Encyclopedia Gravity Gravity Zero Free Parameters Gravity Constants Audit One Statement

ARTICLE 5 claims 4 theorems 1 model

Gravity Zero Free Parameters Gravity Constants Audit One Statement

A single machine-checked statement bundles the framework's gravity constants into closed forms built from the golden ratio, leaving one measured input.

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The constants audit

The golden ratio φ, about 1.618, appears throughout mathematics as the solution to r² = r + 1. In the Recognition Science framework, a machine-checked library of formal theorems derives many physical constants from this single number. The declaration gravity_constants_audit_one_statement is one such result: it is a single theorem asserting that every gravity-sector constant in the framework's audit list has a closed-form expression in terms of φ.

The theorem is a conjunction of ten equalities. It states that the reduced Planck constant ℏ equals φ⁻⁵, the Einstein gravitational coupling κ_E equals 8φ⁵, and the zero-parameter-gravity coupling κ_rs equals the same 8φ⁵. It also fixes the BMV phase coefficient α_RS as φ⁵/(8π), the black hole entropy leading-log coefficient c_RS as −log φ / 2, the per-echo amplitude damping ratio as 1/φ, and the per-rung phase delay as log φ. The statement further asserts that the Hawking temperature for a black hole of mass M is 1/(8πM), the leading Bekenstein-Hawking entropy for horizon area A is A/4, and the baryogenesis rung integer η_B is −44, so η_B equals φ⁻⁴⁴.

These are not empirical fits. Each equality is a theorem in the library, proved with zero axioms beyond the standard three of the underlying type theory. The audit aggregates existing closed-form results into one record, satisfying a clause in the framework's quantum-gravity master plan. The framework's own constants, such as ℏ and G, are derived from the forcing chain rather than chosen freely.

The statement does not claim that any of these constants matches a measured value. The single empirical anchor is the CODATA measurement of G_SI, used only to convert the framework's native units to SI units. The SI-2019-exact values of c, ℏ, and k_B are also used in that bridge. The theorem itself says nothing about whether φ⁵/π, the framework's expression for G, agrees with the measured gravitational constant to any precision. That comparison is a separate empirical check, not part of the audit.

What the audit establishes is structural: within the framework, the gravity sector has zero free dimensionless parameters and one dimensional anchor. Every dimensionless constant is a closed form in φ. This is a statement about the framework's internal derivation, not a claim about the external world. The theorem's value is that it makes the framework's promise of parameter-free gravity precise and machine-checkable.

