Why 1/r, what it takes to rewrite Maxwell for it, the vacuum we chose, and what breaks.
Chemistry here needs very little of the Standard Model, and everything it does use is written down below. Nothing else can creep in: a particle we haven't listed isn't in this universe. Any new one (a light particle that pulls on electrons, say) would have to be added here first, because it would add a force.
| ingredient | what we take, and why it matters |
|---|---|
| electrons | charge −1, spin ½, and the smallest spinors: two states, so two to an orbital (motion and spin). That they're fermions is essential: bulk matter is stable only because of it (below). It's stated, not derived. In our world it follows from relativity and spin together (the spin-statistics theorem), but this universe gives up relativity for the force, so it can't be borrowed from there. |
| nuclei | point charges +Z, thousands of times heavier than electrons, and stable for every Z the table uses, up to 78. That's declared, not derived: nuclear physics would need the strong force in 4D. The heavy ones are the least sure. A 4D nucleus's radius grows as A^¼, so its protons' repulsion grows as Z²/A^¼, faster than our Z²/A^⅓. |
| light | the lens vacuum's photon, three polarizations (light). No other long-range field pulls on charges. |
| two numbers | Atomic units absorb ħ, the electron's mass and the pinned force's strength. What's left is that strength over ħc, the fine-structure constant, taken small, like our 1/137. Every atom here is solved without relativity, which is right only if it's small. Near 1, heavy atoms would need Dirac's equation and pair creation. The lens length ℓ only sets how strongly light pushes charges. |
| gravity | open: under Gauss's law no orbit is stable, so it has to be pinned too, somehow. |
| not needed | quarks, gluons, the weak force, the Higgs, neutrinos. They reach chemistry only through which nuclei exist (declared) and the electron's mass (a unit). |
In our world it's a theorem (Dyson and Lenard, then Lieb and Thirring) that a lump of N atoms has energy proportional to N, so matter doesn't collapse. The proof leans on Newton's shell theorem, which fails here. So here is the evidence instead. Atoms don't collapse: the solver's total energies grow as a fixed power of Z, the 4D Thomas–Fermi law Z^(5/2), where ours grow as Z^(7/3). Many atoms don't either: every force between neutral atoms here (the permanent R⁻⁵, dispersion's R⁻⁶, salt's R⁻⁷) falls faster than R⁻⁴, so summed over 4D space each atom's share is finite. Matter condenses, but it doesn't run away. And it rests on Pauli. The same scaling that gives fermion matter an energy ∝ −N gives boson matter −N^(3/2), a collapse. Ours does the same, as −N^(7/5). This is argued from scaling, not proved.
A charge's field lines spread over the sphere round it. In 3D that sphere's area grows as r², so the force falls as 1/r² and the potential as 1/r. In 4D the sphere is a 3-sphere, its size grows as r³, and the potential falls as 1/r². That sounds like a detail. It isn't: an electron's pull toward the nucleus then falls exactly as fast as its own resistance to being squeezed, the centrifugal term. One of them wins at every distance at once. Too weak a pull and nothing binds. Too strong and the electron falls into the nucleus. There's no distance an atom could settle at: no Bohr radius, no atoms, no chemistry. (Ehrenfest saw the classical version in 1917: no stable orbits either.)
So we pin the force: keep the potential at 1/r in four dimensions and see what happens. Everything in ATOM⁴ lives in that world. The rest of this chapter asks what it costs.
Gauss's 1/r² fails because it matches the squeeze. Confining an electron to a size r gives it momentum about ħ/r, and its energy p²/2m costs 1/r². So there's a second way out: leave the force alone and change the electron. If an electron's energy grew faster with its momentum, as p³ or p⁴, squeezing would cost 1/r³ or 1/r⁴. That outgrows the 1/r² pull close in, so the electron settles at a size, and atoms come back at every strength of pull. With p³ it's the same trick this universe uses, moved: the operator |k|³ that the lens vacuum puts into the field would go into the electron instead.
What it would keep: Gauss's law, so the shell theorem and the Faraday cage. A neutral atom would be invisible from outside, as ours is. There'd be no permanent R⁻⁵ pull, and noble gases would be gases. None of chapter 12's world of liquids would follow.
What it would cost: relativity, for all matter rather than for one force. E = p²/2m is what Galileo's relativity forces on anything slow (and Einstein's, in the slow limit): a moving lab and a still one see the same physics only if energy is quadratic in momentum. Any other power picks out a frame of absolute rest for every massive particle. Momentum would no longer be mass times velocity (v = dE/dp grows as p² or p³), so F = ma, Kepler's orbits and chapter 3 would all change. A heavy body built of such particles wouldn't move like one particle with their summed mass. And |p|³ is nonlocal, like the lens. p⁴ is local, but it keeps every other cost.
The lens vacuum also gives up relativity, but only for how charges push each other. Light, motion and the electron stay as they were. That's why this universe changes the force rather than the electron. It's a judgement, not a theorem: the electron route keeps our Faraday cages and our gases, and the lens keeps our mechanics. This section is argued from scaling, not computed.
Pinning 1/r was a choice of potential. Now let's find a field theory that gives it, and look for what breaks. The starting point: in Maxwell's theory a charge's potential is the Fourier transform of 1/k², in any dimension, and that transform is 1/r in 3D and 1/r² in 4D. For 1/r in 4D we need 1/k³. There are five ways to get it, each with a cost, and a sixth, which is the one this universe takes.
Our 1/r is special in 3D: it's harmonic (∇²(1/r) = 0 away from the charge). That one fact gives Gauss's law, Newton's shell theorem (a charged sphere acts like a point outside and has no field inside) and the Faraday cage. In 4D, 1/r isn't harmonic: ∇²(1/r) = −1/r³. The flux of the field through a sphere round a charge grows with the sphere's radius, as if empty space were charged. A charged sphere doesn't act like a point, and a hollow conductor doesn't shield its inside: the potential at its centre is 3π/8 = 1.18 times the potential at its wall, so a charge inside is pushed toward the wall.
The way real physics gets 1/r from four dimensions is to make the fourth one small: a circle of circumference ℓ. Close to a charge (r ≪ ℓ) the field spreads in four directions and falls as 1/r². Far away it has filled the circle, can only spread in three, and falls as 1/r. That's how extra dimensions could hide in our world. But it doesn't give us 4D chemistry. Atoms much bigger than ℓ are 3D atoms, and atoms much smaller are back to having no ground state.
Coulomb's 1/r isn't a free choice. In 3D, locality, linearity, relativity and positive probabilities together force it. In 4D the same four force 1/r², and then there are no atoms. To keep 4D atoms, something has to go. This universe gives up relativity for electromagnetism: the lens vacuum keeps light exactly as 4D Maxwell has it and keeps every field adding, but its pull reaches across space at once. That's an absolute "now", a preferred frame, set by one new length ℓ. What it does to light is the next chapter. What it does to atoms, through the failed shell theorem, is in the periodic table.
ATOM⁴: the position · how it works · Lumen⁴ · IRIS⁴ · KATABASIS · Four · the arcade