Unique binary digits · strictly increasing powers · full unique bit length · Constructive arithmetic

Constructive binary length and powers of two

BitLen(0,1) · n>0 ⇒ 2^(ℓ−1)≤n<2^ℓ · ∀n ∃!ℓ BitLen(n,ℓ)

Twenty-one independently checked constructive theorems prove unique binary digit decomposition, strict power-of-two growth, and existence and uniqueness of the exact canonical bit length for every natural number.

Open the definition-aware map Read the final theorem See campaign milestone G102

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact certificate

Fully expanded arithmetic

Inspect all 542 native tactic lines and 50 actual proof prerequisites with every definition fully expanded.

Open the exact edition →

Focused route

Final dependency cone

Start at theorem BL0014 and follow only the lemmas and conservative definitions supporting binary_length_exists_unique.

Trace prerequisites →
Zoom between mathematical scales: complete research atlas → research domain → proof family → G101 milestone · G102 milestone → theorem and definition dependencies.
Major independently established statements: BL0011 binary_length_exists · BL0013 binary_length_functional · BL0015 binary_length_power_exact · BL0014 binary_length_exists_unique.
Independently verified Alpha v34 checked-use theorem family: 21 dependency-curried kernel-checked theorem bodies · 50 proof prerequisites · 10 linked definitions · 11 definition-dependency arrows · 542 exact tactic lines · first admitted v22 · not Stable. The unchanged intuitionistic kernel and separately compiled Lean verifier independently accept all 240 bundle nodes; SHA-256 95e5f8a3baef113721d748f9d7071864b4bf9511737a27a1272d2695428fb938.
Exact mathematical boundary: G101 and G102 were OPEN when these BitLen foundations were first admitted in Alpha v22. Both are now CLOSED in Alpha v23: complete canonical exponent digits and both exact logarithmic execution bounds are proved.