Physics Coherence Time From Jcost

Coherence time and cost

In quantum computing, a qubit loses its quantum character through two distinct processes. Energy relaxation, called T1, is how quickly the qubit drops from its excited state to its ground state, losing energy to its environment. Dephasing, called T2, is how quickly the phase relationship between the qubit's states becomes random, without necessarily losing energy. For most superconducting qubits, T1 is noticeably longer than T2, and the ratio T1/T2 is often close to 1.618, the golden ratio φ.

Recognition Science offers a structural reason for that ratio. The framework models physical systems as maintaining a ledger, a discrete record of recognition events, and assigns a cost, a forced penalty, to each possible ratio of two quantities. The cost function J(x) = (x + 1/x)/2 - 1 is the unique function satisfying five plain conditions, a theorem proved in a machine-checked library of formal theorems. When the framework applies this cost to the ratio of a relaxation rate and a dephasing rate, the minimum-cost configuration lands at T1/T2 = φ, meaning amplitude decay is slower than phase decay by exactly the golden ratio.

The module CoherenceTimeFromJCost in the framework's library does not itself prove that physical claim. It proves three general facts about the cost function J(m/e) where m and e are positive real numbers: the cost is zero when m equals e, the cost is never negative, and the threshold φ - 3/2 is positive. The module defines a certificate structure bundling these three facts, and proves that such a certificate exists. What would turn this into a theorem about coherence times is a definition of m and e in the qubit's own terms, which the module does not provide.

The distinction matters. The three proved facts are universal: they hold for any positive inputs, not just for qubits. The connection to T1 and T2 is a research note, a statement of where the idea was meant to go. The framework's own documentation is explicit that the module proves nothing specific to this subject, because the cost is defined without reference to a physical system. The ratio T1/T2 = φ is therefore a prediction with a named falsifier: a superconducting qubit with T1/T2 clearly different from φ would count against it.

What the module does establish is the mathematical groundwork. It shows the cost function behaves as a well-formed measure of asymmetry: it vanishes when two quantities are equal, it never rewards imbalance, and its canonical threshold is a positive number tied to the golden ratio. That is a small but solid foundation. The step from these general facts to a physical law about qubits remains open, waiting for a definition that connects the ledger's quantities to actual relaxation and dephasing times.

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)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module does not prove that actual qubits have T1/T2 = φ. The module does not define what m and e mean for a specific physical system. The five conditions on the cost function are not restated here; they are proved in a separate module.

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

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