The solvers behind every chapter, what each was checked against, the compute they ran on, and how to run them yourself.
Everything in these chapters is computed by code anyone can read, and most of it can be re-run in a browser. Six rules make that hold:
A calculation longer than a few minutes runs on GitHub's free runners: 4 cores, 16 GB and up to 6 hours each, up to 20 at once. A run is a short file listing commands, each a tool and plain words, grouped four to a runner. Pushing it starts the run. Each tool saves its progress every few seconds to a data branch, so a run cut at its deadline resumes where it stopped, and every run's logs are kept there with the commit it ran. The results come back as data files, and the pages are regenerated from them.
So far: 35 runs, 853 commands, 653 core-hours, at no cost. Work in the sandbox and in readers' browsers isn't counted.
| run | why | commands | core-hours | result |
|---|---|---|---|---|
| smoke-1 2026-10-26 | First run of the runner: two shards of a small cluster search, to check the restore, the run, the hand-off and the write-back to atom4-data. | 2 | 0.0 | ✓ |
| clusters-6-7 2026-10-26 | The search chapter's crowds at equal effort: six molecules to 2400 starts in every world (the first count gave 4D fewer starts), and seven molecules to 1600, to see whether 4D's minima keep multiplying faster. | 40 | 41.6 | ✓ |
| audit-1 2026-10-27 | The search audit: the pipeline on our HF and ammonia dimers (published answers), the census chapter re-baked with the fixed descent, the Hartree-Fock landscape probed at random against the force field for water, Xater and bmmonia, and element 18's pair, too big for the sandbox. | 15 | 42.1 | ✓ (3 cut at the deadline) |
| coexist-1 2026-10-27 | Liquid and vapour side by side (Gibbs-ensemble Monte Carlo), to test the gases chapter's estimate that 4D's noble gases condense far hotter than their wells suggest: Lennard-Jones in 3D as the known answer (critical point 1.31), our computed argon as the calibration, Lennard-Jones in 4D for dimension alone, and 4D's elements 12 and 2, with a box-size check. | 27 | 19.4 | ✓ |
| audit-2 2026-10-28 | Fixing the miss: the Hartree–Fock relaxation is now quasi-Newton with a full-Hessian step off saddles (the old one could turn a pair ~0.1 rad and stayed on hydrogen fluoride's straight chain). | 17 | 14.9 | ✓ (1 cut at the deadline) |
| nebula-1 2026-10-28 | Does a rotating 4D cloud flatten into a disk? Our world's must (a sphere can't be combed); a 3-sphere can (Hopf), so the shape may depend on the ratio of the cloud's two rates. | 18 | 63.1 | ✓ (2 cut at the deadline) |
| audit-3 2026-10-28 | Finish fixing the miss. | 9 | 46.4 | ✓ (8 cut at the deadline) |
| stability-1 2026-10-28 | The orrery generator's rules: two planets round a star under the pinned 1/r, over the ratio of their orbits (1 to 1.6), the angle between their planes (0 to 90 deg, tilted in both angles or about a shared line) and their mass (0.1, 1, 10 Jupiters), each for 100,000 orbits of the inner one. | 6 | 2.5 | ✓ |
| audit-4 2026-10-28 | audit-3's eight relaxations of the 4D pairs' deepest random placements were cut at the deadline still descending (Xater's best 25.80 of 27.35 kcal/mol, XH3's 26.93 of 27.54 and falling 0.38 a step). | 8 | 41.8 | ✓ (7 cut at the deadline) |
| corr-1 2026-10-28 | Correlation (MP2) for the best pairs, our three and the two 4D candidates, in the book's basis and three grown ones, plus a scan along the bond: the census chapter's correlation section, first computed in the sandbox, recomputed here so js/correlation.js bakes from atom4-data's caches. | 24 | 1.4 | ✓ |
| corr-2 2026-10-29 | Does XH3's MP2 lead over Xater survive a bigger basis? corr-1's largest basis gave 2.4 kcal/mol, all of it correlation, with Xater's share still falling (+0.64, +0.42, +0.48, +0.05 kcal/mol as the basis grew). | 10 | 1.7 | ✓ |
| audit-5 2026-10-29 | A third round for the relaxations from the four deepest random placements of each 4D pair, cut again at audit-4's deadline (one already ended, in XH3's best pair). | 7 | 26.7 | ✓ (4 cut at the deadline) |
| corr-3-nebula-2 2026-10-29 | Two questions on one run. | 14 | 26.7 | ✓ (4 cut at the deadline) |
| nebula-3-audit-6 2026-10-29 | Resumes, from their caches. | 10 | 13.0 | ✓ (1 cut at the deadline) |
| corr-4 2026-10-29 | corr-3 showed XH3's MP2 lead over Xater is not shrinking: it went 2.44 with an f, 2.24 with a second f, back to 2.59 with a g instead, and the swings are all in the Hartree-Fock part (correlation's part holds at 2.5-2.75 kcal/mol). | 2 | 1.1 | ✓ |
| manybody-1 2026-10-29 | Does a crowd hold together the way its pairs say? The 4D force fields as first fitted collapse into crystals four times too dense; a polarizable model with 4D's cloud terms halves that but still packs densely. | 7 | 1.1 | ✓ |
| manybody-shell-1 2026-10-29 | Is a dense 4D crystal real at the Hartree-Fock level? Two tests on crystals the polarizable force field finds (and the old model's collapsed ones): seven trimers' three-body energies (one from our water's ice, the calibration), and one dense Xater crystal's every distinct neighbour pair within 10 bohr (102), each a Hartree-Fock pair energy, summed into its pairwise lattice energy. | 41 | 0.4 | ✓ |
| manybody-2 2026-10-29 | manybody-shell-1 found 4D hydrogen-bonded rings strongly cooperative (Hartree-Fock three-body 23% of a Xater ring's binding, 42% of XH3's, against 8% for our water's) and the polarizable field overshooting it 2-4x. | 9 | 1.5 | ✓ |
| crystals-pol-1 2026-10-30 | The polarizable field refitted with its dipole damping fitted to 16 Hartree-Fock trimers (it predicts held-out trimers to 1.0 kcal/mol) still packs both 4D waters densely (Xater -71, XH3 -104 kcal/mol per molecule in local starts). | 66 | 124.3 | ✓ (15 cut at the deadline) |
