NO OTHERNESS

Abstract

We propose a unified theoretical framework based on Quantum Contextual Probability (QCP) asserting that both fundamental physical constants (e.g., \alpha, G, \hbarα,G,ℏ) and mathematical constants (e.g., \pi, eπ,e) are dynamically variable, context-dependent entities. Their variability is governed by a background scalar field \phiϕ via an effective theta parameter \theta_{\text{eff}} = \theta + f(\phi)θeff​=θ+f(ϕ). We introduce the Unification Constant \beta = e^{-2 \log(\alpha_0)} (e/\hbar)^3β=e−2log(α0​)(e/ℏ)3 and the Unified Constants Equation (UCE) to model their interrelations. Crucially, we extend QCP to mathematical constants to reinterpret their infinite series convergence as a quantum decoherence process and their non-repeating decimals as a probabilistic entanglement topology. This framework addresses cosmological fine-tuning and provides a foundation for empirically testing constant variation through astrophysical and neutrino oscillation experiments.

Introduction

Modern physics rests on the constancy of fundamental parameters. Yet observations of quasar absorption spectra suggest a potential variation of the fine-structure constant \alphaα over cosmological timescales (\Delta\alpha/\alpha \sim 10^{-5}Δα/α∼10−5 at z \sim 2z∼2) . Meanwhile, mathematical constants, defined through infinite series, possess a fixed, seemingly a-contextual nature.

This paper bridges these two domains through Quantum Contextual Probability (QCP). QCP posits that measurement contexts fundamentally influence outcomes, encompassing entanglement, nonlocality, and the stochastic emergence of classical states.

We use QCP to propose that all constants are field-dependent random variables, with their observed ‘fixed’ values representing a long-term decohered limit defined by the local state of the universe’s scalar field \phiϕ.

This integration culminates in the Unified Constants Equation (UCE), to offer a dynamic and unified perspective on nature’s laws.

The QCP Framework for Constant Variability

1. Principles of Context-Dependent Constants

QCP principles are reinterpreted to model the dynamic evolution of constants:

– Context-Dependence (Scalar Field \phiϕ): The effective value of any constant XX is a probability distribution \mathcal{P}(X | C(\phi))P(X∣C(ϕ), where the context C(\phi)C(ϕ) is defined by the local state of the scalar field \phiϕ (potentially dark energy).

This field directly modulates the vacuum properties through the effective theta parameter:\theta_{\text{eff}} = \theta + f(\phi)θeff​=θ+f(ϕ)

– Decoherence and Classical Stabilization: The observed stability of constants (e.g., d\alpha/dt / \alpha < 10^{-17}dα/dt/α<10−17 per year in laboratory tests ) is a result of decoherence.

Over cosmological time, the quantum state of the vacuum \rho(t)ρ(t) interacts with the environment, leading to a classical, fixed constant value \rho_{\text{classical}}ρclassical​, thus stabilizing XX at its local \theta_{\text{eff}}(\phi)θeff​(ϕ) value.

– Probabilistic Entanglement: Physical constants correlate via a non-local unification operator U(\phi)U(ϕ), which ensures that the probability distributions of distinct constants (e.g., GG and \alphaα) are interdependent, addressing the fine-tuning problem.

2. The Unified Constants Equation (UCE)

We define a dimensionally consistent structure for constant interrelation:

\mathbf{C}_{\text{uni}}’ = c^{\alpha_1} \cdot G^{\alpha_2} \cdot \hbar^{\alpha_3} \cdot e^{\alpha_4} \cdot \beta^{\alpha_5} \cdot m_e^{\alpha_6} \cdot u^{\alpha_7} \cdot \varepsilon_0^{\alpha_8} \cdot \left(1 + h(\theta_{\text{eff}})\right)Cuni′​=cα1​⋅Gα2​⋅ℏα3​⋅eα4​⋅βα5​⋅meα6​​⋅uα7​⋅ε0α8​​⋅(1+h(θeff​))

where \alpha_iαi​ are exponents ensuring dimensional balance, and h(\theta_{\text{eff}})h(θeff​) is a scalar function that introduces the dynamic context derived from the \phiϕ field.

3. The Unification Constant \betaβ

We introduce the Unification Constant \betaβ, which mediates the fractal-like relationships between fundamental parameters:

\beta = e^{-2 \log(\alpha_0)} \left(\frac{e}{\hbar}\right)^3β=e−2log(α0​)(ℏe​)3

The structure of \betaβ, which incorporates the mathematical constant ee as a base for a logarithmic-exponential relation involving the fine-structure constant \alpha_0α0​ and the ratio of elementary charge to reduced Planck constant (e/\hbare/ℏ), formally entangles mathematical and physical constants at the level of the unified vacuum state.

QCP and the Emergence of Mathematical Constants

The defining feature of this framework is the application of QCP to \piπ and ee, viewing them not as static limits, but as dynamic, probabilistic entities whose fixed values emerge through a process analogous to quantum measurement.

