The question presented by this appeal is whether a claim by the Internal Revenue Service to recover an erroneous refund is dischargeable in bankruptcy even if as a consequence of the refund the debt- or underpaid his taxes. The bankruptcy court, seconded by the district court, said yes, it is dischargeable, and the IRS appeals.
When it reviewed the Frontones’ 2000 tax return, the IRS determined that they had overpaid the taxes they owed by more than five thousand dollars, and it mailed them a refund. Within a couple of months the IRS discovered its mistake and made a supplemental assessment. What was assessed was a “deficiency” — the amount by which the tax owed by a taxpayer exceeds the amount reported on his return plus the amount of any “rebates.” 26 U.S.C. § 6211(a). Rebates are refunds or credits awarded when the IRS determines that the taxpayer owed less than what the return reported as the amount due from him. § 6211(b)(2). If a rebated refund (or credit) is made in error, this may increase the amount of taxes that the recipient of the refund owes, in which event the IRS is entitled to issue a supplemental assessment, as it did here. (See § 6204(a), discussed below.) If, for example, the tax owed by the taxpayer was $10,000, the amount reported on his return was $7,000, and he had received an erroneous rebate-refund of $1,000, the deficiency would be $4,000 ($10,000 — $7,000 + $1,000). The refund that the Frontones had received was a rebate, as they concede, so if the refund was in error it could have given rise to a deficiency; the IRS believed it had— hence the supplemental assessment.
The IRS notified the Frontones of the supplemental assessment. Their response was to pay only a small part of it ($492), instead filing for bankruptcy under Chapter 7 of the Bankruptcy Code the following year and receiving from the bankruptcy court a discharge of their (dischargeable) debts. Three days before receiving the discharge, they had filed another petition for bankruptcy, this one under Chapter 13 of the Code (reorganization). That is a common sequence (nicknamed “Chapter 20”). For after receiving a discharge of dischargeable unsecured debts in his Chapter 7 proceeding, the debtor may still be burdened with debts — non-dischargea-ble unsecured debts, plus secured debts— and Chapter 13 enables him to work them off in accordance with an installment payment schedule approved by the bankruptcy court. Johnson v. Home State Bank, *658501 U.S. 78, 87-88, 111 S.Ct. 2150, 115 L.Ed.2d 66 (1991); 2 Thomas D. Crandall, Richard B. Hagedorn & Frank W. Smith, Jr., The Law of Debtors and Creditors § 17:4 (2004); Lex A. Coleman, “Individual Consumer ‘Chapter 20’ Cases After Johnsorr. An Introduction to NonBusiness Serial Filings under Chapter 7 and Chapter 13 of the Bankruptcy Code,” 9 Bankr.Development J. 357 (1992).
The IRS filed a claim for the supplemen-tally assessed taxes in the Frontones’ Chapter 13 case, contending that the tax liability reflected by the assessment had not been discharged by the discharge granted them in their Chapter 7 case. Section 523(a)(1)(A) of the Bankruptcy Code exempts from discharge a tax “of the kind and for the periods specified in section 507(a)(2) or 507(a)(8)” of the Code. Section 507 lists the kinds of claim given priority in the distribution of a bankrupt’s assets, and one of them, which section 523(a)(1)(A) incorporates by reference, is a claim for unpaid income taxes for (roughly speaking) the three years before the taxpayer filed his petition for bankruptcy. 11 U.S.C. § 507(a)(8)(A)(i). Not only are such claims not dischargeable in a Chapter 7 bankruptcy, but a Chapter 13 plan must provide for their payment in full. § 1322(a)(2). If when the plan expires the claim still hasn’t been paid, it remains payable even if the debtor is granted an unconditional — a purportedly “full” — discharge. § 1328(c)(2).
Before coming to the main issue, we must smooth a procedural wrinkle. The Internal Revenue Service is forbidden, with immaterial exceptions, to assess a deficiency until it has issued the taxpayer a notice of deficiency. 26 U.S.C. § 6213(a). It failed to do that here (the errors mount up!). The notice of assessment that the IRS sent the Frontones can’t be treated as a notice of deficiency, because it followed rather than preceded the assessment, which therefore was invalid. Singleton v. United States, 128 F.3d 833, 838-39 (4th Cir.1997); Philadelphia & Reading Corp. v. United States, 944 F.2d 1063, 1072 (3d Cir.1991); Russell v. United States, 774 F.Supp. 1210, 1213-16 (W.D.Mo.1991). It might seem to follow that the government has no tax claim. It does not follow. The IRS had three years from the filing of the Frontones’ 2000 tax return in which to issue the notice of deficiency, and it finally did issue it, earlier this year, the month before the three-year reach-back deadline expired. It could have issued a new assessment but it didn’t have to. All that matters is that, the notice of deficiency having been timely, the deficiency was assessable. For though assessment is a prerequisite to certain remedies that the IRS might seek, it is not a prerequisite to the IRS’s making a claim in a bankruptcy proceeding, because the Bankruptcy Code gives priority to a tax claim that is “assessable” as well as to one that is actually assessed. 11 U.S.C. § 507(a)(8)(A)(iii); see In re Hillsborough Holdings Corp., 116 F.3d 1391, 1394-96 (11th Cir.1997); In re Pacific-Atlantic Trading Co., 64 F.3d 1292, 1301-04 (9th Cir.1995); In re L.J. O’Neill Shoe Co., 64 F.3d 1146, 1149-51 (8th Cir.1995). And remember that it is priority claims that are exempt from discharge.
