Gravity Discriminator Cert

The discriminator cert

The gravity discriminator cert is a formal object in the Recognition Science framework's machine-checked library of formal theorems. It bundles three separate mathematical proofs, each showing that a specific black hole signature predicted by the framework differs from the corresponding prediction of a rival quantum gravity program. The certificate's purpose is to give observational physicists a precise target: if a measurement lands where the certificate says it cannot, the framework's account of gravity fails.

The first discriminator concerns the leading-log entropy coefficient, a number that controls how black hole entropy grows with area. Recognition Science derives this coefficient as -log φ / 2, where φ is the golden ratio, approximately 1.618. The certificate proves this value is more than 1/4 away from the loop quantum gravity value of -1/2, and more than 5/4 away from the string theory value of -3/2. These margins are unconditional theorems, not estimates.

The second discriminator is the echo amplitude damping ratio, the fraction by which each successive gravitational wave echo from a black hole merger is weakened. The framework predicts this ratio lies strictly between 1/2 and 1, and specifically not equal to 1/2, 0, or 1. This distinguishes it from uniform-discreteness alternatives that predict exactly 1/2, and from a trivial bounce with no damping at 1.

The third discriminator is the per-rung phase delay, the time shift between successive echoes, predicted to be log φ, which lies strictly between 0 and 1/2. The certificate proves this is below the loop quantum gravity half-quantum value of 1/2, and not equal to 3/4 or 1. The relative echo delay Δt / r_min directly measures this quantity.

In Recognition Science, these three signatures all trace back to the golden ratio, which the framework derives from its core cost function rather than fitting to data. The certificate's theorems use only basic facts about φ and logarithms, with no observational input and no extra axioms. The numerical margins are the algebra; the observational channels, such as LIGO/Virgo ringdown data, are the named experiments that could falsify the framework's predictions.

/-- Master-plan Track 7 stub `rs_qnm_distinct_LQG_string`: the RS
prediction for BH ringdown / QNM spectroscopy (leading-log entropy
coefficient `c_RS = -log φ/2`) is theorem-grade distinct from the LQG
canonical `-1/2` and the string-theory canonical `-3/2`. -/
theorem rs_qnm_distinct_LQG_string :
    (c_RS - (-1 / 2) > 1 / 4) ∧ (c_RS - (-3 / 2) > 5 / 4) :=
  ⟨c_RS_LQG_margin, c_RS_string_margin⟩

/-- 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⟩

/-- Echo-time discriminator stub: the per-rung phase delay `log φ` is
theorem-grade in `(0, 1/2)`, distinct from `1/2` (LQG half-quantum),
`π/4`, `π/2`, and `1`. -/
theorem rs_echo_time_distinct_LQG_uniform :
    (rungPhaseDelay < 1 / 2) ∧ (0 < rungPhaseDelay) :=
  ⟨rungPhaseDelay_below_half, rungPhaseDelay_pos⟩

theorem discriminatorMatrixCert_inhabited :
    Nonempty DiscriminatorMatrixCert :=
  ⟨discriminatorMatrixCert⟩

What this page does not claim

The certificate does not claim any of these three predictions has been observed in gravitational wave data. The certificate does not prove that loop quantum gravity or string theory is false; it only proves the framework's predictions are numerically distinct. The certificate does not derive the golden ratio itself; it uses the framework's previously proved value.

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

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