# The road to a 4D Earth The goal: **a worked 4D Earth.** What its ocean, rock and air are made of; how dense the rock is, which sets G and the planet's size; and so whether an airplane can fly in its sky. Every link is tagged computed, rough or chosen, as the book tags everything. It started with the airplane: gravity, a planet, air, lift and propulsion need to know what things are made of, and that turned out to need the abundance of the elements, which needed nuclei, which needed the strong force. The nuclear groundwork is now in the nuclei chapter. What the goal needs from it is not a full abundance table but three things: element 10 against element 9 (which water), the few commonest rock-forming elements, and what is volatile (the air). One lesson shapes the route. Anything that depends only on **which stage a star reaches** is robust: a capture rate is so steep in temperature that a millionfold uncertainty in it moves the temperature a stage lights at by about two. Anything that depends on **rate ratios at one temperature** (what a burning core turns into, element by element) is uncertain by orders of magnitude. The route leans on the first kind. ## The physics spine (PI, 2026-11-12) The Earth path below needs the universe's composition, and that needs four things the book has so far borrowed or assumed. They come first, in this order, because each feeds the next. **Every result downstream is carried as a band across the choices it rests on, and any conclusion that flips inside a band is named.** | | objective | today | the gap | |---|---|---|---| | A1 | **the weak force in 4 + 1** | entirely ours: our coupling, neutron lifetime and freeze-out, ¹³N's ft, and 3D lepton phase space | not reasoned at all | | A2 | **the strong force, reasoned** | a strength dial s on a borrowed shape (Minnesota, Malfliet–Tjon), its window set by the deuteron and the diproton | no tensor or spin–orbit force; the three-nucleon repulsion fitted to our matter; Brink's grid and the ladder bracket ours' masses from either side (one model adopted, masses-1), and the one mass they disagreed on (the CNO-like cycle's) decided by a free-cluster variational bound | | B | **it doesn't collapse** | Pauli derived; one atom stable by scaling; a Chandrasekhar mass in pinned 4D | bulk matter (stability of the second kind); nuclear matter saturating without our fitted repulsion; what collapsing cores end as | | C | **4D solar lifetimes** | a convention: k_pp is set so a Sun-like core halves its hydrogen in 4.6 Gyr | an absolute weak rate (A1) through the main sequence (js/mainseq.js) at each G and s | | D | **the universe's composition over its age** | the first minutes (helium where the deuteron binds as ours), a rough "what stars give back", the burning network | yields per stellar mass; a one-zone galaxy enriching over time; the heavy elements (no neutron source, 2026-11-12) | - **A1. The weak force in 4 + 1.** Fermi's theory in d dimensions: the coupling G_d has units of mass^−(d−1), each lepton's phase space is p^(d−1)dp, so β decay goes as ∫p^(d−2)W(W₀ − W)^(d−1)dW. From one parameter, G₄ (and the first minutes' clock, which gravity sets): the neutron's lifetime, freeze-out and the primordial helium, an absolute p + p rate, and every β decay in the network. The deliverable is the **weak window**: the G₄ for which the first minutes leave hydrogen and stars still live long. Days, in the sandbox. `js/weak.js`. **First result (2026-11-13, `js/weak.js`, `tools/weak.mjs`, selftest 75):** checked on ours (neutron 945 s without the Fermi function, measured 879; freeze-out 0.800 MeV and Y = 0.241 against 0.245 with the clock before e± annihilation). **The lower edge of the weak window:** the first minutes leave half their hydrogen only if G₄ > 10^(4.4 ± 1) × G_F/M_W (10^5.43 at a 0.01 s clock, 10^3.43 at 100 s; it moves as clock^(−1/2)). So "as strong as ours at its own scale" is ruled out: it leaves a universe of helium. At the edge G₄m_e³ is 5 × 10⁻¹⁴ – 5 × 10⁻¹², against ours' G_F m_e² of 3 × 10⁻¹²: **4D's weak force must be about as strong as ours at the electron's energies**, which in a gauge theory means a weak scale near 1–30 MeV rather than ours' 80 GeV. There, the neutron lives 1 hour to 2 years and p + p's weak factor is 10⁻⁵–10⁻¹ of ours, so the Sun-pace convention's p + p rate needs C to say whether stars still live long. The upper edge is C's to find. - **A2. The strong force, from a dial to an argument.