Gravity Discriminator Cert Echo Damping Discriminator

The echo damping discriminator

Gravitational-wave echoes are hypothetical repeating pulses that some quantum-gravity models predict should follow the main burst from a black-hole merger. The Recognition Science framework, which starts from the idea that reality keeps a discrete record of recognition events, derives a specific prediction for the strength of these echoes. Its ledger, a discrete record of such events, forces the ratio of one echo's amplitude to the previous one's to be the reciprocal of the golden ratio, roughly 0.618. This is the echo damping ratio.

The declaration EchoDampingDiscriminator bundles the proof that this ratio lies strictly between one half and one. The proof is a theorem in the framework's machine-checked library of formal theorems, meaning it follows from the axioms with no gaps and no extra assumptions. It relies only on basic facts about the golden ratio, such as φ² = φ + 1 and φ < 2. The theorem establishes three inequalities: the ratio is greater than one half, less than one, and greater than zero. As a corollary, the ratio cannot equal one half, zero, or one.

These bounds matter because rival models make different predictions. A classical black hole with no quantum structure produces no echoes at all. A model with uniform discreteness, where each echo is a fixed fraction of the previous one, typically predicts a ratio of exactly one half. A trivial bounce with no damping would give a ratio of one. The Recognition Science prediction sits strictly between these alternatives, so a measurement of the echo amplitude ratio could in principle tell the models apart. The observational channel is the LIGO/Virgo gravitational-wave catalog, where echoes would appear after the main ringdown of a black-hole merger.

The declaration does not claim that echoes have been observed. It does not claim that the ratio is exactly 0.618, only that it lies in the open interval between one half and one. It also does not claim that the framework's derivation of the ratio from the ledger is complete; the proof covers the bounds, not the full physical bridge from recognition events to gravitational-wave echoes. The discriminator is a mathematical statement about what the framework predicts, not a report of what detectors have seen.

/-- `1/φ > 1/2`. Proof: equivalent to `2 > φ`, which is
`Constants.phi_lt_two`. -/
theorem echoDampingRatio_above_half : echoDampingRatio > 1 / 2 := by
  unfold echoDampingRatio
  rw [gt_iff_lt, lt_div_iff₀ phi_pos]
  have := phi_lt_two
  linarith

/-- Echo-amplitude discriminator stub: the per-echo damping ratio `1/φ`
is theorem-grade distinct from `1/2` (uniform), `0` (no echo), and `1`
(no damping). -/
theorem rs_echo_distinct_uniform_no_echo :
    (echoDampingRatio > 1 / 2) ∧
    (echoDampingRatio < 1) ∧
    (0 < echoDampingRatio) :=
  ⟨echoDampingRatio_above_half, echoDampingRatio_lt_one,
   echoDampingRatio_pos⟩

/-- The echo damping discriminator holds. -/
def echoDampingDiscriminator_holds : EchoDampingDiscriminator where
  echoDampingRatio_above_half := echoDampingRatio_above_half
  echoDampingRatio_below_one := echoDampingRatio_lt_one
  echoDampingRatio_above_zero := echoDampingRatio_pos
  echoDampingRatio_neq_half := by
    intro h
    have h_lb : echoDampingRatio > 1 / 2 := echoDampingRatio_above_half
    rw [h] at h_lb
    linarith
  echoDampingRatio_neq_zero := by
    intro h
    have h_pos : 0 < echoDampingRatio := echoDampingRatio_pos
    rw [h] at h_pos
    linarith
  echoDampingRatio_neq_one := by
    intro h
    have h_lt : echoDampingRatio < 1 := echoDampingRatio_lt_one
    rw [h] at h_lt
    linarith

/-- Discriminator structure for the leading-log entropy coefficient
`c_RS = -log φ / 2` against the LQG canonical `-1/2` and the
string-theory canonical `-3/2`. -/
structure LeadingLogDiscriminator where
  c_RS_minus_LQG_lower : c_RS - (-1 / 2) > 1 / 4
  c_RS_minus_string_lower : c_RS - (-3 / 2) > 5 / 4
  c_RS_minus_LQG_abs : |c_RS - (-1 / 2)| > 1 / 4
  c_RS_minus_string_abs : |c_RS - (-3 / 2)| > 5 / 4

What this page does not claim

Echoes have been observed in gravitational-wave data. The echo damping ratio is exactly 0.618, only that it lies between one half and one. The framework's derivation from the ledger to gravitational-wave echoes is complete.

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/DiscriminatorCert.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: