Physics Black Body Radiation From Jcost

The Planck Spectrum

Blackbody radiation is the electromagnetic radiation emitted by an idealized object that absorbs all light falling on it. A hot piece of metal, the Sun, and the cosmic microwave background all approximate this emission. The spectrum, the intensity of light at each frequency, follows a smooth curve that rises to a single peak and then falls. In 1900 Max Planck derived the exact form of this curve, and the peak shifts to higher frequencies as temperature rises, a relationship known as Wien's displacement law. The classical Rayleigh-Jeans law predicted an infinite intensity at high frequencies, a failure that Planck's quantum hypothesis resolved.

In Recognition Science, the framework models each photon mode as a recognition unit, a discrete record of an event, with energy ε = hν. The framework's central cost function J(x) = (x + 1/x)/2 - 1 measures the cost of a recognition event. The key structural claim is that this cost, evaluated at the ratio of photon energy to thermal energy, hν/kT, determines the Planck distribution. The equilibrium condition, where the cost is zero, occurs at hν = kT, which is exactly the Rayleigh-Jeans limit. Off-peak modes, where the ratio is not 1, carry positive cost.

The module in the framework's machine-checked library of formal theorems defines five spectral regions: radio, infrared, visible, ultraviolet, and X-ray. It proves that there are exactly five such regions, that the cost function is zero at equilibrium, and that off-peak modes have positive cost. These three facts are packaged into a certificate structure. The framework's library shows that the number of spectral regions, five, matches the configuration dimension D = 5 from the forcing chain. This is a structural alignment, not a derivation of Planck's law itself.

What this establishes in plain language is a consistency check. The framework's cost function, derived from first principles, reproduces the equilibrium condition of blackbody radiation and the count of spectral regions. It does not derive the full Planck spectrum or the value of Planck's constant. The framework models the known physics, and the known physics matches the framework's structure. The certificate is a formal statement that the framework's core object is compatible with a central result of quantum statistical mechanics.

structure BlackBodyRadiationCert where
  five_regions : Fintype.card SpectralRegion = 5
  rj_equilibrium : Jcost 1 = 0
  off_peak : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r

/-- Rayleigh-Jeans equilibrium: J = 0 at hν = kT (ratio = 1). -/
theorem rayleighJeans_equilibrium : Jcost 1 = 0 := Jcost_unit0

theorem spectralRegionCount : Fintype.card SpectralRegion = 5 := by decide

/-- Off-peak modes have positive J-cost. -/
theorem off_peak_positive {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

What this page does not claim

This module does not derive Planck's law or the value of Planck's constant. The framework's model is a consistency check, not a derivation of blackbody radiation from first principles. The five spectral regions are a definitional choice, not a prediction of the framework.

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

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