Annex 2: Structural Time and the Hidden Origin of Mass and Energy (Main article: Structural Time in Theory F: Emergence Without Chronology)

Annex 2: Structural Time and the Hidden Origin of Mass and Energy Antonio Bern´ardez Gumiel Madrid, June 25, 2025 Abstract English: This annex explores the conditions under which the structural time variable Tappears constant, leading to its misinterpretation as mass, and proposes a structural reinterpretation of Einstein’s equation E=mc2within Theory F. We argue that mass is not a fundamental entity but an emergent measure of the frequency of nodal activation in T, and that energy arises from the flow of structural time. Espa˜nol: Este anexo explora las condiciones bajo las cuales la variable de tiempo estructural Taparece constante, lo que lleva a su err´onea interpretaci´on como masa, y propone una reinterpretaci´on estructural de la ecuaci´on de Einstein E=mc2dentro de la Teor´ıa F. Se argumenta que la masa no es una entidad fundamental, sino una medida emergente de la frecuencia de activaci´on nodal en T, y que la energ´ıa surge del flujo del tiempo estructural. 1 When TAppears Constant: A Historical Misinterpretation In classical physics, mass has traditionally been considered a fundamental and invariant property of matter. However, Theory F proposes that what we perceive as ”mass” is the result of a hidden structural time function T(x), which, under certain conditions, behaves as if it were constant. This apparent constancy has historically misled physicists into interpreting it as a material scalar rather than a temporal phenomenon. We identify three principal regimes where this misinterpretation emerges, each corresponding to observational or structural limitations in resolving the dynamics of T(x): 1. Homogeneous fracture density: ∇µT(x)≈0 In regions where the structural field is in equilibrium, with no net gradients, the spatial and temporal derivatives of Tvanish or become negligible. This static configuration leads to the impression of a persistent and localized mass, though it merely reflects a lack of detectable structural change. 1 2. Resonant structural rhythm: T(x) = T0+ϵsin(ωt)⇒ ⟨T⟩t≈T0 Even when there is an oscillatory behavior in T, the average value perceived by a macroscopic observer over time remains stable. Such quasi-periodic internal rhythms create the illusion of permanence. 3. Instrumental resolution limit: ∆T≪δinstrument If the temporal fluctuations of Tfall below the detection threshold of the available instruments, the variable becomes indistinguishable from a constant. Thus, a structural dynamic escapes empirical recognition and is misclassified as a static quantity. These three conditions define the epistemic boundaries that contributed to the conceptual emergence of mass as a ”thing” rather than a process. 2 The Emergence of Mass The conventional understanding of mass treats it as an intrinsic attribute of particles—immutable, fundamental, and given. In contrast, within the framework of Theory F, mass arises from the structural behavior of the field T(x), which encodes the temporal density of nodal activity within the universal substrate. This reframing moves us from an ontological to a dynamical perspective: mass is not something that ”is,” but something that ”happens” at a specific rate. When the variation of T(x) slows down or stabilizes, its dynamic nature becomes imperceptible, and it is then interpreted as a fixed mass: T(x)≡meff But in the structural ontology of Theory F, mass is more accurately defined through the rate of nodal events over classical time: m≡dNT dt Here, NTrepresents the number of structural activations—microscopic reorganizations or resonances—within the field T. Thus, mass corresponds to the observed rate at which structural time pulses within a given region. It is a frozen temporal flux, a crystallized rhythm, emergent from the discrete quantization of structural time flow. This approach transforms mass into a variable dependent on local structural conditions and the observer’s resolution. It is no longer a static scalar but the statistical manifestation of structural coherence over t. 2 3 Where the Approximation Breaks Down The classical identification of mass as a constant scalar becomes inadequate in several physical regimes where the structural field T(x) reveals its true dynamic nature. These are the frontiers of physics in which the simplifying assumption of a frozen structural time no longer holds. Each of these contexts exposes cracks in the conventional ontology of mass, unveiling its deeper temporal essence as proposed by Theory F. •Relativistic regimes: As a particle accelerates to relativistic speeds, its effective mass increases. This well-documented phenomenon, usually attributed to relativistic corrections to inertia, finds a natural interpretation in Theory F: the structural activation rate dNT/dt increases due to higher internal resonance demands within the field T(x), which must maintain coherence at higher velocities. Thus, mass is seen not as “growing,” but as reflecting an increased rate of temporal structuring. •Quantum field scenarios: In the quantum domain, particles acquire effective masses via interaction with fields, such as the Higgs mechanism. Theory F complements this by proposing that such mass generation corresponds to a modulation of the local structural time flow. When field interactions occur, they reconfigure T(x), locally accelerating or decelerating nodal events, hence modifying the mass. •Black hole thermodynamics: In the case of black holes, the classical notion of mass breaks down dramatically. Through Hawking radiation, mass appears to evaporate—a paradox under classical definitions. But if mass is fundamentally a measure of structural time activity, evaporation corresponds to a redistribution of T(x) via extreme curvature. As coherence dissipates, nodal activity ceases, and the structural mass vanishes naturally. •Symmetry-breaking phenomena: During early cosmological phases or phase transitions in condensed matter systems, mass emerges dynamically. In such contexts, the structural field T(x) shifts phase or symmetry configuration, giving rise to stable patterns of nodal activation. The mass is not inserted externally but is born from the system’s intrinsic structural reorganization over time. In all these scenarios, Theory F provides a unifying perspective: mass is not violated or lost, but revealed as the emergent trace of an underlying temporal geometry. When structural time becomes turbulent, non-uniform, or topologically distorted, the naive approximation T≡mcollapses, and we glimpse the deeper strata of reality. 