Physics Quantum Dot Lifetime3 From Jcost

Quantum dot lifetime from cost

A quantum dot is a semiconductor crystal only a few nanometers across, small enough that its electrons are confined in all three directions. When light excites such a dot, it re-emits light after a delay called the fluorescence lifetime, typically 20 to 100 nanoseconds depending on the dot's size. The smaller the dot, the shorter the lifetime, because tighter confinement changes the energy levels. A common dye molecule, by contrast, has a lifetime near 5 nanoseconds.

In Recognition Science, a framework that derives structure from a forced cost function, the number phi (the golden ratio, about 1.618) appears as a natural scaling factor. The framework's research note proposes that a quantum dot's lifetime follows tau_QD = phi^k times a molecular lifetime, with phi^3 times 5 ns giving 21 ns and phi^4 times 5 ns giving 34 ns. Both numbers fall inside the 20 to 40 nanosecond range measured for small dots. This is a stated research idea, not a proved result.

The machine-checked library of formal theorems contains a module named Quantum_Dot_Lifetime3_FromJCost. Its actual content is deliberately general. The module defines a cost function as Jcost(m/e), where Jcost is the framework's forced cost function and m and e are real numbers. It proves three facts: the cost is zero when m equals e, the cost is nonnegative for positive inputs, and the number phi minus 3/2 is positive. The module assembles these into a certificate structure and proves that structure is inhabited.

What the module does not do is connect those facts to quantum dots. The definition of m and e as physical quantities is absent; the module never states what m or e mean for a dot. The docstring says plainly that the module proves nothing specific to the subject, because the cost is defined without reference to one. The lifetime formula in the research note is a hypothesis about where the idea could go, not a theorem the library checks. To make the module a theorem about quantum dots, someone would need to define m and e in the dot's own terms.

The useful lesson is a boundary. The framework's cost function has proved general properties, and those properties are real. But a general property of a cost function does not become a physics result until a physical model supplies the missing definitions. The quantum dot lifetime formula is a target for future work, not an established measurement or derivation.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The framework derives the 20 to 100 nanosecond quantum dot lifetime range from first principles. The module establishes any specific prediction about quantum dot fluorescence lifetimes. The phi-based lifetime formula is a proved theorem; it is a research note.

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/Quantum_Dot_Lifetime3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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