Essay 02 · Square-root closure
Myrion
square roots
The real-to-complex transition is the only dimensional enlargement required for square-root existence.
One familiar reason for extending the real numbers to the complex numbers is the square root of minus one.
Within the real numbers, √−1 has no solution. Introducing the imaginary unit i, with i² = −1, solves that problem and gives us the complex numbers.
This raises a natural question for myrions: if we move beyond complex numbers into three, four, five or more dimensions, can taking a square root ever force us to introduce yet another dimension?
If we move beyond complex numbers, can a square root ever force a further dimensional extension?
Perhaps surprisingly, the answer is no. Once the complex dimension is available, every finite-dimensional myrion system is square-root closed: every value has at least one square root within the same number of dimensions.
At the same time, moving into a higher myrion dimension can reveal additional square roots that were not present before. Those two facts give myrion square roots an interesting structure of their own.
The familiar jump from real to complex numbers
The real number system is not closed under square roots. For example, x² = −1 has no real solution. Complex numbers repair that by introducing i: i² = −1. More generally, every complex number has a square root that is also complex.
The striking result for myrions is that this remains true when further dimensions are added. If M₂ ≅ C is the two-dimensional myrion system, then for every finite n ≥ 2, every value in Mₙ has at least one square root in that same Mₙ.
In mathematical language, the squaring map Sₙ(x) = x ⋆ x is surjective: every possible output in Mₙ can be reached by squaring some value in Mₙ.
The square-root closure theorem
Square-root closure theorem
For every finite n ≥ 2, and every y ∈ Mₙ, there exists at least one x ∈ Mₙ such that x ⋆ x = y.
The phrase at least one is important. No additional dimension is required to find a square root. It does not say that a larger myrion system cannot contain further roots.
Why the theorem works
A general finite myrion can be written as x = a₀ + a₁e₁ + a₂e₂ + ··· + aₘeₘ, where e₁ = i, e₂ = j, e₃ = k, … and m = n − 1. The important feature of myrion multiplication is its downward structure: multiplying non-scalar basis elements never creates a new component higher than those already present.
For square roots, it is useful to collect the coefficients into cumulative, or tail, sums: sₖ = aₖ + aₖ₊₁ + ··· + aₘ. When x² = x ⋆ x is expressed using these sums, the non-scalar cumulative coefficients take a triangular form:
Cₖ = 2a₀sₖ + s²ₖ₊₁This is the key to the proof. Starting with the highest required component, the equation can be solved one level at a time, working downwards.
- Choose a positive scalar part, a₀ = t.
- Solve for the highest tail coefficient.
- Use that to solve the next one.
- Continue down through the remaining dimensions.
This guarantees all the non-scalar components of the required square. Only the scalar component remains to be matched.
That final condition becomes a continuous real-valued function of t. As t approaches zero, the function becomes negative without bound; for sufficiently large t, it becomes positive without bound. A continuous function passing from negative to positive must cross zero somewhere in between.
At that point, every coefficient matches, giving a square root in the same myrion system. This is an application of the intermediate value theorem.
What happens to purely real values?
There is one simple special case. If the myrion whose square root we want is just a real scalar y = b₀, then if b₀ ≥ 0, its usual real square root works; if b₀ < 0, the complex component provides the root: √−|b₀| = √|b₀| i.
This is why the result starts at two dimensions rather than one. The real numbers alone are not square-root closed. But once i has been introduced, no further dimension is ever required merely to ensure that a square root exists.
Higher dimensions can still reveal new roots
Square-root closure does not mean that all possible roots are confined to their original myrion subset. The simplest example is i. Within the complex plane, it has the familiar roots ±(1 + i) / √2. But the three-dimensional system introduces j, for which j² = i. So in M₃, i also has the roots ±j.
The pattern continues up the ladder: j² = i, k² = j, l² = k, e²ₚ₊₁ = eₚ.
A higher dimension is not required to make a square root exist, but it can reveal additional square roots.
How far outside the original dimension can a root go?
Suppose a myrion x has its highest non-zero component at level eᵣ: x = a₀ + ··· + aᵣeᵣ, aᵣ ≠ 0. When x is squared, its highest possible coefficient is 2a₀aᵣeᵣ.
If a₀ ≠ 0, this component remains, so the square has the same highest level. If a₀ = 0, that highest component disappears - but the next surviving level is forced by the non-zero square of the highest coefficient. Therefore squaring can reduce the highest occupied myrion level by at most one.
Any finite-support square root can lie at most one myrion basis level above the value being rooted.
So if a value genuinely occupies Mₙ, an additional root may appear in Mₙ₊₁, but it cannot require Mₙ₊₂, Mₙ₊₃, or an arbitrarily higher dimension. This explains the examples i ← j, j ← k, k ← l: each extra root appears just one step higher.
Zero is a special case
Could a non-zero higher-dimensional myrion square to zero? No. If a non-zero value has a highest occupied basis level, the same argument guarantees that its square retains a non-zero component either at that level or exactly one level below it. Therefore x² = 0 implies x = 0.
Zero has no hidden non-zero square roots in higher myrion dimensions. This is distinct from the question of zero divisors, which arise through products of two different myrions and are connected with the singular division locus.
Closure and exhaustiveness are different ideas
Square-root closure
A system is square-root closed if every value in it has at least one square root in the same system. For every finite myrion algebra Mₙ, n ≥ 2, this is true.
Exhaustiveness of roots
A system would be exhaustive for a particular value if embedding that value in a larger system revealed no further roots. Myrion subsets are not exhaustive in this sense: i has roots in M₂ but acquires the extra roots ±j in M₃. The hierarchy becomes richer without making its lower levels incomplete.
A particularly interesting hierarchy
The first few myrion dimensions gain a neat interpretation: M₁ = R is not square-root closed; M₂ ≅ C becomes square-root closed; every higher finite myrion system M₃, M₄, M₅, … remains square-root closed.
The transition R → C is the only dimensional enlargement forced by the need for square roots. Beyond that point, dimensions add algebraic structure, including non-associativity and additional roots, rather than repairing a failure of square-root existence.
Square roots are simpler than higher powers
Square roots have an advantage in a non-associative system: there is no ambiguity about what squaring means. For any x, x² = x ⋆ x contains only one multiplication.
With four or more repeated factors, association history can matter. For example, (j ⋆ j) ⋆ (j ⋆ j) = −1, while ((j ⋆ j) ⋆ j) ⋆ j = −j. So shorthand such as x⁴ is potentially ambiguous unless an association convention is specified.
What remains to investigate?
The closure theorem answers the existence question, but opens several others:
- How many square roots does a typical myrion have?
- When do additional roots appear in the next dimension?
- What geometric shape is formed by a complete set of roots?
- How does root multiplicity interact with the singular division locus?
- Can analogous results be obtained for cube roots?
- How should fourth and higher roots be defined when different association trees give different power maps?
The main result
Closure without confinement
Every finite-dimensional myrion system from two dimensions upward contains at least one square root of every one of its elements; larger systems may reveal additional roots, but those roots can extend by no more than one basis level.
The real-to-complex transition appears to be unique within the myrion hierarchy. It is the only point at which adding a dimension is required to guarantee square roots. After that, higher dimensions add possibilities rather than repair a missing operation.