perturbation exist? Why does this dynamic condition manifests itself in the Elementary Field (this "state", as conventional Physics would call it—a term I find unsatisfactory, since it evokes a static condition rather than the dynamic situation I see at the root of natural Phenomena, and thus also at the basis of the "states" of the systems we observe).

Mass

I have already discussed Mass and Mass density above. Here I summarize their essential features from a more distinctly phenomenological standpoint.

Elementary Action, in its simplest form—Perturbation—exists because the internal interaction of Force with the Field, the ceaseless expression and manifestation of the Agency of the Primary Source of Reality, occurs in a way that can be synthesized through the Principle of Uncertainty, in the general form stated in this model: everything varies incessantly, and even variation itself varies, to use a hyperbolic image. The rate of variation also varies, and the result of the variation of variation is a change that does not follow a deterministic law, but instead expresses and manifests chaos—uncertainty itself.

Within the complexity of the interaction among an immense number of event-points, even considering the quadratic decrease of reciprocal interaction intensity with distance, this characteristic of indeterminacy and chaoticity is further amplified. The outcome is an omnipresent turbulence, spanning from the most infinitesimal scales to the most extended cosmic ones, with dimensional resonances that lead to the emergence and manifestation of Vortex type Structures of Information, which we observe as Particles of Ordinary Matter and Halos of Dark Matter/Energy.

This turbulence arises from the spontaneous dynamics of the Elementary Field. Structures emerge from such turbulence, organizing, expanding, contracting. Some are "real"—more or less stable and permanent—while others are "virtual", ephemeral and impermanent. All participate in an incessant cosmic dance of Creation and Evolution, contributing to the Formation of the building blocks of the immense Complexity of Physical Reality.

This omnipresent turbulence, together with the Structures that emerge within it—representing inhomogeneities in the tau-distributions of sub-elementary sigma-fluctuations—produces spontaneous vortex-like dynamics. These, like vortices in a fluid, generate regions of lower density of Elementary Events in their tau-distributions. Under favorable circumstances (when what I call "conditions of dynamic equilibrium" are satisfied), such regions can acquire a degree of temporal permanence, more or less extended (causal as well as variational permanence, with the capacity to produce effects and Phenomena, and to structure and organize the Elementary Field and its turbulence across vast scales).

We may say that these InfoStructures (Vortices appearing as Halos of Dark Matter/Energy or as Particles of Ordinary Matter) possess non-zero Mass, or Mass density, and this non-zero Mass is precisely what allows them to generate gravitational effects, to interact causally with other Structures, and to contribute to the medium-scale (massive Particles) and large-scale (Dark Matter/Energy Halos) organization of the Elementary Field.

Mass arises as a Phenomenon produced by a local, non-zero amount of Perturbation. This amount determines the local density of Mass.

A vortex Structure will exhibit an internal region (the core) characterized by a specific profile of depression in the value of Perturbation , and an external region (the shell) where the density profile is greater than the average value we associate with zero Perturbation—meaning no inhomogeneity in the tau-distributions of the amplitude of the sub-elementary sigma-fluctuations. Even if the global Energy balance is null (due to the cancellation of the equal positive and negative contributions), at the local level, within the inhomogeneous distribution of Perturbation, we will have a non-zero Mass density inside the Structure.

This occurs because, while in the core of the vortex Structure the below-average density is self-sustaining and remains concentrated, the above-average Perturbation in the external region—the shell—tends instead to spread across increasingly larger volumes, progressively reducing its density (its density, not the total amount) until reaching an equilibrium level with the pressure of the natural turbulence of the Field at scales large enough to interact.

