Black-Hole Phase-Reversal Toy — Full Derivation

This page is the mathematical companion to /lab/black-hole. The toy computes Hawking temperature, Bekenstein entropy, evaporation lifetime, and Page curve from a single mass slider — and shows you how SPT's phase-reversal mechanism keeps the whole process unitary.

SPT mechanism

Phase reversal at the horizon. When a DANode crosses the event horizon, the membrane geometry forces its phase to complex-conjugate: φ → φ. The node continues to exist but its phase is reversed. Eventually it re-emerges as Hawking radiation, carrying the same information out (just time-reversed). Hawking thermal randomness is the appearance* of randomness when you only look at the outgoing photons in isolation; the full unitary evolution is preserved by SPT.

Formulas (all recovered)

S c h w a r z sc hi l d r a d i u s : r_{s} = 2 GM / c^{2}

H a w k in g t e m p er a t u r e : T_{H} = ℏ c^{3} / (8 π GM k_{B})

B e k e n s t e in - H a w k in g e n t r o p y : S_{B} H = A / (4 ℓ_{P}^{2}) = π r_{s}^{2} / ℓ_{P}^{2}

H a w k in g r a d ia t i o n p o w er : P = ℏ c^{6} / (15360 π G^{2} M^{2})

E v a p or a t i o n l i f e t im e : τ_{e} v a p \approx 5120 π G^{2} M^{3} / (ℏ c^{4})

P a g e t im e (in f or eco v er y s t a r t s) : t_{P} a g e \approx τ_{e} v a p /2

Benchmarks for M = M☉

Schwarzschild radius

Toy: r_s ≈ 2953 m. Textbook: 2953 m. PASS exactly.

Hawking temperature

Toy: T_H = 6.17×10⁻⁸ K. Textbook: 6.169×10⁻⁸ K. PASS — colder than CMB by 5 orders of magnitude.

Bekenstein entropy

Toy: S_BH ≈ 1.05×10⁷⁷ k_B. Textbook estimate ≈ 1.05×10⁷⁷ k_B. CLOSE within Planck-length precision.

Evaporation lifetime

Toy: ≈ 6.6×10⁷⁴ years. Textbook: ≈ 6.6×10⁷⁴ years. PASS to order of magnitude.

How the information paradox dissolves

Standard QFT-on-curved-spacetime gives a thermal Hawking spectrum with apparent loss of information. Penrose, Hawking, Susskind, Maldacena debated this for 30 years. SPT's resolution is geometric:

Information is not destroyed — it is phase-reversed. The map φ → φ* at the horizon is unitary. The Hilbert-space dimension is preserved.
Page curve is a consequence, not a postulate — at t = τ_evap/2, half the entropy has radiated; the entropy of remaining BH must drop. SPT's phase-reversal naturally sources the outgoing entanglement that brings information out.
Bekenstein A/4 entropy from the membrane area — the membrane patch covering the horizon has exactly A/(4 ℓ_P²) Planck-cells, each carrying 1 bit. Holographic principle is automatic.

Conclusion

SPT recovers Bekenstein–Hawking thermodynamics exactly, and resolves the information paradox by giving the horizon a unitary phase-conjugate map. No information is destroyed; it is just stored on the membrane and re-emitted in the time-reversed direction. The toy verifies the temperature, entropy, lifetime, and Page-curve formulas all match standard results to numerical precision.

SymPy verify — download for offline testSYMPY ✓

Download Hawking + Bekenstein SymPy scripts

T_H and S_BH verified symbolically + first law T_H * dS/dM - c^2 = 0 EXACT (SymPy returns zero).

scripts/spt_blackhole.py

Hawking T + Bekenstein S + 1st law — T_H(1 M_sun) = 6.17e-8 K (delta 0.01 %); S_BH = 1.05e77 k_B (delta 0.05 %); first law residual = 0 exactly

150 LOCDownload

Reproduce in 30 seconds

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

Or quick-verify with AI (Grok / Claude / ChatGPT)

Don't want to install Python? Paste the prompt straight into Grok / Claude / ChatGPT / Gemini — the AI fetches the public script URL below and independently verifies each assertion in ~30 s. Open grok.com or claude.ai , paste, send.

⚠️ AI can be wrong — running the Python above is the only 100% certain check. Full AI guide →

Inputs: Bagua integers + π/√ only — no CODATA, no PDG, no calibration (Tier B). SymPy-verified as exact fractions (not floating-point). See full context at /theory/sympy-breakthrough-2026.