TFNR - Translation - Divergence - Revision history

Paolo at 16:51, 21 May 2026

2026-05-21T16:51:24Z

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Paolo at 16:45, 21 May 2026

2026-05-21T16:45:38Z

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Paolo at 16:00, 4 May 2026

2026-05-04T16:00:31Z

Paolo: /* - Event-points complete perspective: second level of inhomogeneity, pre-temporal of pre-spatial (Level of Elementary Events – pre-temporal) -> Pre-Temporal Proto-Energy */

2026-04-28T15:14:38Z

‎- Event-points complete perspective: second level of inhomogeneity, pre-temporal of pre-spatial (Level of Elementary Events – pre-temporal) -> Pre-Temporal Proto-Energy

Paolo: /* - Event-points complete perspective: second level of inhomogeneity, pre-temporal of pre-spatial (Level of Elementary Events – pre-temporal) -> Pre-Temporal Proto-Energy */

2026-04-28T14:02:48Z

‎- Event-points complete perspective: second level of inhomogeneity, pre-temporal of pre-spatial (Level of Elementary Events – pre-temporal) -> Pre-Temporal Proto-Energy

Paolo: /* - Event-points complete perspective: second level of inhomogeneity, pre-temporal of pre-spatial (Level of Elementary Events – pre-temporal) -> Pre-Temporal Proto-Energy */

2026-04-28T13:59:03Z

‎- Event-points complete perspective: second level of inhomogeneity, pre-temporal of pre-spatial (Level of Elementary Events – pre-temporal) -> Pre-Temporal Proto-Energy

Paolo: /* - Event-points partial perspective: first level of inhomogeneity, purely pre-spatial (Level of Elementary Events – pre-spatial) -> Pre-Spatial Proto-Energy */

2026-04-27T16:02:28Z

‎- Event-points partial perspective: first level of inhomogeneity, purely pre-spatial (Level of Elementary Events – pre-spatial) -> Pre-Spatial Proto-Energy

Paolo: /* From the point of view of the Fundamental Principles of Evolutionary Dynamics */

2026-04-27T14:08:28Z

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

Paolo: /* The variational perspective */

2026-04-17T19:22:49Z

‎The variational perspective

Paolo: /* The causal perspective */

2026-04-17T17:16:32Z

‎The causal perspective

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 ===Formulation in vector calculus===
-As seen, Translation can be understood as '''an inhomogeneity in tau-distributions of orientations of sigma‑fluctuations'''. In vector calculus, such inhomogeneities in a vector field are '''captured by the divergence operator''', which quantifies how much a vector field spreads out (positive divergence) or converges (negative divergence) from a point, acting respectively as a source or a sink—the net flow of a vector field out of an infinitesimal volume. Here I am referring specifically to the '''divergence operator (DIV)'''. '''In this model''', divergence denotes '''any propagation of gradients (inhomogeneities of amplitude, and/or orientation, and/of vorticity, in the tau-distributions of sub-elementary sigma-fluctuations) in the dynamics of the Elementary Field along a direction.'''.
+As seen, Translation can be understood as '''an inhomogeneity in tau-distributions of orientations of sigma‑fluctuations'''. In vector calculus, such inhomogeneities in a vector field are '''captured by the divergence operator''', which quantifies how much a vector field spreads out (positive divergence) or converges (negative divergence) from a point, acting respectively as a source or a sink—the net flow of a vector field out of an infinitesimal volume. Here I am referring specifically to the '''divergence operator (DIV)'''. '''In this model''', divergence denotes '''any propagation of gradients (inhomogeneities of amplitude, and/or orientation, and/of vorticity, in the tau-distributions of sub-elementary sigma-fluctuations) in the dynamics of the Elementary Field along a direction'''.
 Resuming the analogy of a scalar field representing temperature, a positive divergence of the heat flux corresponds to a heat source (heat spreading out), while a negative divergence corresponds to a heat sink (heat concentrating). This is tightly linked to the underlying dynamics of molecular motions, as in a gas: a dynamics built upon a deeper dynamics. More precisely, it is the '''vector dynamics (divergence) of an underlying synthetic scalar representation (temperature gradient) of a vector dynamics (molecular velocities)'''.

