Schrödinger to Dirac: Why This Paper Is Different
Most physics papers start by assuming the rules of the game: space has three dimensions, time is a background parameter, particles have spin, and relativistic quantum mechanics is built by writing down the Dirac equation because we already know it works. This paper takes a very different approach. Instead of starting with the equations of physics, it asks a more basic question: what minimal rules must any consistent universe obey if information evolves locally and reversibly?
From just a few simple principles—discrete updates (“ticks”), preservation of distinguishability, locality, and smooth large-scale behavior—the paper shows that the familiar equations of quantum mechanics are not optional. The Schrödinger equation emerges as the only way reversible local updates can look smooth at large scales. More strikingly, when the same logic is pushed one step further to demand first-order dynamics, the Dirac equation appears automatically. It is not postulated. It is forced.
Spin Isn’t Added — It’s Inevitable
One of the most surprising results concerns spin. In standard physics, spin is treated as a mysterious intrinsic property of particles—something nature simply “decided” to include. In this framework, spin turns out to be unavoidable. The mathematics shows that if you want a first-order equation whose square gives the correct energy–momentum relation, then internal two-component (and in 3D, four-component) structure must appear. There is no alternative.
In other words, particles don’t happen to have spin. Any universe with first-order, local, reversible dynamics must contain spinor degrees of freedom. This reframes spin from an empirical curiosity into a structural necessity.
Why Three Dimensions — Not Assumed, but Derived
Perhaps the most distinctive part of the paper is its treatment of spatial dimension. Most theories quietly assume three dimensions from the outset. Here, dimensionality emerges from the geometry of the most stable local closure structure—a simple hexagon with a central hub (seven vertices in total). This structure has exactly three independent “opposition” axes, and the paper shows—using symmetry, spectral analysis, and topology—that no fourth independent transport direction can survive at large scales.
Crucially, the argument is not “we live in 3D, therefore the math works.” It’s the reverse: if closure is stable and information flows locally and reversibly, three spatial directions are all that can emerge within a single homogeneous phase. Anisotropic or inhomogeneous patterns would correspond to different physical phases, not extra dimensions of the same space.
What This Paper Actually Proves
The paper proves several things at once, and their combination is what makes it unique:
- The Schrödinger equation follows from reversible, local updates.
- The Dirac equation follows when those updates are required to be first-order.
- Spin and spinors are mathematically unavoidable.
- Fermion doubling is resolved by a stiffness term derived from entropy, not added by hand.
- Three spatial dimensions emerge from closure geometry rather than being assumed.
What it does not do is claim to explain everything. Constants like the speed of light and Planck’s constant are not fixed here, and the detailed particle spectrum is left to companion work. But the paper establishes something deeper: the basic structure of relativistic quantum mechanics is not a choice—it is the inevitable consequence of very minimal informational rules.
Why This Matters
If the paper is right, then much of what we treat as fundamental input in physics—spin, relativistic wave equations, even spatial dimensionality—may actually be emergent features of a deeper, discrete informational substrate. That doesn’t replace existing physics; it explains why it had to look the way it does.