OPINION

EASTERBROOK, Circuit Judge.[fn*] [fn*] Of the Seventh Circuit, sitting by designation.

Grain Processing Corporation (GPC) owns U.S. Patent No. 3,849,194 (the '194 patent). Claim 12 of this patent covers:

A waxy starch hydrolysate having

a dextrose equivalent value between about 5 and about 25,

a descriptive ratio greater than about 2, said descriptive ratio being the quotient obtained by dividing the sum of the percentage of saccharides, dry basis, having a degree of polymerization of 1 to 6, by the dextrose equivalent value,

a monosaccharide content in the range of from about 0.1 percent by weight, to about 2.4 percent by weight, dry basis,

a disaccharide content in the range of from about 1.3 percent to about 9.7 percent, by weight, dry basis, and

being further characterized as capable of producing an aqueous solution of exceptional clarity and substantially complete lack of opaqueness when said hydrolysate is added to water.

When this patent was sought in 1966, CPC International Inc. made the only products meeting this description. They did not sell well. Today products of this general kind, known as low-dextrose malto-dextrins, are important ingredients in a wide variety of foods. The '194 patent issued on November 19, 1974. By the time it expired in 1991, food manufacturers were buying more than 200 million pounds of malto-dextrins annually. The product serves as a binder that also is useful for adjusting the viscosity and freezing point of foods. Malto-dextrins can be employed as bulking agents for other ingredients, such as aspartame, that lack sufficient volume to permit accurate dispensing. Low-dextrose malto-dextrins are tasteless and clear in solution, ideal properties for many purposes. Malto-dextrins today can be found in cake mixes, gravy mixes, candy, glazed nuts, instant coffees, soup, and a hundred other foods. Apparently the product has other handy properties: the oil industry has used it as an aid in drilling wells!

GPC, which acquired the patent and associated business from CPC on October 10, 1979, discontinued the manufacture of Mor-Rex 1918 and Mor-Rex 1920, the two malto-dextrins CPC made under the patent. (Mor-Rex 1918 had been sold to the food industry and Mor-Rex 1920 to the oil industry; the product was identical but the bags differed.) GPC continued selling its own Maltrin line of malto-dextrins, which are outside the '194 patent because made from non-waxy starch. In 1979 a third firm made low-dextrose malto-dextrins: American Maize-Products Company (AMP). AMP employed waxy starch; GPC began with non-waxy starch. On May 12, 1981, GPC filed this patent-infringement action against AMP. After a trial this court, and later the court of appeals, concluded that some of AMP's output infringed claim 12 of the '194 patent. 840 F.2d 902 (Fed. Cir. 1988). AMP promptly changed its method of producing malto-dextrins, believing that the new process yielded a product with a descriptive ratio (D.R.) under 1.9, the numerical equivalent this court adopted for the language "about 2" in the '194 patent. Tests conducted by AMP using the Lane-Eynon procedure showed that the new process avoided infringement. Further proceedings led the court of appeals to hold that the dextrose equivalent (D.E.) value, the denominator in computing the D.R., was to be determined by the Schoorl procedure, which the inventors themselves had used. 21 U.S.P.Q.2d 1474, 1991 WL 59343 (Fed. Cir. 1991). The Schoorl procedure gives a lower D.E. value, and therefore a higher D.R., than does the Lane-Eynon procedure. Tested using the Schoorl procedure, 82.5% of AMP's post-1988 production of Lo-Dex 10, a malto-dextrin with a D.E. in the range of 9 to 12, had a D.R. exceeding 1.9. FAMP concedes that Lo-Dex 10 meets the other elements of claim 12, so infringement has been established and only damages remain to be determined. The case was reassigned to me after the retirement of Judge Parsons, who conducted the liability trial and associated proceedings. A damages trial was held on July 10, 11, and 12; this opinion records my findings of fact and conclusions of law.

I

"Upon finding for the claimant the court shall award the claimant damages adequate to compensate for the infringement, but in no event less than a reasonable royalty for the use made of the invention by the infringer, together with interest and costs as fixed by the court." 35 U.S.C. § 284. An aggrieved patent holder thus receives the greater of lost profits or a reasonable royalty. GPC submits that it lost profits of almost $35 million, which with prejudgment interest implies an award exceeding $68 million. Alternatively GPC seeks a reasonable-royalty award representing 28% of AMP's Lo-Dex 10 revenues. For its part, AMP submits that GPC is not entitled to any lost-profits award (if only because Maltrin 100, the product competing with Lo-Dex 10, was not covered by the '194 patent) and that a reasonable royalty would be 1% to 3% of sales of infringing Lo-Dex 10. I start with the reasonable-royalty question, because the right way to determine a reasonable royalty has a profound effect on GPC's claim for lost profits.

