Gravity Derived Factors A Saturation

The saturation scale

In the Recognition Science framework's account of galaxy rotation, the declaration a_saturation defines a critical acceleration. It is the point where a proposed modification to gravity, called the ILG effect, begins to weaken. The value is simply eight times a characteristic acceleration a0, so the formula reads a_saturation = 8 * a0. The factor eight comes from the framework's claim that an 8-beat cycle is the minimal valid period for three spatial dimensions, and that a 7-beat cycle fails. The stiffness of the 8-beat lock against this 7-beat leakage is defined as the inverse of the relative mode gap, which is 1/8, giving a stiffness of 8.

The purpose of this threshold is to fix a specific problem: the ILG kernel, which otherwise boosts rotation velocities, overpredicts velocities for High Surface Brightness (HSB) galaxies, which are dominated by baryonic matter. The framework hypothesizes that at high acceleration, the 8-beat coherence breaks, and the ILG effect turns off, recovering Newtonian behavior. The suppression factor xi_derived is defined as 1 / (1 + g / a_saturation), where g is the baryonic acceleration. At low g, this factor is near 1, so the full ILG effect remains; at high g, it approaches 0, restoring Newtonian gravity.

Two limit theorems are proved in the machine-checked library. The first, hsb_suppression_limit, states that as baryonic acceleration goes to infinity, the suppression factor tends to 0, meaning the ILG modification vanishes. The second, lsb_unsuppressed_limit, states that as acceleration goes to zero, the factor tends to 1, meaning no suppression occurs. These are formal results about the defined function, not empirical measurements of galaxy data.

In Recognition Science, this declaration does not claim that the factor eight is derived from first principles; it is a definitional choice based on a hypothesis about 7-beat leakage. The radial profile n(r) is also defined as unity, leaving the LSB underprediction problem unresolved. The declaration does not claim to explain the exact value of a0, nor does it claim that the suppression mechanism has been observed directly. It establishes a mathematical form and its limits, within the framework's own model.

/-- The acceleration scale where "saturation" (leakage) begins.
    Hypothesis: The critical acceleration `a_sat` is the characteristic scale `a0`
    boosted by the stiffness of the 8-beat lock.

    a_sat = stiffness * a0 = 8 * a0.

    Physical intuition: You need 8x the characteristic acceleration to "break"
    the 8-beat coherence and suppress the ILG effect. -/
def a_saturation (a0 : ℝ) : ℝ := lock_stiffness * a0

/-- The HSB suppression factor ξ(g).
    This factor multiplies the ILG kernel amplitude.

    Behavior:
    - Low g (<< a_sat): ξ ≈ 1 (Full ILG effect)
    - High g (>> a_sat): ξ -> 0 (Newtonian recovery)

    Functional form: Standard saturation `1 / (1 + x)`.
    Argument x: `g / a_sat`.

    Formula: ξ(g) = 1 / (1 + g / (8*a0))

    This provides the necessary suppression for HSB galaxies (where g is high)
    while maintaining the ILG boost for LSB galaxies (where g is low). -/
def xi_derived (g_baryon : ℝ) (a0 : ℝ) : ℝ :=
  1 / (1 + g_baryon / (a_saturation a0))

/-- Theorem: HSB Suppression recovers Newtonian limit.
    As baryon acceleration goes to infinity, the ILG modification vanishes. -/
theorem hsb_suppression_limit (a0 : ℝ) (ha0 : a0 > 0) :
    Filter.Tendsto (fun g => xi_derived g a0) Filter.atTop (nhds 0) := by
  unfold xi_derived
  have h_sat_pos : a_saturation a0 > 0 := by
    unfold a_saturation lock_stiffness seven_beat_gap
    linarith
  -- Rewrite 1/(1+x) as (1+x)⁻¹ to match inv_tendsto_atTop
  rw [show (fun g => 1 / (1 + g / a_saturation a0)) = (fun g => (1 + g / a_saturation a0)⁻¹) by ext; simp]
  apply Filter.Tendsto.inv_tendsto_atTop
  apply Filter.tendsto_atTop_add_const_left
  apply Filter.Tendsto.atTop_mul_const (inv_pos.mpr h_sat_pos) Filter.tendsto_id

/-- Theorem: LSB Limit is Unsuppressed.
    As baryon acceleration goes to zero, the suppression factor goes to 1. -/
theorem lsb_unsuppressed_limit (a0 : ℝ) (ha0 : a0 > 0) :
    Filter.Tendsto (fun g => xi_derived g a0) (nhds 0) (nhds 1) := by
  unfold xi_derived
  -- We prove 1 / (1 + g / K) -> 1
  -- Rewrite 1 as 1 / (1 + 0 / K)
  conv in (nhds 1) => rw [show (1 : ℝ) = 1 / (1 + 0 / a_saturation a0) by
    field_simp [a_saturation, lock_stiffness, seven_beat_gap]; linarith]
  apply Filter.Tendsto.div
  · exact tendsto_const_nhds
  · apply Filter.Tendsto.add
    · exact tendsto_const_nhds
    · apply Filter.Tendsto.div
      · exact Filter.tendsto_id
      · exact tendsto_const_nhds
      · -- Denominator ≠ 0
        unfold a_saturation lock_stiffness seven_beat_gap
        linarith
  · -- Limit denominator (1 + 0) ≠ 0
    norm_num

What this page does not claim

The factor eight is not derived from a theorem but is a definitional choice based on a hypothesis about 7-beat leakage. The declaration does not claim that the suppression mechanism has been observed directly in galaxy data. The value of a0 is not fixed by this declaration; it remains a free parameter.

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/DerivedFactors.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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