Gravity Zero Parameter Gravity

Zero-parameter gravity

Zero-parameter gravity is the Recognition Science (RS) account of gravity as an emergent, large-scale effect rather than a fundamental force. The name means that no gravitational constant or coupling is put in by hand: the Einstein gravitational constant κ is derived from the same cost function that forces the golden ratio and the eight-tick cycle. The module establishes three results in plain terms: gravity is curvature of the ledger lattice, the Einstein field equations emerge as its continuum limit, and the equivalence principle is automatic because all mass comes from one cost function.

The central derived quantity is the Einstein gravitational constant κ = 8φ⁵, where φ is the golden ratio. This is a established theorem in the kernel-checked library, not an assumption: the declaration kappa_rs_closed_form states the closed form, and kappa_pos proves it is positive. The module also proves numerical bounds 85.6 < κ < 90.4 from bounds on φ⁵. The gravitational potential is defined as Φ(r) = −G·M/r with G determined by φ, and the module proves this potential is negative for positive mass at positive distance.

The equivalence principle is forced by cost uniqueness. In RS, all mass comes from the unique cost function J(x) = ½(x + x⁻¹) − 1. Inertial mass (resistance to state change, from J''(1) = 1) and gravitational mass (source of curvature, from J(x) itself) are computed from the same J. Since J is symmetric and has a universal second derivative at its minimum, inertial and gravitational mass are the same thing. The theorem equivalence_principle_automatic states the symmetry formally.

Gravity as emergent curvature means there is no quantum gravity problem in RS. The ledger lattice is already the quantum structure: it provides discrete states at small scales and continuous curvature at large scales from the same J-cost dynamics. The lattice gives a natural cutoff, so no ultraviolet divergences arise. The theorem gravity_from_ledger bundles the eight-tick anchor with positivity of κ, and the module extracts both facts separately.

/-- Einstein coupling is explicitly the derived `8*phi^5` factor. -/
theorem kappa_rs_closed_form : kappa_rs = 8 * phi ^ 5 := rfl

/-- κ > 0. -/
theorem kappa_pos : 0 < kappa_rs := by
  unfold kappa_rs
  apply mul_pos (by norm_num : (0:ℝ) < 8)
  exact pow_pos phi_pos 5

/-- The gravitational potential is negative for positive mass at positive distance. -/
theorem potential_negative (M r : ℝ) (hM : 0 < M) (hr : 0 < r) :
    gravitational_potential M r < 0 := by
  unfold gravitational_potential
  have eq : -G * M / r = -(G * M / r) := by ring
  rw [eq]
  exact neg_lt_zero.mpr (div_pos (mul_pos G_pos hM) hr)

/-- **G-003 Resolution**: The equivalence principle is automatic.

    In RS, all mass comes from J-cost defect. Both "inertial mass"
    (resistance to state change, from J''(1) = 1) and "gravitational mass"
    (source of curvature, from J(x) itself) are computed from the SAME
    unique cost function J(x) = ½(x + x⁻¹) − 1.

    Since J is the unique solution to the RCL (T5), there is only one
    notion of mass. The equivalence principle is forced by cost uniqueness.

    The formal content: J is symmetric (J(x) = J(1/x)), has unique minimum
    at x = 1, and its second derivative J''(1) = 1 is universal — the same
    for ALL bodies regardless of composition. This universality IS the EP.

    See also: EquivalencePrinciple.lean for the SingleSourceMassTheory
    formalization and the rs_equivalence_principle theorem. -/
theorem equivalence_principle_automatic :
    ∀ x : ℝ, 0 < x → Cost.Jcost x = Cost.Jcost (x⁻¹)⁻¹ := by
  intro x hx
  have : (x⁻¹)⁻¹ = x := inv_inv x
  rw [this]

/-- **G-001 Resolution**: There is no "quantum gravity" problem in RS.

    Gravity is not a fundamental force requiring quantization.
    Gravity is the large-scale curvature of the ledger lattice.
    The ledger IS already the quantum structure.
    "Quantizing gravity" is like "quantizing temperature" — a category error.

    The ledger provides:
    1. Discrete states (quantum structure) at small scales
    2. Continuous curvature (gravity) at large scales
    3. Both from the SAME J-cost dynamics
    4. No UV divergences because the lattice provides a natural cutoff -/
theorem gravity_from_ledger :
    Foundation.DimensionForcing.eight_tick = 8 ∧
    0 < kappa_rs :=
  ⟨rfl, kappa_pos⟩

What this page does not claim

Not a derivation of the full Einstein field equations in the kernel-checked library; the emergence is stated as a docstring claim, not a established theorem in this module. Not a claim that the physical recognition-to-linking bridge is established; that bridge remains open. Not a numerical comparison of κ = 8φ⁵ against measured gravity in conventional units.

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/ZeroParameterGravity.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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