What We Observe but Do Not Understand
Every chapter of this treatise has drawn a hard line: physics enters only once Nature has voted, through a reproducible measurement. This closing chapter collects what stands on the far side of that line — established observational facts with no confirmed theoretical explanation — and, separately, names the theoretical programs sometimes proposed to explain them that have not yet crossed it. Every quantity below carries its measured value in The Free Parameters of Physics or the relevant experiment chapter; nothing here restates a number already given precisely elsewhere.
The mass and mixing puzzles
The Standard Model successfully accommodates, but does not explain, three tiers of numbers cataloged in The Free Parameters of Physics:
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The charged-lepton and quark masses span more than five orders of magnitude between the electron and the top quark, with the generation-to-generation ratios (Cabibbo angle) fixed by Yukawa couplings the theory does not predict — the flavor puzzle.
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The strong CP problem (QCD vacuum angle (strong CP problem) (open)): the QCD Lagrangian permits a CP-violating vacuum angle \(\bar\theta_{\mathrm{QCD}}\) of order unity, yet the neutron electric dipole moment bounds it to \(\lesssim 10^{-10}\). Nothing in the confirmed Standard Model forces this suppression.
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Neutrino oscillations establish that neutrinos are massive and mix (Neutrino Oscillations and the PMNS Matrix), but oscillation data alone fix only mass-squared differences, never the absolute mass scale, and whether the neutrino mass term is Dirac or Majorana — decidable only by a direct observation of neutrinoless double beta decay, not yet seen — remains open (Absolute neutrino mass scale and Dirac/Majorana nature (open)).
The hierarchy problem
The Planck scale, fixed purely by dimensional analysis from \(\hbar\), \(c\), \(G\) (Proposition B.1), sits some seventeen to twenty-two orders of magnitude above every confirmed particle-physics mass scale (Gravitational Coupling, Planck Units, and the Hierarchy Problem, entries on \(m_e/M_{\mathrm P}\), \(m_p/M_{\mathrm P}\), \(v/M_{\mathrm P}\)). For the Higgs vacuum expectation value specifically, this separation is a genuine puzzle rather than a mere curiosity: unlike a fermion mass, a fundamental scalar's mass is not protected by any confirmed symmetry from radiative corrections that scale quadratically with whatever ultraviolet physics completes the Standard Model. If that completion sits near the Planck scale, those corrections are technically expected to dwarf the observed electroweak scale by many orders of magnitude, demanding an unexplained, extremely precise cancellation to keep \(v\) as small as it is measured to be. No confirmed dynamical mechanism protecting the electroweak scale in this way has been observed.
Dark matter
The gravitational evidence for non-baryonic matter is overwhelming and multiply cross-checked: galactic rotation curves, gravitational lensing by galaxy clusters, and the acoustic peak structure of the cosmic microwave background all independently require a cold dark matter density \(\Omega_c h^2 = 0.1200(12)\) (Cold dark matter density parameter), roughly five times the baryon density. What dark matter is — a new particle, and if so with what mass and couplings — is not established by any confirmed direct-detection, collider, or indirect-detection signal to date. This treatise records the gravitational evidence as settled and the particle identification as open.
Dark energy and the Hubble tension
The accelerating expansion of the universe is established from Type Ia supernova distances, baryon acoustic oscillations, and the CMB angular-diameter distance, all converging on \(\Omega_\Lambda = 0.6847(73)\) (Dark energy density parameter). Its microphysical origin — a true cosmological constant, vacuum energy, or a dynamical field — is unidentified. A second, independent tension sharpens the puzzle: the Hubble constant inferred from the CMB, \(H_0 = 67.36(54)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\), disagrees with late-universe distance-ladder measurements, \(H_0 \approx 73.0(1.0)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\), at the \(4\)–\(5\sigma\) level (Hubble constant (open)). Whether this Hubble tension signals unrecognized systematics or a genuine incompleteness of the base \(\Lambda\)CDM model is not yet resolved.
