Continuous Universe: Matter–Antimatter Symmetry and Alignment of Effective Dimension in Space-Time

Continuous Universe: Matter–Antimatter Symmetry and Alignment of Effective Dimension in Space-Time Sócrates Georges Petrakis Physicist Licensed (UFMG), S.E.EMG, Belo Horizonte – MG, Brazil E-mail: [email protected].br Abstract The ΛCDM model predicts a continuous expansion of 3D space, provided that dark energy (DE) remains cosmologically dominant. With constant density and equation-of-state parameter w ≈ −1 (well below the threshold w = −1/3), DE drives this expansion. In this scenario, the negative gravitational potential energy (Epg) asymptotically approaches zero. With Planck-scale, |∂Epg / ∂t| can be interrupted by linearizing the effective space. This study explores a linear space that modifies the Universe’s degrees of freedom. The key mechanism associated with this transition is the symmetrical annihilation of matter and antimatter. Assuming spatial flatness as in ΛCDM, and the absence of classical boundaries, this work aims to present a mathematical structure that characterizes and relates the extreme states of expansion and contraction through the conversion between energy and mass. Thus, the cause for an expansive beginning is defined. In this context, natural effective dimensional boundaries become possible without invoking classical spatial edges. The result is an evolução that connects the current Universe to a broader cosmological scenario. Keywords: Universe Linearization of effective space Matter-antimatter Entropy Energy Space ΛCDM 1 I. Introduction The ΛCDM model, currently the leading paradigm in modern cosmology, predicts the continuous and accelerated expansion of a three-dimensional (3D Space) Universe driven by the dominance of dark energy. Although this model successfully accounts for a wide range of observations— such as the cosmic microwave background and the formation of large-scale structures — remains open to extensions that could clarify fundamental aspects such as the origin of the matter–antimatter asymmetry and the physical characterization of extreme boundary states. In this context, this work explores the hypothesis that the Universe may undergo a transition in spatial dimensionality. This transition reduces the number of spatial degrees of freedom. A key element in this process is the maximum annihilation of matter and antimatter, which plays a decisive role in balancing the energy content and allows new topological configurations. The model enables a reinterpretation of cosmic evolution introducing natural dimensional limits in place of classical spatial edges, in which linear expansion and contraction are seen as superimposed and interchangeable states of a broader recurrent process. The main goal of this work is to construct a mathematical framework that characterizes the most extreme potential state of cosmic expansion and contraction through a unified physical mechanism. With a focus on uncovering answers such as the definition of the initial mechanism of expansion, it is crucial to highlight the need to analyze the potential boundary state, i.e., the most extreme one; this state is a well-defined state of maximum expansion, where the camouflaged matter consists of two minimal particles separated to the maximum (maximum linear space), forming a single structure: the fixed Universe, and where the only possible “quantum fluctuation” is the transformation into the superposed and adjacent structure in the time continuum of maximum contraction. 2 Thus, after a process of complete annihilation of 3D structures (complete annihilation between matter and antimatter in a state of maximum expansion of 3D space), it is possible to emerge a state of maximum static 1D spatial expansion of minimum entropy formed by minimal 3D particles separated to the maximum by a linear space filled with Dark Energy (DE) contained in 1D space (from annihilation in 3D space) converting into a state of maximum contraction formed by extended masses filling the entire same linear space separated by a minimum central double space (Minimum DE) in motion, with the interchangeability between mass and energy ((ρ DE = ρₘ), by Heisenberg's uncertainty principle and for the increase of entropy restarting the 3D spatial expansion of a recurring Universe. This proposal is rooted in general relativity, energy conservation, and the possibility of dimensional reduction in extreme gravitational or temporal regimes It builds upon the idea that topological transitions in spacetime, coupled with symmetric energy conversions between dark energy and matter, can give rise to cosmic dynamics consistent with observations. II. Evolutionary Roadmap The current state of accelerated expansion, with dark energy density already exceeding that of matter, indicates that the Universe tends toward indefinite expansion. As the variation in gravitational potential energy (Epg), which is negative, becomes increasingly small and approaches a limit at the Planck scale, the observable 3D space of matter stops absorbing dark energy from a planar interface that separates it from a 3D antimatter region. This dark energy interface is planar because the Universe is flat. Its extinction in this central planar region results from the extinction of linear masses of matter and antimatter through a progressive encounter that has fed this dark energy plane. 