most times your are right.but some older version of sh(like ash , that is usually not Posix compliant on many systems) your syntax is invalid.
[[ *maybe it is various of evulation expression bug of a ash.]]
(so its meant some older versions of (bourne) sh(s) like (0.38) has it, and i cannot test newer version ash(0.4 or newer if there is) so ash is not compability to SUSv3.

actually aritmetic expansion is defined for SUSv3
`Arithmetic Precision and Operations`
in Introduction
`2.6.4 Arithmetic Expansion`
in Shell Command Language]

now lets look at the belows..
[not expanded values before aritmetic expression]

Code:

# a=2;b=1;echo $((a+b))  
arith: syntax error: "a+b"

it is required a `$` before variables

Code:

# a=2;b=1;echo $(($a+$b))  
3

in sh another interesting issue

Code:

# unset x
# x=1

--try 1

Code:

# echo $(( $x=5 ))  
arith: syntax error: " 1=5 "

--try 2

Code:

# echo $(( $x=5 ))   
arith: syntax error: " 1=5 "

--try 3 is like OK , but?

Code:

# echo $(( $x=5 ))
5

but is not.

Code:

# echo $x
1

regards
ygemici