In the physical world, almost everything we observe is the result of a hidden cause acting upon a system. When a leaf spins in a stream, when a planet orbits a star, or when a particle moves through a field, there is an underlying force or condition driving that motion. Scientists often try to work backward: they measure the motion and ask, "What caused this?" This is known as an inverse problem. It is a fundamental challenge in physics because the path from cause to effect is often clear, but the path from effect back to cause is full of traps. Sometimes the information is lost forever; sometimes different causes produce the exact same result; and sometimes the way we choose to look at the data limits what we can see. Understanding exactly where this information disappears is crucial for fields ranging from fluid dynamics to quantum mechanics, yet distinguishing between a missing piece of data and a fundamental limitation of nature has remained difficult.

A researcher at Kansai University has now provided a clear map for navigating these traps. By treating the process of reconstruction as a structured journey, the work separates the limits of what is theoretically possible from the limits of what is actually achievable in a real physical system. The study introduces a three-tiered framework to measure how much of a hidden cause can be recovered from observations. It distinguishes between the structural potential of the system, the effectiveness of the specific method used to analyze it, and the physical reality that often degrades the data before it can even be measured. This approach allows scientists to pinpoint exactly where the ability to reconstruct a cause is lost: is it because the mathematical tools were not designed well enough, or because the laws of physics themselves have erased the information?

The core of this new framework is a hierarchy of three "dimensions," or measures of capacity, that act as upper bounds on how much information can be retrieved. The first level is the structural dimension. This represents the maximum amount of information that could theoretically be recovered if the system were perfect and the mathematical tools were designed with infinite care. It is determined solely by the symmetries of the system—the ways in which the system looks the same after being rotated or shifted. If a system has a certain symmetry, the structural dimension tells us the absolute ceiling of what could be known, assuming no other constraints exist. It is a theoretical limit based on the architecture of the problem itself.

However, a theoretical ceiling is not always reachable. The second level, the effective dimension, accounts for the specific design of the reconstruction method. In practice, scientists must choose how to process the data they collect. They might combine measurements in specific ways or use particular algorithms to extract the cause. The effective dimension measures how well a chosen method actually utilizes the available information. It is possible to design a method that is inefficient, failing to extract all the information that the structural dimension says is there. For instance, if a method combines data points in a way that creates redundancy, it might miss independent directions of information. The gap between the structural and effective dimensions reveals whether the loss of information is due to a poor choice of mathematical tools.

The third and final level is the physical dimension. This is where the real world intervenes. In any actual physical system, the state of the matter is not just a collection of numbers waiting to be read; it is the result of physical processes that evolve over time. These processes can cause the state space to become degenerate, meaning that different starting conditions might collapse into the same observable state. For example, if particles in a fluid are subject to a specific flow, they might all settle into a predictable pattern, losing the unique details of their initial positions. The physical dimension measures the information that remains after these physical processes have done their work. It is the true, practical limit of what can be reconstructed, regardless of how clever the mathematical method is.

The paper illustrates this hierarchy with two distinct examples to show how the framework applies to different types of problems. The first example involves flake-like particles floating in a fluid. Scientists want to know the velocity gradient of the fluid—the way the speed and direction of the water change from point to point—by observing how these tiny flakes rotate. The researchers designed a mathematical tool to reconstruct this gradient from the particle orientations. They found that while the structural dimension suggested a certain amount of information could be recovered, the physical dimension was lower. This was because the physics of the fluid flow caused the particles to align in a specific way, effectively erasing certain details of the flow. The gap between the theoretical limit and the physical limit showed that the information was lost due to the nature of the fluid dynamics, not because the mathematical tool was flawed.

The second example looks at a classic problem in mechanics: two charged particles interacting with each other. Here, the goal was not just to find a known cause, but to define what the cause actually is based on the symmetry of the system. The researchers built a reconstruction map to analyze the motion of the two particles. They discovered that the physical dimension was lower than the effective dimension for one specific component of the motion. This reduction was due to the conservation of angular momentum, a fundamental law that forces the motion to stay within a single plane. This physical constraint meant that some information about the rotation was inherently unavailable, no matter how the data was processed. In this case, the framework helped identify that the missing information was a direct consequence of a conservation law.

By separating these three levels, the work provides a diagnostic tool for scientists. If the gap exists between the structural and effective dimensions, the solution is to redesign the mathematical method to be more efficient. If the gap exists between the effective and physical dimensions, the problem is deeper; the information has been physically erased, and no amount of mathematical cleverness can recover it. This distinction is vital because it prevents researchers from wasting effort trying to solve problems that are physically impossible to solve, and it guides them to improve their methods when the limitation is merely a matter of design.

The study does not claim to solve every inverse problem or to guarantee perfect accuracy in reconstruction. Instead, it offers a way to measure the reach of any reconstruction attempt. It clarifies that the ability to recover a cause is not a single yes-or-no question, but a layered structure where different types of constraints operate at different stages. The findings suggest that by understanding which layer is the bottleneck, scientists can make more informed decisions about how to design experiments and interpret data. The framework applies to a wide range of systems, from the flow of fluids to the motion of celestial bodies, providing a unified language to discuss the limits of knowledge in physical systems. Ultimately, it turns the abstract question of "can we know this?" into a concrete analysis of where the information stops and why.