3 Without the separating plane, the 3D space of matter comes into contact with the opposite antimatter region and undergoes symmetric annihilation; it absorbs dark energy and continues in a planar expansion, perpendicular to this annihilation, until only two minimal remnant particles remain, maximally distant from each other. This configures an effective space (which truly governs evolution) as linear, since it is the only energetic structure that is always defined between masses. As there are only two 3D masses, the effective space is a 3D-linear configuration, because the minimal particles it separates are one of 3D matter and one of 3D antimatter. The effective space is thus linear within a 3D universal space. This leads to the phase of maximum linear expansion, with two static minimal particles separated by a maximally extended linear space filled with dark energy. Since dark energy is confined within this linearity — that is, it is linear dark energy — its density becomes equal to the local density of each mass, making dark energy and matter interchangeable. By energy conservation, this linear state of maximum expansion becomes interchangeable with a state of maximum linear contraction: a mirrored configuration where the presence of dark energy converts into the presence of matter, and the matter at the extremities becomes double central dark energy (avoiding infinite-density singularities and enabling the 3D accelerated expansion state observed currently). Due to the uncertainty principle, the increase in entropy, and energy conservation, the static linear state of maximum expansion transitions into a state of maximum linear contraction, which already emerges in motion toward the formation of 3D space. The only possible motion for linear masses is toward the central space filled with dense dark energy. As these linear masses of matter from one side and antimatter from the other move toward each other, in the initial interaction the dark energy breaks them into particles and disperses them, forming 3D space of matter on one side and 3D space of antimatter on the other. As this encounter continues, the central dark energy density decreases with the formation of 3D space on each side. This not only distributes the masses into 3D space, but—due to the weakening density—also allows parts of the linear masses to meet and annihilate. 4 Thus, each region of matter and antimatter expands while remaining separated by the central dark energy plane, which itself expands with partial annihilation. When the density of this central dark energy plane becomes sufficiently low, our matter-dominated region and the opposite antimatter region cease to gain mass, as the linear masses can only annihilate within the planar region. This is the phase current of the observable Universe, where the 3D Universe gains only dark energy (i.e., spatial volume), while preserving a constant amount of matter. Annihilation radiation is not currently observed because radiation requires a three-dimensional medium to propagate, which the dark energy plane does not provide. In the future, when the linear masses are extinguished, the central dark energy plane will also vanish; dark energy will be absorbed, and the Universe will complete its cycle, returning to a linear state of maximum expansion, with maximal annihilation, once again consisting only of two minimal particles—one of matter and one of antimatter—maximally separated. Due to the variation in effective spatial geometry, linear dark energy has a different density than the dark energy absorbed by observable 3D space, which corresponds to the observed characteristics and density values in our Universe. In the static boundary state of maximum linear expansion, the total energy satisfies: E0 + Epg = 0 , with Epg < 0 , and Ek = 0 This corresponds to a camouflage of mass, yet it implies a potential for motion— as if a state of "nothing" could give rise to a state of "everything". III. Dimensional Equilibrium Principle of Dark Energy This work explores the hypothesis that the Universe may undergo linearization in its effective spatial dimensionality. A key element in this process is the symmetric annihilation of matter and antimatter, which plays a decisive role in balancing the energy content and enabling new effective topological configurations. 5 The model enables a reinterpretation of cosmic evolution introducing natural effective dimensional limits in place of classical spatial edges, in which maximum expansion and contraction are seen as interchangeable linear states of a broader recurrent process. The main goal of this work is to construct a mathematical framework that characterizes and relates the most extreme potential state of cosmic expansion and contraction in the linear effective space through a unified physical mechanism. This study considers that, through a future process of maximum annihilation of 3D structures — occurring between matter and antimatter in a state of maximum expansion of 3D space (where the degrees of freedom are minimized and only transformations are possible, as full 3D expansion has been reached) — it is possible for an effective state to emerge: a maximally extended, static, linear spatial configuration of minimum entropy, formed by mass structures (a positron and an electron) separated to the greatest possible extent by a linear space filled with contained Dark Energy (DE). In the future, after the maximum annihilation between matter and antimatter, the Universe reaches a phase in which it is composed of two minimal particles at the extremities, separated maximally by a linear space filled with dark energy. In this state, there will be no other particles available to annihilate and increase the amount of dark energy or space. When maximally extended, the universal (global) dark energy density must be much higher than the matter density; only locally, at the extremities, could the matter density be higher. According to the principle of linear momentum conservation, with no further physical processes capable of generating new dark energy, the Universe tends to halt its expansion in this asymptotic static or boundary state. However, if the universal (global) dark energy density is higher than the matter density, the Universe would have to continue expanding, since the pressure of dark energy would be much greater than the pressure of matter. 