Reference

The Math Ledger

Every mathematical object in the book: where it was defined, what it was defined for, and everywhere it gets spent.

How to use this

When a symbol turns up three parts later and you want to know where it came from, this is the index. The middle column is the important one — it records the question each object was invented to answer, because that is what you actually need to remember. Entries in grey are planned, not yet built. The ledger grows with every batch.

Part 0 · The Toolkit

Object	Introduced — and what for	Spent in
Limit $\lim_{h\to0}$	0.1 — to make sense of "the rate right now", which naive algebra returns as $0/0$ .	Everywhere. Explicitly re-examined in 4.3 (limits of functions behave worse than limits of numbers).
Derivative $f'(a)$ , $\dv{f}{x}$	0.1 — the coefficient of the best linear approximation: $f(a+h)=f(a)+f'(a)h+o(h)$ . Defined this way rather than as a slope precisely so it survives leaving one dimension and leaving flat space.	0.6 (total derivative = a matrix), 1.2 (varying a whole path), 3.2 (tangent spaces), 3.3 (covariant derivative), 5.7 (functional derivative).
Chain rule $(f\circ g)'=f'(g)\,g'$	0.1 — because linear approximations compose, and linear maps compose by multiplication.	0.6 (product of Jacobians), 2.4 (the transformation law that defines a tensor), 3.2 (changing charts on a manifold).
$\ee$ and the exponential $\dv{}{x}\ee^{x}=\ee^{x}$	0.1 — the unique base for which a function is its own derivative; forced, not chosen. The numerical value is a consequence.	0.8 (every linear ODE), 0.9 (Fourier kernel), 4.6 ( $\ee^{-\ii\hat Ht/\hbar}$ ), 5.8 (the Dyson series), 6.1 (the exponential map from a Lie algebra to its group).
Second derivative as curvature of the approximation $\tfrac12f''(a)h^{2}$	0.1 — the leading error left over by linearizing; noticed numerically before it was derived.	0.3 (Taylor), 0.6 (Hessian, classifying stationary points), 1.2 (second variation and whether an action is minimised), 3.4 (Riemann curvature is the same idea one level up).
Definite integral $\int_a^b f\,\dd x$	0.2 — to recover an accumulated total from a rate; defined as a limit of tagged Riemann sums, with the integrability condition stated honestly.	1.2 (the action $S=\int L\,\dd t$ ), 3.6 (the Einstein–Hilbert action), 5.6 (the path integral, where the thing being summed over is an entire history).
Fundamental Theorem of Calculus	0.2 — the discovery that a local statement (derivative) and a global one (total accumulation) are the same statement.	0.7 (Green, Stokes, divergence — all the same theorem in higher dimensions), 3.5 (the generalised Stokes theorem on manifolds), 5.2 (local conservation ⇒ global conserved charge).
Integration by parts	0.2 — the product rule run backwards; the single most-used manipulation in theoretical physics.	0.3 (Taylor's theorem with remainder), 1.2 (Euler–Lagrange), 5.2 (field equations from a Lagrangian density), 3.6. Its boundary terms are physics, not bookkeeping.
Gaussian integral $\int\ee^{-ax^{2}}\dd x=\sqrt{\pi/a}$	0.2 — derived by squaring and going to polar coordinates, because $\ee^{-x^{2}}$ has no elementary antiderivative and the trick is the only way in.	0.9 (Fourier transform of a Gaussian), 4.6 (normalising a wave packet), 5.7 (the whole perturbative expansion of QFT is Gaussian integrals with corrections), 5.6.
Parameter differentiation $\pdv{}{a}\int\!\ldots$	0.2 — to generate whole families of integrals from one solved case, by differentiating with respect to a constant.	5.8 — where the constant becomes a source $J$ and the technique becomes the generating functional $Z[J]$ , from which every Feynman diagram is extracted.
Taylor series + remainder	0.3 — to keep going past the linear term, with an error term, because a series without an error estimate is a wish.	1.2, 2.5 (recovering Newton from Einstein), 4.10 (the classical limit), 5.8 (perturbation theory), 7.1 (power counting).
Euler's formula $\ee^{\ii\theta}=\cos\theta+\ii\sin\theta$	0.3 — by substituting an imaginary argument into the exponential series; establishes that multiplying by $\ee^{\ii\theta}$ is a rotation.	0.8, 0.9, and all of Parts IV–VII. It is why the $\ii$ in the Schrödinger equation converts decay into oscillation, and why U(1) phase is the seed of electromagnetism (6.3).
