Lossless compression by arithmetic — proven, not learned
What if every binary number had a path
atleast one bit shorter — guaranteed by the math?
Lossless compression has always depended on learned patterns — Huffman trees, LZ dictionaries, statistical models trained on the data itself. Binary Equation Paths is fundamentally different. Standard binary representation of n requires ⌈log₂(n+1)⌉ bits — equivalently, ⌊log₂ n⌋ + 1. BEP produces at most ⌊log₂ n⌋ bits — at least one bit shorter than the standard form, every time. The savings begins with the implicit MSB that binary must store but BEP derives. No dictionaries, no training, no context. The mathematics came from one small but consequential modification to a famous unsolved problem.
The starting point — Collatz, 1937
A useful but broken tool
n / 2 if n is even
3n + 1 if n is odd
3n + 1 if n is odd
The Collatz Conjecture describes a path any integer can walk toward 1 — but the odd step
multiplies n by 3, causing unpredictable growth. It has never been proven to always converge,
making it unusable as a reliable compression mechanism.
❌ unpredictable — unproven
Binary Equation Paths — 2026
One change. A guaranteed path.
n / 2 if n is even
(n − 1) / 2 if n is odd
(n − 1) / 2 if n is odd
Replace multiply with subtract. The value strictly decreases every step, convergence is
guaranteed in exactly ⌊log₂ n⌋ steps, and those steps become the compressed binary —
one bit shorter than n’s standard form, fully reversible, requiring no external data.
✓ proven — converges in ⌊log₂ n⌋ steps
Standard binary stores ⌊log₂ n⌋ + 1 bits.
BEP stores only floor(log₂ n) — atleast one less, every time.
Pure arithmetic — and it’s open to everyone.
BEP stores only floor(log₂ n) — atleast one less, every time.
Pure arithmetic — and it’s open to everyone.
— New Dawn Data (2026)