Physics Quantum Chaos From Jcost

Quantum chaos and the cost bound

Quantum chaos studies quantum systems whose classical counterparts are chaotic. A central result is the Maldacena-Shenker-Stanford (MSS) bound: the Lyapunov exponent λ_L, which measures how quickly nearby trajectories diverge, cannot exceed 2πkT/ℏ in a thermal system. The bound sets an absolute speed limit on information scrambling in quantum matter.

The Recognition Science framework proposes a connection. It defines a cost function, a measure of recognition effort, and asks whether this cost could set the Lyapunov exponent. The specific proposal is λ_L = J(φ) × (2πkT/ℏ), where J(φ) ≈ 0.118, a number derived from the golden ratio φ. This would place the chaotic scrambling rate at about 11.8 percent of the MSS bound at a recognition threshold.

In Recognition Science, the module QuantumChaosFromJCost establishes three general facts about the cost function J(m/e) when m and e are positive real numbers. First, the cost vanishes when m equals e. Second, the cost is never negative for positive inputs. Third, a threshold value φ - 3/2 is strictly positive. These are formal theorems in the framework's machine-checked library of formal theorems.

The module does not, however, establish the quantum chaos claim itself. The cost function is defined as J(m/e) with no reference to a physical system. The Lyapunov exponent formula remains a research note, a statement of where the idea was meant to go, not a result. What would make the module a theorem about quantum chaos is a definition of m and e in the subject's own terms, a step the formal code does not take.

So the plain-language status is this: the framework proves a small set of facts about its cost function, and it records an intention to connect those facts to quantum chaos. The connection itself is open. A reader should see a template for a derivation, not a derivation.

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 Lyapunov exponent formula λ_L = J(φ) × (2πkT/ℏ). The module does not define m and e in physical terms. The framework does not claim the MSS bound itself.

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/QuantumChaosFromJCost.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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