Radius of convergence / analyticity	0.3 — to know when an expansion is entitled to be trusted; the surprise being that singularities off the real axis control behaviour on it.	5.11 (pole and branch-cut structure of amplitudes), 6.5.
Asymptotic series	0.3 — because the most accurate predictions in physics come from series that diverge, and using them correctly is a skill, not a lapse.	5.11 (renormalised perturbation theory), 6.5.
Non-analytic terms $\ee^{-1/g^{2}}$	0.3 — smooth but with an identically-zero Taylor series; invisible to every order of perturbation theory.	6.5 — instantons and confinement live exactly in this invisible sector.
Dimensional analysis	0.3 — to get the form of an answer before doing the calculation, and to know what a calculation cannot tell you.	5.11 (which couplings are relevant), 7.1 (the Planck scale is the only length buildable from $G$ , $\hbar$ , $c$ — which is why quantum gravity has a built-in scale).
Small oscillations $V\approx V_0+\half V''(x_0)\,\delta x^{2}$	0.3 — every stable system near equilibrium is a harmonic oscillator, because the linear term vanishes at a minimum.	0.8, 4.8, 5.3, 7.4. This one fact is why quantum field theory is built out of oscillators.
Vector space	0.4 — because "things that add and scale" is a structure shared by arrows, polynomials, solutions of a linear ODE, and quantum states. "Vector" means element of a vector space; it never meant arrow.	0.8 (superposition of ODE solutions), 4.2 (states), 5.3 (field configurations).
Basis and coordinates	0.4 — to describe a vector by numbers, while keeping straight that the vector is the object and the coordinate list is a description relative to a choice.	2.4 (the entire tensor formalism is this distinction, made systematic), 3.2 (charts), 4.2 (representation of a state).
Linear map, and its matrix	0.4 — a map determined entirely by its action on a basis. Matrix multiplication is derived from composition of maps, not adopted as a convention.	0.6 (the Jacobian is one), 4.2 (observables are operators), 6.2 (representations).
Change of basis $A' = P^{-1}AP$	0.4 — to separate what is physical (the map) from what is conventional (the numbers describing it).	2.4 (an object transforming this way is a tensor), 3.2, 6.2.
Determinant $\det A$	0.4 — built from multilinearity, alternation and normalisation, which are exactly the properties of signed volume. $\det(AB)=\det A\det B$ is then one line: scaling factors multiply.	0.6 (change of variables), 3.3 ( $\sqrt{-g}$ , the volume element of curved spacetime), 5.7 (Gaussian integrals in many dimensions).
Trace $\det(I+\epsilon A)=1+\epsilon\,\mathrm{tr}A+O(\epsilon^{2})$	0.4 — trace is the infinitesimal version of determinant. Basis-independent, because $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ .	0.7 (divergence is the trace of the Jacobian), 1.3 (Liouville's theorem is this identity), 6.1 (why SU( $N$ ) generators are traceless).
Non-commutativity $AB\neq BA$	0.4 — noticed here as the plain fact that composing transformations in the other order gives a different transformation.	4.9 (the commutator is the uncertainty principle), 3.4 (curvature is a commutator of covariant derivatives), 6.4 (gluon self-interaction).
Inner product and Cauchy–Schwarz	0.5 — to give a vector space a notion of length and angle, and hence of overlap between states.	4.9 — Cauchy–Schwarz becomes the Heisenberg uncertainty principle with nothing added but physical interpretation. Also 0.6 (steepest ascent), 0.9, 4.2.
Orthonormal basis, Gram–Schmidt, completeness $\sum_i\ket{e_i}\bra{e_i}=\hat I$	0.5 — because in an orthonormal basis coordinates are just inner products. Dirac notation introduced here, in pure linear algebra, so it isn't mysterious later.	0.9 (Fourier series is this in an infinite-dimensional space), 4.2, 4.3.
Orthogonal projection	0.5 — the nearest point in a subspace; equivalently, least squares and the normal equations.	4.2 (measurement projects the state), 4.19 (density matrices).
