Physics Hawking Radiation From Rs

Hawking radiation in the framework

Hawking radiation is the prediction that black holes emit particles and slowly lose mass, as if they had a temperature. The temperature, named after Stephen Hawking who derived it in 1974, is T_H = ℏc³/(8πGMk_B), where ℏ is the reduced Planck constant, c is the speed of light, G is the gravitational constant, M is the black hole's mass, and k_B is the Boltzmann constant. The formula says smaller black holes are hotter and radiate faster, while larger ones are cooler and nearly inert.

In the Recognition Science framework, the constants ℏ and G are not free parameters. The framework derives them from a forcing chain that begins with a cost function for recognition events, and it sets ℏ = φ⁻⁵ and G = φ⁵/π, where φ is the golden ratio, about 1.618. Substituting those into Hawking's temperature formula gives T_H = φ⁻¹⁰/(8M) in the framework's natural units, so the product T_H × M is a fixed positive number. The framework's machine-checked library of formal theorems proves that this factor is positive and that φ¹⁰ exceeds 100, using the Fibonacci identity φ¹⁰ = 55φ + 34.

The module also catalogs five canonical effects that follow from Hawking radiation: the thermal spectrum of the emitted particles, the information paradox about whether information is lost, black hole evaporation, the Page curve describing entropy over time, and the possible remnant left behind. The library proves that this list has exactly five entries, matching a dimension count in the framework's configuration space. The formal development contains no unproved assumptions and no axioms beyond the standard logical ones.

What this establishes in plain language is that the framework's derived constants, when fed through a standard physics formula, reproduce a known temperature law and organize its known consequences into a five-item structure. It does not prove that Hawking radiation exists in nature, and it does not resolve the information paradox. It shows that the framework's machinery can reach a famous physics result and give it a compact, positive, and internally consistent form.

inductive HawkingEffect where
  | thermalSpectrum | informationParadox | evaporation | pageCurve | remnant
  deriving DecidableEq, Repr, BEq, Fintype

/-- Hawking temperature factor (dimensionless): 1/(8 × φ^10). -/
noncomputable def hawkingFactor : ℝ := 1 / (8 * phi ^ 10)

theorem hawkingFactor_pos : 0 < hawkingFactor :=
  div_pos one_pos (mul_pos (by norm_num) (pow_pos phi_pos 10))

/-- φ^10 = 55φ + 34 > 100. -/
theorem phi10_large : phi ^ 10 > 100 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  have h5 : phi ^ 5 = 5 * phi + 3 := by nlinarith
  have h10 : phi ^ 10 = phi ^ 5 * phi ^ 5 := by ring
  rw [h10]; nlinarith [phi_gt_onePointSixOne]

theorem hawkingEffectCount : Fintype.card HawkingEffect = 5 := by decide

What this page does not claim

The module does not prove that Hawking radiation exists in nature. The information paradox is not resolved by this module. The framework does not derive the Hawking temperature from first principles; it substitutes derived constants into a standard formula.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/HawkingRadiationFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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