Phi-zeta-Zeckendorf Unified Attractor Theory: Complete Framework from Binary Phase Cancellation to 11-Dimensional Euler Coupling
Year: 2025
Type: article
Abstract: This paper establishes a complete mathematical framework from binary phase cancellation to $11$-dimensional Euler coupling, unifying the core structures in the $\Phi$--$\zeta$--Zeckendorf theory system. Through the axiomatic starting point $e^{i\theta} = \cos\theta + i\sin\theta$, we prove the length-2 atomic law for binary phase sequences $a_n = (-1)^n$ and the theorem that the sum of arbitrary binary length blocks equals 0. We define the $\Phi$--Zeckendorf weighted phase $\Theta_\phi(N)$ and $\chi$-phase $\theta_$\\chi(s)$$ = \arg $\\chi(s)$$$, proving linear phase cancellation of phase pairs and the $\Phi$--Zeckendorf $\chi$-phase layer closure theorem. Core achievements include: (1) Established $11$-dimensional Euler coupling network $G=(V,E)$, where $V=\{1,\ldots,11\}$, each edge couples phase $\Theta_{(d,k)} = \sum \phi^{-j} (\theta_\chi(1/2+it) + \theta_\chi(1/2-it))$, total phase $\Theta_{(11D)} = \sum \lambda_{(d,k)} \Theta_{(d,k)}$, $\lambda_{(d,k)} = \phi^{-(d+k)/2}$; (2) Directly proved the $11$-dimensional coupling closure theorem $e^{i\Theta_{(11D)}} = 1$ from χ-phase symmetry; (3) Provided high-precision numerical verification (mpmath dps=100) of $11$-dimensional hierarchical data, including d (dimension), $\lambda_d$ (weight), t[0] (zero offset), $\theta_\chi(+it)$, $\theta_\chi(-it)$ (phase values), $\Theta_d$ (hierarchical total phase), $|e^{i\Theta_d}-1|$ (error), and triadic weights $w_+$, $w_0$, $w_-$; (4) Established connection to black hole information recovery, proving $\Delta S_{total} = 0$, Hawking temperature $T_H = 1/(8\pi) \approx 0.039789$, black hole entropy $S_{BH} = 4\pi \approx 12.566$; (5) Verified consistency with triadic information conservation, $\langle $i_+$ \rangle \approx 0.403$, $\langle $i_0$ \rangle \approx 0.194$, $\langle $i_-$ \rangle \approx 0.403$. All calculations verified residual $|e^{i\Theta_{(11D)}}-1| = 0.0$, confirming the mathematical completeness of the theory. This framework provides new perspectives for quantum gravity, predicts phase noise exponent $\alpha \approx 2/3$ in nano-holographic experiments, and provides the mathematical foundation for island compensation operator $T_\epsilon^{island}$ in black hole information paradox.
Authors: Ma, Haobo, Zhang, Wenlin
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7091336772
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