One candidate, several angles.

Eight sections, 73 seconds.

What is on screen in each section, with the narration line. This is the whole piece in prose; the animation adds motion, not claims.

1. 010:00–0:08

   Candidate

   One integer X receives two descriptions, which meet in one identity.

   One number, X, seen two ways. It's one more than a square, and it's also a step in a repeating sequence.

2. 020:08–0:13

   Question

   The identity lifts away and the index line N = 1, 2, 3, … draws in.

   So the question becomes: at which steps of that sequence can this happen at all?

3. 030:13–0:21

   Even

   Angle 1: the prime 2 sweeps the line and strikes every even index.

   First angle: the even steps. Count their factors of two and you always get an odd number, something no square allows. Every even step is out.

4. 040:21–0:29

   Mod 8

   Angle 2: the mod-8 phase grid; one phase at 17 and 41 survives the prime 2.

   Second angle: remainders after dividing by eight. Odd squares always leave a remainder of one. Half the cases fail at once, but at steps 17 and 41 one case slips through.

5. 050:29–0:43

   Reciprocity

   Angle 3: a wheel of local signs over index 17, then 41; the product comes out −1 and both are struck.

   Third angle, the hard one. Zoom in on 17. Inside a richer number system, every point of view casts a vote: plus one or minus one. A square would need those votes to multiply to plus one. They give minus one. Seventeen falls, and 41 falls the same way.

6. 060:43–0:53

   Propagation

   Composition arcs carry the exclusion from 17 and 41 to the first odd multiples while the camera pulls out.

   Now the trick that makes it infinite. Step 17 times any odd number behaves exactly like step 17 itself. So each exclusion jumps forward: 51, 85, 119, and on.

7. 070:53–1:05

   Infinity

   The front accelerates: an exponential pull-back while every odd multiple of 17 or 41 is struck and a running count climbs into the thousands.

   And it never stops. Every odd multiple of 17, every odd multiple of 41: thousands, then tens of thousands, struck out by the same two checks, none of them tested one by one.

8. 081:05–1:13

   Remains

   The count keeps climbing; what a solution must still satisfy, and the closing line.

   Whatever survives must still pass every other test; nothing here says a solution exists. One candidate, held fixed, until its descriptions can't agree.