Gravity Raremergence Rar Slope Rs Value

The derived slope

The Radial Acceleration Relation (RAR) is an empirical pattern in galaxy dynamics: the observed acceleration at a point in a galaxy correlates tightly with the acceleration expected from the visible baryonic matter alone. The relation is a power law, and on a log-log plot it has a slope, the rate at which the observed acceleration changes as the baryonic acceleration changes. Measurements across many galaxies show a slope near 0.8.

In the Recognition Science framework, this slope is not fitted to data but derived. The framework models the observed acceleration as the baryonic acceleration multiplied by a weight function, recognition (a discrete record of events) that depends on the ratio of a characteristic acceleration scale to the local baryonic acceleration. The weight function has an exponent α/2, where α is a dynamical-time exponent. A theorem in the framework's machine-checked library of formal theorems, rar_power_law, proves that under this model the observed acceleration equals a₀^(α/2) · a_baryon^(1 − α/2).

The slope in log-log space is therefore 1 − α/2. The framework's library also proves a specific value for this slope. The theorem rar_slope_rs_value establishes that the slope equals 1 − (1 − 1/φ)/4, where φ is the golden ratio, approximately 1.618. This gives a slope of about 0.845, a number that emerges from the framework's internal constants rather than from galaxy observations.

The theorem is a statement about the framework's definitions. It proves an algebraic identity: given the framework's choice of α, the slope takes that exact form. It does not prove that this number matches the observed RAR slope. The comparison with data is an empirical check, not a theorem. The framework's library also notes that with a different α, the slope would differ; for α = 0.389, the slope is about 0.8055, closer to the observed value.

The result matters because it shows the framework can produce a concrete, testable prediction for a galaxy-scale phenomenon from its internal structure. The slope is not a free parameter in this account; it is a consequence of the framework's definitions. Whether that consequence agrees with the measured RAR is a separate question, one the theorem itself does not answer.

theorem rar_slope_rs_value : rar_slope_rs = 1 - (1 - 1/phi) / 4 := by
  unfold rar_slope_rs alphaLock
  ring

/-- **RAR Emergence Theorem (exact form):**
    The observed acceleration is a power-law function of baryonic acceleration:

    \(a_{\rm obs} = a_0^{\alpha/2} \cdot a_{\rm baryon}^{1 - \alpha/2}\)

    This is the RAR with exponent \(1 - \alpha/2\).
-/
theorem rar_power_law (a₀ a_baryon α : ℝ) (ha0 : 0 < a₀) (ha : 0 < a_baryon) :
    a_obs_ilg a₀ a_baryon α = a₀ ^ (α / 2) * a_baryon ^ (1 - α / 2) := by
  unfold a_obs_ilg w_accel
  -- (a₀/a)^(α/2) * a = a₀^(α/2) * a^(-α/2) * a = a₀^(α/2) * a^(1 - α/2)
  -- This is straightforward algebra using rpow identities
  have ha0' : 0 ≤ a₀ := le_of_lt ha0
  have ha' : 0 ≤ a_baryon := le_of_lt ha
  -- Split the ratio power: (a₀/a)^(α/2) = a₀^(α/2) / a^(α/2)
  rw [Real.div_rpow ha0' ha' (α / 2)]
  -- Convert a / a^(α/2) into a^(1-α/2)
  have hsub : a_baryon ^ (1 - α / 2) = a_baryon ^ (1 : ℝ) / a_baryon ^ (α / 2) := by
    simpa using (Real.rpow_sub ha (1 : ℝ) (α / 2))
  -- Finish by reassociation.
  calc
    (a₀ ^ (α / 2) / a_baryon ^ (α / 2)) * a_baryon
        = a₀ ^ (α / 2) * (a_baryon / a_baryon ^ (α / 2)) := by ring
    _ = a₀ ^ (α / 2) * (a_baryon ^ (1 : ℝ) / a_baryon ^ (α / 2)) := by
          simp [Real.rpow_one]
    _ = a₀ ^ (α / 2) * a_baryon ^ (1 - α / 2) := by
          -- rewrite the right factor using `hsub`
          simp [hsub]

/-- The observed (effective) acceleration from ILG.
    \(a_{\rm obs} = w(a_{\rm baryon}) \cdot a_{\rm baryon}\) -/
def a_obs_ilg (a₀ a_baryon α : ℝ) : ℝ :=
  w_accel a₀ a_baryon α * a_baryon

What this page does not claim

The theorem does not prove that the derived slope matches the measured RAR slope. The theorem does not establish the value of the dynamical-time exponent α from first principles. The theorem does not claim the RAR is universal without the parameters a₀ and α being global constants.

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/RAREmergence.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.

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