D-Bonacci Values

Abstract

D-bonacci names acquire real values from the order-d Perron root.

A true digit in position i contributes beta_d to the negative power i+1. The prefix enumeration follows the same finite run-budget split that counts admissible names, so its order is canonical rather than chosen.

Definition 1.1 (D-bonacci name value).

$$\forall d\in N,\forall Q\in N,\mathrm{dbonacciNameValue}\left(d,Q\right)$$

Formalization. D5/S0/Tower/DBonacci/Values.dbonacciNameValue (✓ std3).

Source. Repository-derived.

Commentary.

The value is the finite base-beta_d sum over the true positions of an admissible Boolean word.

Definition 1.2 (Indexed d-bonacci name value).

$$\forall d\in N,\forall Q\in N,\mathrm{indexedNameValue}\left(d,Q\right)$$

Formalization. D5/S0/Tower/DBonacci/Values.indexedNameValue (✓ std3).

Source. Repository-derived.

Commentary.

The recursive equivalence lists false-prefix names before true-prefix names at every run-budget state.

Theorem 1.3 (Order-three values agree with Tribonacci).

$$\forall Q\in N,\mathrm{dbonacciNameValue}\left(3,Q,word\right)=\mathrm{tribonacciNameValue}\left(Q,word\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/DBonacci/Values.dbonacciNameValue_three_eq_tribonacciNameValue (✓ std3). ∎

Source. Repository-derived.

Commentary.

The already proved identity beta_3=t makes the two word sums equal term by term; the bridge therefore preserves the underlying word.

References

• Truth anchor: D5/S0/Tower/DBonacci/Values.dbonacciNameValue
• Truth anchor: D5/S0/Tower/DBonacci/Values.dbonacciNameValue_three_eq_tribonacciNameValue
• Truth anchor: D5/S0/Tower/DBonacci/Values.indexedNameValue
• Dependency: D5/S0/Tower/DBonacci/Names
• Dependency: D5/S0/Tower/DBonacci/PerronRoot