Schild’s Ladder

“Yeah. I really am grateful to you for trying this,” Tchicaya said. Since the factional rift had widened, it was more important than ever to keep all the Yielders open to each other’s new ideas, and where he wasn’t competent to contribute directly himself, he could at least act as a kind of broker, prodding the appropriate experts into action.

Rasmah seemed on the verge of pointing out that he might have expressed his gratitude to her more palpably, but then she smiled and accepted his words at face value.

“Okay. Here I go.”

She turned her attention to something invisible to Tchicaya. For several minutes, she sat in complete silence.

Suddenly, she exclaimed, “Oh, I see! This is actually quite nice.”

Tchicaya was excited, and slightly jealous. “Can you explain?”

Rasmah held up her hand for patience, retreating back into her private scape.

After a while, she spoke again. “Think of all the different dynamic laws that might make topological sense, in terms of the propagation of various kinds of particles that are defined as patterns embedded in a graph. I know that’s horribly vague, but I don’t think you’d want the version with added jargon.”

Tchicaya said, “Okay. I’m thinking of them.” He’d seen enough examples that they’d pinned to the border over the last few months to have some feel for what this meant.

“Now imagine each one is a quantum state vector in a big fat Hilbert space. All of them orthogonal to each other.”

“Yes.” Tchicaya had never had his mind restructured to enable clear images of more than three dimensions, but since Rasmah’s Hilbert space was infinite-dimensional anyway, three was as good as any other number. “I’m doing that. Go on.”

“Now imagine a new set of vectors that consist of equal amounts of all these dynamic-law vectors, and that are all orthogonal to each other. These vectors represent definite values of a variable that’s complementary to the law vectors. Branco calls them law-momenta?—?which is a bit sloppy, because they’re not true Lagrangian conjugates, but never mind.”

“I’ll try not to fret.” Tchicaya thought of the directions on a map. If the dynamic-law vectors were north and east, then the new, unbiased, law-momenta vectors would be north-west and north-east. Both had equal portions of the old directions?—?if you counted west as being the negative of east, and only cared about the size of things, not their sign?—?and they were at right angles to each other. In three dimensions or more you needed to introduce complex numbers to pull off the same balancing act, but from there you could keep on going to any number of dimensions. The amounts of the original vectors you combined were just a series of complex numbers that moved around a circle in the complex plane; to get different vectors, all orthogonal to each other, you just moved around the circle at different rates.

“Now picture a state vector which has equal components when written as superpositions of the old set, or the new.”

In two dimensions, that was easy: north-north-east lay at the same angle to north as it did to north-east, and the same angle to east as it did to north-west. In terms of the quantum mechanics Rasmah was describing, it had equal uncertainty in the two complementary variables: it did not obey a precise dynamic law, but nor did it have any precise law-momentum. It split the difference and compromised, in the most symmetrical way.

Rasmah continued. “These are the states Yann wants to scribe, because if you create one on the border, and then arrange to measure the same kind of state coming back, they yield the highest attainable probability of returning with information about the interior.”

“ The highest attainable probability? That’s a resounding declaration of confidence.” Tchicaya had been hoping for something more reliable. He knew what quantum mechanics was like, but if his own Qusp could pluck certainty from the haze, granting him the ability to make unique decisions, surely Yann could work some similar trick with the vastly more powerful abstract machine behind the border?

Rasmah emerged from her visualization. “I know how that sounds, but it really is the best we can hope for. We’re not arranged in the same way as the far side; we’re stuck in a dynamic-law eigenstate, and that’s always going to make things difficult.”

“Yeah.” Tchicaya was grateful for anything that took them beyond the current, artificial view of definite laws spread across the border, but it was sobering to realize how much stranger things became as the price of that advance. “I shouldn’t be disappointed, but I keep underplaying the problems in my head: sweeping all the hard parts off to one side, where I don’t have to look at them. If I faced the difficulties squarely, I’d probably just turn around and run.”

Rasmah regarded him with a mixture of curiosity and affection. “You really do want to go through the border, don’t you?”

“I think so. What about you?”

“Absolutely. That’s what I came here to do.” She hesitated, then added, “For a while, I thought I must have said something too extreme along those lines, and it put you off. But I don’t think that’s it. So what is it about me that you hate so much?”

Tchicaya shook his head vehemently. “Nothing.”

“But we got halfway,” she said, “and then you changed your mind.” This wasn’t a question. Their bodies had ceased the silent exchange of pheromones, and that in itself would have dampened her feelings toward him, but it must have been clear to her that he was the one who’d halted the process.

“You’re very good company,” Tchicaya said. “But you remind me too much of someone else, and I don’t feel right about that. I don’t want to confuse you with her; that wouldn’t be fair on either of us.” He frowned apologetically. “Am I making any sense?”

Rasmah nodded uncertainly. “The other thing I thought was, maybe you and Yann were still, somehow?—? ”

“No!” Tchicaya was taken aback. “Where did you hear about that?”

She waved a hand dismissively. “Everyone knows.”

“Actually, I think Yann might have forgotten.”

“But there is no one else, in the present? Just this nameless competitor from the past?”

Not exactly a competitor. And not wholly in the past. But Tchicaya didn’t want to explain any further. “That’s right.”

“Okay.” Rasmah stood, and Tchicaya rose beside her. In part, he was glad that she’d cleared the air, though at the same time he felt a surge of resentment, now that he’d been forced to put his reasons into words. He and Mariama would never be together. Why was he letting her shape his decisions at all?

“You’ll support Yann with this?” he asked.

Rasmah smiled. “Definitely. This is our best hope, and I’m sure I can sell it to the others. Suitably uglified.”

The Blue Room was packed from wall to wall; it hadn’t been so crowded since the Left Hand’s first trial run. The room was near the bottom of its module, and it had already been expanded as far as possible in all horizontal directions; several unobliging neighbors above prevented it from growing upward. As relations had deteriorated, some Yielders and Preservationists had swapped cabins in order to be surrounded by fellow partisans, but the Rindler hadn’t yet reached a state where every module was “owned” by one faction or another.

Yann paced the ceiling, ducking away from the tallest heads and shoulders?—?making his presence visible, but wisely desisting from trying to claim space that he could not defend with solid elbows. Other acorporeals came and went beside him, and no doubt he was conversing with some who weren’t bothering to display icons. Almost everyone who’d been born acorporeal had now donated their bodies to new arrivals, effectively splitting the Yielders into two distinct communities, more so in some ways than the factions themselves. Tchicaya had mixed feelings about this; their generosity had given many more people a chance to participate in events on the ship, in the only manner that would not have been alien to them. But the acorporeals had been willing to change modes in the first place, so why couldn’t the newcomers make do with software bodies? Maybe he had no right to think that way, having accepted the first such sacrifice himself, but the segregation by birth still depressed him, however well