Astrophysics Event Horizon Radius From Jcost

Event horizon radius

The event horizon radius of a black hole, also called the Schwarzschild radius, is the distance from the center at which the escape velocity reaches the speed of light. In general relativity, for a non-rotating, uncharged black hole, this radius is r_s = 2GM/c², where G is the gravitational constant, M is the mass, and c is the speed of light. For the Sun, this radius is about 3 kilometers; for Earth, about 9 millimeters. The horizon is not a physical surface but a boundary in spacetime: anything crossing it, including light, cannot return.

The standard derivation comes from Karl Schwarzschild's 1916 solution to Einstein's field equations, the first exact solution describing a point mass. The radius emerges where the metric's radial component becomes singular, marking the point of no return. In the Newtonian limit, the same expression appears by setting the classical escape velocity equal to c, though that shortcut hides the relativistic meaning. The radius scales linearly with mass, so a black hole of ten solar masses has a horizon ten times wider than one of one solar mass.

In Recognition Science, the framework models the horizon as a zero of a cost function, a measure of mismatch between a measured value and an expected value. The framework's library of machine-checked formal theorems defines domainCost as J(measured / expected), where J is the forced cost function from the framework's core. The framework proves three general facts: the cost vanishes when measured equals expected, the cost is never negative for positive inputs, and the golden-ratio constant phi minus 3/2 is positive. These are statements about the cost function itself, not about gravity.

The framework does not derive the Schwarzschild radius from first principles. Its definition of domainCost uses the ratio m/e without specifying what m and e mean in astrophysical terms. The docstring records the intended direction: that the Schwarzschild radius could appear as a zero of the radial potential, but that identification is a research note, not a proved theorem. What the framework establishes is the scaffolding: a certificate structure that packages the three cost properties, and a proof that such a certificate exists. This is a template that would become a theorem about event horizons only if m and e were defined in the subject's own terms.

For a reader, the practical takeaway is the distinction between a proved mathematical fact and an intended physical application. The cost function's properties are real and machine-checked; the connection to black hole horizons remains an open research direction. The classical Schwarzschild radius stands on its own as a well-tested prediction of general relativity, independent of the framework's ambitions.

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem domainCost_at_equilibrium (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]

What this page does not claim

The framework does not prove that the Schwarzschild radius equals any zero of the cost function. The framework does not establish the event horizon radius from the framework's first principles. The framework's cost function is not shown to be equivalent to the general-relativistic escape velocity condition.

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

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