With a soundless rush we emerged into a spherical cavity miles wide. Beams of crimson starlight crossed the hollow, bathing its walls with a blood-red glow. At the heart of the chamber was a sphere. A couple of miles across, the sphere gleamed golden and was semi-transparent, like a half-silvered mirror. Platforms bearing Ghost workers hovered over its surface.

Some vast machine moved softly, within the confines of the mirrored sphere.

"Mr. Raoul, welcome to our experiment," the Sink Ambassador said.

"What is that sphere?"

"Nothing material. The sphere is the boundary between our Universe... and another domain, which we have constructed by letting quagma droplets inflate under controlled conditions. Within this domain the ratio you know as Planck's constant is reduced, to about ten percent of its value elsewhere. Other physical constants are identical."

"Why the half-silvered effect?"

"The energy carried by a photon is proportional to the Planck number. When a photon enters the Planck domain the energy it may carry is reduced. Do you understand? It therefore sheds energy at the boundary, in the form of a second photon, emitted back into normal space."

I asked if we were to enter the Planck space.

"I fear not," the Ambassador said. "Our fundamental structure is based on Planck's constant: the spacing of electrons around the nucleus of an atom, for example. If you were to enter the domain, you would be -- adjusted. The device in there -- an artificial mind -- has been constructed to withstand such Planck changes. The device controls the regeneration of the domain from quagma; we are also using it to conduct computational experiments."

The machine in its golden sac turned, brooding, like some vast animal.

"Ambassador, what is your purpose?"

The Ghosts, the Ambassador said, had two objectives. The first was to use the Planck boundary conditions to build a perfect reflective surface, an age-old goal of the energy-hoarding Ghosts.

The second objective was more interesting.

"The capacity of any computing machine is limited by the Uncertainty Principle," the Ambassador said. "The exploration of, say, high-value prime numbers has always been constrained by the fact that energy changes within a device must remain above the uncertainty level.

"With the reduction in Planck's constant we can go further. Much further. For example, we have already managed to find a disproof of an ancient human hypothesis known as Goldbach's conjecture."

Goldbach, it seems, speculated that any even number can be expressed as the sum of two primes. Twelve equals five plus seven; forty equals seventeen plus twenty-three. Centuries of endeavor had neither proved nor disproved the hypothesis.

The Planck machine had found a counterexample, a number in the region of ten raised to the power eighty.

"I guess I'm impressed," I said.

The Ghost rolled gently. "My friend, age-old problems melt before our Planck machine; already several NP-type problems have--"

I told the Ambassador I believed it, and to dump down the details later.