NOTE:
I cannot answer in mathematical terms, but I can venture a physical and philosophical answer. The fifth dimension involves extension into and beyond 4-d. Thus, it involves not just the possibility, but the guarantee of two time dimensions. A general way to extend dimensions is to think first of possibility, and then in the next dimension as reality.
For example,
1-d sometimes exists.
2-d is virtual but always exists.
3-d exists and is never virtual and is not fully aware of time.
4-d exists and is never virtual and controls time.
5-d exists and is never virtual and controls time and potentially has two fundamental types of matter.
6-d exists and is never virtual and controls time and always has two fundamental types of matter.
7-d exists and is never virtual and controls time and always has two fundamental types of matter, and potentially has two concepts of time.
8-d exists and is never virtual and controls time and always has two fundamental types of matter, and always has two concepts of time, and potentially has some other thing.
9-d exists and is never virtual and controls time and always has two fundamental types of matter, and always has two concepts of time, and always has some other thing.
10-d exists and is never virtual and controls time and potentially has three concepts of matter, and always has two concepts of time, and always has some other thing.
11-d exists and is never virtual and controls time and always has three concepts of matter, and always has two concepts of time, and always has some other thing.
12-d exists, and is never virtual and controls time and always has three concepts of matter, and sometimes has three concepts of time, and always has some other thing.
13-d exists, and is never virtual, and controls time, and always has three concepts of matter, and always has three concepts of time, and always has some other thing.
14-d exists, and is never virtual, and controls time, and always has three concepts of matter, and always has three concepts of time, and sometimes has two other things.
15-d exists, and is never virtual, and controls time, and always has three concepts of matter, and always has three concepts of time, and always has two other things.
16-d might not exist, or is partly virtual, or partly chaotic, and always has three concepts of matter, and always has three concepts of time, and always has two other things.
17-d doesn't exist, or is virtual, or is chaotic, and always has three concepts of matter, and always has three concepts of time, and always has two other things.
18-d doesn't exist, or is virtual, or is chaotic, and always has three concepts of matter, and always has three concepts of time, and sometimes has three other things.
19-d doesn't exist, or is virtual, or is chaotic, and always has three concepts of matter, and always has three concepts of time, and always has three other things.
20-d doesn't exist, or is virtual, or is chaotic, and always has three concepts of matter, and always has three concepts of time, and always has three other things, and sometimes has an additional power.
21-d doesn't exist, or is virtual, or is chaotic, and always has three concepts of matter, and always has three concepts of time, and always has three other things, and always has an additional power.
22-d doesn't exist, or is virtual, or is chaotic, and sometimes has four concepts of matter, and always has three concepts of time, and always has three other things, and always has an additional power.
23-d doesn't exist, or is virtual, or is chaotic, and always has four concepts of matter, and always has three concepts of time, and always has three other things, and always has an additional power.
24-d doesn't exist, or is virtual, or is chaotic, and always has four concepts of matter, and sometimes has four concepts of time, and always has three other things, and always has an additional power.
25-d doesn't exist, or is virtual, or is chaotic, and always has four concepts of matter, and always has four concepts of time, and always has three other things, and always has an additional power.
26-d doesn't exist, or is virtual, or is chaotic, and always has four concepts of matter, and always has four concepts of time, and sometimes has four other things, and always has an additional power.
27-d doesn't exist, or is virtual, or is chaotic, and always has four concepts of matter, and always has four concepts of time, and always has four other things, and always has an additional power.
28-d doesn't exist, or is virtual, or is chaotic, and always has four concepts of matter, and always has four concepts of time, and always has four other things, and sometimes has two additional powers.
29-d doesn't exist, or is virtual, or is chaotic, and always has four concepts of matter, and always has four concepts of time, and always has four other things, and always has two additional powers.
30-d is an illusion, or composite, or mind, and always has four concepts of matter, and always has four concepts of time, and always has four other things, and always has two additional powers.
31-d is an illusion, or composite, or mind, and sometimes has five concepts of matter, and always has four concepts of time, and always has four other things, and always has two additional powers.
32-d is an illusion, or composite, or mind, and always has five concepts of matter, and always has four concepts of time, and always has four other things, and always has two additional powers.
33-d is an illusion, or composite, or mind, and always has five concepts of matter, and sometimes has five concepts of time, and always has four other things, and always has two additional powers.
34-d is an illusion, or composite, or mind, and always has five concepts of matter, and always has five concepts of time, and sometimes has five other things, and always has two additional powers.
35-d is an illusion, or composite, or mind, and always has five concepts of matter, and always has five concepts of time, and always has five other things, and always has two additional powers.
36-d is an illusion, or composite, or mind, and always has five concepts of matter, and always has five concepts of time, and always has five other things, and sometimes has three additional powers.
37-d is an illusion, or composite, or mind, and always has five concepts of matter. and always has five concepts of time, and always has five other things, and always has three additional powers.
38-d is an illusion, or composite, or mind, and sometimes has six concepts of matter. and always has five concepts of time, and always has five other things, and always has three additional powers.
