The paper in plain words

What each section of the paper says, why it is there, and which number carries it. No formula is needed to follow it.

Abstract: the question

Electrons, protons and neutrons carry a property that has exactly two values. A beam of silver atoms sent through an uneven magnet splits into two spots, not a smear. Pauli wrote that property down in 1925 and refused to say what it "is". Within months others pictured it as the electron spinning like a top, and the name "spin" stuck. Since then many theories have filled the picture in: a turning body, a charge running in a tiny circle, an axis fixed at birth, motion in a hidden direction, a twist in space-time, a new field. The paper asks one thing of each: does a published measurement contradict a number this picture fixes?

1. Introduction: what is beyond doubt

The two-valuedness itself is not in question. It splits the silver beam into two, it sets the electron's magnetic moment (measured to 0.13 parts per trillion), and it makes a neutron's wave come back to itself only after two full turns. What is in question is which pictures of it survive measurement. The paper says "two-valuedness" for the measured property and keeps "spin" for theory names.

2. Measurements: the evidence table

No large fourth direction. If gravity could leak into a fourth direction of space, it would weaken faster than one over the distance squared. A torsion balance sees the ordinary law down to 52 micrometres, a little thinner than a hair.

No small curled direction, as far as searched. A direction curled up very small would leave gravity alone but would give every particle that moves in it a ladder of heavy copies. The Large Hadron Collider has not seen the first rung in the models searched.

The electron has no size that can be seen. Electron–positron collisions show no structure down to 2.8 × 10⁻¹⁹ m.

Two beams, and a wave that needs two turns. Silver atoms land in two traces. A neutron wave rotated in a magnetic field is back where it started only after 716.8 ± 3.8°, within one error bar of 720° and 94 error bars away from 360°.

Bell's number. If the two particles of a pair each carried their own preset answers, a combination of four correlations called S could not exceed 2. Quantum theory allows up to 2.83. Four experiments, from 1982 to a satellite in 2017, find 2.37 to 2.73. Each apparatus has its own losses, so the number belongs to the apparatus; what they share is that all lie above 2.

No signal. What one end sees on its own never depends on how the other end is set. That is a theorem, and experiments agree.

Torsion, polarised matter, the speed of light, and partner particles. A pendulum carrying about 10²³ aligned electrons feels no pull from the Sun or the Earth's matter; light of different energies from a gamma-ray burst arrived together; no supersymmetric partner appears below about 2 TeV in the simplified models searched.

3. Nine theories against the measurements

A theory is contradicted when a listed measurement contradicts a number it states. It is not contradicted when nothing does, either because the measurements do not reach its predictions, or because it fixes no number, or because it predicts exactly what quantum mechanics predicts. Where a theory gives no number at all, this is said.

• The electron as a turning ball of charge: contradicted. A turning body is back after 360°; the neutron needs 720°. A turning magnet pointed at random spreads a beam into a band; silver gives two spots.
• A charge circling at light speed (zitterbewegung): not contradicted, not decided. The circle would be 1.93 × 10⁻¹³ m across, far larger than anything collisions see, but no one has computed what such an orbit would show in a collision, so the comparison does not decide it.
• Each particle carries its own axis: contradicted. That is exactly the preset-answer case, and it caps S at 2.
• Motion in a closed extra direction: not contradicted. The theory never says how large the direction is, so no measurement contradicts it; the measurements only fence off the sizes that are left.
• Einstein–Cartan torsion: not contradicted. Its effects live inside matter at densities no laboratory reaches.
• "Torsion fields" from rotating bodies: no number stated. No number was ever published to test.
• Spin networks with energy-dependent light speed: not contradicted. The size of the effect is not fixed by the theory; the gamma-ray burst bounds it.
• Supersymmetric partners: not contradicted. The theory fixes no partner mass; the searches bound the masses in the models named.
• The Bohm–Schiller–Tiomno rotating fluid: not contradicted. It reproduces quantum mechanics exactly, so no measurement can separate them.

What survives every measurement is the property as Pauli stated it: two-valued, and back to itself after 720°, not 360°.

4. Entanglement in words

Two facts set the picture. The two ends of an entangled pair agree more often than any pair of preset answers could. And neither end can be used to send a signal to the other. The paper reads the pair as one wave tied at two ends: one object with two places where it can be read, not two objects carrying copies of an answer. Reading the wave once for both ends gives the quantum correlation (taken from quantum mechanics as given, not derived) and S up to 2.83; losses bring it down to the measured 2.37–2.73. Each end on its own sees a fair coin, so nothing usable passes. The correlation survived 1,203 km between a satellite's two ground stations, so the wave is tied to the particles, not to places, and the picture needs no extra direction of space. The Intuition page draws it.

5. Conclusions

Two pictures are contradicted in their stated form: the turning charged electron and the preset local axis. Six are not contradicted, for the reasons listed, and one states no number to test. In every contradicted case the deciding number is a published measurement with its error bar.

Where to check

Every number above is on the Proof page with its source and DOI, and the reproduction script recomputes the distances in standard errors.