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When measuring a set with n members, there are 2n-1 parameters that need to be described. (This is a lot when n isn’t very small.) We could use the λ measure, which is a nonnegative set function and involves a generalization of additivity (i.e. superadditivity and subadditivity—see “Fuzzy Measures”), since it has n – 1 parameters. The thing is, we want fewer parameters, but we also want to be able to look at the low-degree interaction (maybe 2-3 attributes). (See “Touching.”) We want to see the pure interaction between variables. This is done using the Moebius Transformation*, a set function which allows us to catch the pure interaction between or among the attributes in any given set, and something called the k-interactive measure, which allows us to only consider the interaction for up to k attributes. (Because if we looked at all the possible interaction between 100 different variables, from any two of them interaction up through any 99 and all 100 of them interacting, the problem will get very unwieldy. Instead, we may just look at the interaction between, say, any sets of anywhere up to three variables. For large numbers of variables, this saves us a ton of computation.)


* I’m actually theoretically saving this for a “bonus” story, perhaps tomorrow or whenever I have spare time (I’d been thinking tonight, but then this took me til after 5), so I won’t describe it for now. In case I don’t get to the other story, if you want to know more, let me know!



            John wondered whether he ought to move out to the country.

            The thought made him feel old—everything made him feel old these days. Yesterday, he visited Mary at her grave. He was a widower, for god’s sakes. Not that that was anything new, but even after four years he still wasn’t used to it.

            John’s loneliness wasn’t anything new, either. He’d been lonely for much longer than four years. Mary had known it, too; she took it better than John did. Maybe she didn’t understand. Maybe he didn’t understand. Maybe there was nothing to understand. John alternated between theories, when he was feeling masochistic enough to devote much thought to it.

            Most of the time, John was at work. Saving lives was one thing that he could do. Maybe GP work was a bit duller than “saving lives” implied—but it was something. When he wasn’t at work, he read journals. When he wasn’t reading journals, he took walks. Part of the reason he went on walks was to convince himself that even if he did rely on a cane these days, he could still get around.

            The other part of the reason he went on walks was in the hopes he’d find someone to save.

            It was morbid, of course, walking by cafes and outdoor dining areas of restaurants to see if anyone was choking on their food, hanging hopelessly about bridges and bodies of water so he could find out if he still remembered how to swim, should the need arise. He took dark streets because there was always the chance that someone was being pickpocketed, or worse, assaulted, and he could stop the bugger who tried to do it. Cane or not.

            The last time he had seen Sherlock was three years ago—completely by accident. Sherlock hadn’t come to Mary’s funeral, but that was no surprise; John had seen him a few days afterward, but not since then, not until this time. There had been gunshots—gunshots—and John raced down the street to the source, still, after all these years, fearing the worst. It was too late for the victim, though, and the shooter must have slipped out while John scoured the area. Sherlock arrived several minutes later, and confusion carved its way into his face as he recognized John.

            “It wasn’t me,” John had said.

            “John,” Sherlock had said.

            “I couldn’t find him.”


            That was about all he could handle. That was about his limit, these days. It should have gotten better over time, but it didn’t, or it hadn’t then, either way, and John, John the doctor, John the healer, seeing the pain on Sherlock’s face, could only walk way. “Good luck.”

            Sherlock hadn’t followed. John wondered absently if Greg was still working on the force; they hadn’t seen each other for a good long while, and John knew it was because he had never asked Greg about meeting up again after Greg had mentioned the weight Sherlock carried on his shoulders, wishing and hoping for John to come back. It had been especially uncalled-for then; John was still with Mary, Mary was still alive.

            Mary was the most understanding person John had ever known.

            “You should go solve another case with him,” she’d said once, from behind a novel she was reading, as John thumbed through a medical journal.

            “I’m sure he could find another assistant if he needed one.” Of course he couldn’t. He was Sherlock Bloody Holmes. No one was good enough for him.

            “That’s not what I meant, and you know it.” She lowered her book and placed the bookmark, and John set his journal aside. This was going to be one of those discussions.

            “Well, what did you mean, then?”

            “I’ve seen how you get,” she smiled wistfully. “Well, got. John, if there’s any compelling argument in the world for fate, it’s you two.”

            “I don’t think you mean that the way it sounds.”

            She’d just continued on with the wistful smiling. “John, even I miss the two of you being friends, and I didn’t even know you before—”

            John exhaled deeply. “Look, Mary, I—I don’t want to talk about this.”

            “I’m sure he misses your company.”

            “So I’ve been told.”

            Something about his tone must have caught Mary’s attention, because her voice softened. “What happened between you two?”

            Mary, because Mary was good and honest and believed in sharing secrets over a glass of wine and fancy desserts at least once every other month, and because Mary was earnest and sensitive and understood the things that mattered even having never been a doctor or a soldier or almost blown to pieces by a semtex vest, knew almost everything about John’s relationship with Sherlock, those two years of absolute heaven that had ended as suddenly as they started. But she didn’t know about the ending.

