Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Same-rule Defects on Twisted Cycles

Abstract

The same deterministic rule has a defect in every complemented period.

Theorem 1.1 (A pointwise defect in every period).

Lean statement: D5/S3/TotalVariation/TwistedSameRuleDefect.twisted_same_rule_forces_defect

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

Source. Repository-derived.

Commentary.

For every natural period L with L > 0, every base sign word v : Fin L -> Bool, every natural window length R, and every deterministic table f : (Fin R -> Bool) -> Bool, extend v by complementing its absolute sign after each turn of length L. The resulting relation bits are L-periodic. Apply the same table f to each adjacent R-bit relation window. The defect at t is the XOR of the outputs at t+1 and t with the complement of the relation bit at t+R. Then the defect is true at some t < L, and the sum over t in range L of its zero-one defect indicator is at least one. This is a pointwise statement for the fixed word and rule; it asserts no probability law.

References