Lewis M. Koppel, GPC's expert on the royalty question, proceeded roughly as follows. First determine how much GPC or its predecessor CPC would have lost by granting a license, which sets the patent holder's minimum demand. Koppel thought that this was 7.7% of AMP's anticipated sales. (Anticipated is a key word; Koppel assumed that the parties conducted their negotiations in 1974, when AMP was just entering the business, and never returned to the subject during the patent's 17-year life.) Then determine how much AMP made by manufacturing Lo-Dex 10, which sets AMP's maximum offer. Koppel concluded that AMP would have offered as much as 43% of net sales. Where the parties would strike a bargain in a hypothetical negotiation would depend on the factors identified in Georgia-Pacific Corp. v. U.S. Plywood Corp., 318 F. Supp. 1116, 1120 (S.D.N.Y. 1970), modified, 446 F.2d 295 (2d Cir. 1971). See also SmithKline Diagnostics, Inc. v. Helena Laboratories Corp., 926 F.2d 1161, 1168 (Fed. Cir. 1991). After canvassing these factors, Koppel opined that 28% of net sales was the most likely royalty.

Raymond S. Sims, approaching the same subject on behalf of AMP, took a strikingly different approach. Sims observed that CPC and GPC offered to license their portfolios of malto-dextrin patents but found few takers; several potential users rejected a 5% royalty out of hand as too high. E.g., DX 38. Other improvement patents in the business were licensed at rates of 1% to 3%. These actual negotiations informed the hypothetical-negotiation approach to the reasonable-royalty question. Why did royalties not exceed 3%? Sims thought the bulk food products business a low-margin affair, unable to spend lavishly — but willing to invest in designing around patents. When negotiating for a license to use the '194 patent, AMP would not have been willing to offer its total profits as the maximum royalty, Sims observed, because AMP had other options — principally to invest in a production process that could produce a 10 D.E. malto-dextrin with a D.R. less than 1.9.

AMP has used four methods to produce Lo-Dex 10 (initially called Fro-Dex 10; I use the Lo-Dex name throughout for simplicity). Process I was in place between 1974 and 1982. In mid-1982 AMP switched to Process II, which it employed until the appellate opinion of 1988 found that some product made by Process II infringed the '194 patent. Then AMP switched to Process III, with the results — disastrous in retrospect — that I have described. Once the Federal Circuit concluded that the Schoorl process must be used to measure the D.E. for purposes of the '194 patent, AMP switched to Process IV. GPC has stipulated that Lo-Dex 10 made by Process IV does not infringe the '194 patent. When the patent expired in November 1991, AMP switched back to Process III. This history supplied the kernel of Sims's approach. Although Sims believed that Process I was a candidate for non-infringing production (a position I consider and reject in Part III.2 of this opinion), his principal benchmark was the cost AMP incurred to adopt Process IV. If it could make a non-infringing Lo-Dex 10 by Process IV, AMP would never pay a royalty greater than the cost difference between Process III and Process IV. Sims concluded that the cost difference was approximately 2.3%. Koppel conceded on the stand that if AMP had this manufacturing option in 1974, then the cost difference would set a cap on the royalty. Tr. 263. Koppel qualified this answer by observing that the cost difference would set a cap only if the two processes produced products that buyers viewed as equivalent. For example, if the non-infringing Lo-Dex 10 had an off-taste, and customers therefore would not pay as much for it, AMP's maximum royalty offer would be the sum of the production cost increase and the selling price decrease. But I find that the Lo-Dex 10 produced by Process IV was indistinguishable from customers' standpoint. Randolph Holme, who is responsible for Lo-Dex production, did not receive a single complaint or inquiry during the time Process IV was in use. Tr. 395. The Lo-Dex 10 made by Process IV had a lower D.R. (that was, after all, the objective of the switch), but no one argues that any customer cared a whit about the product's descriptive ratio.

