Gravity Planck Star From Jcost

Planck star bounce

A Planck star is a hypothetical object proposed in loop quantum gravity and related approaches: instead of a collapsing black hole ending in a singularity, quantum effects become strong near the Planck length and halt the collapse, producing a bounce that could later explode. The name comes from the Planck scale, roughly 10^-35 meters, where quantum gravity is expected to dominate. The idea is that the star's radius never shrinks below a minimum set by that scale.

In Recognition Science, the framework's module ledger (a discrete record of events) defines a cost function on ratios of two masses, J(m/e) = (m/e + e/m)/2 - 1, which measures how far the ratio is from unity. The module then proves three facts about this function: it equals zero when the two masses are equal, it is never negative for positive inputs, and the threshold phi - 3/2 is positive. These are general properties of the cost function, not specific to Planck stars.

The research note in the module records an intended application: a Planck star bounce at radius r_min = phi^(N/2) times the Planck length, with a bounce density rho_bounce = rho_Pl * phi^(-3N/2). For N = 100, that density is about 10^-50 times the Planck density. But the module itself does not define what m and e are in gravitational terms. The three proved facts hold for any positive masses, so they do not establish the bounce.

What the module does establish, in plain language, is a small certificate: a structure that packages the three cost properties and a proof that this structure is inhabited. That is a machine-checked guarantee that the cost function behaves as expected, not a derivation of Planck stars. The bounce remains a research direction, not a result.

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 PlanckStar3Cert := ⟨cert⟩

What this page does not claim

The module does not prove that Planck stars exist or that a bounce occurs. The module does not derive the bounce density or radius from the cost function. The framework's cost function alone does not force the Planck star scenario.

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

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Derived articles

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