6 The only way to reconcile the pressure exerted by dark energy (DE) with the static condition of the Universe is through Universal (global) equality between the densities: ρ DE = ρ m. Through the process of linearization, which confines DE to an increasingly linear space with fewer degrees of freedom, it becomes possible to achieve such an intense concentration of DE that its density equals the matter density at each location in space, reaching a fully linear boundary state. Dark energy ceases to drive the expansion of the Universe when its global and local density equals that of the remaining matter, which is only possible in a linear boundary state, with reduced spatial degrees of freedom and a static metric. In this state of equal dark energy and matter density, due to the retention of a linear space between the two maximally separated particles, dark energy becomes interchangeable with mass. The boundary state, represented by the state of maximum linear expansion, in which space is maximally filled by the maximum amount of dark energy, is characterized by a minimum rest energy (E₀), corresponding to two minimal masses located at the extremities; a maximum (i.e., least negative) gravitational potential energy (Epg), tending toward zero due to the maximally extended linear space filled with dark energy; zero kinetic energy (Ek), as the Universe is static; and a maximum dark energy (DE) content. The total energy in this state (ET me) is given by: ET me = [(E0 min + (− Epg max)) + Ek = 0] + DE max Since E0 + (−Epg) = 0 and Ek = 0, it follows that: ET me = DE max 7 Considering the maximal conversion of dark energy into matter, along with the corresponding increase in entropy, this state evolves into a state of maximum linear contraction, represented by a mirror-symmetric structure. That is, instead of two minimal masses at the extremities, the configuration now presents two minimal central gaps (filled with dark energy) between two maximally extended linear masses, forming an effective gravitational well and enabling the onset of motion toward a non-linear 3D space. The total energy in this state (ET mc) is composed of: ET mc = [(E0 max + (− Epg min)) + Ek] + DE min From the state of maximum linear expansion to that of maximum linear contraction, the conversion of dark energy into matter results in the massive filling of space, leading to an increase in E0 and a corresponding variation in Epg , which continues to cancel E0 under the linear configuration. Thus, the state of maximum linear expansion evolves into a state of maximum linear contraction plus motion into a non-linear 3D space, driven by entropy growth associated with the difference in dark energy between these two states: ET me = DE max = ET mc = DE min + Ek. The framework's total energy comprises all universal components: ET = [E₀ + Epg] + Ek + DE, where E₀ (rest mass energy), Epg (gravitational potential energy, always negative), Ek (kinetic energy), and DE (dark energy) represent the complete energy inventory. ET is always non-zero and cyclic evolution proceeds through three distinct configurations governed by energy conservation. State 1 — Maximum Linear Expansion (Static): Two minimal particles (electron and positron) at maximal separation establish complete gravitational selfneutralization: E₀ − Epg = 0. With no motion (Ek = 0) and maximal spatial extent, total energy reaches its maximum: ET = DE maximum. 8 Energy resides predominantly in spatial expansion rather than matter. State 2 — Maximum Linear Contraction (Static, Intermediate): Dark energy converts to matter through spatial "thickening” energy density concentrates within linear extent, forming extended mass structures. Gravitational equilibrium persists (E₀ − Epg = 0), motion remains absent (Ek = 0), but dark energy reduces to minimum as most energy now exists in mass form. Total energy decreases to its “minimum” : ET = DE min (< DE max). This configuration represents lowest total energy because energy concentration in gravitationally self-neutralized matter reduces "free" energy compared to spatially distributed dark energy. State 3 — Maximum Linear Contraction + Motion (Dynamic): The static contracted equilibrium (State 2) cannot persist due to thermodynamic instability. Gravitational balance breaks (E₀ − Epg ≠ 0), kinetic energy emerges (Ek > 0), and total energy restores to maximum: ET = [E₀ − Epg] + Ek + DE min = DE maximum. ( State 1) Energy conservation governs the complete sequence: DE maximum = DE min + E k The dark energy reduction between States 1 and 2 precisely supplies the kinetic energy emerging in State 3. This mechanism—where energy stored in spatial expansion converts to mass (State 1 → 2) and subsequently releases as motion (State 2 → 3)—drives cyclic evolution without external input, maintaining rigorous energy balance while enabling perpetual transformation. The system avoids singularities by distributing mass along extended linear structures rather than point collapse, with dynamics arising from inherent energy redistribution rather than spontaneous creation. 9 Coherence of the process: 1.Initial linearized 3D state: • The Universe is already 3D, but matter and antimatter are distributed in a one-dimensional manner along an axis (linear filament). • The residual central dark energy is minimal, yet sufficient to interact with matter. 2. Fragmentation and redistribution at the extremities: • The linear masses at the extremities (matter on one side, antimatter on the other) tend to approach each other. • The central dark energy prevents direct collision, fragmenting the linear matter. • This generates volumetric redistribution of mass, filling the non-linear 3D space. 