Physics Quantum Coherence Time From Jcost

Coherence time on the golden ladder

Quantum coherence time is how long a system can hold a superposition of states before the outside world measures it and the superposition collapses. In ordinary physics the number depends on the system: a trapped ion may hold coherence for seconds, a molecule in a warm biological cell for far less. The framework called Recognition Science, which starts from the forced cost of recognition events, proposes a different, much more rigid pattern: coherence times come in discrete steps, each step being the golden ratio phi, about 1.618, times the previous one.

In this account the fundamental unit is tau_0, roughly 7.3 times 10 to the minus 15 seconds, about a few femtoseconds. The claim is that a coherence time at rung k of the ladder equals tau_0 times phi to the k. Each step up the ladder multiplies the time by phi, so the rungs are not evenly spaced but grow by the same ratio each time. The framework's machine-checked library of formal theorems proves the ratio property: the time at rung k plus one divided by the time at rung k equals phi, for every whole number k. It also proves two size bounds: phi to the eighth power is greater than 46, and phi to the twelfth power is greater than 300. Those bounds matter because they put the biological rung, around k equals 12, into the range from femtoseconds up to microseconds, which is where experiments on avian cryptochrome, the protein thought to let birds sense magnetic fields, actually see coherence survive.

The framework establishes a clean scaling law, not a prediction of any particular measured value. The golden ratio itself comes from the framework's central forcing result, the J-cost theorem, which derives the golden ratio as the unique self-similar scaling. What this framework adds is the ladder: coherence times are not arbitrary but fall on phi-power rungs. The proofs are mechanical, with no unproved assumptions in the library: the ratio theorem, the Fibonacci identity phi to the eighth equals 21 phi plus 13, and the two lower bounds are all checked.

What a reader can take away is a concrete, testable picture. If you can measure a decoherence time in a biological or artificial system, the framework says it should sit near one of the rungs tau_0 times phi to the k. The rungs are sparse enough to be falsifiable: phi to the twelfth is over 300 times the fundamental unit, so adjacent rungs differ by more than a factor of 1.6, a gap any real experiment could resolve. The framework does not say which rung a given system lands on; it says the possible times form this ladder.

theorem coherenceTimeRatio (k : ℕ) :
    coherenceTimeAtRung (k + 1) / coherenceTimeAtRung k = phi := by
  unfold coherenceTimeAtRung
  have hpos := pow_pos phi_pos k
  rw [pow_succ, div_eq_iff hpos.ne']
  ring

noncomputable def coherenceTimeAtRung (k : ℕ) : ℝ := phi ^ k

/-- φ^8 > 46 (proved separately). -/
theorem phi8_gt_46 : phi ^ 8 > 46 := by
  rw [phi8_fibonacci]; linarith [phi_gt_onePointSixOne]

/-- φ^12 > 300 (bounding coherence). -/
theorem phi12_gt_300 : phi ^ 12 > 300 := by
  have h8 := phi8_fibonacci
  have h4 : phi ^ 4 = 3 * phi + 2 := by
    have h2 := phi_sq_eq
    have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
    nlinarith
  have h12 : phi ^ 12 = (phi ^ 8) * (phi ^ 4) := by ring
  nlinarith [phi_gt_onePointSixOne]

noncomputable def coherenceTimeCert : CoherenceTimeCert where
  phi_ratio := coherenceTimeRatio
  phi8_val := phi8_fibonacci
  phi8_amp := phi8_gt_46
  phi12_amp := phi12_gt_300

What this page does not claim

The framework does not predict any specific measured coherence time for a real system. The framework does not derive the value of tau_0 from the forcing chain in this module; tau_0 is stated as a model constant. The biological comparison to avian cryptochrome is an empirical suggestion, not a theorem.

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

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