Gravity Caldeira Leggett Response At Zero

The zero-frequency response

The Caldeira-Leggett model describes how a quantum system loses energy by coupling it to a bath of harmonic oscillators. When the bath is traced out, the system feels damping and random fluctuations linked by the fluctuation-dissipation theorem. This framework, originally built for quantum dissipation, has been adapted in the Recognition Science library to gravity, where a similar oscillator bath gives rise to a frequency-dependent response function.

The central object is the transfer function \(H(i\omega)\), which encodes how the system responds to an oscillatory drive at frequency \(\omega\). The library defines it with a single-pole form: \(H(i\omega) = 1 + \Delta/(1 + i\omega\tau)\), where \(\Delta\) is the DC enhancement and \(\tau\) is the memory timescale. The real part, \(C(\omega) = \text{Re}[H(i\omega)]\), is the response function that measures the system's in-phase reaction.

The theorem response_at_zero states that at zero frequency, the response equals \(1 + \Delta\). This is a direct consequence of the definition: when \(\omega = 0\), the denominator in the correction term becomes 1, leaving only the enhancement. The proof is a simple simplification in the machine-checked library of formal theorems. This result pins down the physical meaning of \(\Delta\): it is the amount by which the static response exceeds the Newtonian baseline of 1.

The same library proves that at infinite frequency the response approaches 1, recovering the Newtonian limit, and that a positive \(\Delta\) means the response is always greater than 1 (passivity, or enhancement rather than suppression). These three facts together characterize the single-pole response completely: it interpolates between an enhanced static value and the Newtonian baseline at high frequencies.

In Recognition Science, this formalization is a structural step, not a complete physical derivation. The action that would produce this transfer function from first principles is sketched but not yet formalized; the library explicitly marks the full derivation as pending. The theorem response_at_zero is therefore a precise statement about a defined object, not an empirical prediction about gravity.

/-- At zero frequency, the response equals \(1 + \Delta = w\). -/
theorem response_at_zero (H : TransferFunction) :
    response_function H 0 = 1 + H.Δ := by
  unfold response_function
  simp

/-- At infinite frequency, the response approaches 1 (Newtonian limit). -/
theorem response_limit_high_freq (H : TransferFunction) (hΔ : H.Δ ≠ 0) :
    Filter.Tendsto (response_function H) Filter.atTop (nhds 1) := by
  -- As ω → ∞, Δ/(1 + (ωτ)²) → 0
  unfold response_function
  -- Show the denominator tends to `+∞` as ω → +∞.
  have hmul : Filter.Tendsto (fun ω : ℝ => ω * H.τ) Filter.atTop Filter.atTop := by
    simpa using ((Filter.tendsto_id).atTop_mul_const H.τ_pos)
  have hsq : Filter.Tendsto (fun ω : ℝ => (ω * H.τ) ^ (2 : ℕ)) Filter.atTop Filter.atTop :=
    (Filter.tendsto_pow_atTop (α := ℝ) (n := 2) (by decide)).comp hmul
  have hmono :
      (fun ω : ℝ => (ω * H.τ) ^ (2 : ℕ))
        ≤ᶠ[Filter.atTop] (fun ω : ℝ => 1 + (ω * H.τ) ^ (2 : ℕ)) :=
    Filter.Eventually.of_forall (fun _ω => by linarith)
  have hden :
      Filter.Tendsto (fun ω : ℝ => 1 + (ω * H.τ) ^ (2 : ℕ)) Filter.atTop Filter.atTop :=
    Filter.tendsto_atTop_mono' Filter.atTop hmono hsq

  have hinv :
      Filter.Tendsto (fun ω : ℝ => (1 + (ω * H.τ) ^ (2 : ℕ))⁻¹) Filter.atTop (nhds 0) :=
    (tendsto_inv_atTop_zero).comp hden

  have hfrac_mul :
      Filter.Tendsto (fun ω : ℝ => H.Δ * (1 + (ω * H.τ) ^ (2 : ℕ))⁻¹) Filter.atTop (nhds 0) := by
    have hΔconst : Filter.Tendsto (fun _ω : ℝ => H.Δ) Filter.atTop (nhds H.Δ) := by
      simpa using (tendsto_const_nhds : Filter.Tendsto (fun _ω : ℝ => H.Δ) Filter.atTop (nhds H.Δ))
    -- H.Δ * (denom)⁻¹ → H.Δ * 0 = 0
    simpa using (hΔconst.mul hinv)

  have hfrac :
      Filter.Tendsto (fun ω : ℝ => H.Δ / (1 + (ω * H.τ) ^ (2 : ℕ))) Filter.atTop (nhds 0) := by
    simpa [div_eq_mul_inv] using hfrac_mul

  -- Add back the constant `1`.
  have hone : Filter.Tendsto (fun _ω : ℝ => (1 : ℝ)) Filter.atTop (nhds 1) := by
    simpa using (tendsto_const_nhds : Filter.Tendsto (fun _ω : ℝ => (1 : ℝ)) Filter.atTop (nhds 1))
  simpa using hone.add hfrac

/-- Passivity (enhancement, not suppression): if \(\Delta > 0\), then \(C(\omega) > 1\). -/
theorem response_enhancement (H : TransferFunction) (hΔ : 0 < H.Δ) (ω : ℝ) :
    1 < response_function H ω := by
  unfold response_function
  have h : 0 < H.Δ / (1 + (ω * H.τ)^2) := by
    apply div_pos hΔ
    have : 0 ≤ (ω * H.τ)^2 := sq_nonneg _
    linarith
  linarith

What this page does not claim

The Caldeira-Leggett action has been fully derived from first principles in the library. The response function has been empirically verified against gravitational measurements. The theorem establishes a physical prediction about gravity, rather than a property of a defined mathematical object.

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

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