Quantum Portfolio Management (QPM): Foundations in Motion

QUANTUM PORTFOLIO MANAGEMENT (QPM): FOUNDATIONS IN MOTION A PREPRINT Kevin Corella Nieto ∗ Independent Researcher IEEE Senior Member Madrid, Spain [email protected] November 25, 2025 ABSTRACT From the earliest geometric structures to the dynamic systems of modern physics, science has continuously reshaped not only how we understand complexity, but also how we reason and decide under uncertainty. Today, fields such as optimal control, reinforcement learning, and more recently quantum cognition illustrate a crucial transition: decision-making is no longer conceived as an isolated logic under fixed rules, but as a process embedded within dynamic systems where structure and evolution are inseparable. This reconfiguration opens a new lens on change itself: before measurement or decision, possible trajectories interfere and overlap, influencing their probability of realization, as if the decision were emerging from a field of coherence not yet collapsed. In this pre-decisional space, the symplectic structure acts as a hidden form of equilibrium, a geometry of action and reaction first intuited by Lagrange in orbital dynamics and later formalized by Weyl under the term symplectic, preserving system coherence beyond the space of its observable variables. From this perspective, decision-making ceases to be a series of local optimizations and becomes a dynamic of structural coherence. Quantum Portfolio Management (QPM) extends this dynamic to the realm of complex decision-making, formalizing the geometry of change within a conceptual decision architecture designed for empirical validation. Modern portfolio systems often lose strategic alignment when feedback is delayed, goals conflict, or decisions are distributed across multiple organizational layers. QPM addresses this gap by replacing reward maximization with a trajectory-level ∗ The author occasionally shares reflections on Quantum AI and symplectic systems via linkedin.com/in/kcorella t.me/AI_QC_Chronicle Template for The Thalesians Magazine A PREPRINT coherence functional (formally introduced in Section 3): C[γ] = Ztf t0 ∥ ∇V(γ(t)) −λ(t)A(γ(t), t)∥2 gdt a geometric measure of the degree to which local actions remain aligned with an evolving strategic intent. 1 Introduction Quantum Portfolio Management (QPM) extends the principle of strategic alignment defined by the PMI Standard for Portfolio Management (3rd Edition) into a framework of dynamic coherence, applied to decision environments where strategic, tactical, and operational dimensions evolve as interdependent and non-separable variables. As stated in the PMI standard, portfolio governance provides the decision-making structures that ensure strategic alignment and enable accountability for portfolio performance. Yet in dynamic environments, where feedback is delayed, incentives diverge, and multi-layered decision structures evolve at different speeds, traditional governance frameworks, anchored in periodic oversight, static prioritization, and control become insufficient to preserve structural coherence between intention and action. This loss of coherence between strategic intent and operational action exposes a deeper question: how can strategic decisions be governed when feedback is delayed, incentives diverge, and adversaries evolve? This question defines the central problem of contemporary governance, the loss of structural coherence, conceptually corresponding to the loss of alignment in PMI terminology, which QPM addresses through a formal decision architecture grounded in the geometry of change, establishing a structural correspondence between dynamic governance instability and geometric deformation. QPM proposes a geometric formulation of multiscale coherence that extends the alignment principle by integrating three complementary dimensions: • Inferring intent through Inverse Reinforcement Learning (IRL) to reconstruct the latent structure of value and constraint. • Anticipating trajectories inspired by quantum principles, where decision-making is understood as a superposition of possible futures before operational collapse. • Preserving coherence across strategic, tactical, and operational levels through multiscale governance mechanisms derived from the Standard for Portfolio Management. These dimensions are articulated within a symplectic geometry of governance, which acts as the structural space in which decision trajectories preserve their coherence under system transformations or perturbations. Within this framework, decision-making ceases to be a sequence of local optimizations and becomes a dynamic of structural coherence, capable of maintaining strategic intent without explicit local control. QPM is based on two interdependent dynamic fields. A(t) projects high-level objectives as a governance vector, while a bounded λ(t) encodes urgency. The optimal trajectory γ(t) emerges 2 Template for The Thalesians Magazine A PREPRINT from the deformation of the decision geometry that reconciles the value gradient ∇V(γ(t)) with the governance field λ(t)A(γ(t), t), preserving the structural coherence of the system. Grounded in symplectic geometry, inverse reinforcement learning, and multiscale governance principles, QPM can be understood as a general architecture for decision-making that includes, as limiting cases, models ranging from Markowitz portfolio theory to classical reinforcement learning. In a synthetic seven-stage benchmark used for preliminary validation, QPM reduces the coherence cost by up to 50 percent for σ= 0.25 and maintains sublinear scalability in execution time up to n= 50 dimensions. Formal constraints for stability are presented (metric bounds, λ -convexity, and stochastic ergodicity), while questions remain open regarding coercivity and extensions to discrete actions. QPM is not a parameter adjustment. It redefines portfolio control as the coherent evolution of decisions within a geometry of governance, maintaining strategic alignment where traditional optimization loses continuity. In doing so, it transforms the very notion of control, not as external imposition but as coherence maintained through change. 2 Dynamic Governance and Multiscale Alignment Before introducing the formal dynamics of Quantum Portfolio Management (QPM), it is useful to recall how the Standard for Portfolio Management (PMI, third edition) structures the discipline around a set of knowledge areas. Among them, two stand as the fundamental coordinates of any portfolio system: Strategic Management and Governance Management. Taken together, they define the canonical basis upon which the coherent dynamics of decisions can be described. Strategic Portfolio Management Defines what the organization seeks to achieve and why the portfolio exists. It ensures that every initiative and investment respond to a strategic purpose, translating long-term intent into a structured set of objectives. In QPM, this area becomes the potential field, where the organization’s intent is formalized as a geometry of value: a space of possible trajectories guided by strategic intention. Ultimately, strategic management establishes the coherence framework toward which all portfolio decisions tend. Portfolio Governance Management Defines how decisions are made, authorized, and controlled throughout the portfolio life cycle. It provides the operational mechanism through which alignment, accountability, and transparency are maintained. In QPM, governance acts as the momentum field, shaping the dynamic response of the system and determining how intent evolves under constraint. This function includes the continuous adjustment of the portfolio, the dynamic redistribution of resources and priorities, understood in QPM as the natural manifestation of the coherence flow between intent and execution. Together, these two knowledge areas constitute the canonical pair (Strategy,Governance) , the structural coordinates through which the coherence of a portfolio can be represented, measured, 3 Template for The Thalesians Magazine A PREPRINT and evolved. The other areas defined by the PMI standard (Performance,Communication, and Risk) are not omitted but emerge naturally as derivative behaviors within this simplified formulation, expressing the secondary effects of the interaction between strategic intent and governance flow. In this sense, QPM does not discard the PMI framework but condenses it into its most essential form, treating strategy and governance as conjugate variables within a single symplectic system of decision dynamics. Multiscale Challenges in Management Processes The challenges faced by contemporary portfolios do not arise only from technical complexity but from the internal limitations of their own management processes. The PMI standard articulates them into three main groups (Definition,Alignment, and Control), and it is within these that the discontinuities QPM seeks to resolve emerge. Table 1: Structural challenges in portfolio governance and their interpretation in QPM. PMI Process Group Associated Structural Challenge QPM Interpretation Definition Dynamic environments and competitors that evolve faster than planning cycles. The portfolio is defined on information that becomes obsolete before materializing. QPM introduces an evolutionary geometry of strategic potential that adapts the decision space to changing environmental conditions. Alignment Temporal decoupling between strategic and operational levels. Decision scales evolve at different rates, generating structural desynchronization. QPM models alignment as a continuous flow of coherence capable of maintaining coupling between levels with different time horizons. Alignment / Control Misaligned local incentives. Tactical units optimize their own objectives, degrading global coherence. QPM formalizes this distortion as a symplectic deformation and proposes rebalancing mechanisms based on the Value Gradient. Control / Supervision Incomplete or incoherent feedback. Performance signals arrive delayed or noisy, preventing accurate adjustment. QPM incorporates the Value Gradient (∇V) as an anticipatory correction