            “It was just a lot of little things. You’ve met him.” He hoped that would be enough to satisfy her. He didn’t want to talk about it more. He didn’t need to talk about it more. At that time, he still saw Sherlock very occasionally, but was frequently unable (or “unable”) to accompany him on cases. Sherlock had stopped asking so often. When John went, he couldn’t let go—not completely. They would get caught up in a chase or a mad search for evidence and then stand in the alley huffing and he’d forget, and for seconds, it would be just as it had been, and Sherlock would smile his widest, most genuine smile, and then John would remember—all the baggage, all the little things, and Sherlock’s unwillingness to change, and Mary waiting at home for him, and he’d realize he was halfway to reaching out to hold onto Sherlock for support and he would stagger awkwardly backwards. Sherlock would watch him, hurt, and they would both soldier on, trying to make it as it had been.

            But John knew it was hopeless. He couldn’t just look at Sherlock and not think of the evenings they’d shared, him subjecting Sherlock to his film collection, Sherlock stretching across the whole of the sofa but the little corner John occupied and then lying his head in John’s lap, like a cat waiting to be petted. He couldn’t watch Sherlock shift his weight from leg to leg as he puzzled through a deduction and not remember a bed full of long, sprawling limbs, so distinctly different from Mary’s warm curves. He couldn’t forget the art of falling asleep with Sherlock’s knee digging into his leg, even though he hadn’t had to employ it for years.

            “There must’ve been something, though,” she said, and John was surprised at her insistence—usually, anything he didn’t want to talk about, she let go. It was part of why he loved her.

            “I don’t think he even remembers it. He seemed so confused.” John shook the thought. Whether it confused Sherlock or not, it still hurt John. “You remember those serial killings in 2015, that whole huge string of them, the restaurant worker?” Mary nodded; she’d been a fan of John’s blog, so if nothing else she remembered what he’d written about it. “After Sherlock helped catch him, he was complaining about what a boring case it had turned out to be, and wouldn’t it have been better if the guy hadn’t mixed up his ingredients and brought out unpoisoned food because then he would’ve gone uncaught for longer.” John buried his face in his hands. “When we were heading back to the flat, he said something about Moriarty at least keeping things interesting, about how he would’ve— After—after all the—”

            “John,” Mary had whispered, and crossed the room to pull him up close to her. “I’m so sorry.”

            John had told Mary that the year and a half Sherlock had been dead was rough on him. He had never detailed the worst of it to her. He wasn’t sure he could.

            “Can we stop talking about Sherlock?” John had finally muttered into her hair. He had tried with Sherlock—he really had. For two years after he’d broken it off, they had made a truly valiant attempt at going back to the way things had been. Sherlock obeyed the rules that John laid out for him—no more touching, no more using my toothbrush, no, I won’t pet your hair—like a repentant puppy. They still watched films. They still laughed, sometimes, although John found it difficult to let himself go the way he used to, for fear of what he’d do if he got too caught up.

            Mary hugged John tighter. “Of course we can.” She pulled back. “How does a movie sound? You can pick.”

            Which was Sherlock’s form of apology, too. No, no, no, backtrack, backtrack. John drew his lips into a smile. “I think I’ve got a better idea. I hear it’s great stress relief.” And she giggled, and he carried her over to their room, and tried not to think of paler skin and darker hair and deeper laughter, and, by and large, succeeded.

            But now his main distraction was gone, and so John was working harder than he ever thought he’d be at his age.

            Maybe it had been a bit stupid. Maybe there was a different way.

            No—no. He remembered that last time he’d seen Sherlock—the unrestrained apology, pleading, on his face. Now it was John who was the foolish one—holding onto that one idiotic thing for so long, being the arsehole for not allowing Sherlock to apologize or say, “Hello,” or whatever it was he wanted these days. And what would he do now? “Oh, good evening, Sherlock, look, about how I exploded at you and never let you talk about it with me and then went and got married and then never let you give me a sodding hug when my wife died, sorry about that, what do you think about Thai for dinner and then, for the evening’s entertainment, watching me try to scrub skin off the plate you put toe samples on?” Sherlock would have every right to shove him out the window for it.

            John felt reasonably experienced in the area of love. He had loved a lot of different people for a lot of different reasons—mates he knew he’d probably see die on the battlefield, who he loved like brothers; Harry, who he hated, but also loved because one didn’t just go through some of the shit they went through together and not love one another, no matter how much other hatred goes along with it; girlfriends of various sorts, who he’d loved for things like legs or football debates or contagious enthusiasm about knitting; Mary, who he loved like angel food cake and sharing umbrellas under rain, a co-conspirator. John was used to carrying love in his chest, and lots of it. He was used to forming attachments to near-strangers, something Sherlock constantly chastised him for. He was empathetic to a fault, and although he was quiet about it, allowed himself to adore unbridled. Perhaps that was the most painful part of what had happened after he had broken it off with Sherlock—holding that in. He had always loved Sherlock, in some way, even before it was romantic or even sexual for him; holding that back nauseated him. It was his psychosomatic limp all over again: a self-imposed crippling.

