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Quantitative Tribonacci Conjugate Separation

Abstract

Distinct admissible Tribonacci digit evaluations separate uniformly at their natural conjugate scale.

Let beta be the existing real Tribonacci constant and let z be any member of the characteristic root multiset other than beta. For every natural Q and every two distinct length-Q admissible names u and v, their complex coordinates have distance at least sqrt(beta - 1) times norm(z)^Q. The constant is positive and independent of Q. Both non-Perron roots are covered.

Theorem 1.1 (Uniform separation of all finite admissible layers).

Lean statement: D5/S0/Tower/Tribonacci/Separation.conjugateCoordinate_separation

Proof. Machine-checked in Lean as D5/S0/Tower/Tribonacci/Separation.conjugateCoordinate_separation (✓ std3). ∎

Source. Repository-derived.

Commentary.

The argument applies the classical Pisot algebraic-norm method; no research novelty is claimed. The existing general multinacci irreducibility theorem is reused directly. To establish nonzero Perron evaluation of the difference polynomial, reversal is proved to preserve no-111 admissibility, and its positive-power polynomial evaluation is related exactly to the frozen negative-power name value. The natural-number decoder is not this polynomial evaluation.

The integer resultant of the Tribonacci cubic and the digit difference is nonzero. At the three embeddings it is the Perron evaluation times the squared complex norm. Its absolute value is therefore at least one. A geometric-sum bound on the Perron factor and beta times norm(z)^2 = 1 yield the stated quantitative bound.

This is an infinite-family arithmetic estimate for every length and every admissible pair. It is consumed by the positive-area packing and synchronized-cluster proof of Tribonacci interior.

References