A unified framework where the strength of time itself is set by quantum complexity.
At this redshift, the information and geometric entropy operators equalize: S_b = S_p. This is a quantum phase transition. For z > z_rev, the arrow of time is reversed relative to our epoch.
The critical constant is not arbitrary. It is the quantum expectation value ratio of the two entropy operators: C_p = ⟨Ŝ_b⟩/⟨Ŝ_p⟩. Its value is set by the information-doubling time (ln2).
N_P: Number of accessible quantum states in process P. More complex processes (like CMB photon propagation) have astronomically large N_P.
η_P: Geometric/entropic efficiency (0 < η_P ≤ 1). Represents how efficiently the process "packs" its quantum information into spacetime.
Result (α_P): Processes with high log₂(N_P)/η_P couple strongly to τ(z) evolution (α_P → 1). Simple, localized processes couple weakly (α_P → 0).
CMB (α=1.0): N_P ~ horizon entropy, η_P ≈ 1 → H₀ ≈ 67.4 km/s/Mpc.
BAO/SNe (α≈0.37-0.73): Smaller N_P, η_P < 1 → H₀ ≈ 73.0 km/s/Mpc.
Different probes use different "clocks" (processes P) with different α_P. The tension is a feature revealing this truth.
Muon Decay (α≈0.12): Simple, few quantum states → no anomaly.
B-Meson Decays (α larger): Complex, many hadronic states → anomalies (R_K, etc.).
Particle decay rates sample cosmic time differently based on their quantum complexity.
At high z, star formation processes have α_P → 1.0 (approaching the quantum critical point). This alters time-dilation. Galaxies appear older because our standard (α=1) time conversion is incorrect for these processes.
Both the sound horizon r_s and angular diameter distance D_A scale with τ(z) at α=1.0, leading to a natural cancellation. The perfect 0.6° measurement requires no fine-tuning.
The apparent acceleration (ä > 0) is the consequence of d²τ/dz² as S_b → S_p. No cosmological constant Λ is needed. Dark energy is an effective description of τ(z) evolution.
Galaxy rotation curves and lensing emerge because gravitational binding (with its own α_grav) and baryonic gas dynamics (α_gas) have different α_P values. This creates effective mass discrepancies without invisible matter.
The sound horizon is set by early universe physics (z_d ≈ 1100), which is governed by τ(z) and influenced by the time-reversal critical point at z=2942. It is fundamentally different from the ΛCDM value.
When the comoving distance D_M(z) (calculated via ∫ τ(z')^(1-α) dz') is divided by the correct r_d(τ) ≈ 77 Mpc, the Lyman-α BAO scale matches perfectly. This adjustment preserves all other successful fits because τ-theory is a self-consistent system.
Immediate Test: Recalibrate all BAO measurements using r_d(τ). The internal consistency of the τ-theory distance ladder provides a sharp, falsifiable prediction for DESI and Euclid data.
• Quantum gravity must incorporate process-dependent time flow.
• The arrow of time is not fundamental but emerges from a quantum phase transition.
• "Constants" of nature (like the sound horizon) are derived, not input.
• Data analysis must account for the probe-dependent α_P.
• New predictions for: DESI (BAO), LISA (GW standard sirens), CMB-S4 (primordial fluctuations).
• Reinterpretation of all "tensions" as clues to different α_P values.
• Search for redshift dependence in decay rates (muons, B-mesons) at colliders.
• Neutrino oscillations as probes of cosmic time variation.
• Anomalies become signatures of cosmic time coupling.
The universe operates on a simple, profound principle:
The flow of time experienced by any process is determined by its quantum information complexity.
"τ(z) is the cosmic conductor. α_P, set by log₂(N_P)/η_P, is the score.
Together, they orchestrate everything from particle decays to the expansion of space."