Adjoint, Hermitian, unitary	0.5 — defined basis-free by $\avg{Au,v}=\avg{u,A^{\dagger}v}$ , then shown to be the conjugate transpose in an orthonormal basis. Unitary maps preserve inner products.	4.2 (observables, and §7 — time evolution is unitary because probability is conserved), 6.1.
Spectral theorem $A=\sum_i\lambda_i\ket{e_i}\bra{e_i}$	0.5 — the centre of Part 0. Hermitian ⇒ real eigenvalues, orthogonal eigenvectors, complete basis. Proved with no physics anywhere in sight.	4.2 — where these three theorems are simply renamed: real measurement outcomes, distinguishable outcomes, superposition. Also 0.6 (Hessian), 4.7, 4.8, and PCA.
Functions of operators $\ee^{\ii A}$ unitary for $A$ Hermitian	0.5 — defined through the spectral decomposition; the finite-dimensional shadow of the exponential map.	4.6 ( $\ee^{-\ii\hat Ht/\hbar}$ — a Hermitian energy generating unitary evolution), 6.1 (Lie algebra → Lie group).
Commuting ⟺ simultaneously diagonalisable	0.5 — proved in both directions, degeneracy handled.	4.9 (incompatible observables); and 4.2 §4.3, which defines a complete set of commuting observables — the list of eigenvalues that is a set of quantum numbers — for 4.11 and 4.13 to spend.
Total derivative $\mathrm{D}f(\vv a)$ , the Jacobian	0.6 — Chapter 0.1's equation with $h$ a vector and the coefficient a linear map. The promised payoff for not defining the derivative as a slope.	1.2, 2.4, 3.2, 5.7. Partials existing is not enough — 0.6 gives the counterexample.
Gradient $\nabla f$	0.6 — the vector representing $\mathrm{D}f$ ; steepest ascent derived from Cauchy–Schwarz, and $\nabla f\perp$ level sets.	0.7, 1.1, 3.6.
One-form / covector $\dd f$ , lower indices	0.6 — the honest object. $\partial_i f$ carries a lower index naturally; turning it into a vector, $(\nabla f)^i=g^{ij}\partial_j f$ , requires a metric. In Euclidean space you never notice. In relativity you always do.	2.4, 3.2, 3.5. This is where the upper/lower index distinction is born.
Hessian	0.6 — the second-order term of the multivariable Taylor expansion. Symmetric, so 0.5's spectral theorem classifies critical points by its eigenvalues.	0.8 (normal modes are eigenvectors of the Hessian), 1.3 (stability), 6.6 (the shape of the Higgs potential).
Lagrange multipliers	0.6 — derived geometrically: at a constrained extremum $\nabla f$ has no tangential component, so $\nabla f=\lambda\nabla g$ . The multiplier is a sensitivity, not a bookkeeping device.	1.2, 1.3 (constraint forces), and gauge constraints later. Used in 0.6 itself to derive the Boltzmann distribution.
Jacobian determinant $\dd V\to\abs{\det \mathrm{D}\Phi}\,\dd V$	0.6 — because the map is locally linear (0.6 §2) and a linear map scales volume by its determinant (0.4 §5). Pays off Chapter 0.2's unjustified $r\,\dd r\,\dd\theta$ .	3.3 ( $\sqrt{-g}$ is this factor), 5.7 (changes of variable inside a path integral).
Vector field	0.7 — a vector attached to every point, which is a different object from a vector.	2.6 onward, everything. Becomes a section of the tangent bundle in 3.2.
Line integral, conservative field, potential	0.7 — work along a path; then the equivalence $\vv F=\nabla\phi \iff$ path-independence $\iff \oint=0$ , valid only on a simply connected domain.	1.1, 2.6, 6.3. The topological exception is the Aharonov–Bohm effect: the potential carries physical information the field strength does not.
Divergence $\nabla\cdot\vv F$	0.7 — defined first as flux per unit volume, then shown to be the trace of the Jacobian, hence (via 0.4) the fractional rate of volume change of a blob carried by the flow.	1.3 (Liouville's theorem becomes a one-liner), 3.6, 4.6, 5.2.
Curl $\nabla\times\vv F$	0.7 — circulation per unit area; structurally, the antisymmetric part of the Jacobian. Repackaging it as a vector works only in three dimensions.	2.6 — in four dimensions the antisymmetric object stays a matrix, which is exactly why the electromagnetic field is the tensor $F_{\mu\nu}$ and not a vector.
Green / Stokes / divergence theorems $\int_M\dd\omega=\oint_{\partial M}\omega$	0.7 — proved by the same interior-face cancellation that gave the FTC in 0.2. They are not three theorems; they are one theorem in different dimensions.	3.5 (made precise with differential forms), 5.2 (local ⇒ global conservation).