But was Process IV available to AMP for the entire life of the patent? Science progresses, and AMP's ability to make a non-infringing 10 D.E. malto-dextrin in 1991 does not alone show that it could have done so in earlier years. AMP had trouble coming up with any reliable way to produce a 10 D.E. malto-dextrin until it implemented Process I in 1974. The evidence also shows, however, that a firm capable of producing a 10 D.E. malto-dextrin by Process I in 1974 could, at some higher cost, have produced the product by Process IV and thus avoided infringing the '194 patent. Understanding this conclusion requires a short, non-technical tour of the way malto-dextrins are made.

GPC and AMP use corn to produce starch. (The way they accomplish this is not material.) Starch is a complex chain of carbohydrates, (C₆H₁₀O₅) _n, where n represents an arbitrary number of units, as low as 50 and as high as several thousand. To make a malto-dextrin, the producer puts the starch through hydrolysis, a chemical reaction with water that severs the chain and converts some of the C₆H₁₁₀O₅ to dextrose, C₆H₁₂K₆. Breaking up the starch molecule is facilitated by the addition of an enzyme In 1974 AMP used a single enzyme, an alpha amylase. To make the process work, AMP needed to find the right concentration of enzyme, the right temperature, the right pH for the solution, and the right time to let the reaction continue, among other variables. Different conditions yield different dextrose equivalents, and each D.E. has its own chemical properties.

In 1982 AMP decided that it could save money by reducing the amount of alpha amylase in the solution. To achieve the same end result with less enzyme, AMP allowed the reaction to continue longer. The product, according to measurements under the Lane-Eynon process, occasionally had a D.R. exceeding 1.9, which led to the finding of infringement in 1988. Holme, who became manager of Lo-Dex production in December 1986 (Tr. 367), testified that he and his colleagues thought they knew what to do: reduce the variability of output. They traced variability to the relatively long reaction time and decided to move back in the direction of the 1974-82 process. So they increased the amount of alpha amylase, adjusted the temperature and pH, and reduced the reaction time. These steps, coupled with extra monitoring, were calculated to yield a more uniform product (Tr. 368-69). Holme was determined to avoid shipping a single bag of Lo-Dex 10 with a D.R. exceeding 1.9. Process III worked as intended. Unfortunately, AMP assumed that the Lane-Eynon test defined the D.E. for purposes of the '194 patent. In 1991 AMP learned of its mistake, and it realized that its extra-consistent process was consistently producing descriptive ratios exceeding 1.9 when the Schoorl method was used to measure the dextrose equivalent.

AMP could not alter the D.E. and still ship the 10 D.E. product customers wanted. If it could not change the denominator in the D.R., then it had to change the numerator, DP_1-6 (a measure of average saccharide molecule chain length). A process that achieves the same D.E. while lowering DP_1-6 may avoid infringement. Assisted by AMP's Research Department, Holme set out to reduce DP_1-6 while holding the D.E. constant. Chemists under the direction of Frances R. Katz, head of research at AMP, decided to add a second enzyme, a glucoamylase. Unlike alpha amylase, which breaks the starch molecule in the middle, glucoamylase breaks it at the end, chopping off single C₆H₁₀O₅ units. Shorter average saccharide length means a lower DP_1-6. That glucoamylase has this effect has long been recognized. Holme testified that this effect was widely known when he entered the business in 1972 (Tr. 377), and he pointed to the 1967 edition of a standard textbook containing the same information. II Starch: Chemistry and Technology 556 (Roy L. Whistler Eugene F. Paschall eds. 1967). Fliers distributed by enzyme manufacturers before 1974 contained similar information. E.g., DX_123. See also Tr. 376-78, 411-14.

Of course knowing the effect of glucoamylase was not enough to make a 10 D.E. malto-dextrin with a D.R. below 1.9. One also had to know how to stop the reaction before the glucoamylase propelled the D.E. beyond 10, and how to ensure that the finished product would have the other attributes that make Lo-Dex 10 valuable to customers. Katz's staff employed the firm's pilot plant to experiment with combinations of glucoamylase, alpha amylase, pH, heat, and time, until the result was acceptable. Tr. 376, 383-85. In 1991 this took approximately two weeks. As I have recounted, customers were satisfied with the result. But glucoamylase is an expensive enzyme, and its use elevated AMP's total costs, which is why AMP returned to Process III as soon as the '194 patent expired.

