Cost Real Trace Root Mul Dalembert Duplication

The doubling identity

The declaration mulDAlembert_duplication establishes a specific algebraic consequence of a broader functional equation. The equation in question is a multiplicative version of the classical d'Alembert equation, which in its additive form dates to Jean le Rond d'Alembert's 1747 work on vibrating strings. The multiplicative form reads g(x*y) + g(x/y) = 2 * g x * g y for all nonzero x and y, together with the normalization g 1 = 1. The theorem proves that any function satisfying this law must also satisfy the doubling identity g(x*x) = 2 * (g x)^2 - 1 for every nonzero x.

The proof is a short, direct manipulation. Setting y = x in the law gives g(x*x) + g(1) = 2 * (g x)^2, since x/x = 1. The normalization g 1 = 1 then yields the identity by subtracting 1 from both sides. This is a purely algebraic consequence; it requires no continuity, no differentiability, and no assumption about the function beyond the law itself and its value at 1.

The identity matters because it is the first step toward characterizing the function's behavior at products. A companion theorem, mulDAlembert_prod, applies the law to the pair (x*y, x/y) and derives a relation between values at squares and values at products and quotients. Together these results constrain the possible forms of g, but they do not by themselves single out a unique function. The full classification requires additional structure, such as the trace-root construction defined elsewhere in the library, which uses the principal root of X² - t X + 1 = 0 to build explicit solutions.

In Recognition Science, this algebraic lemma feeds into the framework's account of how a forced cost function behaves under composition. The framework models recognition events as entries in a discrete ledger, and the cost of a recognition is forced by five plain conditions. The doubling identity is a small but necessary step in deriving the golden ratio as the unique self-similar scaling, which in turn leads to an eight-tick recognition cycle and, ultimately, to three spatial dimensions. None of that larger chain is proved by this declaration alone; the theorem is a local algebraic fact, not a statement about physics.

What the declaration does not claim is equally important. It does not assert that the law has a unique solution, nor that any particular function satisfies it. It does not mention continuity, differentiability, or any regularity condition. It does not connect the identity to the golden ratio, to the fine-structure constant, or to any empirical measurement. It is a lemma about real-valued functions, proved in a machine-checked library of formal theorems, and its role in the broader framework is one of many such links in a longer chain.

/-- **Multiplicative duplication.** From the product law at `(x, x)` with `g 1 = 1`. -/
theorem mulDAlembert_duplication {g : ℝ → ℝ}
    (hd : ∀ x y, x ≠ 0 → y ≠ 0 → g (x * y) + g (x / y) = 2 * g x * g y)
    (h1 : g 1 = 1) :
    ∀ x, x ≠ 0 → g (x * x) = 2 * (g x) ^ 2 - 1 := by
  intro x hx
  have h := hd x x hx hx
  rw [div_self hx, h1] at h
  linarith

/-- **Multiplicative duplication.** From the product law at `(x, x)` with `g 1 = 1`. -/
theorem mulDAlembert_duplication {g : ℝ → ℝ}
    (hd : ∀ x y, x ≠ 0 → y ≠ 0 → g (x * y) + g (x / y) = 2 * g x * g y)
    (h1 : g 1 = 1) :
    ∀ x, x ≠ 0 → g (x * x) = 2 * (g x) ^ 2 - 1 := by
  intro x hx
  have h := hd x x hx hx
  rw [div_self hx, h1] at h
  linarith

What this page does not claim

The declaration does not prove that the multiplicative d'Alembert law has a unique solution. The declaration does not connect the doubling identity to the golden ratio or to any physical constant. The declaration does not assert any regularity condition such as continuity or differentiability for the function g.

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/RealTraceRoot.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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