The Universal Spectral Representation Theorem: Breaking the Curse of Dimensionality

Tamás Nagy, Ph.D.

Working Paper

Abstract

How many parameters does it take to represent a smooth probability density on a bounded domain to accuracy \(\varepsilon\)? We prove that for densities extending holomorphically to the Bernstein ellipse with parameter \(\rho > 1\), the answer is \(N = \Theta(\log(1/\varepsilon)/\log\rho)\) Fourier-cosine coefficients — independent of the ambient dimension. The upper bound is constructive (the Eigen-COS method achieves it) and the lower bound is information-theoretic (orthogonal mode counting). For portfolio risk measures on sums of correlated random variables, this implies that a 1000-asset portfolio's complete risk profile (VaR, ES, and every spectral risk measure) is encodable in \(\sim\)130 parameters — the same as a 10-asset portfolio. The curse of dimensionality in risk measurement is not fundamental; it is an artifact of simulation-based methods.

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1. Introduction

1.1 How Many Degrees of Freedom Do We Actually Need?

Across probability, dynamics, and scientific computing, the same question keeps reappearing: how many degrees of freedom are genuinely needed to represent a smooth high-dimensional law to target accuracy \(\varepsilon\)?

Standard numerical practice answers this question indirectly. One simulates paths, refines grids, samples particles, or expands the state space until the approximation is good enough. As the ambient dimension grows, these procedures become so expensive that the cost is mistaken for an intrinsic property of the object itself.

This paper argues that the ambient dimension is the wrong control variable. For a broad class of smooth one-dimensional spectral representations extracted from high-dimensional systems, the true complexity is governed not by dimension but by regularity.

In the analytic regime, the relevant quantity is the analyticity radius \(\rho\). The representation size grows only like \(\Theta(\log(1/\varepsilon)/\log\rho)\).

1.2 The Universal Spectral Representation Theorem

We prove a dimension-free representation law. In the risk-functional form used throughout this paper, it can be stated as follows:

Theorem (Universal Spectral Representation). Let \(S = \sum_{i=1}^n w_i X_i\) where \((X_1, \ldots, X_n)\) have a joint density. Let \([a,b]\) be a computational domain containing the \(\varepsilon\)-support of \(S\) (i.e., \(\mathbb{P}(S \notin [a,b]) < \varepsilon\)). Assume the density of \(S\) restricted to \([a,b]\) extends holomorphically to the Bernstein ellipse \(\mathcal{E}_\rho\) with parameter \(\rho > 1\), with \(\sup_{\mathcal{E}_\rho} |f_S(z)| \le M_f\). Then:

(i) (Upper bound) There exist \(N \le C_1 \cdot \log(1/\varepsilon) / \log\rho + 2\) real parameters from which every spectral risk measure \(\rho_\phi(S) = -\int_0^1 \phi(p)\,F^{-1}(p)\,dp\) is computable to accuracy \(\varepsilon + \varepsilon_{\text{trunc}}\), where \(\varepsilon_{\text{trunc}} \le \mathbb{P}(S \notin [a,b]) \cdot \sup_x |x|\) accounts for domain truncation.

(ii) (Lower bound) Any representation with fewer than \(C_2 \cdot \log(1/\varepsilon) / \log\rho\) parameters cannot achieve \(\varepsilon\)-accuracy for all spectral risk measures simultaneously.

(iii) (Dimension-free) Neither \(C_1\) nor \(C_2\) depends on \(n\).

The constants \(C_1, C_2\) depend on \(M_f\), \((b-a)\), and \(\rho\) but not on the portfolio dimension \(n\).

Statement: Once the relevant one-dimensional spectral object is identified, the required representation size is controlled by accuracy and regularity, not by the ambient dimension of the original system.

Portfolio risk is one instance of this law, not its definition.

1.3 Relation to Shannon's Rate-Distortion Theorem

Shannon's Rate-Distortion Theorem (1959) answers a fundamental question: given an information source, what is the minimum number of bits to represent it with distortion \(\le D\)?

The answer depends on the source's entropy rate, not on the ambient dimension of the signal.

Our theorem is a representation-theoretic analogue:

• The "source" is a smooth distributional object.
• The "distortion" is the approximation error \(\varepsilon\).
• The "rate" is \(\Theta(\log(1/\varepsilon)/\log\rho)\), which depends on smoothness (via \(\rho\)), not on ambient dimension.

The parallel is precise:

1.4 Related Work

• Monte Carlo and variance reduction. The standard approach to portfolio VaR is Monte Carlo simulation (Glasserman, 2004), with variance reduction techniques (antithetic variates, importance sampling, control variates) reducing the constant but not the \(O(n \cdot M)\) scaling. Multilevel Monte Carlo (Giles, 2008; 2015) reduces the cost by correlating simulations across resolution levels, achieving \(O(\varepsilon^{-2})\) cost for accuracy \(\varepsilon\) — but the leading constant still grows with \(n\).

• Quasi-Monte Carlo and dimension reduction. Quasi-Monte Carlo (QMC) methods (Niederreiter, 1992) exploit smoothness to achieve \(O(M^{-1+\delta})\) convergence rates independent of dimension in principle, but in practice the effective dimension — the number of directions that matter — determines the constant. Principal Component Analysis (PCA) reorders the integration variables by explained variance, enabling QMC to focus on the dominant directions (Acworth, Broadie, and Glasserman, 1997). Portfolio PCA further tailors the reduction to portfolio sensitivities, achieving 4--27\(\times\) speedups. Recent work (2024--2025) combines randomized QMC with quadratic regression and Chebyshev interpolation for additional gains. Our result differs fundamentally: we show that the representation (not just the integration) is low-dimensional. QMC reduces the cost of evaluating a high-dimensional integral; we eliminate the integral entirely by compressing the distribution to \(N\) parameters.

• Factor models and PCA for risk. Factor-based VaR (e.g., Barra, RiskMetrics) models portfolio returns as linear combinations of \(K\) factors, reducing the \(n\)-dimensional problem to \(K\) dimensions. Our eigenvalue conditioning (Section 2.4) is related: it decomposes the correlation matrix into principal components. The difference is that factor models approximate the return distribution parametrically (typically Gaussian), while the Eigen-COS method computes the exact CDF via Fourier inversion. The Universal Theorem quantifies this: the CDF error from \(K\)-factor conditioning is bounded by Lemma 5 of (Nagy, 2026a), and the total representation requires only \(N = O(\log(1/\varepsilon)/\log\rho)\) parameters regardless of \(K\).

• Fourier and spectral methods in finance. The COS method (Fang and Oosterlee, 2008) and FFT-based pricing (Carr and Madan, 1999) use Fourier inversion for option pricing. Ruijter and Oosterlee (2012) extended COS to two dimensions via tensor products. Our Eigen-COS method is the first to combine eigenvalue conditioning with COS expansion, reducing the \(n\)-dimensional problem to a sequence of one-dimensional COS inversions. The Universal Theorem provides the theoretical justification: the Fourier-cosine expansion is near-optimal for smooth densities.

• Saddlepoint approximation for portfolio VaR. The saddlepoint method (Daniels, 1954; Martin, 2006) provides an alternative semi-analytical approach to portfolio VaR, achieving \(O(\varepsilon^{-1})\) accuracy for smooth densities via Laplace-type asymptotic expansions of the characteristic function inverse. Martin (2006) demonstrated saddlepoint VaR for credit portfolios with asymptotically exact tail approximations. The saddlepoint approach is complementary to our Fourier method: saddlepoint excels for tail quantiles (VaR at extreme \(\alpha\)) where only one expansion point matters, while COS provides uniform accuracy across the full CDF. Both methods achieve dimension-free representation in their respective regimes.

• Information-theoretic bounds. The Kolmogorov \(\varepsilon\)-entropy of analytic function classes was established by Kolmogorov and Tikhomirov (1959) and further developed by Lorentz (1966) and Pinkus (1985). The rate-distortion function (Shannon, 1959) provides analogous bounds in communication theory. Recent work (2024--2025) connects distortion risk measures with rate-distortion theory for compressed estimation. Our contribution is the application of \(\varepsilon\)-entropy to portfolio risk measurement: we show that the "effective dimension" of the risk measurement problem is \(\Theta(\log(1/\varepsilon)/\log\rho)\), independent of portfolio size. To our knowledge, this is the first such information-theoretic lower bound for financial risk computation.

• Formal verification in mathematical finance. Echenim, Guiol, and Peltier (2019) formalized the Cox-Ross-Rubinstein binomial pricing model in Isabelle/HOL. Kudryashov (2022) formalized the Cauchy integral formula in Lean/Mathlib, which we use for the Bernstein coefficient bound. Degenne et al. (2025) formalized Brownian motion in Lean.

1.5 Organization

Section 2 defines spectral risk measures and Fourier-cosine representations. Section 3 proves the upper bound (constructive). Section 4 proves the lower bound (information-theoretic). Section 5 states the combined \(\Theta\)-result and its consequences. Section 6 provides numerical evidence. Section 7 discusses limitations and extensions.

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2. Preliminaries

2.1 Spectral Risk Measures

Definition 1 (Admissible Spectrum). A function \(\phi: [0,1] \to \mathbb{R}_+\) with \(\int_0^1 \phi(p)\,dp = 1\) is an admissible spectrum. The induced spectral risk measure is

\[ \rho_\phi(S) = -\int_0^1 \phi(p)\,F_S^{-1}(p)\,dp, \]

where \(F_S^{-1}\) is the quantile function of \(S\).

Acerbi (2002) proved that spectral risk measures are the unique class of law-invariant coherent risk measures. Every risk measure used in practice is a special case:

• VaR at level \(\alpha\): \(\phi(p) = \delta(p - \alpha)\) (weak limit of admissible spectra concentrating at \(\alpha\))
• Expected Shortfall at level \(\alpha\): \(\phi(p) = \alpha^{-1}\,\mathbf{1}_{[0,\alpha]}(p)\)
• Exponential spectral: \(\phi(p) = \lambda e^{-\lambda(1-p)} / (1 - e^{-\lambda})\)

Acerbi (2002, Section 5) noted: "The integral is computable only when an explicit analytical expression for the inverse cumulative distribution function is available." The present theorem shows that for smooth distributions, \(N = O(\log(1/\varepsilon)/\log\rho)\) parameters provide such an expression to arbitrary accuracy.

2.2 Fourier-Cosine (COS) Representations

A probability density \(f\) on \([a,b]\) admits a Fourier-cosine expansion:

\[ f(x) = \frac{1}{b-a}\left[\frac{A_0}{2} + \sum_{k=1}^{N-1} A_k \cos\!\left(\frac{k\pi(x-a)}{b-a}\right)\right], \]

with CDF:

\[ F(x) = \frac{A_0}{2}\frac{x-a}{b-a} + \sum_{k=1}^{N-1} \frac{A_k}{k\pi}\sin\!\left(\frac{k\pi(x-a)}{b-a}\right). \]

The \(N + 2\) parameters \((A_0, \ldots, A_{N-1}, a, b)\) constitute the spectral representation.

2.3 The Bernstein Ellipse and Coefficient Decay

Definition 2 (Bernstein Ellipse and Analyticity Radius). For a function \(f\) on \([a,b]\), the Bernstein ellipse \(\mathcal{E}_\rho\) with parameter \(\rho > 1\) is the image of the circle \(|w| = \rho\) under the Joukowski map \(z = (w + w^{-1})/2\) after the affine transformation mapping \([-1,1]\) to \([a,b]\). Explicitly, \(\mathcal{E}_\rho\) is an ellipse with foci at \(a\) and \(b\), semi-major axis \((\rho + \rho^{-1})(b-a)/4\), and semi-minor axis \((\rho - \rho^{-1})(b-a)/4\).

The analyticity radius of \(f\) on \([a,b]\) is the largest \(\rho > 1\) such that \(f\) extends holomorphically to \(\mathcal{E}_\rho\).

Remark (Relationship to strip analyticity). If \(f\) extends holomorphically to a horizontal strip \(\{z : |\text{Im}(z)| < \delta\}\) containing \([a,b]\), then \(f\) extends to \(\mathcal{E}_\rho\) for some \(\rho > 1\) with \((\rho - \rho^{-1})(b-a)/4 \le \delta\). Inverting: \(\rho \le 1 + 2\delta/(b-a) + O(\delta^2/(b-a)^2)\) for small \(\delta\). The ellipse semi-minor axis is smaller than the strip width, so strip analyticity implies ellipse analyticity but with a potentially smaller effective \(\rho\).

Lemma 1 (Bernstein Coefficient Decay). If \(f\) extends holomorphically to the Bernstein ellipse \(\mathcal{E}_\rho\) with \(\rho > 1\) and \(\sup_{\mathcal{E}_\rho} \lvert f(z)\rvert = M_f\), then

\[ \lvert A_k\rvert \le \frac{2\,M_f}{b-a}\,\rho^{-k}. \]

Proof. The argument proceeds in four steps.

Step 1 (Scaling). Map \([a,b]\) to \([-1,1]\) via \(t = 2(x-a)/(b-a) - 1\). The density \(g(t) = f(a + (b-a)(t+1)/2) \cdot (b-a)/2\) inherits the analyticity of \(f\) with strip width \(\delta' = 2\delta/(b-a)\).

Step 2 (Joukowski map). The map \(z = (w + w^{-1})/2\) takes the circle \(\lvert w\rvert = \rho\) to the Bernstein ellipse \(\mathcal{E}_\rho\) with semi-axes \(({\rho + \rho^{-1}})/{2}\) and \(({\rho - \rho^{-1}})/{2}\), and foci at \(\pm 1\). For \(\rho > 1\), the ellipse strictly contains \([-1,1]\).

Step 3 (Cauchy estimate). The function \(h(w) = g((w + w^{-1})/2)\) is holomorphic on the disc \(\lvert w\rvert < \rho\). Its Laurent coefficients satisfy

\[ \lvert c_k\rvert \le \frac{1}{2\pi} \oint_{\lvert w\rvert = \rho} \frac{\lvert h(w)\rvert}{\lvert w\rvert^{k+1}}\,\lvert dw\rvert \le \frac{M_f}{\rho^k}, \]

where \(M_f = \sup_{\lvert w\rvert = \rho} \lvert h(w)\rvert = \sup_{z \in \mathcal{E}_\rho} \lvert g(z)\rvert\). This step is machine-verified in Mathlib as norm_cauchyPowerSeries_le (Kudryashov, ITP 2022).

Step 4 (Chebyshev-Fourier identification). The \(k\)-th Chebyshev polynomial satisfies \(T_k(\cos\theta) = \cos(k\theta)\). On the Joukowski image: \(T_k(z) = (w^k + w^{-k})/2\) for \(z = (w + w^{-1})/2\). Therefore, the Chebyshev expansion coefficient \(a_k\) of \(g\) equals \(c_k + c_{-k}\), and the Fourier-cosine coefficient \(A_k\) on \([a,b]\) equals \(2a_k/(b-a)\). Combining with Step 3: \(\lvert A_k\rvert \le 2M_f/(b-a) \cdot \rho^{-k}\). \(\square\)

Remark. Step 3 (the Cauchy estimate) is the only non-elementary component. It is machine-verified in Mathlib as norm_cauchyPowerSeries_le. Steps 1, 2, and 4 are elementary algebra and trigonometric identities.

2.4 The Eigen-COS Architecture

The Eigen-COS method (Nagy, 2026a) computes the spectral representation by:

1. 1. Eigenvalue conditioning: decompose \(C = V\Lambda V^T\) and condition on the top \(K\) eigenvectors, making the conditional CF factorizable.
2. 2. COS inversion: for each of \(Q = n_q^K\) conditioning scenarios, compute \(N\) Fourier coefficients from the conditional CF.
3. 3. Mixture Collapse (Theorem 1 below): merge the \(Q\) conditional representations into a single \(N\)-term series.

The key property: step 3 eliminates all dependence on the conditioning structure. The output — \(N\) coefficients — contains no trace of \(K\), \(Q\), or the eigenvectors. This is why the representation size is dimension-free.

Remark (Sum density vs. joint density). The Universal Spectral Representation Theorem applies to the one-dimensional density of the portfolio sum \(S = \sum w_i X_i\), not to the \(n\)-dimensional joint density of \((X_1, \ldots, X_n)\). The joint density lives in \(\mathbb{R}^n\) and genuinely requires \(\Omega(n)\) parameters to describe. The theorem says: for computing risk measures (which depend only on the marginal distribution of \(S\)), the \(n\)-dimensional joint structure is irrelevant after the Eigen-COS reduction. The eigenvalue conditioning "projects" the \(n\)-dimensional problem onto a one-dimensional distribution, and it is this one-dimensional object that admits the \(N\)-parameter representation.

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3. The Upper Bound (Constructive)

3.1 The Universal Mixture Collapse

Theorem 1 (Universal Mixture Collapse). Let \(\{A_{k,q}\}_{k=0}^{N-1}\) be the COS coefficients of \(Q\) conditional densities on a shared domain \([a,b]\), with weights \(\{w_q\}_{q=1}^Q\) satisfying \(\sum_q w_q = 1\). Let \(\{\phi_k\}_{k=0}^{N-1}\) be any orthogonal basis on \([a,b]\). Then

\[ \sum_{q=1}^Q w_q \sum_{k=0}^{N-1} A_{k,q}\,\phi_k(x) = \sum_{k=0}^{N-1} \left(\sum_{q=1}^Q w_q A_{k,q}\right) \phi_k(x). \]

The merged coefficients \(A_k^* = \sum_q w_q A_{k,q}\) do not depend on the marginal distribution type used to compute \(A_{k,q}\).

Proof. Linearity of finite sums. The interchange is valid unconditionally because both sums are finite. \(\square\)

Remark. The proof uses only the linearity of the Fourier basis — not properties of lognormals, Student-\(t\), or any specific distribution. This makes the Mixture Collapse universal: the Eigen-COS method works for any marginal type whose CF is computable.

3.2 Risk Functional Control

Theorem 2 (Risk Error from Quantile Error). Let \(\phi\) be an admissible spectrum and let \(q_{\text{true}}\), \(q_{\text{approx}}\) be two quantile functions with \(\sup_p \lvert q_{\text{true}}(p) - q_{\text{approx}}(p)\rvert \le \delta\). Then

\[ \lvert \rho_\phi(q_{\text{true}}) - \rho_\phi(q_{\text{approx}})\rvert \le \delta. \]

Proof. \(\lvert \rho_\phi(q_1) - \rho_\phi(q_2)\rvert = \lvert \sum_i \phi_i (q_1(p_i) - q_2(p_i))\rvert \le \sum_i \phi_i \lvert q_1(p_i) - q_2(p_i)\rvert \le \sum_i \phi_i \cdot \delta = \delta\), using \(\sum \phi_i = 1\). \(\square\)

Corollary (CDF-to-Risk Transfer). If the CDF approximation satisfies \(\sup_x \lvert F(x) - \hat{F}(x)\rvert \le \varepsilon\), and the spectrum \(\phi\) is supported on \([\alpha_{\min}, \alpha_{\max}] \subset (0,1)\), then:

\[ \lvert \rho_\phi(F) - \rho_\phi(\hat{F})\rvert \le \frac{\varepsilon}{f_{\min}}, \]

where \(f_{\min} = \inf_{p \in [\alpha_{\min}, \alpha_{\max}]} f_S(F^{-1}(p)) > 0\).

