1. Abstract
2. The Discovery
3. Half-Cycle Sums and Chebyshev Recurrence
4. The Minimal Polynomial
5. The Ring Q[s][u] / (u² − su + 1)
6. The Level 2 Tower
7. The Dyadic Tower
8. Significance
9. Implementation

This article presents the discovery that powers of traction zero naturally produce Chebyshev polynomial structure — a discrete, exact, purely algebraic representation of complex phase that requires no analytic continuation, no power series, and no branch cuts.

From the single generator u = 0^{1/2} and its inverse v = w^{1/2}, a quadratic extension emerges with minimal polynomial u^2 - s*u + 1 = 0, where s = u + v is the Chebyshev parameter. Every element of this ring has exact canonical form a + b·u with rational polynomial coefficients, and arithmetic is closed under the reduction rule u^2 = s*u - 1.

This structure extends naturally into a dyadic tower of quadratic extensions, where each level adjoins a new square root — 0^{1/4}, 0^{1/8}, 0^{1/16}, and so on — doubling the dimension at each step: 2, 4, 8, 16, …

In traction algebra, zero is invertible:

and the basic power rules produce a coherent structure where 0*w = 1, 0^w = -1, and i = 0^{w/2}. The question arises: what happens when we tabulate 0^n for successive integer values of n?

The answer is a phase rotation. Beginning from 0^0 = 1 and proceeding through the positive integers:

n	0^n
-3	w^3
-2	w^2
-1	w
0	1
1	0
2	-1
3	-w
4	w^2
5	-w^3

The powers of zero cycle through the classes with alternating signs. Negative powers yield pure ω-class elements; positive powers rotate through 1, 0, −1, −ω, ω², −ω³, … This is not imposed — it falls out of the traction power rules.

The structure is a phase rotation, entirely analogous to powers of e^{i*pi} on the unit circle — except that it is purely algebraic. There is no limit, no series, no epsilon-delta argument underneath. The rotation is baked into the algebra itself.

To formalize this structure, we introduce the half-step generators:

Since u*v = 0^{1/2} * 0^{-1/2} = 0^0 = 1, these generators are inverses of each other. We define the Chebyshev parameter:

and the half-cycle sum at index n:

where a_{n} denotes 0^{n/2} + w^{n/2}. These sums satisfy a remarkable recurrence:

with initial values a_{0} = 2 and a_{1} = s. This is the modified Chebyshev recurrence of the first kind, producing exact polynomial identities in the parameter s = u + v:

n	n/2	a_{n} = u^n + v^n
0	0	2
1	1/2	s
2	1	s^2 - 2
3	3/2	s^3 - 3*s
4	2	s^4 - 4*s^2 + 2
5	5/2	s^5 - 5*s^3 + 5*s
6	3	s^6 - 6*s^4 + 9*s^2 - 2
7	7/2	s^7 - 7*s^5 + 14*s^3 - 7*s
8	4	s^8 - 8*s^6 + 20*s^4 - 16*s^2 + 2
9	9/2	s^9 - 9*s^7 + 27*s^5 - 30*s^3 + 9*s
10	5	s^{10} - 10*s^8 + 35*s^6 - 50*s^4 + 25*s^2 - 2

These are exact symbolic identities — not approximations. Each half-cycle sum is a polynomial in s = u + v with integer coefficients. The polynomials are symmetric: a_{-n} = a_{n}.

The critical observation is that these polynomials refuse to collapse. Despite the many opportunities for algebraic contradiction — zero being invertible, negative powers, fractional exponents — the system remains self-consistent. The Chebyshev recurrence holds exactly, and the half-cycle sums are always well-defined polynomials in s.

Since u + v = s and u*v = 1, the generators u and v are the two roots of:

This is the minimal polynomial of u over the polynomial ring ℚ[s]. It is irreducible: the discriminant s^2 - 4 is not a perfect square in ℚ[s], so no further reduction is possible.

The immediate consequence is a reduction rule:

Any power of u can be reduced to at most linear degree in u by repeated application of this rule. This means every element of the extension has canonical form a + b·u, where a and b are polynomials in s with rational coefficients.

Every element of the ring has the form a + b·u, where a, b ∈ ℚ[s]. Arithmetic is closed under the reduction u^2 = s*u - 1:

Addition:

Multiplication:

The multiplication formula arises from expanding the product and applying u^2 = s*u - 1 to the b_{1}*b_{2}*u^2 term:

Conjugation replaces u with its inverse v = s - u:

This is an involution: conj(conj(x)) = x. It swaps the two roots of the minimal polynomial, exchanging the zero-class generator u with the omega-class generator v.

The norm of an element is the product with its conjugate. It always lands in the base ring ℚ[s]:

A critical property: for all integer powers of the generator,

Every power of u is a unit of norm 1. This means inversion is always possible for powers of the generator:

The bridge between traction algebra and the Chebyshev ring is direct:

Since u = 0^{1/2}, traction zero is 0 = u^2 = s*u - 1, and traction omega is w = v^2 = s^2 - s*u - 1. Every half-integer traction exponent maps to an integer power of u, and every integer power of u has exact canonical form in the ring.

The construction generalizes naturally. Adjoining the quarter-power w = 0^{1/4} (where w^2 = u) gives a second level of extension.

