If you have evidence that the explanation for your termination is not factually correct, then you may be successful in your application for unemployment benefits. But the failure of HR to be frank with you as to the reason for your termination does not invalidate or make unlawful your termination. And nothing that you have stated here supports a claim for wrongful termination.
If you do not have an employment contract or a union agreement, then you are an at will employee. At will employees can be terminated at any time for any reason not prohibited by law. Prohibited reasons typically include unlawful bias based on race, gender, religion, etc.
Your employer does not have to be correct in its factual reasons for terminating employees, nor is the employer obligated to make an investigation or obtain any tests or proof. And your employer is not required to tell you the truth about the reasons for termination -- although a claim for unemployment benefits will ordinarily cause disclosure of that information.
If you were represented by a union, contact your union rep right away. Consult with a local employment attorney for a fuller discussion and analysis of the facts of this situation.
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