THEOREM gravity_constants_audit_one_statement · IndisputableMonolith/Gravity/ZeroFreeParameters.lean
gravity_constants_audit_one_statement · IndisputableMonolith/Gravity/ZeroFreeParameters.lean:167 
/-- **ONE-STATEMENT AUDIT** (Track 5.B form): a single conjunction
listing the closed-form φ-rational expressions for every gravity-sector
constant. -/
theorem gravity_constants_audit_one_statement :
    (Constants.hbar = Constants.phi ^ (-(5 : ℝ))) ∧
    (Constants.kappa_einstein = 8 * Constants.phi ^ (5 : ℝ)) ∧
    (ZeroParameterGravity.kappa_rs = 8 * Constants.phi ^ 5) ∧
    (NoGraviton.UnitBridge.alphaRS =
      Constants.phi ^ (5 : ℝ) / (8 * Real.pi)) ∧
    (BlackHoleEntropyFromLedger.c_RS =
      -(Real.log Constants.phi) / 2) ∧
    (BlackHoleEchoesFromBounce.echoDampingRatio = 1 / Constants.phi) ∧
    (BlackHoleEchoesFromBounce.rungPhaseDelay = Real.log Constants.phi) ∧
    (∀ M : ℝ, HawkingTemperatureFromRung.T_hawking M =
      1 / (8 * Real.pi * M)) ∧
    (∀ A : ℝ, BlackHoleEntropyFromLedger.S_lead A = A / 4) ∧
    (Cosmology.PhiRungLadder.eta_B_rung_val = (-44 : ℤ)) :=
  ⟨Constants.hbar_eq_phi_inv_fifth,
   Constants.kappa_einstein_eq,
   ZeroParameterGravity.kappa_rs_closed_form,
   rfl, rfl, rfl, rfl,
   HawkingTemperatureFromRung.T_hawking_def,
   BlackHoleEntropyFromLedger.S_lead_eq_BH,
   rfl⟩
THEOREM gravity_constants_audit_one_statement · IndisputableMonolith/Gravity/ZeroFreeParameters.lean
gravity_constants_audit_one_statement · IndisputableMonolith/Gravity/ZeroFreeParameters.lean:167 
/-- **ONE-STATEMENT AUDIT** (Track 5.B form): a single conjunction
listing the closed-form φ-rational expressions for every gravity-sector
constant. -/
theorem gravity_constants_audit_one_statement :
    (Constants.hbar = Constants.phi ^ (-(5 : ℝ))) ∧
    (Constants.kappa_einstein = 8 * Constants.phi ^ (5 : ℝ)) ∧
    (ZeroParameterGravity.kappa_rs = 8 * Constants.phi ^ 5) ∧
    (NoGraviton.UnitBridge.alphaRS =
      Constants.phi ^ (5 : ℝ) / (8 * Real.pi)) ∧
    (BlackHoleEntropyFromLedger.c_RS =
      -(Real.log Constants.phi) / 2) ∧
    (BlackHoleEchoesFromBounce.echoDampingRatio = 1 / Constants.phi) ∧
    (BlackHoleEchoesFromBounce.rungPhaseDelay = Real.log Constants.phi) ∧
    (∀ M : ℝ, HawkingTemperatureFromRung.T_hawking M =
      1 / (8 * Real.pi * M)) ∧
    (∀ A : ℝ, BlackHoleEntropyFromLedger.S_lead A = A / 4) ∧
    (Cosmology.PhiRungLadder.eta_B_rung_val = (-44 : ℤ)) :=
  ⟨Constants.hbar_eq_phi_inv_fifth,
   Constants.kappa_einstein_eq,
   ZeroParameterGravity.kappa_rs_closed_form,
   rfl, rfl, rfl, rfl,
   HawkingTemperatureFromRung.T_hawking_def,
   BlackHoleEntropyFromLedger.S_lead_eq_BH,
   rfl⟩
THEOREM gravity_sector_zero_free_parameters · IndisputableMonolith/Gravity/ZeroFreeParameters.lean
gravity_sector_zero_free_parameters · IndisputableMonolith/Gravity/ZeroFreeParameters.lean:147 
/-- **GRAVITY-SECTOR ZERO-FREE-PARAMETERS THEOREM** (master plan §4
Track 5.B closure form).

Every gravity-sector dimensionless constant has a closed-form φ-rational
expression. The dimensional bridge is anchored on the SINGLE CODATA
measurement `G_SI` (plus the SI-2019-exact `c_SI`, `ℏ_SI`, `k_B_SI`).
Zero free dimensionless parameters; one dimensional anchor.

This is one of the master-theorem-template clauses
(`gravity_sector_zero_free_parameters`). It is theorem-grade in this
module via the named anchor theorems in
`Constants`, `ZeroParameterGravity`, `NoGraviton.UnitBridge`,
`BlackHoleEntropyFromLedger`, `BlackHoleEchoesFromBounce`,
`HawkingTemperatureFromRung`, and `Cosmology.PhiRungLadder`. -/
theorem gravity_sector_zero_free_parameters :
    Nonempty GravitySectorConstantsClosedForm :=
  ⟨gravitySectorConstantsClosedForm⟩
MODEL GravitySectorConstantsClosedForm · IndisputableMonolith/Gravity/ZeroFreeParameters.lean
GravitySectorConstantsClosedForm · IndisputableMonolith/Gravity/ZeroFreeParameters.lean:85 
/-- **GravitySectorConstantsClosedForm**: every gravity-sector constant
listed in the master plan §4 Track 5.B audit has a closed-form φ-rational
expression, anchored on a named existing theorem. The fields are
populated by the corresponding `rfl` or named theorem.

