Gravity Gravitational Entropy2 From Jcost

Gravitational entropy from cost

Black hole thermodynamics assigns entropy to a black hole's horizon, with the famous Bekenstein-Hawking formula S = A/(4 l_Pl^2), where A is the horizon area and l_Pl the Planck length. Quantum corrections add a logarithmic term; the framework's research note proposes the coefficient -log(phi)/2, with phi the golden ratio. The module GravitationalEntropy2FromJCost builds a bridge from the framework's cost function to this correction, but the formal content stops well short of the physics.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 measures the price of a recognition event, a discrete record in the framework's ledger. The module defines domainCost as J(m/e), a ratio of two masses. The machine-checked library proves three general facts: the cost vanishes when the masses are equal, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive. These are properties of the cost function itself, holding for any positive ratio.

What the module does not do is define what m and e mean for a black hole. The docstring states plainly that the paragraph proposing the entropy correction is a research note recording where the idea was meant to go, not a result. The formal theorems would become statements about gravitational entropy only if someone supplied a definition of m and e in that subject's own terms. Until then, the module is a template instantiated with the cost function, shared verbatim with 2383 sibling modules.

The distinction matters for reading the framework's claims. The certificate structure bundles the three proven facts into a single object, but the certificate's name, GravEntropy2Cert, does not make it a theorem about gravity. The honest summary: the library proves the cost function's basic analytic properties, and the entropy application remains an unformalized hypothesis awaiting a physical definition of its inputs.

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]

What this page does not claim

The module does not prove the black hole entropy correction formula. No physical meaning for the masses m and e is established. The certificate GravEntropy2Cert does not establish a gravitational result.

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/GravitationalEntropy2FromJCost.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.

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