4 Reinterpreting Einstein’s Equation Einstein’s iconic relation, E=mc2, is traditionally seen as a cornerstone of modern physics, establishing a direct equivalence between mass and energy via the universal constant c2. Yet this formulation conceals a deeper structural interpretation when viewed through the lens of Theory F. It compresses dynamic processes into static symbols, masking the origin of both mass and energy as emergent phenomena. 3 If we adopt the structural definition of mass proposed in the previous section, namely, m=dNT dt , where dNT/dt is the local rate of nodal structural events, then the energy Emust also be reconceived. Substituting into Einstein’s equation yields: E=dNT dt c2, suggesting that energy is not merely ”contained” in mass, but is the visible manifestation of the flow of structural time—scaled by the square of the speed of light, which in Theory F may itself be an emergent structural quantity. Taking a further step, we express energy as a field function: E(x) = c2·dT(x) dt , where T(x) represents the structural time field at position x, and its temporal derivative reflects the intensity of structural transformation. In this view, energy is not a stored substance but a rate of change of structural geometry, modulated by the background metric. This reformulation leads to a series of profound reinterpretations: •Energy is a local temporal gradient in structural organization. •Mass is a special case where this flow becomes regular and statistically frozen. •The quantity c2acts as a geometric translation factor between temporal flux and spatial propagation. The elegance of Einstein’s original equation is preserved, but its ontological meaning is inverted: instead of mass generating energy, it is the structured flux of time that generates both mass and energy as secondary statistical effects. This structural reading dissolves the classical dichotomy between matter and process, pointing instead to a unified temporal substrate as the true source of physical reality. 5 Structural Time Across Perceptual Layers A fundamental insight of Theory F is that the structural time field T(x) does not manifest uniformly across all scales or observational frameworks. What an observer interprets as “mass,” “energy,” or even “causality” depends not only on the value of T(x), but on the resolution and mode through which Tis perceived. Thus, the epistemic access to structural time is layered and filtered by the observational apparatus—be it instrumental, cognitive, or theoretical. We distinguish at least three primary perceptual layers: 1. Microscopic (quantum) perception: At the smallest scales, fluctuations in T(x) become dominant. Here, Tis no longer smooth or continuous but exhibits stochastic resonances, coherence breakdowns, and topological bifurcations. Mass and energy in this regime become probabilistic, and the structural time field behaves as a quantized lattice of temporal activations. This is the realm of particle emergence, virtual states, and field superposition. 4 2. Mesoscopic (classical) perception: At human or laboratory scales, T(x) appears stable and differentiable. Its local gradients define mass and energy in the traditional sense. The classical world emerges as a coarse-grained average over fluctuations in T, filtered through measurement devices and bounded by decoherence. In this regime, mass appears permanent because the observer’s window averages out the subtle flows of structural time. 3. Macroscopic (cosmological) perception: On the largest scales—such as galaxies, gravitational waves, or the early universe—the structure of T(x) reveals global patterns: expansion fields, anisotropies, and large-scale coherence. Variations in Tat cosmological distances translate into evolving mass-energy densities, metric curvature, and shifts in fundamental constants. What appears as cosmological dynamics may, in fact, be reorganizations of the global field T. Each of these layers imposes its own limitations and interpretative schemes. Yet, Theory F insists that all of them are projections of a single structural continuum. The differences lie not in Titself, but in how its flow is sampled and interpreted at each scale. This perspective dissolves the boundaries between quantum, classical, and relativistic regimes, offering a unified substrate through which all physical phenomena can be understood as expressions of structural temporality. 6 When Structural Time Reactivates: Mass Becomes Dynamic Although mass may appear constant under stable structural conditions, this stability is not guaranteed. Structural time T(x) can be reactivated by environmental or topological disruptions, leading to a breakdown of the frozen configuration associated with mass. This reactivation results in a renewed flow of structural events and thus a transformation of mass into other forms of energy or structure. The transition from static to dynamic mass occurs under several physically relevant circumstances: •Phase transitions: In cosmology and condensed matter, phase transitions reconfigure the symmetry of the vacuum or medium. These events are accompanied by sharp gradients in T(x), resulting in the restructuring of nodal activation patterns. What was once a fixed mass may dissolve or reform under a new structural regime, giving rise to novel particles or collective excitations. •High-energy collisions: Particle accelerators, such as those at CERN, create energy densities sufficient to melt the structural configuration of mass. In these extreme events, the frozen flow of T(x) is ”melted,” leading