arXiv subjects

João P. da Cruz

Publications and source records attributed to João P. da Cruz.

11 recordsLinked to original sources

arXiv2026

Directional Liquidity and Geometric Shear in Pregeometric Order Books

We introduce a structural framework for the geometry of financial order books in which liquidity, supply, and demand are treated as emergent observables rather than primitive market variables. The market is modeled as a relational substrate without assumed metric, temporal, or price coordinates. Observable quantities arise only through observation, implemented here as a reduction of relational degrees of freedom followed by a low-dimensional spectral projection. A one-dimensional projection induces a price-like coordinate and a projected liquidity density around the mid price, from which bid and ask sides emerge as two complementary restrictions. We show that directional liquidity imbalances decompose naturally into a rigid drift of the projected density and a geometric shear mode that deforms the bid--ask structure without inducing price motion. Under a minimal single-scale hypothesis, the shear geometry constrains the projected liquidity to a gamma-like functional form, appearing as an integrated-gamma profile in discrete data. Empirical analysis of high-frequency Level~II data across multiple U.S. equities confirms this geometry and shows that it outperforms standard alternative cumulative models under explicit model comparison and residual diagnostics.

q-fin.TR↗

arXiv2026

Pregeometric Origins of Liquidity Geometry in Financial Order Books

We propose a structural framework for the geometry of financial order books in which liquidity, supply, and demand are treated as emergent observables rather than primitive economic variables. The market is modeled as an inflationary relational system without assumed metric, temporal, or price coordinates. Observable quantities arise only through projection, implemented here via spectral embeddings of the graph Laplacian. A one-dimensional projection induces a price-like coordinate, while the projected density defines liquidity profiles around the mid price. Under a minimal single-scale hypothesis -- excluding intrinsic length scales beyond distance to the mid and finite visibility -- we show that projected supply and demand are constrained to gamma-like functional forms. In discrete data, this prediction translates into integrated-gamma cumulative profiles. We test these results using high-frequency Level~II data for several U.S. equities and find robust agreement across assets and intraday windows. Explicit comparison with alternative cumulative models using information criteria demonstrates a systematic preference for the integrated-gamma geometry. A minimal simulation of inflationary relational dynamics reproduces the same structure without invoking agent behavior or price formation mechanisms. These results indicate that key regularities of order-book liquidity reflect geometric constraints induced by observation rather than detailed microstructural dynamics. Supplementary Material is available at the arXiv submission.

q-fin.TR↗

arXiv2026

Dimensional Constraints from SU(2) Representation Theory in Graph-Based Quantum Systems

We investigate dimensional constraints arising from representation theory when abstract graph edges possess internal degrees of freedom but lack geometric properties. We prove that such internal degrees of freedom can only encode directional information, necessitating quantum states in $\mathbb{C}^2$ (qubits) as the minimal representation. Any geometrically consistent projection of these states maps necessarily to $\mathbb{R}^3$ via the Bloch sphere. This dimensional constraint $d=3$ emerges through self-consistency: edges without intrinsic geometry force directional encoding ($\mathbb{C}^2$), whose natural symmetry group $SU(2)$ has three-dimensional Lie algebra, yielding emergent geometry that validates the hypothesis via Bloch sphere correspondence ($S^2 \subset \mathbb{R}^3$). We establish uniqueness (SU($N>2$) yields $d>3$) and robustness (dimensional saturation under graph topology changes). The Euclidean metric emerges canonically from the Killing form on $\mathfrak{su}(2)$. A global gauge consistency axiom is justified via principal bundle trivialization for finite graphs. Numerical simulations verify theoretical predictions. This result demonstrates how dimensional structure can be derived from information-theoretic constraints, with potential relevance to quantum information theory, discrete geometry, and quantum foundations.

math-ph↗

arXiv2026

Asymptotic Vanishing of the Success Probability in Shor's Algorithm

Shor's factoring algorithm guarantees a success probability of at least one half for any fixed modulus N = pq with distinct primes p and q. We show that this guarantee does not extend to the asymptotic regime. As N -> infinity, the multiplicative groups Omega_N = (Z/NZ)^x form a non-tight family of probability spaces, and the probability weight associated with successful bases, proportional to p(success | a', N) p(a' | N), decays as 1/phi(N). The ensemble of uniform measures {mu_N} therefore admits no weak limit, implying an asymptotic loss of ergodicity. Monte Carlo simulations up to N <= 10^6 confirm this decay and the absence of a stationary success probability. These results demonstrate that the "expected polynomial time" in order finding is only locally defined: no global expectation exists once the arithmetic domain expands. The asymptotic vanishing of success probability explains the empirical absence of large-N implementations of Shor's algorithm and sets a fundamental limit on the scalability of quantum factoring.

quant-ph↗

arXiv2015

On the multiplicative effect of government spending (or any other spending for that matter)

