"If there is no unbroken retraction of the ball to its boundary, then there is no unbroken map from the ball to itself without a fixed ord."

-- The Cosmogon Codex, Leaf 6, Branch 1, Verse 3.

I had gone as far as I could. The quest was over. We trudged our way back to the warm dens. I occasionally gave a side-long glance to the empty halls and tunnels we had investigated already.

There was one dead-end hall that was different from all the others, though we didn't spend much time there. But this particular hall had a very unusual structure: a smooth organic shape to the walls, unlike the standard cubic and cylindrical shapes to rooms of the bedestow.

I insisted that we take one more look inside that peculiar hall, though Dune Song whinged of hunger. Nonetheless, we intread it again. I took the torch and swept it over every nook. Bringing it close to the wall, I felt the surface: metallic stone, like everything in the bedestow. Then I brought the torch low, following the flow of the smooth floor.

There was a glint in the firelight. It was on the floor, near the center of the hall. As I drew my torch nearer, it glowed brighter--seemingly more than what the torch-backshine would permit. I handed the torch to Drum Storm, and at the firelight's farther distance, the markings on the floor kept their gleam. It was very similar to the happening we had met on the wall of the Shaft.

These markings had a different blee, however. There was a thin ring, about twice as wide as Dune Song's head. Next to it was a square filled with a dozen neatly arrayed symbols which I acknew from The Codex, each within their own little ring. Below this was another square, completely blank. I have sketched the arrangement as best I can.

I touched one of the symbols. It jiggled under my fingertip, and I found I could drag it around to any spot on the surface. I dragged a symbol into the lower box.

Drum Storm offered that perhaps these symbols controlled a heathen machine, as in the Shaft. I agreed with him, and added that the symbols were from The Codex. Perhaps, said I, we could arrange the symbols in the exact manner of a branch or verse from The Codex.

Like a theorem, Dune Song said. Except none of us could remember what theorem from which these might be. I said that it would help if we could understand the meaning behind this layout.

Drum Storm objected. The secret meaning behind the theorems and their proofs were for the gods to know, not we intellectually lacking wights. Our job is merely to worship them by copying their symbols through the ages, memorizing the leaves, singing the verses. The intellectual task of wights was to become proficient in symbol manipulation by following the rules of syntax.

I asked him why this was the case. Why can't we understand the meaning of the verses?

Drum Storm bristled at this, and accused me of dancing with heresy. I asked if he was certain he is not the heretic by refusing to listen to what the gods are saying.

He lunged at me, but Dune Song held him back. I commented to Dune Song that Drum Storm has run out of excuses--a physical attack to shut me up was the only way he could make himself feel comfortable again.

This statement produced another lunge, but Dune Song held him fast. Dune Song dragged him back to the entrance of the hall; I thought they would leave, but instead they stood in the doorway and tossed words. I shook my head and returned my heed to the control panel.

I tried dragging one of the symbols over to the big ring. The symbol disappeared from the ring and appeared back in its original position; a red line of text appeared above, written in the House Tongue. I acknew some of the words but could not understand the meaning of the sentence; I supposed it was telling me I was not allowed to put that symbol into the ring.

I noticed an interesting symbol in the shape of a small spiral. I dragged this one into the ring, and something happened. Everything inside the ring changed. I say "everything" because I had not noticed the ring was filled with points. There were so many, and they were so tiny, I had perceived them as a fuzz. Now they looked to me like the surface of thick saguaro juice in a mug--and using the symbol, I had just stirred it. I tried a different symbol near the spiral--all the points moved again.

After playing with more nearby symbols, I relayed my discovery. I announced that all the symbols in a particular area performed actions that changed the interior of the ring. I looked up. No one had heard me; my fellows were still in heated discussion in the doorway. Dune Song was waving the torch around in his gesturing.

I returned my heed to the floor and yondthought about The Codex--if there were anything I could recall that was similar to what was laid out here. Of course, the arrow symbols appeared often in proofs. Then I recalled what the Mouth had told me: the old machines in the Shaft formed a proof. But it had never occurred to me to figure out which proof. I thought how the cones could be moved up and down, towards the center of the Shaft and back to the rim, and rotated around the rimlength--and during all of these movements, the red streak remained, connecting the two cones in a perfect line.

Every streak is incident with at least three distinct ords.

That was the verse from The Codex which came to mind. Yes, that made sense, because you could always pick out three points on the red line, no matter how you moved the cones. Each apex of the cones was a point, and of course you could pick out any other location in the light to make the third.

I had a growing feeling that this hall was also a demonstration in the same manner as the Shaft. Again I dug through my memories of the verses. Was is something to do with rotating ords?

I dragged another symbol into the ring. At the spot where I dropped the symbol, a black dot appeared and expanded. All the points moved to the rim of the ring until the interior was entirely black. I used another nearby symbol, and the points spread out from the rim again, filling the ring space. I dropped in the mirrored versions of the symbols I used earlier, and the points rotated back again.

A phrase from the Codex popped into my head.

Retraction to the boundary.

What did that mean? I closed my eyes. How did the verse go?

If there is no unbroken retraction of the ball to its boundary, then there is no unbroken map from the ball to itself without a fixed ord.

Or something like that. But like everyone else, I copy and repeat the verses without understanding. I remembered that the proof involved arrows between symbols. So, some of the symbols were labels for things. That made sense, because I already learned some of the symbols were labels for actions, or changes, that could be made to the illustration.

Perhaps I could use other symbols to label other parts. I dragged them into the blank square, mentally assigning meaning to them: a label for the ball; a label for the exterior ring; a label for the change that would move all the points to the rim. And then what? I could draw lines between the points inside the ring, and realized that this was a way to distinguish between distinct points, as the cones of the Shaft was still fresh on my mind.

But I still had symbols left over in the top box. Why were they needed? I spent some short-logs rearranging the arrows and symbols, trying to match a vague memory of a Codex diagram. When I put the last piece to the diagram, it changed color from silver to green. All the points inside the ring retracted to the rim.

And the hall itself retracted. Startled, I lept to my feet as a low hum rang the air, and all the wall protrusions receded until I was standing at the bottom of a smooth sphere. My fellows, who had been just inside the entrance, shouted as they lost their balance then slid down toward me. The entrance was now up the side of the sphere--and directly across from it, a new tunnel mouth was revealed. I climbed towards it, not checking to see if my fellows followed.

I was glad I found a way forward, but I knew I had gotten lucky. I had not really figured out anything. Symbols without meaning are not worth writing down. I failed to live up to my side of the argument against Drum Storm. Somehow I would have to do better.

Green lights glimmered from the new tunnel.

~ Path Dust

Upon the Hour of the Hawk, Aloe Day, Third Moon, in this the 17,622nd Year A.R. ~