Marek K

Write down the numbers from 1 to 143. Cross out every one that shares a factor with 144 (so no evens, no multiples of 3). Exactly 48 survive. Multiply any two survivors, divide by 144, and keep the remainder. You always land on another survivor. That closed system is the star of this video, drawn as 4 rings of 12. THE THEOREM Pick a step d where d divides 144 and 144 divides d x d. That works for d = 12, 24, 36, 48 and 72. Then 1, 1+d, 1+2d... is a tiny closed system where multiplying just means adding steps. Example, d = 24: 1, 25, 49, 73, 97, 121. Multiplying by 25 moves you one notch around a hexagon (watch the white dot hop). THE MIRROR Flip the hexagon end to end and pair each number with its opposite: 1 + 121 = 25 + 97 = 49 + 73 = 122 1 x 121 = 25 x 97 = 49 x 73 = 121 (remainder after dividing by 144) One flip gives a constant sum AND a constant product. Each step on the ladder has its own pair: 134/133, 122/121, 110/109, 98/97, 74/73. The ladder is complete, and a number has a mirror only if it isn't squarefree. THE STRING ART All 48 survivors pair up under flips like x -> 121/x. Each flip draws its own web of arcs. The white dots are numbers that are their own mirror. CONSTELLATIONS Fibonacci, Lucas, Pell and others land on tidy shapes in this grid. THREE WAYS TO PAIR Split 1, 5, 7, 11 into two pairs, three different ways, and multiply the pair sums: 6 x 18 = 108 8 x 16 = 128 12 x 12 = 144 THE PRIME TWIST Swap 144 for p x p and the same system turns into the Fermat quotient. The two known Wieferich primes, 1093 and 3511, sit exactly at the zero point. Its mirror partner, the point at -1, has no name and no census that I can find. That's the open question. (This restates the Wieferich problem in one picture. It doesn't solve it.) THE LIFT The same hexagon appears at 5040: 121 -> 4201 and 122 -> 4202. WHY IT MATTERS Messy multiplication becomes rotations and reflections you can see, and a 170-year-old object (the Fermat quotient) lands in the same picture as 144. Universal Mirror Theory Square-Zero Cyclic Structures, the Unique 144 Mirror, Prime-Power Coordinates, and the 7129 Application zenodo.org/records/22948064 ctftheory.com/one-mathematic…