mechanism that restores alignment between intent and action. These limitations are not marginal; they constitute the core of the multiscale challenge that runs through contemporary governance. In the face of processes that fragment coherence, QPM proposes an architecture capable of preserving structural alignment even when temporal, informational, or organizational scales become decoupled. Addressing these challenges requires abandoning the discrete view of processes and describing governance as a continuous flow of structural interaction. From Governance Structure to the Dynamic Decision Space Reducing the PMI system to the canonical pair (Strategy,Governance) allows portfolio management to be reinterpreted not as a set of sequential processes but as a dynamic space of interaction between intent and execution. In this space, strategy defines a potential field that directs the system toward its goals, while governance acts as the response mechanism that modulates organizational energy, imposes limits, prioritizes resources, and corrects trajectories. 4 Template for The Thalesians Magazine A PREPRINT When both fields, the strategic and the governance, interact coherently, the portfolio maintains structural stability: local decisions reflect global intent, and the system preserves its form through change. However, when the interaction becomes unbalanced, due to delayed feedback, divergent incentives, or structural and contextual constraints, the system loses coherence and gradients emerge, natural directions of correction along which strategic value tends to restore equilibrium. These directions define what QPM calls the Value Gradient (∇V) . It is not a static measure of benefit but a vector field that expresses how the perceived value of the system changes under small variations in its structural decisions. In governance terms, the value gradient represents the restorative force that arises when strategy and execution fall out of alignment. Thus, the Value Gradient becomes the dynamic manifestation of the principle of coherence, a natural adjustment mechanism that drives the system to recover equilibrium between intent and action. From this point, the mathematical formalization of QPM will describe how this gradient interacts with the governance field λ(t)A(t), generating the coherent evolution of decision trajectories γ(t). From a Static Framework to a Dynamic Surface of Coherence In the PMI standard, the relationship between strategy and governance materializes in a matrix (see Table 2) that distributes the knowledge areas across process groups Definition, Alignment, Authorization, and Control. From the perspective of QPM, this structure can be reinterpreted as a flow space between two fundamental poles: Strategic Management, which defines purpose and direction (potential field), and Governance Management, which regulates operational momentum and preserves structural coherence (momentum field). Table 2: Portfolio Management Process Groups and Knowledge Areas Mapping (adapted from the PMI Standard for Portfolio Management, 3rd Edition). Knowledge Areas Defining Process Group Aligning Process Group Authorizing and Controlling Strategic Definition of the portfolio’s strategic purpose and of the criteria guiding future value creation. Continuous adjustment of strategic direction to preserve coherence between intent and an evolving context. — Governance Establishment of the governance framework, decision rules, and mechanisms that sustain structural coherence. Synchronization between governance and operations through dynamic readjustment of priorities and thresholds. Oversight of system integrity and authorization of interventions required to preserve traceability and control. Performance Definition of metrics that characterize the portfolio’s global performance behavior. Performance adjustments through balancing mechanisms across capacity, demand, and generated value. — Communication Design of the structural information system. Alignment of perceptions and coordination of informational flows. — Risk Definition of the portfolio’s anticipation, exposure, and risk-tolerance framework. Dynamic adaptation of the risk posture in response to emerging signals and environmental changes. — 5 Template for The Thalesians Magazine A PREPRINT The remaining domains (Performance,Communication, and Risk) manifest as dynamic projections of this exchange between strategy and governance: measurements of circulating value, propagation of information, and perturbations that the system either absorbs or dissipates. Thus, what the PMI presents as a static map of processes becomes, under QPM, a dynamic surface of coherence where each cell represents a form of value transfer between intent and action. It is precisely on this surface that the Value Gradient (∇V) emerges: the local measure of that transfer, the vector that indicates how and in what direction the system tends to restore its strategic alignment when internal governance forces and decision flows fall out of sync. 3 Mathematical Framework & Modular Architecture What the