** The long-range tail from pion exchange in 4D (Yukawa's d-dimensional form); a tensor force; whether 4D nuclear matter saturates without our three-nucleon repulsion; the Brink against ladder disagreement settled. Then the default s = 1.5152 justified or replaced. Weeks, with runner jobs. **First result (2026-11-13, run `masses-1`, `fusion/tools/nuc4.mjs` `MASS`):** on ours against measured masses the ladder underbinds α clusters and Brink's grid overbinds them, more with size, so neither is "the" answer: the adopted model keeps the ladder's nuclei and scales the grid's separations by the two methods' helium-4 separation on the same core (ours to a median 30%), and the grid is the band's other edge. Six helium-4s are bound either way. The one result the band flipped was the CNO-like cycle on ceryllium-8 (5-11 + p → ⁴He + ⁸Ce): open at s ≤ 1.6 in the adopted model, closed in the grid, and ours couldn't decide it. **Decided by a third method (2026-11-13):** a variational upper bound on its Q with every helium-4 and the triton-like cluster free to move (`brink.js` `freeClusterGCM`, `nuclei.mjs qbound`, `fusion/qbound4.json`): ≤ −15.7 to −69.5 MeV over s (ours ≤ 12.25 against 8.59 measured, so the bound holds on ours). No cycle at any strength; the two mass models now agree on every star in the network. Still open in A2: the pion tail, the tensor force, saturation. - **B. Doesn't collapse** (beside A): energy per atom extensive in growing 4D crystals (the plane-wave code) and a Lieb–Thirring-style bound; binding per nucleon levelling off at finite density with A2's force (the liquid drop peaks at mass ~300–2000: checked against runaway giant nuclei); white-dwarf and neutron-star limits and what cores collapse to. Whether pinned gravity lets a universe expand and form structure is recorded as an assumption, not computed. - **C. 4D solar lifetimes** (needs A1): the main sequence with absolute rates and the 4D network's energy release, across G and s: lifetime against mass and the Sun-analogue's, computed. Then core growth, the ielium/ceryllium flash, which masses explode. **First cut (2026-11-13, run `lifetimes-1`, `fusion/eps4.json`, `fusion/tools/lifetimes4.mjs`):** the network's own burning in the main sequence, G 10⁻¹¹–10⁻²³ × s. Long-lived stars are generic (cores taking 10³–10⁵ Gyr at the low-mass end); the Sun analogue (zero-age core t½ = our own zero-age Sun's 11.1 Gyr) sits at 0.2–5 M_Ch for G 10⁻¹⁴–10⁻²⁰. Real 4D cores run hotter than the old Sun-pace convention, so helium-3 stranding at high s gives way to helium as G falls, and s = 1.65 climbs past helium: gravity joins the hydrogen-burning window. The kill criterion's "stars still live long" half holds wherever stars were computed. - **D. Composition over age** (needs C): the first minutes with 4D weak rates; yields per stellar mass; a one-zone galaxy (star formation, a mass function, C's lifetimes, the yields), giving composition against time as a band. This is what step 3 below wants. **Kill criterion:** if no G₄ gives both long-lived stars and hydrogen surviving the first minutes, 4D needs a finely tuned weak force, and that is the result. ## Where we are (done) - The strong force: a stronger one than ours is needed, inside a window set by the deuteron and the diproton, checked with two forces of different shape. - Helium-4 exact; nuclear matter held by the crowd; mass 8 bound and mass 5 unbound, the latter microscopically with the Pauli principle exact, checked on our own helium-5 and lithium-5. - The first minutes stop at helium where the deuteron binds as ours. - Stars climb the helium-4 ladder a rung per step in core mass; the rungs to element 10 are computed, and run at or below the barrier-only temperature. ## The path | step | question | why it's on the path | method | status | |---|---|---|---|---| | 1. Odd against even | Can element 9, and the other odd elements, be made and survive about as easily as pxygen? | Decides the ocean. Ours make fluorine ~30,000× rarer than oxygen: odd elements sit off the helium ladder, and protons destroy fluorine. | A nucleon on a ladder nucleus, microscopically (Brink, as mass 5), checked on ours; then the routes in and out. | done. The "+1" nuclei bind (element 9 at mass 17 by 11–153 MeV over