an interactive paper · parallax engineering · 2026

Proofs of a
Stranger Universe

Five results from quantum mechanics that sound like mysticism — superposition, interference, nonlocal entanglement, tunneling, uncertainty — demonstrated not with metaphor but with live computation. Every figure on this page is being calculated in your browser right now: a Schrödinger equation is being integrated, Bell inequality trials are being tallied, wavefunctions are collapsing. Nothing is a video. Everything is math.

The claim of this document is simple: the mystical-sounding parts of quantum mechanics are its most rigorously proven parts. Scroll down and check the numbers.

▽ begin the argument
Proposition I

A thing can be two things,
until you ask.

A qubit is the simplest quantum object: a system with two answers, $|0\rangle$ and $|1\rangle$. Its full state is a point on a sphere — the Bloch sphere —

$$|\psi\rangle = \cos\tfrac{\theta}{2}\,|0\rangle + e^{i\varphi}\sin\tfrac{\theta}{2}\,|1\rangle$$

The poles are the definite answers. Everywhere else on the sphere is a superposition — a state with genuine amplitude for both outcomes at once. The mystical claim "it's both until observed" is literally the geometry: almost every state on the sphere is neither pole.

Drive the state yourself. Apply gates — real unitary rotations — and then measure. Measurement samples $P(0)=\cos^2\frac{\theta}{2}$ and snaps the state to a pole. Run it repeatedly from the equator and watch the 50/50 statistics accumulate: the superposition was real, and the collapse is real too.

Fig. 1 — Live qubit state. Gates rotate; measurement collapses. Drag to orbit.
Proposition II

One particle takes
both paths.

Fig. 2 — Electrons arriving one at a time, sampled from $|\psi_1+\psi_2|^2$. Toggle the detector to sample $|\psi_1|^2+|\psi_2|^2$ instead.

Fire electrons at two slits, one at a time, so no electron can possibly interfere with another. Each arrives as a single dot. And yet the dots accumulate into interference fringes — the signature of a wave that went through both slits:

$$P(x) \propto \Big|\psi_1(x) + \psi_2(x)\Big|^2 = |\psi_1|^2 + |\psi_2|^2 + \underbrace{2\,\mathrm{Re}\,\psi_1^*\psi_2}_{\text{interference}}$$

That cross-term is the whole mystery. If the electron had secretly picked one slit, probabilities would just add — no fringes. The fringes are a measurement of the fact that it didn't pick.

Now the twist the mystics love, correctly: put a which-path detector at the slits. Learn which slit each electron used — don't block anything, just know — and the cross-term dies. The pattern becomes two boring humps. Information, not force, changes the physics.

Proposition III

The universe is provably
not locally real.

This is the strongest result on the page — a 2022 Nobel Prize, and the closest physics comes to proving something spooky. Prepare two particles in the singlet state $\;|\Psi^-\rangle = \tfrac{1}{\sqrt2}\big(|01\rangle - |10\rangle\big)$, separate them, and measure each along chosen axes.

Bell's theorem (1964): if each particle carried a pre-existing answer — any local hidden plan whatsoever, however clever — then the CHSH statistic obeys

$$S = \big|E(a,b) - E(a,b') + E(a',b) + E(a',b')\big| \;\le\; 2$$

Quantum mechanics predicts $E(a,b) = -\cos(a-b)$, which at the optimal angles reaches $S = 2\sqrt{2} \approx 2.828$. Both can't be right.

Below, two experiments run head-to-head: a genuine quantum singlet sampler, and the best possible local hidden-variable model, each measured at the same four angle pairs. Run the trials. The classical model saturates at 2 and stops. The quantum one walks straight through the wall — with error bars.

Fig. 3 — Live CHSH trial tally. Quantum $S$ vs. the classical bound $S\le2$ and the Tsirelson limit $2\sqrt2$.
Proposition IV

Walls are a matter
of probability.

Fig. 4 — $|\psi(x,t)|^2$ integrated live by split-step Fourier. Phase shown as hue.

This figure is a real numerical solution of the time-dependent Schrödinger equation,

$$i\hbar\,\partial_t \psi = -\tfrac{\hbar^2}{2m}\,\partial_x^2\psi + V(x)\,\psi,$$

computed ~60 times a second by split-step Fourier (the kinetic term is applied in momentum space via FFT — the same algorithm used in research codes).

A wave packet with energy $E$ hits a barrier of height $V_0 > E$. Classically the transmission is exactly zero: the particle does not have the energy, full stop. Quantum mechanically, the wavefunction decays inside the barrier instead of stopping, and leaks through with probability $\;T \approx e^{-2\kappa L},\; \kappa = \sqrt{2m(V_0-E)}/\hbar.$

Watch the transmitted probability tick upward on the right of the wall. That number is why the sun shines (proton fusion is classically forbidden), why flash memory works, and why alpha decay happens. "Walking through walls" isn't a metaphor here — it's a boundary condition.

Proposition V

Vagueness is woven
into the fabric.

The uncertainty principle is routinely mis-told as "measuring disturbs." The truth is stranger and cleaner: position and momentum are Fourier transforms of each other. They are two readings of one object, and a function cannot be narrow in both readings:

$$\sigma_x\,\sigma_p \;\ge\; \frac{\hbar}{2}$$

This is a theorem about waves, older than quantum mechanics — the same reason a short audio click has no definite pitch. The particle doesn't hide its momentum; a particle with a sharp position does not possess a sharp momentum, any more than a chord possesses a single note.

Squeeze the position distribution with the slider. The momentum distribution — computed live as the actual Fourier transform of $\psi(x)$ — flares out in exact compensation. The product $\sigma_x\sigma_p$ is printed below; for a Gaussian it pins the theoretical floor $\hbar/2$ precisely. Try the box state: hard edges in $x$ cost you dearly in $p$.

Fig. 5 — $|\psi(x)|^2$ (top) and its live discrete Fourier transform $|\tilde\psi(p)|^2$ (bottom).