The Fractal Murders

by Mark Cohen

“First,” I said, “get into counseling. Once a week for at least a year. You pick the therapist, then send me a canceled check or receipt every so often so I know you’re sticking with it.” He nodded. Hatred and shame were turning to respect and relief.

  “What’s the other condition?” he asked.

  “Don’t get between me and Jayne Smyers.”

  I spent the afternoon at Troy’s gym and worked off some aggression. The heavy bag was Polk’s body, the speed bag Finn’s face. I was still mad at Finn because of my problems with Jayne. My brother hadn’t seen me work that hard in a long time. “You coming out of retirement?” he joked.

  “Why not?” I said. “I’m younger than Foreman and Holmes.”

  When I returned home I had a message to call Dick Gilbert. It was just after five. I punched in his number. “What’s up?” I asked.

  “You’re a smart son of a bitch,” he said in his gravel-edged voice.

  “Mind if I quote you in my next yellow pages ad?”

  “We hit the jackpot,” he said. I could tell he was working on a cigarette.

  “Tell me about it,” I said.

  “I had a technician from the state patrol check Fontaine’s home computer and the one he had in his campus office. They used some type of utility program to recover deleted files.”

  “Keep talking.”

  “The hard drive on the office computer contained portions of letters he’d apparently written to the other two concerning some type of project they were working on.”

  “That’s great,” I said.

  “It gets better.”

  “Yeah?”

  “After the state patrol guy told me what he’d found, I got to thinking about your theory. I figured if these three had developed some sort of model to predict the stock market, and if it was good enough that someone was willing to kill them for it, maybe Fontaine would’ve kept another set of documents or disks or whatever. So I drove out to his parents’ farm and spoke with that kid Bartels. Turns out Fontaine had a little office at the farm. Just a small room in one of the barns where they keep some of the combines and tractors. We found all sorts of correspondence between Fontaine, Underwood, and Carolyn Chang. It looks like she developed the model and shared it with the other two. I don’t really understand the technical stuff, but I sent you copies of everything by overnight mail. You’ll have it tomorrow.”

  “That’s fantastic,” I said. “We’re closing in on this thing.”

  “There’s been one other development,” he said.

  “What’s that?”

  “The bureau requested the gun.”

  “You didn’t give it to them?”

  “Hell no.”

  “Where’d the request come from?”

  “Denver.”

  “Who requested it?”

  “Polk.”

  “It didn’t come from the agent in charge?”

  “It came from Polk,” he said.

  “They’ll probably hit you with a lot of paperwork real soon,” I said. “We need to wrap this thing up before some federal judge threatens to hold you in contempt.”

  “It would be a welcome vacation,” he said.

  “I’ll wait until I get the documents tomorrow,” I said, “then I guess we have to think about going to the bureau.”

  “You really think this guy Polk is capable of this?”

  “We found out he flew to Boston just before Underwood’s death.”

  “Christ.”

  “The weird thing is, he flew under his own name and paid with a government credit card.”

  “When was that?” he asked.

  “Just before Valentine’s Day,” I said.

  “Hang on a second,” he said. He put down the phone and I heard some background noise. The rustling of papers, the opening and closing of file cabinet drawers. “Here it is,” he said. “There was a big conference on sex crimes in Boston that week. I was gonna go myself, but I canceled because of my daughter’s health problems.”

  “Polk goes to a sex crimes seminar. Underwood’s death is set up to be autoerotic death.”

  “I still don’t have enough to charge Polk with Fontaine’s murder,” he said.

  “No,” I said. “You don’t have his prints on the weapon and you can’t place him in your area at the time of the offense.”

  “How’d you find out he was in Boston?” Gilbert asked. I told him. “I’ll see what I can do on my end,” he said. “Maybe I can place him here or in Nebraska.”

  33

  MORNIN’, PEPPER,” WANDA SAID. She reached for my Foghorn Leghorn mug.

  “Mornin’, Wanda,” I said. As always, the place smelled wonderful, the aroma of just-baked treats saturating the air.

  “You look happy,” she said.

  “Been waiting a long time for something,” I said, “and it looks like it’s finally going to happen.”

  “Glad to hear it,” she said. I treated myself to a hot pecan roll, poured some coffee, left three dollars on the glass counter, and found an empty booth. It was ten-thirty A.M. and I’d just come from the post office.

  Gilbert had sent the documents in one of those large bubble envelopes. I tore it open and began studying its contents. The letters, memos, graphs, and other papers had been jammed into the package in no particular order. It took nearly two hours to arrange them in chronological order and review them. I ignored some passages that were simply over my head, but I understood enough to learn what had brought Paul Fontaine, Carolyn Chang, and Douglas Underwood together.

  At about the time she started dating Dale Hawkins, Carolyn Chang developed an interest in the possible applications of fractal mathematics in the business world. She immersed herself in the literature and learned that many academics and investors were already using fractal mathematics and related concepts to predict market behavior.

  As Carolyn became more familiar with the subject, she realized that everyone who had written on the topic had treated the concept of time in the same fashion. A second was a second, a minute was a minute, and so on. Identical units of time each received the same weight. This was the conventional approach, but Carolyn thought it simplistic.