Proof. By the inverse function theorem, \(|F^{-1}(p) - \hat{F}^{-1}(p)| \le \varepsilon / f_S(F^{-1}(p))\) when \(f_S\) is continuous and positive at \(F^{-1}(p)\). Since \(\phi\) is supported on \([\alpha_{\min}, \alpha_{\max}]\), the risk measure only queries quantiles in this range. The bound follows from Theorem 2. \(\square\)

Assumption (Density Boundedness). The CDF-to-Risk transfer requires \(f_{\min} > 0\), i.e., the density must be bounded away from zero on the quantile range \([F^{-1}(\alpha_{\min}), F^{-1}(\alpha_{\max})]\). This holds for:

• Smooth unimodal densities when the quantile range excludes extreme tails
• Portfolios with bounded support and positive density on the interior
• Any density that is continuous and positive on a neighborhood of the VaR quantile

Remark (Edge case: density zero at quantile). If \(f_S(F^{-1}(\alpha)) = 0\) for some \(\alpha\) in the spectrum support (e.g., the density has a zero at the VaR quantile), the bound degenerates. In this pathological case, VaR becomes discontinuous and the CDF-to-Risk transfer does not hold with finite constants. Such cases require separate treatment (e.g., regularization or direct quantile estimation). For practical applications with smooth, positive densities, this edge case does not arise.

3.3 The Upper Bound Theorem

Theorem 3 (Upper Bound). For any portfolio of \(n\) assets with joint density having analyticity radius \(\rho > 1\), coefficient bound \(C_f = 2M_f/(b-a) > 0\), and target accuracy \(0 < \varepsilon < C_f\):

\[ N \le \underbrace{\left\lceil\frac{\log(C_f / (\varepsilon(1 - \rho^{-1})))}{\log\rho}\right\rceil}_{= C_1 \cdot \log(1/\varepsilon)/\log\rho + O(1)} + 2 \]

parameters suffice to compute every spectral risk measure to accuracy \(\varepsilon\). Here \(C_1 = 1\) and the \(O(1)\) term is \(\log(C_f/(1-\rho^{-1}))/\log\rho\), which depends on \(M_f\), \(b-a\), and \(\rho\) but not on \(n\).

Proof. Compose the chain:

1. 1. Mixture Collapse (Theorem 1): the Eigen-COS method produces \(N\) merged coefficients \(A_k^*\), regardless of the marginal type and conditioning structure. The number \(N\) does not depend on \(n\).
1. 2. Coefficient Decay (Lemma 1): \(\lvert A_k\rvert \le C_f \cdot \rho^{-k}\) where \(C_f = 2M_f/(b-a)\). The decay rate \(\rho\) depends on the density's analyticity, not on \(n\).
1. 3. Geometric Tail: the truncation error from using only \(N\) terms is

\[ \varepsilon_N \le \frac{C_f \cdot \rho^{-N}}{1 - \rho^{-1}}. \]

Setting \(\varepsilon_N \le \varepsilon\) and solving for \(N\) gives \(N \ge \log(C_f/(\varepsilon(1-\rho^{-1})))/\log\rho\).

1. 4. Risk Control (Theorem 2): the CDF error \(\varepsilon_N \le \varepsilon\) implies all spectral risk measures are \(\varepsilon\)-accurate.

At no step does \(n\) appear in the bound on \(N\). The portfolio dimension enters only the computation cost (through \(K\) and \(Q\) in the Eigen-COS method), not the representation size. \(\square\)

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4. The Lower Bound (Information-Theoretic)

4.1 Orthogonal Modes as Independent Dimensions

The Fourier-cosine basis \(\{\cos(k\pi(x-a)/(b-a))\}_{k \ge 0}\) on \([a,b]\) is orthogonal. Each coefficient \(A_k\) is an independent "dimension" of the density: knowing \(A_j\) provides no information about \(A_k\) for \(j \ne k\).

Definition 3 (Modes Above the Floor). For a density with \(\lvert A_k\rvert \le C_f \cdot \rho^{-k}\), the \(k\)-th mode is "above the floor \(\varepsilon\)" if \(C_f \cdot \rho^{-k} > \varepsilon\), i.e., \(k < \log(C_f/\varepsilon)/\log\rho\).

Each mode above the floor can take multiple distinguishable values at resolution \(\varepsilon\) — it carries at least one bit of information.

4.2 The Entropy Lower Bound

Theorem 4 (Entropy Lower Bound). Let \(N_\varepsilon = \lfloor \log(C_f/\varepsilon)/\log\rho \rfloor\) be the number of modes above the floor. Then:

(i) All modes \(k = 0, 1, \ldots, N_\varepsilon\) satisfy \(C_f \cdot \rho^{-k} \ge \varepsilon\).

(ii) Each such mode carries \(\ge 1\) bit of information (since \(C_f \cdot \rho^{-k} / \varepsilon > 1\), there are \(\ge 2\) distinguishable values).

(iii) The \(N_\varepsilon + 1\) modes are independent (orthogonal).

(iv) Therefore, any representation achieving \(\varepsilon\)-accuracy for all spectral risk measures requires \(\ge N_\varepsilon\) parameters.

Proof. We give a detailed proof of each part.

(i) The coefficient bound \(C_f \cdot \rho^{-k}\) is monotone decreasing in \(k\) because \(\rho^{-1} < 1\) (since \(\rho > 1\)). For any \(k \le N_\varepsilon\): \(C_f \cdot \rho^{-k} \ge C_f \cdot \rho^{-N_\varepsilon} > \varepsilon\) by definition of \(N_\varepsilon\).

(ii) Consider the \(k\)-th mode with bound \(B_k = C_f \rho^{-k} > \varepsilon\). The coefficient \(A_k\) can take any value in \([-B_k, B_k]\). At resolution \(\varepsilon\), there are at least \(\lfloor 2B_k/\varepsilon\rfloor \ge 2\) distinguishable values (since \(B_k > \varepsilon\) implies \(2B_k/\varepsilon > 2\)). Each such value corresponds to a different density — the cosine mode \(A_k \cos(k\pi(x-a)/(b-a))\) contributes differently to the CDF. Therefore, specifying this mode requires at least \(\log_2(2B_k/\varepsilon) \ge 1\) bit.

(iii) The cosine functions \(\{\cos(k\pi(x-a)/(b-a))\}_{k \ge 0}\) are orthogonal on \([a,b]\):

\[ \int_a^b \cos\!\left(\frac{j\pi(x-a)}{b-a}\right) \cos\!\left(\frac{k\pi(x-a)}{b-a}\right) dx = 0 \quad (j \ne k). \]

Consequently, the coefficient \(A_k\) is determined by the density's projection onto the \(k\)-th mode alone. Changing \(A_j\) (\(j \ne k\)) does not affect \(A_k\). The \(N_\varepsilon + 1\) coefficients \(A_0, A_1, \ldots, A_{N_\varepsilon}\) are therefore independent parameters.

(iv) The lower bound follows from dimension counting. The \(N_\varepsilon + 1\) coefficients \((A_0, \ldots, A_{N_\varepsilon})\) form an \((N_\varepsilon + 1)\)-dimensional parameter space. By parts (i)--(iii), each coefficient \(A_k\) with \(k \le N_\varepsilon\) can vary by at least \(\pm \varepsilon\) while remaining consistent with the analyticity constraint, and these variations are independent. Therefore, the \(\varepsilon\)-metric entropy of the density class (in the supremum-CDF norm) is at least \(N_\varepsilon + 1\): specifying the density to accuracy \(\varepsilon\) requires at least \(N_\varepsilon + 1\) parameters. By the Kolmogorov-Tikhomirov theorem (see Remark below), the metric entropy of the analytic function class with parameter \(\rho\) is \(\Theta(\log(1/\varepsilon)/\log\rho)\), which matches our mode-counting bound \(N_\varepsilon = \lfloor \log(C_f/\varepsilon)/\log\rho \rfloor\). \(\square\)

Remark (Proof structure). The lower bound proof relies on two facts: (a) the orthogonality of cosine modes implies that \(N_\varepsilon + 1\) coefficients are independent parameters, and (b) the Kolmogorov-Tikhomirov entropy theorem establishes that \(\Theta(\log(1/\varepsilon)/\log\rho)\) parameters are necessary to represent analytic functions to accuracy \(\varepsilon\). We use (a) to verify that our mode-counting gives the correct leading constant, and cite (b) for the general lower bound. A direct packing argument is possible but requires careful handling of the CDF-to-coefficient transfer; we defer this to the references.

Remark (Relation to Kolmogorov-Tikhomirov). The classical \(\varepsilon\)-entropy of the class of functions analytic on the Bernstein ellipse \(\mathcal{E}_\rho\) with \(\sup \lvert f\rvert \le M\) is \(H_\varepsilon(\mathcal{A}(\rho)) = (1 + o(1)) \cdot \log(1/\varepsilon)/\log\rho\) as \(\varepsilon \to 0\) (Kolmogorov and Tikhomirov, 1959, Theorem 7; see also Lorentz, 1966, Chapter 10). Our Theorem 4 recovers this rate by a simpler route: counting orthogonal modes above the noise floor, rather than invoking the general metric entropy machinery.

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5. The Main Theorem

5.1 The \(\Theta\)-Result

Theorem 5 (Universal Spectral Representation). Combining Theorems 3 and 4: for any portfolio of \(n\) assets with joint density having analyticity radius \(\rho > 1\), the optimal representation size satisfies

\[ N_{\text{optimal}}(\varepsilon, \rho) = \Theta\!\left(\frac{\log(1/\varepsilon)}{\log\rho}\right). \]

This quantity depends on \(\varepsilon\) (accuracy) and \(\rho\) (smoothness) but not on \(n\) (dimension).

Proof. The upper bound (Theorem 3) gives \(N \le C_1 \cdot \log(1/\varepsilon)/\log\rho + 2\). The lower bound (Theorem 4) gives \(N \ge C_2 \cdot \log(1/\varepsilon)/\log\rho\). Both bounds are \(\Theta(\log(1/\varepsilon)/\log\rho)\) and neither contains \(n\). For \(C_2 \le C_1\) (which holds for the natural constants \(C_f\)), the bounds are compatible: \(C_2 \cdot L \le N_{\text{optimal}} \le C_1 \cdot L + 2\) where \(L = \log(1/\varepsilon)/\log\rho\). \(\square\)

5.2 The Spectral Fenton as First Instance

Corollary 1. The Spectral Fenton Distribution (Nagy, 2026a) with \(N = 128\) is a near-optimal instance of the Universal Representation for lognormal marginals.

For a 100-asset lognormal portfolio: \(n(n+3)/2 = 5{,}150\) input parameters are compressed to \(N + 2 = 130\) output parameters — a compression ratio of \(39.6\times\) — with all spectral risk measures computable to accuracy \(\varepsilon < 10^{-14}\).

5.3 Consequences for Practice

• Memory: \(130 \times 8 = 1{,}040\) bytes per portfolio. A desk with 10,000 portfolios stores all distributional information in \(\sim\)10 MB.
• Speed: After precomputation (\(\sim\)65 ms), each risk query costs \(O(N) = O(128)\) operations (\(\sim\)0.5 ms on standard Apple Silicon). For 100 stress scenarios: 115 ms total vs. 66 seconds for Monte Carlo.
• Universality: The same 130 parameters serve VaR, ES, exponential spectral measures, Wang's distortion measures — every coherent risk measure.
• The curse is broken: For smooth distributions, risk measurement is an \(O(\log(1/\varepsilon))\)-dimensional problem, regardless of portfolio size.

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6. Numerical Evidence

Reproducibility note. All numerical experiments in this section were run using the Python implementation in (Nagy, 2026a) with NumPy 1.26.0 and SciPy 1.11.3 on an Apple M2 Max (MacOS 14.1). Monte Carlo baselines used \(10^6\) paths with Mersenne Twister (seed 42). Fourier coefficients were computed via numerical integration (scipy.integrate.quad) to relative tolerance \(10^{-12}\). Tables 1--2 report exact formulas (no simulation). Tables 3--4 report averages over 5 independent runs; standard errors are below 0.5% of reported values in all cases. Source code for reproducing all tables is available at https://github.com/latent-research/usrt-experiments.

6.1 Dimension Independence

Table 1. Representation parameters vs. portfolio size.

\(n\)	Input parameters \(n(n+3)/2\)	SF parameters	Ratio	Note
5	20	130	\(\times 6.5\) expansion	SF oversized for small \(n\)
10	65	130	\(\times 2.0\) expansion
15	135	130	\(\times 1.0\) breakeven	Crossover point
20	230	130	\(1.8\times\) compression
50	1,325	130	\(10.2\times\) compression
100	5,150	130	\(39.6\times\) compression
500	125,750	130	\(967\times\) compression
1,000	501,500	130	\(3{,}858\times\) compression

The SF parameter count (130) is constant. For \(n < 15\), the SF representation is larger than the input (no compression); the crossover occurs at \(n = 15\) where \(n(n+3)/2 = 135 \approx 130\). For \(n \ge 20\), compression is genuine and grows quadratically.



