Gravity Derived Factors Lsb Unsuppressed Limit

The low-density limit

The declaration lsb_unsuppressed_limit is a formal statement about a specific mathematical function, not a physical measurement. The function, called xi_derived, is a proposed suppression factor that multiplies the amplitude of a gravity modification in the Recognition Science framework. The theorem says that as the baryonic acceleration g approaches zero, the value of xi_derived approaches 1. In plain terms, at very low acceleration, the modification is fully active, not suppressed, and the factor does not reduce the effect.

The function itself is defined as xi_derived(g, a0) = 1 / (1 + g / (8 * a0)), where a0 is a characteristic acceleration scale and the factor 8 comes from a stiffness parameter derived from the gap between an 8-beat cycle and a 7-beat cycle. This is a definitional choice, not a derived physical law. The theorem is a mathematical limit: as g goes to zero, the denominator goes to 1, so the whole expression goes to 1. The proof in the machine-checked library of formal theorems is a direct application of limit arithmetic.

In the context of galaxy rotation curves, this limit is meant to address a known discrepancy. The framework's gravity modification overpredicts rotation velocities for high surface brightness galaxies, which are baryon-dominated, and underpredicts for low surface brightness galaxies, which are dark-matter-dominated. The suppression factor xi_derived is designed to turn off the modification at high acceleration, recovering Newtonian behavior, while leaving it fully on at low acceleration. The lsb_unsuppressed_limit theorem confirms the low-acceleration end of that design: the suppression factor does not vanish, it stays at 1.

What the theorem does not claim is more important than what it proves. It does not say that the function xi_derived is correct physics. It does not say that the 8-beat cycle or the stiffness factor 8 is derived from first principles; those are hypotheses in the source module. It does not say that the low surface brightness underprediction is resolved; the module explicitly notes that the radial factor n(r) is set to unity and that the underprediction might require tuning a0 or C. The theorem is a mathematical fact about a chosen function, not an empirical validation.

/-- 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

/-- 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))

/-- The stiffness of the 8-beat lock against 7-beat leakage.
    Stiffness = 1 / Gap = 8. -/
def lock_stiffness : ℝ := 1 / seven_beat_gap

What this page does not claim

The theorem does not validate xi_derived as a correct physical model. The theorem does not resolve the low surface brightness underprediction. The 8-beat stiffness factor is a definition, not a derived consequence.

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.

Derived articles

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