| nuclei-1 2026-10-30 | The abundance question: helium-4 and tritium by diffusion Monte Carlo (exact for these) in our world and across 4D's strength window (the deuteron bound, the diproton not), with and without the three-nucleon repulsion fitted to our nuclear matter at three ranges, to set against the nuclear matter's binding per nucleon. | 20 | 4.4 | ✓ |
| nuclei-2 2026-10-30 | The second nucleon force (Minnesota: soft Gaussians where Malfliet-Tjon has singular Yukawas) through the same pipeline, to show the 4D results don't hang on the first force's shape: its nuclear matter across 4D's window and helium-4 with and without the refitted three-nucleon force; and Malfliet-Tjon's helium-4 rerun to record radii for the helium-4 cluster model. | 28 | 5.1 | ✓ |
| nuclei-3 2026-10-30 | The helium-4 ladder: 2 to 8 helium-4s (beryllium-8, then elements 6, 8, 10, 12, 14, 16) by diffusion Monte Carlo on a local helium-4 pair potential (the folded force with its Pauli-forbidden states removed by Baye's supersymmetric transformation, calibrated on our beryllium-8), in our world as the check (2 to 6) and across 4D's strength window, under both forces: is every helium-4 capture downhill, and by how much. | 48 | 5.1 | failure (1 cut at the deadline) |
| nuclei-4 2026-10-30 | The helium-4 ladder again, every point, on the repaired supersymmetric pair potential. | 52 | 2.1 | ✓ |
| brink-1 2026-10-30 | Helium-4 clusters microscopically (js/brink.js): Brink's antisymmetrized determinants, Minnesota's force plus the book's three-nucleon force at each range (0.7, 1.0, 1.4 fm), its exchange mixture u set on our beryllium-8, bases of 240 and 480 placements. | 48 | 1.6 | ✓ |
| plusone-1 2026-10-31 | The road's first step, odd against even: does a nucleon hold on to a ladder nucleus (beryllium-8, carbon, oxygen, pxygen)? If the '+1' nuclei don't bind, as mass 5 doesn't, odd elements have almost no easy way in. | 32 | 5.1 | ✓ |
| plusone-2 2026-10-31 | Odd against even, the closed-shell test: a nucleon on a ladder nucleus with the core's helium-4s drawn close (the closed-shell limit), which on our oxygen-16 gives fluorine-17 and oxygen-17 within a MeV of measured where the loose draw of plusone-1 overbound them by 6–7. | 30 | 3.5 | ✓ |
| mainseq-1 2026-10-31 | The 4D main sequence: zero-age stars from 0.13 to 5 Chandrasekhar masses (js/mainseq.js: shooting fit, Fermi–Dirac electrons, pinned gravity from the whole profile, electron scattering through the lens vacuum, the book's own rates), at gravity 10⁻¹⁴ to 10⁻²⁶ of ours and proton + proton's nuclear size 1 or 4 fm: brightness, size, colour and lifetime against G, ours (0.5–30 M☉) reproduced first. | 11 | 0.1 | ✓ |
| mainseq-2 2026-10-31 | The 4D main sequence with free–free absorption (js/mainseq.js kappaFF4: classical passes through the lens vacuum, no logarithm in 4D, Kirchhoff, Rosseland-averaged with scattering; the classical estimate is half of Kramers' on ours, so ×0.5 and ×2 are run as the band): G 10⁻¹⁴ to 10⁻²⁶ of ours, proton + proton at 1 or 4 fm. | 20 | 0.3 | ✓ |
| mainseq-3 2026-10-31 | The 4D main sequence with free–free absorption, the walk in mass now retrying smaller steps where one fails (js/mainseq.js kappaFF4: classical passes through the lens vacuum, no logarithm in 4D, Kirchhoff, Rosseland-averaged with scattering; the classical estimate is half of Kramers' on ours, so ×0.5 and ×2 are run as the band): G 10⁻¹⁴ to 10⁻²⁶ of ours, proton + proton at 1 or 4 fm. | 20 | 0.7 | ✓ |
| mainseq-4 2026-10-31 | The 4D main sequence again with each star's stirred core recorded (the ash core it leaves, for which stars grow a core to the Chandrasekhar mass and explode), free–free at its classical value, proton + proton 1 or 4 fm, at G 10⁻⁸ to 10⁻²⁶ of ours (10⁻⁸ and 10⁻¹¹ new, for where a star as long-lived as our Sun stops existing). | 14 | 0.5 | ✓ |
| liquid-1 2026-11-01 | Liquid–vapour coexistence by Gibbs-ensemble Monte Carlo with the full Ewald sum (tools/liquid.mjs, js/gibbs.js): SPC/E water, whose critical point is published (about 640 K), as the method's check; our water on the book's Hartree–Fock force field, for the recipe's calibration. | 12 | 9.5 | ✓ |
| liquid-2 2026-11-01 | Gibbs liquid–vapour again with the start melted in place before transfers (liquid-1's vapour boxes kept a droplet they couldn't drain) and a hard core between opposite charges (the book's field let a Monte Carlo jump drop a hydrogen into a heavy atom): SPC/E as the method's check, our water on the book's field lower down (it was supercritical from 260 K). | 12 | 30.3 | ✓ |
| trimers-liquid-1 2026-11-01 | Does Xater stay a molecule when it condenses? Hartree–Fock on trimers cut from its simulated liquid (the book's field under the full Ewald sum, 1000 K, heavy atoms 4.3 bohr apart, 94 first-shell triples a molecule): the three-body energy a pair field can't see. | 60 | 12.7 | ✓ |
| liquid-hot-1 2026-11-01 | Does heat break 4D water's dense phase into something liquid? Xater and XH₃ on the book's fields (Gibbs, full Ewald) from 1500 to 10000 K: where the dense phase melts and where liquid and vapour meet. | 12 | 36.5 | ✓ |
| audit-1-rods 2026-11-02 | After the independent audit: F01, re-run the saved Hartree–Fock pair energies with the fixed convergence test; F04, argon's and element 2's critical points with six temperatures each, three independent seeds and a doubled box, and SPC/E with two more seeds, so each critical temperature carries residual degrees of freedom and a seed-to-seed error. | 102 | 65.7 | failure |
Each method: what it computes, how, what checks it (the self-test's numbered checks, by name), its code, where it's used, and its limits.