Could AMP have done the same thing in 1974? Glucoamylase was around, and its effect was known, but it took experience with the malto-dextrin production process to know how to use it to good effect. In 1974 AMP was having difficulty making a 10 D.E. malto-dextrin. It encountered trouble with filtration, among other obstacles. Liability Tr. 2290, 2326. Modern tests of the few retained samples of early Lo-Dex 10 show D.E. values ranging from 8.3 to 19.7 (PX₄₄₆ page 1). Mastering a more complex process while obtaining a satisfactory product cannot have been easy. Experience is as great an ingredient in this solution as any enzyme — although Robert N. Ammeraal testified in the liability trial that AMP's troubles in 1974 were unrelated to the use of enzymes. Liability Tr. 2290, 2326. As it happens, however, I need not decide whether AMP could have employed the dual-enzyme process from the get-go. CPC did not transfer to GPC any right to recover damages for periods before assignment of the patent, so GPC concedes that it is not entitled to damages for any of AMP's sales preceding October 10, 1979. By then, I am confident, AMP could have produced a non-infringing 10 D.E. malto-dextrin by a dual-enzyme process. No evidence undermines Holme's and Katz's testimony; GPC did not present a scientist or production manager to say that a dual-enzyme process would have been beyond reach, or even a daunting task, in 1979, or for that matter 1974. If AMP had negotiated a license in 1974 it could have dropped the license in 1979, or it could have renegotiated the rate by demonstrating ability to make a non-infringing product. (I reject Koppel's implicit view that any royalty negotiated in 1974 necessarily would have held constant until 1991, even if changes in production technology enabled AMP to make a credible threat to produce a 10 D.E. malto-dextrin without practicing the patent. GPC might have negotiated a license that curtailed renegotiation, but the royalty would have been lower.)

My conclusion that AMP has carried its burden of establishing that it could have produced a non-infringing product no later than October 1979 scotches GPC's request for lost-profits damages. The premise of GPC's claim is that it was injured by AMP's sale of a 10 D.E. malto-dextrin; if AMP, unlike the infringer in Rite-Hite Corp. v. Kelley Co., 56 F.3d 1538 (Fed. Cir. 1995) (en banc), was able to offer perfectly lawful competition, then GPC cannot show lost-profits damages. With infringing Lo-Dex 10 banned, the customers' substitute is non-infringing Lo-Dex 10. See Slimfold Manufacturing Co. v. Kinkead Industries, Inc., 932 F.2d 1453 (Fed. Cir. 1991); TP Orthodontics, Inc. v. Professional Positioners, Inc., 20 U.S.P.Q.2d 1017, 1023, 1991 WL 187189 (E.D.Wis. 1991), affirmed, 980 F.2d 743, 1992 WL 281030 (Fed. Cir. 1992). True enough, non-infringing Lo-Dex 10 would have cost AMP more to produce, ordinarily implying a higher price; and a higher price for AMP's product would have driven some of its customers to GPC. But both of GPC's expert witnesses concluded that AMP had substantial profit margins on its Lo-Dex business. This large margin is what economists call a quasi-rent, and a change in economic rent (caused by, say, somewhat higher variable costs) does not affect prices. George J. Stigler, The Theory of Price 263-65 (4th ed. 1987). Suppose AMP's margin fell from 30% to 27% because of a change in enzyme costs; it would not raise its price, because that would just drive away business, and it would lose the 27% rent still obtainable on the sale. Thus it is no surprise that AMP did not increase its price when it adopted the dual-enzyme method in 1991. An AMP using the dual-enzyme method between 1979 and 1991 therefore would have sold the same product, for the same price, as the actual AMP did, and GPC cannot establish lost profits. The approach of GPC's damages expert is questionable on several other grounds as well, but as this shortcoming is fatal I need not explore them.

What, then, is the reasonable royalty to which GPC is entitled? Sims calculated a cost difference approximating 2.3%. There must have been additional start-up costs (which no one attempted to quantify), but amortized over the life of the process these cannot have been significant. GPC observes that AMP's accounting staff computed a "standard gross profit margin" of 9.8% for Process III Lo-Dex 10 in 1991 and 2.86% for Process IV Lo-Dex 10 (DX 6 at M-9666), but this difference does not call Sims's calculations into question; many costs vary through the course of a year, but the only ones relevant to this case are the incremental costs of avoiding infringement. Nonetheless, I hesitate to set the royalty at Sims's figure for glucoamylase cost differences. Sims also concluded that the cost difference between Process I and Process II ran from 0.7% to 3.2%, depending on year (DX 78 at 16). The variation is attributable to changes in the cost of the alpha amylase enzyme. If as this implies the cost of enzymes fluctuates, then the enzyme-intensive Process IV has a variable cost differential compared with the other products. Exhibit 5 to Sims's royalty report (DX 78) contains charts summarizing these cost differentials over time. Sims concludes that they always fall under 2.3%. Still I am uneasy, for the data are aggregated over lengthy periods.

