Gravity Discriminator Matrix Discriminator Matrix Full Inhabited

The discriminator matrix

A discriminator matrix is a table that lets one theory be told apart from its rivals by numbers, not by taste. Recognition Science's version has four rows for rival approaches to quantum gravity: loop quantum gravity, string theory, causal dynamical triangulations, and Bohmian approaches. It has three columns for observable sectors: the leading-log coefficient of black hole entropy, an echo-damping ratio, and a rung-phase coefficient. Each cell records an inequality that separates the framework's prediction from the rival's. The declaration discriminatorMatrixFull_inhabited establishes that such a complete matrix exists: a machine-checked library of formal theorems contains a certificate object filling every cell.

The matrix is not a wish list. Each filled cell is a theorem-grade inequality, meaning it is a proved statement in the machine-checked library with no unproved assumptions and no empirical input. Against loop quantum gravity, the leading-log margin is greater than one quarter; against string theory, greater than five quarters. Against causal dynamical triangulations and Bohmian approaches, the cells assert positive existence: these rivals predict no quantum-gravity signal in the relevant sector, so any positive recognition-science signal discriminates. The full certificate also bundles a per-rival distinguishability record, so the matrix satisfies the track's binding success criterion: at least one cell per rival shows an unambiguous distinction.

The declaration does not claim that any experiment has been run. The matrix organizes theorem-grade discriminators into the required format; it does not replace the dataset-tied falsifier register, which needs specific sensitivity numbers from LIGO/Virgo, LISA, NANOGrav, and similar instruments. The cells give inequalities, not measured values. The matrix also does not claim that the rival theories are false; it claims that, if the recognition-science predictions hold, the rivals' predictions differ by the stated margins. The certificate is a structural statement about the library's theorems, not a physical measurement.

theorem discriminatorMatrixFull_inhabited :
    Nonempty DiscriminatorMatrixCert :=
  ⟨discriminatorMatrixFull⟩

/-- **DISCRIMINATOR MATRIX ONE-STATEMENT** (Track 6.D closure form).

The 4 × 3 discriminator matrix has at least one theorem-grade
distinguishing inequality per rival:

* **LQG row**: `c_RS - (-1/2) > 1/4` (leading-log sector).
* **String row**: `c_RS - (-3/2) > 5/4` (leading-log sector).
* **CDT row**: `0 < echoDampingRatio` (echo-damping sector; CDT
  predicts no echoes).
* **Bohmian row**: `0 < echoDampingRatio` (echo-damping sector;
  Bohmian/DP do not produce φ-rational signals).

Plus the RS prediction bands:
`c_RS ∈ (-1/4, 0)`, `1/φ ∈ (0.617, 0.622)`, `log φ ∈ (0, 1/2)`. -/
theorem discriminator_matrix_one_statement :
    (c_RS - (-1 / 2) > 1 / 4) ∧
    (c_RS - (-3 / 2) > 5 / 4) ∧
    (0 < echoDampingRatio) ∧
    (-1 / 4 < c_RS ∧ c_RS < 0) ∧
    ((0.617 : ℝ) < echoDampingRatio ∧ echoDampingRatio < 0.622) ∧
    (0 < rungPhaseDelay ∧ rungPhaseDelay < 1 / 2) :=
  ⟨c_RS_LQG_margin, c_RS_string_margin, echoDampingRatio_pos,
   ⟨c_RS_gt_neg_quarter, c_RS_neg⟩,
   echoDampingRatio_band,
   ⟨rungPhaseDelay_pos, rungPhaseDelay_below_half⟩⟩

/-- (LQG, LeadingLog): RS leading-log coefficient is at least `1/4`
above LQG's `-1/2`. -/
theorem cell_LQG_LeadingLog : c_RS - (-1 / 2) > 1 / 4 :=
  c_RS_LQG_margin

/-- (String, LeadingLog): RS leading-log coefficient is at least `5/4`
above string-theory's `-3/2`. -/
theorem cell_String_LeadingLog : c_RS - (-3 / 2) > 5 / 4 :=
  c_RS_string_margin

def perRivalDistinguishability_holds : PerRivalDistinguishability where
  LQG_distinct := c_RS_LQG_margin
  String_distinct := c_RS_string_margin
  CDT_distinct := echoDampingRatio_pos
  Bohmian_distinct := echoDampingRatio_pos

What this page does not claim

No experiment has been run; the matrix gives inequalities, not measured values. The rivals are not claimed to be false, only that their predictions differ by stated margins. The certificate does not replace the dataset-tied falsifier register with specific sensitivity numbers.

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/DiscriminatorMatrix.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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