Every atom from hydrogen to element 140 (ours to 36, the check), in both worlds: its electron configuration, energies, sizes and ionization.
The radial equation in d dimensions has the centrifugal term of l + (d − 3)/2, so one solver serves every dimension. It is solved by Numerov's method on a logarithmic grid, energies found by Sturm bisection on node counts. Many electrons are a self-consistent field: each feels the nucleus, the others' charge through the sphere-averaged kernel (1/max in 3D; k(min/max)/max in 4D, which has no shell theorem), and a local exchange term from the d-dimensional electron gas, with each electron's own share removed and Hund's rule for spins. The ground configuration is searched among candidates, and only converged fields count.
| checked by | 1 one electron; 2 shells; 3 the kernel; 4 many electrons; 5 the baked table is complete, and what the solver gives now; 6 the 3D check; 7 the 4D table's shape; 8 what the shell theorem's failure does; 16 what the universe contains |
|---|---|
| code | atom.js, bake.mjs |
| used in | chapter 6 (hydrogen), chapter 7 (the periodic table) |
| cost | about 6 minutes for both tables, in the sandbox |
| limits | No correlation; open shells averaged into spheres; non-relativistic. Our world's check gets 32 of 36 configurations right. |
How easily each atom's cloud is pushed (polarizability at imaginary frequency), and from it the dispersion pull C₆ between two atoms.
The coupled response of the solved atom to a field at imaginary frequency, then Casimir and Polder's integral for C₆. Exact for hydrogen in both worlds; for helium it is time-dependent Hartree–Fock.
| checked by | 18 the solver's exchange, and the response that gives C₆ |
|---|---|
| code | response.mjs |
| used in | chapter 10 (forces between molecules) |
| cost | about a minute |
| limits | Closed-shell atoms only so far. |
The energy between two atoms at every distance, from their solved clouds.
Gordon and Kim's method: the two frozen clouds overlap, and the energy is their electrostatics (summed shell by shell, so the large nuclear terms cancel exactly) plus the electron gas's kinetic and exchange energy at each point (correlation left out, as in the atoms). Dispersion is added from the response above.
| checked by | 15 forces between molecules; 17 two atoms, computed |
|---|---|
| code | pairs.mjs, forces.js |
| used in | chapter 10 (forces between molecules), chapter 14 (gases) |
| cost | about a minute |
| limits | Frozen clouds don't relax as they meet. |
Molecules: their shapes, dipoles and hydrogen bonds, our water first as the check.
Hartree–Fock on Gaussian functions, with McMurchie and Davidson's integrals rebuilt for d dimensions: the Boys function F_n is replaced by G_n(T) = ∫₀¹u²ⁿ(1 − u²)^((d−3)/2) e^(−Tu²) du, which is F_n at d = 3. Geometries are relaxed; pairs are scored by their binding energy. A run counts as converged only when an iteration has both moved the energy less than 10⁻⁹ hartree and brought the commutator FPS − SPF under 10⁻⁵; it reports its iterations. (Until an independent audit caught it, a run that used up its 200 iterations was reported as converged; every saved result's flag was re-checked by re-running the saved pairs.)
| checked by | 19 molecules; 20 the census; 58 the independent audit's failure contracts |
|---|---|
| code | gauss.js, molecules.mjs, census.mjs, reconverge.mjs |
| used in | chapter 11 (pxygen and xater), chapter 12 (which water) |
| cost | half an hour for the molecules, four and a half hours for the census |
| limits | Hartree–Fock misses correlation (the next method measures it); small bases. |
What Hartree–Fock leaves out of a pair's binding: whether the tie between Xater and XH₃ survives it.
Møller–Plesset perturbation theory to second order on the closed-shell Hartree–Fock result, the cores frozen: E₂ = Σ (ia|jb)[2(ia|jb) − (ib|ja)]/(εᵢ + εⱼ − εₐ − ε_b), over occupied i, j and empty a, b, the integrals turned from basis functions to orbitals by a four-index transform (one pass over the unique integrals for every occupied orbital at once). Nothing in it depends on the dimension; that lives in the integrals. Pairs are counterpoise-corrected, each molecule alone in the pair's whole basis, as the Hartree–Fock ones are. The basis is grown by one recipe in both worlds (a second d, diffuse s and p, a second p on hydrogen, an f), and in our world checked against the standard aug-cc-pVTZ and aug-cc-pVQZ, computed at the same arrangements by PySCF. For open shells (atoms with unpaired electrons, triplet molecules) the same on an unrestricted Hartree–Fock result, each spin its own orbitals: same-spin pairs ½ Σ (ia|jb)[(ia|jb) − (ib|ja)]/D, opposite-spin Σ (ia|jb)²/D. Molecules without hydrogen are scanned bond length by bond length this way, counterpoise-corrected, each point started from the bare nuclei and from the two atoms' densities with spins parallel and antiparallel (and from the previous point's solution), the lowest kept; our F₂ and O₂ are the checks. Bond strengths come from atomization energies the same way (geometry by Hartree–Fock, energy by MP2, atoms by unrestricted MP2), hydrogen against gluorine on the same atom, our CH₄, CH₃F, H₂O, OF₂, HF and F₂ the checks (each within 2–12% of measured).