Continuity equation $\pdv{\rho}{t}+\nabla\cdot\vv J=0$	0.7 — what "conserved" means locally: the amount inside changes only by flowing through the boundary.	4.6 (probability current), 5.2 (Noether currents), 3.6 ( $\nabla_\mu T^{\mu\nu}=0$ ).
$\nabla\times\nabla\phi=0$ , $\nabla\cdot(\nabla\times\vv A)=0$	0.7 — from equality of mixed partials. The second is why $\nabla\cdot\vv B=0$ permits a vector potential at all.	3.5 — both are shadows of the single statement $\dd^{2}=0$ . Also 2.6, 6.3.
Laplacian, Poisson equation $\nabla^{2}\phi=-\rho/\epsilon_0$	0.7 — the Laplacian measures how much a field at a point differs from its average on a small surrounding sphere.	3.6 (the Newtonian limit of Einstein's equations is Poisson's equation), 4.6 (the Schrödinger equation's kinetic term is a Laplacian).
Linear ODE as an eigenvalue problem	0.8 — the solution set is a vector space (0.4) whose dimension is the order; $\ee^{\lambda t}$ works because the exponential is the eigenfunction of $\dd/\dd t$ . Repeated roots are the Jordan block of 0.5's warn box.	0.9 (Fourier is this observation, industrialised), 4.6 to 4.8 (energy eigenstates), 5.3 (field modes).
Harmonic oscillator $\ddot x+\omega^{2}x=0$	0.8 — solved three ways; phase-space ellipse; the shape every stable system takes near equilibrium (0.3).	4.8 (ladder operators), 5.3 (particles as excitations), 7.4 (string modes). The most reused equation in the book.
Damping, $Q$ , resonance, the Lorentzian	0.8 — the driven damped oscillator's power response, with FWHM $=2\gamma$ exactly.	5.9 — the Breit–Wigner lineshape of an unstable particle. A resonance bump in a cross-section is a driven damped oscillator, and its width is its inverse lifetime.
Normal modes	0.8 — eigenvectors of a symmetric matrix (0.5) that decouple a coupled system; identically, the principal axes of the potential's Hessian (0.6). Beats are two modes interfering.	5.3, 7.4.
Wave equation, and $N\to\infty$	0.8 — $N$ coupled masses give $N$ modes; let $N\to\infty$ and the chain becomes continuous. A field is the infinite- $N$ limit of coupled oscillators.	2.1 (Maxwell and the appearance of $c$ ), 5.3 — quantising a field is quantising infinitely many oscillators. The spine of Parts V–VII.
Nonlinearity	0.8 — superposition is a privilege of linear equations, and most of the interesting physics is not linear.	3.6 (gravity gravitates), 6.4 (gluons carry colour), 5.8 (which is why perturbation theory is the only general tool).
Fourier series and transform	0.9 — a change of basis (0.4) in an infinite-dimensional inner-product space (0.5). Plancherel says the transform is unitary — a rotation in function space.	4.3, 4.6 (which is why it conserves probability), 5.3, 5.8.
Differentiation diagonalised $\widetilde{f'}=\ii k\tilde f$	0.9 — the entire reason Fourier analysis exists: in this basis a linear differential equation becomes an algebraic one.	5.4 (momentum space), 5.10. This is 0.5's spectral theorem applied in infinite dimensions, which 4.5 has to work to legitimise.
Convolution and Green's functions	0.9 — convolution in position space is multiplication in momentum space. 0.8's Duhamel solution was a convolution all along.	5.4 — the Green's function of a wave operator is the propagator, and Feynman diagrams are products of propagators because of this theorem.
Dirac delta $\delta(x)$	0.9 — not a function; a distribution, defined by $\int\delta f=f(0)$ . Its Fourier representation is 0.5's completeness relation in continuous disguise.	4.5 (normalising continuous spectra), 5.3, 5.10 — where $\delta(0)$ and $\delta^{2}$ appear and signal that regularisation is needed.
Bandwidth theorem $\Delta x\,\Delta k\ge\tfrac12$	0.9 — derived from 0.5's Cauchy–Schwarz, saturated by the Gaussian. Proved with no physics in it whatsoever.	4.9 — quantum mechanics adds exactly one substitution, $p=\hbar k$ . The inequality was never quantum; what is quantum is that a particle's momentum is a wavenumber.
Probability, characteristic function, CLT	0.9 — the characteristic function is the Fourier transform of the density, independent sums convolve, so the CLT is a Fourier argument. Variance adds, so error scales as $1/\sqrt n$ .	4.19, 5.11. The same $\sqrt N$ governs trial power calculations and $5\sigma$ discovery thresholds.

