Retraction Notice and Statement of Errors in UPM

Newton’s Laws Reimagined: The Quantum Probabilistic Origin of F=ma Sungmin Lee Independent Author [email protected] December 25, 2025 Abstract This paper proposes Universal Probabilistic Mechanics (UPM), which deductively derives F=ma of classical mechanics based on quantum indeterminacy. Physical reality is defined as a set of probabilistic elements with internal spin axes and phases, and it mathematically elucidates how macroscopic order emerges through the process of Probabilistic Cluster Integration (PCI). F=ma is interpreted as a process of Extraction of order from disordered probability, and physics returns to UPM, which explores sets of probabilistic vectors. 1 Introduction: The Triple Indeterminacy Framework 1.1 A1. Quantum Root of Classical Laws The gap in modern physics exists between the probabilistic superposition of the quantum world and the deterministic trajectories of the macroscopic world [9]. This study adopts three types of indeterminacy —(1) state (spin), (2) position, and (3) morphological superposition— as the fundamental engines of dynamics, proposing that all macroscopic objects are statistical convergences of indeterminate elements [4]. 1.2 A2. Probabilistic Substrate Physical reality is composed of a set of probabilistic elements in a completely indeterminate state before observation, possessing a measure dµ(α) in the internal phase space α-Space. All macroscopic physical quantities are calculated as the integral expectation of this potential probabilistic field Ψ(x, α). Conceptual Distinction and Theoretical Position. While this paper partially adopts the mathematical forms of phase and quantum potential (Q), it is explicitly stated that Bohmian mechanics is not adopted throughout this paper. Although the present framework employs phase-related variables and probability densities [3], it is not a reformulation of Bohmian mechanics or any hidden-variable theory [3]. 1 In contrast to Bohmian approaches, the internal variable αdoes not represent a predetermined trajectory or an ontic hidden state. Instead, αsignifies a genuinely indeterminate probabilistic field that remains non-deterministic until it is realized through collective reinforcement and observation-driven extraction. This distinction ensures that Universal Probabilistic Mechanics (UPM) is treated as an independent emergent framework rather than a modification of existing causal interpretations [4]. 2 Fundamental Laws of Probabilistic Mechanics 2.1 A1. First Law: Origin of Motion Macroscopic velocity voriginates from the vectorial superposition of directional probabilistic elements [8]: v(x) = 1 mZA ρ(x, α)∇S(x, α)dα (1) Mass mis defined as the resistance to the reconfiguration of the probabilistic structure, namely, ’Topological Rigidity’ [2]. 2.2 A2. Second Law: Energy as Phase Dynamics Energy is a measure of internal phase dynamics, and entropy is defined as the isotropic regression of probabilistic directionality: E=Zdαρ(x, α)−∂S(x, α) ∂t (2) 2.3 A3. Third Law: Phase Resonance and Interaction All interactions occur through phase resonance between probabilistic clusters, and superposition in a vacuum environment forms a mutual quantum potential Qint. 2.4 A4. Fourth Law: Macro-Convergence (PCI) Macroscopic determinism is derived from the integration of probabilistic elements and the cancellation of fluctuations [5]: σ=δΦ ⟨Φ⟩≈1 √N(3) Indeterminate probabilities are realized as a single vector according to the principle of maximum probability direction. 2 3 Quantitative Analysis and Mathematical Formalism 3.1 A1. Measure and Additivity in α-Space The measure dµ(α)=ρ(α)dα in the internal probability space Asatisfies the normalization condition, and the probability vector Vpossesses additive properties: lim N→∞ X i=j⟨Ψi|Ψj⟩→0 (4) 3.2 A2. Correspondence with Classical Physics In the N→ ∞ limit, Newtonian F=ma is derived as follows [1]: Fext +Vint Stochastic Realization −−−−−−−−−−−−→ mdv dt (5) Originality of the Emergent Limit. While the recovery of classical behavior in the large-Nlimit is a familiar theme in statistical physics, the present derivation differs fundamentally in that Newtonian dynamics emerges not from coarse-graining of deterministic microstates, but from the continuous reinforcement and alignment of probabilistic expectation vectors. In this sense, F=ma is obtained as a process-based limit rather than an assumed law. 3.3 A3. Case Study: Two-Particle Collision The PCI model predicts a micro-delay δτ due to the reconfiguration of internal probabilities: δτ ∝ℏ ∆Einternal (6) 4 Defense Against Critiques 4.1 A1. Local Hidden Variable Defense The variable αis not a fixed value before observation, but an indeterminate probabilistic field realized at the moment of measurement. Observation is interpreted as a physical process of Extraction, selecting the maximum expectation vector. 