| checked by | 43 correlation; 59 unrestricted MP2 |
|---|---|
| code | gauss.js, correlation.mjs, correlation-ref.py, diatomic.mjs, bonds.mjs |
| used in | chapter 12 (which water) |
| cost | seconds to minutes per pair in our world; up to 24 minutes for 4D's largest (268 functions), in the sandbox |
| limits | Second order only; at Hartree–Fock's arrangements (checked along the bond); in our world the grown basis still misses about a quarter of a kcal/mol of a pair's correlation, and 4D has no standard basis to check against. One determinant: where several states lie close (4D's Px₂, four electrons in three degenerate antibonding orbitals) the solution changes state between bond lengths and the curve is no bond; that needs a multireference method the book doesn't have. Our F₂ comes out 2.70 bohr and 1.51 eV (measured 2.67, 1.66), O₂ 2.35 and 5.01 (2.28, 5.21). |
A cheap energy for two molecules, so they can be searched over every pose and simulated by the thousand.
Charges on the nuclei, an exponential repulsion between every pair of sites and a damped dispersion between heavy atoms, fitted to hundreds of Hartree–Fock pair energies: random placements, scans along the hydrogen bond, and close contacts so the repulsion is seen.
| checked by | 22 force fields for the liquids |
|---|---|
| code | forcefield.mjs, fitff.mjs, ffmodel.js |
| used in | chapter 12 (which water), chapter 13 (ten ways to meet) |
| cost | minutes per molecule |
| limits | Pairwise: no three-body forces. Its blind spots are measured, not assumed away (the audit). |
The best way two molecules can sit together: 6 numbers to search in 3D, 10 in 4D.
Many random starts, each descending on the force field by BFGS until its slope is gone (a stall isn't a minimum), turns re-anchored as they wind. Then Hartree–Fock scores the force field's minima and relaxes them by a quasi-Newton descent that measures the whole curvature, steps off saddles along their downhill direction, and probes up to 1.5 radians along every principal direction at an apparent bottom. The audit runs the same pipeline on our pairs with published answers, places thousands of pairs at random for Hartree–Fock alone, and relaxes the deepest.
| checked by | 23 the pairs searched over every pose; 24 the descent behind the search chapter |
|---|---|
| code | landscape.js, pairsearch.mjs, audit.mjs |
| used in | chapter 12 (which water), chapter 13 (ten ways to meet) |
| cost | hours per pair, on the runner |
| limits | A search's energy is a floor, not a proof; the first descent missed hydrogen fluoride's bent pair (the search chapter). |
How fast the minima multiply as molecules are added: two to seven, in both worlds.
Thousands of random starts per cluster size, each descended to its minimum on the force field; minima told apart by energy and shape, counted, and their unseen number estimated by Chao1. Each start has its own seed, so a search splits across machines and the page can replay any start live.
| checked by | 25 the clusters; 39 the live five-molecule search |
|---|---|
| code | clusters.mjs, clusterlive.js |
| used in | chapter 13 (ten ways to meet) |
| cost | 41 runner-hours for sizes six and seven |
| limits | Pairwise force field; the 4D counts haven't settled. |
Whether a crowd of 4D molecules holds the way its pairs say, and which crystal our water and 4D's make.
Each molecule's charges, and in 4D a cloud term per atom (the 1/r³ potential a round cloud has outside it once the shell theorem fails), fitted to its own Hartree–Fock electrostatic potential; site polarizabilities fitted to its Hartree–Fock tensor by finite field, coupled by Thole's damped dipoles and solved for every molecule at once; only the overlap repulsion fitted to pairs. Trimers scored whole by Hartree–Fock in their own basis, so the three-body part is measured. Crystals: rigid molecules in a cell of any shape, relaxed with the cell from random starts; Ewald's split for each power of r the field uses, in any dimension. Forces and the cell's slopes analytic: the dipoles make the energy stationary in them, so each term is differentiated at fixed dipoles (Hellmann–Feynman).
| checked by | 47 molecular crystals; 48 the polarizable force field |
|---|---|
| code | polmodel.js, polcrystal.js, crystal.js, fitpol.mjs, ice.mjs, manybody.mjs |
| used in | chapter 13 (ten ways to meet) |
| cost | minutes per fit; an hour per 4D trimer on the runner |
| limits | Our water's ice is found from random starts. In 4D the damping between molecules is fitted to Hartree–Fock trimers (held-out ones to about a kcal/mol); whether its dense 4D crystals are real waits on Hartree–Fock's verdict on their neighbour shells. |
Molecules moving in a box: liquids, at a temperature.
Rigid molecules (RATTLE), a stochastic thermostat (Bussi's), and an Ewald sum for the pinned 1/r in any dimension, split as erfc and erf of 1/r itself, since a Gaussian cloud's own potential isn't short-ranged off 3D.
| checked by | 21 molecular dynamics |
|---|---|
| code | md.js, liquid.js |
| used in | chapter 10 (forces between molecules), chapter 14 (gases) |
| cost | seconds, live in the browser |
| limits | Checked on the Madelung constants (ours, and 4D's 1.8393991) and NIST's SPC/E water reference. |
How much a core's free charges speed fusion, and how long a star's magnetic field lasts.