1. Infinite Series as Contextual Decoherence

The definition of \piπ and ee through infinite series (e.g., \pi = 4 \sum \frac{(-1)^n}{2n+1}π=4∑2n+1(−1)n​) is reinterpreted:

– The Quantum Superposition of Terms: The sum of all terms is initially a quantum superposition state (\pi_{\text{superposition}}\rangle = \sum_{n=0}^\infty c_n |a_n\rangle∣πsuperposition​⟩=∑n=0∞​cn​∣an​⟩, existing in a high-dimensional mathematical Hilbert space.

– Contextual Summation (Measurement): The Context CC is the number of terms summed (NN). The act of taking the partial sum S_NSN​ is a context-dependent measurement that yields a value X=S_NX=SN​. The probability of the remainder being R_NRN​
 is \mathcal{P}(R_N | N)P(RN​∣N).

– Decoherence Time (t_{\mathcal{D}}tD​): The convergence of the series (N \to \inftyN→∞) is a process of stochastic coherence loss (\rho_{\text{series}} \xrightarrow{\mathcal{D}} \rho_{\text{Fixed}}ρseries​D​ρFixed​).

The number of terms NN required to achieve a classical, stable precision 10^{-k}10−k acts as the mathematical decoherence timescale (t_{\mathcal{D}}tD​).

t_{\mathcal{D}}(e) \ll t_{\mathcal{D}}(\pi)tD​(e)≪tD​(π): The rapid convergence of ee’s factorial series is a fast decoherence to the classical value. The slow convergence of \piπ’s series is a long, persistent quantum state of high uncertainty/entropy.

2. Transcendental Decimals and Probabilistic Entanglement

QCP augments our concept of infinite decimals by relating them to quantum phenomena:

Repeating Decimals: A repeating decimal is a classical, fully decohered system. The sequence of digits is defined by a simple, finite, and local rule. The Information Gain (Entropy, S_{\text{info}}Sinfo​) from calculating the next digit is zero after the repeating block is established.

Transcendental Decimals (\pi, eπ,e): These numbers exhibit probabilistic entanglement between their digits. Though the digits may appear statistically random (normality), their specific sequence is constrained by the number’s ultimate definition, implying a nonlocal correlation that is not derivable from a local, finite algorithm.

Each newly calculated digit d_kdk​ is a Bayesian Probability Update (derived from the initial \mathcal{P}(X|C)P(X∣C)), where the prior probability is derived from the contextual \theta_{\text{eff}}(\phi)θeff​(ϕ) state of the universe.

Unified Energetics and Phenomenology

1. Comprehensive Energy and Entropy

The dynamic nature of the constants is captured by the Comprehensive Energy Equation (CEE) and Unified Entropy Equation (UEE):

H = \sum_{\text{matter}} T_m + \sum_{\text{DM}} T_{\text{DM}} + \sum_{\text{DE}} T_{\text{DE}} + \frac{1}{2} \sum_n \hbar \omega_n \quad \text{(CEE)}H=matter∑​Tm​+DM∑​TDM​+DE∑​TDE​+21​n∑​ℏωn​(CEE)

S_{\text{total}}(t) = S_{\text{quantum}}(t) + S_{\text{macro}}(t) + S_{\text{cosmo}}(t) + S_{\text{info}}(t) \quad \text{(UEE)}Stotal​(t)=Squantum​(t)+Smacro​(t)+Scosmo​(t)+Sinfo​(t)(UEE)

Constant variations h(\theta_{\text{eff}})h(θeff​) represent the coupling between the scalar field energy (T_{\text{DE}}TDE​) and the other energy components. Furthermore, constant evolution directly impacts the total entropy S_{\text{total}}Stotal​, linking fundamental physics to the flow of cosmic information and the principles of Landauer.

2. Experimental Hypotheses

This framework yields testable predictions:

– Fine-Structure Constant  (\alpha)(α) Modulation: Variation in \alphaα
is a direct probe of \theta_{\text{eff}}(\phi)θeff​(ϕ). High-precision quasar absorption spectra measurements remain the primary test bed.

– Generalized Seesaw Mechanism: Neutrino mass matrices, MM, are functions of \theta_{\text{eff}}θeff​. Temporal or spatial shifts in \theta_{\text{eff}}θeff​ would alter the neutrino oscillation parameters, a phenomenon testable in long-baseline experiments like DUNE .

– Gravitational Waves: Signatures of GG variation may be imprinted on gravitational wave propagation, offering a unique test of the UCE via LIGO observations .

Conclusion

The Quantum Contextual Probability framework yields a profound revision to the concept of fundamental constants. By unifying their variability under the influence of the contextual scalar field \theta_{\text{eff}}θeff​, we bridge theoretical physics and abstract mathematics. The constants \piπ and ee are recast as emergent, decohering probabilistic amplitudes, their non-repeating digits a manifestation of fundamental probabilistic entanglement.

Future research will focus on formally deriving the functions f(\phi)f(ϕ) and h(\theta_{\text{eff}})h(θeff​) and leveraging high-precision empirical data to validate the dynamic, unified reality described by the UCE and \betaβ.

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