39-d is an illusion, or composite, or mind, and always has six concepts of matter, and always has five concepts of time, and always has five other things, and always has three additional powers.
40-d is an illusion, or composite, or mind, and always has six concepts of matter, and sometimes has six concepts of time, and always has five other things, and always has three additional powers.
41-d is an illusion, or composite, or mind, and always has six concepts of matter, and always has six concepts of time, and always has five other things, and always has three additional powers.
42-d is an illusion, or composite, or mind, and always has six concepts of matter, and always has six concepts of time, and sometimes has six other things, and always has three additional powers.
43-d is an illusion, or composite, or mind, and always has six concepts of matter, and always has six concepts of time, and always has six other things, and always has three additional powers.
44-d is an illusion, or composite, or mind, and always has six concepts of matter, and always has six concepts of time, and always has six other things, and sometimes has four additional powers.
45-d is an illusion, or composite, or mind, and always has six concepts of matter, and always has six concepts of time, and always has six other things, and always has four additional powers.
46-d is relative, and broken, and magical, and always has six concepts of matter, and always has six concepts of time, and always has six other things, and always has four additional powers.
47-d is absolute, and undifferentiated, and universal, and always has six concepts of matter, and always has six concepts of time, and always has six other things, and always has four additional powers.
48-d is absolute, and undifferentiated, and universal, and sometimes has seven concepts of matter, and always has six concepts of time, and always has six other things, and always has four additional powers.
49-d is absolute, and undifferentiated, and universal, and always has seven concepts of matter, and always has six concepts of time, and always has six other things, and always has four additional powers.
50-d is absolute, and undifferentiated, and universal, and always has seven concepts of matter, and sometimes has seven concepts of time, and always has six other things, and always has four additional powers.
51-d is absolute, and undifferentiated, and universal, and always has seven concepts of matter, and always has seven concepts of time, and always has six other things, and always has four additional powers.
52-d is absolute, and undifferentiated, and universal, and always has seven concepts of matter, and always has seven concepts of time, and sometimes has seven other things, and always has four additional powers.
53-d is absolute, and undifferentiated, and universal, and always has seven concepts of matter, and always has seven concepts of time, and always has seven other things, and always has four additional powers.
54-d is absolute, and undifferentiated, and universal, and always has seven concepts of matter, and always has seven concepts of time, and always has seven other things, and sometimes has five additional powers.
55-d is absolute, and undifferentiated, and universal, and always has seven concepts of matter, and always has seven concepts of time, and always has seven other things, and always has five additional powers.
56-d is absolute, and undifferentiated, and universal, and sometimes has eight concepts of matter, and always has seven concepts of time, and always has seven other things, and always has five additional powers.
57-d is absolute, and undifferentiated, and universal, and always has eight concepts of matter, and always has seven concepts of time, and always has seven other things, and always has five additional powers.
58-d is absolute, and undifferentiated, and universal, and always has eight concepts of matter, and sometimes has eight concepts of time, and always has seven other things, and always has five additional powers.
59-d is absolute, and undifferentiated, and universal, and always has eight concepts of matter, and always has eight concepts of time, and always has seven other things, and always has five additional powers.
60-d is absolute, and undifferentiated, and universal, and always has eight concepts of matter, and always has eight concepts of time, and sometimes has eight other things, and always has five additional powers.
61-d is absolute, and undifferentiated, and universal, and always has eight concepts of matter, and always has eight concepts of time, and always has eight other things, and always has five additional powers.
62-d is absolute, and undifferentiated, and universal, and always has eight concepts of matter, and always has eight concepts of time, and always has eight other things, and sometimes has six additional powers.
62-d is absolute, and undifferentiated, and universal, and always has eight concepts of matter, and always has eight concepts of time and always has eight other things, and always has six additional powers.
63-d is impermanent or finite or fatal, and always has eight concepts of matter, and always has eight concepts of time, and always has eight other things, and always has six additional powers.
64-d is permanent, infinite, and eternal, and always has eight concepts of matter, and always has eight concepts of time, and always has eight other things, and always has six additional powers.
65-d is permanent, infinite, and eternal, and sometimes has nine concepts of matter, and always has eight concepts of time, and always has eight other things, and always has six additional powers.
66-d is permanent, infinite, and eternal, and always has nine concepts of matter, and always has eight concepts of time, and always has eight other things, and always has six additional powers.
67-d is permanent, infinite, and eternal, and always has nine concepts of matter. and sometimes has nine concepts of time, and always has eight other things, and always has six additional powers.
68-d is permanent, infinite, and eternal, and always has nine concepts of matter, and always has nine concepts of time, and always has eight other things, and always has six additional powers.
69-d is permanent, infinite, and eternal, and always has nine concepts of matter, and always has nine concepts of time, and sometimes has nine other things, and always has six additional powers.
70-d is permanent, infinite, and eternal, and always has nine concepts of matter, and always has nine concepts of time, and always has nine other things, and always has six additional powers.
71-d is permanent, infinite, and eternal, and always has nine concepts of matter, and always has nine concepts of time, and always has nine other things, and sometimes has seven additional powers.
