Therefore, this circuit can only work if all elements are conducting current and as the definition of class A is that the no output device stops conducting during any part of the output waveform, this circuit must be a class A. If the name of circuit had just been the "Circle amplifier," I think everyone would understand the necessary class A nature of the circuit. Am I wrong? If so, where?
-Will

Will, you are wrong, but in a truly elegant way that I admire greatly. As for the Macarthur Grant selection committee, it does not know that this webzine exists; but if it did, who knows what it would think of it. Besides, as Jean Paul Sartre wrote in his essay, The Puritans of the Rialto, "And let no one come up and say that he is aware of his genius, for a genius, this is ironical but true, knows his courage but not his worth."

The Current Circle
The Circlotron circuit at idle does define one continuous current path, ending where it began, like a snake biting its own tail. And like that snake, it may twisted or folded on itself without the "circuit" being broken. In the schematic below, if one device is removed, the circuit is broken and no current flows.

So far no controversy. The circuit defines one current loop. But what good is it? As configured, its only use is to drain the batteries or task the power supply. It is not an amplifier or a unity gain buffer any other practical circuit. "Well of course," you might be thinking, the load attaches across the cathodes. What's the big deal?" The deal is that the load counts.
The addition of a load radically changes the circuit, as it increases the number of current loops from one to two. Let us start with one extreme example: a load impedance of zero ohms. Lest some balk at such a low impedance, Remember that some speaker loads come close (I was once owned ribbon speakers with an impedance of 0.5 ohms) and that here we are more concerned with theory than practicability.

With the two cathodes bridged we clearly see two current paths, which flow independently from the other. These two current paths need not share an equal current flow. And any one element may be removed without ceasing all current flow in the total circuit. Working into this load, Class A, AB, C are all possible, as the load completes a sub-current path with each tube independent of the other, two current circles or, if you wish, current triangles.
Now, zero ohms is indeed an extremely low  load impedance. Is there some trick in this low impedance? What if the load impedance were 8 ohms? The only answer in terms of current paths is "No difference." The load could be as high as a 1 meg resistor, but the two potential current paths remain.

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