Part 0 is complete

Nine chapters, fifty-eight objects, one accumulating toolkit. You have the linear approximation, the integral, the series expansion, the vector space, the spectral theorem, the total derivative, the field theorems, the oscillator, and the Fourier transform.

Chapter 1.1 begins the physics — and the first thing that happens is that most of this gets used at once.

Part I · The Action Principle

Object	Introduced — and what for	Spent in
Conservative force and potential energy	1.1 — 0.7's four equivalent conditions, cashed: curl-free ⟺ path-independent ⟺ $\vv F=-\nabla V$ ⟺ $\oint=0$ , and hence $E=T+V$ conserved.	1.2, 1.3, 1.4. The failure of the third law for moving charges is the crack that 2.6 repairs by giving the field its own momentum.
Functional $S[x]=\int L\,\dd t$	1.2 — a map from an entire function to a number. The domain is infinite-dimensional; the "variable" is a whole path.	3.6, 5.2, 5.6, 6.4, 7.2 — every theory in this book is specified by one.
Variational derivative $\delta S/\delta x(t)$	1.2 — Chapter 0.1's linear approximation with the displacement $h$ promoted to an entire function $\eta$ . The single largest payoff of having defined the derivative as a linear map rather than a slope.	3.6, 5.2, 5.6, 5.7.
Euler–Lagrange equation	1.2 — derived by integration by parts (0.2) plus the fundamental lemma of the calculus of variations, which is proved, not quoted. In Cartesian coordinates it reads $\dot p=F$ : Newton's second law is a special case.	2.5, 2.6, 3.3 (geodesics), 3.6, 5.2, 6.4, 7.2.
Form invariance	1.2 — Euler–Lagrange keeps its shape under any invertible change of coordinates. $\vv F=m\vv a$ does not; in polar coordinates it grows centrifugal and Coriolis terms out of nothing.	3.2, 3.6 — general relativity has no global Cartesian coordinates, so a law whose shape depends on having them cannot survive. This property is why Part III is possible.
Total-derivative (gauge) freedom $L\to L+\dv{F}{t}$	1.2 — changes $S$ only by boundary terms, so the equations of motion never notice. The first appearance of gauge freedom in the book.	2.6, 6.3, 6.4. Also needed in 1.4, where quasi-invariance up to $\dv{F}{t}$ is what makes Galilean boosts a symmetry at all.
Legendre transform $H=\sum p_i\dot q_i-L$	1.3 — the dictionary between describing a convex function by its points and by its tangent slopes. Same transform as the thermodynamic potentials.	4.2, 5.3. $H=T+V$ is a theorem with hypotheses, not a definition.
Canonical momentum $p_i=\pdv{L}{\dot q_i}$	1.3 — not always $m\dot q$ . In a magnetic field $\vv p=m\vv v+q\vv A$ .	6.3 — minimal coupling, arriving three parts early. It is $\vv p$ , not $m\vv v$ , that becomes the quantum operator.
Hamilton's equations, phase space	1.3 — $2n$ first-order equations; one point fixes the entire future, so trajectories cannot cross. The oscillator's ellipse, the pendulum's separatrix.	4.10 (Bohr–Sommerfeld quantises exactly this enclosed area), 5.3.
Liouville's theorem	1.3 — the phase-space flow has zero divergence by equality of mixed partials, so by 0.7 (divergence = trace of the Jacobian) volume is exactly conserved. A blob can be stirred beyond recognition but never compressed.	4.6 (unitarity is the same statement), statistical mechanics, and 7.9 — the black-hole information problem is what it looks like when this appears to fail.
Poisson bracket $\{q_i,p_j\}=\delta_{ij}$	1.3 — with $\dv{f}{t}=\{f,H\}+\pdv{f}{t}$ , the equation of motion for any observable.	4.9 — quantum mechanics replaces $\{\,,\}$ with $\tfrac{1}{\ii\hbar}[\,,]$ and changes nothing else. Canonical quantisation is a one-line substitution into this structure. Also 5.3.
Generators $\delta f=\epsilon\{f,G\}$	1.3, completed in 1.4 — $H$ generates time translation, $\vv p$ generates spatial translation, $\vv L$ generates rotation.	4.2, 4.11, 6.1 — the bracket algebra of the generators is the Lie algebra of the symmetry group.
Noether's theorem $Q=\sum p_iK_i-H\tau-F$	1.4 — every continuous symmetry of the action yields a conserved charge, on shell. Energy, momentum and angular momentum are three substitutions into one formula. Momentum is conserved because space is uniform; energy because time is.	2.5, 3.5 (Killing vectors are this theorem geometrised), 3.6, 5.2, 6.3, 6.8. The tool by which Parts V–VII construct theories instead of discovering them.
Noether current $\partial_\mu j^{\mu}=0$	1.4 — the field version, which is 0.7's continuity equation. Stronger than the mechanics version because conservation is local: nothing vanishes here and reappears there.	5.2, 6.3. A global U(1) phase symmetry gives electric charge — and asking what happens when the phase is allowed to vary from point to point is the gauge principle.
Hidden symmetry Laplace–Runge–Lenz	1.4 — the $1/r$ potential has a conserved vector beyond angular momentum, which is why Kepler orbits close.	4.14 — the same $SO(4)$ is why hydrogen's levels depend only on $n$ and not $\ell$ . The moral: an unexplained coincidence in physics is usually an unrecognised symmetry.

Part I is complete

Four chapters ago, mechanics was a list of forces. It is now a single scalar $L$ , a stationarity principle, a bracket, and a theorem converting symmetry into conservation.

Every remaining part of this book is that same package applied to a different symmetry group — Lorentz in Part II, diffeomorphisms in Part III, unitary phase rotations in Parts IV–VI, worldsheet conformal symmetry in Part VII. You now have the whole method. What you need next is the groups.

