Gravity Caldeira Leggett Response Enhancement

The response enhancement theorem

The Caldeira-Leggett formalism is a standard tool in physics for describing how a quantum system loses energy to its surroundings. It works by coupling the system to a bath of harmonic oscillators; when the bath is traced out, the system experiences damping and fluctuations linked by the fluctuation-dissipation theorem. In the Recognition Science framework, this formalism is adapted to gravity, modeling the gravitational response to baryonic matter as a frequency-dependent transfer function H(iω).

The central object is the response function C(ω) = Re[H(iω)] = 1 + Δ/(1 + (ωτ)²), where Δ is a DC enhancement parameter and τ is a memory timescale. The theorem response_enhancement proves that if Δ > 0, then C(ω) > 1 for all frequencies ω. In plain terms: if the model's enhancement parameter is positive, the gravitational response is always an amplification of the Newtonian baseline, never a suppression. At zero frequency the response equals 1 + Δ = w, and at infinite frequency it approaches 1, recovering the Newtonian limit.

The proof is a short argument from the definitions: the denominator 1 + (ωτ)² is always positive, so a positive numerator Δ forces the fraction to be positive, and therefore the whole response exceeds 1. The theorem is checked in the machine-checked library of formal theorems, with no gaps in the reasoning. It does not, however, prove that Δ is positive; that is an input assumption. The theorem also does not show that the Caldeira-Leggett action actually gives rise to this transfer function; that derivation is left as a placeholder for future formalization.

What the theorem establishes is a conditional statement about the model's behavior: given a positive enhancement parameter, the response is enhanced at all frequencies. This matters because it gives a precise, checkable consequence of the framework's gravitational adaptation. It tells a reader what the model predicts under a stated condition, and it does so with the confidence of a machine-checked proof. The open work is the derivation from the action to the transfer function, which remains a target for future formalization.

/-- 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

/-- 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

What this page does not claim

The theorem does not prove that the enhancement parameter Δ is positive. The theorem does not derive the transfer function from the Caldeira-Leggett action. The theorem does not establish any empirical claim about observed gravitational phenomena.

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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