In late summer, with the capital still half-asleep, a stream of the curious and the already hooked approach the windows of the slick, lozenge-shaped booth that houses 502 titles old and new. You can't browse the shelves, it's too compact for that, but you can consult a screen list, or take away a folded paper catalogue.
"We see all ages, all races, people who never read, and those who are very cultured but don't have time to go to a library. Some ask us to recommend something good, others have their catalogue ready in their pocket, all marked up," Ms Gomez says.
Our conversation is interrupted every few moments by newcomers eager to sign up - more than 1,000 have joined in the month since it opened. It must be Spain's least bureaucratic procedure: you show your ID card and receive your card and choice of book on the spot.
"We carry classics and contemporary popular literature and poetry, but not the latest bestsellers," co-librarian Gustavo Fernandez says. Most popular are Patrick Susskind's Perfume, and HG Wells's War of the Worlds. Sixty per cent are Spanish or Latin American writers, the rest are translations, from Bob Dylan to Catullus, Stefan Zweig to George Orwell.
"Sometimes people tell us how much they enjoyed a book, sometimes they just chat. They are happy and surprised to find us in their path," Mr Gomez says. Your borrowed book is stamped with the return date a fortnight on. And the penalty for defaulters? To be banned from borrowing for the number of days late.
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As one modest but pleasing cultural attraction opens, another one sadly closes. The Luna cinema, one of the few in Madrid that showed "non-Hollywood" foreign films, has shut, to widespread dismay. The Life of Brian played here for months, making it a pilgrimage spot. Tucked behind the Telefonica building off the Gran Via, in full-on drug dealers' and prostitutes' territory, the Luna was no place to wait in the doorway for your viewing companion. The area has apparently become too seedy, even for this offbeat venue.
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One area that had come up in the world, meanwhile, is under threat. Residents of Madrid's working-class neighbourhood of Eugenia de Montijo are up in arms over plans to drive a four-lane motorway through their only patch of green parkland. Bulldozers have carved a trench through a stretch of grass and trees, savaging a leafy haven in this ferociously built-up area.
Stickers attached to surviving pines, birches and poplars read: "RIP: this tree is condemned to death for a motorway." When Pilar Robledo, 72, first moved here in 1970, it was just scrubland. "We bought saplings from a roadside nursery, one each for my three children," she said. "They planted them, and named them, and brought a bucket to water them every day. That was Felipe." The feisty lady points to a sawn-off stump, and wipes away a tear.
She indicates a deep gash in the earth where, she says, a lustrous tree once shaded 40 people seated beneath it. "They said it was sick, but the machine couldn't even pull the roots out."
Ms Robledo, who leads residents' weekly protests, walks to the park's edge, and shows me an unmade road. "That's where we'd planned to put the main road, all those years ago. Why didn't they dig there?"