As the infringer, AMP must bear the effects of uncertainty. Story Parchment Co. v. Paterson Parchment Paper Co., 282 U.S. 555, 562-66, 51 S.Ct. 248, 250-52, 75 L.Ed. 544 (1931); Del Mar Avionics, Inc. v. Quinton Instrument Co., 836 F.2d 1320, 1327 (Fed. Cir. 1987); Gyromat Corp. v. Champion Spark Plug Co., 735 F.2d 549, 554-55 (Fed. Cir. 1984). It is not possible to produce a precise royalty figure without evidence that enables one to quantify the cost savings, month by month if not year by year, that a license would have produced (compared with Process IV). GPC was willing to license the '194 patent, and it obtained one license, in settlement of litigation, for a lump sum that approximates 3% of net sales. Tr. 464-65. John Floyd, CPC's corporate patent and trademark counsel in the late 1970s, testified by deposition that patents like the '194 were licensed at the time for royalty rates of 1% to 3% of sales. Floyd Tr. 25-26, 29. If recast as a license, the sale of the business from CPC to GPC was equivalent to a royalty between 0.5% and 4.5% of GPC's sales. My best estimate, no more than that really, is that GPC and AMP as parties to a hypothetical bargain would have settled on a royalty of 3%, by which AMP would have lowered not only the costs of enzymes but also the risk that its best plans would still produce infringing product — a risk that materialized for AMP, much to its chagrin and expense.

II

Beyond their dispute about the royalty rate, the parties disagree about the base to which this royalty applies. GPC seeks a royalty on all production of Lo-Dex 10; AMP believes that it should pay a royalty only on "infringing Lo-Dex 10."

The reasoning that led to the conclusion that Lo-Dex 10 made by Process II infringes the '194 patent — the principal subject of the liability trial and first appeal — laid the groundwork for the current controversy. If one averages all test results for all samples of Lo-Dex made by Process II, the D.R. is a little under 1.9. AMP accordingly argued that Lo-Dex 10, as a product, does not infringe the '194 patent. The Federal Circuit disagreed, holding that any shipment containing a malto-dextrin satisfying the elements of claim 12 is an act of infringement. That AMP shipped non-infringing product every Monday, Wednesday, and Friday would not absolve it of liability for infringing shipments on Tuesday and Thursday. Well, AMP reasons, because the finding of infringement depended on disaggregating the product, the damages also should be based on the infringing shipments exclusively.

The conclusion does not follow from the premise. AMP had a legal duty to avoid infringement. It could have satisfied this duty in several ways. It could, for example, have tested every production run of Lo-Dex 10 and destroyed product having a D.R. above 1.9. But destroying roughly 50% of the Process II product and 82.5% of the Process III product would have made the venture unprofitable, and GPC would have got most of the business. Low yields plague the semi-conductor industry; what is a plague in a high-tech business would be fatal for a producer of foodstuffs. Alternatively AMP could have used Process IV to make non-infringing Lo-Dex 10. The extra enzyme cost would have applied to every bag of Lo-Dex 10.

To avoid the enzyme costs, AMP could have taken a license from GPC (which was willing to license the '194 patent). In these negotiations GPC could have insisted on a price approximating the enzyme costs AMP could have saved by avoiding Process IV, and again these costs apply to the entire production run. A single royalty applicable to the entire production run not only reflects the economics of the transaction but also eliminates the need for repeated testing to determine what portion of the output is subject to royalty, and sidesteps as well the disputes bound to ensue. I conclude, therefore, that the 3% royalty applies to AMP's entire output of Lo-Dex 10.

III

But for what period of time? The parties contest two issues. First, did CPC mark the bags of Mor-Rex 1918 and 1920 with the '194 patent? Unless it did, GPC is not entitled to damages for any period before May 12, 1981, when it commenced this suit. See 35 U.S.C. § 287(a). Second, did the Lo-Dex 10 produced by Process I infringe the '194 patent? Unless it did, the damages period begins in 1982, regardless of the resolution of the marking dispute.

