Law 45 — Entropy + Arrow of Time from DA Coset Decoherence (Đợt 15 · 11/05/2026 v3.17)

🎯 Law 45 — Entropy + Arrow of Time = Q_7 → Q_3 Coset Decoherence. Boltzmann 1872 H-theorem assumes molecular chaos (Stosszahlansatz) — but Loschmidt 1876 paradox: microscopic dynamics is time-reversal symmetric, so where does the arrow of time come from? SPT closes this 150-year question via Law 44's regime-switch mechanism: Entropy structure: each DANode has 16 Q_3 cosets (Law 44) as macrostates. Boltzmann entropy S = -k_B Σ p_i log(p_i) over these 16 states. Maximum: S_wave = k_B · log(16) ≈ 2.77 k_B nats (uniform delocalised wave regime). Minimum: S_particle = 0 (locked to 1 coset). 2nd law from environment dilution: microscopic unitarity preserves S_total exactly. But the environment has ~10¹⁰⁴ virtual DA modes (Law 41 density). Phase info transferred to env spreads over these modes — spontaneous recoherence requires phase-matching all ~10¹⁰⁴, probability < exp(-10¹⁰⁴) ≈ 0. Practically observed (coarse-grained) entropy increases monotonically. Bekenstein S_BH recovered: each ℓ_Pl² horizon patch ↔ 1 yin-yang Z₂ bit. N = A/ℓ_Pl² bits; gravity coupling provides 1/4 prefactor (Law 12 cross-check) → S_BH = A/(4ℓ_Pl²). Same Bagua bit-counting as Law 12. Cosmological arrow: cascade depth d_0(t) increases with cosmic time (Law 6 anchor √7/4 fixed by initial conditions). Real-DA cluster count N_clusters(t) grows monotonically post-recombination. S_universe(t) = N(t) · log(16) increases monotonically → thermodynamic arrow. Boltzmann brain paradox dissolved: the cascade direction is INTRINSIC, not statistical fluctuation.

pip install sympy numpy && python3 scripts/spt_entropy.py

Importance: VERY HIGH — the 2nd law of thermodynamics is the foundational law of irreversibility, but its origin has been a 150-year puzzle (Loschmidt 1876 paradox, Zermelo 1896 recurrence). Boltzmann's H-theorem assumed molecular chaos; Gibbs gave it ensemble dressing; modern decoherence theory (Zurek 1981) provides the practical mechanism. SPT Law 45 closes the structural question: entropy IS Q_7 coset-count, and the arrow of time IS the cascade direction (Law 6 anchor). Cross-link saturation: same Bagua substrate that gives 1/α_em = 137 (Law 5), Ω = 1 cosmology (Law 11), forces (Law 42), and wave-particle duality (Law 44) now ALSO defines entropy structure. Boltzmann brain paradox dissolved: spontaneous low-entropy fluctuations cannot reverse the cascade direction because d_0 anchor is fixed structurally (√7/4 from Q_6 Laplacian spectral gap).

✅ Entropy = -k_B Σ p_i log p_i over 16 Q_3 cosets of Q_7. Arrow of time = cascade direction (Law 6 anchor). 2nd law emerges from practical irreversibility of Q_7 → Q_3 decoherence into ~10¹⁰⁴ virtual DA modes (Law 41). Bekenstein S_BH = A/(4ℓ_Pl²) recovered from Bagua bit-counting. Loschmidt 1876 paradox dissolved: cascade direction is structural, not statistical fluctuation. Cross-links: Law 6 cascade anchor · Law 12 Bekenstein S_BH · Law 41 virtual DANode · Law 44 wave-particle duality · Law 46 entanglement.