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Arcadia Tech, Inc. (Arcadia), a corporate taxpayer, has a 2024 deficiency assessment of PHP 45,000,000 (exclusive of interest and penalties). Arcadia also has a currently collectible receivable of PHP 50,000,000 from Zenith Solutions, Inc. In a single transaction, Zenith withholds PHP 45,000,000 from a payment due to Arcadia and remits that amount directly to the BIR to satisfy Arcadia’s tax deficiency, arguing that private set-off against a tax debt should be allowed. The remaining PHP 5,000,000 of Arcadia's receivable remains with Zenith to be paid to Arcadia, and Arcadia’s tax liability is deemed settled to the extent of the withheld amount. The BIR refuses to recognize a private set-off against Arcadia’s tax debt, citing the Lifeblood Doctrine. In light of the Lifeblood Doctrine, address: (a) whether Zenith’s withholdings and remittance constitute a valid private set-off against Arcadia’s tax debt; (b) the implications of the Lifeblood Doctrine for private compensation and set-off in tax collection; and (c) the impact on Zenith as a third party and the proper remedy for Arcadia. Include the controlling rule, apply it to the facts, and state a concise conclusion.