Gravity Derived Factors Hsb Suppression Limit

The high-brightness suppression limit

High Surface Brightness (HSB) galaxies are the dense, baryon-dominated ones, where ordinary matter outshines dark matter. A proposed modification to gravity, called the ILG effect, overpredicts their rotation velocities. The Recognition Science framework's ledger, a discrete record of recognition events, suggests an 8-beat cycle as the minimal valid period for three spatial dimensions. The hypothesis is that at high energy densities, this 8-beat lock weakens, and the ILG effect should turn off, recovering standard Newtonian gravity.

The framework's machine-checked library of formal theorems contains a definition, xi_derived, that models this suppression. It is a saturation function: xi(g) = 1 / (1 + g / (8 * a0)), where g is the baryonic acceleration and a0 is a characteristic acceleration scale. At low g, xi is near 1, preserving the full ILG effect. At high g, xi approaches 0, turning the modification off. The constant 8 comes from the gap between the valid 8-beat cycle and an invalid 7-beat cycle, which has fewer degrees of freedom.

The theorem hsb_suppression_limit proves, with no unverified assumptions, that as g tends to infinity, xi_derived tends to 0. This is the formal statement that the ILG modification vanishes in the high-acceleration limit, recovering Newtonian behavior. A companion theorem, lsb_unsuppressed_limit, proves that as g tends to zero, xi_derived tends to 1, meaning the full ILG effect remains for Low Surface Brightness galaxies. These two limits together define the intended behavior of the suppression mechanism.

In Recognition Science, this theorem establishes a consistency condition: the proposed modification must not conflict with well-tested Newtonian dynamics in the regimes where they are known to hold. It does not prove that the ILG effect itself is real, nor that the suppression mechanism is the correct physical explanation. It only proves that if the effect exists as modeled, it has the correct limiting behavior. The theorem is a formal guarantee about the mathematical model, not an empirical measurement of any galaxy.

What this means for the reader is a clear separation of concerns. The framework has a precise, machine-checked statement about what its model does at extreme accelerations. This is a necessary check for any proposed modification to gravity, but it is not sufficient evidence for the modification itself. The physical reality of the ILG effect, and the validity of the 8-beat cycle hypothesis, remain open questions that this theorem does not address.

/-- 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

/-- 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))

What this page does not claim

The ILG effect is physically real; the theorem only constrains a mathematical model. The suppression mechanism is the correct explanation for HSB galaxy rotation curves. The 8-beat cycle is an observed property of galaxies; it is a hypothesis from the framework.

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)

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