PMI represents as a structured map of processes is reinterpreted, from the perspective of QPM, as a dynamic surface of coherence where each cell, understood as a local unit of value transfer between intention and action, represents a dynamic form of coherence within the system. On this surface emerges the Value Gradient (∇V) : the local measure of that transfer, the vector that indicates how and in what direction the system tends to restore its strategic alignment when internal governance forces and decision flows lose synchrony. Coherence, understood as structural stability, cannot be described only qualitatively. If the Value Gradient expresses the instantaneous direction of readjustment, a deeper principle is required to measure its temporal persistence. In QPM, that measure takes a variational form: coherence ceases to be a state and becomes a trajectory. The governance field and the strategic gradient are integrated into a unified mathematical formulation where each trajectory γ(t) is evaluated by its degree of structural alignment with strategic intent. QPM introduces its formal framework here: a symplectic dynamics that quantifies coherence through the structural coherence functional C[γ] , the starting point of its modular architecture. 3.1 Structural Coherence Functional Structural coherence is defined as the integral of the mismatch between the Value Gradient and the governance field (state-dependent), measured under the metric g: C[γ] = Ztf t0 ∥ ∇V(γ(t)) −λ(t)A(γ(t), t)∥2 gdt. where: •M : differentiable manifold of possible portfolio configurations, each point representing an attainable strategic state; •ω: closed, non-degenerate symplectic 2-form (dω = 0); •J: compatible almost-complex operator; •g: Riemannian metric compatible with (ω, J). Decision trajectories γ(t)⊂ M evolve under a Hamiltonian flow: dγ dt =XH(γ(t), t), ιXHω=dH, 6 Template for The Thalesians Magazine A PREPRINT where the Hamiltonian H(x, t)governs the conservative part of the strategic value dynamics. Governance enters either as a control input u(t) (for instance, a policy based on λ ) or as a scalar coupling in H: dγ dt =XH(γ(t), t)+B(γ(t), t)u(t), or H(·, t;u)=H0(·, t) + Φ(·, t)⊤u(t). Coherent trajectories are the solutions of the flow that minimize C[γ] with respect to admissible policies λ(·)and fields A(·,·): ˙γ(t)=XH(γ(t), t) (or its controlled variant), γ∗∈arg min γadmissible C[γ]. 3.1.1 Structural Components of the Decision Geometry Table 3: Structural components of the QPM decision geometry. Symbol Name Nature / Domain Structural Function in QPM MDecision space Differentiable manifold Possible portfolio configurations; each point is a feasible strategic state. ωSymplectic form Closed 2-form (dω = 0) Preserves structure and defines canonical intention–governance relations. JAlmost-complex operator Endomorphism with J2= −Id Geometric compatibility; bridge between ωand g. gCoherence metric Positive-definite tensor Measures deviations and stability; induces the norm ∥·∥g. H(x, t)Coherence Hamiltonian Scalar on M Generates XH ; governs the conservative component. A(x, t)Governance field Vector in TxM Directs the response; organizational constraints and priorities. λ(t)Urgency factor Scalar in [0, λ] Modulates the intensity of A ; temporal pressure or executive energy. ∇V(x)Value gradient Vector in TxM Direction of maximum recovery of strategic coherence. γ(t)Decision trajectory Smooth curve in M Continuous evolution of decisions; solution of the flow. XHHamiltonian field Vector in TxM Structural flow preserving coherence invariants. C[γ]Coherence functional Scalar integral Dynamic discrepancy between intention and governance; coherence criterion. Hamiltonian dynamics and minimal coherence jointly govern the system under laws of conservation and strategic alignment. 7 Template for The Thalesians Magazine A PREPRINT 3.2 Modular Architecture of QPM The geometric framework of QPM is not an abstract formulation but an operational architecture distributed across interdependent modules. Each module performs a function within the global symplectic flow: propagate, evaluate, infer, superpose, and couple. The architecture acts as a bridge between theory and governance (PMI), transforming the geometry of change into verifiable operations. Table 4: Modular components of the QPM architecture. Module Function in the System Mathematical / Physical Core PMI Interpretation Symplectic Flow Engine (SFE) Propagates decisions as coherent flows in (M, ω, J, g) ; preserves invariants. ˙γ=XH(γ, t) and, if applicable, +B(γ, t)u(t);ιXHω=dH. Strategic alignment flow: ensures that tactical decisions preserve portfolio intent. Coherence Controller (CC) Evaluates coherence and regulates deviations between ∇Vand λA. C[γ] = R∥∇V−λA∥2 gdt. Restricted minimization, not isolated δC= 0. Governance and performance control: consistency between objectives and execution. Intent Inference Unit (IIU) Reconstructs latent intent from data or experience; updates A(x, t). IRL: potential update V←V+ ηR ; or redefinition of H(·, t;u) . Organizational learning: inferring real intent from operational behavior. Superposition Explorer (SE) Maintains and evaluates a set Γ={γi(t)} before decision collapse. Selection by coherence: C[γ∗]< ε. Scenario and strategic option management. Governance Interface (GI) Couples internal dynamics with external governance artifacts. A(x, t) and λ(t) derived from traceable policies. Portfolio Governance Management: traceability between decision and governance. In QPM, superposition designates a state of coexistence of strategic trajectories maintained before decision collapse. The term is quantum-inspired: it translates into the governance domain the logic of coherence and selection that, in physics, governs the evolution of states. The SFE establishes the fundamental motion; the CC measures the tension between ∇V and λA and feeds back the flow to preserve structure; the IIU infers intent and updates A(x, t) , reconfiguring the effective dynamics; the SE maintains alternative trajectories when coherence degrades; and the GI ensures institutional traceability (PMI). Together they form a closed circuit of distributed coherence: the flow generates motion, the controller stabilizes, the inference corrects intent, the superposition provides adaptability, and the interface anchors the system in organizational reality. Operational Flow 1. Governance Interface (GI) provides the traceable policies (A(t), λ(t)). 2. Intent Inference Unit (IIU) intervenes when such policies are absent or outdated, inferring latent intent by updating V,A(x, t), or redefining H(·, t;u). 3. Symplectic Flow Engine (SFE) propagates decision trajectories γ(t) under the Hamiltonian flow defined by (H, ω, g), preserving structural invariants. 8 Template for The Thalesians Magazine A PREPRINT 4. Coherence Controller (CC) evaluates coherence through C[γ] = Z∥∇V−λA∥2 gdt, and, when C[γ]≥ε, emits a collapse flag indicating structural decoherence. 5. Superposition Explorer (SE) maintains the bundle Γ={γi(t)} and, upon receiving the collapse flag, selects the coherent trajectory satisfying C[γ∗]< ε. Decisions are not optimized; they evolve coherently within a robust architecture. They are structural flows that preserve strategic intent even under uncertainty or transformation. Technical Assumptions ∇V is smooth; A(·, t) is measurable and bounded; λ(t)∈[0, λ] . The trajectories considered are those following the reference flow defined by the system’s Hamiltonian field (as introduced, for instance, in Section 3.1), which prevents pathological minimization and ensures structural preservation in practice. A regularized version, incorporating dynamic compliance and control penalty, may be presented in future work while maintaining backward compatibility with C[γ]. Note on Compatibility with Hilbert Space QPM operates on a symplectic decision manifold (M, ω, J, g) , structurally compatible with the Kähler geometry of quantum state spaces. This compatibility ensures that the present classical formulation remains fully extensible toward a Hilbert-space representation, where coherence, as a structural invariant, may acquire an amplitude–phase interpretation analogous to quantum evolution. It is important to emphasise that the Hamiltonian structure introduced in QPM does not minimise the coherence functional C[γ] . Instead, the Hamiltonian flow defines the admissible family of trajectories on the decision manifold, while the optimisation of C[γ] is performed over that flowgenerated set. In this sense, the dynamics specify the feasible geometric evolution, and the coherence functional selects the trajectories that are structurally most coherent. 4 Stage-wise Benchmark and 2D Example After establishing the geometric and modular framework of QPM, this section introduces its first structural validation: the Stage-wise Benchmark, a sequential experiment that subjects the symplectic dynamics to progressive coherence tests. Here, the model transcends its abstract formulation to become a verifiable operational system, where each stage examines a fundamental property of the functional C[γ] : its stability, convexity, and ability to preserve strategic alignment under controlled conditions. The benchmark consists of seven stages that reproduce, at reduced scale, the evolution of a portfolio subjected to structural perturbations, from the initial deterministic alignment to scenarios of stochastic noise, multi-agent composition, and dimensional scalability. Each phase corresponds to a mathematical principle validated by conditions M-1 through M-8 (see Appendix A), ensuring that coherence remains well-defined, finite, and recoverable. As a culmination, the 2D Symplectic Flow Example offers the first tangible representation of coherence motion: a two-dimensional plane where value and governance intertwine in a continuous 9 Template for The Thalesians Magazine A PREPRINT the penalty is smooth, the system remains differentiable and stable; if abrupt, it acts as a reflecting boundary for risk. Evolution Dynamics: Symplectic Flow. The system’s temporal evolution is governed by Hamilton’s equations: ˙q=∂H ∂p ,˙p=−∂H ∂q . These equations are integrated using the Störmer–Verlet method, a second-order symplectic integrator that exactly conserves the form ω=dq ∧dp , ensuring area preservation and thus structural coherence. The integration scheme is: pn+1 2=pn−∆t 2∇qV(qn) qn+1 =qn+ ∆t pn+1 2 pn+1 =pn+1 2−∆t 2∇qV(qn+1) This scheme preserves the symplectic structure even under significant perturbations, allowing the observation of how coherence is maintained within the allowed risk limits. Experimental Parameters. Table 8: Experimental parameters for the 2D symplectic flow simulation. Parameter Value Meaning zα2.33 99% quantile. B0.30 Risk budget (VaR). k1.0 Structural stiffness. λ80 Risk penalty. Σ0.04 0.03 0.03 0.09Moderate correlations. q0(0.28, 0.10) Initial position near VaR boundary. p0(0.00, 0.06) Small initial momentum. ∆t0.01 Time step. Iterations 7000 Extended evolution. Structural Result. Figure 3 displays the potential landscape with VaR penalisation and the coherent trajectory generated by the symplectic flow. The dashed black curve marks the VaR boundary, while the surrounding contours represent the penalised potential. The blue trajectory shows how the system approaches the risk boundary, detects it, and redirects its motion without collapsing, thereby maintaining structural coherence. The Hamiltonian trace (right panel) confirms that the energy remains nearly conserved along the trajectory, evidencing coherent reconduction and stability under the symplectic integrator. 16 Template for The Thalesians Magazine A PREPRINT 0.5 0.0 0.5 q 1 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 q 2 2D symplectic flow with VaR penalisation VaR boundary trajectory 2D symplectic flow with VaR penalisation. Coherence preserved while avoiding the VaR boundary. 0 2000 4000 6000 step 0.0020 0.0015 0.0010 0.0005 0.0000 relative energy drift Hamiltonian along trajectory Energy conservation along the trajectory. Minimal relative drift with a symplectic integrator. Figure 3: Symplectic dynamics and energetic stability. The representation shows structural coherence and strategic energy preserved under VaR constraints with correlations. In QPM terms, this simulation illustrates how strategic coherence acts as a geometric invariant. The flow does not seek static equilibrium but a dynamic sustainability among value, risk, and governance. Even in a two-dimensional plane, the principle of minimum coherent action manifests as the structural persistence of value. 5 Structural Scope and Constraints The following section updates the original scope presented in Zenodo “Quantum Portfolio Management (QPM): A Symplectic Decision Architecture for Adversarial and Uncertain Environments”, reflecting the empirical validation achieved up to Stage 4. While QPM remains in an early theoretical phase, its structural behavior has now been partially demonstrated through simplified simulations, confirming the stability and coherence of its symplectic formulation under stochastic perturbations. To understand QPM’s contribution, it remains essential to delineate the contexts in which its architecture is most applicable, the boundaries it currently faces, and the assumptions embedded within its design. Unlike traditional optimization frameworks, QPM introduces a structural paradigm: decisions evolve not through scalar reward maximization but through geometric coherence across time, intent, and uncertainty. This section clarifies the types of systems where QPM may offer a measurable advantage, articulates the limits of its current form, and outlines the next steps for translating its theoretical structure into operational models. 5.1 Strategic Application Scenarios Although QPM’s implementation is still confined to controlled experiments, several system classes exhibit structural affinity with its governing principles. 17 Template for The Thalesians Magazine A PREPRINT Adversarial decision environments. QPM’s coherence-based evolution supports resilient planning under distorted feedback or adversarial interference, where classical reward-driven policies tend to collapse. Autonomous mission governance. In long-horizon operations such as distributed monitoring or mission-critical robotics, QPM provides a means to maintain policy coherence even in the absence of centralized control or immediate feedback. Portfolio-level decision alignment. Drawing from the PMI Standard for Portfolio Management, QPM naturally supports multi-scale alignment across concurrent projects or strategic initiatives, ensuring that decision flows remain structurally consistent under changing conditions. Multi-agent misalignment resolution. For systems with locally conflicting incentives, QPM enables the inference of collective intent and the enforcement of structural coherence without explicit coordination, thus reducing systemic drift. 5.2 Structural Boundaries and Current Limitations While the validation of Stages 1 to 4 confirms that QPM maintains stability and coherence under stochastic noise, several intrinsic constraints remain. Partial computational instantiation. QPM has been implemented only in simplified environments (toy models). Further scaling is required to test its feasibility in high-dimensional or mission-critical systems. Dependence on intent estimation. The inference of strategic direction vectors A(t) still assumes either structured expert input or inverse-reinforcement frameworks. Environments with ambiguous or shifting goals may challenge this assumption. Continuous-state dependency. QPM’s symplectic core operates over differentiable manifolds. Integrating discrete or hybrid decision spaces will require further geometric reformulation or manifold approximation. 5.3 Risks and Theoretical Assumptions Primacy of coherence over reward convergence. QPM’s design presupposes that maintaining structural alignment is more fundamental than optimizing scalar rewards. This may not hold in domains dominated by short-term utility metrics. Functional alignment sensitivity. The effectiveness of the coherence functional C[γ] depends on accurate calibration of value fields and intent vectors. Poor calibration can distort or prematurely collapse the decision flow. Limited benchmarking. No formal comparison has yet been performed against classical control, reinforcement learning baselines, or hierarchical planning systems. Such analysis will be critical for external validation. 18 Template for The Thalesians Magazine A PREPRINT 5.4 Evolutionary Challenges and Future Work The central challenge ahead lies in extending QPM from its current stage-wise validation into a unified decision framework that preserves theoretical fidelity while ensuring computational tractability. The forthcoming research efforts will focus on the following structural dimensions. Formal coercivity and lambda-convexity analysis. Empirical results from the Stage-4 grid sweep confirmed local convexity and coercive behavior of the functional C[γ](λ, k) . A formal proof of global coercivity without reliance on lambda clipping will be included in the full version. This extension will not alter the algorithmic results obtained so far but will strengthen theoretical guarantees regarding the existence and attainment of the minimizer (λ∗, k∗). Ergodicity and long-horizon coherence. Future formulations will investigate the ergodic properties of the QPM flow, assessing whether its symplectic dynamics preserve invariant measures under bounded stochastic perturbations. Establishing ergodicity would confirm that QPM’s structural coherence remains stable over extended horizons, even under correlated risk or fluctuating decision landscapes. Advanced IRL formulation for intent inference. A refined integration of Inverse Reinforcement Learning will be developed to estimate the intent fields A(t) that govern QPM’s decision evolution. The objective is to replace fixed expert trajectories with learned manifolds of strategic intent, extending QPM’s adaptability to complex or partially observable environments. Hybrid integration and scalability. The next iteration will embed QPM’s symplectic core within hierarchical planning architectures and hybrid discrete–continuous environments, validating its scalability across multi-agent and portfolio-level systems. Extended validation roadmap. Subsequent phases will: 1. Extend the symplectic formulation to hybrid discrete–continuous environments. 2. Design scalable coherence functionals for high-dimensional and multi-agent decision systems. 3. Embed QPM into hierarchical planning and reinforcement learning architectures. 4. Establish formal benchmarks against standard optimization and control frameworks. 5. Validate structural resilience through ablation studies and adversarial stress testing. Ultimately, these developments aim to consolidate QPM as a stable, ergodic, and coercive decision framework, one that sustains structural coherence under uncertainty and bridges theoretical geometry with practical governance. 6 Conclusions and Outlook The present work has consolidated the first empirical validation of Quantum Portfolio Management (QPM) as a coherent decision architecture governed by symplectic dynamics. Through the progressive stages (1 to 4), the model demonstrated cost reduction, stability, and scalability under stochastic perturbations, confirming that geometric coherence can effectively replace reward convergence as the organizing principle of decision evolution. 