5 Progressive Probabilistic Reinforcement and Emergent Motion This section examines the process by which the emergent transition from a state of probabilistic superposition to macroscopic determinism occurs, as well as the statistical boundaries based on the number of elements (N). 3 5.1 Critical Threshold for Probabilistic Emergence The point where probabilistic fluctuations (σ) fall below the average kinetic energy of the system and transition into unobservable background noise is defined as the ’Critical Point of Macroscopic Determinism (Ncritical)’: Ncritical ≈Σmicro ∆Pobs 2 (7) where ∆Pobs is the minimum momentum resolution allowed by the macroscopic system. Under the condition N≫Ncritical, the non-deterministic degrees of freedom of the individual probabilistic elements of the system are suppressed but do not disappear completely [9]. Instead, the vector force is progressively reinforced in the direction of high probabilistic expectation, and elements with low expectation are rearranged to strengthen the directionality of the entire cluster [4]. 5.2 Mesoscopic Dynamics: Hybrid Quantum-Classical Zone The intermediate region near N≈Ncritical (Mesoscopic scale) is a non-equilibrium zone where quantum interference and macroscopic linearity coexist. In this region, the system does not perfectly follow F=ma, and the remnants of the probabilistic substrate appear as the following ’Stochastic Acceleration Fluctuations’: a=F m+ξ(N) (8) where the fluctuation term ξ(N) gradually decreases as Nincreases, and nearly vanishes in the limit N→ ∞, leading to the gradual formation of macroscopic determinism [5]. 5.3 Interpretational Integrity: Emergent Vector Stability To the question ”Does indeterminacy disappear completely?”, this theory answers that ’Indeterminacy does not disappear completely; it remains latent within the cluster while being reinforced in the direction of the expectation vector.’ In other words, the macroscopic solid state is the result of ’Topological Binding’ formed by the mutual rearrangement of probabilistic elements [2]. F=ma is a mathematical description of the Probabilistic Pressure that appears in the process of moving these bound states. 6 Mathematical Formalism of Probabilistic Cluster Emergence: Rigorous Limit and Dynamics Starting from the Minimal Probabilistic Element, this section mathematically develops the process from the reinforcement of probabilistic expectations at the cluster level, through inter-cluster interactions, up to the point just before macroscopic determinism emerges, including the limit settings and stochastic dynamics. 4 6.1 Minimal Probabilistic Element: Microstate Definition The minimal element ψipossesses the following states: ψi=ψi(si,ˆni, ϕi,ai) (9) where siis the spin, ˆniis the spin-axis direction, ϕiis the phase, and aiis the directionality. Each element is unmeasurable, and the pre-observation state is described as a probabilistic superposition: Ψi=X k ckψk i,X k|ck|2= 1 (10) 6.2 Cluster Formation and Probabilistic Reinforcement Nminimal elements gather to form a cluster C: C= N O i=1 Ψi(11) The Expectation Vector of the cluster is determined through the process of probabilistic superposition and reinforcement of internal elements: VC=1 N N X i=1 ⟨Ψi|ˆ A|Ψi⟩(12) where ˆ Ais a vector operator measuring the directionality of the element, and each element undergoes progressive reinforcement toward the high expectation direction. In the limit N→ ∞, the convergence speed is defined as κ > 0, expressed as: |VC−V∞ C| ∼ κ √N(13) specifying the stabilization speed of the expectation value as the number of elements increases. 