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 This dynamic within an event‑point—namely, '''a pre-temporal prevalence of expansion—will induce in the adjacent event‑points a reactive dynamic aimed at containing and compressing''' the behavior of the first event‑point. '''This can be interpreted as a propagation dynamic, a form of translation''', which stands in opposition to the Translation expressed in the expansive dynamic. A symmetric situation occurs for event‑points that exhibit a pre-temporal prevalence of contraction phases as a consequence of the expansion of surrounding event‑points.
-'''In other words''', let us imagine '''an oscillator''' (which has a random component and exhibits a chaotic attractor that draws its dynamics toward a mean value with a given variance, dependent on various internal and external factors) oscillating with an amplitude that falls within a sufficiently narrow range. If this range, for most of the Time, is positioned far from the "standard" point in the spectrum of possible oscillation amplitudes (standard for that type of oscillator for that event-point), we may say that it possesses '''an Energy linked to the discrepancies between the two pre-temporal distributions''' (always relative to a single event-point). This is the form of Energy that we can identify in the context of the pre-temporal dynamics of event‑points.
+'''In other words''', let us imagine '''an oscillator''' (which has a random component and exhibits a chaotic attractor that draws its dynamic amplitude toward a mean value with a given variance, dependent on various internal and external factors) oscillating in absolutely random directions. If this range, for most of the Time, is positioned far from the "standard" point (perfect randomness) in the spectrum of possible oscillation orientations (standard for that type of oscillator for that event-point), we may say that it possesses '''an Energy linked to the discrepancies between the two pre-temporal distributions''' (always relative to a single event-point). This is the form of Energy that we can identify in the context of the pre-temporal dynamics of event‑points.
 This applies to Perturbation, as we have seen above and recalled here. '''But it also applies to Translation. What changes is the object of the pre-temporal distributions'''. For Perturbation, the relevant quantities are the amplitudes of the elementary pre-spatial fluctuations—that is, of the Elementary Events. For Translation, instead, the relevant quantities are '''the orientations of those pre- spatial fluctuations'''.

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 Symmetrically to what was stated for the Perturbation Mode, here too we can identify '''a second form of Energy'''. It can be conceived as '''a deformation / inhomogeneity in the pre-temporal distribution of the pre-spatial orientation of pulsation process''' described above, which corresponds to the Elementary Action in the Mode/Component I call Translation.
-Here too, we are dealing with a sui generis form of Energy, one that nonetheless represents a bridge to the third form we will encounter shortly. '''At this stage, a form—albeit still embryonic—of relation appears for the first time'''. The pre-temporal distribution (or rather, in what will emerge as the temporal dimension) of the elementary fluctuations, which '''in this expression focuses on the prevailing pre-spatial orientation in the vorticity of those pre-spatial fluctuations to which the corresponding pre-temporal amplitudes are associated''', is '''a relation dominated by the Principle of Action and Reaction, Action of the Force and re-Action of the Field'''.
+Here too, we are dealing with a sui generis form of Energy, one that nonetheless represents a bridge to the third form we will encounter shortly. '''At this stage, a form—albeit still embryonic—of relation appears for the first time'''. The pre-temporal distribution (or rather, in what will emerge as the temporal dimension) of the elementary fluctuations, which '''in this expression focuses on the prevailing pre-spatial orientation of those pre-spatial fluctuations to which the corresponding pre-temporal amplitudes are associated''', is '''a relation dominated by the Principle of Action and Reaction, Action of the Force and re-Action of the Field'''.
 In this context, '''Energy can be identified with the asymmetry / inhomogeneity / polarization of such pre-temporal distributions'''. If the pre-spatial amplitude and pre-temporal duration of the fluctuations are not uniformly distributed across the pre-temporal spectrum in terms of duration, the dynamics of the Field will tend to restore homogeneity. This tendency, which manifests itself as a re‑Action of the Field, can be viewed as a "counter‑Force" in Action — a Force that may be interpreted as the expression of a system possessing Energy ("potential" in the phase in which the deformation, the inhomogeneity, is still present, and "effective or expressed" characterizing the reaction that tends toward the restoration of homogeneity).

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 Symmetrically to what was stated for the Perturbation Mode, here too we can identify '''a second form of Energy'''. It can be conceived as '''a deformation / inhomogeneity in the pre-temporal distribution of the pre-spatial orientation of pulsation process''' described above, which corresponds to the Elementary Action in the Mode/Component I call Translation.
-Here too, we are dealing with a sui generis form of Energy, one that nonetheless represents a bridge to the third form we will encounter shortly. '''At this stage, a form—albeit still embryonic—of relation appears for the first time'''. The pre-temporal distribution (or rather, in what will emerge as the temporal dimension) of the elementary fluctuations, which '''in this expression focuses on the prevailing pre-spatial orientation of those pre-spatial fluctuations to which the corresponding pre-temporal amplitudes are associated''', is '''a relation dominated by the Principle of Action and Reaction, Action of the Force and re-Action of the Field'''.
+Here too, we are dealing with a sui generis form of Energy, one that nonetheless represents a bridge to the third form we will encounter shortly. '''At this stage, a form—albeit still embryonic—of relation appears for the first time'''. The pre-temporal distribution (or rather, in what will emerge as the temporal dimension) of the elementary fluctuations, which '''in this expression focuses on the prevailing pre-spatial orientation in the vorticity of those pre-spatial fluctuations to which the corresponding pre-temporal amplitudes are associated''', is '''a relation dominated by the Principle of Action and Reaction, Action of the Force and re-Action of the Field'''.
 In this context, '''Energy can be identified with the asymmetry / inhomogeneity / polarization of such pre-temporal distributions'''. If the pre-spatial amplitude and pre-temporal duration of the fluctuations are not uniformly distributed across the pre-temporal spectrum in terms of duration, the dynamics of the Field will tend to restore homogeneity. This tendency, which manifests itself as a re‑Action of the Field, can be viewed as a "counter‑Force" in Action — a Force that may be interpreted as the expression of a system possessing Energy ("potential" in the phase in which the deformation, the inhomogeneity, is still present, and "effective or expressed" characterizing the reaction that tends toward the restoration of homogeneity).

