Physics Decoherence Timescale From Jcost

Decoherence timescale from cost

Decoherence is the gradual loss of quantum coherence, the property that lets a particle be in two states at once. When a quantum system interacts with its environment, this superposition leaks away, and the system behaves more like a classical object. The timescale for this process, often called τ_D, is a measure of how quickly that leakage happens. In the standard picture, a faster interaction with the environment means a shorter decoherence time.

A common way to estimate this timescale is to compare the system's energy scale with the thermal energy of its surroundings. The formula τ_D = ħ / (J · k_B · T) expresses this: ħ is the reduced Planck constant, k_B is Boltzmann's constant, T is the temperature, and J is a dimensionless factor that sets the rate. A larger J means faster decoherence, so a smaller τ_D. This is the kind of relation that appears in studies of quantum measurement and open quantum systems.

In Recognition Science, the framework proposes a specific value for that factor J. It uses a cost function, a mathematical tool that measures how far a system is from a balanced state. The framework's cost function is J(x) = (x + 1/x)/2 - 1, which is zero when x equals 1 and grows as x moves away from 1. The framework models the decoherence timescale by setting J to J(φ), where φ is the golden ratio, about 1.618. This choice is not derived from first principles in the module; it is a definitional choice about how to map the framework's cost onto a physical rate.

The machine-checked library of formal theorems that accompanies this framework contains a module for this decoherence timescale. What that module actually proves is more modest than the physics might suggest. It proves three general facts about the cost function: it vanishes when the two inputs are equal, it is never negative for positive inputs, and the quantity φ - 3/2 is positive. These are properties of the cost function itself, not of decoherence. The module does not prove that decoherence follows this timescale; that would require a definition of the system's mass and energy in terms of the framework's own concepts, which the module does not provide.

So the plain-language takeaway is this: the framework offers a formula for a decoherence timescale, and its formal library verifies the mathematical hygiene of the cost function used in that formula. But the bridge from the cost function to actual physics remains open. The module is a template, not a finished theory of decoherence.

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 that decoherence follows the proposed timescale. The choice of the golden ratio as the rate factor is not derived in the module. The formula is not yet connected to any specific physical system or measurement.

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

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