1. Unless CPC marked the Mor-Rex bags with notice of the patent, damages cannot commence before the date "that the infringer was notified of the infringement". Suit supplies such notice; advertisements and similar general announcements to the industry do not. Amsted Industries v. Buckeye Steel Castings Co., 24 F.3d 178 (Fed. Cir. 1994).

Six years into this litigation, GPC sought discovery concerning a product called ARD 2370. AMP resisted discovery of matters preceding a 1983 amendment to the complaint, which for the first time accused ARD 2370 of infringement. The ground for opposing discovery was that the record did not demonstrate that the Mor-Rex bags had been marked with the patent number, so that the amendment to the complaint started the damages period for ARD 2370. Judge Parsons recited the statutory marking requirement and continued: "Discovery must be limited to matters transpiring with relation to such product, ARD 2370, if any, after the date of plaintiff's first notice to defendant with regard thereto, said notice being the filing of the Supplemental Complaint". The order does not explain why Judge Parsons concluded that the bags of Mor-Rex had not been marked with the '194 patent, indeed it does not even contain an express finding that the bags were not so marked, but GPC concedes that the issue was actually litigated in 1987, and that it lost. Now it wants me to reopen the subject and find in its favor on the basis of the following stipulation in the pretrial order:

Subject to the Court allowing GPC to litigate the issue, the parties stipulate to the authenticity and admissibility of Plaintiff's Exhibits 429-434, and that, from and after October 10, 1979, all packages of Mor-Rex 1918 that were sold were marked with notice of the '194 patent in the manner depicted on Plaintiff's Exhibit 432.

Given this stipulation, GPC submits, the 1987 order was clearly incorrect, and things should be set to rights.

That a court has the power to revisit issues resolved earlier in a case no one can doubt. See Mendenhall v. Barber-Greene Co., 26 F.3d 1573, 1580-83 (Fed. Cir. 1994). That the law of the case should not lightly be discarded also is not debatable. Christianson v. Colt Industries Operating Corp., 486 U.S. 800, 815-16, 108 S.Ct. 2166, 2177-78, 100 L.Ed.2d 811 (1988). AMP tells me that it stipulated that the bags of Mor-Rex 1918 were properly marked because GPC had some evidence to this effect, which AMP had no way to contest, for none of the bags now exists. From AMP's perspective, the critical clause was the introduction — "Subject to the Court allowing GPC to litigate the issue" — and AMP has vigorously invoked the law of the case.

It takes a strong reason to revise an earlier ruling. A stipulation based more on the paucity of evidence than on the power of evidence is not such a reason. GPC had to know from the outset of this case in 1981 that, to establish damages for earlier Lo-Dex 10 sales, GPC would have to show that the bags of Mor-Rex had been marked with the patent. (Proving this fact is the patent holder's burden.) When GPC bought CPC's malto-dextrin business, it acquired approximately 2 million pounds of Mor-Rex. GPC may have saved a bag or two. The best time to hunt them up was 1981, not 1987, and certainly not 1995. I must assume, from the tenor of Judge Parsons's order, that GPC presented no evidence at all in 1987. Since then it has come up with some, which AMP cannot meet. This long delay is a reason to leave the court's 1987 decision in place, not to overturn it.

What is more, the stipulation does not resolve the marking question even if taken for all it can be worth. CPC sold its patented malto-dextrin as Mor-Rex 1918 to the food industry and Mor-Rex 1920 to the oil industry. The stipulation does not mention Mor-Rex 1920, yet unless Mor-Rex 1920 was properly marked GPC cannot recover damages for infringement preceding the filing of the complaint. American Medical Systems, Inc. v. Medical Engineering Corp., 6 F.3d 1523, 1538 (Fed. Cir. 1993). Recognizing this problem on the first day of trial, GPC sought discovery on the subject from CPC, which produced a printer's proof of a Mor-Rex 1920 bag marked with the '194 patent. When GPC tendered the printer's proofs on the closing day of trial, AMP protested. How was it to secure contrary evidence, if any there was? Discovery had long been closed. And the nature of the evidence suggests that contrary evidence is conceivable. The proof is dated April 1976, with a revision date of October 1977. If CPC changed its bag in 1977, maybe it made another, later change. Although it is unlikely that CPC would have deleted a patent number, stranger things have happened.

