Multiplicative digital root

In number theory, the multiplicative digital root of a natural number formula_1 in a given number base formula_2 is found by multiplying the digits of formula_1 together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of formula_1. Multiplicative digital roots are the multiplicative equivalent of digital roots.
Definition.
Let formula_1 be a natural number. We define the digit product for base formula_6 formula_7 to be the following:
where formula_9 is the number of digits in the number in base formula_2, and
is the value of each digit of the number. A natural number formula_1 is a multiplicative digital root if it is a fixed point for formula_13, which occurs if formula_14. 
For example, in base formula_15, 0 is the multiplicative digital root of 9876, as 
All natural numbers formula_1 are preperiodic points for formula_13, regardless of the base. This is because if formula_21, then
and therefore
If formula_24, then trivially 
Therefore, the only possible multiplicative digital roots are the natural numbers formula_26, and there are no cycles other than the fixed points of formula_26. 
Multiplicative persistence.
The number of iterations formula_28 needed for formula_29 to reach a fixed point is the multiplicative persistence of formula_1. The multiplicative persistence is undefined if it never reaches a fixed point.
In base 10, it is conjectured that there is no number with a multiplicative persistence formula_31: this is known to be true for numbers formula_32. The smallest numbers with persistence 0, 1, ... are:
The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be sorted, and, except for the first two digits, all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits.
Based on these restrictions, the number of candidates for formula_33-digit numbers with record-breaking persistence is only proportional to the square of formula_33, a tiny fraction of all possible formula_33-digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.
Extension to negative integers.
The multiplicative digital root can be extended to the negative integers by use of a signed-digit representation to represent each integer.
Programming example.
The example below implements the digit product described in the definition above to search for multiplicative digital roots and multiplicative persistences in Python.
def digit_product(x: int, b: int) -> int:
 if x == 0:
 return 0
 total = 1
 while x > 1:
 if x % b == 0:
 return 0
 if x % b > 1:
 total = total * (x % b)
 x = x // b
 return total
def multiplicative_digital_root(x: int, b :int) -> int:
 seen = []
 while x not in seen:
 seen.append(x)
 x = digit_product(x, b)
 return x
def multiplicative_persistence(x: int, b: int) -> int:
 seen = []
 while x not in seen:
 seen.append(x)
 x = digit_product(x, b)
 return len(seen) - 1