Sign-Product Invariant in SO(3), Algebra & Probability

• Sign-product invariance is a property where an object remains unchanged under arbitrary coordinate sign flips, with deep implications in symmetry-driven systems.
• In SO(3) alignment, the PASI formalism leverages proper signed permutations to resolve axis ambiguities and achieve efficient, robust registration.
• In probabilistic and algebraic frameworks, this invariant underpins universal laws in random walks and enables orthogonal decompositions in non-associative algebras.

A sign-product invariant is a structural property—across algebraic, geometric, and probabilistic settings—wherein an object, law, or functional remains unchanged under the action of arbitrary coordinate-wise sign flips. This invariance emerges as a discrete symmetry (often along with permutation invariance) critical for characterizing robust correspondences, orthogonal decompositions, and universal probabilistic laws in domains ranging from SO(3) alignment algorithms to the fluctuation theory of sums of random variables, and the algebra of higher-dimensional hypercomplex numbers.

1. Formal Definition and Contexts

Sign-product invariance, also referred to simply as sign-invariance, denotes invariance under simultaneous reversal of the sign of each coordinate or element in a collection. Given $n$ real variables $x = (x_1, \ldots, x_n)$, an object $F(x)$ is sign-product invariant if $$F(\epsilon_1 x_1, \ldots, \epsilon_n x_n) = F(x_1, \ldots, x_n)$$ for all $(\epsilon_1, \ldots, \epsilon_n) \in \{-1, +1\}^n$. This action forms the hypercube group and is central in discrete symmetry theory.

Such invariance arises in several contexts:

• In SO(3) pattern alignment, it enables robust registration independent of axis conventions or reflectional ambiguities.
• In the law of random walks, it underpins universal persistence properties beyond the i.i.d. case.
• In non-associative algebras (e.g., octonions), it organizes tensorial decompositions by projecting onto symmetry sectors under various discrete involutions (Sarker et al., 29 Nov 2025, Berger et al., 2023, Kharinov, 2018).

2. SO(3) Alignment and the PASI Formalism

The alignment of two rotation sets on $SO(3)$ in calibration and registration can be fundamentally obstructed by unknown axis relabels and arbitrary sign flips. This problem is addressed via the Permutation-and-Sign Invariant (PASI) wrapper, which explicitly instantiates sign-product invariance (Sarker et al., 29 Nov 2025).

• Transformed Basis Vectors (TBVs): Any $R \in SO(3)$ is represented as a triple $(TBV_x, TBV_y, TBV_z) \in (S^2)^3$, where $TBV_i(R)$ is the $i$-th axis (row) of $x = (x_1, \ldots, x_n)$0 (as a unit vector), isolating axis and sign ambiguities as discrete symmetries.
• Proper Signed Permutations: The 24-element group of proper signed permutations $x = (x_1, \ldots, x_n)$1 ($x = (x_1, \ldots, x_n)$2 a permutation, $x = (x_1, \ldots, x_n)$3 a diagonal sign-matrix, $x = (x_1, \ldots, x_n)$4) captures all possible axis relabels and per-axis sign flips.
• Scoring and Selection: For each $x = (x_1, \ldots, x_n)$5, summed per-axis spherical correlation scores are computed (using fast matchers). The maximizing $x = (x_1, \ldots, x_n)$6 over all 24 possibilities identifies the correct sign-product transformation. The corresponding per-axis rotations are then fused into a single $x = (x_1, \ldots, x_n)$7 estimate.
• Algorithmic Efficiency: The PASI pipeline maintains $x = (x_1, \ldots, x_n)$8 overall complexity, compared to $x = (x_1, \ldots, x_n)$9 for classical spherical correlation, and achieves robust performance under high outlier ratios and axis ambiguity (Sarker et al., 29 Nov 2025).

This framework ensures fully correspondence-free, sign-flip–invariant registration.

3. Sign-Invariance in Probabilistic Laws

Sign-product invariance plays a foundational role in the structure of certain universal probabilistic laws, notably in the context of symmetric random walks and additive functionals of Markov chains (Berger et al., 2023).

• Exchangeable, Sign-Invariant Laws: For a collection of random variables $F(x)$0, sign-invariance (under arbitrary coordinate flips) and exchangeability suffice to yield the symmetrical persistence probability formula

$F(x)$1

in the absence of atoms, as established in the Sparre Andersen fluctuation theorem.

• Generality: Independence is not required; the law remains dictated solely by permutation and sign symmetries. This permits applications to symmetric birth-death chains and integrated random walks, yielding precise persistence exponents and bounds for these processes (Berger et al., 2023).
• Structural Implications: Sign-product invariance determines the ordering symmetries that govern recurrence relations and convolution equations, establishing the universality of certain asymptotic distributions.

4. Algebraic Decomposition and Invariants in Non-Associative Algebras

In the algebra of octonions, sign-product invariance enters through the study of triple products under pairs of reflection-like involutions—Hermitian conjugation and order reversal—each of which induces a sign change on specific terms (Kharinov, 2018):

• Involutive Symmetries: The operations $F(x)$2 (Hermitian conjugation with respect to the central slot) and $F(x)$3 (multiplicative order inversion) generate a Klein-four group of symmetries on the triple product $F(x)$4.
• Canonical Decomposition:

$F(x)$5

where the triple anticommutator $F(x)$6 is invariant under both involutions (the unique sign-product invariant sector), the associator $F(x)$7 is odd under conjugation but even under order inversion, and the triple commutator $F(x)$8 is odd under both.

• Structural Role: The sign-product invariant (fully even) combination forms an orthogonal component in the norm-squared decomposition, and uniquely characterizes the algebraic roles of commutator and associator terms.

This decomposition generalizes the classical Hermitian framework for symmetric operators to the non-associative, higher-dimensional setting.

5. Broader Applications and Implications

Sign-product invariance provides a unifying structural principle in several domains:

• In $F(x)$9 pattern alignment, it guarantees robust axis and sign identification, critical for outlier-resistant, correspondence-free rotation set registration (Sarker et al., 29 Nov 2025).
• In probability theory, it captures the universality of persistence properties for a wide class of stochastic processes, transcending classical independence assumptions (Berger et al., 2023).
• In non-associative algebra, it organizes orthogonal decompositions and the intrinsic symmetries of hypercomplex products, allowing systematic separation of invariant, partially invariant, and antisymmetric sectors (Kharinov, 2018).

A plausible implication is that advances in efficient discrete search or convex projection under sign-product symmetry may generalize to other computational problems with analogous invariance requirements.

Paper Title	Focus Domain	arXiv ID
Fast, Robust, Permutation-and-Sign Invariant SO(3) Pattern Alignment	SO(3) alignment/registration	(Sarker et al., 29 Nov 2025)
An application of Sparre Andersen's fluctuation theorem for exchangeable & sign-invariant random variables	Probability theory, random walks	(Berger et al., 2023)
Symmetry of the triple octonionic product	Algebraic decomposition, octonions	(Kharinov, 2018)

These results collectively underscore sign-product invariance both as a computational tool and as a structural invariant, central to the organization of algorithms, probabilistic laws, and algebraic decompositions across contemporary research.