Probability is commonly misunderstood.
It is often treated as a statement about belief, luck, or chance.
In mathematics, however, probability is a statement about measure.
The Arc Theory of Certainty formalizes a simple but frequently misinterpreted fact:
In an unbounded system, any event with non-zero probability is inevitable.
This is not a philosophical claim.
It is a mathematical one.
1. Informal Statement
Let an event E exist in a probability space S.
If:
P(E) > 0
the system continues to evolve without bound
then:
The occurrence of E is not merely possible — it is certain.
No assumption of intent, retries, or external intervention is required.
2. Formal Theorem (Arc Certainty Theorem)
Let:
(S, F, P) be a probability space
E in F such that P(E) = p > 0
{X_n}_{n=1}^{infty} be independent realizations over S
Define the event:
A_n = { E occurs at least once in n realizations }
Then:
lim_{n -> infty} P(A_n) = 1
3. Proof
The probability that E does not occur in a single realization is 1 - p.
The probability that it does not occur in n independent realizations is:
(1 - p)^n
Since 0 < p le 1:
lim_{n -> infty} (1 - p)^n = 0
Therefore:
P(A_n) = 1 - (1 - p)^n -> 1
QED.