So the question here is whether a claim for taxes based on an erroneous refund is — a claim for taxes. Clearly the general answer is yes. Section 6204(a) of the Internal Revenue Code authorizes the IRS, “at any time within the period prescribed for assessment, [to] make a supplemental assessment whenever it is ascertained that any assessment is imperfect or incomplete in any material respect.” That describes this case. The initial assessment was “imperfect or incomplete in [a] materi*659al respect” because it understated the Frontones’ tax liability. So the IRS made a new, accurate assessment, which showed that they owed additional income tax for the year 2000. It is true that they had paid this amount (with some discrepancies that we needn’t get into) when they first filed their return; but the money had been returned to them by mistake, and so they had to pay it again. The additional payment due was a payment of taxes due, as the case law recognizes in allowing the IRS to use its deficiency procedures, which are intended for the recovery of taxes, to recapture erroneous refunds. Brookhurst, Inc. v. United States, 931 F.2d 554, 555-57 (9th Cir.1991); Beer v. Commissioner, 733 F.2d 435 (6th Cir.1984) (per curiam); Warner v. Commissioner, 526 F.2d 1 (9th Cir.1975); United States v. C & R Investments, Inc., 404 F.2d 314 (10th Cir.1968). This is provided, however, that the error affects the tax liability of the person who received the refund; the significance of this qualification will appear shortly.
The Frontones argue that while it may be true in general that the IRS can treat an erroneous rebate-refund as an underpayment of taxes, it is false in bankruptcy — and “characterizations in the Internal Revenue Code are not dispositive in the bankruptcy context.” United States v. Reorganized CF & I Fabricators of Utah, Inc., 518 U.S. 213, 224, 116 S.Ct. 2106, 135 L.Ed.2d 506 (1996). Had the IRS not given the Frontones an erroneous refund, they wouldn’t have a nondischargeable tax debt facing them, because they wouldn’t have owed taxes. And having a nondischargeable debt, as both the bankruptcy judge and the district judge noted, makes it harder for the debtor to get back on his feet after bankruptcy.
The judges believed that both “equity” (bankruptcy proceedings are part of the equity jurisdiction of the federal courts) and the “fresh start” rationale of bankruptcy supported the Frontones’ claim. The appeal to equity rings a little hollow. The IRS’s error did not “cause” the Fron-tones to emerge from their Chapter 7 bankruptcy with a nondischargeable tax debt. Had they paid the supplemental assessment — which, remember, they had been notified of before they declared bankruptcy — they would have emerged from Chapter 7 with no dischargeable debts, though the record is silent on whether their assets when they were notified were sufficient to enable them to pay it in full.
And while the Bankruptcy Code is indeed a code of debtors’ rights (hence the “fresh start” rationale), it is equally a code of creditors’ remedies. Bankruptcy maximizes the repayment of an insolvent debt- or’s debts by overcoming the collective-action (or musical-chairs) problem that arises when each of the debtor’s unsecured creditors races to seize the debtor’s assets, when a more orderly liquidation, or a reorganization, would yield a larger total recovery. Aiello v. Providian Financial Corp., 239 F.3d 876, 879 (7th Cir.2001); In Matter of Milwaukee Cheese Wisconsin, Inc., 112 F.3d 845, 847-48 (7th Cir.1997); Covey v. Commercial Nat’l Bank, 960 F.2d 657, 661-62 (7th Cir.1992).
But what is most important is that it is the Code itself — not bankruptcy judges, district judges, circuit judges, or even Supreme Court Justices, exercising a free-wheeling “equitable” discretion — that strikes the balance between debtors and creditors. Cohen v. de la Cruz, 523 U.S. 213, 222, 118 S.Ct. 1212, 140 L.Ed.2d 341 (1998); Grogan v. Garner, 498 U.S. 279, 286-87, 111 S.Ct. 654, 112 L.Ed.2d 755 (1991); In re Kmart Corp., 359 F.3d 866, 871 (7th Cir.2004); In re Stoecker, 179 F.3d 546, 551 (7th Cir.1999); In re Mayer, 51 F.3d 670, 673-74 (7th Cir.1995); cf. Raleigh v. Illinois Dept. of Revenue, 530 *660U.S. 15, 24-25, 120 S.Ct. 1951, 147 L.Ed.2d 13 (2000). “The fact that a [bankruptcy] proceeding is equitable does not give the judge a free-floating discretion to redistribute rights in accordance with his personal views of justice and fairness, however enlightened those views may be.” In re Chicago, Milwaukee, St. Paul & Pacific R.R., 791 F.2d 524, 528 (7th Cir.1986).
