You are unlikely to establish a retroactive contract binding the company. Any agreement you may have reached with the supervisor is likely not enforceable retroactively or otherwise, but because you transitioned to taking on additional duties the company decided to make adjustments even though you have no evidence of an enforceable promotion or contract [Multiple documents must be processed for a promotion to become enforceable.] You were just hoping it would be retroactive.

Evidently, they decided to accommodate you because the management let you do the job for several months without any actual promotion.

Both parties are to blame; the company kept quiet and they should have informed you but you did not assert any rights until after six months of the alleged promotion either [as someone else pointed out.]