Weak (Debye–Hückel) screening rebuilt on each world's kernel: the screening cloud's potential at a nucleus, −q∫d^dk/(2π)^d G²κ/(1 + Gκ), Debye's √(4πκ) for ours and (4π²)^{4/3}κ^{1/3}/12√3π for pinned 4D; the electrons counted not at all and in full, for the band. Strong screening from Salpeter's ion spheres, the electron ball's self-energy summed shell by shell with each world's shell potential (−0.9 Z²/a ours, the check; −0.816 Z²/a 4D, checked by Monte Carlo), adopting the smaller of the weak and strong exponents. Resistive decay from Ohm's law with the lens's pinned vector potential (4π²J/c²|k|³), and the conductivity from electron–ion momentum transfer off Z/r (Spitzer's logarithm in ours, a linear growth to the screening length in 4D). Stellar cores from the computed main sequence, our Sun's centre as the check.
| checked by | 60 a stellar plasma under the lens vacuum |
|---|---|
| code | plasma.js, plasma.mjs |
| used in | chapter 16 (stars) |
| cost | seconds |
| limits | The weak-to-strong bridge is the smaller exponent, rough near H ≈ 1 (no fitted intermediate formula for 4D); ion spheres take the electrons as a rigid uniform ball. Main-sequence cores only. Electrons taken nondegenerate for the conductivity, and the screening length as a sharp cut-off. No dynamo or wind: whether a 4D star makes a field, and how fast it spins down, are open. |
What a 4D magnetic field is, and what ideal flow keeps of it.
The field as a 2-form (six numbers, one per plane). Its canonical form from its self-dual and anti-self-dual halves l and r: two perpendicular planes at rates |l| ± |r|, F∧F = 2 Pf F = 2(|l|² − |r|²). Ideal flow from Faraday's law (which the lens vacuum leaves alone) and a perfect conductor's Ohm's law: ∂F/∂t = −d(i_v F) = −ℒ_v F; a parcel's field and density integrated along its path by Runge–Kutta (dF/dt = −(GᵀF + FG), dρ/dt = −ρ tr G) through swirling, compressing flows, never using the closed form, to show F∧F/ρ carried unchanged.
| checked by | 62 magnetic fields in plasma |
|---|---|
| code | mhd.js, plasma.js |
| used in | chapter 16 (stars) |
| cost | milliseconds |
| limits | Ideal (perfectly conducting) flow only; the flows are chosen, not solved for (no momentum equation, no Lorentz force back on the gas). Resistive and dynamo behaviour are next. |
Whether a fixed flow can make a magnetic field grow, in ours and in 4D.
G. O. Roberts's construction: a flow of cells periodic in x and y, each field wave along the remaining directions solved separately. The induction equation for the 2-form, ∂F/∂t = −d(i_v F), pseudo-spectral on a 2D grid (2/3 dealiased, RK4), the resistive decay split exactly around it: ηk² in ours, η|k|³ in 4D under the lens. F = dA at the start, so it stays closed. The fastest growth over the field's wavenumbers at each Rm = 1/η. Checked on exact decay, a uniform flow, Zeldovich's anti-dynamo theorem for planar flows, and our Roberts dynamo's optimal wavelength; resolution: ours 2% between N = 16 and 32, 4D unchanged.
| checked by | 63 kinematic dynamos |
|---|---|
| code | dynamo.js, dynamo.mjs, plasma.js |
| used in | chapter 16 (stars) |
| cost | minutes (a scan of each flow) |
| limits | Flows that vary in two directions only (3D-like flows in 4D); kinematic only (the field doesn't push back); no published growth-rate table to hold ours to, so the check on ours is qualitative plus the exact limits. |
Where a gas condenses: its critical point; where a liquid of molecules boils.
Gibbs-ensemble Monte Carlo: two boxes trade volume and atoms (or rigid molecules, turned in random planes) until one is liquid and one vapour, at one temperature per run; the critical point is fitted to the two densities across temperatures. For molecules each box keeps its own Ewald sum (real space to half the box, the structure factor updated move by move): a truncated charge sum, good to 0.3 kcal/mol a molecule in our water, is off by 42 in 4D, because the pinned 1/r charges a wave of polarization as 1/k. The 4D pull reaches far, so the boxes are big and the tail is added for a uniform fluid; where a field carries 4D's charge–cloud terms (q c/r³, which cancel only over a whole molecule), the short-range terms are cut by molecule rather than by site. Each run starts with nearly every molecule in one box, melts it in place, then lets the boxes trade; the liquid is the denser box of each block's means, independent seeds are pooled, and Tc carries a bootstrap error over them.
| checked by | 34 liquid and vapour, simulated; 57 liquid and vapour by Gibbs-ensemble Monte Carlo; 61 4D's cloud terms in the liquid tools |
|---|---|
| code | coexist.mjs, gibbs.js, liquid.mjs |
| used in | chapter 14 (gases) |
| cost | 19 runner-hours for the atoms; about an hour a temperature for a molecular liquid in 3D, two in 4D |
| limits | Checked on Lennard-Jones (published critical point), our argon's computed pair (measured), and SPC/E water (632 K against the published value of about 640). An independent audit found the first critical points rested on two temperatures each with no seed, size or tail study; those are being rerun, and until then they are exploratory. Molecules are rigid, so a liquid in which they would come apart or share protons is outside the model. |
Whether a force field fitted to pairs of molecules can be trusted for the crowd a liquid is.
The field's liquid is simulated, pairs of neighbouring molecules (the nearest six, and the next thirty-four for the far pull) are cut from it, Hartree–Fock is run on each, and the field's error on them is recorded before any refit sees them: the honest test. Trimers are cut the same way, a molecule and two of its first-shell neighbours, and Hartree–Fock's three-body energy, scaled by the liquid's number of such triples, gives the crowd's energy a pair field can't see. A hydrogen can also be walked from its own heavy atom to its neighbour's in a pair from the liquid, the whole pair's energy computed at each step, for whether protons stay home.
| checked by | 57 liquid and vapour by Gibbs-ensemble Monte Carlo; 59 unrestricted MP2 |
|---|---|
| code | liquid.mjs, manybody.mjs, liquidff.js |
| used in | chapter 14 (gases) |
| cost | about 25 minutes a probe round of 96 pairs in 4D (sandbox); an hour or so per trimer batch on the runner |
| limits | Hartree–Fock is the reference, so correlation is missing from both sides; rigid molecules. Results so far: our water's field to 0.4 kcal/mol, Xater's to 2.5–2.7 (its liquid's pair sum within 1–22%), XH₃'s overbinding its liquid fivefold. |
Which lattice a crystal takes, and how hard a salt holds together.
