Gravity Bhecho Amplitudes Echo Amplitude Pos

The amplitude theorem

Black-hole echoes are hypothetical repeating ripples in gravitational waves, thought to arise if a black hole's event horizon is replaced by a reflective surface. In the Recognition Science framework, the ledger, a discrete record of recognition events, predicts that each successive echo is quieter than the last by a fixed factor. The declaration echoAmplitude_pos is the formal anchor for one piece of that prediction: for every echo number n, the amplitude is strictly greater than zero.

The theorem itself is almost trivial once the amplitude is defined. The framework defines the amplitude of echo n as phi^(-n), where phi is the golden ratio, approximately 1.618. Since a positive number raised to any integer power is positive, the positivity claim follows immediately. The machine-checked library of formal theorems records this as a proved statement, with no unproved assumptions. The certificate structure BHEchoAmplitudeCert bundles this positivity result with three companions: the primary echo has amplitude exactly 1, each step multiplies the amplitude by 1/phi, and the amplitudes strictly decrease.

What the theorem does not claim is that black-hole echoes exist. The framework's amplitude prediction is a conditional statement: if an echo appears in the data, this is its expected strength relative to the previous echo. The docstring names a falsifier: post-processing of any high-SNR merger event that shows either no echo or an amplitude ratio systematically different from 1/phi between successive echoes. The positivity theorem is a mathematical fact about a defined sequence, not an observation of gravitational waves.

The practical consequence is narrow but real. When the framework's catalog cert assigns an amplitude to each echo in a hypothetical event, the positivity theorem guarantees that no computed amplitude is zero or negative, which would be physically meaningless. It also supplies the nonzero denominator needed to define the signal-to-noise ratio between successive echoes. Without this theorem, the ratio echoAmplitude(n+1) / echoAmplitude(n) would not be well-formed in the formal system.

theorem echoAmplitude_pos (n : ℕ) : 0 < echoAmplitude n :=
  zpow_pos Constants.phi_pos _

theorem echoAmplitude_one : echoAmplitude 0 = 1 := by
  simp [echoAmplitude]

/-- Each echo is attenuated by 1/φ relative to the previous. -/
theorem echoAmplitude_succ_ratio (n : ℕ) :
    echoAmplitude (n + 1) = echoAmplitude n * phi⁻¹ := by
  unfold echoAmplitude
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  have : phi ^ (-((n : ℤ) + 1)) = phi ^ (-(n : ℤ)) * phi⁻¹ := by
    rw [show (-((n : ℤ) + 1)) = -(n : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((n + 1 : ℕ) : ℤ) = (n : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]

/-- Amplitudes are strictly decreasing. -/
theorem echoAmplitude_strictly_decreasing (n : ℕ) :
    echoAmplitude (n + 1) < echoAmplitude n := by
  rw [echoAmplitude_succ_ratio]
  have hn : 0 < echoAmplitude n := echoAmplitude_pos n
  have : phi⁻¹ < 1 :=
    inv_lt_one_of_one_lt₀ (by have := Constants.phi_gt_onePointFive; linarith)
  linarith [mul_lt_iff_lt_one_right hn |>.mpr this]

What this page does not claim

The theorem does not assert that black-hole echoes have been observed in gravitational-wave data. The theorem does not derive the existence of a reflective surface from general relativity. The theorem does not claim the amplitude ratio matches any measured echo, only that the sequence is well-defined and positive.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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