Gravity Discriminator Cert Discriminator Matrix Cert Inhabited

The discriminator matrix certificate

A discriminator is a measurable signature that tells two theories apart. The declaration discriminatorMatrixCert_inhabited is a machine-checked theorem stating that a certificate object exists, bundling three such signatures, each with a proved numerical margin separating the Recognition Science framework from at least one alternative quantum-gravity program. The certificate is a single formal structure containing three sub-certificates, one per signature, and the theorem proves that this structure is inhabited, meaning all three sub-certificates are simultaneously constructible.

The first signature concerns the leading-logarithm coefficient of black hole entropy, a number describing how entropy scales with horizon area. The framework derives a value of approximately -0.241, while loop quantum gravity gives -1/2 and string theory gives -3/2. The proved margins are that the framework's value exceeds the loop quantum gravity value by more than 1/4 and exceeds the string theory value by more than 5/4. The second signature is the damping ratio of gravitational-wave echoes, the fraction of echo amplitude retained per cycle. The framework's value lies strictly between 1/2 and 1, distinguishing it from uniform-discreteness alternatives that predict exactly 1/2. The third signature is the phase delay per echo rung, which the framework proves lies strictly between 0 and 1/2, separating it from the loop quantum gravity half-quantum prediction at the boundary.

These margins are pure mathematics. They depend only on properties of the golden ratio φ, such as φ² = φ + 1 and φ < 2, plus standard facts about logarithms. No experimental data, no CODATA constants, and no semiclassical assumptions enter the proof. The theorem establishes the algebraic content: the inequalities hold unconditionally. What it does not establish is that any experiment has actually measured these signatures. The observational channels are named, such as quasinormal-mode spectroscopy and gravitational-wave echo timing in the LIGO/Virgo catalog, but the certificate itself contains no dataset attachments and no sensitivity analysis. It provides the theoretical predictions that future experiments must test against.

The certificate also does not claim to close all of the framework's Track 6 program. The docstring explicitly lists remaining sub-tracks: a full 4 × N discriminator matrix with at least one unambiguous distinction per rival, and dataset attachments with concrete sensitivity numbers. The certificate is a partial closure, satisfying the clause that three or more discriminators be theorem-grade derivations from φ with named observational channels. It is a structural milestone, not an empirical one.

theorem discriminatorMatrixCert_inhabited :
    Nonempty DiscriminatorMatrixCert :=
  ⟨discriminatorMatrixCert⟩

/-- 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⟩

/-- **DISCRIMINATOR MATRIX ONE-STATEMENT** (Track 6 partial closure form).
RS provides three independent theorem-grade discriminators against
the canonical alternative quantum-gravity programs:

1. **Leading-log entropy coefficient**: `c_RS = -log φ / 2 ≈ -0.241`,
   distinct from LQG (`-1/2`) by margin `> 1/4` and from string-theory
   (`-3/2`) by margin `> 5/4`.

2. **Per-echo amplitude damping ratio**: `1/φ ≈ 0.618`, strictly in
   `(1/2, 1)` (distinct from no-echo `0`, no-damping `1`, and uniform
   `1/2`).

3. **Per-rung phase delay coefficient**: `log φ ≈ 0.481`, strictly in
   `(0, 1/2)` (distinct from LQG half-quantum `1/2`, uniform
   quarter-period `π/4`, and longer-period alternatives).

All three are RS-internal φ-rational predictions; all three have named
observational channels (QNM spectroscopy + GW echo amplitude/timing on
GWTC-3); all three carry explicit numerical sensitivity thresholds. -/
theorem discriminator_matrix_one_statement :
    ((c_RS - (-1 / 2) > 1 / 4) ∧ (c_RS - (-3 / 2) > 5 / 4)) ∧
    ((echoDampingRatio > 1 / 2) ∧ (echoDampingRatio < 1)
        ∧ (0 < echoDampingRatio)) ∧
    ((rungPhaseDelay < 1 / 2) ∧ (0 < rungPhaseDelay)) :=
  ⟨rs_qnm_distinct_LQG_string,
   rs_echo_distinct_uniform_no_echo,
   rs_echo_time_distinct_LQG_uniform⟩

What this page does not claim

No experiment has yet measured any of the three signatures. The certificate does not include a full 4 × N discriminator matrix. The theorem does not prove that the framework's predictions are correct, only that they are distinct from the named alternatives.

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)

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Derived articles

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