Gravity Discriminator Cert Rs Qnm Distinct Lqg String

The black hole ledger

Black holes, when disturbed, ring like bells. The exact frequencies of that ringdown are called quasinormal modes, and the way the ring fades encodes the entropy of the hole. In the Recognition Science framework, a ledger (a discrete record of recognition events) underlies the hole's physics, and from that ledger a specific prediction emerges for the leading logarithmic coefficient of the entropy: c_RS = -log φ / 2, where φ is the golden ratio, approximately -0.241.

The theorem rs_qnm_distinct_LQG_string states that this RS coefficient is separated from the predictions of two rival quantum gravity programs by explicit margins. Against loop quantum gravity's c_LQG = -1/2, the RS value is larger by more than 1/4. Against string theory's c_string = -3/2, the RS value is larger by more than 5/4. The declaration proves both inequalities as a conjunction, with no observational input and no free parameters; it is a pure mathematical result derived from φ's defining equation φ² = φ + 1 and basic properties of the logarithm.

The practical meaning is a falsifiable signature. If a future gravitational-wave observatory measures the leading-log entropy coefficient of a black hole ringdown with sensitivity finer than 0.25, the result will distinguish RS from loop quantum gravity. A measurement finer than 1.25 separates RS from string theory. The theorem does not itself predict which theory is correct; it only certifies that the three programs make numerically distinct, testable claims in this channel.

This is one of three theorem-grade discriminators in the framework's Gravity Track 6, alongside an echo amplitude damping ratio and a per-rung phase delay. The declaration bundles the leading-log inequality into a larger matrix certificate, but the theorem itself concerns only the entropy coefficient. It does not claim that any measurement has been made, that any theory is observationally favored, or that the other two discriminators are proved by this same statement. The mathematical margins are unconditional; the empirical test remains open.

/-- 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⟩

/-- 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

/-- The leading-log discriminator holds via the Session 90 margins. -/
def leadingLogDiscriminator_holds : LeadingLogDiscriminator where
  c_RS_minus_LQG_lower := c_RS_LQG_margin
  c_RS_minus_string_lower := c_RS_string_margin
  c_RS_minus_LQG_abs := c_RS_LQG_margin_abs
  c_RS_minus_string_abs := c_RS_string_margin_abs

theorem discriminatorMatrixCert_inhabited :
    Nonempty DiscriminatorMatrixCert :=
  ⟨discriminatorMatrixCert⟩

What this page does not claim

No measurement has been made; the theorem is purely mathematical. The theorem does not say which theory is correct, only that they are numerically distinct. The declaration does not prove the other two discriminators (echo damping and phase delay); it only bundles them in a matrix certificate.

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.

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