The theory, explained four ways

The fair way to compare two masses, whatever units or scale you use, is the relational distance

Only this form stays the same if every mass is multiplied by the same number, so it is the unique scale-fair distance (record M3). Applied to the 19 known particles it gives 171 pairs, plus the antiparticle and decay matrices (download the workbook). For the electron, muon and tau, the relationships form one 3×3 matrix with equal "self" parts, links of 1/√2 and a twist of 2/9. That single matrix gives all three masses from one scale, to 0.003%.

The lepton matrix is a circulant square-root mass matrix. It is equivalent to Koide's relation (1981) in Foot's geometric form and Brannen's parametrisation, and here it is read as relations between generations. The same matrix, applied to neutrinos, predicts normal ordering with m₁ = 0.36, m₂ = 8.7, m₃ = 50.1 meV. A philosophy of relations first also appears in Rovelli's relational quantum mechanics (1996).

Measure the universe in proton units. Its mass is about 5.5 × 10⁷⁹ protons, it is about 6.5 × 10⁴¹ proton lengths across, and gravity is weaker than the quantum scale by about 1.7 × 10³⁸. Every particle's view gives its own set of numbers; the Scale_Views sheet in the workbook lists them all.

These are the "large-number coincidences" noticed by Eddington (1930s), Dirac (1937) and Weinberg (1972). They are real and still unexplained. The scale-view idea says why the proton's numbers are the ones we notice. What remains to be tested is whether the proton is special or one rung on a smooth ladder.

"Always full" means the shares add up to one: Ω_matter + Ω_radiation + Ω_Λ = 1. Put that into the expansion equation and the age of the universe comes out at 13.79 billion years (measured 13.787 ± 0.020).

This is the flat Friedmann–Lemaître model with Planck 2018 parameters. Treating the vacuum as a medium has a real research line (Volovik, The Universe in a Helium Droplet, 2003; analogue gravity, Barceló, Liberati and Visser).

For stars the meter is compactness, C = GM/(Rc²): the Sun is about 0.000002, a neutron star about 0.17, the last stable star 4/9, a black hole 1/2. For the proton the balance is the bag pressure B against the quarks' motion. Fitting only the measured proton and Delta masses gives B = 60 (57 to 64) MeV/fm³, a quark core of 0.725 fm, and, carried to neutron stars, stars of 2.0 to 2.2 solar masses that pass the gravitational-wave test, provided quarks in their cores pair up with a gap of 70 to 90 MeV.

Stars: Tolman–Oppenheimer–Volkoff equations, Buchdahl's limit. Proton: the MIT bag model (Chodos, Jaffe, Johnson, Thorn and Weisskopf, 1974) with the cloudy-bag pion cloud (Théberge, Thomas and Miller, 1980). The pairing gap is colour superconductivity (Alford, Rajagopal and Wilczek; Rapp, Schäfer, Shuryak and Velkovsky, 1998). The measured pressure inside the proton (Burkert, Elouadrhiri and Girod, 2018) shows the push-pull balance directly.

This adds explanation to the containment picture; it is not separate physics. Fusion in the gas is measured (Borexino solar neutrinos), and a small dense central object is not ruled out (studies of "Hawking stars", Bellinger and others, 2023). The test is whether helioseismology near the core shows its fingerprint.

A resistance n = 1 + 2GM/(rc²) around a mass bends starlight passing the Sun by 1.752″ (measured 1.75″) and gives the radar echo delay exactly. If the resistance acted only on time, the bending would be half (0.876″); if it had no direction, there would be none. So it must act on time and space equally.

This is the optical-metric form of weak-field general relativity (Eddington; Gordon), tested by Shapiro's radar delay and Cassini. Feynman, Weinberg and Deser showed that a long-range, universal, tensor field must be Einstein's gravity, so the factor 2 is general relativity's.

Entangled pairs give the correlation E(a,b) = −cos(a − b), stronger than any hidden-note model allows. In a filled medium, entanglement across a boundary grows with the boundary's area (checked on a grid; 80 times better fit than volume). Add the rule heat = temperature × entropy at every horizon, and Einstein's equations follow. Uncertainty is a property of waves: Δx·Δk ≥ ½.

Schrödinger (1935); Einstein, Podolsky and Rosen; Bell; Aspect, Clauser and Zeilinger. Gravity from entanglement: Jacobson (1995), Ryu and Takayanagi, Van Raamsdonk (2010). Touching through the geometry: ER = EPR (Maldacena and Susskind, 2013), which sends no signal faster than light. The area law: Bombelli, Koul, Lee and Sorkin; Srednicki.

The simplest crossing wave, z = cos x + cos y, has one peak, two saddles and one trough in every cell. Its total curvature over a cell is exactly zero, so it is flat on average, as Planck measures for our universe.

The zero total curvature is the Gauss–Bonnet theorem; counting peaks, saddles and troughs is Morse theory.

Homogeneous expansion with no special centre is the Friedmann–Lemaître–Robertson–Walker picture. Universes that begin as a bounce inside a black hole have been proposed (Popławski, 2010).