72-d is permanent, infinite, and eternal, and always has nine concepts of matter, and always has nine concepts of time, and always has nine other things, and always has seven additional powers.
73-d is permanent, infinite, and eternal, and sometimes has ten concepts of matter, and always has nine concepts of time, and always has nine other things, and always has seven additional powers.
74-d is permanent, infinite, and eternal, and always has ten concepts of matter, and always has nine concepts of time, and always has nine other things, and always has seven additional powers.
75-d is permanent, infinite, and eternal, and always has ten concepts of matter, and sometimes has ten concepts of time, and always has nine other things, and always has seven additional powers.
76-d is permanent, infinite, and eternal, and always has ten concepts of matter, and always has ten concepts of time, and always has nine other things, and always has seven additional powers.
77-d is permanent, infinite, and eternal, and always has ten concepts of matter, and always has ten concepts of time, and sometimes has ten other things, and always has seven additional powers.
78-d is permanent, infinite, and eternal, and always has ten concepts of matter, and always has ten concepts of time, and always has ten other things, and always has seven additional powers.
79-d is permanent, infinite, and eternal, and always has ten concepts of matter, and always has ten concepts of time, and always has ten other things, and sometimes has eight additional powers.
80-d is permanent, infinite, and eternal, and always has ten concepts of matter, and always has ten concepts of time and always has ten other things, and always has eight additional powers.
81-d is destruction, art, and purpose, and always has ten concepts of matter, and always has ten concepts of time, and always has ten other things, and always has eight additional powers.
82-d is time, jest, and transformation, and always has ten concepts of matter, and always has ten concepts of time, and always has ten other things, and always has eight additional powers.
83-d is time, jest, and transformation, and sometimes has eleven concepts of matter, and always has ten concepts of time, and always has ten other things, and always has eight additional powers.
84-d is time, jest, and transformation, and always has eleven concepts of matter, and always has ten concepts of time, and always has ten other things, and always has eight additional powers.
85-d is time, jest, and transformation, and always has eleven concepts of matter, and sometimes has eleven concepts of time, and always has ten other things, and always has eight additional powers.
86-d is time, jest, and transformation, and always has eleven concepts of matter, and always has eleven concepts of time, and always has ten other things, and always has eight additional powers.
87-d is time, jest, and transformation, and always has eleven concepts of matter, and always has eleven concepts of time, and sometimes has eleven other things, and always has eight additional powers.
88-d is time, jest, and transformation, and always has eleven concepts of matter, and always has eleven concepts of time, and always has eleven other things, and always has eight additional powers.
89-d is time, jest, and transformation, and always has eleven concepts of matter, and always has eleven concepts of time, and always has eleven other things, and sometimes has nine additional powers.
90-d is time, jest, and transformation, and always has eleven concepts of matter, and always has eleven concepts of time, and always has eleven other things, and always has nine additional powers.
91-d is time, jest, and transformation, and sometimes has twelve concepts of matter, and always has eleven concepts of time, and always has eleven other things, and always has nine additional powers.
92-d is time, jest, and transformation, and always has twelve concepts of matter, and always has eleven concepts of time, and always has eleven other things, and always has nine additional powers.
93-d is time, jest, and transformation, and always has twelve concepts of matter, and sometimes has twelve concepts of time, and always has eleven other things, and always has nine additional powers.
94-d is time, jest, and transformation, and always has twelve concepts of matter, and always has twelve concepts of time, and sometimes has twelve other things, and always has nine additional powers.
95-d is time, jest, and transformation, and always has twelve concepts of matter, and always has twelve concepts of time, and always has twelve other things, and always has nine additional powers.
96-d is time, jest, and transformation, and always has twelve concepts of matter, and always has twelve concepts of time, and always has twelve other things, and sometimes has ten additional powers.
97-d is time, jest, and transformation, and always has twelve concepts of matter, and always has twelve concepts of time, and always has twelve other things, and always has ten additional powers
98-d is time, jest, and transformation, and sometimes has thirteen concepts of matter, and always has twelve concepts of time, and always has twelve other things, and always has ten additional powers.
99-d is time, jest, and transformation, and always has thirteen concepts of matter, and always has twelve concepts of time, and always has twelve other things, and always has ten additional powers.
100-d is time, jest, and transformation, and always has thirteen concepts of matter, and sometimes has thirteen concepts of time, and always has twelve other things, and always has ten additional powers.
Etc.
AFTERWORD
It is possible to conceive of other typological systems, however, in which these typological dimensions are not reached, or in which one characteristic, such as material complexity, time complexity, or sheer complexity develops quickly while other characteristics do not.
My conclusion for the fifth dimension based on the study of characteristic dimensions is that it objectively involves the most advanced conceivable concept of five properties (one for each dimension): (1) nature, (2) materials, (3) structure, (4) properties / culture, and (5) evolution. These are sort of the Qubits of 5-d space. And it can be added that 5-d may be a different spatial or metaphysical dimension, with the assumption that simpler experiences are analogous to Flatland, and that more advanced experiences can be approximated.
The above is also available as an academic article HERE: https://www.academia.edu/22558908/How_to_Build_a_100-Dimensional_Universe
