We don't live together 'nor do we publicly state that we are married.
What you have explained does not meet the legal requirements for a 'common law' marriage.
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This probably does not make you married. If you are concerned about it, you might want to consider having a lawyer draft a Cohabitation Agreement that sets out both of your property rights and that you do not agree to be married.
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Not by itself. If you want to make sure you do not "accidentally" get married, talk to a lawyer about how to avoid that result.
To be common law married, you have to agree to be married and hold yourself out to others as married.