Coding Spectrum Fingerprint

Abstract

The first-place Binet main term divided by its decoded value distinguishes three coding systems.

Definition 1.1 (Scale-independent coding fingerprint).

$$codingFingerprint=\frac{leadingMainTerm}{decodedValue}$$

Formalization. D5/S0/Tower/Champions/CodingFingerprint.codingFingerprint (✓ std3).

Source. Repository-derived.

Commentary.

The fingerprint is the leading first-place expansion main term divided by the value decoded from that first place.

Theorem 1.2 (Common rescaling does not change the fingerprint).

$$\forall leadingMainTerm\in R,decodedValue\in R,scale\in R,scale\ne 0\Rightarrow \frac{scale\cdot leadingMainTerm}{scale\cdot decodedValue}=\frac{leadingMainTerm}{decodedValue}$$

Proof. Machine-checked in Lean as D5/S0/Tower/Champions/CodingFingerprint.coding_fingerprint_scale_invariant (✓ std3). ∎

Source. Repository-derived.

Commentary.

Multiplying both the Binet main term and the decoded value by one nonzero scale cancels exactly in the quotient.

Theorem 1.3 (The shifted Tribonacci coefficient is the frozen coefficient times t).

$$\frac{t^2}{3\cdot t^2-2\cdot t-1}=a\cdot t$$

Proof. Machine-checked in Lean as D5/S0/Tower/Champions/CodingFingerprint.tribonacci_binet_normalization_bridge (✓ std3). ∎

Source. Repository-derived.

Commentary.

The derivative-form coefficient for powers t to n minus one equals the frozen coefficient for powers t to n multiplied by t. The proof uses the frozen Tribonacci cubic equation.

Theorem 1.4 (Binary fingerprint).

$$rBinary=1$$

Proof. Machine-checked in Lean as D5/S0/Tower/Champions/CodingFingerprint.binary_coding_fingerprint_value (✓ std3). ∎

Source. Repository-derived.

Commentary.

The first binary main term and its decoded positional weight are both one.

Theorem 1.5 (Zeckendorf fingerprint).

$$rZeckendorf=\frac{{phi}^2}{\mathrm{sqrt}\left(5\right)}$$

Proof. Machine-checked in Lean as D5/S0/Tower/Champions/CodingFingerprint.zeckendorf_coding_fingerprint_value (✓ std3). ∎

Source. Repository-derived.

Commentary.

The first Zeckendorf position decodes to Fib two, hence to one, while its exact Perron main term is phi squared over square root five.

Theorem 1.6 (Tribonacci fingerprint).

$$rTribonacci=\frac{t^2}{3\cdot t^2-2\cdot t-1}\cdot t^2$$

Proof. Machine-checked in Lean as D5/S0/Tower/Champions/CodingFingerprint.tribonacci_coding_fingerprint_value (✓ std3). ∎

Source. Repository-derived.

Commentary.

The occupied first Tribonacci digit decodes to one through the frozen representation carrier, leaving the shifted Binet coefficient times t squared.

Theorem 1.7 (The three coding fingerprints are pairwise distinct).

$$rBinary\ne rZeckendorf\wedge \left(rBinary\ne rTribonacci\wedge rZeckendorf\ne rTribonacci\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/Champions/CodingFingerprint.coding_fingerprint_values_pairwise_distinct (✓ std3). ∎

Source. Repository-derived.

Commentary.

Exact ordered-field estimates give one below the Zeckendorf value, the Zeckendorf value below two, and two below the Tribonacci value.

References

• Truth anchor: D5/S0/Tower/Champions/CodingFingerprint.binary_coding_fingerprint_value
• Truth anchor: D5/S0/Tower/Champions/CodingFingerprint.codingFingerprint
• Truth anchor: D5/S0/Tower/Champions/CodingFingerprint.coding_fingerprint_scale_invariant
• Truth anchor: D5/S0/Tower/Champions/CodingFingerprint.coding_fingerprint_values_pairwise_distinct
• Truth anchor: D5/S0/Tower/Champions/CodingFingerprint.tribonacci_binet_normalization_bridge
• Truth anchor: D5/S0/Tower/Champions/CodingFingerprint.tribonacci_coding_fingerprint_value
• Truth anchor: D5/S0/Tower/Champions/CodingFingerprint.zeckendorf_coding_fingerprint_value
• Dependency: D5/S0/Tower/GoldenNames
• Dependency: D5/S0/Tower/Tribonacci/Binet
• Dependency: D5/S0/Tower/Tribonacci/Representation