            Falling in love with Sherlock had been fantastic. The time they’d had together after Sherlock returned, the couple of years before things went to shit—they were brilliant. John couldn’t remember being happier. He couldn’t remember caring less what anyone thought about him (Anderson had said something particularly tasteless; John had practiced a few rude hand gestures). Their relationship had been like their friendship, but more. It took a while for him to get used to the idea of wanting to have sex with (and kiss and cuddle with and make love to) a man; it was made easier by Sherlock’s general lack of experience. They learned together.

            John wondered if Sherlock had spent the time since then alone, and felt a guilty pang. When they had just been trying it out—this whole “having sex with another man” thing—John had asked Sherlock if he enjoyed it, or was just pretending to in order to collect data. Was he all right with this position or that position? Did he like doing that or was John maybe just not his cup of tea, sexually?

            “Of course I enjoy it,” Sherlock had breathed. “It’s you.”

            Of course Sherlock had spent the time since then alone. John had been lonely, even with Mary. If it weren’t for her—if Mary hadn’t been so spectacular an individual, so well-suited to John—he would probably have stayed alone the whole time, too. Or maybe they would have tried to continue being “regular flatmates.” There was no telling.

            There was no telling anything, really. After all, John hadn’t seen Sherlock in three years. The last conversation they’d had—besides the few short words three years ago—had been several days after Mary’s funeral.

            “You must be upset,” Sherlock had said.
            “Yes, thanks for the reminder. Why weren’t you at the funeral?”

            “I didn’t want to intrude.” He seemed smaller as he said it.

            “So you come to my flat while I’m trying to have a quiet evening in instead?”

            “I’m sorry. I won’t do it again.”

            John sighed.

            “I was wondering if you might like a hug. I hear it’s helpful.” Sherlock had attempted a smile, and held his hands outward in an open, unassuming gesture.

            “No thanks,” John had said.

            “Is there anything I can do for you?”

            “Look.” John turned completely toward Sherlock. “I know you’ve never been around when somebody near me has just died, so this may come as a shock to you, but I tend to prefer to be left alone, all right?”

            Sherlock had clearly caught the other death to which John had been referring. “Of course.” He had glanced around the flat for another minute, and opened his mouth several times, as if prepared to make a handful of deductions about John and his day-to-day life, but held back. With one more glance at John, he’d turned, put his coat back on, and left with a quiet, “Goodbye, John.”

            John shifted over to that same chair he’d been in for those words as he thought. Today, he could go to a gym in case anyone had an asthma attack. He could hang around by a pool.

            Maybe he should move out to the country.




            Four days later, his mobile rang. He startled at the name that came up and scrambled to answer. “Sherlock? Is something wrong?”

            “No,” came the quiet answer. “No, nothing like that.” No, of course not. But then…what the hell was it?

            “What is it?”

            “Just calling.” A pause. John considered telling him to piss off, to stop yanking his heart around, but that seemed a little childish. “Mrs. Hudson died last month.”

            A heavy, icy weight slammed into John’s chest. “Oh, god.” He ran one hand down the side of his face. “Was it bad? Why didn’t you tell me?” It was Mrs. Hudson. It was—and now she was gone. She was just—gone. Mrs. Hudson. John couldn’t even remember the last time he’d seen her. They’d wound up having tea and biscuits together some day he’d stopped by 221B to help Sherlock with a case.

            “An illness,” Sherlock answered. “It was sudden. And quick. And she…she didn’t die alone.”

            John felt a shudder in his chest. Death. Dying. Mrs. Hudson. Mrs. Hudson, dead. “You were there?”


            “Well, I’m glad someone was. Christ. Mrs. Hudson. How…I mean…did she…” he trailed off, not sure what he was trying to ask. He couldn’t shake the image of Sherlock kneeling down beside a hospital bed, holding Mrs. Hudson’s hand, whispering deductions about stupid little things, the sort of thing he’d only do for people he cared for—people he loved. He’d sit close to her and tell her he had performed an experiment and found irrefutable proof that she was the best landlady ever to inhabit London. He’d tell her that he could still smell her perfume and would spritz it about the flat now and again when he wanted to imagine she was there being not-his-housekeeper and cleaning up after his mess. If he still lived in 221B, anyway—it was difficult to imagine Sherlock anywhere else.

            “I think she must’ve put half of every one of our rent payments in savings,” Sherlock spoke again. Our, John thought, he still thinks ‘our.’ “She…she left me 221.”

