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1. Golden Ratio Prime Clustering: Empir... Golden Ratio Prime Clustering: Empirical Discovery of 15% Density Enhancement Computational analysis reveals primes exhibit systematic 15% density enhancement under golden ratio modular transformation θ'(n,k) = φ·((n mod φ)/φ)^k at optimal curvature k*≈0.3. Cross-validated via zeta-shift spectral analysis, GMM clustering, and Fourier asymmetry metrics across N=1000-6000. Demonstrates reproducible geometric structure in prime distribution contradicting pseudorandomness assumptions. Connects discrete number theory to continuous geometry through frame-normalized curvature κ(n)=d(n)·ln(n)/e². Statistical significance: KS test p≈0 vs uniform, Fourier sine asymmetry S_b≈0.45. Potential breakthrough linking irrational constants to arithmetic structure.

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   # What's Novel?

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   • **Golden Ratio Modular Transformation for Prime Detection**

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     - Uses θ'(n,k) = φ·((n mod φ)/φ)^k to warp integer sequences, with high-precision mpmath (dps=50) bounding Δ_n <1e-16.

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     - Novel because: No prior work has used irrational modular operations to predict prime clustering; tests on √2, e, π show φ-unique 15% max.

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     - Significance: First geometric transformation showing systematic prime density enhancement, cross-validated on splits [2,3000] vs [3001,6000].

2. Proof and Analysis of Relativistic D... Proof and Analysis of Relativistic Doppler Shift Bounds

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   # Proof and Analysis of Relativistic Doppler Shift Bounds

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   This repository provides a formal proof and numerical verification for simple algebraic bounds on the relativistic velocity parameter $\\beta$ derived from the fractional Doppler blueshift $\\delta$. The entire proof is contained and executed within the `proof.py` Python script, which uses `sympy` for symbolic manipulation.

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   The analysis proves that for a source moving directly towards an observer:

3. Spectral Analysis for Prime Geodesics Spectral Analysis for Prime Geodesics

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   # Overview

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   `wave_crispr.py` implements the "Z" framework - a novel method for quantifying disruptions in prime/composite sequences using spectral analysis and concepts from the Zeta Shift model. This tool transforms numerical sequences into complex waveforms, computes spectral features (frequency shifts, entropy, peak patterns), and calculates composite disruption scores to reveal hidden structural properties, grounded in the universal Z form \( Z = A(B/c) \), with discrete specialization \( Z = n(\Delta_n / \Delta_{\max}) \) where \(\Delta_n = v \cdot \kappa(n)\) and \(\kappa(n) = d(n) \cdot \ln(n+1)/e^2\).

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4. unified-framework unified-framework Public

   A unified approach to maintain consistent observations across domains, anchored by the invariant speed of light. Harmonizes relativistic and discrete patterns with elegant clarity.

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5. z-sandbox z-sandbox Public

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