Integral Planes and Unit-Norm Polytopes

Technical Summary: Integral Planes and Unit-Norm Polytopes

Problem Statement
The paper addresses the geometric and algebraic structure arising from integral orders within real composition algebras (specifically $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$). While the arithmetic skeletons of these algebras (e.g., Gaussian integers, Hurwitz quaternions, Coxeter–Dickson octonions) are well-studied as root systems, the author investigates a complementary question: what spherical polytopes emerge if one treats the integral order $\mathcal{O}$ not as an algebra in isolation, but as the components of an integral plane $\mathcal{O}^2$ equipped with the quadratic form $Q(x, y) = N(x) + N(y)$?

Specifically, the work seeks to:

1. Classify the unit-norm shells of these integral planes for both crystallographic orders (over $\mathbb{Z}$) and non-crystallographic orders (over $\mathbb{Z}[\phi]$, where $\phi$ is the golden ratio).
2. Determine whether an indecomposable rank-eight "golden octonion" order exists in the non-crystallographic setting, analogous to the Coxeter–Dickson order in the crystallographic setting.
3. Clarify the relationship between the algebraic Hopf map and the discrete structure of the "balanced shell" within these planes, particularly in the context of Moufang loops.

Methodology
The author employs a unified construction based on the integral plane $\mathcal{O}^2 = \mathcal{O} \oplus \mathcal{O}$ with the quadratic form $Q(x, y) = N(x) + N(y)$. The methodology proceeds through three main technical pillars:

1. Shell Decomposition: The unit shell $S_1(\mathcal{O}^2)$ is analyzed by distinguishing two sub-shells:

   • The Axis Shell $A(\mathcal{O}) = (S_1(\mathcal{O}) \times \{0\}) \cup (\{0\} \times S_1(\mathcal{O}))$.
   • The Balanced Shell $D(\mathcal{O}) = \frac{1}{\sqrt{2}} S_1(\mathcal{O}) \times S_1(\mathcal{O})$.
     A "no-splitting criterion" (Lemma 3.6) is proven, showing that for the specific orders considered, the only solutions to $N(x) + N(y) = 1$ are trivial ($N(x)=1, N(y)=0$ or vice versa). Consequently, the full unit shell coincides with the axis shell, $S_1(\mathcal{O}^2) = A(\mathcal{O})$.

2. Root-Shell Doubling: Using the orthogonality of the two factors in $\mathcal{O}^2$, the paper establishes that the root system of the axis shell is the orthogonal direct sum $R \oplus R$ of the root system $R$ associated with the original order $\mathcal{O}$. The convex hulls of these shells are identified as free sums (or tegums) of the original polytopes.

3. Coxeter-Theoretic Obstruction: To address the existence of higher-rank non-crystallographic orders, the paper applies the classification of finite irreducible real reflection groups. It restricts attention to "root-shell systems" (Definition 2.3), which require all roots to have the same length and Cartan coefficients to lie in $\mathbb{Z}[\phi]$.

4. Fibration Analysis: The algebraic Hopf map $h_A(a, b) = (2a\bar{b}, N(a) - N(b))$ is restricted to the balanced shell. The paper demonstrates that on this shell, the second coordinate vanishes, and the map factors through the equator to coincide with the principal fibration $\pi(u, v) = u\bar{v}$ of the unit loop. This is proven for both associative algebras and the alternative (Moufang) case using Artin's two-generated associativity theorem.

Key Results

• Uniform Classification: The construction yields a uniform classification of unit-norm polytopes for ten crystallographic and two non-crystallographic orders:

  • Crystallographic: The orders $\mathbb{Z}, \mathbb{Z}[i], \mathbb{Z}[\omega]$, Hamilton, Hurwitz, Cayley–Graves, and Coxeter–Dickson produce the orthogonal-direct-sum root systems $2A_1, A_2 \oplus A_2, 4A_1, D_4 \oplus D_4, 16A_1$, and $E_8 \oplus E_8$. Geometrically, these correspond to the tegums of the square, hexagon, 16-cell, 24-cell, 8-cross polytope, and Gosset polytope ($4_{21}$).
  • Non-Crystallographic: The cyclotomic order $\mathbb{Z}[\zeta_{10}]$ and the icosian ring $I$ produce the root-shell systems $H_2 \oplus H_2$ and $H_4 \oplus H_4$ over $\mathbb{Z}[\phi]$. Their convex hulls are the decagonal and 600-cell tegums.

• Rank Obstruction Theorem: The paper proves that no indecomposable non-crystallographic root-shell system over $\mathbb{Z}[\phi]$ can have a rank greater than 4.

  • Corollary: Consequently, no indecomposable rank-eight golden octonion order exists. Any such order would require a root-shell system of rank 8, which is impossible under the axioms of single-length root-shell systems over $\mathbb{Z}[\phi]$. This result is derived purely from Coxeter theory, independent of the octonion multiplication structure.

• Finite Principal Fibrations: The paper identifies the restriction of the algebraic Hopf map to the balanced shell as a finite principal fibration of the unit loop $U = S_1(\mathcal{O})$ over itself.

  • In the associative case, $U \times U \to U$ is a trivial principal $U$-fibration.
  • In the alternative (Moufang) case (specifically for the Coxeter–Dickson order), the map remains a finite principal fibration due to Artin's theorem, despite the lack of full associativity. The paper explicitly distinguishes this discrete fibration from the topological Hopf fibration $S^{2d-1} \to S^d$.

Significance and Claims
The paper claims to provide a "single construction" that unifies the study of crystallographic and non-crystallographic arithmetic skeletons within real composition algebras. Its primary contributions are:

1. Geometric Unification: It organizes the doubling of root systems and polytopes (e.g., $E_8 \to E_8 \oplus E_8$) into a single framework based on the integral plane $\mathcal{O}^2$, separating the algebraic root-system data from the geometric polytope data.
2. Resolution of an Existence Question: It unconditionally closes the question of the existence of an indecomposable rank-eight golden octonion order, proving its non-existence via a purely Coxeter-theoretic argument.
3. Clarification of Hopf Structures: It precisely delineates the relationship between the continuous algebraic Hopf map and discrete principal fibrations on unit loops, correcting potential terminological conflations in the literature.
4. Distinction of Polytopes and Root Systems: The work emphasizes the distinction between the root system $nA_1$ (algebraic) and the cross-polytope $\beta_n$ (geometric), noting that while they share the same vertex set, they represent different structural concepts (e.g., $4A_1$ vs. the 16-cell).

The author notes that while the rank obstruction closes the door on indecomposable rank-8 systems under the current axioms (single-length, full reflection closure), the question remains open if these axioms are weakened (e.g., allowing multiple root lengths or relaxing reflection closure). The paper positions itself as establishing the "map" of this specific corner of composition algebra geometry, providing a foundation for future investigations into weaker structures or projective companions.