A symphony of crystalline blue, a pool can transform any backyard into a personal paradise. But amidst the dreams of languid lounging, we must face the practicality of the process: building a pool is a significant investment that demands a well-strategized budget. As such, this discourse aims to provide prospective pool owners with a step-by-step guide to crafting a comprehensive budget for a pool construction project.

To begin, we must elucidate the primary cost determinants in pool construction. These are primarily the size, shape, and material of the pool, but also include aspects such as the depth, finishing, heating, lighting, filtration system, and additional features. Each of these elements, while crucial to the final product, come with specific cost implications that must be incorporated into the budget.

From a mathematical perspective, the size and shape of the pool represent the initial major cost variables. Larger pools necessitate more materials and labor, thereby increasing costs, while complex shapes often require specialized design and construction methods. In terms of materials, pools are often constructed from concrete, vinyl, or fiberglass, each with distinct price points and maintenance requirements. In the economic theory of trade-offs, these material choices represent a balance between initial costs and long-term maintenance expenses. Concrete, for instance, has a higher upfront cost but offers greater flexibility in design, compared to the lower initial cost but higher maintenance demands of vinyl.

The pool's depth also dramatically influences the budget. Deeper pools require more excavation work and materials, thereby escalating costs. Yet, the benefits such as enabling diving or enhancing swimming experiences must be weighed against these additional expenditures.

The finishing, an often-overlooked aspect, plays a pivotal role in aesthetics and durability. Plaster finishes are more economical, but aggregate and tile finishes, while more costly, offer superior resilience and visual appeal. Again, this is an application of the economic principle of cost-benefit analysis, where the longer lifespan and aesthetic benefits of aggregate and tile finishes may justify their higher cost.

Heating, lighting, and filtration systems may not directly contribute to the pool’s structural integrity, but they significantly impact its utility and maintenance. Pool heaters, ranging from gas to solar types, differ in their initial and operational costs. The trade-off here lies in balancing an upfront investment in renewable energy sources such as solar heaters with their potential to reduce long-term energy costs. The quality of lighting and filtration systems also influences operational efficiency and maintenance costs, necessitating careful consideration in the budgeting process.

Additional features such as waterfalls, slides, pool covers, and automated cleaners further augment overall expenses. However, they often enhance the pool's usability or reduce maintenance exertions, demonstrating again the delicate balance between initial outlay and long-term value.

Beyond these direct costs, one must consider ancillary expenses such as landscaping, fencing, decking, and increased insurance premiums. The law of unintended consequences, a fundamental theory in economics, essentially posits that actions, in this case, pool construction, often have effects that are unanticipated or unintended. As such, a comprehensive budget must account for these potential 'hidden' costs.

To construct such a budget, begin by clearly defining the features and specifications of the desired pool. Subsequently, conduct research or consult with professionals to determine the costs associated with each aspect. Incorporate a contingency allowance of 10-20% to cover unforeseen expenses, as recommended by financial risk management strategies. At the same time, ascertain the potential increase in property value following the pool installation, which is an important aspect of return on investment. Lastly, consider financing options available and their implications on overall costs and repayment strategies.

In conclusion, constructing a budget for a pool project is a complex task demanding a keen understanding of the various cost determinants and their interrelationships. By using principles from mathematics, economics, and finance, prospective pool owners can craft a comprehensive budget that strikes the ideal balance between their dreams and their wallet’s realities. As with any significant investment, diligent planning and judicious decision-making are key to achieving the best value for your money, ensuring that your personal paradise is a joy, not just for the moment, but also for the long term.