Holography Seam Cycle Carrier Holonomy Correlator Reflection

The reflection symmetry

In mathematics and physics, a two-point function measures how two events or insertions are correlated as a function of their separation. For systems on a circle, such a function is periodic: shifting the separation by the circle's circumference leaves it unchanged. A stronger property, reflection symmetry, means the function looks the same whether you read time forward or backward from the middle of the period. The classical example is the cosine function, which satisfies cos(β − τ) = cos(τ) for any period β.

The relevant object here is a cycle flow: a one-parameter family of translations that composes additively, has an identity at time zero, and returns to its starting point after one full period. When the flow also carries a symmetric pairing between two insertions, the resulting correlator, defined as the pairing of one insertion with the other after time τ, is automatically periodic and even. The evenness follows because translating one insertion forward by τ equals translating the other backward by τ, and the pairing is unordered. This is a general structural fact, not a special choice of function.

The declaration holonomy_correlator_reflection instantiates this general fact for a specific flow on the complex plane: the holonomy flow U(t)z = exp(iκt)·z, where κ is a positive real number and the period is the deficit-free return time 2π/κ. The theorem states that for any complex number z and any real time τ, the correlator satisfies C(β − τ) = C(τ), where β is that period. In closed form, the correlator evaluates to |z|²·cos(κτ), which visibly has the reflection symmetry. The theorem is machine-checked in the framework's library, meaning its proof has been verified by a computer.

What the theorem does not claim is that this holonomy flow is the correct physical description of any real system. The identification of the physical seam correlator with this pairing correlator remains a modeling premise, not a proved statement. The theorem establishes a symmetry property of a mathematical structure; it does not establish that the structure itself is the one nature uses. That identification is a separate, unproved assumption, and the theorem's status as a proof does not extend to it.

The consequence is that reflection symmetry, a property often assumed as a separate analytic premise in thermal systems, here emerges from the more basic structure of a cycle flow with a symmetric pairing. This reduces the number of independent assumptions needed to derive the symmetry, and it links the symmetry to the same reciprocity principle that forces the unique recognition cost J(x) = (x + 1/x)/2 − 1. The theorem does not, however, prove that any physical system realizes this flow, nor that the downstream consequences such as detailed balance or the Bekenstein bound follow.

/-- **The instantiated headline: reflection symmetry of the U(1) seam correlator at
the deficit-free period (THEOREM, no premises beyond `κ > 0`).** Carrier → periodic
+ even → reflection, fully discharged on the forced U(1) carrier. -/
theorem holonomy_correlator_reflection (kappa : ℝ) (hk : 0 < kappa) (z : ℂ) :
    ∀ τ, (holonomyFlow kappa hk).correlator z (euclideanPeriod kappa - τ)
        = (holonomyFlow kappa hk).correlator z τ :=
  (holonomyFlow kappa hk).reflection z

/-- **Closed form of the holonomy correlator: `C(τ) = ‖z‖²·cos(κτ)`.** The lowest
Matsubara harmonic emerges from the flow. -/
theorem holonomyFlow_correlator (kappa : ℝ) (hk : 0 < kappa) (z : ℂ) (τ : ℝ) :
    (holonomyFlow kappa hk).correlator z τ
      = Complex.normSq z * Real.cos (kappa * τ) := by
  show (z * (starRingEnd ℂ) (holonomy kappa τ * z)).re = _
  have hmain : z * (starRingEnd ℂ) (holonomy kappa τ * z)
      = (starRingEnd ℂ) (holonomy kappa τ) * (z * (starRingEnd ℂ) z) := by
    rw [map_mul]; ring
  rw [hmain, Complex.mul_conj, holonomy_conj]
  simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, mul_zero, sub_zero]
  rw [holonomy_re, mul_neg, Real.cos_neg]
  ring

/-- **The instantiated headline: reflection symmetry of the U(1) seam correlator at
the deficit-free period (THEOREM, no premises beyond `κ > 0`).** Carrier → periodic
+ even → reflection, fully discharged on the forced U(1) carrier. -/
theorem holonomy_correlator_reflection (kappa : ℝ) (hk : 0 < kappa) (z : ℂ) :
    ∀ τ, (holonomyFlow kappa hk).correlator z (euclideanPeriod kappa - τ)
        = (holonomyFlow kappa hk).correlator z τ :=
  (holonomyFlow kappa hk).reflection z

What this page does not claim

The holonomy flow is the correct physical description of any real system. The carrier identification, that the physical seam correlator equals this pairing correlator, is proved. Downstream consequences such as detailed balance or the Bekenstein bound follow from this theorem alone.

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/SeamCycleCarrier.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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