ATOM⁴

1 · The force

• Gauss's law in 4D gives a 1/r² potential, and under it no atom has a ground state. We pin the potential at 1/r.
• The universe holds electrons (fermions), stable nuclei, the lens vacuum's light and a small fine-structure constant, and nothing else. Matter is stable, by scaling, because electrons are fermions.
• Maxwell's 1/k² can't give 1/r in 4D: it needs 1/k³. Every way to get it costs one of locality, linearity, relativity, positive probabilities, or four large dimensions.
• Our vacuum is the lens vacuum, ε(k) = ε₀ℓk and μ(k) = μ₀/ℓk. Light stays Maxwell's, fields still add, energy stays positive. The cost is relativity for the force: there's an absolute now.
• 1/r isn't harmonic in 4D, so there's no shell theorem and no Faraday cage. A charged shell's centre sits at 3π/8 of its wall's potential.
• The other way out is to change the electron: an energy growing as p³ or p⁴ makes squeezing cost more than Gauss's pull and gives atoms back, keeping the Faraday cage. It costs relativity for all matter (F = ma goes), where the lens costs it only for the force. Argued, not computed.

2 · Light

• The light that travels is Lumen⁴'s, exactly: 1/r³, three polarizations, the wake, 1.43 λ/D, ν⁴ and T⁵, the rainbow's conics.
• Matter feels light through the lens, c/ℓω: long waves push harder, and a flash is felt faintly before it arrives.
• Small particles scatter as ω³, so the sky is blue, though less steeply than ours.

3 · Motion and spin

• F = ma. Orbits lie in a plane, and under 1/r they're Kepler's closed ellipses. Under Gauss's force none is stable.
• Turns happen in planes: six components, double rotations, left and right isoclinic halves (SU(2) × SU(2)).
• Electrons pair up, two to an orbital, and their spin is handed: it turns with the left half of every rotation and ignores the right. Only half of a magnetic field can flip it.
• Rigid bodies tumble by Euler's equations in six components, still exactly solvable. Every single bond twists in three angles.

4 · Hydrogen

• E = −Z²/2(n + ½)²: bound by 6.05 eV, twice our size. Shells hold 2, 10, 28, 60.
• Its fall to n = 2 glows red to orange, ending at 570 nm. Fine structure splits each l into (l + 1)(l + 2) and l(l + 1).

5 · The periodic table

6 · The shapes

• Six regular polytopes. The 24-cell has no cousin of ours. 4D can be tiled regularly three ways.
• 4D crystals may turn by 5, 8, 10 and 12, not only 2, 3, 4, 6.

7 · Darbon

• Element 7: five sp⁴ bonds at 104.48°, DH₅, and chains of 3n + 2 hydrogens.
• Our chiral molecules aren't chiral in 4D, since half a turn through w mirrors them. Darbon's are.
• Darbon's diamond is two A₄ lattices, five-fold, and can't sit on a hypercube.

8 · Forces between molecules

• Neutral atoms attract permanently, as −3 S_A S_B / 64 R⁵ (S = Σ⟨r²⟩). For element 12 at neon's spacing that's about 14 times neon's whole well.
• Dispersion stays −C₆/R⁶ (London's 3/2 becomes 7/4), computed by coupled response, within 11% on our noble gases. In 4D it's weaker than the permanent pull for every noble gas but the first.
• Computed end to end, 4D noble-gas pairs from element 12 on sit 4.5 to 5.25 bohr apart. Simulated, element 12 condenses below 4,700 to 8,900 K: they would hardly be gases.
• Bond springs and angles carry over. A bond twists three ways, and staggered means inverted through the bond.
• Salt water screens by a power law: an ion is overscreened, with a −r⁻⁷ tail.

9 · Pxygen and Xater

• Pxygen, element 10: 1s² 2s² 2p⁶, two short of a noble gas, so two bonds and three lone pairs. 4D hydrogen holds its electron with 6.05 eV, against Pxygen's 13.1, so their bond is lopsided.
• Xater, H₂Px, is bent at about 120°, not water's 104.5°, and soft: straightening it costs 0.21 eV (water 1.42). Its lone pairs make a ring around the molecule's plane.
• Its dipole is 2.1× water's and its hydrogen bond 3.1× as strong, by a Hartree–Fock solver written for any dimension and checked on our water. Its hydrogen bonds are short, halfway to shared, and its liquid should hold together far past our water's boiling point.

10 · Which water

• Almost every 4D hydride has positive hydrogen: iydrogen holds its electron with only 6.05 eV. Mithium and todium are the exceptions (Mi⁺H⁻, like our LiH).
• Every hydride of rows 2 and 3 computed: their shapes are what counting says, except that row 2's XH₄ comes out flat, with no dipole.
• On chemistry, two tie for the water: element 9's XH₃ and Xater, searched over every arrangement: each binds a pair at about 28 kcal/mol, 6 times our water's, and not in a line. Their best pairs lie antiparallel, dipole to dipole. Which fills the oceans is the abundance question, still open.

11 · Ten ways to meet

• Two molecules meet in six ways in our world and ten in 4D: d moves and d(d − 1)/2 turns. Turning grows as d², so in 4D most of the ways are turns.
• Chapter 10 first searched two of Xater's ten numbers and missed its best pair by a factor of two. Searched over all ten, a pair's landscape is a funnel: every one of 400 random starts finds the same pair.
• A search's energy is a floor: more searching can only bind harder. Whether it's finished has to be argued separately, and the audit has run. Our ammonia's and water's pairs are found. Hydrogen fluoride's bent pair is missed for a straight one 7% weaker: a search goes where its force field points. 4,568 random Hartree–Fock placements found nothing deeper than the best pairs, and chapter 10 stands.
• Crowds are a maze, and more so in 4D: from two molecules to seven, our water's minima go 1, 5, 5–7, 8–9, 37–45, 138–152, Xater's 1, 2–3, 8–10, 20–25, 148–268, 342–765, and the 4D counts haven't stopped climbing. At seven, 2% of starts find our water's best and 0.9% Xater's. Which crystal a 4D water makes will be a hard search, and its freezing point is predicted low for its cohesion.

12 · Gases

• PV = NkT holds in any dimension. A 4D atom carries 2kT, γ = 3/2, and sound runs at 0.61 of the rms speed (ours 0.75).
• 4D's noble gases are held by the crowd. R⁻⁵ over r³ of space adds up far out: element 12's 24 nearest neighbours give 31% of its crystal's hold, argon's 12 give 80%. Its Boyle temperature is 28 times its well depth (ours 2.8).
• A throttle cools them up to tens of thousands of kelvin: element 12 below 31,000 to 56,000 K, against argon's 794 K. Even 4D's helium cools at room temperature, so a 4D liquefier needs no precooling. Counted by the whole pull, not the well, the gas is ours rescaled.
• Element 12 condenses below 4,700 to 8,900 K, simulated (liquid and vapour side by side, checked on Lennard-Jones atoms and our argon). That's where the crowd's rules put it, 6 times the well-depth rule. Element 2 is the one 4D noble gas that might be a gas at room temperature.
• A hypothesis: a world of liquids. Gas only when hot, wide liquid ranges, glass more often than crystal: element 12's two best crystals nearly tie. Salts are the exception. Computed end to end, 4D's todium gluoride holds together 1.4 times as hard as our sodium fluoride, which the same method gets to 5%.