both draws of the core; plusone-1, plusone-2), and proton-in, helium-4-out reactions are uphill by ≥ 10 MeV, so odd elements survive. Where hydrogen remains, ladder nuclei catch protons, not helium-4s (≥ 10⁵×, protonflow), and climb a rung per four protons through the odd elements, as our nitrogen is made; unlike ours the climb has no closed-shell bottleneck (pxygen-20 holds a nucleon by 70–216 MeV, plusone-2). What stops it is the electric barrier; where, is step 2. | | 2. The ash | Does a helium-burning core's product pile up just below the first rung it can't light, whatever the uncertain rates do? And how hot are the zones where hydrogen and helium burn together, which step 1 left to decide element 9 against element 10? | If so, what each core mass makes is robust, and step 3 needs no precise rates; the zone temperatures settle the water. | A network of captures (protons, helium-4s, nuclei by charge) at each temperature, every rate a ratio to proton + proton's so density drops out, checked on our Sun's centre (hburn). | first estimate: in 4D helium-4 + helium-4 is electromagnetic and proton + proton weak, so above ~7–10 MK helium burns as fast as hydrogen and every core that hot burns both. A zone then turns wholly into the element where the protons' climb stops: element 8 at 10–12 MK, 9 at ~12–17, pxygen at ~17–22, 12–15 by 40; below ~8 MK hydrogen goes to helium and helium cores take the ladder by core mass. A core with the Sun's pace runs at 15–21 MK. **Proton + proton is now computed** from the strong force (the deuteron's overlap with the protons' scattering state, the weak conversion ours; tools/ppfusion.mjs, 2026-11-11): where the deuteron binds as ours it equals the old barrier at 3.7 fm, so a Sun-paced core burns at 15–17 MK into element 9 (oitrogen) 36–79% and pxygen 18–64%, by the core's draw. Element 9 isn't rare; which leads goes to step 3. What decides proton + proton now is the strong force's strength: below a deuteron binding of 1.80 MeV (the window's lowest fifth) it makes no deuteron at all, and at the window's top it runs ~10⁷× faster. | | 3. What stars give back | The rough proportions of the commonest elements in the gas planets form from. | The rock's elements, and element 9 against pxygen. | Declared choices, reported as a band: stars that finish before a Sun-like one forms, down to 10³–10⁴ times shorter-lived; our mass function or a top-heavy one; lifetime ∝ M^−2 to M^−3; a fixed share of each star given back as core ash, its make-up from step 2 (giveback). | first estimate, for the hydrogen-and-helium ash: the gas comes back rich in gluorine (4D's fluorine), oeon (its neon) and pxygen; oitrogen at 0–20%, metals a few percent. Pxygen outnumbers oitrogen in every case (oitrogen 0.06–0.41 of it at 4 fm, almost none at 1 fm), so Xater's element leads on abundance; gluorine's hydride, our HF's counterpart, joins the candidates. With the main sequence (starmix): G decides the mix — at 10⁻¹⁴ todium and nagnesium come back with pxygen, gluorine and oeon; at 10⁻²⁰ mostly two of pxygen, gluorine, oeon; pxygen ahead of oitrogen throughout; and below ~10⁻²² of our gravity nothing comes back for tens of billions of years (even the heaviest star light holds, ~7 M_Ch, takes 3.5 × 10¹⁰ years at 10⁻²³): the stars set a floor on G. Open: tilicon, from the heavy cores' later stages (step 4). | | 3b. The main sequence, by G | How bright, big, hot and long-lived is a 4D star of each mass, at each strength of gravity? | G is open, and 4D starlight thins as 1/r³: brightness and G together set where a planet can sit and how long its star lasts, and a star's core temperature sets what it makes. | A main-sequence solver for any d (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; ours reproduced first (mainseq-1). | first estimate: L ∝ M⁵ at the bright end, burning per kilogram ∝ G^(2/3) at a fixed share of M_Ch; 4D opacity worked out (lens-vacuum scattering, and free–free with no Gaunt logarithm, run ×0.5–2 as its band), which dims the faint end 16–40× and halves its surface temperature. A star as long-lived as our Sun is blue-hot (27,000–37,000 K) at G 10⁻¹⁴–10⁻²⁰ of ours, none in the window that brief at 10⁻²³ and below; a sunlight-coloured star lives 10¹⁴–10¹⁶ years; years at 288 K from weeks to ~75 years (mainseq-3). Open: surfaces below ~5,000 K, where 4D's iydrogen recombines. | | 