The baryon asymmetry
The observed universe is made overwhelmingly of matter, with essentially no primordial antimatter, quantified by the baryon-to-photon ratio \(\eta = 6.12(4)\times 10^{-10}\) (Baryon-to-photon ratio) — independently confirmed by Big Bang nucleosynthesis and the CMB. Generating any matter-antimatter asymmetry from an initially symmetric state requires baryon-number violation, C and CP violation, and a departure from thermal equilibrium (the Sakharov conditions). The Standard Model contains CP violation (CKM CP-violating phase), but its magnitude, via the Jarlskog invariant (Jarlskog invariant), is many orders of magnitude too small to account for the observed \(\eta\). No confirmed mechanism supplies the shortfall.
Confinement and the Yang–Mills mass gap
No isolated quark or gluon has ever been observed, in accelerator debris, in matter, or in cosmic rays, although the \(SU(3)\) gauge Lagrangian of Quantum Chromodynamics contains only those fields; and the observed hadron spectrum begins at the pion mass rather than at zero, although the gluons of that Lagrangian are massless. Both statements are experimentally secure and neither has been derived from the theory. Lattice computation reproduces the hadron spectrum and exhibits the confining area law at strong coupling (Quantum Chromodynamics), but a proof from the continuum theory — that four-dimensional Yang–Mills theory exists as a mathematically well-defined quantum field theory and that its spectrum has a mass gap \(\Delta > 0\) — is missing. The statement is one of the recognized open problems of mathematical physics, posed as a Clay Millennium Prize problem [Jaffe:2006]. What is open here differs in kind from the other entries of this chapter: the physics is not in doubt and no new particle or parameter is called for; what is absent is the derivation that the editorial rules of this treatise would demand of any other observed phenomenon, which is why the QCD chapter carries it as a named, pending derivation rather than as a result.
What lies outside this treatise
The puzzles above are observational facts awaiting explanation; they are evidence-based by construction, since each is a measured discrepancy or absence rather than a theoretical assertion. Several theoretical programs have been proposed, at various times, to address one or more of them: supersymmetry (a protective symmetry for the hierarchy problem), the Peccei–Quinn mechanism and its axion (for the strong CP problem), grand unified theories (for the apparent convergence of gauge couplings at high energy), string theory and its variants (for a unification including gravity), and speculative approaches to quantum gravity such as loop quantum gravity or asymptotic safety. As of this writing, none of these programs has direct experimental confirmation: no supersymmetric partner has been observed at any collider to date, no axion has been detected by any haloscope or helioscope search, proton decay has not been observed at the rates generic GUTs predict, and string theory makes no confirmed, falsified, or falsifiable prediction yet tested against data. Per the scope rule stated in Epistemology and the Scientific Method, this treatise accordingly excludes them from its physics: they are named here only so that the boundary of current evidence is stated honestly, not silently. Should any of them — or some other mechanism — acquire direct experimental support, it belongs in a future edition of this chapter, exactly where the puzzle it resolves is recorded above.
A second category needs distinguishing from the first, because the two are easily confused and the scope rule treats them differently. The programs named above are candidate physical theories without evidence. Alongside them runs a large body of work that is mathematics about structures physics uses — classifying the symmetry algebras a spacetime could carry, extending them, and building gauge theories from the extensions. That work is not excluded from this treatise: its results are theorems, they are true whether or not nature uses them, and the treatise states them as such. The classification of kinematical algebras (Theorem 14.90) is in the book for exactly this reason, together with an explicit column recording which of the eleven possibilities a measurement has ever probed (Table 14.1) — three of them.
What the treatise declines to do is let the second category be read as the first. An algebra that admits a consistent extension, a Lagrangian that can be written for it, and a gauge theory that can be constructed from it are three mathematical facts and zero physical ones. Current examples include the Maxwell and non-Lorentzian extensions of the kinematical algebras and the three-dimensional Chern–Simons gravities built from them [Concha:2026]; gravity in three spacetime dimensions has no propagating degrees of freedom, no such theory has an observational test, and the papers concerned claim none. They are cited where their mathematics is used and nowhere else. The rule this implements is the one of Epistemology and the Scientific Method: a derivation is not evidence, and an existence proof for a structure is not a measurement of it.