Native Scaled Ternary Lucas Units
Abstract
Native ternary Lucas layers have exact three-adic valuations and alternating scaled units.
Let phi be the golden integer with phi squared equal to phi plus one. The integer L_n is goldenLucas n, the trace of phi^n. For every natural j, put A_j=L_(3^j)^2+2. The symbol v_3 denotes padicValInt 3 on these positive integers. Powers have natural exponents.
Theorem 1.1 (Exact depth and signed integer quotient).
Proof. Machine-checked in Lean as D5/S3/Arith/GoldenCubicScaledTernaryUnit.golden_cubic_scaled_ternary_unit (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every natural j at least one, A_j has three-adic valuation j+1. The displayed fraction is exact integer division in Z: 3^(j+1) divides A_j, and the resulting integer has residue (-1)^j modulo three.
The native cubic Lucas identity gives A_(k+1)=A_k(A_k^2-3), starting from A_0=3. Induction constructs an integer u_k with A_k=3^(k+1)u_k and u_k congruent to (-1)^k modulo three. The successor is u_(k+1)=u_k(3(3^k)^2u_k^2-1), so its residue is the negative of the preceding residue. Exact division by the positive power 3^(j+1) recovers this same u_j.
For the valuation, Fibonacci and Lucas doubling give F_(43^j)=F_(3^j)L_(3^j)A_j. The first two factors are nonzero and indivisible by three. The Fibonacci valuation at index 43^j is j+1, hence the valuation of A_j is j+1 as well.