3. Formation of separated regions: • Two predominant volumetric regions emerge, one of matter and one of antimatter, separated by a central 2D plane of dark energy. (that does not propagate radiation). • This dark energy acts as a separation barrier, preventing total annihilation. 4. Generation of space and 3D expansion: • Partial annihilation of matter at the edges near the center generates energy, which acts as a repulsive force, expanding the non-linear 3D volume. 16 • The result is a three - dimensional Universe with separated regions of matter and antimatter, consistent with the physics of energy gradients, density, and the negative pressure of dark energy, which in the non-linear space can return to its currently observed density. When linear matter and linear antimatter are annihilated in the linear space, the regions of matter, antimatter, and the space separating them cease to be supplied (corresponding in the standard model to the full emergence of the currently observed amount of matter in our Universe). As the predominant regions of matter and antimatter expand three-dimensionally after the initial separation, their mass concentrations increase toward the center. This configuration leads to a growth of gravitational attraction between the centers of the two regions, where spatial separation is minimal and gravitational density is maximal. Consequently, the residual dark energy in this central zone becomes unable to maintain the separation, initiating a progressive collapse of the space separating these regions from the center toward the extremities (corresponding in the standard model to the progressive “absorption” of space [dark energy] from the central plane into our Universe, equivalent to the current accelerated expansion). As the annihilation between matter and antimatter initially occurs in the central zones and propagates symmetrically outward, energy is released again for a new cycle, triggering the transition of the non-linear three-dimensional Universe back into an effective linear state. At the end of the process, after the spatial plane is completely annihilated, all volumetric mass is either converted or annihilated, leaving only two fundamental particles — a positron and an electron — positioned at opposite extremities of the new linear space, marking the complete restoration of the state of maximum linear expansion and restarting the cosmological regeneration cycle. 17 In summary: • Tμν (matter)= 0 (total energy of matter — rest mass — canceled by Epg). • Tμν (total) = Tμν (DE). • This leads to Gμν ≠ 0, Rμν ≠ 0, and gμν ≠ gμν. • Even with zero spatial curvature, space-time is accelerated by dark energy. The result is a spatially configured, static linear Universe, dynamically unstable and ready to initiate the next cycle. In the boundary state: • The rest energy of the masses, E₀, is canceled by (− Epg). • Kinetic energy is zero (Ek = 0). Thus, Tμν (matter) = 0 globally, and Tμν (total) = Tμν (DE) — pure, constant, and dominant dark energy. • The linearity of space is perfect in this state (purely linear space). • Dark energy is spatially constant but acts by curving time, not space. Therefore: Gμν ≠ 0, but the curvature is purely temporal, associated with the expansion/acceleration of space, not spatial deformation. Ricci tensor and scalar (Rμν and R): • Rμν ≠ 0 solely due to dark energy, not due to residual masses since they have Tμν = 0. • The Ricci scalar R ≠ 0 because space-time has temporal curvature. The space is perfectly linear (spatial curvature = 0), but space-time is not flat due to the contribution of dark energy, which affects temporal dynamics (instability: “expansion” [filling] into the only configured space — the internal space existing in the boundary state / acceleration). The space is linear (by construction of the boundary state). 18 There is no spatial deformation due to masses, since their energetic contribution has been canceled. However, g μν ≠ η μν because of temporal curvature induced by dark energy. → The metric can take the form: ds² = – dt² + a(t)² dx² Even though dx² represents a 1D linear space, the scale factor a(t) can vary due to the negative pressure of dark energy, generating temporal curvature. The metric is nearly Minkowskian, but with temporal deformation caused by dark energy. This deformation is not local (not arising from matter) but rather global and smooth. With a focus on uncovering answers such as the definition of the initial mechanism of expansion, it is crucial to highlight the need to analyze the potential boundary state, i.e., the most extreme one. This state is a well-defined linear space of maximum expansion, where the “camouflaged” matter consists of two particles separated to the maximum (linear space), forming a single structure: the fixed Universe, and where the only possible fluctuation by Heisenberg's uncertainty principle is the transformation into the adjacent structure to the time continuum of maximum contraction. In linear space, radiation ceases to propagate; the remaining particles cease to absorb radiation and remain static in position. With the interchangeability between mass and energy [ρm = ρΛ (ρ DE)], arises a state of maximum contraction formed by two extended masses (matter and antimatter) filling the entire linear space separated by a minimum central double space (Minimum DE) in motion, by Heisenberg's uncertainty principle and for the increase of entropy restarting the 3D spatial expansion of a recurring Universe. This proposal is rooted in general relativity, energy conservation, and the possibility of dimensional linearization in extreme gravitational or temporal regimes [1,2]. 