  In real life, Carolyn knew, activity frequently occurs in clusters. Little happens when people sleep, for example, but much happens during the day. She wondered whether models designed to predict market behavior might be improved if time periods filled with activity received greater weight than those during which nothing happened. She called this approach “intrinsic time” and formalized her idea in an unpublished paper entitled “The Use of Intrinsic Time in Predicting Market Behavior.” In explaining her concept, she wrote:

  Quite simply, intrinsic time compresses time when little happens and expands time when much happens: Seconds consume less time during the Asian lunch break than during the American lunch break, for example, because American traders eat lunch at their desks while continuing to trade.

  She shared her idea with Fontaine because she knew him and knew of his interest in the stock market. Fontaine liked Carolyn’s concept of intrinsic time and helped her refine it. He urged Carolyn to contact Underwood because of Underwood’s ability to develop software to implement her theoretical ideas.

  Working together the three developed mathematical models and corresponding software designed to predict the behavior of various economic markets. For months they monitored various markets and sent highly technical papers back and forth. Their preliminary studies demonstrated that Carolyn’s concept of intrinsic time had tremendous potential. They continued to refine the idea and finally decided to subject it to a more rigorous “real world” test.

  To test Carolyn’s idea the three decided to focus on one particular market index. Fontaine suggested they use the S&P 500. Using Fontaine’s theoretical ideas about market behavior, Carolyn’s concept of intrinsic time, and Underwood’s skill in creating computer programs, the three developed a model designed to predict the behavior of the S&P 500. Underwood created a sixteen-gig
abyte database containing every tick in the S&P 500 dating back to 1983. Then he put his neural networks to work and started looking for fractal patterns.

  He found them. In one year of testing, the Chang-Fontaine-Underwood model had beaten the S&P 500 threefold. Intrinsic time worked. That fact established, they began working on a paper intended for publication. After again explaining the concept of intrinsic time, they wrote:

  Having redefined time in this way, the computer then draws a series of graphs incorporating assumptions about how the different traders in the market will react to a price change in intrinsic time. Each graph has the same overall shape, but with different slopes according to each trader’s time horizon or risk profile. The model assumes each trader will react in a nonlinear way: little at first to a price rising above its moving average, then with increasing interest, and finally to slacken off as he thinks he has invested enough. The computer merely adds up these models and arrives at an estimate of how the market as a whole will react to an event. It does not work with normal time, but it works quite well in intrinsic time.

  Nothing in the documents suggested that the three had ever attempted or even considered selling their idea to a brokerage house or consulting firm. I remembered asking Russ Seifert how one who had developed a good model could make money if it wasn’t really possible to sell the model. “You go into business for yourself,” he had said. “Either that or you publish and hope to win the Nobel Prize.” But what if making money had never been their goal?

  Nothing indicated that the three had ever considered going into business for themselves. They had intended to publish their work for all the world to see. Their correspondence contained no mention of seeking the Nobel Prize or any other award. Maybe they never felt their contribution was that significant.

  But someone had learned of their work and concluded that the idea was valuable. Worth killing for. I didn’t know who that someone was, but as an economist and Carolyn Chang’s occasional lover, Dale Hawkins seemed a good bet.

  “Hey, Wanda,” I said from my booth, “can I use your phone?” My cell phone does not work in Nederland because there are no cell phone towers up here.

  “Sure,” she replied. I dialed Scott.

  “McCutcheon,” he said.

  “You got any time this afternoon?” I asked.

  “I have to help Bobbi put down some mulch, then I’m free.”

  “Two o’clock at Moe’s?”

  “I’ll be there,” he said.

  “It’s an interesting concept,” Scott said as he finished reviewing the stack of documents. He wore tan shorts, a white T-shirt with grass stains on it, leather sandals, and a baseball cap with the National Rifle Association’s logo on it. He’s not really a member, but he gets a kick out of wearing things like that in the liberal enclave of Boulder. I’d summarized the chronology of events for him, then allowed him to study the model and test data while I read the News. We were seated at an outside table on a warm but overcast afternoon. “How much do you think you could get for something like this if you sold it to one of these consulting firms?” he asked.

  “You probably wouldn’t sell it,” I said. “You’d probably go into business for yourself.” I told him what Russ Seifert had told me.

  “Doesn’t look like they were planning to go into business for themselves.”

  “Looks like they planned to publish their research in some academic journal,” I said. “I’m guessing Carolyn shared her work with Hawkins. He saw dollar signs and decided to claim the idea as his own. We find the connection between Hawkins and Polk, we solve the case.”

  “What about Underwood?” he asked. “He worked for an economic consulting firm, didn’t he?”

  “New Paradigm Systems.”

  “Maybe he presented the model to them and they decided they wanted it.”

  “I thought about that,” I said, “but the guy who runs that company didn’t strike me that way. And I’m not aware of any connection between New Paradigm Systems and Polk. We know there’s a connection between Polk and the Koch Group.”

  “So in addition to connecting Hawkins and Polk, we have to connect Hawkins with the Koch Group.”

  “There’s a connection,” I said. “Koch got real fidgety when I asked if he knew Dale Hawkins.”