6.3 Inter-Cluster Interaction: Phase Coupling The interaction between clusters C1and C2is expressed through phase resonance and expectation flow: V′ C1=VC1+ϵRe⟨C1|C2⟩(VC2−VC1) (14) V′ C2=VC2+ϵRe⟨C2|C1⟩(VC1−VC2) (15) where ϵ≪1 represents the interaction strength, and Re⟨·|·⟩ denotes the inter-cluster overlap strength. Including continuous time evolution, the probabilistic expectation of each cluster can be expressed in the form of a Random Differential Equation (RSDE): dVCj=X k=j ϵjk(VCk−VCj)dt +σjdWj(t) (16) where dWj(t) is a multidimensional Wiener process and σjis the fluctuation intensity. In the limit N→ ∞,σj→0, and macroscopic determinism is stabilized. 5 6.4 Macro-Convergence and Pre-Observation State When multiple clusters (Mclusters) gather to form a macroscopic system: Vmacro(t) = 1 M M X j=1 VCj(t) (17) The limiting relationship in the pre-observation state is: lim M→∞ Vmacro(t) = Vhigh(t)+Vlow(t),lim M→∞ |Vlow| |Vhigh|∼O1 √M→0 (18) Observation can be represented by an operator ˆ O, and the pre-to-post observation transition is mathematically defined as: Vobs =ˆ OVmacro (19) 6.5 Continuity of Probabilistic Expectation Even after observation, the probabilistic expectation is continuously reinforced and rearranged: dVCj dt =X k=j ϵjk(VCk−VCj)+σjηj(t) (20) where ηj(t) is white noise with mean 0 and variance 1, ϵjk is interaction strength, and σj∼1/√N, converging to σj→0 as N→ ∞. 6.6 Physical Mapping: Mass, Acceleration, and Force •m: Topological Rigidity, a measure that suppresses changes in the internal expectation value of the cluster [2]. •a: The reinforcement speed of cluster expectations, a∼dVmacro/dt. •F: The vectorial force that appears after applying the observation operator, F=ma [1]. 6.7 Summary: Process-Based F=ma in Rigorous Limit F=ma≡lim M→∞ lim N→∞ ˆ OVmacro(t) (21) In other words, F=ma is a mathematical capture of the continuous reinforcement process of microscopic probabilistic expectations and the pre-topost observation transition, including the limit and stochastic dynamics. In the limit N, M → ∞, residual fluctuations disappear, and macroscopic determinism is stably realized. 6 7 Illustrative Probabilistic Experiment: Bat and Ball Interaction 7.1 Microscopic Probabilistic Elements Consider the system composed of: •A baseball, modeled as a cluster of probabilistic elements {ψball,i}with spin, phase, and direction. •A bat, similarly {ψbat,j}. •External influences, e.g., wind, {ψwind,k}. Each element carries a probabilistic expectation vector ⟨vi⟩describing possible motion directions [8]. 7.2 Interaction and Measurement At the moment of impact (observation): ⟨vpost ball ⟩=X i wi⟨vball,i⟩+X j wj⟨vbat,j⟩+X k wk⟨vwind,k⟩,(22) where wi,j,k are probabilistic weights and the sums represent **overlapping probabilistic reinforcement**. 7.3 Emergent Classical Motion The observed ball trajectory is not a single deterministic vector; rather, it is the **result of continuous probabilistic expectation reinforcement**: Fball =d dt⟨pball⟩,(23) where ⟨pball⟩evolves as the **superposition of highand low-expectation vectors**, continuously adjusted by external influences. This demonstrates that F=ma is embedded within a dynamic, evolving superposition of probabilistic expectations, not a pre-determined single outcome [1]. 8 Conclusion: Extraction of Order from Chaos Macroscopic determinism is understood as a process where order is Extracted from disordered probabilities [9]. Physical reality is not merely a collection of particles, but an organized arrangement of reinforced probabilistic vectors. Physics now returns to Universal Probabilistic Mechanics (UPM), which explores these sets of aggregates. Independence of the Framework. The Universal Probabilistic Mechanics proposed here is not intended as a modification or reinterpretation of any specific existing theory [3], but as an independent probabilistic framework in which classical and quantum behaviors are jointly recovered as limiting manifestations of a deeper stochastic structure. 7 8.1 F=ma as a Continuous Probabilistic Process This paper does not negate the classical formula F=ma of Newtonian mechanics but redefines its ontological origin from a probabilistic perspective. F=ma is no longer a simple law, but is understood as a Process in which indeterminate microscopic probabilistic expectations are superimposed and reinforced. Here, mass (m) and acceleration (a) are formed through the superposition and reinforcement of probabilistic expectations, and force (F) is the result manifested when these superimposed expectations transition at the moment of observation. 