Foundation Operator Core Complex Structure Forcing Mode Cost Phase Invariant

Phase invariance of mode cost

The declaration mode_cost_phase_invariant is a theorem in the framework's machine-checked library of formal theorems. It states that the cost assigned to a mode of an eight-tick signal does not change when the mode's phase is shifted. In plain terms, the framework models a signal as eight discrete ticks, and a mode is one of the eight frequency components obtained by a discrete Fourier transform. The cost is a number derived from the amplitude of that component, and the theorem proves that sliding the component's phase, meaning changing when its peak occurs within the eight ticks, leaves that number alone.

This is a formal consequence of the framework's definition of cost. The cost function, written JcostC, depends only on the magnitude of the complex amplitude, not on its argument. The phase shift is implemented by an operator that multiplies each component by a unit complex number, which changes the argument but not the magnitude. The theorem is an abbreviation for a longer proof in the library, and it is tagged as a theorem because it is derived from definitions and proved in the machine-checked system.

What the declaration does not claim is broader. It does not say that phase is physically unobservable or that all costs are phase invariant. It concerns only the specific cost function defined in the framework, and only for the eight-tick discrete signal. It does not claim that the framework's cost function is the only possible one, nor that the phase invariance extends to other cost definitions or to continuous signals. The theorem is a structural fact about the framework's model, not a statement about any measured physical system.

abbrev mode_cost_phase_invariant := IndisputableMonolith.Foundation.ComplexStructureForcing.mode_cost_phase_invariant

abbrev jcost_phase_invariant := IndisputableMonolith.Foundation.ComplexStructureForcing.jcost_phase_invariant

What this page does not claim

The theorem does not claim that phase is physically unobservable in any general sense. It does not claim that the cost function is the only possible cost function. It does not claim that the phase invariance holds for continuous signals or other cost definitions.

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

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