19 Template for The Thalesians Magazine A PREPRINT The results suggest that QPM’s formulation, rooted in Hamiltonian flows and adaptive vector alignment, maintains structural integrity even in adversarial or uncertain environments. Unlike conventional frameworks that optimize scalar rewards or expected utilities, QPM preserves a field of coherent trajectories that evolve according to internal consistency rather than external feedback, allowing decision systems to remain dynamically stable despite noise or conflicting incentives. From a theoretical standpoint, the experiments provide strong empirical evidence of λ -convex–like behavior and practical coercivity of the coherence functional C[γ](λ, k) , offering initial indications of the existence and stability of coherent optima. These empirical properties, together with the preservation of area in phase space, support the interpretation of QPM as a Hamiltonian governance system that conserves structural information across time, uncertainty, and scale. The next phase will address the formal proof of coercivity, the study of ergodic invariants within the QPM flow, and the integration of Inverse Reinforcement Learning for intent inference. These extensions aim to transform QPM from a validated theoretical construct into a fully operational model capable of supporting long-horizon governance, multi-agent decision alignment, and adaptive portfolio control. Ultimately, QPM aspires to offer more than an optimization method; it proposes a reformulation of decision-making itself. In replacing convergence with coherence, and control with structural geometry, it opens a path toward systems that do not merely react but maintain form. 7 Acknowledgements This work would not have been possible without the unconditional support of my family. To my wife and daughter, thank you for your patience, affection, and understanding during countless hours of research, reflection, and iteration. Much of the conceptual foundation of Quantum Portfolio Management (QPM) was forged in long solitary sessions at the library and further developed in my home laboratory, where ideas could grow freely during late nights and early mornings. To those who, long before metrics, rewards or observable signals existed, were able to sense structure in a world that seemed governed by noise. To those early architects of thought, who understood that form precedes calculation, belongs the conceptual root of this work. In continuity with that tradition, I want to acknowledge those who, today, have built the frameworks that made QPM possible. I do not conceive of them as isolated influences, but as a structural ensemble: Alexander Lipton, Valentino Zocca, Igor Halperin, Paul Bilokon, Marcos Lopez de Prado, Constantin Gonciulea, and Theophano Mitsa. Each of them, whether from quantitative finance, physics, information geometry, portfolio theory or algorithmic design, provides a distinct coordinate of a single conceptual space. Their contributions have not been models to imitate, but tensions that force deeper clarity: to build without shortcuts, to justify every structure, to preserve coherence even when no observable confirms it, and to develop a framework that does not derive from any single school, but from a structural necessity. To them I owe not only inspiration, but challenge. The kind of rigor that does not allow rest and transforms an intuition into a system. 20 Template for The Thalesians Magazine A PREPRINT To all those who build in silence, question structure, and believe that coherence matters, this work is also for you. This study was conceived, developed, and completed independently, without funding, institutional affiliation, or directive. It reflects entirely personal research and authorship, and does not represent or relate to any current or past employer or organization. A Appendix A: Structural and Numerical Conditions for QPM Simulation Validity To ensure that the structural coherence functional C[γ] = Ztf t0 ∥∇V(γ(t)) −λ(t)A(γ(t), t)∥2 gdt remains well-posed, convex enough to optimise, and numerically stable under all benchmark stages, QPM enforces a set of eight technical conditions, denoted M-1 to M-8. These conditions act as the structural safety frame of the simulator: they guarantee that the symplectic flow remains bounded, differentiable, and geometrically consistent throughout the stage-wise experiments. Table 9: Technical conditions (M-1 to M-8) ensuring structural and numerical validity of QPM simulations. # Condition Mathematical statement Why it matters M-1 Value-landscape regularity V∈C2(M)and ∥∇2V∥≤LV Ensures Lipschitz gradient; prevents gradient explosions near saddle points. M-2 Riemannian metric bounds mminI⪯g(γ)⪯mmaxI Avoids ill-conditioned norms; defines upper and lower energy bounds. M-3 Governance-field integrability A∈L2([t0, tf]; TγM) . Guarantees R∥A∥2dt<∞. Ensures bounded energy input and smooth control propagation. M-4 Gain positivity and boundedness 0≤λ(t)≤λmax , with λ∈W1,∞ . Keeps control inputs causal and physically realisable; enables Grönwall bounds. M-5 Coherence convexity For fixed A , the map λ7→ C[γ] is Lλ -convex (empirically, Hessian ≥ 0). Required for a unique λ∗ ; facilitates grid-search and SGD optimisation. M-6 Stochastic extension In SDE form dγt=−∇V dt + λA dt +σ dWt , require σσ⊤≻0 and ∥σ∥ ≤ σmax. Ensures ergodicity and finitevariance Monte Carlo simulation. M-7 Multi-stakeholder composability Let A=PK k=1 αkA(k) with αk≥ 0,Pkαk= 1. Enables composition of governance vectors and adaptive re-weighting across agents. M-8 High-dimensional scalability For dim(M)=n , require ∥∇V∥,∥A∥=O(n) and runtime O(n1+ε)with ε≤0.3. Prevents numerical blow-ups; ensures sub-linear growth of runtime for n≤50. 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