So it is critical that nothing in the Bankruptcy Code provides the slightest footing for the Frontones’ position. On the contrary, it is apparent from section 523(a)(1)(A) that the IRS — that most importunate of creditors — has won from Congress a broad exemption from the dis-chargeability of tax claims. The result is that if a debtor lacks sufficient assets to cover his tax debts, those debts will follow him out of bankruptcy and impede his “fresh start.”
That is what Congress has decreed and how likely is it that Congress would have wanted the courts to carve an exception for the case in which the IRS had mistakenly refunded a portion of the taxes due from the debtor? Such cases are common, given the enormous volume of refunds. Warner v. Commissioner, supra, 526 F.2d at 2. Congress knew, moreover, and did not object, that the IRS would sometimes attempt to recover in a bankruptcy proceeding an erroneous refund, for it provided in section 507(c) of the Bankruptcy Code that “a claim ... arising from an erroneous refund or credit of a tax has the same priority as a claim for the tax to which such refund or credit relates.” The preferred status of tax claims, including claims based on an erroneous refund, is not an inference from the Internal Revenue Code; it is express in the Bankruptcy Code.
The Frontones argue that their supplemental debt to the IRS arose not from their 2000 income but from the IRS’s mistake. There is no either-or; it is both-and. The ultimate source of the debt is the Frontones’ 2000 income. It is as truthful to say that the claim for those taxes is based on the Frontones’ 2000 tax return as it is to say that it is based on the IRS’s error in making a refund to them. Nor was the IRS the only maker of mistakes in this case; the Frontones made a serious mistake in not paying the supplemental assessment before they declared bankruptcy, assuming they had some assets beyond the $492 that they did send to the IRS in (very) partial payment of the assessment.
What is true — though not in this case— is that a mistaken refund can give rise to a claim that is not a tax claim and so would not enjoy the priority that the Bankruptcy Code gives such claims and would therefore be dischargeable. Thus in United States v. Reorganized CF & 1 Fabricators of Utah, Inc., supra, the Supreme Court ruled that a “tax” on employers who underfund pension plans is really a fine, and similarly the IRS’s claim for the erroneous rebate issued to someone who actually owed no taxes would be a claim for restitution rather than a tax. Suppose the Fron-tones had no income and therefore paid no income taxes, but the IRS made a mistake and mailed them a check. The government would be entitled to the return of the money, but not because the Frontones owed it any taxes. The ground would be unjust enrichment, since the Frontones would have no right to retain money paid them by mistake. Lawyers Title Ins. Corp. v. Dearborn Title Corp., 118 F.3d 1157, 1163-65 (7th Cir.1997); First Wisconsin Trust Co. v. Schroud, 916 F.2d 394, 400-01 (7th Cir.1990); Leasing Service Corp. v. Hobbs Equipment Co., 894 F.2d 1287, 1291-92 (11th Cir.1990). The government would proceed by suing for restitution under federal common law, United States v. Wurts, 303 U.S. 414, 415-16, 58 *661S.Ct. 637, 82 L.Ed. 932 (1938); Old Republic Ins. Co. v. Federal Crop Ins. Corp., 947 F.2d 269, 275 (7th Cir.1991); United States v. Domino Sugar Corp., 349 F.3d 84, 88 (2d Cir.2003); Bechtel v. Pension Benefit Guaranty Corp., 781 F.2d 906 (D.C.Cir.1985) (per curiam), rather than proceeding by assessing a deficiency; there would be no deficiency.
In a case such as the present, however, in which the ultimate source of the IRS’s claim is a tax owed—the refund or credit having resulted in the taxpayer’s underpaying his taxes—the IRS can proceed either by the assessment route, as it did here, which would enable it if it wanted to utilize the summary procedures for tax collection authorized by 26 U.S.C. §§ 6320, 6330, or by a suit under 26 U.S.C. § 7405, which, in the subsection that would be applicable to such a case, authorizes the government to sue to recover “any portion of a tax imposed by this title which has been erroneously refunded.” § 7405(b). Either way, the government would be prosecuting a tax claim. Only if the refund went to someone who owed no tax would the government be able to proceed only by suing for restitution, because in that case the refund would not have resulted in the recipient’s paying less than his full taxes.
To vary the case slightly, suppose the taxpayer requests a refund on the basis of a deduction and the IRS sends him the requested refund but later discovers that the taxpayer isn’t entitled to it. The mistaken refund would still be a rebate because it would have been “made on the ground that the tax imposed by [the Tax Code] was less than the excess of the amount specified in subsection (a)(1) over the rebates previously made,” § 6211(b)(2), after the error was noticed. And so in United States v. C & R Investments, Inc., supra, 404 F.2d at 315-16, an erroneous-refund case in which the error was based on the taxpayer’s claiming a credit to which he was not entitled, the court held that the IRS was entitled to recover the refund either by a suit under section 7405 or by the assessment route.
The case on which the Frontones principally rely, O’Bryant v. United States, 49 F.3d 340 (7th Cir.1995), was similar to C & R Investments insofar as the IRS had credited the tax payment of the taxpayers twice and as a result had erroneously refunded them the entire payment. But there was a critical though subtle difference, which led us to conclude that the assessment route was closed to the IRS. “[T]he money the O’Bryants have now is not the money that the IRS’ original assessment contemplated, since that amount was already paid. Rather, it is a payment the IRS accidentally sent them. They owe it to the government because they have been unjustly enriched by it, not because they have not paid their taxes.” 49 F.3d at 346. In C & R Investments, the taxpayer had received a refund on the ground that by virtue of its claimed tax credit it owed less tax than would otherwise have been due. In O’Bryant, the refund was based not on any calculation of tax due, but rather on an accounting error, the double posting of the credit. See Michael I. Saltzman, IRS Practice & Procedure ¶ 10.03[1][c], pp. 19-20 (2004). Remember that a deficiency is the difference between the amount of tax due and (1) the amount of tax due that is reported on the taxpayer’s return plus (2) the amount of any rebates awarded because the IRS has determined that the taxpayer owed less than his return showed him as owing. A deficiency can thus arise as a result of a determination that the rebate in (2) was in error. But in O’Bryant there was no recalculation of the taxpayer’s liability. The IRS’s accounting error was akin to writing *662a refund check to the wrong person, 49 F.3d at 345-46, an example we gave earlier of a refund that is not a rebate.
A different conclusion would, by extending the availability of supplemental-assessment procedures to cases involving posting errors, create tension with 26 U.S.C. § 6204. That section, as we know, authorizes “a . supplemental assessment whenever it is ascertained that any assessment is imperfect or incomplete in any material respect”; and a posting error does not involve an imperfect or incomplete assessment. We acknowledge the tension between O'Bryant’s conception of when assessment is available and the broader conception suggested by Bilzerian v. United States, 86 F.3d 1067, 1069 (11th Cir.1996), and Clark v. United States, 63 F.3d 83, 88 and n. 7 (1st Cir.1995), but this case is unlike O’Bryant; it is on the C & R Investments side of the ledger.
The Frontones point finally to a change Congress made in 1984 in section 507(c) of the Bankruptcy Code, the section that gives a claim arising from an erroneous refund “the same priority” as a claim for the tax to which such refund relates. Before 1984, the section provided that a claim arising from an erroneous refund “shall be treated the same” as a claim for the underlying tax, and the Frontones argue that the effect of the change is that a claim arising from an erroneous refund affects only priority and not discharge. (A similar argument was accepted in In re Jackson, 253 B.R. 570 (M.D.Ala.2000), though in the context of a nonrebate refund.) The legislative history, however, describes the change as “stylistic and clarifying,” S.Rep. No. 65, 98th Cong., 1st Sess. 79 (1983), and the description is apt. Section 507 is about priorities, and the point of section 507(c) is that the priority of a claim based on a refund is the same as that of the underlying tax. Thus when Congress in the earlier version of the section said that the claim based on the refund “shall be treated the same” as a claim for the underlying tax, it meant “shall be treated the same for the purpose of determining priority.” Our ease has nothing to do with priority. Section 507 is about the order in which claims are paid when, as is usually the case, the bankrupt’s liabilities exceed his assets. Section 523(a)(1)(A) incorporates the list of tax claims from section 507 and pronounces them nondischargeable, but the conferral by the latter section of priority upon those claims is irrelevant to this appeal, which is about whether the IRS has a claim in the Frontones’ Chapter 13 bankruptcy or whether the claim has already been discharged in their Chapter 7 bankruptcy. We have just held that IRS does have a claim in the Chapter 13 bankruptcy, and so section 507 will determine its priority vis-a-vis the other claims in that bankruptcy; but, to repeat, that has no bearing on this appeal.
In sum, we cannot find any legal basis for the Frontones’ contention that the IRS’s tax claim founded on the mistaken refund is dischargeable and was therefore discharged in the Chapter 7 proceeding. The judgment' affirming the dismissal of the claim is therefore reversed and the case is remandéd to the bankruptcy court with directions to reinstate the claim in the current, Chapter 13 proceeding.
REVERSED AND REMANDED, WlTH DIRECTIONS.