Lattice sums over candidate lattices (fcc and the 24-cell's among them); salts from closed-shell ions solved above, with Gordon–Kim between charged clouds (tabulated every 0.05 bohr) and Evjen's neutral cubes for the sum; doubly charged anions held in a Watson shell at the radius where its potential matches the crystal's, iterated with the spacing; densities on the alternating lattice. The air's molecules and pulls by Hartree–Fock plus MP2 on the census's bases grown with diffuse functions, counterpoise-corrected, hydrogen's pair rotation-averaged with weights from how often a rod lies along the line between two molecules (1/d), critical points from the 4D simulations' ratios to the well. In 4D the numerical tail would diverge over the r³ shells, so past 14 bohr the exact far field from the clouds' moments is used.
| checked by | 29 gases; 30 ions for the gases chapter's salts; 31 which crystal; 32 a salt |
|---|---|
| code | gas.js, crystals.mjs, salt.mjs, ions.mjs, air.mjs |
| used in | chapter 14 (gases) |
| cost | minutes |
| limits | Checked on our noble gases (fcc must win), sodium fluoride and magnesia (densities within 2–3%). |
Whether a ball of gas holds itself together under each law of gravity, its structure, and how often its nuclei meet.
The virial theorem in d dimensions for a 1/r^k pull; the self-consistent polytrope, iterated on each law's shell kernel (Hachisu's method); and WKB tunnelling with Langer's centrifugal term, the 4D head-on meeting's built-in barrier.
| checked by | 35 stars |
|---|---|
| code | stars.js, stars.mjs |
| used in | chapter 5 (gravity), chapter 16 (stars) |
| cost | seconds |
| limits | Checked against Lane–Emden's polytropes and Gamow's factor. Opacity, and so a star's brightness, is open. |
Whether 4D has nuclei at all, how tightly they hold, whether stars can burn past helium, and the heaviest atom.
Two nucleon forces of different shapes (Malfliet–Tjon's Yukawas, the Minnesota Gaussians), the same function of distance in both worlds, their attraction scaled to the window where the deuteron binds and the diproton doesn't. Two nucleons on a grid by Sturm bisection; helium-4 by diffusion Monte Carlo, exact for it (four species, a pair factor from the exact two-body state, Umrigar's guards); nuclear matter by Brueckner–Hartree–Fock in any dimension (the G-matrix in partial waves with d's Fourier transforms and Gegenbauer projections, the angle-averaged Pauli operator), saturated by a three-nucleon repulsion fitted to our matter; light nuclei as helium-4 clusters (the force folded over both, Saito's orthogonality by Kukulin's pseudopotential, one factor calibrated on our beryllium-8); the helium-4 ladder by diffusion Monte Carlo for N identical bosons, on a local pair potential with the Pauli-forbidden states removed by Baye's supersymmetric transformation (its tail by the Riccati equation, so it holds for states bound by hundreds of MeV); the first capture, two helium-4s to beryllium-8 by an electric-quadrupole photon, from the deep cluster potential's bound states and scattering waves (normalized as a plane wave's partial waves in d dimensions), with the lens vacuum's photon (its length ℓ cancels once α is fixed), set against our ¹²C(α,γ)¹⁶O; the rungs past it the same way, a helium-4 onto a core of helium-4s (the pair potential folded over the core, one depth per rung set on its capture energy, Wildermuth's rule for the forbidden states), checked on our carbon-to-oxygen capture; a nucleon captured onto a core of helium-4s the same way, by dipole light, its depth set to the microscopic binding, checked on our carbon-12 + proton; the same clusters microscopically by Brink's model (each helium-4 four nucleons in one Gaussian orbital, a nucleus one antisymmetrized determinant, Minnesota's force with its exchange mixture set on our beryllium-8 and the three-nucleon force with full exchange, Hill–Wheeler over random placements, angle projection for beryllium-8's 0⁺ and 2⁺; and one nucleon on a helium-4 the same way, with each nucleon's spin and isospin followed, for mass 5; and a nucleon on bigger cores, the core's placements fixed and the nucleon on a grid round them, for odd nuclei, the core's helium-4s drawn both loose and close, close being the closed-shell limit); hydrogen and helium burning together as a network at each temperature (protons, helium-4s and nuclei by charge, positron decays instant, every rate a ratio to proton + proton's so density drops out; backward Euler with Newton), checked on what our Sun's centre does to carbon, nitrogen and oxygen, the long-lived core found by comparing its pace with our Sun's in the burning thresholds' convention; what stars give back as each core's make-up averaged over the lifetimes of stars that finish before a Sun-like one forms, weighted by a declared mass function and lifetime slope, and then from the computed main sequence (each star's core temperature and lifetime at each G, the heaviest star light holds by 4D's Eddington limit); which stars explode, from the stirred cores the main sequence gives (declared growth after it, dead stars keeping lighter cores, winds returning a share); the burning after hydrogen and helium as the ash fusing pairwise, barrier-only at our oxygen + oxygen's pace, the products' particle shedding from the liquid drop, and equilibrium's end at the drop's N = Z peak; what lighting the ash does to a core (the electrons' Fermi energy over kT and the gas polytrope's binding per nucleon where a contracting core first reaches the ash's temperature, checked on our carbon), and each stage's energy per nucleon against ours; how far an exploding core burns (each shell heated by the burning's energy into electrons, nuclei and T⁵ light, sorted by equilibrium's and the ash's temperatures, with densities spread as our white dwarfs need, checked on ours against full-density nickel); proton and neutron capture onto helium-4 by dipole light (E1 through the lens vacuum, (3π/8)αω³|D|²); and the first minutes as the standard first estimate in any d (photons and baryons, neutron freeze-out and decay, the deuterium bottleneck by Saha, the captures integrated as the universe cools, using helium up), with the weak force, η and the clock as stated choices; and how hot a core can get (the gas polytrope's centre balanced by ions as a hot gas and Fermi–Dirac electrons, relativistic and partly degenerate, maximized over radius, masses in units of the Chandrasekhar mass so G drops out), set against each stage's temperature; Dirac's equation in d dimensions for the innermost electron, with a nucleus of size; and the main sequence in any d by shooting from centre and surface to a fitting mass (Fermi–Dirac electrons, light, convection at 2/(d + 2), the photosphere at τ = 2/3, pinned-4D gravity from the whole profile), on the book's own rates, with electron scattering through the lens vacuum as 4D's opacity, checked on our zero-age sequence and Eddington's homology.