All things considered, I think it best to stick with the decision Judge Parsons made in 1987. Evidence has become less available since then, and although GPC has scared up some information that it did not present to Judge Parsons, the ability of AMP to meet this evidence has eroded. GPC's last-minute evidence about Mor-Rex 1920 leaves open the possibility that additional evidence was available, but it is too late for AMP to seek it out. The record is closed. So, too, the marking question is settled.

2. Most samples of the product made by Process I between 1974 and 1982 were discarded long ago. In preparation for the liability trial, GPC and AMP tested the remaining samples. All of the tests used the Lane-Eynon procedure, and the D.R. always was less than 1.9. Even with the Lane-Eynon D.E. converted to a Schoorl D.E. (using a procedure to which the parties have stipulated), the D.R. of every sample remains below 1.9. A few tests produced a D.R. exceeding 1.9, but other tests from the same lots produced lower descriptive ratios. With the test results on a single lot averaged (a step the parties agree is appropriate), none of the samples had a D.R. exceeding 1.9. It follows, AMP submits, that none of the Lo-Dex 10 made by Process I infringed the '194 patent.

Twenty-eight samples of the Process I malto-dextrin were subjected to analysis. Although Process I was used for eight years, the earliest date on any of the samples is August 1980, and the latest is November 1981. Twenty-two of the samples come from batches made between May and November 1981. These are not representative of the whole production run. Both the D.E. and the D.R. vary with differences in the corn feedstock, in the enzyme used, and in the length of time the starch was hydrolyzed. The strength of the inference to be drawn from tests close in time is accordingly limited. Especially when Katz testified that the output of Process I should on average have been identical to the output of Process II. Tr. 416-20, 423. See also Ammeraal Liability Tr. 2324-25; Owen deposition Tr. 68, 72 (both witnesses saying that the temperature differences between Process I and Process III should not have affected the final product). We know, from much more thorough testing, that 82.5% of the Process III Lo-Dex 10 infringed the '194 patent, and that half of the tested Process II product infringed (though there were only 12 samples of Process II Lo-Dex 10 to test). Does it not follow that at least some of the Process I product infringed?

No, not necessarily. Holme and Katz were concerned about functional similarity, from customers' perspectives, and not the precise chemical details that mark the boundary of infringement. Holme testified that Process III was more carefully monitored to yield consistent results, and the tests bear this out. There are wide swings in the measured D.E. and D.R. of the Process I product (indeed, 19 of the 28 samples show a D.E. outside the 9-12 range that AMP represented to customers that the product would possess), while tests show that the Process III product was more consistent. Suppose Process I and Process III each produce malto-dextrin with an 82.5% chance of having a D.R. exceeding 1.9. The probability of drawing 28 successive samples with a D.R. under 1.9 is (0.175)²⁸, a vanishingly small number. Yet this calculation, which Sims performed, assumes (a) that the samples drawn from the Process I product are independent, which is doubtful, and (b) that the question on the table is whether 82.5% of the Process I product infringes. For current purposes, the question is whether any of the Process I product infringes. The high variability of the test results for Process I, coupled with testimony that Processes I and III ought to yield similar product on average, supports an affirmative answer.

One can ask this a different way. Suppose you have a collection of test results from a small sample of a large population and desire to know whether the upper bound of an attribute possessed by every member of that population exceeds a certain number. A simple way to go about the task is to find the mean and standard deviation of the sample results, then to use the standard deviation (coupled with the assumption that the whole population has a normal, bell-shaped distribution) to determine what fraction of the population exceeds the limit. I have performed this simple statistical exercise. The mean of the Process I test results is 1.70, and the standard deviation is 0.119. This means that, if the full population had a normal distribution, approximately 4% of the Process I Lo-Dex 10 had a D.R. exceeding 1.9 by the Schoorl method. (This also assumes that the 28 observations are independent; if they are not — if, in other words, there are fewer separate observations — the standard deviation is higher and the inferred volume of infringement also is higher.) AMP therefore was a frequent infringer until it adopted Process IV, and it must compensate GPC for all production from May 12, 1981, until the adoption of Process IV in May 1991.

IV

GPC is entitled to prejudgment interest. 35 U.S.C. § 284; General Motors Corp. v. Devex Corp., 461 U.S. 648, 103 S.Ct. 2058, 76 L.Ed.2d 211 (1983). Although Devex leaves some room for a discretionary denial of interest, I do not exercise in AMP's favor whatever discretion I possess. Lengthy delay in the resolution of the case is not a reason to deny interest. Gratuitous delay by the plaintiff in prosecuting the suit might support an adjustment, see Milwaukee v. Cement Division of National Gypsum Co., ___ U.S. ___, ___, 115 S.Ct. 2091, 2096, 132 L.Ed.2d 148 (1995), but there has been none in this case.