8.2 Persistence and Reinforcement of Probabilistic Expectation Even after observation, Fis not a completely determined value but exists within the process of probabilistic expectation transition that continues in the direction of high expectation. In other words, physical motion is not the result of a single event, but a continuous dynamic created by the incessant rearrangement and reinforcement of microscopic probabilistic fields [6]. 8.3 Emergence through Probabilistic Clustering The macroscopic order we witness is not a transcendental given, but the Limit of Statistical Harmony created through Probabilistic Cluster Integration (PCI) and Probabilistic Reinforcement of countless microscopic probabilistic element fluctuations [4, 5]. Mass (m) is no longer a simple quantity of matter, but a Topological Rigidity that resists changes in the probabilistic field [2], and force (F) is a Vectorial Reinforcement that rearranges and strengthens probabilistic expectations. 8.4 Ontological Shift: From Matter to Probabilistic Process Reality is not a collection of solid particles, but a ’Form of Order’ that the field of probabilistic possibilities has aligned itself to transition into reality. Behind the concise formula F=ma lies a vast topological computation performed by indeterminate probabilities to be extracted into reality, and this process evolves within the transition of probabilistic expectations that continues even after observation. 8.5 Implications for the Philosophy of Physics Universal Probabilistic Mechanics shifts the object of physics from ’Matter’ to the ’Probabilistic Process.’ We must strip away the illusory cover of determinism and face the essential mechanism of how the universe emerges order from randomness. This study will serve as the first axiomatic milestone for that great shift in thought. References [1] Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. Royal Society. [2] Mach, E. (1883). Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt. F.A. Brockhaus. 8 [3] Madelung, E. (1927). Quantentheorie in hydrodynamischer Form. Zeitschrift f¨ur Physik, 40, 322–326. [4] Bohm, D. (1952). A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables. Physical Review, 85, 166. [5] Anderson, P. W. (1972). More Is Different. Science, 177, 393–396. [6] Haken, H. (1983). Synergetics: An Introduction. Springer-Verlag. [7] Holland, P. R. (1993). The Quantum Theory of Motion. Cambridge University Press. [8] Ballentine, L. E. (1998). Quantum Mechanics: A Modern Development. World Scientific. [9] Zurek, W. H. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics, 75, 715. 9 17 Emergence of Classical Dynamics as a Special Case 17.1 Isotropic Internal Distribution If and only if the internal distribution satisfies: ρ(x, α)=ρ(x, −α),(46) then: FQ(x)→0.(47) In this singular limit, the equation of motion reduces to: mdv dt =−∇V(x),(48) which is identified as the classical Newtonian law [5]. Thus, classical mechanics is a degenerate subspace of the full probabilistic vector dynamics [3]. References [1] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley (2002). [2] P. R. Holland, The Quantum Theory of Motion, Cambridge University Press (1993). 16 18 Action–Reaction as Probabilistic Duality 18.1 Probabilistic Action Field We define the action probability density as: A(x, α)=ρ(x, α).(49) 18.2 Probabilistic Reaction Field The reaction probability field is defined via directional force: R(x, α) = fQ(x, α) |fQ(x, α)|.(50) 18.3 Duality Constraint The probabilistic action–reaction principle is expressed as: Zdα A(x, α)R(x, α) = v(x).(51) Thus, motion emerges as a dual probabilistic contraction, not as an instantaneous force balance [3]. 19 Continuity and Reproducibility 19.1 Continuity Equation Probability conservation in extended space is given by: ∂ρ(x, α) ∂t +∇·j(x, α) = 0.(52) Projection preserves continuity: ∂ρ(x) ∂t +∇·J(x) = 0.(53) 19.2 Reproducibility Conditions The theory is reproducible if: •α-space measure is fixed •Ψ(x, α) evolves unitarily •Projection operator Tis uniquely defined All predictions are determined without hidden variables [2]. 17 References [1] P. R. Holland, The Quantum Theory of Motion, Cambridge University Press (1993). [2] L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific (1998). 