This natural distribution across wide volumes, with the decreasing density of positive Perturbation in the external region, obviously occurs only if there is no compression from adjacent Structures. In such cases, distribution into the surrounding Space is not possible. The interaction among the shells of adjacent Structures, especially within densely populated regions, produces a significant increase of the density of positive Perturbation (positive Mass density, or simply positive Mass), which produces repulsion among the Structures themselves. At intermediate scales—the scales where we find the Particles of Ordinary Matter—this repulsion results in the intangibility of the Structures themselves, as exemplified by what is called the Pauli exclusion principle. This is a fundamental principle of quantum mechanics stating that two identical fermions cannot occupy the same quantum state simultaneously. In the terms of my model, this means that two Vortex Structures cannot interpenetrate; they cannot occupy the same region of the Elementary Field, the same Space at the same Time. And the reason for this impenetrability becomes evident once one considers their Essence/Form and the micro dynamics that make their Existence possible.

So, to summarize: Perturbation—and therefore Mass, or more precisely Mass density—can be negative or positive, corresponding respectively to Ordinary and Dark Matter (attractive) and Dark Energy (repulsive). In this view, Mass is not an intrinsic substance but a manifestation of localized perturbative density within the sub-elementary and elementary dynamism of the Elementary Field.

Spacetime Curvature

Mass and the Phenomena it produces, or those associated with it, are Phenomena generated by inhomogeneities in the spatial correlation of the Mode/Component of Elementary Action that I call Perturbation. In analogy with what is described in conventional physics—especially in the relativistic domain—we can observe these Phenomena from two complementary perspectives: from the standpoint "of" the Elementary Field and from that of the InfoStructures "within" the Field.

The first perspective leads us to observe what is called the "curvature of Spacetime", thus the Metric and the associated metric Phenomena (curvature, dilation and contraction of Time, etc.).

The second perspective leads us to the observation of Mass and gravitational Phenomena—that is, the Forces that Structures, Forms, and objects of Physical Reality experience in their mutual interactions, arising from their respective Masses, their distributions, and their reciprocal Motions. I have discussed this above, and I will return to it in a later chapter.

In the next section, dedicated to Translation, we will see how kinetic Phenomena—such as Motion, and acceleration in particular—are in fact indistinguishable from gravitational attraction. We will see how this equivalence is not truly an equivalence, but an identity, since the underlying Phenomena—the dynamics of Elementary Action and its Modes / Components—are not merely identical, but are indeed the very same. We will also see how certain Phenomena produced by gravitation (e.g., Time dilation/contraction) are likewise produced by Motion, when the velocities involved become relativistic, approaching the limit of the speed of light.

A short remark on Spacetime curvature. This important Phenomenon can be described within two fundamentally different physical models:

• Curved Spacetime with constant speed of light (General Relativity): lengths and durations vary with Mass-Energy density.
• Flat Space with Curved Time and locally variable light speed (the present model): spatial extensions remain fixed, while durations and the local value of the speed of light vary as a function of Mass density.

In both models, Mass-Energy density determines the curvature of Spacetime.

But in my model, a positive Mass density corresponding to a positive Perturbation profile—tau-distributions of amplitudes of Elementary Events above the universal average or a significant neighborhood (which we identify as Dark Energy, manifesting in the shells of Dark Halos or in the overpressure regions of dark turbulence)—produces repulsive effects, especially in the interstitial regions of cosmic areas crowded by large Dark Structures, effects that mimic what we call cosmic expansion. These effects, however, do not constitute an actual expansion of spatial distances, nor a true curvature of Space. In these regions, time should flow more rapidly (increasing its rhythm above the average value of pre-temporal flow). One might be tempted to use the expression "contraction of Time", but great care is required with such terminology, as it can easily create confusion between the language of General Relativity and that of everyday or pseudo scientific discourse.

And, symmetrically, always as hypothesized in my model, a negative Mass density corresponding to a negative Perturbation profile—tau-distributions of amplitudes of Elementary Events below the universal average or a significant neighborhood (which we identify as Dark Matter, manifesting in the cores of Dark Halos or in the depressions of dark turbulence)—produces attractive effects due to the concentration in the cores of large Dark Structures. And of course, the same attractive effects occur in the case of Structures composing Ordinary Matter (Vortices/Particles) with enormously higher densities. In these regions of negative Perturbation, Time effectively dilates (as predicted by GR), decreasing its rhythm below the average value of pre-temporal flow.