There is, among the economist ecosystem, the idea of virtuous public spending as a form of promotion of economic growth. If we think on the way GDP is measured, it is not possible to get that conclusion because it becomes circular: measuring the money flow obviously will detect directly the public spending but always mixed with the flow of money from other sources. The question is how virtuous is public spending per se? Can it promote economic growth? Is there multiplicative effect in GDP bigger than 1? In this paper, we make use of the first principles of Economics to show that government spending is, at the most, as virtuous as private consumption and can be a source of economic depression and inequality if it is not restricted to fundamental services.

q-fin.GN↗

arXiv2014

Stochastic Evolution of Stock Market Volume-Price Distributions

Using available data from the New York stock market (NYSM) we test four different bi-parametric models to fit the correspondent volume-price distributions at each $10$-minute lag: the Gamma distribution, the inverse Gamma distribution, the Weibull distribution and the log-normal distribution. The volume-price data, which measures market capitalization, appears to follow a specific statistical pattern, other than the evolution of prices measured in similar studies. We find that the inverse Gamma model gives a superior fit to the volume-price evolution than the other models. We then focus on the inverse Gamma distribution as a model for the NYSM data and analyze the evolution of the pair of distribution parameters as a stochastic process. Assuming that the evolution of these parameters is governed by coupled Langevin equations, we derive the corresponding drift and diffusion coefficients, which then provide insight for understanding the mechanisms underlying the evolution of the stock market.

q-fin.ST↗

arXiv2014

A thermostatistical approach to scale-free networks

We describe an ensemble of growing scale-free networks in an equilibrium framework, providing insight into why the exponent of empirical scale-free networks in nature is typically robust. In an analogy to thermostatistics, to describe the canonical and microcanonical ensembles, we introduce a functional, whose maximum corresponds to a scale-free configuration. We then identify the equivalents to energy, Zeroth-law, entropy and heat capacity for scale-free networks. Discussing the merging of scale-free networks, we also establish an exact relation to predict their final "equilibrium" degree exponent. All analytic results are complemented with Monte Carlo simulations. Our approach illustrates the possibility to apply the tools of equilibrium statistical physics to study the properties of growing networks, and it also supports the recent arguments on the complementarity between equilibrium and nonequilibrium systems.

physics.soc-ph↗

arXiv2013

The bounds of heavy-tailed return distributions in evolving complex networks

We consider the evolution of scale-free networks according to preferential attachment schemes and show the conditions for which the exponent characterizing the degree distribution is bounded by upper and lower values. Our framework is an agent model, presented in the context of economic networks of trades, which shows the emergence of critical behavior. Starting from a brief discussion about the main features of the evolving network of trades, we show that the logarithmic return distributions have bounded heavy-tails, and the corresponding bounding exponent values can be derived. Finally, we discuss these findings in the context of model risk.

q-fin.ST↗

arXiv2013

The dynamics of financial stability in complex networks

We address the problem of banking system resilience by applying off-equilibrium statistical physics to a system of particles, representing the economic agents, modelled according to the theoretical foundation of the current banking regulation, the so called Merton-Vasicek model. Economic agents are attracted to each other to exchange `economic energy', forming a network of trades. When the capital level of one economic agent drops below a minimum, the economic agent becomes insolvent. The insolvency of one single economic agent affects the economic energy of all its neighbours which thus become susceptible to insolvency, being able to trigger a chain of insolvencies (avalanche). We show that the distribution of avalanche sizes follows a power-law whose exponent depends on the minimum capital level. Furthermore, we present evidence that under an increase in the minimum capital level, large crashes will be avoided only if one assumes that agents will accept a drop in business levels, while keeping their trading attitudes and policies unchanged. The alternative assumption, that agents will try to restore their business levels, may lead to the unexpected consequence that large crises occur with higher probability.

q-fin.RM↗

arXiv2012

Self-organized criticality in a network of economic agents with finite consumption

We introduce a simple model for addressing the controversy in the study of financial systems, sometimes taken as brownian-like processes and other as critical systems with fluctuations of arbitrary magnitude. The model considers a collection of economical agents which establish trade connections among them according to basic economical principles properly translated into physical properties and interaction. With our model we are able to reproduce the evolution of macroscopic quantities (indices) and to correctly retrieve the common exponent value characterizing several indices in financial markets, relating it to the underlying topology of connections.

q-fin.GN↗

arXiv2012

Heavy-tails in economic data: fundamental assumptions, modelling and analysis

The study of heavy-tailed distributions in economic and financial systems has been widely addressed since financial time series has become a research subject.After the eighties, several "highly improbable" market drops were observed (e.g. the 1987 stock market drop known as "Black Monday" and on even more recent ones, already in the 21st century) that produce heavy losses that were unexplainable in a GN environment. The losses incurred in these large market drop events did not change significantly the market practices or the way regulation is done but drove some attention back to the study of heavy-tails and their underlying mechanisms. Some recent findings in these context is the scope of this manuscript.

q-fin.CP↗