19 IV. From Accelerated Expansion to Universal Regeneration 1. Evolution and Foundations Presenting a recurring Universe that solves the problem of heat death through effective dimensional linearization and energy regeneration processes, this model proposes that the Universe alternates between effective states of maximum three-dimensional expansion and linear configurations, maintaining permanent existence through mass-energy conversions in extreme states. As will be clearly explained, the current 3D space Universe is composed of distinct regions of matter and antimatter, well isolated from each other by a region of space (with dark energy), which sustains both regions in expansion, and at the same time is sustained by a progressive degradation of a primordial state of linear space. In the future of an extremely expanded Universe, with the complete extinction of this linear space, the intermediate region will progressively disappear from its center outward. This occurs because the central parts of both the matterdominated and antimatter-dominated regions will have a higher mass concentration, leading to stronger gravitational attraction. This will promote a more intense convergence between the distinct regions, culminating in the maximum annihilation between matter and antimatter, and generating the energetic transformation that results in the transition from the current 3D space to an effective linear space. 1.1 Current Cosmological Context The ΛCDM model, supported by galactic redshift observations, cosmic microwave background (CMB), and type Ia supernovae, describes a spatially flat Universe (Ω ≈ 1) dominated by dark energy (w ≈ – 1) that drives accelerated expansion. With constant dark energy density already exceeding matter density, the Universe tends to expand indefinitely, leading to heat death in the standard model. 20 1.2 Fundamental apparent problems During accelerated expansion, the global gravitational potential energy, being negative, tends asymptotically toward zero, decreasing progressively. When this variation becomes smaller than the Planck scale, it becomes imperceptible, suggesting the need for new physical mechanisms to preserve a variation in subsequent evolution. 2. Model Development 2.1 Baryon Asymmetry Question Initial Objection: The standard model assumes global baryon asymmetry based only on local observations. Proposed Solution: All collider experiments and controlled observations show symmetric matter-antimatter pair production. This fundamental symmetry suggests that locally observed asymmetry indicates the existence of distinct regions with antimatter predominance, preserving global balance. The burden of proof falls on explaining asymmetry "from nothing" versus natural spatial separation. 2.2 Separation and Isolation Objection: Absence of observational evidence for antimatter regions and gamma radiation at boundaries. Solution: Matter and antimatter regions are effectively separated by a dark energy–dominated plane that does not interact electromagnetically. Thus, in the current stage of evolution, with the progressive extinction of this central plane (region) — absorbed by our observable Universe and promoting the current accelerated expansion — the annihilation energy resulting from the encounter between matter and antimatter, or the emergent residual radiation in this plane that separates our matter Universe from the antimatter Universe, since it occurs within this plane (due to lack of degrees of freedom) remains confined, as it is not capable of propagating into the 3D space (non-planar and non-linear). 21 Thus, it is absorbed as dark energy (in space), and becomes incorporated into our observable Universe, manifesting as accelerated expansion rather than as radiation. In an earlier stage of evolution, still supplied by the linear masses, this separation prevents direct contact, avoiding boundary annihilation, which also explains the absence of detectable large-scale annihilation signals. The radiation from partial annihilation at the center, due to its linear nature, is not able to propagate into non-linear space, and its energy is therefore converted into dark energy (DE), which in turn fuels the expansion of the flat separation boundary. 3. Transition Mechanisms 3.1 Annihilation and Absorption Process During extreme expansion, when variations in gravitational potential energy are minimized on the Planck scale, linear space, which maintains central space (DE), disappears; all DE is absorbed by distinct regions of matter and antimatter predominance, and thus, these regions attract each other gravitationally. Thus, the matter and antimatter of these previously separate regions gradually annihilate each other. The resulting radiation : 1. Progressive separation of particles 2. Creation of greater spatial volume 3. Proportional increase in total dark energy 3.2 Final state of maximum expansion The process continues until only two fundamental particles remain—an electron and a positron. These particles, being the last two remaining, configure the linear space; thus, they remain in this static state and as far apart from each other as possible, creating an effectively one-dimensional spatial configuration. 22 Explanation for why electrons and positrons are the most likely particles in the extreme regime: • Smallest charged particles. • Stable fundamental leptons. • Represent perfect matter-antimatter symmetry. • Do not decay spontaneously. 3.3 Effective Dimensional Transition Objection: No mechanism exists for 3D → linear space transition. Solution: The concept of "effective space" distinguishes between fundamental spacetime geometry and matter configuration. With only two particles separated by maximum possible distance, relevant dynamics become effectively onedimensional along the connecting line, without violating 3D+1 spacetime structure. Clarification on Effective Dimensional Transition 3.4 Effective Dimensional Transition: From 3D to linear System Dynamics It is crucial to clarify what is meant by "dimensional transition" in this model to avoid misinterpretation and guide proper mathematical formulation. 