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He imagined the ghost travelling into town, walking down Market Street, the khaki-coloured buildings and tramways and clean sunlight. It would be weary, T-shirted, happy. He imagined walking up over towards Chestnut, and surprising Carey at Muffin Tops while she was pouring coffee for a customer. Then – blank. Nothing. He wasn’t imagining Muffin Tops. He was imagining Central Perk from Friends. And he wasn’t imagining Carey. He was imagining Jennifer Aniston. And his images of the city were a mash-up of Carey’s postcards and Google Street View.

He couldn’t imagine Muffin Tops. He couldn’t imagine Carey. That future was illegible. Instead he was in Atlanta, in a motel room with its brown curtains drawn against the daylight, while his more purposeful ghost flew across America to surprise his girlfriend.

He couldn’t stay, and he couldn’t go. He didn’t feel sad, or scared, or anything at all. He’d flown here on an impulse, and now the impulse had left him and no other impulses had arrived to take its place. He examined the feeling, or lack of it. He felt like if he stabbed a knife into his leg it would make a dull thunk, and then stick out as if he’d driven it into a wooden table leg.

He took out the small cherrywood box and thumbed it open in his lap. The ring was inside, a half-moon of silver metal standing proud of its little velvet cushion. Where there might have been a solitaire diamond, there was instead a double loop in the metaclass="underline" the lemniscate.

He shut the box with a snap, opened it again, shut it again. What a small thing it had been to decide to change his life. He tried to remember whether he had decided to ask Carey to marry him before he’d seen the ring in the antique shop, or whether the idea had come fully formed into his head when he’d seen its object expression in the world. He couldn’t.

He remembered wanting the ring, and knowing he wanted the ring, and knowing what it was for. The ring had cost about half what he had in the bank. The infinity symbol. He’d thought it was cool. Now he thought it was tacky.

He didn’t know if it would fit. He didn’t know if Carey would say yes. He didn’t know what he’d say if she did. He didn’t feel hungry. Eventually, as the afternoon slid into the evening, he turned on the television.

‘Isla Holderness,’ Red Queen continued. Hands was more or less at his ease, now. And he was curious to know how this story would unfold. A weapon? That sounded wrong. Banacharski was a fierce pacifist.

He knew Holderness’s story well. It had done the rounds in the mathematical world. She was the woman who had found Banacharski. She’d started as a disciple and they’d exchanged some letters. She wanted to see if she could talk him out, talk him back in. She’d schlepped through tiny villages in the Pyrenees armed with an old photograph of the mathematician. She’d found him, living in a shack in the hills, living like a monk.

Banacharski, to her surprise, had been friendly. The shack was a mess, by all accounts. Banacharski slept on a pallet on the floor, and – in one version Hands heard – lived on grass. Hands wasn’t actually sure that was possible.

Somehow, though, he took to Holderness. He had even been flattered to hear that, since his disappearance, one or two of his conjectures had been proved, but he said he’d stopped doing mathematics. Then he’d started ranting about the devil and the physics of free will. He believed that an agency – he called it the devil, though Holderness hadn’t been sure whether or not he meant it as a metaphor – was interfering in measurements, making precise knowledge impossible, minutely bending space time.

After she returned to England, they began to correspond. It went well at first. He indicated, so the story went, that he intended to make her the custodian of his legacy – that he’d pass her his findings. Then one day a letter arrived, apparently, demanding the answer to a question: ‘What is a metre?’

Holderness had no idea how to answer it. A sheaf of further letters arrived before she had even finished composing her reply. The first was incoherently angry, filled with scrawled capitals and obscenities. It accused her of being in league with the Enemy. It arrived on the same day as another that appeared to threaten suicide. The third arrived a day after the other two. It was addressed to ‘The Supposed Isla Holderness’. Every mention of her name in this letter was surrounded by bitterly sarcastic inverted commas. ‘Since you are not who you say you are, you know that I cannot be who you say I am,’ it opened. It was signed ‘Fred Nieman’. Holderness had set off to find him. She had found the shack burned to the ground and Banacharski, again, gone.

‘He thought Isla Holderness was a spy,’ said Red Queen.

‘Why would he have thought that?’ said Hands.

‘He was intensely paranoid,’ said Red Queen. ‘Also, he was quite right. She was a spy. She worked for us. We’d been monitoring all her correspondence with him. His letters dropped hints of what he was working on – I think he thought that was what was keeping her interested; after all these years of isolation, the human contact was welcome, and he was flattered by her interest. But you have to remember this was a deeply, deeply paranoid man. The hints were purposely fragmentary. We had to stay at arm’s length, and she was deep cover.

‘We thought we had the breakthrough, but “What is a metre?” completely threw us. It threw her too. But she waited to respond.

‘That was when it became clear something was very wrong. Banacharski vanished. Then we thought Holderness had gone rogue. We knew enough about what this man was like to know that the longer she left her response, the more likely it was that he’d flip his wig. We’d sent the clearest possible instructions on receipt of the “What is a metre?” letter that she should go immediately, in person, to see him and find a way of talking him round. But instead she spent a week trying to work out the answer to his riddle.

‘She was responding to none of our signals to come in. She didn’t use the dead-letter drops, and the messages she was sending stopped making any sense: newspaper buying, for instance. There was a complex series of codes surrounding what paper she bought, at what time of day, and how much change she used.

‘All of a sudden she seemed to be buying newspapers completely at random. It took us a while to cotton on to what had actually happened.’

‘What had happened?’

‘It turned out that – actually – Banacharski was wrong. Isla Holderness was just what she professed to be: an academic mathematician who was interested in his work.’

‘I don’t follow you.’

‘He was right but he was wrong. Isla Holderness was a spy. But that was a different Isla Holderness. We made a huge mistake. This woman with Banacharski wasn’t our Isla Holderness, and she wasn’t spying on him. It was another woman with the same name. And she’d been buying newspapers completely at random all along.’

Red Queen looked a little bitter about this.

It was hard enough keeping track of all the DEI’s agents, and the organisation’s institutional reluctance to commit anything to paper for fear of being counter-surveilled made it more or less impossible.

Many DEI field agents didn’t even know they were working for the DEI – and a good few of them actually thought they were spying on it for one of the several fictitious cover agencies that it ran. Red Queen occasionally pretended to be unsure whether the DEI was a real organisation or, itself, a red herring.

Hands looked mystified. ‘So where was your Isla Holderness while the other Isla Holderness was with Banacharski?’

‘Thousands of miles away in a cave network in northern Pakistan,’ said Red Queen, gloomily. ‘She had spent the previous three years infiltrating with unprecedented success – not that we knew about how she was getting on at the time, since we’d forgotten she was there – bin Laden’s inner circle.’