When they entered the hospital ward, Agata caught sight of Tarquinia and called out excitedly, ‘Ah, you’re free! Congratulations! Come and hear some great news!’
As they approached the sand bed, Ramiro could see that Agata had gained some flesh since his last visit, but she was still limbless. The doctors had told him that she would need every scrag of tissue to support her recovering digestive tract.
‘What’s the news?’ Tarquinia asked her.
‘I just had a visit from Lila and her student Pelagia,’ Agata replied. ‘The innovation block is well and truly over!’
‘Yeah?’ Tarquinia had probably been expecting to spend the whole visit trying to put the record straight about the inscription, but Agata’s mind was on something else entirely.
‘Pelagia’s settled the topology question,’ Agata proclaimed. ‘The cosmos is a four-dimensional sphere. It’s not a torus – it can’t be!’
‘This is from your work?’ Ramiro asked. ‘Pelagia found a way to complete the calculations?’
‘Not so much complete them as see into my blind spot,’ Agata explained. ‘Listen, it’s simple. A luxagen is described by a wave that changes sign when you rotate it by a full turn. That has no effect on any probability you calculate – so long as you apply the same rotation to everything in sight – because the probability comes from squaring the value of the wave. Minus one squared is one, so the change of sign makes no difference. It only shows up in more complicated experiments where you rotate some things and leave others unaltered.’
Tarquinia said, ‘I follow that much.’
‘Pelagia’s idea just replaces rotations with trips around the cosmos. Suppose the cosmos were a four-dimensional torus, and you carried a luxagen all the way around it in a giant loop. What happens to the sign of its wave? Does it come back unchanged, or does it come back reversed?’
Ramiro frowned. ‘I can tell that you want us to say “reversed” by analogy, but I thought there had to be perfect agreement around a loop.’
Agata buzzed. ‘I didn’t want you to give either answer. There could be perfect agreement, or the sign could be reversed: nothing rules out either possibility. If the sign’s reversed, that will be undetectable: everything you can measure locally will still be in perfect agreement.’
Tarquinia said, ‘Hang on, if the sign changes . . . where exactly does it change? What’s this special place on the torus where it flips over?’
‘There is no special place,’ Agata insisted. ‘It’s like cutting open a band and rejoining the ends with a twist: once you’ve glued them together, there’s really nothing special going on at the join. The twist isn’t located there – or anywhere. It’s a property of the whole band.’
Agata began to form a sketch, but Ramiro saw that she was having trouble so he drew what she’d described on his own chest.
‘So you’re talking about the cosmos being some kind of . . . twisted torus?’ Tarquinia asked.
‘No, not the cosmos,’ Agata replied. ‘The two bands, twisted or not, both have identical circles as their midlines—’
Ramiro added the midlines to his diagram.
‘—and you should think of those circles as the cosmos. What happens with the bands is an additional structure that the topology of the cosmos doesn’t fix, one way or the other. It’s all about the luxagens, not space itself.’
Tarquinia said, ‘All right. I think I’ve got it.’
‘Then the next step is to remember that we’re talking about a four-dimensional torus,’ Agata continued. ‘So there are four completely different ways you can travel in a loop. There’s nothing that compels those four routes to have the same effect – it would be perfectly consistent to have a luxagen whose sign changed around some of those loops but not the others. So there are sixteen possibilities altogether: for each loop, the sign might change or it might not.’
Ramiro understood the counting argument, but he couldn’t see where it was leading. ‘Aren’t these distinctions all invisible, though? They have no effect on any probabilities.’
‘They have no effect on probabilities,’ Agata agreed. ‘But if there were sixteen times more choices for the state of every luxagen, that would multiply their contribution to the vacuum energy by a factor of sixteen. Photons give a positive vacuum energy, but luxagens make the vacuum energy negative, and a factor of sixteen would be enough to guarantee that the total energy density in the cosmos was negative, absolutely everywhere.’
Ramiro struggled to recall the implications of this, but Tarquinia beat him to it.
‘A negative energy density means positive curvature,’ she said tentatively. ‘But you can’t have a torus that’s positively curved everywhere.’
Agata chirped. ‘Exactly! You end up with a contradiction. So the cosmos can’t be a torus. But in a four-sphere, every route you might travel can be shrunk down gradually to a tiny circle, and then to a point: a path that goes nowhere. The sign of the wave can’t change along a path that goes nowhere, so there are no extra modes for the luxagens. The vacuum energy stays positive, which means the curvature will mostly be negative – but it also has to change from place to place, because you can’t have uniform negative curvature on a sphere. And because the curvature depends on the entropy of matter, that has to change too. That’s why the cosmos isn’t in a state of equally high entropy everywhere. That’s why there’s a gradient. That’s why we exist at alclass="underline" with a history, with memories, with an arrow of time.’
Watching her as she spoke, Ramiro couldn’t help sharing her joy. Perhaps the discovery changed nothing tangible, but it vindicated all her years of effort – and it proved that the Peerless was back on course. New ideas were possible again. The paralysis was over.
‘And that settles everything?’ he asked. ‘Cosmology is complete now?’
‘Not at all!’ Agata replied gleefully. ‘There are still dozens of open questions. People will be working on this until the reunion, and beyond.’
Tarquinia said, ‘I have some news of my own that you should hear.’
Ramiro had been afraid that the change of subject would go down badly, but Agata listened to the revised version of the last day on Esilio with no sign of hostility.
When Tarquinia was finished, Agata said mildly, ‘I’m glad you weren’t lying to me, after all.’ She glanced over at Ramiro. ‘And I’m glad you weren’t either, even if you meant to.’ It was an infinitely gentler barb than he’d expected.
A doctor approached and suggested that they let Agata rest. Agata glanced down at her shrivelled torso, as if she’d forgotten the state of it while they’d been talking. ‘Not one person has said that I look like I’ve shed twice,’ she complained. ‘I’ve been ready to tell them the names of the children, but the joke’s just not happening.’
Tarquinia placed a hand gently against her cheek. ‘Get strong. We’ll see you again soon.’
Ramiro shared a meal with Tarquinia in the food hall, then they retired to his apartment.
‘What is it that’s troubling you?’ Tarquinia asked. ‘I thought it was the inscription, but Agata was fine about that.’
Ramiro didn’t reply. Better to offer no denials or explanations, and she’d come to her own conclusions about the cause.
‘We survived,’ she said. ‘We might have been fools to go along with Giacomo . . . but if we hadn’t, what would have caused the disruption?’