Therefore, to summarize: from scales immediately above the Planck scale up to the most extended scales, Space is flat, and the effects of expansion and curvature are apparent. Whereas Time, at all scales, is "curvable"—curved (contraction or dilation) according to the local value of Perturbation, and thus of Mass density (positive or negative). Indeed, in the context of this research, the hypothesis I propose is that Time curvature and Mass density are the same thing—the same Phenomenon seen from two different but complementary perspectives: that of the Metric (the viewpoint of the Elementary Field) and that of Mass (the viewpoint of Structures within the Field).

In other words, my hypothesis is that at scales even slightly above the Planck scale, Space is flat, and therefore what actually curves is only Time. It follows that the speed of light—or more precisely, the limiting speed of the propagation of Causality / Action / Information—while being a limit, is not constant. It varies locally to the extent that the rhythm of Time varies, which is determined by the local value of Perturbation, itself determined by the inhomogeneities in the dynamism of the Elementary Field as described above.

The two models (mine and the relativistic one) seem to produce the same Phenomena. It would be interesting to investigate further in order to identify discordant phenomenological domains that could be subjected to testing. Yet my approach, unlike the relativistic one, is compatible with the broader vision of the fundamental nature of Reality that I propose in this work, and it also satisfies a criterion of plausibility which, though not appearing strictly scientific, I consider indispensable for achieving a deeper understanding of Nature and its Phenomena.

Personally, it seems highly implausible to me that Space itself should curve, expand and contract, or fold inwards and around neutron stars and black holes, etc. Such behavior would imply potentially destructive structural effects on what is called the fabric of Spacetime—effects that we do not, in fact, observe. The curvature of Time is far easier to accept. It appears plausible, close to the psychological sense of Time that we experience in our ordinary lives. Moreover, the curvature, dilation, and contraction of Time seem entirely compatible with observations both of the microcosm and the macrocosm.

The deformation of Time does not evoke any destructive structural effects. And the variability of the speed of light—which would retain the character of a local and universal "limit", but not of an absolute and universal "constant"—I know may be unsettling and may seem to contradict more than a century of physics and (apparently) observations. But since everything (or almost everything) we observe is indeed observed directly or indirectly through electromagnetic radiation, I believe that, until proven otherwise, it is not possible to have direct experimental evidence of this variability.

All this is to say that, in my view, these two models could be equivalent for all, or almost all, aspects of Reality. But for me, and within the context of the model I am outlining here, a structure that allows for Time to be curved, Space to remain flat at scales larger than the Planck scale, and the limiting speed of light to be variable (not constant across Space and Time), seems more plausible, more adequate, and better able to adapt to the Dynamics of Elementary Action that I have placed at the foundation of the functioning of Physical Reality.

I reiterate once again that everything presented in this work consists of hypotheses and proposals—non conventional ways of thinking about and imagining Physical Reality, and new lines of inquiry to be explored. Throughout this work I explicitly try to distinguish empirical facts from theoretical constructions. This methodological transparency is essential, especially in fields such as cosmology and gravitation, where many widely accepted models rely on assumptions that are not directly verified by experiment.

Unfortunately—and far too often—contemporary academic science, especially in cosmology and fundamental physics, relies on highly theoretical models, on hypotheses that cannot be directly verified, on parameters adjusted to make the equations work, on narrative interpretations of data, and on metaphysical assumptions disguised as mathematical necessities. This does not mean that such models are "wrong"; it simply means that they are not facts, but conceptual constructions.

Some emblematic examples include:

• Dark energy - Never measured. Nature unknown. Introduced to account for cosmic acceleration.
• Dark matter - Never directly observed. Introduced to account for galactic dynamics.
• Cosmic inflation - Never observed. Introduced to solve problems of the Big Bang model.
• The CMB as fossil radiation - A coherent interpretation, but not independently verified.
• Spacetime curvature produced by energy - Never measured directly. Assumed because General Relativity requires it.
• Singularities, horizons, extreme metrics - Mathematical constructions, not observations.