3.4.1 Nature of the Transition The proposed dimensional transition is not a change in the fundamental geometry of spacetime, which remains 3D+1 throughout the entire process. Rather, it represents an effective reduction in the degrees of freedom of the physical system, leading to a fundamental change in system dynamics. 3.4.2 Initial State: 3D Effective System In the initial configuration: • Multiple masses are distributed throughout 3D space. • The system has three spatial degrees of freedom (x, y, z) • Particles can move and interact in all three spatial dimensions. 23 • The effective space is defined as the region between masses where energy transformations occur. • System dynamics are genuinely three-dimensional. 3.4.3 Final State: linear Effective System After the annihilation and absorption process: • Only two 3D masses are still (electron and positron) • These masses occupy discrete 3D points in space. • However, the system's degrees of freedom are reduced to one. • The only possible motion/transformation occurs along the linear direction connecting the two masses. • The effective space becomes the one-dimensional line segment between the particles. • All energy and dynamics are constrained to this linear configuration. 3.4.4 Mathematical Interpretation From a mathematical perspective: • Spacetime metric: Is still 3D+1 (no geometric dimensional change) • Mass objects: Continue to be 3D entities. • System dynamics: Reduce from 3D to 1D effective behavior. • Degrees of freedom: Transition from 3 → 1 spatial degree • Energy distribution: Becomes confined to linear geometry. This is analogous to a pendulum in 3D space: the pendulum bob is a 3D object in 3D space, but its dynamics are effectively one-dimensional due to the constraint imposed by the string length. 3.4.5 Physical Significance This effective dimensional reduction has profound physical consequences: 1. Electromagnetic propagation: In the linear effective space between the particles, electromagnetic waves cannot propagate normally, as they require perpendicular field components that need at least two spatial dimensions. 24 2. Energy confinement: All available energy becomes concentrated along the linear axis, maximizing the separation between the two remaining masses. 3. System stability: The reduction to one degree of freedom creates a configuration where further energy absorption is impossible. 4. Entropy considerations: The system reaches its minimum entropy state given the constraint of two remaining particles (by annihilation). 3.4.6 Distinction from Geometric Dimensional Change It is important to emphasize that this model does not propose: • Change in the fundamental dimensionality of spacetime • Alteration of the 3D+1 metric structure • Modification of general relativistic geometry • Creation of genuinely lower-dimensional space regions Instead, it describes: • Configuration-induced dimensional reduction of system dynamics • Effective space is defined by matter distribution and energy flow. • Constraint-imposed reduction in degrees of freedom by massive reduction • Physical system evolution within unchanged spacetime geometry 3.4.7 Implications for Mathematical Formulation This clarification guides mathematical development by showing that: • The metric tensor is still 3D+1. • Mass-energy tensors describe 3D objects. • System dynamics can be reduced to 1D effective equations. • Energy flow and transformations follow linear effective geometry. • Conservation laws apply in the context of reduced degrees of freedom. This effective dimensional linearization provides the foundation for the cyclical regeneration mechanism while maintaining full consistency with established spacetime physics. 25 • Epg is not generated dynamically — it is intrinsic to the system configuration and cancels E0 of the extreme boundary state. • Cancellation is the very condition of physical existence of the linear space Universe. • The extreme state is not a vacuum, but a perfect balance between existence and apparent non-existence. To respect Tμν (matter) = 0 (required by GR in linear space), the system self-imposes:E0+ (−Epg) = 0. • Tuv ( Total ) = Tuv (linear space) = Tuv ( DE) ≠ 0. • Maximum expansion: maximum Tuv ( DE). Maximum contraction: minimum Tuv ( DE). Physical Necessity of the Extreme State in linear Space: Cancellation is a physical, non-arbitrary condition. Epg is an energy potential necessary in linear Space to avoid physical inconsistencies. Thus, matter and gravity Even if camouflaged, configure a linear space between them or Dark Energy, avoiding an inconsistent “static void,” which does not generate the existential development of the known Universe. Matter (m₀) exists, but its effect only appears in the overall balance. The “camouflage” of EPG does not violate GR; it is a limiting solution where gravity and matter cancel each other out. Cancellation is a physical condition for the existence of the linear universe. The extreme state is not a vacuum, but a perfect balance between existence and apparent non-existence. VI. The State of Maximum Contraction (linear space) and Energy Reversal In summary: 1) The Universe (3D space) expands, increasing entropy. 32 2) This expansion work (fueled by dark energy) consumes “effective energy” (the ability to perform work). 3) Dark energy (central space) is totally absorbed by distinct regions of matter and antimatter predominance. Space between these regions is extinguished. 4) Matter (3D) and antimatter (3D) interact through gravitational attraction and maximum annihilate each other. 5) A state of maximum linear space expansion with minimum entropy emerges; this process is not a motion in space, but a transformation of the structure of space itself. Annihilation destroys the 3D structure, causing dimensional collapse to a fundamental linear state. 