This is the **constants-from-φ audit** required by Track 5.B. Together
with the SI bridge of `Foundation.SIBridgeClosure` (single CODATA
`G_SI` anchor), it establishes that the RS gravity sector has ZERO free
dimensionless parameters and ONE dimensional anchor. -/
structure GravitySectorConstantsClosedForm where
  /-- `ℏ` (RS-native) = `φ^{-5}`. -/
  hbar_closed_form : Constants.hbar = Constants.phi ^ (-(5 : ℝ))
  /-- Einstein gravitational coupling `κ_E` = `8·φ^5`. -/
  kappa_einstein_closed_form :
    Constants.kappa_einstein = 8 * Constants.phi ^ (5 : ℝ)
  /-- Zero-parameter-gravity Einstein coupling `κ_rs` = `8·φ^5`. -/
  kappa_rs_closed_form :
    ZeroParameterGravity.kappa_rs = 8 * Constants.phi ^ 5
  /-- BMV phase coefficient `α_RS` = `φ^5/(8π)`. -/
  alphaRS_closed_form :
    NoGraviton.UnitBridge.alphaRS = Constants.phi ^ (5 : ℝ) / (8 * Real.pi)
  /-- BH entropy leading-log coefficient `c_RS` = `-log φ / 2`. -/
  c_RS_closed_form :
    BlackHoleEntropyFromLedger.c_RS = -(Real.log Constants.phi) / 2
  /-- Per-echo amplitude damping ratio = `1/φ`. -/
  echoDampingRatio_closed_form :
    BlackHoleEchoesFromBounce.echoDampingRatio = 1 / Constants.phi
  /-- Per-rung phase delay = `log φ`. -/
  rungPhaseDelay_closed_form :
    BlackHoleEchoesFromBounce.rungPhaseDelay = Real.log Constants.phi
  /-- RS-native bounce radius at rung gap `N` = `φ^N`. -/
  bounceRadius_closed_form :
    ∀ N : ℕ, BlackHoleEchoesFromBounce.bounceRadius N = Constants.phi ^ N
  /-- RS-native Hawking temperature `T_H(M)` = `1/(8πM)`. -/
  T_hawking_closed_form :
    ∀ M : ℝ, HawkingTemperatureFromRung.T_hawking M = 1 / (8 * Real.pi * M)
  /-- RS-native Bekenstein-Hawking leading entropy `S_lead(A)` = `A/4`. -/
  S_lead_closed_form :
    ∀ A : ℝ, BlackHoleEntropyFromLedger.S_lead A = A / 4
  /-- Baryogenesis η_B rung integer = `-44`, so η_B = `φ^{-44}` as a φ-rational
  power. -/
  eta_B_rung_eq_neg_44 :
    Cosmology.PhiRungLadder.eta_B_rung_val = (-44 : ℤ)
THEOREM gravity_sector_zero_free_parameters · IndisputableMonolith/Gravity/ZeroFreeParameters.lean
gravity_sector_zero_free_parameters · IndisputableMonolith/Gravity/ZeroFreeParameters.lean:147 
/-- **GRAVITY-SECTOR ZERO-FREE-PARAMETERS THEOREM** (master plan §4
Track 5.B closure form).

Every gravity-sector dimensionless constant has a closed-form φ-rational
expression. The dimensional bridge is anchored on the SINGLE CODATA
measurement `G_SI` (plus the SI-2019-exact `c_SI`, `ℏ_SI`, `k_B_SI`).
Zero free dimensionless parameters; one dimensional anchor.

This is one of the master-theorem-template clauses
(`gravity_sector_zero_free_parameters`). It is theorem-grade in this
module via the named anchor theorems in
`Constants`, `ZeroParameterGravity`, `NoGraviton.UnitBridge`,
`BlackHoleEntropyFromLedger`, `BlackHoleEchoesFromBounce`,
`HawkingTemperatureFromRung`, and `Cosmology.PhiRungLadder`. -/
theorem gravity_sector_zero_free_parameters :
    Nonempty GravitySectorConstantsClosedForm :=
  ⟨gravitySectorConstantsClosedForm⟩

What this page does not claim

The theorem does not claim that any framework constant matches a measured value. The theorem does not claim that the framework's expression for G equals the measured gravitational constant. The theorem does not claim that the framework's gravity sector is physically correct.

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/ZeroFreeParameters.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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