Navigating the labyrinth of home renovation necessitates an understanding of various components, one of which is the task of selecting the appropriate interior painter. This crucial step, if executed with precision, can take your home renovation project from satisfactory to extraordinary.

Let us initiate our exploration by decoding the term 'interior painter'. An interior painter is a professional contractor who applies paint to the inside of buildings, usually residential homes. They are skilled artisans who not only apply paint but also repair wall surfaces and understand the scientific aspects of paint composition, application, and durability. The relevance of an interior painter in the home renovation process is thus integral, akin to a knight on the chessboard, maneuvering strategically to protect the King (your renovation project) and contribute to the ultimate goal of Checkmate (your dream home).

The mathematical theory of optimization will guide us in this quest for the right interior painter. This theory essentially involves choosing the best element from some set of available alternatives. We are faced with a similar scenario where our decision-making process will involve evaluating various painters based on different parameters. It's imperative to create a list of potential interior painters who have a reputation for quality work and a proven track record. The optimal painter, much like the solution in an optimization problem, will be the one who maximizes quality and minimizes cost.

The Pareto efficiency principle, derived from economics, can also be a useful tool in our decision-making process. In simple terms, Pareto efficiency, or Pareto optimality, is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off. Applying this principle in our case, the ideal painter would be one where we cannot improve one aspect (e.g., cost) without worsening another aspect (e.g., quality).

The next step is interviewing potential candidates. This is where principles from social sciences come into play, particularly those from psychology and communication studies. The interview should not be merely a fact-checking exercise but an opportunity to assess the painter's understanding of your vision. This can be facilitated by open-ended questions that encourage detailed responses, such as asking them to explain how they would approach a particular room or specific design element.

Asking for references and examining previous work is vital. Akin to peer review in scientific research, this helps ensure that the work meets established standards. Photos and testimonials from past clients provide insights into the painter's consistency, attention to detail, and ability to meet deadlines and budgets.

The contractual agreement, driven by the principles of contract law, is another crucial step. The contract should clearly state what work will be done, when it will be completed, and how much it will cost. It should also include clauses for potential contingencies, such as changes in the scope of work or unforeseen obstacles.

Finally, let us consider the question of 'when'. Timing is a critical factor that often gets overlooked in the renovation hustle. The best time to hire a painter is before you need them. This is derived from the Just-In-Time (JIT) system used in manufacturing, where materials are ordered and delivered only when they are needed, thus reducing costs and improving efficiency. By hiring a painter in advance, you will have ample time for planning, preparation, and any necessary revisions.

To conclude, the task of selecting the right interior painter for a home renovation project involves a delicate interplay of various concepts from mathematics, economics, social sciences, and law. An understanding and application of these principles can not only simplify the task but also result in an outcome that is both economically efficient and aesthetically pleasing. Remember, the right painter can transform the canvas of your home into a masterpiece.