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