            Oh, god. Oh, Sherlock. John tried not to think of him there, alone but for Mrs. Hudson all these years he’d been with Mary and then all by himself. He tried not to imagine him there now, for this past month, all on his lonesome, Mrs. Hudson dead, Mrs. Hudson gone.

            “She told me to say ‘hello’ to you. And ‘goodbye.’”

            The last was a word John never wanted to hear from Sherlock over the phone again. What had he been doing, all alone all this time? Not drugs, John hoped, not anything—not anything terribly stupid or—well—surely he wasn’t so careless as to get himself killed, but… “For her, I hope, not you,” he whispered.

            “Yes, of course. For her.”

            “I’m glad you were there, Sherlock,” John said after a few moments. “I really am. And I’d bet if she could’ve had anyone by her bedside, she still would’ve chosen you.” Mrs. Hudson dying by herself would be too much—would be—god. John wondered, if he were struck down with a terrible illness today, and were going to die tomorrow—  “I know I…well.” No point in saying it.

            “Yes,” he heard Sherlock say back through a thick voice. “Me too.” John’s heart dropped heavy. Of course he would. Here he was, soon to be old, all alone, and the person who loved him more than anyone in the world was an ex-boyfriend—not that that described the half of it—he’d spent the past fifteen-odd years trying to get by without.

            “So you’re still at 221B, then?” John finally asked, when he was fairly certain he had his voice back.

            “I am.”

            “You still got that bloody skull on the mantelpiece?”

            Sherlock was his friend. To hell with it all—he could deal with it being weird. He could fight the impulse to touch, if it came. He could be happy to have his friend back.

            “I still have about half of your crap film collection that you never picked up. And I still have a television.”

            Movie night. Apology night. “God, you don’t watch television now, do you?” Of course not. But he had to play along.

            “Hardly.” John wondered if Sherlock was actually choked up, or if there was something off with the line. “I watch your crap film collection.”

            Oh. Actually choked up, then. John gulped a lump down his throat. He gulped again. He’d forgotten—well—he’d remembered, of course, but he’d still forgotten how it was, to have Sherlock in his life. How long since they’d had a civil conversation, or had fun.

            “I was wondering where You Only Live Twice went,” John finally answered. “Perhaps I’ll come get it back.” He only hoped Sherlock knew that he really didn’t want to—

            “I’m not sure I can part with it.”

            “And I suppose you couldn’t do without any of the others, either?”

            “Most definitely not.” When he spoke again, he sounded brighter, “You shall have to stop by if you want to watch them.”


            “Crime in London isn’t what it used to be, so I’m often sitting about.” John fought past a sudden spasm in his chest at the statement. Was that Sherlock trying to bring it up again, or—no, no, he didn’t remember. And—he seemed—accepting of it, now, if his tone was anything to be believed. “It may be a nice distraction from my latest work on concealable explosives.”


            “In fact, I ought to get back to that right now. Now that I don’t owe anything if I destroy, say, a wall or two…”

            John smashed his finger onto the phone’s screen to hang up and grab his coat. As he put it on, he sent a text, and then another as an afterthought.

            set your sorry arse on the sofa and wait for me to get there –JW


            As John dashed out the door to hail a taxi, he got a response. Excepting, I suppose, the sofa? –SH

            “Smartarse,” he muttered to himself, and then he suddenly wondered if Sherlock had gotten an idea and was going to—

            if you are even thinking about blowing up the sofa –JW
            Wouldn’t dream of it. –SH

            But if I do it’ll be your side. –SH

            John grinned. His side was really about a quarter of the sofa. His side. His side. My side. It was all he could think of for the cab ride there.

            After Sherlock had, as John probably should have expected, tossed his cane aside and invited him up, John chose a film and took a seat on his side. “God, I missed this.”

            “Me too,” Sherlock answered, his voice quiet, as if with awe. Half an hour later, as if to Sherlock the span of time had never existed, he added, “I missed you John.” John couldn’t think of anything to say. He pretended to be focused on the film. “Please don’t go away.”

            That was it.

            That was it.

            Friendship be damned.

            “You know, something about this doesn’t quite feel like it should.”

            “I’ve forgotten to tell you to make us tea.”

            “No, not that, not that…”

            “I may have shifted the sofa away from the wall slightly.”

            “Not that either.”

            “We typically didn’t watch movies together until after seven o’clock and it’s only five fifty-three?”

            “No…no, none of that’s what’s throwing me. Ah! I know.”


            “You haven’t laid your head down on my lap yet.”

            John smiled at the glowing smile that overcame Sherlock’s features, and he felt warmth melt out of him and into the room, and into his fingers as he stroked Sherlock’s hair, and into Sherlock as they cuddled close, and he let go and loved and loved and loved, and it was then that John returned to Baker Street.