Tribonacci Periodic Generator

Abstract

Five certified branches and exact cubic arithmetic generate every periodic fixed-point equation.

The three normalized gap types have one, two, and two legal outgoing branches. Affine compositions are evaluated exactly in Q(t), using t cubed equal to t squared plus t plus one.

Theorem 1.1 (Branch targets match the frozen substitution).

$$\forall g\in TribonacciGap,\mathrm{mapTargets}\left(\mathrm{tribonacciStepsFrom}\left(g\right)\right)=\mathrm{gapLetterSubstitution}\left(g\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicGenerator.tribonacci_steps_from_targets (✓ std3). ∎

Source. Repository-derived.

Commentary.

Mapping each legal edge to its target gap gives exactly the frozen three-letter Tribonacci gap substitution.

Theorem 1.2 (Periodic points return to generated cubic codes).

$$\forall p\in N,s\in TribonacciPeriodicState,\left(\mathrm{iterate}\left(tribonacciPeriodicTransition,p,s\right)=s\wedge \mathrm{fixedPointDenominator}\left(p,s\right)\ne 0\right)\Rightarrow \left(\exists c\in TribonacciCubicCode,c\in \mathrm{tribonacciFixedPointCodes}\left(p\right)\wedge s=\mathrm{decodeTribonacciState}\left(c\right)\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicGenerator.tribonacci_periodic_point_enumeration_complete (✓ std3). ∎

Source. Repository-derived.

Commentary.

Reading the actual legal branch at every iterate constructs a closed symbolic word. When its exact fixed-point denominator is nonzero, the original real state is the decoding of the generated code.

References

• Truth anchor: D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicGenerator.tribonacci_periodic_point_enumeration_complete
• Truth anchor: D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicGenerator.tribonacci_steps_from_targets
• Dependency: D5/S0/Tower/DBonacci/OrbitAlgebra