Cost Classical Results Complex Norm Exp I Mul

The unit circle

The complex exponential function extends the familiar real exponential to complex numbers. For a real number θ, the value exp(iθ) is a point in the complex plane. The theorem complex_norm_exp_I_mul states that the norm (distance from the origin) of this point is always exactly 1. In symbols: ‖exp(iθ)‖ = 1 for every real θ.

This is the standard fact that exp(iθ) traces the unit circle in the complex plane. It is a classical result from complex analysis, proved by Euler in the 18th century. The theorem is not new physics; it is a formal restatement of a well-known mathematical identity, checked by the machine-checked library of formal theorems.

The declaration is part of a module that records standard mathematical facts as axioms pending full formalization. The theorem itself is proved in the library using the existing fact that the norm of exp(iθ) equals 1. It does not claim anything about the value of θ, the argument of exp(iθ), or any physical quantity. It only establishes the magnitude.

In Recognition Science, this theorem is a small piece of infrastructure. It is used in derivations involving complex exponentials, which appear in the framework's cost functions. The theorem itself is not a statement about recognition, cost, or any physical constant. It is a purely mathematical fact, imported from classical analysis.

The theorem does not claim that exp(iθ) is the only point on the unit circle, nor that the map θ ↦ exp(iθ) is one-to-one. It does not assert any property of the real exponential function. It only states the norm of the complex exponential for purely imaginary arguments.

theorem complex_norm_exp_I_mul (θ : ℝ) : ‖Complex.exp (θ * I)‖ = 1 := by
  simpa using Complex.norm_exp_ofReal_mul_I θ

What this page does not claim

The theorem does not claim that exp(iθ) is the only point on the unit circle. The theorem does not assert any property of the real exponential function. The theorem is not a statement about recognition, cost, or any physical constant.

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/Cost/ClassicalResults.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

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