Gravity Eight Tick Resonance Interpolation Cost Zero At Integer

The interpolation cost

The interpolation cost is a way to measure how far a real number is from being an integer. For any real number r, the cost is the distance to the nearest integer, written as min(fract r, 1 - fract r), where fract r is the fractional part of r. If r is exactly an integer, the fractional part is zero, so the cost is zero. If r is a half-integer like 3.5, the fractional part is 0.5 and the distance to the nearest integer is also 0.5, the maximum possible value. The cost therefore ranges from 0 at perfect alignment to 1/2 at maximal misalignment.

The theorem interpolation_cost_zero_at_integer states this fact formally: for any integer n, the interpolation cost of n is zero. The proof is a direct calculation using the definition of the fractional part. The same file also proves the cost is never negative and never exceeds 1/2, so the range is exactly the closed interval from 0 to 1/2. These are elementary facts about the distance-to-nearest-integer function, checked line by line in a machine-checked library of formal theorems.

In Recognition Science, this cost appears inside a model of resonance. The framework defines a resonant weight w_resonant(r) = 1 + C_lag * interpolation_cost(r), where C_lag is a positive constant equal to phi^(-5). At an integer ratio, the cost is zero, so the weight equals 1, the minimum possible. Away from integers, the cost is positive, so the weight exceeds 1. The theorem w_at_resonance states this directly: for any integer n, w_resonant(n) = 1. The structure EightTickResonanceCert packages three facts: the weight is minimal at resonance, it exceeds 1 off resonance, and resonance therefore reduces the weight compared to any off-resonance ratio.

The theorem does not claim that real physical systems actually exhibit this resonance, nor that the constant C_lag has any measured value. It establishes only a mathematical property of a defined function. The connection to gravity, if any, is a modeling choice within the framework, not a consequence of this theorem. The theorem also does not say anything about which specific integer ratios are physically preferred; it treats all integers equally as points of zero cost.

/-- At integer ratios, interpolation cost is zero — perfect synchronization. -/
theorem interpolation_cost_zero_at_integer (n : ℤ) :
    interpolation_cost (n : ℝ) = 0 := by
  unfold interpolation_cost
  simp [Int.fract_intCast]

theorem interpolation_cost_nonneg (r : ℝ) : 0 ≤ interpolation_cost r := by
  unfold interpolation_cost
  exact le_min (Int.fract_nonneg r) (by linarith [Int.fract_lt_one r])

theorem interpolation_cost_le_half (r : ℝ) : interpolation_cost r ≤ 1/2 := by
  unfold interpolation_cost
  rcases le_or_gt (Int.fract r) (1/2) with h | h
  · exact min_le_of_left_le h
  · exact min_le_of_right_le (by linarith [Int.fract_lt_one 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]

What this page does not claim

The theorem does not claim that any physical system actually exhibits this resonance behavior. The theorem does not claim that the constant C_lag has a measured or empirically verified value. The theorem does not claim that any particular integer ratio is physically preferred over another.

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)

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: