EASTERBROOK, Chief Judge.

After Congress reduced from 100:1 to 18:1 the ratio between crack and powder cocaine for purposes of statutory minimum and maximum sentences, see Dorsey v. United States, ___ U.S. ___, 132 S.Ct. 2321, 183 L.Ed.2d 250 (2012), the Sentencing Commission made corresponding changes to the Guideline ranges for crack. Amendment 750 makes the changes; Amendment 759 authorizes retroactive application. Adolfo Wren and Anthony Moton asked the district court to cut their sentences. The applications went to different judges; both said no.

Wren and Moton are serving sentences below the normal statutory floor, often but misleadingly called a "mandatory minimum." The word "mandatory" is imprecise because judges may give lower sentences to nonviolent first offenders (the "safety valve" provision in 18 U.S.C. § 3553(f)) and defendants who help with the investigation and prosecution of their confederates (18 U.S.C. § 3553(e); 28 U.S.C. § 994(n); see also U.S.S.G. § 5K1.1). Wren and Moton provided valuable assistance and received sentences lower than the presumptive floor of 120 months' imprisonment. 21 U.S.C. § 841(a)(1), (b)(1). The original sentencing range for each was 121 to 151 months; each original sentence was 100 months. Amendment 750 affects them differently, because they are responsible for different quantities of cocaine. The new range for Wren is 100 to 125 months, and the new range for Moton is 84 to 105 months.

A district judge may reduce a sentence if a change to the Sentencing Guidelines allows. 18 U.S.C. § 3582(c)(2). The district judges concluded that U.S.S.G. § 5G1.1 prevents Wren and Moton from receiving lower sentences: it provides that, when all or part of a Guideline range lies below a statutory minimum sentence, the statutory minimum becomes the lower bound of the range. Thus the "range" becomes a point if the whole range otherwise would be less than the statutory floor. If § 5G1.1 applies, then the amended range for Moton is 120 months, and the amended range for Wren is 120-125 months. The prosecutor argued, and both district judges concluded, that the ranges for Wren and Moton had not really been reduced by Amendment 750 — and only defendants who are beneficiaries of a lower range can receive lower sentences. See § 1B1.10(a)(2)(B).

But does § 5G1.1 apply? The prosecutor says yes because, to know whether any given defendant's range has been cut, the district court must work through the sequence prescribed in § 1B1.1. That Guideline includes (see § 1B1.1(a)(7)) a reference to Chapter Five, which contains § 5G1.1 and sets the new range at 120 months. The old range for both Wren and Moton was 121 to 151 months, so it has declined by one month even on the prosecutor's understanding. That's not enough to help these defendants, the prosecutor insists, because both are serving below-range sentences. Guideline 1B1.10(b)(2)(B) provides that a defendant whose sentence is below the original Guideline because of the prosecutor's substantial-assistance motion may receive a new sentence that is "comparably less than the amended guideline range". The original 100-month sentences of both Moton and Wren were 17% below the bottom of their original ranges; take 17% off the new range of precisely 120 months and you still get 100 months (since the result, 99.6 months, would be rounded up). The two district judges followed this approach and left each defendant's sentence unchanged.

The prosecutor's preferred approach relies on § 1B1.1(a)(7), the pointer to § 5G1.1. Section 1B1.1 is the starting point for sentencing — yet the application of a retroactive Guideline is not supposed to be a new sentencing. See Dillon v. United States, ___ U.S. ___, 130 S.Ct. 2683, 177 L.Ed.2d 271 (2010); § 1B1.10(a)(3). Guideline 1B1.10 (entitled "Reduction in Term of Imprisonment as a Result of Amended Guideline Range") is the right starting point when a prisoner seeks a lower sentence. And § 1B1.10(b)(1) tells a court not to work through the sequence in § 1B1.1 as if it were sentencing the prisoner afresh. It provides (emphasis added):

In other words, the court is supposed to start not with § 1B1.1, but with its own original calculation, then swap the amended Guideline range into that calculation without making any other change. See also § 1B1.10 Application Note 2. This means that, if § 5G1.1 did not affect the original calculation, it does not come into play when a court considers the effect of a retroactive change to the Guidelines.

Guideline 1B1.10(b)(2)(B), which we have already mentioned, reinforces this understanding. It says that prisoners whose original sentences are below the presumptive statutory minimum are eligible for a reduction "comparably less than the amended guideline range". Under the prosecutor's approach, however, the range of prisoners who have received a lower sentence to reward substantial assistance would be reset at the statutory minimum, and they could lose all benefit of the retroactive change to the Guidelines, as Wren and Moton did. The prosecutor does not suggest any reason why the Sentencing Commission would have disabled persons who provided substantial assistance from receiving the benefit of the lower penalty for crack cocaine. Nothing in the revised Guidelines, or the explanations for them in Amendments 750 and 759, hints at a goal of giving uncooperative defendants greater sentence reductions than those available for cooperative defendants. Reading § 1B1.10 the way we have done allows both types of defendants to gain and preserves the reward for cooperation, a reward that the prosecutor's reading would diminish or even abolish.

According to the prosecutor, several other courts of appeals have held that a recalculation of the Guideline range after a retroactive change requires the district court to return to § 1B1.1, and thus to § 5G1.1, which resets the range at the presumptive statutory minimum. See United States v. Doe, 564 F.3d 305 (3d Cir.2009); United States v. McClain, 691 F.3d 774 (6th Cir.2012); United States v. Glover, 686 F.3d 1203 (11th Cir.2012); United States v. Williams, 549 F.3d 1337 (11th Cir.2008). What these decisions have in common is that the original Guideline calculation entailed the use of § 5G1.1 to tie the range to the statutory minimum sentence. Then a straightforward application of § 1B1.10 means that the statutory minimum continues to govern — for § 1B1.10(b)(1) says to plug in the revised Guideline and "leave all other guideline application decisions unaffected." One of the "unaffected" decisions is the use of a statutory minimum sentence to supersede the otherwise-applicable range. Guideline 1B1.10(a)(2)(B) provides that a district court cannot reduce a sentence when "[a]n amendment ... does not have the effect of lowering the defendant's applicable guideline range" — and Application Note 1(A) observes that the operation of a statutory minimum term of imprisonment produces one of the situations in which the amendment does not lower the applicable range.

The original calculation for Wren and Moton did not include the use of § 5G1.1 to set the range at a statutory minimum, so the command in § 1B1.10(b)(1) to use the new range and "leave all other guideline application decisions unaffected" means that the new range must not be reset to equal the presumptive statutory minimum. For most prisoners the statutory minimum continues to limit the district court's authority, because a statute prevails over an inconsistent Guideline. See Dorsey, 132 S.Ct. at 2329, 2335; Neal v. United States, 516 U.S. 284, 116 S.Ct. 763, 133 L.Ed.2d 709 (1996); United States v. Robinson, 697 F.3d 443 (7th Cir.2012); United States v. Cannon, 429 F.3d 1158 (7th Cir.2005). But when a district court is authorized (by the prosecutor's substantial-assistance motion or a safety-valve reduction) to give a sentence below the presumptive statutory floor, that authority is equally applicable to a sentence-reduction motion after a change in the Guideline range.

Only one decision we have found deals with the situation in which Wren and Moton found themselves — an original Guideline range above the statutory floor, a sentence below that floor because of substantial assistance to the prosecutor, and a retroactive change to the Guidelines that (apart from § 5G1.1) permits a reduction in the sentence. United States v. Liberse, 688 F.3d 1198 (11th Cir.2012), holds that in these circumstances the district court may grant a motion under § 3582(c)(2) without resetting the Guideline range at the statutory minimum. We agree with that conclusion, for the reasons we have given.

The Sentencing Commission may want to take a close look at the way § 1B1.10(b)(1) works when the original sentencing range is at a presumptive statutory minimum. It is difficult to see why prisoners in that situation who received a substantial-assistance or safety-valve sentence should be excluded from a retroactive Guideline reduction, while prisoners whose original ranges were just slightly above the statutory floor are eligible for the benefit of the retroactive change. That is how the Guidelines work as currently written, however. Wren and Moton are entitled to seek relief under § 3582(c)(2) as the Guidelines stand, and we remand so that the district judges may exercise the discretion they possess.

VACATED AND REMANDED.