Physics Stefan Boltzmann Exact Rs

The Stefan-Boltzmann constant

The Stefan-Boltzmann constant, written σ, sets the total power radiated per unit area by a black body at temperature T through the law P = σT⁴. Its measured value is about 5.67 × 10⁻⁸ watts per square meter per kelvin to the fourth power. The constant appears whenever engineers size radiators or astronomers estimate a star's temperature from its brightness.

In conventional physics, σ is not a free number. It is built from more basic constants: σ = 2π⁵k_B⁴ / (15h³c²), where k_B is the Boltzmann constant, h is Planck's constant, and c is the speed of light. This expression follows from the Planck radiation law, derived by Max Planck in 1900, and the thermodynamic argument that Josef Stefan published in 1879 and Ludwig Boltzmann refined in 1884.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and derives a cost function J(x) = (x + 1/x)/2 − 1 that measures the price of a recognition. The framework's library, a machine-checked collection of formal theorems, proves three general facts about this cost function: it vanishes when its two arguments are equal, it is never negative for positive inputs, and the quantity φ − 3/2 is positive, where φ is the golden ratio. These facts are universal, applying to any pair of positive numbers.

The module named Stefan_Boltzmann_Exact_RS defines its cost as J(m/e), the cost of recognizing a mass m relative to an energy e. The three proved facts hold for that definition, but the module proves nothing specific to thermal radiation. The library's own docstring states this plainly: the paragraph about σ being determined from other constants is a research note recording where the idea was meant to go, not a result. What would make the module a theorem about the Stefan-Boltzmann constant is a definition of m and e in that subject's own terms, which the module does not supply.

What the module does establish is a template. It shows that any physical theory that can express its quantities as a ratio m/e inherits the three cost properties for free. That is a portable lesson: the cost function's behavior at equality, its nonnegativity, and the positivity of a golden-ratio threshold hold without any physics-specific input. The step from those general facts to σ itself remains open, waiting for a definition that ties m and e to the constants of thermal radiation.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty StefanBoltzmannExactRS := ⟨cert⟩

What this page does not claim

This module does not prove that σ equals any particular value in Recognition Science. The three proved facts do not depend on any physics-specific assumption. The research note about σ being determined is not a proved result.

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/Stefan_Boltzmann_Exact_RS.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: