Formal research model integrating geometry, probability, and multiverse financial structure modeling.
This paper introduces a new mathematical–financial structure called a Hedgehog. A hedgehog represents a portfolio in $\mathbb{R}^n$, where each financial instrument is modeled as a direction (a spine), and its weight represents position along that direction. Both positive and negative values are allowed, corresponding to long and short positions.
The framework avoids projection onto 2D risk–return space and instead preserves full $n$-dimensional structure. We extend the idea to parallel hedgehogs, existing across multiple hypothetical financial time lines inspired by multiverse physics. We prove that merging two hedgehogs results in another hedgehog with expanded dimensionality while preserving linear, probabilistic, and geometric structure.
Traditional portfolio models such as Markowitz oversimplify high-dimensional financial systems into a 2D optimization domain. Hedgehog Theory preserves full dimensionality, describing portfolios as points in $\mathbb{R}^n$ with vector geometry corresponding to investment exposure.
$\mathcal{W} = \mathbb{R}^n$ represents all portfolio states.
$L_i = \{ \alpha e_i \mid \alpha \in \mathbb{R} \}$ defines infinite long/short capacity per instrument.
With constraint $\sum w_i = 1$, the feasible portfolio exists on the hyperplane:
$H = \{ w \in \mathbb{R}^n \mid \sum w_i = 1 \}$
Let $R_1, \dots, R_n$ be random financial returns. The induced random space is:
$\mathcal{V} = \mathrm{span}\{R_1, \dots, R_n\}$
Mapping: $T(w) = \sum w_iR_i$ is linear.
Expected return: $\mathbb{E}[R(w)] = \mu^\top w$
Variance: $\mathrm{Var}(R(w)) = w^\top \Sigma w$
If $\mathcal{W}_1 = \mathbb{R}^n$ and $\mathcal{W}_2 = \mathbb{R}^m$, then:
$\mathcal{W}_\oplus = \mathcal{W}_1 \oplus \mathcal{W}_2 \cong \mathbb{R}^{n+m}$
The combined hedgehog inherits structure and becomes a multiverse-capable geometric risk framework.
Example portfolio: $w=(0.5,0.3,0.2)$. Expected return: $10.6\%$ Variance: $0.0299$ → $\sigma \approx 17.3\%$
The hedgehog allows optimization through linear functionals and quadratic energy expressions. Gradient, convex, heuristic, and hybrid systems are naturally compatible.
For multiverse hedgehogs we define the optimization problem:
$\max_{\tilde{w} \in \mathbb{R}^{n+m}} \; U\big( \mathbb{E}[\tilde{R}(\tilde{w})], \mathrm{Var}(\tilde{R}(\tilde{w})] \big)$
where $U$ represents decision preferences under parallel-world uncertainty. Optimization algorithms include:
Conceptually, the optimal hedgehog is the portfolio whose geometry performs best across divergent timeline economies.
Hedgehog Theory aligns with:
The structure allows AI systems to reason about uncertainty that is not only stochastic—but also structurally multivariate in reality space.
Future directions include:
The theory suggests that portfolios may not be flat numerical entities — but cosmological geometric objects evolving through time and possibility.
Classical portfolio theory, particularly Markowitz mean–variance optimization, projects high-dimensional financial information into a two-dimensional representation of expected return and variance. This approach compresses structural relationships between assets, eliminating geometric and algebraic information inherent in the full space of exposures.
In contrast, Hedgehog Theory models a portfolio as a point in an $n$-dimensional Euclidean space $\mathcal{W} = \mathbb{R}^n$, where each asset defines a direction (spine) and the associated weight determines its activation. Both positive and negative coordinates are allowed, representing long and short positions. This work develops the basic algebraic structure of Hedgehogs (Abelian group), a dynamic extension (Lie algebra) and a Riemannian metric derived from the risk structure.
Let $S = \{S_1, \dots, S_n\}$ be a finite set of financial instruments.
The weight space of the Hedgehog is defined as
$\mathcal{W} = \mathbb{R}^n.$
Each vector $w = (w_1, \dots, w_n) \in \mathbb{R}^n$ represents a portfolio state of the Hedgehog, where $w_i$ is the weight of instrument $S_i$. We allow $w_i \in \mathbb{R}$, so both long ($w_i > 0$) and short ($w_i < 0$) positions are possible.
For each instrument $S_i$, define the standard basis vector
$e_i = (0,\dots,0,1,0,\dots,0),$
with a $1$ in the $i$-th coordinate and $0$ elsewhere. The line
$L_i = \{ \alpha e_i \mid \alpha \in \mathbb{R} \}$
represents all possible portfolios that only involve instrument $S_i$. Geometrically, each line $L_i$ is a spine of the Hedgehog, and a point on this line is the length and direction (sign) of that spine.
In many settings, a budget (normalization) constraint is imposed:
$\sum_{i=1}^n w_i = 1.$
The set of all portfolio states that satisfy this constraint is the affine hyperplane
$H = \{ w \in \mathbb{R}^n \mid \sum_{i=1}^n w_i = 1 \}.$
We call $H$ the head of the Hedgehog. It is an $(n-1)$-dimensional affine subspace living inside $\mathbb{R}^n$.
Let $R_1, \dots, R_n$ be random variables representing returns of instruments $S_1, \dots, S_n$ on a given time horizon. We define the random return space as
$\mathcal{V} = \mathrm{span}\{R_1,\dots,R_n\}.$
Any portfolio state $w \in \mathcal{W}$ induces a portfolio return via the linear combination
$R(w) = \sum_{i=1}^n w_i R_i.$
This defines a mapping
$T : \mathcal{W} \to \mathcal{V}, \quad T(w) = \sum_{i=1}^n w_i R_i.$
Theorem 1. The mapping $T$ is linear.
Proof. For any $w, v \in \mathcal{W}$ and $\lambda \in \mathbb{R}$,
$T(w + v) = \sum_i (w_i + v_i) R_i = \sum_i w_i R_i + \sum_i v_i R_i = T(w) + T(v),$
$T(\lambda w) = \sum_i (\lambda w_i) R_i = \lambda \sum_i w_i R_i = \lambda T(w).$
Hence, $T$ is linear. $\square$
Let
$\mu = (\mathbb{E}[R_1], \dots, \mathbb{E}[R_n])^\top$
be the vector of expected returns, and let $\Sigma \in \mathbb{R}^{n \times n}$ be the covariance matrix of $(R_1,\dots,R_n)$. Then for any $w \in \mathcal{W}$:
$\mathbb{E}[R(w)] = \mu^\top w, \qquad \mathrm{Var}(R(w)) = w^\top \Sigma w.$
Thus, expected return is a linear functional on the Hedgehog space, while risk is represented as a quadratic form.
Let $\mathcal{H}$ denote the set of all Hedgehog states (e.g., $\mathcal{H} = \mathcal{W}$ or $\mathcal{H} = H$ with affine structure). Consider vector addition as the group operation.
Theorem 2. $(\mathcal{H}, +)$ forms an Abelian group under vector addition.
Sketch of proof.
Therefore, $(\mathcal{H}, +) \cong (\mathbb{R}^n, +)$ is an Abelian group representing static combinations of portfolios.
Static Hedgehogs capture only the combinatorial structure of portfolios. To describe dynamical behavior, sequencing of transformations, and interacting markets, we introduce differential operators on Hedgehog space and a Lie algebra structure.
For each coordinate $w_i$, define the partial derivative operator:
$X_i = \frac{\partial}{\partial w_i}.$
Portfolio dynamics can be written as
$\dot{w}_i = f_i(w,t),$
which leads to a vector field on $\mathcal{W}$:
$\mathcal{X} = \sum_{i=1}^n f_i(w,t)\,\frac{\partial}{\partial w_i}.$
For two such vector fields $X$ and $Y$, define their Lie bracket as
$[X,Y] = XY - YX.$
In financial terms, $X$ and $Y$ can represent different rebalancing rules, risk constraints, or market-driven transformations. When the order in which transformations are applied matters (e.g. due to leverage, liquidity or contagion effects),
$[X,Y] \neq 0.$
This non-commutativity motivates the introduction of a Lie algebra generated by such operators.
Define
$\mathfrak{H} = \mathrm{span}\{X_1, \dots, X_n\}$
with the Lie bracket $[\cdot,\cdot]$ as above. Under standard regularity conditions, $(\mathfrak{H}, [\cdot,\cdot])$ satisfies:
Thus $\mathfrak{H}$ forms a Lie algebra describing the infinitesimal generators of Hedgehog dynamics in interacting markets.
Risk and correlation between instruments can be interpreted as curvature of the Hedgehog space. This suggests endowing $\mathcal{W}$ with a Riemannian metric induced by the covariance matrix.
Define the metric tensor $g$ on $\mathcal{W}$ via
$g_{ij} = \Sigma_{ij}.$
Then, for an infinitesimal portfolio displacement $dw = (dw_1,\dots,dw_n)$, the squared line element is
$ds^2 = \sum_{i,j=1}^{n} \Sigma_{ij} \, dw_i \, dw_j.$
This provides a notion of distance in portfolio space shaped by variance and covariance. Directions with high variance become “longer”, while strong correlations distort angles and distances.
Given this Riemannian structure, natural rebalancing paths correspond to geodesics of the metric:
$\frac{d^2 w^k}{ds^2} + \Gamma^k_{ij}(w)\,\frac{dw^i}{ds}\frac{dw^j}{ds} = 0,$
where $\Gamma^k_{ij}$ are Christoffel symbols determined by $g$. Intuitively, optimal portfolio transitions follow “shortest” or least-risk paths in a curved risk space rather than straight lines in $\mathbb{R}^n$.
To model multiple financial timelines or parallel markets, we extend the Hedgehog space as a direct sum of component spaces:
$\mathcal{W}_\oplus = \mathcal{W}_1 \oplus \mathcal{W}_2 \oplus \dots \cong \mathbb{R}^{n_1 + n_2 + \dots}.$
Each block $\mathcal{W}_k$ represents a distinct market, economic regime or hypothetical world. The Riemannian metric generalizes to a block structure:
$g = \begin{pmatrix} \Sigma^{(1)} & C_{12} & \dots \\ C_{21} & \Sigma^{(2)} & \dots \\ \vdots & & \ddots \end{pmatrix},$
where $\Sigma^{(k)}$ is the covariance matrix within world $k$ and $C_{ij}$ encodes cross-world covariances. If $C_{ij} = 0$, markets are independent; if $C_{ij} \neq 0$, the worlds are probabilistically entangled.
Hedgehog Theory reframes portfolio modeling as a geometric, algebraic and differential problem:
This structure opens a path towards quantum extensions, Hamiltonian formulations of portfolio dynamics, and deep integration with AI-based optimization in curved, multi-world financial geometries.