4. Rock | Which rock-forming elements the later stages of stars make (burning after hydrogen and helium are gone, and what massive stars and dead ones throw out), what they make with pxygen and gluorine, and how dense it is. | Density gives G and the planet's size (js/gscale.js). | The ash fusing pairwise, barrier-only checked on our carbon + carbon, with the liquid drop's energies and what the hot product sheds; equilibrium's end at the drop's N = Z peak (heavy). Then the book's Hartree–Fock and crystal search on the abundant elements' compounds. | under way: only cores at the Chandrasekhar mass burn on (the ash fuses at 2.6–9 billion K); the products shed 5–9 nucleons, never a helium-4, and end as elements 13–23, the rock row (todium, bluminium, tilicon, qhosphorus, tulfur, dhlorine). From ~11–16 billion K equilibrium takes over and ends at element 103–137, this world's iron. The ash lights with the electrons far more squeezed than ours at carbon (Fermi/kT 11–115 for cores up to 1.1 M_Ch, 2–5 at 1.5–2; ours 0.5–3) on 6–11 MeV per nucleon, 1.2–40× the core's binding (runaway): cores near M_Ch most likely explode, throwing out the rock row and the iron. An exploding core burns almost wholly (explode: each layer heated by the burning, densities spread 30–100× first as our white dwarfs need, ours checked against Arnett's full-density nickel): to iron (97–100%) up to 1.01 M_Ch, mixed at 1.05, mostly the rock row at 1.1 for oitrogen or pxygen ash (a mix for gluorine or oeon), a few percent unburnt. Which stars get there (cores, from the main sequence's stirred cores, 3–19% of a star, grown ×1.5–2 after; dead stars keep lighter cores, winds return 3–30% of their ash): stars from ~2.8–3.8 M_Ch explode, the first after 18 years at G 10⁻⁸ to 1.8 × 10⁹ at 10⁻²⁰, none within 10 Gyr below ~10⁻²²; the gas gets back this world's iron 10–68% and the rock row 8–68%, the hydrogen-and-helium ash 1–75%: rock isn't short where heavy stars finish. Open: core growth after the main sequence (with none, only stars past the solved 5 M_Ch could explode), and winds. Rock, first estimate: 4D's rock salts computed (tools/salt.mjs, our NaF and MgO within 2–3% in density; doubly charged anions held in a Watson shell): nagnesium pxide 2.3 × 10¹³, todium gluoride 1.8 × 10¹³ kg/m⁴. An Earth-g, Earth-sized planet of that rock needs G = 3.2–4.0 × 10⁻¹⁷ of ours (down to 2 × 10⁻¹⁷ with an iron core), inside the stars' window; there a Sun-long star is 1.1 M_Ch, ~31,000 K, and a 288 K planet sits at 1.1 × 10¹¹ m with a year of ~1.2 of ours. Open: bluminium pxide and tilicon's compounds (structure searches: the ionic model, js/ionic.js with the electron gas between Watson-held ions plus dispersion, finds corundum lowest at two formula units a cell, but at four it ranks a corundum-like arrangement 0.02 eV below it, which periodic PBE puts 0.38 above: the model's energies can be off by ~0.6 eV per formula between arrangements, too much to pick a structure. It proposes well, PBE decides correctly every time, and 4D has no periodic quantum code to decide with. Bl₂Px₃'s best, every Bl 13-coordinated at 3.6 × 10¹³ kg/m⁴, is a candidate, its runner-up 0.36 eV away), this world's iron (elements past 78). | | 5. Ocean and air | Xater, XH₃, hydrogen gluoride, or a molecule without hydrogen (PxGl₂, Gl₂); at what temperature (see the hot-world hypothesis below); what the air is (oeon, a noble gas, is among the commonest elements given back); how dense. | Closes the planet. | Step 3's proportions and the census's chemistry; the liquid simulations; the volatile molecules (air.mjs: H₂ and the He and H₂ pairs by MP2, ours the check). | under way: 4D hydrogen makes a long, weak molecule (3.0 bohr, 1.1 eV; ours 1.4 bohr, 4.4 by the same method); wells by MP2, He–He 61–71 K and H₂–H₂ 48 K (ours 5.1 and 15.5 by the same method, about half of measured). With 4D's simulated critical-point ratios, helium condenses below ~210–1,200 K and hydrogen below ~140–900 K: both straddle 288 K, and oeon is solid. The air is hydrogen and helium near critical, or a surface wet with liquid hydrogen. Next: Gibbs-ensemble runs on these pulls, a larger basis, vapour pressure at 288 K; then which liquid the ocean is. | | 6. The airplane | Lift, drag and thrust in 4D air on that planet. | The first question. | 4D aerodynamics. | | ### Step 5's working hypothesis: a hot world What step 5 has found so far, put together (2026-11-02; each piece first-estimate, the falsifiers below): - **4D hydrogen is weak.