  “I can work on the connections,” he said, “but you’re talking about phone records and bank records. The bureau can do that a lot easier than we can.”

  “I know,” I said, “but let’s see what we can do on our own in the next few days.” He gave me a skeptical look; he knew I had a long history with Polk and wanted to go as far as I could on my own. I went inside to refill my drink.

  “Oh, by the way,” Scott said as I resumed my seat, “I think I’ve about exhausted the airline reservation angle. I can’t find any evidence that Polk flew to Nebraska before Carolyn was murdered. If he did, he didn’t use his own name and he didn’t use any of his credit cards.”

  “He might have driven to Nebraska,” I said. “I can’t believe I forgot to tell you this.”

  “What’s that?”

  “There’s a retired homicide detective who lives near Jayne. He told me the federal agents all drive Crown Victorias now. He saw a dark blue one, brand new, park near Jayne’s town house the day she called me about the break-in. The guy driving it fits Polk’s description. Guess what the plate prefix was.”

  “A-M-K.”

  “Yeah. He thinks it’s a dummy plate—the kind they use for undercover ops—but he promised to run it for me.”

  “Why would he drive to Lincoln?” Scott asked.

  “Gives him mobility while he’s there,” I said. “And it eliminates the risk that someone on a commercial flight might remember him.”

  “If he flashes his badge at Carolyn, that might explain why she’d get into his car so willingly.”

  “It would also explain how the killer got into Fontaine’s house and Underwood’s apartment without breaking anything and without a struggle.” Scott nodded, went inside, and emerged with a glass of iced tea and another bagel. To the west I noticed the afternoon clouds thickening. Thunderstorms were certain.

  “So how are things with you and the math professor?” Scott asked.

  “Not so good,” I said. I told him about Finn’s disclosure of my manslaughter arrest to Jayne. I also told him about Finn’s relationship with Amanda and how I’d handled Finn the previous morning.

  “You let him off easy,” Scott said. I shrugged. Thinking about my problems with Jayne was bringing me down and Scott could see it.

  “What are you doing this weekend?” he asked.

  “No plans,” I said.

  “I’m thinking of going camping. Bobbi bought me some night-vision goggles for my birthday, if you can believe that, and I’m dying to try ’em out.”

  “That sounds like a plan,” I said. “I could use a little rest and relaxation.”

  “We can leave tonight if you want.”

  “No, I want to talk with Jayne. Let’s leave tomorrow.”

  Jayne wasn’t home, so I parked the truck and sat on the cement slab leading to her front door. I stared at the Russian olive trees beside the creek and wondered what idiot had brought them to the United States in the first place. My cell phone rang. It was Tom Hammond.

  “It’s a dummy plate,” he said. “The car belongs to the Denver office of the FBI.”

  “I guess that’s no surprise,” I said, “but thanks for your efforts.”

  “Sure thing,” he said. I almost hung up, but decided to ask one more question.

  “Hey, Tom?”

  “Yeah.”

  “If a law enforcement agency in Nebraska had asked the Colorado DMV to provide a listing of all vehicles meeting that description with an A-M-K prefix, would the DMV have given out the information about the car registered to the bureau?”

  “Not right away,” he said. “The DMV gets these requests all the time. The clerks who handle them don’t even have access to that i
nformation, so the decision on disclosure has to be made at a higher level.”

  “Okay,” I said. “Thanks again.” I couldn’t blame Amanda for sloppy police work. She’d had no reason to believe the FBI might own the mystery vehicle.

  It was three-thirty P.M. My butt was sore from sitting on the concrete. Jayne showed up around five. She wore tan slacks and a white sleeveless shirt. She wasn’t happy to see me.

  “Hello,” she said. I stood. She walked past me and unlocked the door, but I didn’t follow her in and she didn’t invite me. I saw her set her briefcase and purse down on the kitchen table.

  “Let’s take a walk,” I said from the entrance. “I’ll tell you about it.” She removed a pitcher of water from the refrigerator and poured herself a glass. When she had finished drinking it, she walked toward me and pulled the door shut behind her.

  We started walking east on Pearl Street. “I thought I’d bring you up to date on a few things,” I said. She didn’t respond, but she continued walking with me, so I summarized the latest developments in the case. She listened patiently and, despite her anger toward me, I sensed a certain satisfaction. We hadn’t pieced it all together yet, but we had effectively established that the three deaths were related. I told her we’d have to turn it over to the bureau sooner or later.

  “You’ve done a good job,” she said. “Do I owe you any more money?”

  “No.” We turned right on Ninth Street and walked south to Boulder Creek. A pedestrian path follows the creek, and some high school kids were tubing in the clear water. I knew they’d have to call it a day soon because of the approaching storm.

  “You remember my dog Wheat?” I asked.

  “Yes.” She didn’t look at me.

  “I adopted him a few years ago,” I said. “He’d been abused and I read about him in the paper.” I pulled a photocopy of the article from my shirt pocket and handed it to her. She stopped to read it.

  Denver—A Denver man was arrested on charges of cruelty to animals when his roommates reported him to the police after they found his puppy whimpering and unable to walk.