Foundation Operator Core Complex Structure Forcing Jcost Phase Invariant

Phase symmetry of the cost

In the Recognition Science framework, a ledger (a discrete record of recognition events) represents each of its eight ticks as a point in the complex plane. The framework's cost (the forced expense of recording a recognition) assigns a real number to each tick. The theorem named jcost_phase_invariant states that this total cost is unchanged when every tick is multiplied by the same complex number of unit magnitude, a rotation of the whole signal. In plain terms, the cost of a signal does not depend on its overall phase, only on its shape.

The statement is an abbreviation for a theorem in the framework's machine-checked library of formal theorems. That library proves the invariance as a consequence of the discrete Fourier transform, the standard operation that separates a signal into its frequency components. The transform preserves the inner product that defines the cost, and the phase rotation acts as a symmetry of that inner product. The result is therefore not an assumption; it is derived from the definitions of the transform and the cost.

What the theorem does not claim is that the cost is invariant under arbitrary changes to the signal. It applies only to global phase rotations, not to changes in the relative phases between ticks or to changes in their magnitudes. The theorem also does not say that phase is physically meaningless; it says that the total cost cannot distinguish a signal from its rotated version. The framework's account of phase as a meaningful quantity comes from other statements, not from this one.

The practical consequence is that the framework's cost function respects a symmetry that any reasonable measure of a periodic signal should have. A signal and its rotated copy are the same pattern viewed from a different starting point, and the cost treats them as equal. This invariance is a necessary condition for the cost to serve as a stable measure of recognition, one that does not depend on an arbitrary choice of phase reference.

abbrev jcost_phase_invariant := IndisputableMonolith.Foundation.ComplexStructureForcing.jcost_phase_invariant

abbrev dft8_preserves_inner := IndisputableMonolith.Foundation.ComplexStructureForcing.dft8_preserves_inner

What this page does not claim

The cost is invariant under changes to the relative phases or magnitudes of ticks. The theorem assigns physical meaning to phase; it only establishes a symmetry of the cost. The discrete Fourier transform is the only way to define the cost; it is one of several definitions in the framework.

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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