Physics Schwarzchild Interior From Jcost

A formal placeholder

The Schwarzschild interior is the region inside a non-rotating, uncharged black hole's event horizon, where the standard solution to Einstein's field equations describes matter collapsing under its own gravity. In general relativity, the pressure and density of that matter grow without bound as the center approaches, a singularity that marks the limit of the classical theory. The module named "Schwarzschild Interior from J-Cost" in the Recognition Science library was intended to apply the framework's cost function to this problem, but its actual content is far narrower.

The file defines a quantity called domainCost, a discrete record of how much recognition a system pays, as the cost function J applied to the ratio of two real numbers, m over e. It then proves three facts about this quantity: it equals zero when m equals e, it is never negative when both inputs are positive, and the number phi minus 3/2 is positive. The final structure, SchwIntCert, packages these three statements together and shows they are consistent. All of this is proved in the machine-checked library of formal theorems, but none of it mentions mass, energy, pressure, or gravity.

The docstring records the intended physics: interior pressure at the Schwarzschild radius would be P = (3/8π) × (c^4/G^2 M^2) × (1-J(φ)) for a corrected Tolman-Oppenheimer-Volkoff equation. That formula, however, is a research note, not a theorem. The Lean code never defines m or e in terms of black hole physics, so the three proved facts apply to any positive real numbers, not to a star's interior. The module is a template, shared nearly verbatim with 2383 sibling files, that proves generic properties of the cost function and nothing specific to this subject.

What the module does establish, in plain language, is a small set of formal truths about the cost function J: it vanishes at unity, it is nonnegative for positive arguments, and a particular threshold involving the golden ratio is positive. These facts are real and machine-checked, but they are not physics. The bridge from the abstract cost function to a physical quantity like pressure remains open; this file does not cross it.

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

No physical claim about the Schwarzschild interior is proved in this module. The pressure formula in the docstring is not a theorem and is not derived from the Lean code. The cost function J is not shown to relate to any gravitational quantity in this file.

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

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