Holography Circle Correlator Cos Witness Even

The cosine witness

In mathematics, an even function is one that satisfies f(-x) = f(x) for every input x. The ordinary cosine function is the classic example: cos(-θ) equals cos(θ) for every angle θ. This property means the function's graph is symmetric about the vertical axis, a fact known since the earliest tables of chords in ancient Greek astronomy.

The declaration cosWitness_even in the framework's machine-checked library of formal theorems establishes precisely this property for a scaled cosine used in a physical context. It proves that for any real numbers beta and tau, the expression cos(2π(-τ)/beta) equals cos(2πτ/beta). The proof is immediate: multiplying the argument by -1 changes the sign inside the cosine, and the cosine function's evenness absorbs that sign. The declaration is a theorem, not an assumption, and it is checked by a computer.

In Recognition Science, this cosine is a witness, a concrete example proving that a proposed set of conditions is not vacuous. The conditions in question describe a seam correlator, a function on a circle that represents the thermal two-point function of a quantum field. The framework's core result, named periodic_even_reflection, states that any function which is both periodic (returns to its value after one full circuit of the circle) and even (symmetric under reflection) must also be symmetric about the half-period point, meaning C(β - τ) = C(τ). The cosine function satisfies both premises, and cosWitness_even is the formal certificate for the evenness half of that requirement.

The importance of this witness is that it rules out a trivial reading of the framework's central claim. Without an example, one might suspect that the premises of periodicity and evenness could only be satisfied by constant functions, making the reflection theorem an empty statement. The cosine witness is genuinely τ-dependent, so it demonstrates that the theorem applies to a real, non-trivial family of functions. This is a standard move in mathematics: before proving a general theorem, one checks that its hypotheses are actually satisfiable.

What cosWitness_even does not claim is any physical content about the actual universe. It does not assert that any real physical correlator is even, nor does it derive the value of any coupling constant or mass. It is purely a logical statement about a specific mathematical function. The framework's broader project of deriving physics from recognition costs is a separate matter; this declaration is a small, self-contained step within that project, establishing that one particular example fits the required shape.

theorem cosWitness_even (beta τ : ℝ) : cosWitness beta (-τ) = cosWitness beta τ := by
  unfold cosWitness
  rw [show 2 * Real.pi * (-τ) / beta = -(2 * Real.pi * τ / beta) by ring, Real.cos_neg]

theorem cosWitness_even (beta τ : ℝ) : cosWitness beta (-τ) = cosWitness beta τ := by
  unfold cosWitness
  rw [show 2 * Real.pi * (-τ) / beta = -(2 * Real.pi * τ / beta) by ring, Real.cos_neg]

theorem cosWitness_periodic (beta : ℝ) (hbeta : beta ≠ 0) :
    Function.Periodic (cosWitness beta) beta := by
  intro τ
  unfold cosWitness
  have harg : 2 * Real.pi * (τ + beta) / beta = 2 * Real.pi * τ / beta + 2 * Real.pi := by
    field_simp
  rw [harg, Real.cos_add_two_pi]

What this page does not claim

This declaration does not assert that any real physical correlator is even; it only proves a property of a specific mathematical function. This declaration does not derive any physical constant or coupling. This declaration does not prove the reflection theorem itself; it only provides an example satisfying its premises.

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/Holography/CircleCorrelator.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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