The finite 2-adic complexity of a finite binary sequence measures the minimum size of a rational number
representing its digits modulo a power
of 2. For a string of 
bits 
, 
, ..., 
, let
 |
(1)
|
Thus 
is the least significant bit. Define the
rational complexity
 |
(2)
|
where the minimum is over integers 
and 
with 
positive and odd. The finite 2-adic complexity is then the binary
logarithm
 |
(3)
|
This convention uses the unrounded logarithm. The odd denominator makes
a p-adic integer for 
, and the congruence fixes
its first 
binary digits (Lin et al. 2026).
For example, when 
, the residues 0, 1, 2, and 3 have respective values 
of 1, 1, 2, and 1.
Lin et al. (2026, Question 3.1) asked about the mean and variance of this complexity for uniformly chosen strings. With
each 
in 
equally likely, the expectation value is
![mu_N=E[L_N]=2^(-N)sum_(r=0)^(2^N-1)log_2R_N(r),](/images/equations/Finite2-AdicComplexity/NumberedEquation4.svg) |
(4)
|
and the variance is
 |
(5)
|
For example, 
and 
.
Babanskyy (2026) reported exact finite distribution formulas and the bound
 |
(6)
|
for every 
,
together with the asymptotic formula
 |
(7)
|
where 
is as in little-O notation, 
is the Riemann zeta
function, and 
is the natural logarithm.
The reported limiting variance is
 |
(8)
|
The author reports assistance from OpenAI Codex and GPT-6.1 Sol. As of Oct. 5, 2026, finite certificate checks for 
through 14 had been reproduced, but independent specialist
review of the arguments for all 
and the limiting variance had
not been reported. No Lean formalization was supplied (Babanskyy 2026, VibeMathed
2026).
