Gravity Raremergence Rar Power Law Emergence

The Radial Acceleration Relation

The Radial Acceleration Relation (RAR) is an empirical pattern in galaxy dynamics. For any galaxy, the acceleration inferred from its observed rotation speed correlates tightly with the acceleration expected from its visible baryonic matter alone. The relation is a power law: observed acceleration equals a constant times baryonic acceleration raised to a power near 0.8, with intrinsic scatter of about 0.1 dex.

Astronomers have measured this relation across thousands of galaxies since the late 20th century, and it holds regardless of galaxy mass, size, or morphology. The relation is surprising because baryonic matter should not fully determine dynamics if dark matter dominates. The tightness of the correlation, with scatter under 0.1 dex, makes it a sharp test for any theory of gravity or dark matter.

In Recognition Science, the framework models gravity through a weight function that modifies baryonic acceleration. The framework defines the observed acceleration as the baryonic acceleration multiplied by a weight that scales as a power of the ratio of a characteristic acceleration scale to the baryonic acceleration. Under this model, the framework proves a theorem named rar_power_law_emergence: the observed acceleration equals the characteristic scale raised to half the dynamical-time exponent, times the baryonic acceleration raised to one minus half that exponent. This is a pure algebraic identity, derived from the definitions of the weight function and observed acceleration.

The theorem establishes that the RAR emerges as a mathematical consequence of the model's assumptions, not as an independent postulate. The exponent in the power law is one minus half the dynamical-time exponent. When the framework's locked value for that exponent, about 0.191, is used, the predicted slope in log-log space is about 0.9045. The framework also proves the relation is universal: for any two galaxies with the same global parameters, the ratio of their observed accelerations equals the ratio of their baryonic accelerations raised to that same power.

What the theorem does not claim is as important as what it proves. It does not prove that the model's assumptions are true of the real universe. The weight function is a definitional choice, and the characteristic scale and exponent are parameters, not derived constants. The theorem shows that if the model holds, then the RAR follows; it does not show that the model holds. The empirical agreement with measured galaxy data is a separate check, not part of the theorem. The framework's library proves the algebra; the universe must still supply the physics.

/-- Same result as `rar_power_law`, but stated directly as an algebraic identity
    in the "weight times baryonic acceleration" form. -/
theorem rar_power_law_emergence
    (a₀ a_baryon α : ℝ) (ha0 : 0 < a₀) (ha : 0 < a_baryon) :
    (a₀ / a_baryon) ^ (α / 2) * a_baryon = a₀ ^ (α / 2) * a_baryon ^ (1 - α / 2) := by
  simpa [a_obs_ilg, w_accel] using (rar_power_law a₀ a_baryon α ha0 ha)

/-- The RAR is universal (same function for all galaxies) when α and a₀ are global. -/
theorem rar_is_universal
    (a₀ α : ℝ) (ha0 : 0 < a₀)
    (galaxy1_a_baryon galaxy2_a_baryon : ℝ)
    (h1 : 0 < galaxy1_a_baryon) (h2 : 0 < galaxy2_a_baryon) :
    -- The ratio of observed accelerations equals the ratio of baryonic accelerations
    -- raised to the power (1 - α/2)
    a_obs_ilg a₀ galaxy1_a_baryon α / a_obs_ilg a₀ galaxy2_a_baryon α =
    (galaxy1_a_baryon / galaxy2_a_baryon) ^ (1 - α / 2) := by
  rw [rar_power_law a₀ galaxy1_a_baryon α ha0 h1]
  rw [rar_power_law a₀ galaxy2_a_baryon α ha0 h2]
  -- The a₀^(α/2) factors cancel, leaving (a1/a2)^(1-α/2)
  set p : ℝ := 1 - α / 2
  have ha0_ne : (a₀ ^ (α / 2)) ≠ 0 := by
    exact ne_of_gt (Real.rpow_pos_of_pos ha0 (α / 2))
  have hcancel :
      (a₀ ^ (α / 2) * galaxy1_a_baryon ^ p) / (a₀ ^ (α / 2) * galaxy2_a_baryon ^ p)
        = (galaxy1_a_baryon ^ p) / (galaxy2_a_baryon ^ p) := by
    simpa [mul_assoc, mul_left_comm, mul_comm] using
      (mul_div_mul_left (galaxy1_a_baryon ^ p) (galaxy2_a_baryon ^ p) ha0_ne)
  rw [hcancel]
  have h1' : 0 ≤ galaxy1_a_baryon := le_of_lt h1
  have h2' : 0 ≤ galaxy2_a_baryon := le_of_lt h2
  have hdiv : (galaxy1_a_baryon / galaxy2_a_baryon) ^ p =
      galaxy1_a_baryon ^ p / galaxy2_a_baryon ^ p := by
    simpa [p] using (Real.div_rpow h1' h2' p)
  -- Rewrite the RHS using `hdiv`.
  simp [hdiv]

What this page does not claim

The theorem does not prove the model's weight function is physically true. The theorem does not establish that the characteristic scale or exponent are derived constants rather than fitted parameters. The theorem does not claim the RAR holds for real galaxies; that is a separate empirical check.

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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