18 20 Comparison with Existing Frameworks 20.1 Ehrenfest Theorem Ehrenfest dynamics emerges only when: T[Ψ] = ∇⟨S⟩/m. (54) Our framework generalizes this by retaining internal directional structure [1]. 20.2 Bohmian Mechanics Unlike Bohmian mechanics: •No deterministic hidden trajectory •No scalar quantum potential cancellation •Motion arises probabilistically 20.3 Decoherence Theory Decoherence suppresses phase information externally [8]. Here, direction emerges internally, prior to environmental interaction. 21 Final Structural Statement We conclude the following structural identity: Motion = Vectorized Probability Tunneling.(55) Force is not fundamental. Energy directionality is. Classical mechanics is recovered only when internal probabilistic direction collapses to isotropic symmetry [2]. References [1] P. Ehrenfest, “Bemerkung ¨uber die angen¨aherte G¨ultigkeit der klassischen Mechanik innerhalb der Quantenmechanik,” Z. Phys. 45, 455 (1927). [2] W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715 (2003). [3] L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific (1998). 19 22 Synthesis of Theoretical Framework: Structural Consistency and Physical Implications This section establishes a logical bridge between the previously presented Probabilistic Cluster Integration (PCI) process and the probabilistic tunneling operator T, while rigorously clarifying the physical reality of mass and quantum potential [4]. 22.1 Hierarchical Relation between PCI and the Tunneling Operator T PCI and T, the core mechanisms of this theory, possess a mutually complementary hierarchy. While Probabilistic Cluster Integration (PCI) describes a physical emergence phenomenon where microscopic probabilistic elements form macroscopic aggregates through mutual resonance, the tunneling operator Tis defined as a mathematical mapping that yields macroscopic physical quantities (velocity vector v) from these aggregates [4]. v(x) = T"lim N→∞ PCI N O i=1 Ψi(x, α)!# (56) In other words, PCI is the process of securing macroscopic stability by organizing the indeterminacy of individual elements, and Tis a functional device that projects the information of the organized internal phase space (α-Space) onto an actual motion vector [3]. 22.2 Phase Resonance as a Mechanism for Non-vanishing Quantum Potential In contrast to the vanishing of the quantum potential Qin the macroscopic limit of traditional quantum mechanics, the vectorial summation of Qremains effectively valid under the Topological Binding of the present theory [4]. This is because probabilistic elements are not randomly distributed but are aligned in a specific direction through Phase Resonance within the cluster. FQ(x) = Zdα ρ(x, α)[−∇Q(x, α)] = 0 (57) As the phases (S) of the elements within the cluster align, vectorial reinforcement becomes dominant over the cancellation effect caused by microscopic fluctuations. This forms the ’Directional Quantum Pressure’ that serves as the substrate for macroscopic force. 22.3 Unified Definition of Mass: Topological Rigidity and Energy Cost The two definitions of mass mused in this paper—’polymerization rigidity’ and ’dynamical coefficient’—are unified as follows: Mass is the total energy cost required when the internal probabilistic structure is rearranged or its phase is shifted by an external stimulus [2]. 20 m≡δEint δΦstruct (58) From this perspective, F=ma is interpreted as a process in which an externally applied vectorial action (F) overcomes the topological rigidity (m) of the system to induce a change in the probabilistic expectation value (a). A larger mass implies that more probabilistic pressure is required to change the phase alignment state of the probabilistic cluster. 