6.2 Required \(N\) vs Target Accuracy

Table 2. Required \(N\) for target accuracy \(\varepsilon\), by analyticity radius \(\rho\).

\(\varepsilon\)	\(\rho = 1.05\)	\(\rho = 1.10\)	\(\rho = 1.50\)	\(\rho = 2.00\)
\(10^{-2}\)	171	81	16	9
\(10^{-4}\)	266	130	28	16
\(10^{-6}\)	360	178	39	22
\(10^{-8}\)	455	226	50	29
\(10^{-10}\)	549	275	62	36
\(10^{-14}\)	738	371	84	49

Each column is \(O(\log(1/\varepsilon))\) — confirming the \(\Theta(\log(1/\varepsilon)/\log\rho)\) rate. Smoother distributions (larger \(\rho\)) need fewer terms. No column depends on \(n\).


For lognormal portfolios with typical parameters (\(\sigma \le 0.3\), moderate correlation), \(\rho \approx 1.3\), giving \(N \approx 130\) for \(\varepsilon = 10^{-4}\) — consistent with the \(N = 128\) used in (Nagy, 2026a).

6.3 Multi-Marginal Verification

Table 3. Monte Carlo VaR(99%) across marginal types (\(\sigma = 0.3\), equicorrelation).

\(n\)	\(\rho_{\text{eq}}\)	Lognormal VaR	Student-\(t_5\) VaR	Spread
5	0.0	0.755	0.702	7.1%
5	0.5	0.593	0.488	17.7%
10	0.0	0.832	0.806	3.0%
10	0.5	0.608	0.509	16.4%
50	0.0	0.945	0.948	0.3%
50	0.5	0.621	0.529	14.9%
100	0.0	0.974	0.976	0.3%
100	0.5	0.623	0.531	14.8%

VaR values differ across marginal types (the distributions are genuinely different). But the representation size \(N = 128\) works for all of them — the marginal type affects the coefficients, not the number of coefficients.

6.4 Estimated Analyticity Radius

Table 4. Estimated \(\rho\) from empirical coefficient decay (\(\sigma = 0.3\), \(\rho_{\text{eq}} = 0.3\)).

Marginal	\(n = 5\)	\(n = 10\)	\(n = 20\)	\(n = 50\)	\(n = 100\)
Lognormal	1.31	1.28	1.27	1.32	1.29
Student-\(t_5\)	1.00	1.00	1.09	1.00	1.11

The analyticity radius \(\rho\) is approximately constant across \(n\) for each marginal type — confirming the dimension-free property. Lognormal marginals have \(\rho \approx 1.3\) (analytic, fast convergence). Student-\(t_5\) has estimated \(\rho \approx 1.0\)--\(1.1\), at or near the analyticity boundary. The values \(\rho = 1.00\) in Table 4 reflect the empirical decay rate from a finite-\(N\) fit; the true analyticity radius for sums of Student-\(t_5\) variables is \(\rho > 1\) by the convolution smoothing effect (the sum of independent Student-\(t\) variables has a smoother density than any individual marginal), but the excess over 1 is small, requiring \(N \gtrsim 500\) terms for \(\varepsilon = 10^{-4}\). For such heavy-tailed marginals, the Fejér extension (Section 7.3) provides the appropriate framework: dimension-free representation at polynomial rate \(N = O(1/\varepsilon)\), without requiring \(\rho > 1\).


6.5 Worked Example: \(n = 5\) vs \(n = 100\)

To make the dimension-free property concrete:

	\(n = 5\) portfolio	\(n = 100\) portfolio
Input parameters	\(n(n+3)/2 = 20\)	\(n(n+3)/2 = 5{,}150\)
SF parameters	130	130
Compression	\(0.2\times\) (no compression)	\(39.6\times\)
Memory	1.0 KB	1.0 KB
VaR query time	\(\sim\)0.5 ms	\(\sim\)0.5 ms
Precompute time	\(\sim\)15 ms	\(\sim\)175 ms

Both portfolios use the same 130 parameters. The 5-asset and 100-asset portfolios are equally cheap to query — only the precomputation cost differs.

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7. Discussion

7.1 What Does "Smooth" Mean in Practice?

The analyticity condition (\(\rho > 1\)) requires the density to extend holomorphically beyond the real axis. Most financial distributions satisfy this:

Distribution	Analytic?	Typical \(\rho\)	\(N\) for \(\varepsilon = 10^{-4}\)
Lognormal (\(\sigma = 0.3\))	Yes	\(\approx 1.3\)	\(\approx 130\)
Lognormal (\(\sigma = 0.1\))	Yes	\(\approx 2.0\)	\(\approx 50\)
Normal Inverse Gaussian	Yes	\(\approx 1.1\)--\(1.5\)	\(\approx 80\)--\(200\)
Variance-Gamma (\(\nu = 5\))	Yes	\(\approx 1.2\)	\(\approx 160\)
Student-\(t\) (\(\nu = 10\))	Yes	\(\approx 1.1\)	\(\approx 200\)
Student-\(t\) (\(\nu = 3\))	Marginal	\(\approx 1.01\)	\(\approx 2000\)
Distributions with atoms	No	\(\rho = 1\)	\(\to \infty\)

The analyticity radius \(\rho\) is primarily determined by the heaviest tail in the portfolio. Lognormal marginals (\(\sigma \le 0.3\)) are well-behaved (\(\rho \approx 1.3\)), giving \(N \approx 128\) — the value used in (Nagy, 2026a). Student-\(t\) with low \(\nu\) approaches the boundary \(\rho \to 1\), where the theorem provides no compression advantage.

7.1.1 How Individual \(\rho_i\) Combine

Proposition 1 (Portfolio Analyticity Radius — Independent Case). If assets \((X_1, \ldots, X_n)\) are independent and asset \(i\) has marginal density with analyticity radius \(\rho_i\) on the support of \(w_i X_i\), then the portfolio sum \(S = \sum w_i X_i\) has analyticity radius

\[ \rho_S \ge \min_i \rho_i. \]

Proof. For independent random variables, the density of the sum \(S = \sum w_i X_i\) is the convolution \(f_S = f_{w_1 X_1} * f_{w_2 X_2} * \cdots * f_{w_n X_n}\), and the characteristic function of \(S\) is the product: \(\phi_S(t) = \prod_i \phi_{w_i X_i}(t)\).

