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octonion._types

Abstract base class for normed division algebras (R, C, H, O).

NormedDivisionAlgebra

Bases: ABC

Abstract base class for the Cayley-Dickson tower: R, C, H, O.

All normed division algebras share: conjugation, norm, inverse, multiplication. Concrete implementations provide algebra-specific multiplication rules.

Source code in src/octonion/_types.py 
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class NormedDivisionAlgebra(ABC):
    """Abstract base class for the Cayley-Dickson tower: R, C, H, O.

    All normed division algebras share: conjugation, norm, inverse, multiplication.
    Concrete implementations provide algebra-specific multiplication rules.
    """

    @abstractmethod
    def conjugate(self) -> "NormedDivisionAlgebra":
        """Return the conjugate (negate all imaginary components)."""
        ...

    @abstractmethod
    def norm(self) -> torch.Tensor:
        """Return the norm (sqrt of sum of squared components)."""
        ...

    def norm_squared(self) -> torch.Tensor:
        """Return the squared norm (avoids sqrt for precision).

        For all normed division algebras: |x|^2 = sum of squared components.
        """
        return torch.sum(self.components ** 2, dim=-1)

    @abstractmethod
    def inverse(self) -> "NormedDivisionAlgebra":
        """Return the multiplicative inverse: x^{-1} = conj(x) / |x|^2.

        Raises ValueError if the element has zero norm.
        """
        ...

    @abstractmethod
    def __mul__(self, other: object) -> "NormedDivisionAlgebra":
        """Algebra-specific multiplication."""
        ...

    @property
    @abstractmethod
    def components(self) -> torch.Tensor:
        """Raw tensor of shape [..., dim] where dim is 1/2/4/8."""
        ...

    @property
    @abstractmethod
    def dim(self) -> int:
        """Algebraic dimension: 1 for R, 2 for C, 4 for H, 8 for O."""
        ...

components abstractmethod property

Raw tensor of shape [..., dim] where dim is 1/2/4/8.

dim abstractmethod property

Algebraic dimension: 1 for R, 2 for C, 4 for H, 8 for O.

conjugate() abstractmethod

Return the conjugate (negate all imaginary components).

Source code in src/octonion/_types.py 
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@abstractmethod
def conjugate(self) -> "NormedDivisionAlgebra":
    """Return the conjugate (negate all imaginary components)."""
    ...

norm() abstractmethod

Return the norm (sqrt of sum of squared components).

Source code in src/octonion/_types.py 
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@abstractmethod
def norm(self) -> torch.Tensor:
    """Return the norm (sqrt of sum of squared components)."""
    ...

norm_squared()

Return the squared norm (avoids sqrt for precision).

For all normed division algebras: |x|^2 = sum of squared components.

Source code in src/octonion/_types.py 
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def norm_squared(self) -> torch.Tensor:
    """Return the squared norm (avoids sqrt for precision).

    For all normed division algebras: |x|^2 = sum of squared components.
    """
    return torch.sum(self.components ** 2, dim=-1)

inverse() abstractmethod

Return the multiplicative inverse: x^{-1} = conj(x) / |x|^2.

Raises ValueError if the element has zero norm.

Source code in src/octonion/_types.py 
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@abstractmethod
def inverse(self) -> "NormedDivisionAlgebra":
    """Return the multiplicative inverse: x^{-1} = conj(x) / |x|^2.

    Raises ValueError if the element has zero norm.
    """
    ...

__mul__(other) abstractmethod

Algebra-specific multiplication.

Source code in src/octonion/_types.py 
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@abstractmethod
def __mul__(self, other: object) -> "NormedDivisionAlgebra":
    """Algebra-specific multiplication."""
    ...