Gravity Metric From Defect

Metric perturbation from ledger strain

General relativity describes gravity as curvature of spacetime. The metric tensor g encodes distances and angles, and in weak fields it splits into a flat background η plus a small perturbation h: g = η + h. The perturbation h is what carries the gravitational effect, such as the bending of light or the slowing of clocks near a mass. In the standard Newtonian limit, the time-time component h₀₀ equals -2Φ, where Φ is the gravitational potential. The potential satisfies Poisson's equation ∇²Φ = κρ, with ρ the mass density and κ the coupling constant.

Recognition Science starts from a different source. Its core object is a ledger, a discrete record of recognition events, and each event carries a cost, a nonnegative number measuring how far the event is from perfect symmetry. The cost function J(x) = ½(x + x⁻¹) - 1 is forced by five plain conditions, a theorem checked in the framework's machine-checked library of formal theorems. Where the ledger has many events with high cost, it accumulates strain, and that strain is what curves spacetime.

The module MetricFromDefect makes this precise in the weak-field limit. It defines a defect field, a function giving the J-cost density at each point of space, and it defines the metric perturbation h as proportional to that density. The proportionality constant is κ = 8φ⁵, where φ is the golden ratio, a value that emerges from the framework's zero-parameter gravity rather than being fitted. Formally, the perturbation is hᵢⱼ = -κρ δᵢⱼ, with ρ the defect density. The module proves three things: the perturbation is symmetric, zero defect density gives zero perturbation (flat space), and the perturbation is exactly proportional to κ times the density.

In Recognition Science, the defect density plays the role of the stress-energy source T⁰⁰. The framework models the source of gravity not as mass but as ledger strain. The weak-field condition, κρ ≪ 1, keeps the perturbation small and the linearized equations valid. The module does not derive the full Einstein field equations; it establishes the first step, the dictionary between defect density and metric perturbation. That dictionary is what connects the ledger picture to familiar gravitational language.

What this changes is the meaning of the source term. In the standard theory, mass density is an input; here it is a derived quantity, the density of recognition cost. The metric perturbation is not an independent field but a response to ledger strain. The module gives a concrete, checkable bridge: if you know the defect distribution, you know the weak-field metric. The next step, left open, is to show that this perturbation satisfies the full Einstein equations, not just the Newtonian limit.

/-- The metric perturbation is proportional to kappa (= 8*phi^5). -/
theorem perturbation_proportional_to_kappa (d : DefectField) (r : ℝ) :
    (metric_perturbation_from_defect d r).components 0 0 =
    -ZeroParameterGravity.kappa_rs * d.density r 0 0 := by
  simp [metric_perturbation_from_defect]

/-- Zero defect density gives zero metric perturbation (flat space). -/
theorem zero_defect_flat_space (r : ℝ) :
    let d : DefectField := ⟨fun _ _ _ => 0, fun _ _ _ => le_refl 0⟩
    (metric_perturbation_from_defect d r).components 0 0 = 0 := by
  simp [metric_perturbation_from_defect, ZeroParameterGravity.kappa_rs]

/-- The metric perturbation is symmetric by construction. -/
theorem metric_perturbation_symmetric (d : DefectField) (r : ℝ) (i j : Fin 3) :
    (metric_perturbation_from_defect d r).components i j =
    (metric_perturbation_from_defect d r).components j i :=
  (metric_perturbation_from_defect d r).symmetric i j

What this page does not claim

The full Einstein field equations are not derived in this module. The proportionality constant κ = 8φ⁵ is not derived here; it is taken from ZeroParameterGravity. The module does not prove that the defect density equals the standard mass density in all regimes.

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/MetricFromDefect.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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