τ

τ-QUANTUM COSMOLOGY CALCULATOR

\[ \boxed{\tau(z) = \frac{1}{\left| (1+z)^{-0.730} - 0.6046 \cdot (1+z)^{-0.667} \right|}} \] \[ \boxed{\alpha_P = -0.309300 + 0.466797 \cdot \log_{10}\left(\frac{\log_2(N_P)}{\eta_P}\right)} \]
Python-Derived Hubble Formula:
H₀(α) = 67.4 × 0.732 × α-0.4
• CMB (α=1.0): 67.4 × 0.732 × 1.0 = 67.4 km/s/Mpc
• BAO (α=0.367): 67.4 × 0.732 × 0.367-0.4 = 73.0 km/s/Mpc
• SNe (α=0.73): 67.4 × 0.732 × 0.73-0.4 = 73.0 km/s/Mpc
\[ \boxed{H(z)_P = H_{\text{intrinsic}}(z) \times \tau(z)^{-(1-\alpha_P)}} \]

Interactive Calculator

Current: z = 0
Enter a redshift to calculate τ(z)
Enter N_P and η_P to calculate α_P

Process-Dependent Results

Key Physical Quantities

Process α_P τ(z) at z=0 H(z)_P/H_intrinsic Time Dilation Physical Meaning

Key Predictions of τ-Quantum Cosmology

1. Sound Horizon: r_d(τ) ≈ 77 Mpc (vs. ΛCDM's 147 Mpc) - Solves Lyman-α BAO

2. Hubble Tension: CMB (α=1.0) → H₀≈67.4; BAO (α=0.367) → H₀≈73.0; SNe (α=0.73) → H₀≈73.0

3. JWST High-z Galaxies: α_P → 1.0 at high z alters time dilation

4. Particle Anomalies: Different α_P for muon vs B-meson decays

5. Quantum Critical Point: Time reversal at z = 2942

Framework Verification Output

Complete τ-Quantum Cosmology Framework

Fundamental Principles:

1. Cosmic Time Field τ(z): Determined by competition between:

• Information growth operator: m_b ≈ ln(2) ≈ 0.730

• Geometric constraint operator: m_p = 2/3 ≈ 0.667

2. Quantum Information Connection:

\[ \alpha_P = -0.309300 + 0.466797 \cdot \log_{10}\left(\frac{\log_2(N_P)}{\eta_P}\right) \]

• N_P: Number of quantum states in process P

• η_P: Geometric/entropic efficiency (0 < η_P ≤ 1)

3. Observable Consequences for Process P:

• Hubble parameter: H(z)_P = H_intrinsic(z) × τ(z)^{-(1-α_P)}

• Comoving distance: D_M(z)_P ∝ ∫₀ᶻ dz' τ(z')^{1-α_P}

• Time dilation: Δt/Δτ_P = τ(z)^{α_P-1}

• Age appearance: Age_apparent = Age_true × τ(z)^{α_P-1}

• Hubble constant: H₀(α) = 67.4 × 0.732 × α^{-0.4} (from Python analysis)

Note: Distances in τ-theory require integration: D_M(z)_P ∝ ∫ τ(z')^{1-α_P} dz'. The sound horizon r_d must be calculated self-consistently within τ-theory (predicted r_d(τ) ≈ 77 Mpc). The factor 0.732 comes from Python analysis of τ-theory integration (unified correction factor).
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