Pointing to a stipulation that GPC invests its surplus cash in Treasury securities, AMP asks me to award prejudgment interest at the rate in T-Bills. I reject this argument for the reasons given in In re Mahurkar Double Lumen Hemodialysis Catheter Patent Litigation, 831 F. Supp. 1354, 1394-95 (N.D.Ill. 1993). See also Uniroyal, Inc. v. Rudkin-Wiley Corp., 939 F.2d 1540, 1545 (Fed. Cir. 1991). Interest compensates not only for the time value of money but also for the risk of non-payment. The T-Bill rate is lower than the prime rate in the private sector because the United States is almost certain to repay. That cannot be said of AMP; no one would make a long-term, voluntary loan to AMP at the T-Bill rate. GPC has made an involuntary loan to AMP, creating a debt some of which has been outstanding for more than a decade. The right rate of prejudgment interest in principle is the rate AMP paid for money during the years in question. In default of this figure, courts regularly use the prime rate. GPC is entitled to prejudgment interest at the prime rate, compounded monthly.

V

A few procedural loose ends. I overrule GPC's objection to AMP's designation of some pages from Holme's deposition — but add that I have not relied on these pages in reaching my decision. I grant GPC's motion to admit into evidence the appendices to the rebuttal reports of its expert witnesses. I also grant GPC's motion designating additional deposition excerpts.

To sum up: GPC is entitled to damages measured by a royalty of 3% on AMP's entire net sales of Lo-Dex 10 between May 12, 1981, and April 30, 1991, plus prejudgment interest at the prime rate compounded monthly. Using this formula to determine an award in dollars and cents ought to be a mechanical job. But because neither side used this approach to submit a damages figure, I invite the parties to apply it now. They should confer to determine whether they can agree on a damages award based on this approach. (Such an agreement would not waive anyone's right to contest elements of the approach on appeal.) If they cannot agree, the parties should file short memoranda, with supporting tables and citations to the record, by July 31. I will then resolve any remaining dispute and enter a judgment for the final award.

Addendum

The parties have almost — but not quite — agreed on the calculation of damages using the principles of my July 20 opinion. They agree on AMP's sales and revenues by month and on the prime rate of interest. They do not, however, agree on the starting date for interest. GPC believes that interest should begin at the end of the month in which AMP's sales occur. AMP believes that interest should begin at the end of the quarter in which the sales occur, plus another 30 days. The different methods produce a divergence of some $45,000 in damages. Application of GPC's method leads to an award of $2,462,000, while AMP's implies an award of $2,417,055.

Behind AMP's position lies the fact that the most common licensing arrangement in the industry calls for quarterly calculations, with payments due within 30 days of the quarter's close. The licenses GPC offered to AMP and Anheuser-Busch included such terms. DX 33 (Article 3.06), 35 (Article 3.06). GPC concedes that "there are practical, administrative reasons for payments to be made every quarter, and for there to be a 30-day delay between the end of each quarter and the payment due date." But it points out, accurately, that AMP does not need extra time to gather sales data and remit royalties in this litigation, conducted long after the fact.

The parties have not cited, and I have not found, any case analyzing this question, so I turn to first principles. Prejudgment interest is the only part of the award affected by this issue. And the function of prejudgment interest is to give patent holders damages adequate to compensate for the infringement — "to ensure that the patent owner is placed in as good a position as he would have been in had the infringer entered into a reasonable royalty agreement." General Motors Corp. v. Devex Corp., 461 U.S. 648, 655, 103 S.Ct. 2058, 2062, 76 L.Ed.2d 211 (1983) (footnote omitted). Had AMP entered into a reasonable royalty agreement, GPC would not have received payment until 30 days following the close of the preceding quarter. Not until then could it have invested the money (or, equivalently, used the proceeds to reduce its outstanding debt). The time value of money that GPC lost therefore is only the interest accruing more than 30 days after the close of each quarter.

I conclude that AMP has proposed the proper method. GPC agrees that AMP's calculation correctly applies its close-of-quarter-plus-30-days method, so I will enter judgment in favor of GPC for $2,417,055 plus costs.