Operation-Aware Arithmetic VII (Revised): Equations Can Always Be Formed but Not Always Solved: Resolving the Fundamental Paradox of Degree-Five Equations

A polynomial equation of degree n can always be constructed from its roots

r1, . . . , rn by expanding (x−r1)· · ·(x−rn) = 0. Yet for n ≥ 5, no general algebraic

formula exists for recovering those roots from the equation. The roots exist; the

equation is formed from them; but the path back is blocked. This apparent para-

dox has been known since Abel and Ruffini (1824) and explained structurally by

Galois theory (1832), but the mechanism behind it — why the two directions are

asymmetric — has lacked an elementary account.

The present paper resolves this paradox within the bijective command framework

of Operation-Aware Arithmetic VI. The key insight is that forming an equation from

its roots and solving an equation for its roots are not symmetric operations: they

are traversals of the same command sequence in opposite directions.

We define the imaginary power operator I

n

(rotation by n × 90Γ in the complex

plane) and prove that its inverse J

n

exists within any system respecting the bijec-

tivity axiom if and only if n ≤ 4. For n ≥ 5, the periodicity i

4 = 1 forces I

n = I

r

for some r < n, making J

n

impossible.

The paradox is then resolved as follows. Forming the degree-n equation from

its roots always uses only defined algebraic operations, including I

n acting on the

solution formula a. Solving the equation requires traversing this sequence in reverse.

For n ≥ 5, the step involving I

n acting on a cannot be reversed, because J

n

is

undefined. The equation is formed by a one-way operation: the path from solution

to equation is open, but the path back is permanently closed.

This single asymmetry resolves the paradox, recovers the Abel–Ruffini theorem

as a corollary, and provides the mechanistic explanation that complements Galois

theory’s structural description: non-solvable Galois groups correspond precisely to

equations whose construction requires an irreversible operation on the solution for-

mula.