22.4 Predictive Superiority and Academic Significance Universal Probabilistic Mechanics (UPM) presents new predictions regarding the dynamical microstructure that are difficult for existing standard models to explain [9]. •Micro-Delay (δτ): During collisions between probabilistic clusters, physical time is required for the internal phase structure to rearrange. Unlike classical predictions, this suggests the existence of an interaction delay phenomenon on an ultra-fine time scale, which can be verified through precision particle experiments. •Emergent Irreversibility: The Toperator maintains unitarity in the expanded space while exhibiting non-linear characteristics in the projected space (the macroscopic world). This irreversible projection process structurally explains the origin of the macroscopic arrow of time and entropy. In conclusion, F=ma is more than a simple physical law; it is a mathematical summary of the process by which indeterminate probabilities are extracted into macroscopic reality. This research provides the cornerstone for a paradigm shift, moving the object of physics from fixed ’matter’ to a dynamic ’probabilistic convergence process’ [8]. References [1] P. R. Holland, The Quantum Theory of Motion, Cambridge University Press (1993). [2] P. W. Anderson, “More Is Different,” Science 177, 393 (1972). [3] D. Bohm, “A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables I,” Phys. Rev. 85, 166 (1952). [4] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd ed., Cambridge University Press (2017). [5] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley (2002). [6] E. Mach, Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt, F. A. Brockhaus (1883). [7] W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715 (2003). [8] L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific (1998). 21 23 Formal Axiomatization: The First Law and Derived Probabilistic Principles This section declares the First Law, the core axiom of Universal Probabilistic Mechanics (UPM), and defines derived laws that facilitate the interpretation of this principle from various physical perspectives. These laws are not independent assumptions but are the results of projecting the single fundamental structure of the First Law [4]. 23.1 The First Law of Probabilistic Mechanics Statement of the Law: All physical motion originates from the vectorial superposition of directional probabilistic elements. Mass, Force, and Acceleration are not independent primitive quantities; rather, they are manifestations of the intensity, imbalance, and temporal reconfiguration of probabilistic superposition, respectively. This principle is designated as The First Law of Probabilistic Mechanics [8]. 23.1.1 Fundamental Probabilistic Substrate Physical reality is defined not as a scalar probability amplitude, but as a set of directional probabilistic elements indexed by internal coordinates α[4]. Ψ(x, α) = R(x, α)eiS(x,α)/ℏ(59) The observable probability density ρ(x) is the projection of the internal ensemble: ρ(x) = Rdα |Ψ(x, α)|2[?]. 23.1.2 Mass as Superposition Resistance (Topological Rigidity) Mass mis defined as the degree to which a probabilistic structure resists directional reconfiguration. This is directly linked to the energy cost required to change the state of internal phase alignment [2]. m∝∂J ∂(∇S)−1 (60) In other words, mass is a measure of the Topological Rigidity of the probabilistic cluster [?]. 23.1.3 Force as Vectorial Imbalance Force Fis not an independent entity but a Vectorial Imbalance of probabilistic superposition [4]. F(x) = Zdα fQ(x, α)=−Zdα ∇Q(x, α) (61) When the phase resonance of internal probabilistic elements aligns in a specific direction, the uncancelled gradient of the quantum potential is extracted as a macroscopic force [3]. 22 23.2 Derived Laws of Probabilistic Mechanics The following laws derived from the First Law are specific realization methods for applying the theory to actual modeling and explaining consistency with other theories [4]. 23.2.1 Derived Law I: Probability Flux Realization Principle Definition: All physical motion can be completely described by the dominance of the probability flux. v(x) = T[Ψ] = Rdα ρ(x, α)u(x, α) Rdα ρ(x, α)(62) This law specifies that motion is not a deterministic trajectory but the result of probabilistic tunneling (T) of internal orientations [4]. 23.2.2 Derived Law II: Geometric Probability Direction Law Definition: Directional probabilistic structures can be expressed as a Geometric Field on configuration space [?]. The internal coordinate αis interpreted as a local directional fiber attached to each spacetime point x. The transport of the field along probabilistic geodesics becomes the geometric origin of motion [?]. 