For a hundred years, quantum mechanics has relied on a mysterious rule: to get the probability of an outcome, you square the wave function. Everyone knows the rule works. No one has been able to explain why it must be squared. Textbooks simply tell us, “that’s how nature does it,” leaving the most fundamental part of quantum theory feeling arbitrary and almost magical.
This paper breaks that spell. It shows that the Born rule—the rule that says probability equals the square of the amplitude—is not a bolt-on assumption at all. It is the only probability law the universe can have if it is built from information, if reversible processes preserve structure, and if irreversible events (like measurements) create new distinguishability in the world. Instead of assuming quantum mechanics, the paper rebuilds it from the ground up using just information, geometry, and the difference between reversible and irreversible change.
The key discovery is surprisingly simple and incredibly deep: when the universe makes a choice—when a quantum system “collapses” into an actual outcome—it doesn’t select an individual history. It selects a pattern of relationships between pairs of possible histories. Those relationships are the things that carry the system’s phase information and interference potential. Because these correlations come in pairs, the math naturally becomes quadratic. And that’s where the mysterious squaring comes from. The Born rule isn’t an arbitrary law—it’s the mathematical fingerprint of a universe where relationships matter more than isolated pieces.
From this single principle, the entire machinery of quantum mechanics unfolds automatically. Complex numbers, wave functions, interference patterns, Hilbert space, unitarity, the Schrödinger equation—none of it needs to be assumed. It all emerges from the geometry of distinguishability and the pairwise nature of irreversible processes. Even classical probability appears as a special case, where interactions with the environment wash out all the phase-based relationships and leave only simple counting behind.
What makes this work especially exciting is that it doesn’t stop at elegance—it makes testable predictions. The paper shows that decoherence rates should depend not just on mass or environment, but on something far more subtle: the gradient of distinguishability in a system’s state space. Two systems with the same mass and environment could decohere at different rates simply because the geometry of their information is different. No existing theory predicts this.
In short:
The Double Square Rule finally explains why quantum mechanics squares amplitudes—because nature counts correlations, not individual paths.
It turns one of the deepest mysteries in physics into a clear, geometric inevitability, and in doing so, it brings us a step closer to understanding the informational foundations of reality itself.