A greater degree of epistemological transparency would be necessary.

Academic science—through cultural inertia, and even more so scientific popularization—often presents models as “descriptions of reality”, confuses predictive success with truth, uses language that suggests certainty where only hypotheses exist, and turns assumptions into operational dogmas.

Unfortunately...

One distinctive feature of the present model, I believe, is precisely that it does not claim to be "reality", but rather a coherent, plausible, and explicitly hypothetical proposal. At least, that is my intention."

Gravity and gravitational attraction / repulsion

This is the point of conjunction between Perturbation and Translation.

In this model, by Gravity we must understand the Translation produced in the Field as an effect of the presence of regions with non‑zero values of Perturbation. This Translation manifests as gravitational acceleration (like other forms of acceleration, produced through different phenomena like the electric and magnetic ones, we can see gravitational acceleration as "potential Motion", forms of unmanifested Action and Energy expressing in the micro-dynamism of the Field). It interacts with the Translation, moment by moment, associated with InfoStructures in Motion within the Field (or at rest—it is the same), and this interaction, through the rules of vector composition, produces the variation of Translation that characterizes the aforementioned Structures, which therefore change their Motion (speed and direction/verse).

Thus, a Perturbation greater than the average value that we consider corresponding to zero Mass density (a region of negative Mass—the Mass we normally speak of, attractive) will produce a more or less intense reorientation of the sub-elementary sigma-fluctuations (Elementary Events) toward the Mass center of the massive Structure that we say generates a "gravitational field", or a gravitational potential in the Field. A stronger correlation of Translation arises in the neighborhood of the massive Structure, oriented toward its Mass center, which translates into attraction—hence Motion toward, acceleration, potential Motion.

If we place nearby another massive Structure, or even an Electromagnetic Wave, a photon, it will interact gravitationally (Translationally, we should say) and modify its Motion according to the interaction of its own Translation (effective Motion, the scheme of Translation correlation that characterizes the test Structure at each instant—inertial Motion, kinetic Energy) with the Translation present in the Field (potential Motion), produced by the Mass of the first Structure (the Translation correlation present in the Field around said Structure). The scheme, intensity, etc. of Translation correlation of the test Structure will undergo a continuous variation, which we observe as acceleration, variation of magnitude, direction and/or orientation of Motion, of its kinetic Energy, etc.

The orientation direction of the Translation produced in the Field by a Structure with Mass (with predominantly negative Perturbation, thus generating attractive effects—the type of Mass we normally identify in Physical Reality, apart from the repulsive effects attributed to the so‑called Dark Energy) will point toward the Mass center of the Structure that produces such orientation of the Field dynamics (negative divergence, or convergence, a "well").

Conversely, if we consider a shell (the external region of ultra-low density positive Mass/Dark Energy, surrounding the internal negative Mass core, as said) of a large Dark Halo (a giant void, a galactic bubble, the Dark Halo encompassing the visible part of a galaxy cluster, or a region of dark turbulence with positive Perturbation), the orientation would extend outward from said shell—which we identify as a region of Dark Energy (characterized by repulsive effects, interpreted as cosmic expansion due to the separation of the involved Structures)—toward the surroundings of that region (positive divergence, a "source").

Conventionally, gravity is seen as a Phenomenon arising from Mass and the dynamics of massive Structures. In this model, from a causal perspective, I propose that Perturbation constitutes the origin of Gravity, its direct "causal" origin, while Translation represents its "variational" effect—expressing how perturbative inhomogeneities govern macroscopic gravitational Phenomena.

Depending on the sign of Mass density, gravitational effects manifest as attraction or repulsion. These opposing tendencies shape cosmic dynamics, balancing the interplay between Matter (both Dark and Ordinary) and Dark Energy.

Links to the tables of contents of TFNR Paper

• Go to the contents list of the paper "The Fundamental Nature of Reality"

• Go to Chapter 2 "A first look"