Gravity Hawking Rad3 From Jcost

The module's actual scope

Hawking radiation is the theoretical prediction that black holes emit particles and slowly lose mass, with a power that depends on the black hole's mass, the speed of light, the gravitational constant, and Planck's constant. The standard formula, P = hbar c^6 / (15360 pi G^2 M^2), describes this emission for a black hole of mass M. In the Recognition Science framework, the idea was to express this same power using the framework's cost function, which measures the forced expense of a recognition event. The module gravity hawking rad3 from jcost was meant to be that bridge, but the file itself proves something far more modest.

The file defines a function called domainCost, which takes two real numbers m and e and returns the cost function J applied to their ratio, J(m / e). The three theorems it proves are general properties of this cost function, not specific to gravity or Hawking radiation. First, when m equals e, the cost is zero. Second, for positive m and e, the cost is never negative. Third, the golden-ratio-derived constant phi minus 3/2 is positive. These are true for any positive inputs, because the cost function J has these properties universally. The file's own docstring states this plainly: it proves nothing specific to Hawking radiation, because the definition of domainCost does not reference mass, energy, or any physical quantity.

The intended physics, recorded as a research note rather than a theorem, was that at the Planck mass, the Hawking power would equal J(phi) times the Planck power, about 0.118 times it. That would be a striking numerical link between the recognition cost and black hole thermodynamics. But the file as written contains no definition of m and e in terms of mass and energy, so the connection remains an unproved aspiration. The machine-checked library of formal theorems, which verifies every proof, certifies only the three general facts about J, not the Hawking power formula or any gravitational claim.

What this means in practice is that the file is a template, shared with 2383 sibling modules, that happens to carry a gravity-themed name. The general facts about J are real and checked, but they are not about black holes. To make this file a theorem about Hawking radiation, someone would need to define m and e in the subject's own terms, for example as a ratio of the black hole mass to the Planck mass. Until that definition exists, the honest statement is that the file establishes the cost function's basic properties and nothing more.

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]

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

What this page does not claim

This file does not prove the Hawking radiation power formula or any gravitational claim. The intended connection between the recognition cost and Hawking power at the Planck mass is not established by this file. The constant phi minus 3/2 has no demonstrated physical meaning in this file beyond being positive.

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/Gravity/HawkingRad3FromJCost.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

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