Part II · Special Relativity

Object	Introduced — and what for	Spent in
$c$ from Maxwell $c=1/\sqrt{\mu_0\epsilon_0}$	2.1 — derived by taking the curl of Faraday and substituting Ampère–Maxwell. Two constants measured with capacitors and wires produce $2.998\times10^{8}\,$ m/s, which is how anyone knew light was an electromagnetic wave.	2.6 — where Maxwell's equations turn out to be exactly Lorentz invariant, which is the resolution of the crisis rather than a coincidence.
Failure of Galilean invariance	2.1 — the wave equation, transformed by chain rule under $x'=x-vt$ , grows a cross term and its characteristic speeds become $c-v$ and $-(c+v)$ . Either relativity fails for electromagnetism or the Galilean transformation is wrong.	2.2 — which takes the second option.
The two postulates	2.1 — after Michelson–Morley predicts 0.373 fringes and measures under 0.01. Note that Lorentz had the transformation equations first; what 1905 added was the claim that they describe space and time rather than the behaviour of matter drifting through an ether.	2.2, and structurally everything after.
Lorentz transformation $ct'=\gamma(ct-\beta x)$	2.2 — derived from the postulates plus three usually-silent assumptions, each named: homogeneity (which forces linearity), isotropy, reciprocity. Galileo is recovered as the leading term.	2.3, 2.4, 2.5, 2.6, 3.1.
Relativity of simultaneity $\Delta t'=-\gamma\beta\Delta x/c$	2.2 — presented first, because time dilation and length contraction are its consequences and readers who meet them in the other order never recover. It is first order in $\beta$ where dilation is second — which is why GPS notices and Michelson did not.	Every apparent paradox in relativity is this one fact, unrecognised.
Rapidity $\beta=\tanh\phi$	2.2 — the parameter that adds. Velocities fail to add because velocity was the wrong variable.	2.3 (it is the hyperbolic angle), 6.1 (a one-parameter Lie group). Constant proper acceleration gives $\phi=a\tau/c$ : rapidity grows without bound while speed asymptotes to $c$ .
Invariant interval $\Delta s^{2}=c^{2}\Delta t^{2}-\Delta x^{2}$	2.3 — proved invariant, then promoted: rotations are defined as the maps preserving $x^2+y^2$ , so Lorentz transformations are defined as the maps preserving $\Delta s^2$ . Special relativity is Euclidean geometry with one sign flipped.	2.4, 2.5, 3.3 — where $\eta_{\mu\nu}$ becomes position-dependent and the subject becomes gravity.
Minkowski metric $\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$	2.3 — with $\Lambda^{\mathsf T}\eta\Lambda=\eta$ as the defining equation of the Lorentz group. Not positive-definite, so it is not a distance in the sense of 0.5: one axiom has been removed on purpose.	2.4, 3.3, 6.1.
Boosts as hyperbolic rotations	2.3 — $\cosh^2-\sinh^2=1$ against $\cos^2+\sin^2=1$ . The orbits of rotation are circles; the orbits of boosts are hyperbolae, which is why boosted axes look skew and why their units need calibrating.	2.4, 6.1, 7.3.
Causal structure timelike / null / spacelike	2.3 — the classification is invariant; timelike order is absolute; spacelike order is always reversible by some boost. Hence: a faster-than-light influence lets you build a closed causal loop, explicitly constructed.	3.7, 5.1 (why fields must commute at spacelike separation), 7.9. This is why $c$ is a causal limit, not merely light's speed.
Proper time, and its maximisation $\tau=\int\sqrt{1-\beta^{2}}\,\dd t$	2.3 — the length of a worldline, and what a carried clock physically reads. Among timelike paths the straight one maximises it — the exact opposite of the Euclidean case, and the minus sign is the whole reason.	2.5, 3.3. The travelling twin ages less because a bent timelike path is shorter. Keep this variational principle, replace $\eta$ by $g_{\mu\nu}(x)$ , and the extremal paths are called gravity.
Contravariant vector $V'^\mu=\Lambda^\mu{}_\nu V^\nu$	2.4 — defined by its transformation law, taking $\dd x^\mu$ as the prototype. The chapter opens by exhibiting four numbers that are not a four-vector and watching them produce frame-dependent nonsense.	2.5, 2.6, 3.2 — where $\Lambda$ stops being constant and nothing else changes.