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 Even for Translation, if we consider only the orientation of the sub-elementary pre-spatial fluctuations and not their pre-temporal distributions, we may regard the most elementary form of Energy—which, strictly speaking, we should call '''"virtual or proto" Energy—as an "elementary deformation": the change in pre-spatial orientation of the expansive-contractive dynamics of event‑points''', of the imaginary infinitesimal volumes whose dynamics we ideally wish to analyse. A deformation that varies incessantly within the fluctuations, the elementary pulsations.
-Here too, '''we can distinguish between potential Energy''' (the characteristic Energy of each instant of the expanded and contracted state—altered, inhomogeneous, deformed in any case) '''and the Energy expressed in the process of expansion and contraction (which we may refer to as effective Energy)'''.
+Here too, '''we can distinguish between potential proto-Energy''' (the characteristic proto-Energy of each instant of the expanded and contracted state—altered, inhomogeneous, deformed in any case) '''and the proto-Energy expressed in the process of expansion and contraction (which we may refer to as effective proto-Energy)'''.
 </blockquote>

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 Translation is '''an instantiation of the [[Operators of Reality|Operator of Reality]] Propagation''' that expresses the [[Fundamental Principles of Evolutionary Dynamics|Fundamental Principle]] I call [[Principle of Propagation]]: the mutations described by the Principle of Variation—the incessant variations in the dynamic conditions of Entities, Events, and the Relations among them—'''always tend to propagate'''.
-The way Translation manifests itself in the construction of Reality is '''described by the Principle of Action and Reaction, which is itself an instantiation of the Principle of Propagation'''.
+The way Translation manifests itself in the construction of Reality is '''described by the Principle of Action and Reaction, which is itself a specification of the more general  Principle of Propagation'''.
 This principle, in particular, '''describes how the Elementary Field responds to the manifestation of Perturbation'''—to the inhomogeneities in the tau-distributions of the sigma-fluctuations—'''in a natural tension toward restoring homogeneity in the dynamics of the Field, and how this Process generates propagation, spreading, fluxes, and divergence'''.

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 ===The variational perspective===
-Symmetrically, from the variational perspective, Translation manifests as the '''re-Action of the Field to Perturbation''', to gradients (Micro-gradients) in the temporal distributions of the minute elementary spatial fluctuations resonating near the Planck scale.
+Symmetrically, from the variational perspective, Translation manifests as the '''re-Action of the Field to Perturbation''', to gradients (Micro-gradients) in the temporal distributions of the sub-elementary pre-spatial  fluctuations, resonating near the Planck scale.
 When, in these pages, I speak of [[Micro-gradients]], [[Micro-flows]], [[Micro-torsions]] (or Micro-vorticities), [[Micro-states]], I am referring to gradients, flows (divergence), torsions (rotor, curl), states, '''within the dynamics of the Elementary Field at the level of individual "event-points"—infinitesimal portions of the Field''' which, for the sake of treatment and visualization, we may consider as point-like. But '''they are not truly point-like'''. As I have already emphasized on several occasions, in this model there is no place for point-like Entities, Events, Relations, Processes or Structures.  Everything possesses a spatial and temporal extension, however infinitely small. We must therefore not forget that these dimensional resonances, like micro-bubbles forming within the dynamism of the Field, though extremely small, '''always represent extended regions in the sub-elementary pre-spatial and pre-temporal dimensions, the causal and variational precursor of what we, from our perspective, observe as three-dimensional Space, with an evolution in Time, representable within the four-dimensional continuum''' so central to relativistic physics.

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 *the Operators are '''Entities not in the classical sense''', but in their function of producing Existence, a Phenomenon—something that occurs, something that exists in the very act of generating Existence, a product of the dynamics of the inner interaction of the two aspects of the Source of Reality.
-From the causal perspective, Translation can be represented as the '''Operator Translation''', a functional derivation of the active aspect of the [[Source of Reality]] ''' that '''organizes the re-Action of the Elementary Field produced by [[Perturbation|Perturbation]] into reorientation of the sub-elementary pre-spatial fluctuations (micro-flows), directed toward the restoration of homogeneity and isotropy''' within the Elementary Field.
+From the causal perspective, Translation can be represented as the '''Operator Translation''', a functional derivation of the active aspect of the [[Source of Reality]] that organizes the re-Action of the Elementary Field produced by [[Perturbation|Perturbation]] into reorientation of the sub-elementary pre-spatial fluctuations (micro-flows), directed toward the restoration of homogeneity and isotropy''' within the Elementary Field.
 ===The variational perspective===