Scorecard
Green: derived from the axioms with specific, checkable predictions. Amber: mechanism derived with quantitative open questions remaining.
Eight derived. Two with mechanisms derived and quantitative details open. All from four axioms about the vacuum's entropy field.
These are not ten separate solutions. The same ℤ3 topology that solves the strong CP problem also produces three generations, the fermion masses, and the Higgs mass. The same Projection Theorem that resolves the information paradox also derives the Born rule and the arrow of time. The same entropy field that replaces dark matter also recovers general relativity, derives Newton's constant, and produces the dark energy fraction to within 1% of observation.
ℤ3 topology that solves the strong CP problem also produces three generations, the fermion masses, and the Higgs mass. The same Projection Theorem that resolves the information paradox also derives the Born rule and the arrow of time. The same entropy field that replaces dark matter also recovers general relativity and derives Newton's constant.The Problems
1. Quantum Gravity
DerivedGeneral relativity describes the very large — planets, stars, galaxies. Quantum mechanics describes the very small — atoms, particles, forces. Both work spectacularly well in their own domains, but they are mathematically incompatible. Unifying them has been the central problem of theoretical physics for nearly a century.
They don't need to be unified — they were never separate. Both are outputs of the same projection. The faithful part of the projection (where no information is lost) produces general relativity: the field equations, the Einstein limit, geodesic motion. The non-faithful part (where information IS lost) produces quantum mechanics: interference, the Born rule, the uncertainty principle, measurement. They are two faces of one process, not two theories to be reconciled.
2. Dark Matter
DerivedGalaxies rotate too fast. The visible matter in a galaxy doesn't have enough gravitational pull to hold it together at the speeds observed. Physicists hypothesise an invisible "dark matter" making up 27% of the universe. Despite decades of searching, it has never been directly detected.
There is no dark matter. The vacuum's entropy field produces an additional gravitational effect that looks exactly like dark matter but requires no invisible particles. The flat rotation curves of galaxies emerge geometrically from the PLES equation in disc geometries. Tested against 55+ galaxies from the SPARC database with a median fit quality of χ² = 0.5 — better than most dark matter models — using one universal scale parameter derived from the de Sitter horizon.
3. Dark Energy
DerivedThe expansion of the universe is accelerating. Something is pushing spacetime apart, and it accounts for about 68% of the universe's energy content. Nobody knows what it is. Two separate mysteries: why is the cosmological constant so extraordinarily small (120 orders of magnitude smaller than quantum field theory predicts), and why is its energy density comparable to the matter density right now (the coincidence problem)?
The dark energy is not a cosmological constant — it is the residual potential energy of the same entropy field that produces galaxy rotation curves. The cosmological constant's extreme smallness is decomposed into two independently small, kernel-determined ratios: ζSM ≈ 2×10−6 (the vacuum's intrinsic modular weakness) and ρcrit/(3ρ0) ≈ 3×10−6 (cosmic dilution below the screening threshold). The coincidence problem is resolved structurally: the entropy field freezes during radiation and matter domination and thaws when the Hubble rate drops to the cosmological entropy mass mcos = H0 — which is the present epoch by a structural identity, not by tuning. The dark energy fraction is derived from the ℤ3 landscape: the maximum displacement within one vacuum sector is one ℤ3 period (2π/3), giving ΩΛ = 2π²/27 ≈ 0.731, reduced by thawing to ΩΛ ≈ 0.69 — within 1% of the observed 0.685, with zero free parameters. The prediction wa > 0 (thawing-class equation of state) distinguishes MEG from a true cosmological constant and is testable by stage-IV surveys.
4. Matter-Antimatter Asymmetry
Mechanism DerivedThe Big Bang should have created equal amounts of matter and antimatter, which would have annihilated each other completely. Yet the universe is made almost entirely of matter. Where did all the antimatter go?
Matter and antimatter correspond to opposite-sign topological defects of the entropy field in the ℤ3 vacuum. The PLES functional's coherence stiffness (χ > 0) makes the symmetric state unstable: sign-polarised vacua (all-matter or all-antimatter) have lower entropic stress than mixed-sign vacua. Below a critical defect density, the symmetric state spontaneously breaks. A small CP-violating bias — derived from the same ℤ3 tunnelling that produces the CKM matrix (within 0.12°) — selects the matter branch. No additional fields or beyond-Standard-Model interactions are required. The quantitative baryon-to-photon ratio ηB remains to be computed from the freeze-out dynamics.
5. The Hierarchy Problem
DerivedGravity is 1036 times weaker than the other forces. In quantum field theory, the Higgs boson's mass should be pushed up to the Planck scale by radiative corrections, making the electroweak scale unnaturally small. Why is the Higgs mass 125 GeV and not 1019 GeV?
The electroweak hierarchy is derived with no free parameters: ln(μGUT/vEW) = 72π²ln3/25 ≈ 31.23, giving vEW = 246.8 GeV. The hierarchy is topological — it comes from ℤ3 winding in the vacuum — not radiative, so it is naturally stable against quantum corrections. The Higgs mass is predicted in the window 124.9–128 GeV (measured: 125.25 GeV).
6. The Black Hole Information Paradox
DerivedWhen something falls into a black hole, quantum mechanics says the information about it must be preserved. But Hawking showed that black holes radiate thermally, which seems to destroy information. This contradicts a fundamental principle of quantum mechanics.
Information is never destroyed — it lives in the non-faithful channel of the projection. At the event horizon, the projection transitions from faithful to non-faithful: the deficit at the causal boundary IS the Hawking spectrum. The Bekenstein-Hawking entropy formula S = A/4 is derived non-circularly, and the entropy field universally saturates at S = ½ at the horizon.
7. The Strong CP Problem
DerivedThe strong nuclear force has a parameter θ̄ that could violate the symmetry between matter and antimatter. Experiments show |θ̄| < 10−10, but there is no known reason why it should be so small. Most proposed solutions require a new particle (the axion) that has never been found.
θ̄ = 0 exactly, from two independent arguments. First, the QCD vacuum angle vanishes at every ℤ3 minimum. Second, the quark mass matrices are Hermitian (they are tunnelling Hamiltonians), so their determinants are real. No axion is needed. The solution comes from the same ℤ3 topology that produces three generations.
8. QCD Mass and Confinement
Mechanism DerivedProtons and neutrons are made of quarks, but the quarks account for less than 2% of the proton's mass. The rest comes from the energy of the strong force binding the quarks together. Proving that QCD confines quarks and generates a mass gap is one of the Clay Millennium Prize problems.
Confinement is derived from the ℤ3 kink topology: quarks are confined because a coloured asymptotic state would break the ℤ3 vacuum democracy, and isolated colour charges are connected by flux tubes of wrong-vacuum whose domain wall tension produces a linear confining potential. The derived string tension is √σ ≈ 440 MeV, consistent with the observed QCD value. The Yang-Mills mass gap is derived separately. The quantitative confinement scale ΛQCD from dimensional transmutation of MEG's derived ultraviolet coupling (g² = 1/3) lands in the right neighbourhood, with a single group-theoretic normalisation factor (the SO(8) → SU(3) embedding index) as the residual gap. The barrier is a deferred computation, not a structural wall.
9. Grand Unification
DerivedElectromagnetism, the weak force, and the strong force are described by three separate mathematical structures. Physicists have long sought a single framework that unifies all three. Traditional approaches predict proton decay at rates not observed.
The three forces are not unified at a high energy scale — they are unified at the level of the vacuum kernel. The gauge group SU(3)×SU(2)×U(1) is derived from the SO(8) triality structure, which itself is derived from the axioms' two compact cycles. The ten-link chain from axioms to the QED Lagrangian closes with no open cases. The Standard Model's ~20 free parameters — fermion masses, mixing angles, the Higgs mass — become outputs.
10. The Nature of Time
DerivedThe laws of physics are symmetric in time — they work equally well forwards and backwards. Yet time clearly has a direction: eggs break but don't unbreak. Why does time have an arrow?
The arrow of time is the fifth corollary of the Projection Theorem. The deficit — the information the projection loses — grows monotonically under forward evolution. Time moves forward because the projection's non-faithfulness only increases.