Encyclopedia Gravity Gravity Zero Free Parameters

ARTICLE 4 claims 4 theorems

Gravity Zero Free Parameters

A machine-checked audit shows every constant in one theory of gravity is a fixed power of the golden ratio, with a single measured input.

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Gravity's constants audit

In physics, a theory's constants are the numbers that must be measured before the theory can predict anything. The gravitational constant G, the speed of light c, and Planck's constant hbar are usual examples. A theory with many free parameters is weak: it can be adjusted to fit almost anything. A theory with none is strong: it either matches nature or it fails.

Recognition Science (RS) is a framework that starts from the idea that reality keeps a discrete record of events, a ledger, and that the cost of recording an event is forced by a few plain conditions. From that starting point, its machine-checked library of formal theorems derives a chain of results, including the golden ratio phi as the unique self-similar scaling. The gravity zero free parameters audit gathers the results of that chain for gravity.

The audit's claim is precise. It states that every dimensionless constant in the RS gravity sector has a closed-form expression as a power of phi. For example, the reduced Planck constant hbar is phi to the minus fifth power, and the Einstein gravitational coupling kappa is eight times phi to the fifth. Other constants in the audit include the Hawking temperature, the black hole entropy coefficient, and the baryon asymmetry parameter, each fixed to a phi-rational value.

The single exception to the pure phi-power rule is the dimensional bridge. To land any of these constants in SI units, the audit needs one measured input: the CODATA value of G. This is the one dimensional anchor. The speed of light, Planck's constant, and the Boltzmann constant are exact by the 2019 SI definition, so they are not free parameters either. The result is that the gravity sector has zero free dimensionless parameters and exactly one dimensional anchor.

This is a structural theorem, not a numerical prediction. The audit does not claim to have measured anything new. It aggregates existing, already-proved expressions into a single audit record, and that record is inhabited by a proof in the library. The consequence is that the framework's gravity sector is not adjustable: every constant is fixed by the golden ratio, and the only empirical input is a single measurement of G.

THEOREM gravity_sector_zero_free_parameters · gravity_constants_audit_one_statement · 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⟩
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_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 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)

What this page does not claim

This audit does not derive the numerical value of G itself; it uses one measured value as the dimensional anchor. This audit does not prove that the framework's gravity sector is physically correct; it proves that the constants are fixed by phi, not that they match experiment. This audit does not claim to have closed the remaining open tracks of the master plan, which are still targets.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:

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