Binary Geometric Baseline

Abstract

The binary geometric baseline has order one, a one-dimensional solution space, one characteristic root, and fingerprint one.

Theorem 1.1 (The binary geometric recurrence is first order).

$$\mathrm{order}\left(Rbinary\right)=1\wedge \mathrm{IsSolution}\left(Rbinary,{\left(2^n\right)}_{n\in N}\right)$$

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

Source. Repository-derived.

Commentary.

The recurrence has order one, and the geometric sequence with nth term two to the n is a solution. Thus each next term depends only on the immediately preceding term.

Theorem 1.2 (The binary recurrence solution space is one-dimensional).

$$\dim \left(C,\mathrm{Sol}\left(Rbinary\right)\right)=1$$

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

Source. Repository-derived.

Commentary.

The standard initial-value basis identifies the solution space with one complex initial coordinate, so its finite dimension is one.

Theorem 1.3 (The binary characteristic polynomial has exactly one root).

$$\mathrm{roots}\left(\mathrm{charPoly}\left(Rbinary\right)\right)=\left\{2\right\}$$

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

Source. Repository-derived.

Commentary.

The characteristic polynomial is X minus two, and its root multiset is the singleton containing two. This is the formal no-hidden-face assertion.

Theorem 1.4 (Binary baseline package).

$$\left(\mathrm{order}\left(Rbinary\right)=1\wedge \mathrm{IsSolution}\left(Rbinary,{\left(2^n\right)}_{n\in N}\right)\right)\wedge \left(\dim \left(C,\mathrm{Sol}\left(Rbinary\right)\right)=1\wedge \left(\mathrm{roots}\left(\mathrm{charPoly}\left(Rbinary\right)\right)=\left\{2\right\}\wedge binaryCodingFingerprint=1\right)\right)$$

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

Source. Repository-derived.

Commentary.

The order-one recurrence, one-dimensional solution space, and singleton characteristic root are conjoined with the frozen binary coding fingerprint value one.

The singleton root is the precise no-hidden-face content. The source phrase collapse back to zeta itself is not formalized: the imported S0 interfaces provide no corresponding zeta-layer object or two-sided construction, so no zeta claim appears in this package.

References

• Truth anchor: D5/S0/Tower/Champions/BinaryBaseline.binary_baseline_package
• Truth anchor: D5/S0/Tower/Champions/BinaryBaseline.binary_characteristic_roots
• Truth anchor: D5/S0/Tower/Champions/BinaryBaseline.binary_geometric_recurrence_first_order
• Truth anchor: D5/S0/Tower/Champions/BinaryBaseline.binary_recurrence_solution_space_finrank
• Dependency: D5/S0/Tower/Champions/CodingFingerprint