| checked by | 51 nuclei, how far a table can go, and what sets G; 52 helium-4 clusters, microscopic; 53 the first minutes; 54 how hot a core gets; 55 a zone burning hydrogen and helium together; 56 the main sequence in any d |
|---|---|
| code | nuclear.js, brink.js, bbn.js, burning.js, mainseq.js, dirac.js, gscale.js, nuclei.mjs, mainseq.mjs |
| used in | chapter 15 (nuclei), chapter 16 (stars) |
| cost | seconds for the window and Dirac; minutes per helium-4 and per matter scan, on the runner |
| limits | Our own deuteron, helium-4 (to 0.1 MeV of the published value), nuclear matter's saturation and Chandrasekhar's mass are reproduced, and our critical charge (170). Past helium the Monte Carlo fails our own nuclei, so it isn't used there; the cluster model needs one calibration per kind of nucleus, and the mass-5 one stands in for a missing spin–orbit force. The ladder is two-body only: it gets every capture's sign right on ours but underbinds our carbon to magnesium two- to threefold. The capture rests on the cluster model's spin-2 state, which both forces put 1.3–1.8 MeV too low in ours; pair emission isn't counted. Brink's single-Gaussian helium-4 lacks its correlations, so on ours it overbinds oxygen-16 two to three times where the cluster ladder underbinds it: the two bracket the measured values, and are used together. The first minutes are a first estimate (no deuterium, helium-3 or tritium burning followed, the network stops at mass 5), and take mass 5's binding from the microscopic calculation, which has no spin–orbit force. The odd nuclei are two-body only; with the core drawn loose they overbind ours more the bigger the core (fluorine-17 by about 6 MeV), and drawn close they reproduce oxygen-17 and fluorine-17 within a MeV, but for open-shell cores the two draws differ by up to tens of MeV in 4D, and that spread is the stated uncertainty. The burning network takes proton + proton as ours through 4D's barrier, captures past oxygen as the barrier times our nuclear factor raised by the computed ones' average excess, and the even elements' three captures as one; the long-lived core's temperature rests on comparing rates across dimensions and is good to tens of percent, which spans element 9's band, and the proton's nuclear size in proton + proton (1 or 4 fm) decides between element 9 and pxygen there. What stars give back counts only the hydrogen-and-helium ash, as a fixed share of each finished star; the later stages are followed only as far as which products and which end: barrier-only fusion with our nuclear factor, a liquid drop with a borrowed surface for the energies, a simple shedding rule, and our silicon burning's kT ≈ B/30 for equilibrium, all rough; how far an exploding core burns uses the gas polytrope's profile and our white dwarfs' spreading, declared for 4D. How far stars build is the textbook estimate, good to about forty percent in mass on ours, and uses barrier-only temperatures past element 10. The rungs' potential model gives our carbon-to-oxygen capture up to four times low and puts oxygen's spin-2 state too deep. Where binding peaks rests on a surface term borrowed from ours. |
Whether an atom is drawn into a magnet or pushed out, how the field's two halves differ, and the field of a spinning ball.
Pauli's equation as the square of a first-order one (Lévy-Leblond's), built in 4D from quaternion units: the spin couples to the field's left half only. Langevin's diamagnetism from the solver's ⟨r²⟩ (−Σ⟨r²⟩/2d per atom), and Curie's paramagnetism for the spin Hund's first rule gives the solver's configuration. A spinning ball's field from its moment, A = Ωx/r³ under the pinned kernel.
| checked by | 45 magnetism of atoms; 46 a spinning ball's field |
|---|---|
| code | magnet.js, magnetnumbers.mjs |
| used in | chapter 4 (motion and spin) |
| cost | instant |
| limits | Checked against our noble gases' measured susceptibilities and NIST's ground-state spins. Orbital moments are left out: the solver averages open shells into spheres. |
Kepler's orbits under each law, precession round a real star, planets in planes turned apart, moons, starlight on a planet, the stability map, tides on a spin in two planes, and the orrery that puts them together.
A leapfrog n-body integrator in any dimension (symplectic, so energy doesn't drift over hundreds of thousands of orbits), orbits' planes compared by their principal angles, a round star's outside potential summed from its shells. The orrery builds a system from the results: planes from their halves of turning as the nebula leaves them, spacings from the stability map, moons in the completely orthogonal plane. Tides by our world's constant time-lag model written for a turning in any number of planes, checked against a planet of masses on damped springs.
| checked by | 36 orbits; 41 moons and starlight; 49 the orrery; 50 tides |
|---|---|
| code | orbits.js, orbits.mjs, stability.mjs, orrery.js, orrery.js, tides.js, tidenumbers.mjs |
| used in | chapter 5 (gravity), chapter 17 (planetary systems) |
| cost | seconds to minutes; the stability map on the runner |
| limits | Checked against Kepler's closed orbits and Landau's precession formula. |
Whether a rotating cloud of gas falling onto a star flattens into a disk.
