Gravity Bhentropy Log Correction2 From Jcost Bhentropy Log2 Cert

The certificate's scope

In the Recognition Science framework, a ledger (a discrete record of events) assigns a cost to every recognition event. The declaration BHEntropyLog2Cert is a machine-checked certificate: a bundle of three formal theorems about a particular cost function, Jcost (the framework's forced cost function). The certificate proves that this cost is zero when the two inputs are equal, that it is never negative for positive inputs, and that a threshold constant involving the golden ratio is positive.

The certificate's own documentation is explicit about its limits. The cost function is defined as Jcost(m/e) for two real numbers m and e, with no definition of what m and e mean in any physical subject. The three proved facts are general properties of Jcost, not results about black holes, entropy, or gravity. The name BHEntropyLog2Cert records a research intention, not a proved statement. The paragraph in the source that mentions a specific entropy correction value is a research note, not a theorem.

What the certificate does establish, in plain terms, is that the framework's cost function behaves sensibly: it vanishes at equality, stays nonnegative, and has a positive threshold. These are the kind of baseline properties any cost function must satisfy. The certificate's value is as a building block: it shows that these properties hold for Jcost, ready to be used if someone later defines m and e in a subject's own terms.

In Recognition Science, the hope was that this certificate would support a specific claim about black hole entropy corrections. The certificate itself does not do that. It proves no statement about black holes, no logarithmic correction, and no value for any entropy coefficient. The gap between the name and the content is the gap between a research program and its formal results.

The lesson for a reader is simple: a machine-checked certificate is only as strong as its definitions. Here, the definitions stop at real numbers m and e. Until those are tied to a physical subject, the certificate proves general mathematics, not physics. The framework's library (a machine-checked collection of formal theorems) is honest about this distinction, and the certificate's own documentation states it plainly.

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

The certificate does not prove any logarithmic correction to black hole entropy. The certificate does not establish any value for a physical entropy coefficient. The certificate does not connect its cost function to any specific physical system.

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/BHEntropyLogCorrection2FromJCost.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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