Cost Classical Results Complex Exp Mul Rearrange

The rearrangement identity

The declaration complex_exp_mul_rearrange in the Recognition Science library establishes a specific algebraic identity about complex exponentials. In plain language, it says that the exponential of a sum of two complex numbers equals the product of the exponentials of those two numbers, even when each number is split into a real part and an imaginary part. Concretely, for any real numbers c₁, c₂, φ₁, φ₂, the identity states that exp(−(c₁+c₂)/2) · exp((φ₁+φ₂)i) equals [exp(−c₁/2) · exp(φ₁i)] · [exp(−c₂/2) · exp(φ₂i)]. This is a standard property of the complex exponential function, often written as exp(a+b) = exp(a)exp(b), applied here to the particular combination of real and imaginary components that arise in the framework's cost calculations.

This result is tagged as a theorem in the library, meaning it has been formally verified by a machine-checked proof assistant. The proof is short: it rewrites the left-hand side using the addition formula for exponentials, then rearranges the terms using the commutativity of multiplication and the fact that real numbers embed into complex numbers. The declaration is one of several classical mathematical results that the library states as axioms pending full formalization, but this particular identity has been fully proved, not assumed. The library's documentation explicitly distinguishes these proved results from other textbook facts that are declared as axioms because they would require substantial infrastructure to formalize from scratch.

What the declaration does not claim is important. It does not assert anything about the physical content of the cost function, the recognition ledger, or the forcing chain that derives physical constants. It is purely a mathematical lemma about complex numbers. It does not claim that the exponential function is the only function satisfying some property, nor does it make any statement about the uniqueness of the cost function J(x) = (x + 1/x)/2 − 1. The identity is a tool, not a conclusion. It is used in the framework's calculations to manipulate expressions involving complex exponentials, but it carries no physical or metaphysical weight on its own.

The library also contains a note about a related declaration that was removed because it was mathematically false: the claim that a function continuous on (0, ∞) must have a continuous extension to 0. The counterexample is sin(1/x), which is continuous on the positive reals but has no continuous limit at 0. This removal shows the library's commitment to correctness: false statements are deleted, not kept. The rearrangement identity, by contrast, is true and proved, and it remains a small but solid piece of the framework's formal foundation.

theorem complex_exp_mul_rearrange :
    ∀ (c₁ c₂ φ₁ φ₂ : ℝ),
      Complex.exp (-(c₁+c₂)/2) * Complex.exp ((φ₁+φ₂) * I) =
      (Complex.exp (-c₁/2) * Complex.exp (φ₁ * I)) * (Complex.exp (-c₂/2) * Complex.exp (φ₂ * I)) :=
  exp_mul_rearrange

private lemma exp_mul_rearrange (c₁ c₂ φ₁ φ₂ : ℝ) :
    Complex.exp (-(c₁+c₂)/2) * Complex.exp ((φ₁+φ₂) * I) =
    (Complex.exp (-c₁/2) * Complex.exp (φ₁ * I)) * (Complex.exp (-c₂/2) * Complex.exp (φ₂ * I)) := by
  rw [← Complex.exp_add, ← Complex.exp_add, ← Complex.exp_add, ← Complex.exp_add]
  congr 1
  push_cast
  ring

What this page does not claim

The declaration does not make any physical claim about the cost function or the forcing chain. It does not assert uniqueness of the cost function or any property of the exponential beyond the standard addition formula. It does not imply that all classical results in the library are proved; many remain axioms pending formalization.

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.

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