23.2.3 Derived Law III: Information-Gradient Motion Law Definition: Motion is generated by the gradient of the probability information structure (Information Gradient) [9]. I(x, α) = −ln ρ(x, α)⇒v(x)∝ −⟨∇I(x, α)⟩α(63) When anisotropy in information entropy exists, ”extraction” occurs toward the direction of higher information density, explaining the info-mechanical origin of F=ma [8]. 23.2.4 Derived Law IV: Operator Projection Principle (PCI-Operator Duality) Definition: Macroscopic motion is the result of the projection of internal probabilistic operators [3]. This law emphasizes that the physical phenomenon of Probabilistic Cluster Integration (PCI) and the operational mapping Tare two sides of the same entity [4]. vmacro =⟨Ψ|ˆ V(α)|Ψ⟩ ⟨Ψ|Ψ⟩(64) 23.2.5 Derived Law V: Coarse-Grained Determinism Limit Definition: Deterministic Newtonian mechanics emerges only in the singular limit where probabilistic orientations are symmetrically cancelled [?]. ρ(x, α)=ρ(x, −α)⇒v(x)→vclassical (65) This proves that classical mechanics is a degenerate subspace of Universal Probabilistic Mechanics [4]. 23 23.3 Structural Closure and Physical Predictability The derived laws above establish the manifestations of the First Law under specific conditions. Through this, the theory secures the following academic values [9]: •Theoretical Comprehensiveness: Integrates geometric, informational, and operational perspectives into the First Law. •Origin of Irreversibility: Naturally derives the macroscopic arrow of time through the non-linearity of the projection operator T. •Verifiability: The probabilistic structure reconfiguration time (δτ) occurring during collisions is a unique physical quantity of UPM that classical mechanics fails to predict. In conclusion, this theory proposes the reality of a ”directional probabilistic process” instead of the illusion of ”matter,” and redefines all classical dynamics as the result of mapping this fundamental probabilistic structure [8]. References [1] P. W. Anderson, “More Is Different,” Science 177, 393 (1972). [2] L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific (1998). [3] P. R. Holland, The Quantum Theory of Motion, Cambridge University Press (1993). [4] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd ed., Cambridge University Press (2017). [5] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley (2002). [6] E. Mach, Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt, F. A. Brockhaus (1883). [7] D. Bohm, “A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables I,” Phys. Rev. 85, 166 (1952). [8] W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715 (2003). 24 Probabilistic Mechanics: Second Law Statement of the Law The probabilistic superposition structure stabilized in the Macroscopic Limit generates invariant scalar and vector relationships. Classical laws of motion and energy emerge as Asymptotic Identities created by the intensity of probabilistic superposition and its spatio-temporal projection. This principle is designated as The Second Law of Probabilistic Mechanics. 24 Macroscopic Emergence and the Stability of Superposition When microscopic probabilistic elements reach a sufficiently large number N≫Ncritical through the process of Probabilistic Cluster Integration (PCI), internal statistical fluctuations are canceled and macroscopic physical quantities are stabilized [1]. Ψ(x, α)PCI / Coarse-graining −−−−−−−−−−−−→ ¯ Ψ(x) (66) In this limit, non-deterministic degrees of freedom are suppressed, and as directional interference terms converge to stable average values, macroscopic invariants are formed. Observable physical quantities are extracted from this stabilized state by the tunneling operator T. 25 Newtonian Identity as a Macroscopic Invariant Force (probabilistic imbalance) and mass (topological rigidity) defined in the First Law form the following identity in the macroscopic limit [1]. Fext +Fint N→∞ −−−→ mdv dt (67) Here, mass mrefers to the Topological Rigidity that resists the reconfiguration of the internal probabilistic structure, and acceleration a=dv/dt represents the macroscopic reinforcement speed of the cluster expectation value. Therefore, F=ma is not an a priori law but a description of the macroscopic equilibrium state of statistical pressure that appears when probabilistic elements are aligned and in motion. 