6) The result is a highly ordered and static state with annihilation energy / dark energy spatially confined in linear space; In this state where radiation does not propagate, the conservation of spatial energy allows the radiation resulting from the annihilation of 3D masses (from the preceding 3D spatial state) to be converted into intrinsic spatial energy. 7) Mass and dark energy become interchangeable, forming two superposed states (same total energy): maximum expansion with (ρm = ρΛ, and linear space) and maximum contraction + motion. 8) Once the structure of the maximum expansion state is defined, the maximum contraction state tends to have an opposite structure: linear space filled massively to the maximum, with a minimum double central space (DE)]. 9) The total energy in state of linear maximum expansion with (ρm = ρΛ) is represented by the following energy: TEME (linear space)=[Energy inherent in mass]+[Energy inherent to space] 33 TE ME (linear space) = [(E0 + (− Epg) = 0) + (Ek = 0)] + [DE (Dark energy)] TE ME (linear space) = DE (maximum) 10) With ρm = ρΛ (interchangeable mass and energy), TE ME (linear space) = [(E0 + (− Epg) = 0) + (Ek = 0)], and TE ME (linear space) = DE (maximum). (Heisenberg's uncertainty principle and the tendency toward increased entropy, the state of maximum contraction emerges: State of Maximum Expansion (linear) ≡ State of Maximum Contraction + Motion 11) Maximum expansion structure: ET (ME) = DE (max.) → Maximum contraction structure: ET (MC) = DE (min.) + energy variation. ET (ME) = DE (max.) = ET (MC) = (E0 + (− Epg)) ≠ 0) + Ek (≠ 0) + DE (min.) The gravitational camouflage (maximum expansion) implies an effective vanishing of the matter energy–momentum tensor, T μν matter = 0. Consequently, the total energy–momentum tensor is dominated by the dark energy contribution: Tμν (total) = Tμν (matter) + Tμν (Λ) Tμν (total) = 0 + Tμν (Λ) Tμν (total) = Tμν (Λ) Substituting this into Einstein's field equations yields: Gμν + Λ gμν = κ Tμν(Λ) Gμν + Λ gμν = − Λ gμν Gμν = − 2 Λ gμν In summary, the framework is built on six conceptual pillars: 1. Gravitational Camouflage : E0 + (− Epg) = 0 hides the mass from classical gravity. 2. Dark Energy as Fuel: ΔE = DE max – DE min drives the transition. 3. Dimensional Transition (linear space → 3D) initiates a new expanding phase of the Universe. 34 This extension leaves Einstein's equations unaltered in regular domains, preserves covariance and energy conservation, and is consistent with general relativity at its extreme geometric boundary. VII. Mechanism for the Onset of Expansion Maximum expansion (linear space) = Maximum contraction (linear space) + motion (Transition to 3D space). Entropy is controlled geometrically and thermodynamically, allowing for transition without physical violations. With the conversion Expansion Maximum (linear space) → Contraction Maximum + Motion, even with the change in spatial dimensionality (effective 3D →linear space), the number of physical constituents (quantized by the Planck scale) remains constant — only their geometric manifestation changes, redistributing all universal constituents between space-time and motion configurations. This divergence in dark energy: DE(ME) − DE(MC) > 0, generated by the interchangeability between mass and energy with the increase in entropy in linear space, not random quantum fluctuations, is the driver of the primordial dimensional transition. Effective dimensional reduction has been suggested in several approaches to quantum gravity, including causal dynamical triangulations and spontaneous dimensional reduction. This linear spatial configuration provides a natural setting for extreme cosmic evolution. The Universe thus becomes recurrent, with a natural cause for the onset of expansion and the regeneration of 3D space. Thus, the structure presented shows that the linear space → 3D spatial transition itself generates a primordial expansion. This occurs because: (i) The DEME > DEMC asymmetry creates a ‘cosmological imbalance. (ii) The gravitational potential (Φ) reaches a critical threshold. (iii) Geometric entropy forces the restoration of dimensions. 35 At the same time: • Adapts the ΛCDM scenario by addressing the asymptotic limits of ΔEpg and Planck time (Tₚ). • Introduces a physically motivated dimensional transition as a natural continuation beyond ΛCDM. The model data are consistent with: Establishes a deterministic, continuous model of the Universe. VIII. Toward a Complete Evolutionary Dynamics The Universe, in its primordial state, consists of two interchangeable structures: Maximum Expansion: two static particles (2 m0), separated by the maximum distance allowed by the effective linear space they define. This arrangement forms an extensive linear structure, representing the entire Universe. • Dark Energy: radiation from prior 3D particle annihilations cannot propagate in the static linear space; the dark energy density equilibrates with the matter density in linear space. • Transition to the Contractive State: Entropy increase redefines the system as: Maximum expansion state ≡ Maximum contraction state + motion. All radiation / dark energy variants convert to mass (E0 = m0 .c2), yielding the maximum possible matter content. A minimal double space emerges at the center, initiating the transition to a 3D Universe. Transition to 3D space: • Fragmentation and Clustering: Linear masses attempt to fill the central space but are fragmented by minimal dark energy. • Through gravitational and kinetic interactions, symmetric regions are formed: 3D Matter (one side) , 3D Antimatter (opposite side). Dark matter: Ungrouped particles (0D) are as non-baryonic components. 