Gravity Eight Tick Resonance Weight Reduction At Resonance

Weight reduction at resonance

The declaration weight_reduction_at_resonance is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states a simple inequality about a quantity the framework calls weight, a positive real number that the framework models as a measure of a process's cost in its ledger, a discrete record of events. The theorem says that when a frequency ratio is an integer, the weight equals 1, and when the ratio is not an integer, the weight is strictly greater than 1.

The framework defines the weight as a function of a frequency ratio. The ratio is the frequency of some periodic process divided by the frequency of a base clock. The function is w(r) = 1 + C * d(r), where C is a positive constant and d(r) is the distance from r to the nearest integer. The distance is 0 at integers and 1/2 at half-integers. The theorem w_at_resonance proves that at an integer, the weight is exactly 1. The theorem w_off_resonance proves that when the distance is positive, the weight is greater than 1. The declaration weight_reduction_at_resonance combines these two results.

The statement is a formal consequence of the definitions. It does not depend on any empirical measurement or on any unproved assumption about the physical world. The framework's library proves the inequality from the definitions of w_resonant and interpolation_cost using standard real-number arithmetic. The theorem is part of a larger structure, EightTickResonanceCert, which packages three related results: the weight is minimal at resonance, it exceeds 1 off resonance, and the weight at resonance is less than the weight off resonance.

The theorem does not claim that any physical object actually has a lower weight at resonance. It is a statement about a mathematical model. The framework models gravity as a ledger of recognition events, and the weight is a cost assigned to a process. The theorem shows that within this model, a process synchronized with the ledger clock has the lowest possible cost. The framework's resonant_frequency definition gives a formula for frequencies that are integer multiples of a base frequency divided by powers of the golden ratio, but the theorem itself does not identify any specific physical frequency.

/-- WEIGHT REDUCTION AT RESONANCE: An object at a resonant frequency
    has strictly lower effective weight than one at a non-resonant frequency. -/
theorem weight_reduction_at_resonance (n : ℤ) (r_off : ℝ)
    (hr_off : 0 < interpolation_cost r_off) :
    w_resonant (n : ℝ) < w_resonant r_off := by
  rw [w_at_resonance]
  exact w_off_resonance r_off hr_off

/-- The resonance-aware ILG weight kernel.
    w(r) = 1 + C_lag · interpolation_cost(r)
    At resonance (integer r): w = 1 (minimum).
    Off resonance: w > 1. -/
def w_resonant (r : ℝ) : ℝ :=
  1 + C_lag * interpolation_cost r

theorem eight_tick_resonance_certified : EightTickResonanceCert where
  minimum_at_resonance := w_at_resonance
  exceeds_off_resonance := w_off_resonance
  resonance_reduces_weight := weight_reduction_at_resonance

What this page does not claim

The theorem does not state that any physical object has a lower weight at resonance. The theorem does not identify any specific physical frequency as resonant. The theorem does not derive the value of the constant C from first principles.

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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