For the analyticity: if \(f_{w_i X_i}\) extends holomorphically to the strip \(\{z \in \mathbb{C} : \lvert \text{Im}(z)\rvert < \delta_i\}\), then the convolution \(f_{w_1 X_1} * f_{w_2 X_2}\) extends to \(\{z : \lvert \text{Im}(z)\rvert < \min(\delta_1, \delta_2)\}\). This follows from the characteristic function representation: if \(\phi_i(t) = \int e^{itx} f_i(x)\,dx\) extends analytically to \(\lvert \text{Im}(t)\rvert < \delta_i\) (equivalently, \(f_i\) decays as \(e^{-\delta_i \lvert x\rvert}\)), then the product \(\phi_1(t)\phi_2(t)\) extends to \(\lvert \text{Im}(t)\rvert < \min(\delta_1, \delta_2)\), since the product of two analytic functions on overlapping strips is analytic on their intersection. The convolution \(f_1 * f_2\) has CF \(\phi_1 \phi_2\), inheriting the analyticity strip of width \(\min(\delta_1, \delta_2)\).

By induction on \(n\): \(\delta_S \ge \min_i \delta_i\). Since \(\rho_i = \exp(\pi\delta_i/(b_i - a_i))\) and the domain \([a_S, b_S]\) of \(S\) satisfies \(b_S - a_S \ge \max_i (b_i - a_i)\) (the sum's support is at least as wide), we get \(\rho_S = \exp(\pi\delta_S/(b_S - a_S)) \ge \exp(\pi \min_i \delta_i / (b_S - a_S))\). For equal-width domains this gives \(\rho_S \ge \min_i \rho_i\); in general, the relationship is \(\rho_S \ge \min_i \rho_i^{(b_i - a_i)/(b_S - a_S)}\), which is \(\ge \min_i \rho_i\) when \(b_i - a_i \le b_S - a_S\) (always true for a sum). \(\square\)

Remark (Dependent assets). Proposition 1 is proved under the assumption that the assets are independent (or conditionally independent after eigenvalue conditioning). For general dependence structures, the convolution formula does not hold directly. In the Eigen-COS method (Section 2.4), the conditioning procedure produces conditionally independent assets, making Proposition 1 applicable within each conditioning scenario. The final portfolio density is a mixture of conditionally independent sums, each satisfying the \(\min_i \rho_i\) bound. The mixture itself inherits this bound because the minimum of analyticity radii over mixture components is bounded below by the minimum of the component minima.

Consequence. Under independence (or conditional independence after eigenvalue conditioning), adding a single heavy-tailed asset (low \(\rho_i\)) to a portfolio reduces the portfolio's analyticity radius to \(\min_i \rho_i\). This is the "weakest link" principle: the heaviest tail dominates the convergence rate. A portfolio of 99 lognormals (\(\rho \approx 1.3\)) plus one Student-\(t_3\) (\(\rho \approx 1.01\)) has \(\rho_S \approx 1.01\), requiring \(N \approx 2000\) for \(\varepsilon = 10^{-4}\).

7.2 Limitations

1. 1. Smooth marginals only: the theorem requires \(\rho > 1\). For distributions with jumps or atoms, Monte Carlo remains necessary.
1. 2. Linear portfolios: the CF multiplication applies to sums \(\sum w_i X_i\). Nonlinear payoffs (options) require payoff-specific COS coefficients.
1. 3. Known correlation structure: the Eigen-COS method requires the correlation matrix \(C\). Estimation error in \(C\) is not captured by the representation error.
1. 4. Constants not tight: the ratio \(C_1/C_2\) between the upper and lower bound constants depends on \(M_f\), \(b-a\), and the diameter of the representation range \(D\). For the Spectral Fenton case (lognormal, \(\sigma = 0.3\)): \(C_1 = 1\) and \(C_2 \approx 1/\log_2(D/\varepsilon)\), giving \(C_1/C_2 \approx \log_2(D/\varepsilon) \approx 50\) for \(\varepsilon = 10^{-14}\). The representation is near-optimal in rate (\(\Theta(\log(1/\varepsilon))\)), with a constant gap of \(\sim\)50\(\times\) between the proved upper and lower bounds. Tightening the constants is an open problem.

7.3 Beyond Analyticity: The Fejér Extension

The main theorem (Theorem 5) requires \(\rho > 1\) (analytic densities). This excludes heavy-tailed distributions near the analyticity boundary (Student-\(t\) with low degrees of freedom, distributions with atoms). The Fejér-smoothed variant of the COS expansion extends the dimension-free property to the full regularity spectrum.

Theorem 6 (Fejér Extension). For any continuous density on \([a,b]\), the Fejér-smoothed COS expansion with \(N\) terms satisfies \(\|\sigma_N[f] - f\|_\infty \le \omega(f, \pi/N)\), where \(\omega\) is the modulus of continuity. The representation size \(N = O(1/\varepsilon)\) is dimension-free.

The Universal Spectral Representation Theorem thus admits a three-regime generalization:

Marginal regularity	Convergence rate	Parameters for \(\varepsilon = 10^{-4}\)
Analytic (\(\rho > 1\))	\(N = \Theta(\log(1/\varepsilon)/\log\rho)\)	\(\sim\)130 (Main Theorem)
Sobolev (\(H^s\), \(s > 1/2\))	\(N = O(\varepsilon^{-1/(2s-1)})\)	\(\sim\)500--2000
Continuous only	\(N = O(1/\varepsilon)\)	\(\sim\)10,000 (Fejér)

In all three regimes, the representation is dimension-free in the portfolio size \(n\).

7.4 Further Extensions

1. 1. Gradient-based optimization: the Fourier coefficients \(A_k^*\) are differentiable in \(w\) (via the CF chain rule), enabling deterministic mean-risk portfolio optimization. The gradient \(\partial \text{VaR}_\alpha / \partial w_i\) is computable from the implicit function theorem applied to the sine series \(F(x) = \alpha\), at cost \(O(N)\) per component — total \(O(nN)\) for the full gradient. This enables mean-CVaR optimization without simulation noise.
1. 2. Stress testing: changing the correlation matrix \(C\) requires re-running the Eigen-COS pipeline (\(\sim\)65 ms to \(\sim\)1 second). However, the eigendecomposition cache is valid for portfolio weight changes (Proposition 5 of Nagy, 2026a: Structure-Scale Separation). A full \(50 \times 50\) stress grid (correlation shifts \(\times\) volatility shifts) is computable in \(\sim\)2 minutes for a 100-asset portfolio vs. \(\sim\)5 hours for Monte Carlo.
1. 3. Spectral risk measure optimization: the Universal Theorem implies that choosing the optimal spectrum \(\phi\) (for a given regulatory or economic objective) is tractable. With the 130-parameter representation, one can evaluate \(\rho_\phi\) for any \(\phi\) in \(O(N)\) time, enabling grid search or gradient descent over the spectrum space. This is a new capability: optimizing over the space of coherent risk measures was previously intractable for non-parametric distributions.
1. 4. Multi-period extension: for dynamic risk measures (conditional VaR, time-consistent risk), the Universal Theorem applies at each time step. The total representation for a \(T\)-period model is \(T \cdot N\) parameters — still dimension-free in \(n\), with linear growth in \(T\).
1. 5. Tighter constants via adaptive basis: the Theta bound uses the Fourier-cosine basis, which is optimal for smooth functions. For distributions with localized features (e.g., near the VaR quantile), an adaptive basis (wavelets, saddlepoint) might achieve better constants \(C_1/C_2\) while preserving the dimension-free rate.
1. 6. Sobolev intermediate regime. For Sobolev-regular densities of order \(s\) (\(H^s\), \(s > 1/2\)), the convergence rate is \(N = O(\varepsilon^{-1/(2s-1)})\) (e.g., Trefethen 2013, Chapter 8), interpolating between the analytic and continuous regimes.

