Gravity Eight Tick Resonance W Resonant Bounded Above

The bounded weight

The declaration w_resonant_bounded_above is a proved theorem in the framework's machine-checked library of formal theorems. It states that for any real number r, the value of a function called w_resonant is always less than or equal to 1 + C_lag / 2. Here C_lag is a positive constant defined as phi⁻¹ raised to the fifth power, where phi is the golden ratio, approximately 1.618. The theorem is a bound: it does not say what the weight equals, only that it cannot exceed this ceiling.

To see what the weight measures, consider the idea of a ledger, a discrete record of events that the framework uses as its basic picture of reality. The function w_resonant takes a frequency ratio r and returns a number that is 1 when r is an integer, meaning the frequency is perfectly synchronized with the ledger's clock. When r is not an integer, the weight rises above 1, and the amount of rise is controlled by interpolation_cost, which measures how far r is from the nearest integer. The bound says that even at the worst possible desynchronization, halfway between two integers, the weight still cannot exceed the ceiling set by C_lag.

The constant C_lag is not arbitrary. It is defined as phi⁻¹ to the fifth power, which is about 0.09017. This means the ceiling 1 + C_lag / 2 is approximately 1.045. The theorem is a companion to other results in the same file: w_resonant is always at least 1, it equals 1 exactly at integer inputs, and it is strictly greater than 1 off resonance. The bounded-above theorem completes this picture by giving the upper end of the range.

What the theorem does not claim is just as important. It does not say that the weight actually reaches this ceiling, only that it never exceeds it. It does not identify the physical meaning of the weight or connect it to any measured force or particle mass; those connections, if they exist, would be separate claims. The theorem is purely a mathematical statement about a defined function, proved in the framework's library, and it carries no empirical content on its own.

/-- The resonance weight is bounded above by 1 + C_lag/2 (since
    interpolation cost ≤ 1/2). -/
theorem w_resonant_bounded_above (r : ℝ) :
    w_resonant r ≤ 1 + C_lag / 2 := by
  unfold w_resonant
  have hic := interpolation_cost_le_half r
  have hcl := le_of_lt C_lag_pos
  nlinarith [mul_le_mul_of_nonneg_left hic hcl]

/-- The resonance weight is bounded above by 1 + C_lag/2 (since
    interpolation cost ≤ 1/2). -/
theorem w_resonant_bounded_above (r : ℝ) :
    w_resonant r ≤ 1 + C_lag / 2 := by
  unfold w_resonant
  have hic := interpolation_cost_le_half r
  have hcl := le_of_lt C_lag_pos
  nlinarith [mul_le_mul_of_nonneg_left hic hcl]

theorem C_lag_pos : 0 < C_lag := by
  unfold C_lag
  exact pow_pos (inv_pos.mpr phi_pos) 5

theorem w_resonant_ge_one (r : ℝ) : 1 ≤ w_resonant r := by
  unfold w_resonant
  linarith [mul_nonneg (le_of_lt C_lag_pos) (interpolation_cost_nonneg r)]

/-- At resonance, the weight kernel equals 1 (minimum). -/
theorem w_at_resonance (n : ℤ) : w_resonant (n : ℝ) = 1 := by
  unfold w_resonant
  rw [interpolation_cost_zero_at_integer, mul_zero, add_zero]

/-- Off resonance, the weight kernel exceeds 1. -/
theorem w_off_resonance (r : ℝ) (hr : 0 < interpolation_cost r) :
    1 < w_resonant r := by
  unfold w_resonant
  linarith [mul_pos C_lag_pos hr]

What this page does not claim

The theorem does not claim the weight ever reaches its upper bound. The theorem does not connect w_resonant to any measured physical quantity. The theorem does not establish the physical meaning of the ledger or the interpolation cost.

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/EightTickResonance.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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