36 • Region Expansion: As matter and antimatter accumulate, their regions expand proportionally absorbing the central spatial plane between them. They are separated by a dark energy space. This space does not propagate radiation (energy is confined to a dark reservoir). No preferred direction: The space is symmetrically pressed by radiation from both 3D regions. Collapse and Recurrence Mass Depletion: When linear masses cease to feed matter/antimatter regions, the dark energy space will begin to dissipate. Final Annihilation: Without the spatial barrier, matter and antimatter annihilate symmetrically, converting all 3D mass to “radiation” nonpropagable ( = DE). Residue and Recurrence: The last annihilations leave two remnant particles, one at each extreme. Residual radiation, unable to propagate or be absorbed = dark energy in space; the particles are separated as much as possible, and the cycle restarts. Matter/Antimatter Symmetry: • The regions are identical in content but separated by a central space. • No global asymmetry: "Our side" (matter dominance) is a topological result of the initial clustering process. Dimensions as Effective Degrees of Freedom: • Dark matter consists of 0D particles that failed to cluster into 3D structures. • The dark energy space is a collective phenomenon (not a fundamental object). 37 Energy Conservation and Dark Energy Flow, Initial State (3D Regions Formation), mass clusters to form fixed quantities of: • Baryonic matter (3D). • Residual radiation. • Dark matter Initial dark energy is stored in the central 3D space from boundary annihilations. Central space Dynamics: Temporary energy reservoir fed by: • Residual annihilations at matter/antimatter boundaries. • Radiation decay. Gradual absorption: Dark energy is symmetrically absorbed by both regions, increasing dark energy but not creating new matter / radiation. Cycle End: Central space vanishes when depleted → Matter-antimatter annihilation → Cycle restarts. IX. Conclusion: A Continuous Universe Governed by linear Extremal States. Traditional cosmological models, confined to three-dimensional (3D) Space non-linear, fail to describe the true physical limits of cosmic expansion and contraction. This model considers the opposite, that the Universe evolves between two extreme states — maximum contraction and maximal expansion [with (ρm = ρΛ)] — both effectively linear space. These linear configurations define the natural dimensional boundaries of cosmic evolution, eliminating the need for speculative mechanisms while preserving the total sum: energy + mass + motion. Key Arguments Supporting the Model 1. Dimensional Necessity: • The linear limit provides a physically meaningful boundary condition for cosmic evolution, unlike undefined singularities or ad hoc assumptions. 38 2. Cyclical Dynamics: • The transition between linear and 3D nonlinear states ensures selfconsistent evolution without requiring external influences or "uncaused" events. . 3. Parsimony & Physical Robustness: • The model adheres to Occam’s razor, relying only on gravity, kinetic energy, and dark energy—no speculative constructs. • It avoids the conceptual pitfalls of infinite energy or external interactions by treating the Universe as a closed, self-transforming system. 4. Implications for Cosmology Any complete cosmological model must account for: • The dimensional transition between linear and 3D Space is a fundamental process. • The internal conservation laws that drive perpetual evolution without external dependencies. By framing cosmic evolution as a cycle between linear extremal states, this model provides a minimalist yet rigorous alternative to complement the standard cosmological model — one that is consistent with observations while avoiding unphysical assumptions. This model is falsifiable due to its intrinsic connection to the asymmetry observed between matter and antimatter — a phenomenon that strongly contradicts the universal symmetry of physical laws. In all controlled experiments and astrophysical observations, matter and antimatter are created and annihilated in perfect balance, highlighting cosmic asymmetry as a central mystery rather than a minor anomaly. 39 The main evidence for this symmetry includes: • Pair production in particle accelerators (e.g., LHC), where particles and antiparticles (such as electron-positron, quark-antiquark) are always produced in equal numbers. • Annihilation events in the laboratory, where matter and antimatter are completely converted into photons, confirming time reversal symmetry. • Spectroscopy of antihydrogen (e.g., ALPHA experiment at CERN), showing identical spectral lines for hydrogen and antihydrogen, reinforcing CPT invariance. • Composition of cosmic rays, where the near absence of primordial antimatter (such as anti-helium nuclei) suggests a global, rather than local, asymmetry. This model proposes that the current observable Universe dominated by matter is not the result of an unexplained initial bias, but the remnant of a separation of distinct regions that exist and are physically effective for the apparent breaking of standard symmetry. If future theory or data were to prove that no such symmetry-breaking processes can occur, the core mechanism of this model would be falsified. Finally, the model explains the survival of this residual matter through the prior expansion of dark energy, which spatially separated matter and antimatter domains. References [1] Carlip, S. Dimension and Dimensional Reduction in Quantum Gravity. Classical and Quantum Gravity, 34(19), 193001. (2017). [2] Ambjørn, J., Jurkiewicz, J., & Loll, R. Reconstructing the universe. Physical Review D, 72(6), 064014 (2005). [3] Misner, C. W., Thorne, K. S.,Wheeler, J. A. Gravitation. W. H. Freeman (1973). [4] Olive, K. A., et al. (Particle Data Group). Review of Particle Physics. Chin. Phys. 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