Gap Endpoints

Abstract

The three gap right endpoints form a transition cycle of period three, each coordinate being exactly its own gap’s length.

The cycle was found while reconciling two enumeration counts that disagreed. The combinatorial closed-itinerary count and a real-coordinate filter differed by exactly three at period nine, and the three words were the rotations of this cycle. The middle step is where the Tribonacci relation enters: the image coordinate is the square of the constant less the constant less one, which is the constant’s inverse, which is the small gap’s length.

Theorem 1.1 (The gap endpoints form a three-cycle).

$$\left(\mathrm{tribonacciPeriodicTransition}\left(largeEndpoint\right)=combinedEndpoint\wedge \mathrm{tribonacciPeriodicTransition}\left(combinedEndpoint\right)=smallEndpoint\right)\wedge \mathrm{tribonacciPeriodicTransition}\left(smallEndpoint\right)=largeEndpoint$$

Proof. Machine-checked in Lean as D5/S0/Tower/EndpointCycle/GapEndpoints.gap_endpoints_form_a_three_cycle (✓ std3). ∎

Source. Repository-derived.

Commentary.

A state on this cycle sits on the boundary of its gap, so whether it belongs is a matter of taking gaps closed or half open, not a matter of computing more precisely. Since the cycle has period three it recurs exactly at periods divisible by three, which is where the counts were observed to be unstable.

References

• Truth anchor: D5/S0/Tower/EndpointCycle/GapEndpoints.gap_endpoints_form_a_three_cycle
• Dependency: D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration