Conjugate Valuation

Abstract

An exact code read at the conjugate is driven by the same digits, so repeating digits leave the conjugate orbit only finitely many values.

The greedy expansion of one is already carried exactly, as integer pairs standing for a value in the ring generated by the base, together with the multiplication that the base’s own quadratic relation reduces. Nothing of that is rebuilt here.

What is added is a second reading of the same pairs, at the conjugate root rather than at the base. The reduction inside the multiplication is the base’s relation, and the conjugate satisfies the same relation, so the reading intertwines with it. That is the whole content: the conjugate orbit is not a separate object to be tracked, it is the same integer stream read elsewhere, and it obeys the same digits.

Two consequences follow without further work. Once the codes repeat, so does the conjugate reading; and then every value the conjugate orbit ever takes is one it already took before the period closed. That is boundedness in the form the escape estimate can be set against, and it needs no measure of how large those values are.

Theorem 1.1 (The conjugate reading of the expansion).

$$\forall n\in N,\mathrm{conjugateRemainder}\left(n+1\right)=betaThirteenConjugate\cdot \mathrm{conjugateRemainder}\left(n\right)-beta13GreedyDigit\left(n\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/NonPisotFrontier/ConjugateValuation.conjugate_reading_of_the_expansion (✓ std3). ∎

Source. Repository-derived.

Commentary.

The displayed half is the recurrence; the second half of the statement is the collapse to finitely many values under repeating codes. Neither says the digits do repeat, and neither supplies the contradiction; they supply the side of it that periodicity would force.

References

• Truth anchor: D5/S0/Tower/NonPisotFrontier/ConjugateValuation.conjugate_reading_of_the_expansion
• Dependency: D5/S0/Tower/NonPisot/Beta13Infinite
• Dependency: D5/S0/Tower/NonPisotFrontier/BaseIdentification