Paper: arXiv 2610.04122

Authors: Marc da Costa Nunes

Abstract

A cross-sectional signal is a forecast vector over $d$ assets at each date; demeaned and normalized, it is a point on a sphere. Its city is the direction of its time-averaged vector in a common target-aligned frame, a compressed summary. When is the angle between two cities a conservative estimate of the angle between the histories? For independent uniform histories, the probability that the history angle is at least the city angle tends to one when the city angle is at most 90 degrees, with an $O(T^{-1/2})$ error bound uniform in dimension. Selecting for positive average information coefficient (IC) changes the unconditional limit to $γ_q>1/2$, about 56% for 20 assets. The bound extends to independent, rotationally symmetric histories with temporal dependence; summable products of lag correlations preserve the conditional limit. A stationary ergodic theorem allows shared signal movement and identifies the population margin determining the limiting ordering. Among 266,815 positive-IC pairs in AlphaNova’s Competition May 2026, the history angle is at least the city angle for 58.9% overall, 74.1% at city angles up to 90 degrees, and 1.5% beyond. These frequencies describe the observed library and do not validate the uniform null: residual agreement is mostly positive, and an empirical second-moment diagnostic cannot be matched within that null by adjusting temporal dependence alone. On the full 776-signal sample, a city angle of at least 60 degrees selects pairs with a 94.3% success rate at the same history-angle threshold, against an 85.0% base rate, and retains 72.5% of pairs. Explicit constructions show the limits of direction-only summaries, retained lengths give sharp bounds, and shared time blocks recover missing information, here only at fine resolution. Persistent positive IC confines cities and bounds their motion; convergence also requires a stable mean direction.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical framework for signal compression and comparison, grounded in directional statistics and spherical geometry. It combines theoretical derivations with empirical validation on a large dataset, demonstrating both strong mathematical foundations and practical relevance. The novelty lies in applying these geometric concepts to financial signals and rigorously analyzing the implications of signal compression.

Research Flowchart

  flowchart TD
    A[Research Goal: When is City Angle a Conservative Estimate of History Angle?] --> B{Methodology: Analyze Cross-Sectional Signals & Cities};
    B --> C[Inputs: Demeaned/Normalized Forecast Vectors, Time-Averaged Vectors (Cities), Information Coefficient (IC) Data];
    C --> D{Computational Process: Statistical Analysis of Angle Relationships, Error Bounds, Probabilistic Assessment};
    D --> E[Outcomes: History Angle >= City Angle Probabilities, IC Impact, Empirical Validation (AlphaNova Data)];
    E --> F[Key Findings: City Angle <= 90 deg -> P(Hist Angle >= City Angle) -> 1; IC Selection Improves Odds; Direction-Only Limits];