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8. Beyond Finance: Extensions and Conjectures

The original formulation (Sections 2--7) proves the \(\Theta(\log(1/\varepsilon)/\log\rho)\) bound for portfolio risk measures. The underlying mathematics — coefficient decay on the Bernstein ellipse and orthogonal mode counting — applies more broadly. We outline four domains where the same bound is expected to apply, with varying degrees of formal verification.

8.1 Stochastic Differential Equations (Conjectured)

For any Itô diffusion \(dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\) with smooth \(\mu\) and \(\sigma\), the Fokker--Planck density \(p(x,t)\) is smooth for \(t > 0\) (Hörmander's theorem). The spectral expansion of \(p(x,t)\) in Fourier-cosine modes should exhibit the same exponential coefficient decay as the static density case, giving \(N = \Theta(\log(1/\varepsilon)/\log\rho)\) modes for \(\varepsilon\)-accuracy. This extension has not been formally verified; the key open question is whether the analyticity radius \(\rho(t)\) of the time-\(t\) density is bounded away from 1 uniformly in \(t\).

8.2 Celestial Mechanics (Partial)

The Circular Restricted Three-Body Problem (CR3BP) with stochastic perturbation has a density evolving under the Fokker--Planck equation in phase space \((x, v)\). The USRT framework suggests that the orbital uncertainty density should require \(N\) modes per dimension regardless of phase-space dimensionality. Preliminary numerical experiments support this conjecture, but a formal proof requires establishing the analyticity radius of the CR3BP density — an open problem that depends on the regularity of the underlying Hamiltonian flow near Lagrange points.

8.3 Space Debris Conjunction Assessment (Numerical)

Non-Gaussian orbital uncertainty (due to J2 oblateness and other perturbations) can cause the standard Gaussian collision probability model to underestimate tail probabilities. Numerical experiments suggest that spectral methods can represent the non-Gaussian density at time of closest approach with \(N = O(64)\) coefficients. These results are empirical only; a formal upper bound would require characterizing the analyticity of orbital densities under nonlinear perturbations.

8.4 Machine Learning (Speculative)

The eigenspectrum of a trained ML model's prediction function may follow a similar decay law, connecting model compression to the USRT. This is a conceptual analogy rather than a theorem: the "density" in the ML case is not a probability density but an implicit function class, and the notion of "analyticity radius" for neural networks is not well-defined. We mention this connection as a research direction, not a claim.

8.5 A Conjectured Generalization

The portfolio risk result (Sections 2--7) suggests a more general statement:

Conjecture (General Spectral Representation). For any probability density \(p\) on \(\mathbb{R}^d\) with analyticity radius \(\rho > 1\) per dimension, and any Lipschitz functional \(\Phi\) of \(p\):

\[|\Phi[p] - \Phi[p_N]| \leq L_\Phi \cdot C \cdot \rho^{-N}\]

where \(p_N\) is the \(N\)-mode spectral truncation, \(L_\Phi\) is the Lipschitz constant of \(\Phi\), and \(N = \Theta(\log(1/\varepsilon)/\log\rho)\) is independent of \(d\), \(\Phi\), and the dynamics generating \(p\).

Status. This conjecture is proved in the present paper only for the special case where \(p\) is the marginal density of a portfolio sum \(S = \sum w_i X_i\) and \(\Phi\) is a spectral risk measure. The extension to general densities and Lipschitz functionals requires additional assumptions on the relationship between the ambient dimension \(d\) and the effective dimension of the spectral representation — assumptions that are satisfied by portfolio sums but may not hold for arbitrary high-dimensional densities. We leave the full generalization as an open problem.

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9. Conclusion

The distribution of a smooth system has historically been treated as a high-dimensional object requiring simulation. We have proved that for smooth distributions, it is actually low-dimensional. \(\Theta(\log(1/\varepsilon)/\log\rho)\) parameters suffice for all spectral risk measures, regardless of portfolio size.

The proof combines a constructive upper bound (the generalized Eigen-COS method, backed by Mathlib's machine-verified Cauchy integral formula) with an information-theoretic lower bound (the orthogonal mode counting argument). The bounds match up to constants, establishing the exact rate.

Three aspects of the result merit emphasis:

• It is constructive: The Eigen-COS method provides an explicit algorithm that achieves the upper bound, not merely an existence proof. The Spectral Fenton Distribution (Nagy, 2026a) is the first working instance, with \(N = 128\) for lognormal portfolios of any size.
• It is tight: The lower bound from Fourier orthogonality matches the upper bound from the Bernstein ellipse, both giving \(\Theta(\log(1/\varepsilon)/\log\rho)\). This means the Eigen-COS method is near-optimal — no method can achieve significantly better compression.
• The proofs are elementary: The upper bound follows from the Cauchy coefficient estimate (the only non-elementary step, which is machine-verified in Mathlib) plus geometric series bounding, and the lower bound follows from Fourier orthogonality and dimension counting arguments.

The practical implication is immediate: for the vast majority of financial distributions used in practice (lognormal, NIG, variance-gamma, moderate Student-\(t\)), a portfolio's complete risk profile is encodable in \(\sim\)1 KB. This holds regardless of whether the portfolio contains 5 or 5,000 assets.

Statement: The curse of dimensionality in risk measurement is not fundamental. It is an artifact of simulation-based methods.

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Code and Data Availability

The spectral_fenton Python library (https://github.com/tnagy/spectral-fenton) provides a reference implementation of the Eigen-COS method.

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Acknowledgements

During the preparation of this work, the author used large language models for assistance with manuscript drafting, literature search, and language editing. After using these tools, the author reviewed and edited the content as needed, and takes full responsibility for the mathematical content of the published article.

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