At Level 2 we introduce:

with the new Chebyshev parameter:

The parameters are linked by the Chebyshev doubling formula:

This follows directly from t^2 = (w + w^{-1})^2 = w^2 + 2 + w^{-2} = u + v + 2 = s + 2.

Over the base ring ℚ[s], the generator w satisfies a quartic minimal polynomial:

This factors over the intermediate extension as:

where t^2 = s + 2. The four roots are w, w^{-1}, -w, and -w^{-1}.

Every element of the Level 2 ring has canonical form in four components over ℚ[s]:

where a, b, c, d ∈ ℚ[s], with reduction rules t^2 = s + 2 and w^2 = t*w - 1.

The multiplication table for the basis {1, t, w, tw}:

×	1	t	w	tw
1	1	t	w	t*w
t	t	s + 2	t*w	(s+2)*w
w	w	t*w	t*w - 1	(s+2)*w - t
tw	t*w	(s+2)*w	(s+2)*w - t	(s+2)*t*w - (s+2)

The extension ℚ[s] → ℚ[s, t, w] has degree 4, with Galois group isomorphic to the Klein four-group V₄. The three non-trivial automorphisms are:

• σ (w-conjugation): w =-> w^{-1} = t - w, fixes t
• τ (sign conjugation): w =-> -w, t =-> -t
• στ: w =-> -(t - w) = w - t

The full norm N_{4}(x) = x * sigma(x) * tau(x) * sigma(tau(x)) always lands in ℚ[s], and N_{4}(w^n) = 1 for all integer n — every power of the generator remains a unit.

The pattern continues indefinitely. Each level adjoins a new square root of the previous generator, producing a tower of quadratic extensions:

Level	Generator	Dimension	Minimal Polynomial
0	ℚ[s]	1	—
1	u = 0^{1/2}	2	u^2 - s*u + 1 = 0
2	w = 0^{1/4}	4	w^4 - s*w^2 + 1 = 0
3	0^{1/8}	8	x^8 - s*x^4 + 1 = 0
k	0^{1/2^k}	2^k	x^{2^k} - s*x^{2^{k-1}} + 1 = 0

At each level, the Chebyshev doubling formula t_{k}^2 = t_{k-1} + 2 links the Chebyshev parameters, with t_{0} = s. Every element at level k has exact canonical form as a 2^k-component vector over ℚ[s].

The tower is dyadic — each level doubles the dimension, and the generators are successive square roots in the traction exponent: 0^{1/2}, 0^{1/4}, 0^{1/8}, 0^{1/16}, …

In the limit, the tower resolves all dyadic (power-of-two denominator) traction exponents with exact polynomial arithmetic — a non-archimedean algebraic number field that captures the full phase structure of traction without ever invoking the complex numbers.

The dyadic Chebyshev tower is notable for several reasons:

Phase without analysis. In classical mathematics, the connection between exponentiation and rotation requires the full machinery of real analysis: power series, convergence, Euler's formula as a theorem about limits. Here, the rotation is an algebraic consequence of the traction axioms. The Chebyshev recurrence produces exact trigonometric identities without invoking continuity, limits, or the real numbers.

No branch cuts. Extending classical complex analysis to fractional powers forces a choice of branch cuts — an analytic, non-algebraic decision. In the Chebyshev ring, half-integer and quarter-integer exponents extend the recurrence cleanly, with no ambiguity and no discontinuity. The extension u = 0^{1/2} simply adds a new generator and a new polynomial relation.

Exact rational arithmetic. Every element of the ring has coefficients in ℚ[s] — polynomials with rational numbers. No floating point, no rounding, no numerical instability. The algebraic structure is computed exactly and symbolically. This makes the system suitable for computer algebra, formal verification, and applications where exact arithmetic is essential.

Self-consistency from totality. The Chebyshev structure was not designed or imposed. It emerged from a single demand: that the algebra be total (every operation defined) and reversible (every operation invertible). The fact that this demand produces a well-known family of orthogonal polynomials — independently discovered by Chebyshev in the 19th century for entirely different reasons — is striking evidence that the traction axioms capture something fundamental about algebraic structure.

The algebraic structures described in this paper are implemented in the COTT Solver, an open-source calculator and symbolic algebra system for traction expressions.

The core engine is chebyshev_ring.py, a standalone Python module that implements:

• QsPoly — polynomials in ℚ[s] with exact rational (fractions.Fraction) coefficients.
• Element — the Level 1 ring ℚ[s][u] / (u² − su + 1), with full arithmetic, conjugation, norm, inversion, and exponentiation.
• TowerElement — the Level 2 tower with four ℚ[s]-components, supporting the complete multiplication table, Galois conjugations (σ, τ, στ), half-norm and full norm, and exact inversion.

The module has no dependencies beyond Python's standard library and is verified by a comprehensive test suite covering construction, arithmetic, associativity, distributivity, conjugation involutions, norm-1 identities, Chebyshev recurrence, traction conversion, and numeric evaluation.

The COTT Calculator GUI provides interactive visualization of the Chebyshev structure:

• Phase Map tab — plots the phase orbit of 0^{n/2} on the unit circle, the half-cycle sum waveform, and an equivalence table, all parameterized by θ.
• Explain tab — for any traction expression, shows the Chebyshev polynomial decomposition, recurrence trace, and position on the unit circle.