Twenty-six people had gathered in the Calculation Chamber. Roi had looked in on Zak on her way; he’d offered her encouragement, but he’d been too tired to come and observe, let alone participate.
By consensus, Roi and Tan had been appointed lead calculators for this session. They would work independently of each other, while the remainder of the team, split in two, would act as their checkers. Only if both groups reached the same answer would it be trusted.
To save scratching out mathematical templates on skin, a wasteful and physically tiring process, Gul had devised an ingenious system for representing and manipulating templates by sliding stones around on a wire frame. It had taken Roi many shifts to master the system, but now she couldn’t imagine working any other way. When each frame full of templates was completed, she copied the last template to a new frame, then passed the full frame to the first of the checkers.
The team had calculated and recalculated the consequences of Neth’s idea many times, and the new templates had a very similar structure, so Roi made rapid progress, and each time she glanced around the chamber the checkers seemed to be keeping pace with her. The familiarity of the calculations also brought its perils, though; with the old version still fresh in their minds, the minor variations that Roi was introducing looked “wrong”, like small mistakes that needed correcting. Several times Roi caught herself nearly reverting to the old templates.
She reached a template describing a connection that respected the new definition of space-time length, and whose geometry was symmetrical about the Hub. That she had come this far without any new problems emerging was an encouraging sign, but as yet it told her nothing concrete, because everything was still expressed in terms of two unknown templates that remained to be found.
Roi used the connection to analyze the possible circular motions around the geometry’s central point. In space-time, circular motion became a helix, constantly advancing in time as it wound its way around the Hub. Only if the pitch of this helix was correct would the connection declare that it was natural motion: the path of a weightless, free-falling body.
Given the shape of a helix that constituted natural motion, she could find the period of any circular orbit. Since the geometry was symmetrical about the Hub, the period depended only on the size of the orbit, and two stones following two identically sized orbits inclined at a slight angle to each other would come together and move apart with exactly the same period as the orbit itself. In other words, she now knew the period of the shomal-junub cycle, and from that the shomal-junub weight.
Next, Roi calculated how the connection carried directions in space along the helix of the Splinter’s orbit. The speed at which the garm or sharq direction was turning—relative to the frame of the Rotator—gave the strength of the hidden spin weight which canceled the rarb-sharq weight.
For the third weight, she considered a point tied to the Splinter, but displaced from its center in the garm or sard direction. Such a point would follow a helix in space-time that would wind around the Hub with the same period as the center of the Splinter, but unlike the center its path would not constitute natural motion. The connection told her the weight it would feel; this included the spin weight, but she could subtract that from it easily enough to obtain the true garm-sard weight.
Roi summed the weights that arose in all three directions, and imposed Zak’s principle, requiring the sum to be zero.
This imposed a relationship between the two unknown templates, but it was still not enough to determine them individually. So Roi carried out a similar calculation for a different set of circumstances, summing the weights that would have been felt within the Splinter had it been plunging straight toward the Hub, rather than circling it. Again, Zak’s principle required the sum to be zero.
Now she could finally solve for the unknown templates, making all of her previously abstract results concrete.
The chamber was almost silent, save for the clicking of stones. Roi had been afraid that her “sign error” would simply cancel itself out, but it seemed to be propagating nicely, spreading its subtle changes throughout the calculations.
She found the template for the orbital periods. They obeyed the square-cube rule approximately, but closer to the Hub they now grew shorter than that simple rule implied, which was the opposite behavior to Neth’s scheme.
She calculated the ratio of the weights. It started out close to three, but on approaching the Hub it now grew smaller, not greater.
Roi passed the frame to Neth for checking and let the final template float in her mind. The ratio of the garm-sard weight to the shomal-junub weight was equal to three, minus six divided by the size of the orbit. That size was measured in units that couldn’t be determined, because among other things it depended on Neth’s unknown scale for converting time into distance. A small enough orbit would certainly have a ratio of two and a quarter, though; all you needed to do was to make its size “eight units”, whatever that came to in spans.
The same template made it equally clear that for an orbit of “six units”, the ratio would shrink further, to a value of two. The garm-sard weight, then, would be double the shomal-junub weight.
Zak’s principle, and the fact that the spin weight canceled the rarb-sharq weight exactly, made it easy to calculate another ratio: that of the garm-sard weight to the spin weight. When the first ratio fell to two, the second would rise to four. And when the second ratio hit four, as Neth had shown, orbits around the Hub would no longer be stable.
If the Splinter’s orbit was ever reduced to three-quarters of its present size, from that moment on the slightest garmwards nudge would send it spiraling into the Hub.
“No errors,” proclaimed Ruz, the last of Roi’s checkers. Shortly afterward, Kal gave the same verdict for Tan’s calculations. Their conclusions, reached separately, were brought together and compared.
“No discrepancies,” Ruz declared.
Everyone in the chamber, Roi knew, would understand what the results implied, but for a dozen heartbeats there was silence. Roi herself felt wrapped in a kind of protective skepticism: just because their esoteric calculations had finally been prodded into yielding some ratios that weren’t at odds with measurements in the real world, that didn’t make the whole theory correct.
Ruz was the first to speak. “Orbits of size six are unstable, but I’m puzzled by what happens even closer to the Hub. It looks as if orbits of size three or less are impossible, because the object would need to be moving faster than the critical speed.”
“At size two,” Gul added, “everything becomes peculiar. The direction in space that points in toward the Hub acquires a negative space-time length, as if it were a direction in time.”
Tan said, “There are many things here that we’ll need to study more carefully. And even if this geometry is correct, we haven’t shown that it’s unique. There might be other geometries that satisfy Zak’s principle and also give the kind of ratios we’ve seen.”
Roi was relieved; she couldn’t argue with any of Tan’s observations. They had made progress of a kind, but it was not yet proof of anything.
The shift wasn’t over but the calculations had been tiring, so some people started moving out of the chamber to eat. Roi joined them; she’d promised Zak that she’d tell him the outcome as soon as possible, and she could gather some food for him along the way.
As she was leaving, Neth approached her. “Sixth time successful,” she said. “Congratulations.”
“Congratulations yourself. It was your idea, I just twisted it slightly.”
Neth made a sound expressing modesty and suggesting that they desist from further mutual praise. “I’ve enjoyed working with you, Roi. I hope we meet again.”