Gravity Raremergence Rar Emergence Direct

The RAR as a power law

The Radial Acceleration Relation (RAR) is an empirical pattern in galaxy dynamics. For a wide range of galaxies, the acceleration actually observed in rotation curves, a_obs, correlates tightly with the acceleration expected from ordinary baryonic matter alone, a_baryon. The relation is not a straight line; it curves, and the standard way to describe it is as a function a_obs = f(a_baryon), with a scatter of about 0.1 dex.

The Recognition Science framework derives this relation from its own model of gravity. In this account, the observed acceleration is the baryonic acceleration multiplied by a weight function, w(a). The weight function is a power law: w(a) = C * (a_0 / a)^(α/2), where a_0 is a characteristic acceleration scale and α is a dynamical-time exponent. The factor 1/2 comes from a bridge between acceleration and time.

In Recognition Science, the declaration rar_emergence_direct proves a specific algebraic identity. It states that, for positive a_0 and a_baryon, the product of the weight function and the baryonic acceleration equals a_0^(α/2) * a_baryon^(1 - α/2). In plain language, the framework proves that the RAR takes the form of a power law, with the observed acceleration proportional to the baryonic acceleration raised to the power (1 - α/2).

The exponent is the key number. In log-log space, the slope of the RAR is exactly 1 - α/2. For the framework's locked value of α, which is approximately 0.191, this slope is about 0.9045. The same theorem also shows that if the exponent α were to approach 1, the slope would approach 0.5, which is the deep MOND limit. The framework's derivation is a mathematical consequence of its chosen weight function, not an independent physical postulate.

What the declaration does not claim is just as important. It does not prove that the RAR is universal across all galaxies; that is a separate theorem in the same module, which assumes the parameters a_0 and α are global constants. It does not derive the value of the characteristic acceleration scale a_0 or the exponent α from first principles; those are parameters of the model. And it does not explain the observed scatter in the RAR, which the framework attributes to observational errors and variations in galaxy morphology.

/-- Alias for the main RAR emergence statement, kept for stable downstream references. -/
theorem rar_emergence_direct
    (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) :=
  rar_power_law a₀ a_baryon α ha0 ha

/-- **RAR slope (log-log):**
    In log-log space, the RAR has slope \(d(\log a_{\rm obs})/d(\log a_{\rm baryon}) = 1 - \alpha/2\).

    For \(\alpha = 0.389\): slope ≈ 0.8055
    For MOND-like (\(\alpha \to 1\)): slope → 0.5 (deep MOND)
-/
def rar_log_slope (α : ℝ) : ℝ := 1 - α / 2

/-- **RAR slope (log-log):**
    In log-log space, the RAR has slope \(d(\log a_{\rm obs})/d(\log a_{\rm baryon}) = 1 - \alpha/2\).

    For \(\alpha = 0.389\): slope ≈ 0.8055
    For MOND-like (\(\alpha \to 1\)): slope → 0.5 (deep MOND)
-/
def rar_log_slope (α : ℝ) : ℝ := 1 - α / 2

What this page does not claim

The declaration does not prove the universality of the RAR across all galaxies. It does not derive the values of the parameters a_0 and α from first principles. It does not explain the observed scatter in the RAR.

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)

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