Gravity Zero Free Parameters Gravity Sector Zero Free Parameters

The zero-free-parameter audit

In physics, a theory's free parameters are the numbers that must be measured before it can predict anything: the electron mass, the fine-structure constant, the cosmological constant. A theory with zero free parameters would be one where every constant is fixed by the theory's own structure. Recognition Science (RS) has a formal claim, gravity_sector_zero_free_parameters, that its gravity sector achieves this: every dimensionless constant in that sector is a closed-form expression built from the golden ratio φ, and only one measured value, Newton's constant G, is needed to connect the framework to SI units.

The claim is not a vague ambition but a theorem in the framework's machine-checked library of formal theorems. The theorem gravity_sector_zero_free_parameters asserts that a structure exists containing closed-form φ-rational expressions for all gravity constants in the audit list. That list includes the reduced Planck constant as φ⁻⁵, the Einstein gravitational coupling as 8φ⁵, the Hawking temperature as 1/(8πM), and the Bekenstein-Hawking entropy as A/4. These are not approximations; they are exact equalities in the framework's native units, with the golden ratio appearing as the fundamental scaling constant.

The single empirical anchor is the CODATA measurement of G, used to assign SI units to the framework's dimensionless expressions. The framework also treats the speed of light, the reduced Planck constant, and the Boltzmann constant as exactly defined SI values from the 2019 redefinition. This means the framework's gravity sector, in its own account, has zero free dimensionless parameters and one dimensional anchor. The theorem is a structural statement about the framework's internal consistency, not a claim about measured physics.

What the theorem does not claim is equally important. It does not claim that the framework's predictions match experiment; that is a separate empirical check. It does not claim that the framework's derivation of the golden ratio is complete, only that the constants are expressed in terms of it. It does not claim that the framework's approach to quantum gravity is correct, only that its constants are fixed. The theorem is an audit of internal structure, a statement that the framework's gravity sector is not a collection of fitted numbers but a single, self-consistent expression.

The practical consequence is that, within RS, gravity is not a theory with adjustable knobs. The strength of gravity, the mass scale of particles, and the temperature of black holes are all supposed to follow from the golden ratio. This is a bold claim, and the theorem is the framework's way of making it precise and checkable. The reader can now see what the framework means by a zero-free-parameter theory: not a theory that predicts everything, but a theory where the constants are all derived from one number.

/-- **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⟩

/-- **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

The theorem does not claim that the framework's gravity predictions match experimental measurements. The theorem does not claim that the framework's derivation of the golden ratio is complete. The theorem does not claim that the framework's approach to quantum gravity 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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