to fragmentation, reactivation, and recombination of nodal patterns. The observed transformation of mass into energy—and vice versa—becomes fully intelligible within the dynamics of T. •Gravitational collapse and curvature: In regions of intense gravitational curvature, such as near black holes or neutron stars, the structural coherence of Tmay break down. Gradients of T(x) become steep, causing internal nodal reactivation 5 and potential mass evaporation or accretion. The structural time field aligns or diverges violently, modifying the local metric and redefining what counts as mass. •Quantum tunneling and entanglement: At the smallest scales, quantum processes can trigger local surges in T(x), even in regions where it was previously stable. Entanglement, in particular, may be seen as a transference of structural rhythm across spatial domains, reactivating mass-like configurations in distant locations without classical transmission. In all these cases, mass is no longer a passive quantity but a dynamic expression of structural evolution. Its constancy is conditional, not fundamental. Reactivation of T(x) reveals that mass is just a phase of structural time—one that can dissolve, reconfigure, or propagate under the right conditions. This dynamical view dissolves the artificial barrier between mass and process, allowing us to understand matter not as a substance, but as a mode of temporally coherent structure—an eddy in the river of time. 7 Energy as the Differential Flow of Structural Time In the structural framework of Theory F, energy is not a mysterious primitive substance that material bodies possess. Rather, it emerges from the temporal dynamics of structure—the evolution of the field T(x) over classical time t. This reconceptualization offers a powerful bridge between geometry and dynamics, aligning with and extending the spirit of general relativity while introducing a fundamentally temporal substrate. At the heart of this approach lies the identification: E(x) = c2·dT(x) dt Here, E(x) is the local energy density at point x, directly proportional to the time derivative of the structural field T(x). Unlike traditional field theories where energy arises from potentials or force carriers, in Theory F energy is a rate—a flow, a differential, a transition across structural configurations per unit of t. This has multiple profound consequences: •Energy is geometric: Since T(x) is itself a geometric field—encoding nodal curvature, frequency, and coherence—the energy becomes a measure of the ”tilt” or ”shear” in the temporal landscape. A flat T(x) implies structural rest; a steep gradient in time implies energy flux. •Energy is local and relational: The value of energy at a point is not absolute, but depends on the relative rate of change of T(x) with respect to classical time. Two observers measuring the same region may detect different energy values depending on their alignment with the structural flow, much like differing velocities alter observed kinetic energy. •Energy is not conserved globally, but structurally reallocated: Because T(x) is not static and may undergo reconfigurations, energy may disappear locally and reappear elsewhere—not through violation of conservation laws, but via redistribution of structural flow. This opens a path for understanding phenomena like quantum jumps, entanglement energy shifts, and cosmological energy dilution. 6 In summary, energy is the tempo of structure. It is the visible surface of invisible change. Where classical physics posited energy as a conserved scalar, Theory F reframes it as a manifestation of temporally structured activity—a wave riding the deep river of T. 8 From Hidden Time to Foundational Structure The journey from interpreting T(x) as a hidden background parameter to recognizing it as a foundational structural field represents a paradigm shift. Theory F elevates Tfrom a passive bookkeeping device (as in thermodynamics or general relativity) to an ontological generator of mass, energy, and all physical form. What was once misidentified as ”mass” is revealed as the projection of frozen structural time—a residue of deeper flows invisible to classical analysis. This shift transforms not only our understanding of particles and forces, but also the nature of physical laws themselves. Several foundational implications emerge: •Mass is contingent, not fundamental: It arises when T(x) stabilizes into temporally regular patterns. When this regularity dissolves, mass transforms into energy or structure. This reframing permits the reinterpretation of massless particles, dynamic mass generation, and even mass evaporation under curvature. •Energy is dynamic temporality: The classical notion of energy conservation is subsumed under a deeper conservation of structural coherence. Energy is the emergent signal of structural time gradients—not a substance, but a symptom of temporal deformation. •The field T(x)is primary: All observable phenomena—particles, waves, interactions—are projections of the dynamic geometry of T. The equations of physics become expressions of how structure bends and evolves through this temporal field. •Classical time tis a statistical envelope: It emerges as a macroscopic average over structural events, just as temperature arises from microscopic motions. Classical causality, simultaneity, and continuity are reinterpreted as effective phenomena derived from the coherence of T(x). In this view, the universe is not made of things but of rhythms—nested layers of structured time. Matter is time that has folded onto itself. Energy is time in motion. And mass is the frozen song of time, stabilized into a measurable beat. Theory F opens a new window onto physical reality, not by discarding the successes of modern physics, but by revealing their deeper structural roots. The reinterpretation of E=mc2within this framework is not a correction, but a completion—a return to the temporal origin from which all structure flows. All mass is a frozen clock. The mistake was to believe it was a stone. Felicidades Jos´e Antonio en tu 25 cumplea˜nos, tu tiempo es siempre el m´ıo. 7