** Bonds to it are 0.46–0.80 of ours (darbon's less than half); bonds without it match ours (0.99–1.08). Hydrides are salt-like: Xater behaves like lithium oxide, hydrogens as cations round pxygen. - **4D matter is sticky.** The missing shell theorem pulls neutral molecules together even at the Hartree–Fock level, and 4D atoms are very polarizable: Gl₂ molecules hold each other about 25× harder than F₂'s; element 12 condenses far above our neon; Xater's liquid is held by >100 kcal/mol a molecule. At 300 K nearly everything is solid. - **Xater is a liquid from at most 1500 K to near 9000 K** (on its Hartree–Fock-fitted field, rigid molecules): dense, ionic-like, as fluid at 1500 K as our water at room temperature; its crowd adds nothing (three-body −10 ± 19 kcal/mol). It stops flowing below about 1000 K (caged in 27-ps runs at 500–1000 K, cooled at constant volume). Its crystal search has **not** converged: cells of 4 reach −170.6 kcal/mol a molecule and cells of 6 −168.6, each lowest found once (an earlier note here said it was converging on −168.8; run 45 showed otherwise). Our water's field on the same method condenses at 250 K against 647 measured, so the 9000 K could be low by up to that factor, or not (Xater's correlation weakens its pair). Run 50: three-body from 176 liquid triples is −9 ± 12 / −10 ± 19 kcal/mol a molecule (a few percent); 12000 crystal starts in cells of 4 found 1084 minima, the lowest once; Hartree–Fock on the crystals' closest pairs says the field overbinds them (−59 vs −8 summed over 24), while in the liquid it is unbiased. So the liquid stands to the field's rms; the crystal energies don't. - **It's a hydrogen problem.** Bonds without hydrogen match ours (0.99–1.08); heavy-atom molecules bond to each other (Gl₂ crossed −0.85 eV at 4 bohr and falling; PxGl₂ pairs bound down to Gl–Gl 3.7 bohr). Their condensed phases are likely networks. The MP2-pair → field → liquid chain's OF₂ check: no liquid in the D basis; with the d split (pairs 1.72× deeper) OF₂ condenses but with Tc ≈ 117–124 K against 258 measured (0.45×), as our water on a Hartree–Fock field (0.39×). Fitted pair fields underbind liquids; read their liquids as lower bounds. The well's bottom was fine; the basis wasn't: the book's Dd equals aug-cc-pVDZ, and all 92 pairs redone in aug-cc-pVTZ (PySCF) bind 1.19× deeper. The refitted field's liquid: Tc ≈ 163–178 K (0.63× measured, from 0.45×). The basis limit and CCSD(T) add ~13% more to the pair; the rest is three-body, which no pair field carries. - **Deep burning ends at element 101–136** (NSE, trusted drops; ours, checked, at Ti-44 vs measured Ni-56). - **4D rock stays solid hotter.** 4D's salts bind 10–40% harder than ours; scaled by our MgO and NaF, nagnesium pxide (4D's mantle oxide) melts about 3600–6100 K, ToG (4D's NaF) about 1000–1800 K. - **So: a planet near 1500–2500 K** with solid pxide rock, a molten-Xater ocean, molten salts, and an air of what is still gas there (4D helium, hydrogen partly broken into atoms, perhaps Gl₂ vapour). At G = 10⁻¹⁷ of ours a Sun-long star (1.25 M_Ch, 32,800 K) puts a 2000 K planet at 0.058 AU, 63 star-radii out, with a 7.5-day year; at 10⁻²⁰, 1.6 AU and 210 days. Close in, so tidally locked (the planets chapter: the star then never moves in the sky). What would break it: Xater's crystal melting above about 2500 K (the melting bracket is next); Xater coming apart or sharing protons in its own liquid at those temperatures (rigid molecules can't show it); 4D rock's melting point computed rather than scaled; and the PxGl₂ and Gl₂ liquids (pair scans running) giving a cooler ocean instead. Step 1 comes first because it is cheap beside everything after it, it uses tools just built, and on its own it may decide the ocean. If element 9 turns out easy to make, chemistry decides instead (the census has XH₃ ahead by its bond), and step 1 still says so. Steps 4 and 5 can start in parallel as soon as step 3 names even a rough top five. ## Off the path, on purpose Where the most massive stars stop (4D's iron) matters for a planet's core; the stages before it now matter for the rock (step 4). The first minutes' clock and gravity's law: G comes out of step 4. The spin–orbit force in mass 5.