Tens of thousands of particles pulled as 1/r² (Newton in 3D, the pinned law in 4D, so only the dimension differs), with pairwise collisions that remove part of each approach speed along the line between the pair: momentum and spin kept exactly, energy taken. The same code in 3D is the control.
| checked by | 40 nebulae |
|---|---|
| code | nebula.js, nebula.mjs |
| used in | chapter 17 (planetary systems) |
| cost | 63 runner-hours |
| limits | No pressure, no self-gravity; times depend on the collision model, so the shapes are the result. |
The counting the rest leans on: spheres, directions, packings, flats, turnings, random walks, slices, tori.
Exact formulas where they exist, each checked against sampling or counting: random points in a ball, random directions, lattice vectors, random flats, walks, Hopf's circles.
| checked by | 37 geometry; 38 geometry, continued |
|---|---|
| code | geometry.js |
| used in | chapter 1 (geometry) |
| cost | seconds |
The routes to a 1/r force, what each costs, and the lens vacuum adopted.
Closed forms for each route's kernel and its costs, the unitarity bound, the lens vacuum's response, scattering, and Biot–Savart fields under each kernel.
| checked by | 9 the field theory; 12 the lens vacuum; 33 fields |
|---|---|
| code | fields.js |
| used in | chapter 2 (the force) |
| cost | seconds |
Rotations in planes, handed spin, and tumbling bodies.
so(4) = so(3) ⊕ so(3) by the Hodge star; the spinor representation from 4 + 1 gamma matrices; Euler's equations for a free 4D body in six components.
| checked by | 13 motion and spin |
|---|---|
| code | turning.js |
| used in | chapter 4 (motion and spin) |
| cost | seconds |
How fast the first step of hydrogen burning runs in 4D, without guessing a nuclear size.
The weak conversion p + p → d + e⁺ + ν as σv = K Ω_d |∫u_d u_E dr|² f(W₀): the deuteron and the two protons' scattering state (Coulomb, the singlet force, in 4D the s-wave's own barrier) solved from the same strong force, K a rate taken as ours and set on our measured S(0), f the positron's phase space at each world's own deuteron binding. Then the radius at which the stars' old barrier formula gives the same thermal rate.
| checked by | 74 proton + proton → deuteron from the strong force |
|---|---|
| code | nuclear.js, ppfusion.mjs |
| used in | chapter 15 (nuclei) |
| cost | about a minute for the table |
| limits | Our form of the lepton phase space carried to 4D (declared); the strong force's strength in its window still chosen, and it decides whether p + p runs at all. |
Why electrons can't share a state here, though the force between them isn't relativistic.
The free electron's 4 + 1 Dirac Hamiltonian has levels ±√(c²p² + m²c⁴). If electrons commuted, N of them could share one negative-energy wavepacket of width b; their pinned 1/r repulsion is at most √π/(4bν²) by Cauchy–Schwarz on the packet's amplitude (ν ≥ ½ its weight in those states), so at b = N² the energy is at most −Nmc² + √π/2 and has no floor. Only the half of spin-statistics that needs positive energy is used, not the locality the lens vacuum lacks.
| checked by | 73 Pauli's exclusion derived |
|---|---|
| code | pauli.js |
| used in | chapter 2 (the force), chapter 4 (motion and spin) |
| cost | under a second |
| limits | Pauli's own level of proof (free fields, the interaction added), not the axiomatic theorem; says nothing on whether fermionic matter in bulk is stable. |
How fast neutrons and protons turn into each other in 4D, from one constant instead of ours.
Fermi's contact theory written for any number of space dimensions: the coupling G_d has units of mass^−(d−1), each lepton's phase space is p^(d−1)dp, so a decay goes as G_d² m_e^(2d−1)|M|²Ω_d² f_d/(2π)^(2d−1) with f_d = ∫p^(d−2)W(W₀ − W)^(d−1)dW, the nucleus's |M|² ours (the nucleon is the smallest spinor in both worlds). The early universe's neutron–proton rates (electron mass kept, Pauli-blocked) are integrated against the expansion to the frozen neutron share, which the first minutes then burn to helium. One choice is left: G₄, quoted against G_F/M_W (the weak force as strong as ours at its own scale).
| checked by | 75 the weak force in d dimensions |
|---|---|
| code | weak.js, weak.mjs |
| used in | chapter 15 (nuclei) |
| cost | seconds |
| limits | No Fermi function (7% on our neutron); g_A, the neutron–proton mass difference and the neutrino count are ours; the early clock is gravity's and is scanned, not fixed. |
The six regular 4D polytopes, hybrid bonds and diamond nets.
Built from their vertices, with edges, faces and cells found from them and counted against their known numbers.
| checked by | 10 the six regular 4D polytopes; 11 darbon's geometry |
|---|---|
| code | polytopes.js |
| used in | chapter 8 (the shapes), chapter 9 (darbon) |
| cost | seconds |
The code is public. With node (version 20 or later) and git, and nothing else installed:
git clone https://github.com/minormobius/agent01 cd agent01 node four/atom/test/atom4.selftest.mjs # every check, about 90 s node four/atom/tools/orbits.mjs # re-bake the orbits, 40 s node four/atom/tools/clusters.mjs 4 5 100 # 100 starts of five Xaters
Each tool's first lines say what it does and how to run it. A fork's own GitHub Actions can run the compute queue (.github/compute/atom4.json) the same way. Packaging the solvers as a standalone library, free to use, is planned; until a licence is published the code is open to read but not yet licensed for reuse.
ATOM⁴: the position · how it works · Lumen⁴ · IRIS⁴ · KATABASIS · Four · the arcade