Gravity Derived Factors Seven Beat Gap

The seven-beat gap

The seven-beat gap is a number, exactly 1/8, that measures the relative difference between two ways of dividing a cycle. The framework's recognition cycle, a discrete record of events, has eight beats. Eight beats give seven active modes plus a constant. A seven-beat cycle, by contrast, has only six degrees of freedom because of a neutrality constraint. The gap is the difference in active modes, seven minus six, divided by the eight slots: (7 - 6) / 8 = 1/8. This number is a definition, not a derived theorem.

The framework uses this gap to define a stiffness factor. Stiffness is the inverse of the gap, so 1 / (1/8) = 8. The idea is that the eight-beat cycle resists breaking into seven-beat modes with a strength of eight. This stiffness then sets a saturation acceleration scale: a_sat = 8 * a0, where a0 is a characteristic acceleration. The physical picture is that you need eight times the characteristic acceleration to break the eight-beat coherence.

This saturation scale feeds into a suppression factor for galaxy rotation. The factor xi(g) = 1 / (1 + g / (8*a0)) multiplies the ILG kernel amplitude. At low acceleration, xi is near 1, so the full effect remains. At high acceleration, xi approaches 0, recovering Newtonian behavior. The framework's machine-checked library of formal theorems proves two limits: as baryon acceleration goes to infinity, xi goes to 0; as it goes to zero, xi goes to 1. These are theorems about the defined function, not measurements of galaxies.

In Recognition Science, the gap is a hypothesis about why high surface brightness galaxies show less of the ILG effect than expected. The suppression mechanism is a proposal, not a proven consequence of the recognition cycle. The number 1/8 itself is a definition; the physical story around it is a model.

/-- The relative "mode gap" between the valid 8-beat cycle and the invalid 7-beat cycle.
    8-beat has 7 active modes (plus DC).
    7-beat has 6 degrees of freedom (neutrality constraint on 7 slots).
    Relative gap = (7 - 6) / 8 = 1/8.
    Alternatively, using `p_steepness` logic: 1 - 1/8.
    Let's use the inverse gap as a stiffness factor. -/
def seven_beat_gap : ℝ := 1 / 8

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

/-- 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 suppression mechanism is a proven consequence of the recognition cycle. The value 1/8 is derived from first principles rather than defined. The radial profile n(r) is derived; it is set to unity.

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$ 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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