Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Twisted Rotation and Parry Window Defects

Abstract

Complemented rotation preserves full path weights and forces a uniform same-rule defect bound.

Theorem 1.1 (Uniform bound for each fixed deterministic rule).

Lean statement: D5/S3/TotalVariation/TwistedRotationDefect.parry_window_lower_bound

Proof. Machine-checked in Lean as D5/S3/TotalVariation/TwistedRotationDefect.parry_window_lower_bound (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every natural k >= 2, R >= 1 and G >= 1, and every fixed deterministic table f : (Fin R -> Bool) -> Bool, put L = R+1+G. The actual twisted prefix defect probability twistedDefect k R G f is at least 1/L. The actual stationary Parry defect probability stationaryDefect k R f is at least 1/L - 2L(3/4)^(G/3) - Real.goldenRatio^(2-L). Here G/3 is natural division and 2-L is an integer exponent. The same table is applied to both adjacent R-bit relation windows. Complete signed state prefixes have the existing twistedLaw distribution: full transition products are divided by loopMass, with no initial stationary factor. Splitting a full path into its prefix and closing continuation identifies the window event exactly, including windows that cross the complemented seam after rotation. Each signed cycle has a same-rule defect; rotation makes the weighted expectations equal. The stationary comparison then gives the second inequality, without a small-error assumption. Moreover, there exists a deterministic table fmin whose actual stationary defect is no greater than that of any table of the same window length, and its attained minimum satisfies the same lower bound.

References