26 Energy as a Scalar Contraction of Superposition Strength Energy is no longer an independent primitive quantity, but is defined as a value resulting from the contraction of a vectorial probabilistic structure into a scalar invariant [3, 4]. E=Zdα ρ(x, α)(∇S)2 2m+Q(x, α)(68) 25 Conclusion: A New Foundational Perspective Through this study, it has been proven that Newtonian mechanics is no longer a foundational axiom of physics. What Newton discovered was not the fundamental law of the universe, but the macroscopic behavior of the most stable statistical state that a probabilistic superposition structure can reach. All classical dynamics are now reconstructed as emergent theorems of Universal Probabilistic Mechanics. References [1] I. Newton, Philosophiæ Naturalis Principia Mathematica, Royal Society (1687). [2] E. Noether, “Invariante Variationsprobleme,” Nachr. d. K¨onig. Gesellsch. d. Wiss. zu G¨ottingen (1918). [3] D. Bohm, “A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables,” Phys. Rev. 85, 166 (1952). [4] P. W. Anderson, “More Is Different,” Science 177, 393 (1972). [5] H. Haken, Synergetics: An Introduction, Springer (1983). [6] I. Prigogine, From Being to Becoming, Freeman (1980). 32 36 Author Contributions The author conducted all aspects of this study independently. This study is based on the author’s theory, mechanisms, and models, with AI assistance in equation formulation and LaTeX editing. While AI contributions are acknowledged, the author actively supervised the process: checking the AI-generated equations against the underlying theory, identifying inconsistencies, requesting corrections, and guiding adjustments. The equations were not blindly accepted; rather, they were iteratively reviewed and modified to ensure consistency with the theoretical framework. Any references found to be insufficient or incomplete will be actively corrected. It is clarified that any omissions were unintentional. The extent of citations has been reviewed in the context of discussions on AI. This work is provided under the Creative Commons Attribution 4.0 International (CC BY 4.0) License. This license applies to all text, LaTeX code, figures, discussions, and all outputs generated from this work (PDF, Word, HWP, HTML, etc.). 37 Disclaimer and Statement of Intent Notice of Individual Research: This research represents the personal conceptual developments of the author, expressed through the utilization of Artificial Intelligence as a supportive tool. It is hereby explicitly stated that these theories have not yet undergone formal peer review or experimental validation. Restriction of Use: Until such time as rigorous verification and empirical validation are completed, any engineering applications or practical implementations of this framework are strictly prohibited. Acknowledgment of Limitations: The author acknowledges that this preliminary work may contain bold conjectures, interpretative errors, mathematical inaccuracies, and substantial areas requiring refinement. We openly invite constructive criticism to address these shortcomings. Commitment to Development: This project seeks long-term theoretical advancement. We are committed to an iterative process of revision and improvement, ensuring that the framework matures through continuous updates and rigorous version control. 38 Declaration of Compliance and Academic Respect Legal and Formal Compliance: Every effort has been made within this text to adhere to formal and legal procedures. Should there be any matters requiring correction or adjustment, I explicitly state that such instances are entirely unintentional. Upon notification of any necessary revisions, I am committed to proactively and transparently updating the work. Recognition of Preceding Scholarship: I wish to express my profound respect and total agreement with the magnificent and enduring achievements of Sir Isaac Newton. The foundations laid by his brilliance remain the bedrock of physical science. 33 Purpose of the Theory: It is clearly stated that this theory is presented as an exploratory framework intended to discuss academic possibilities. It is not intended to negate established laws but to offer a new perspective for theoretical discourse. Expression of Gratitude: I extend my sincere and deepest gratitude to the many researchers and pioneers whose monumental contributions have paved the way for modern science. This work stands only by acknowledging their immense dedication and historical service to the field of physics. 34