Covector, and index height $\partial_\mu$	2.4 — picking up 0.6 §4. Transforms with the inverse Jacobian, so the pairing $V^\mu\omega_\mu$ is invariant. Up and down indices are two different species of object; you only fail to notice in Cartesian coordinates because the metric is the identity there.	3.2, 3.3, 3.5.
Raising and lowering $V_\mu=\eta_{\mu\nu}V^\nu$	2.4 — the metric as the dictionary between the two species. Not "moving a letter" — it is a specific linear map, and forgetting that is the commonest source of sign errors in relativity.	Everywhere in Parts III and V–VII.
Tensor, and the invariance theorem	2.4 — type $(p,q)$ defined by its transformation law; contraction proved to lower the type. Then the theorem the chapter exists for: a tensor equation true in one frame is true in every frame, because the transformation is linear and homogeneous.	Every fundamental law from here on is written as a tensor equation for exactly this reason. $\vv F=m\vv a$ is not one, and 1.1 already showed the damage.
Symmetric / antisymmetric split	2.4 — the same decomposition 0.7 performed on the Jacobian, now proved Lorentz-invariant, so it is a real property rather than a coordinate accident.	3.3 (a symmetric $4\times4$ has 10 components — the metric), 2.6 (an antisymmetric one has 6 — three of $\vv E$ and three of $\vv B$ ).
Four-velocity $u^\mu=\gamma(c,\vv v)$ , $u\!\cdot\! u=c^2$	2.5 — because $\dd x^\mu/\dd t$ is not a four-vector: $t$ is frame-dependent. Differentiating by the invariant $\tau$ fixes it. Differentiating $u\!\cdot\! u=c^2$ then gives $u\!\cdot\! a=0$ — four-acceleration is always orthogonal to four-velocity.	2.6 (the covariant Lorentz force), 3.3 (geodesics).
Four-momentum $p^\mu=mu^\mu=(E/c,\vv p)$	2.5 — selected, not defined. Conservation of a three-vector is not preserved by a boost (2.5 shows it failing by 36% in a worked collision), but 2.4's theorem makes conservation of a four-vector automatic in every frame. Demanding frame-independent conservation forces $p^\mu$ — and having taken three components you are stuck with the fourth.	2.6, 3.6, 5.1, 5.9. The chapter's whole argument.
$E=\gamma mc^{2}$ , $E_0=mc^{2}$	2.5 — the fourth component of $p^\mu$ , identified by Taylor expansion (0.3): $mc^2+\half mv^2+\tfrac38mv^4/c^2+\ldots$ . Rest energy is what you are stuck with; the $v^4$ term is the leading relativistic correction to atomic levels.	4.1, 5.1, 6.5.
Mass shell $E^{2}=p^{2}c^{2}+m^{2}c^{4}$	2.5 — from $p\!\cdot\! p=m^2c^2$ . Geometrically it is 2.3's invariant hyperbola, drawn in momentum space; as $m\to0$ it degenerates onto its own asymptote $E=pc$ , which is the light cone.	5.1, 5.3, 5.4, 5.9. Also $\vv v=\vv pc^{2}/E$ , which is what makes the massless case work.
The relativistic action is proper time $S=-mc^{2}\!\int\!\dd\tau$	2.5 — obtained by solving for the Lagrangian that reproduces $\dot{\vv p}=\vv F$ . Then "extremise $S$ " and 2.3's "maximise proper time" are the same sentence, and the minus sign is what converts one into the other.	3.3 — where only the definition of $\dd\tau$ changes, and the extremal paths become gravity. Entry two in 1.2's table of actions.
Wave four-vector $k^\mu=(\omega/c,\vv k)$	2.5 — a four-vector because the phase $k\!\cdot\! x$ must be invariant: counting wave crests is not frame-dependent. Boosting it gives Doppler, including the transverse shift, which is pure time dilation and has no classical counterpart.	4.1 ( $k^\mu=p^\mu/\hbar$ is the de Broglie relation waiting to happen), 5.3.
Invariant mass, Mandelstam $s$	2.5 — the mass of a system, which is not the sum of its parts. Two back-to-back photons have $s\gt0$ despite both being massless.	5.9, 6.5. Also the reason colliders beat fixed targets: $\sqrt s$ grows as the square root, so matching the LHC on a fixed target needs $1.45\times10^{4}$ times the beam energy.
Four-current $j^\mu=(c\rho,\vv J)$ , $\partial_\mu j^\mu=0$	2.6 — built by requiring that it reproduce 0.7's continuity equation. Charge conservation becomes one manifestly invariant line.	5.2, 6.3 — where Noether produces this current from a symmetry rather than assuming it.
Field tensor $F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$	2.6 — 2.4 counted six independent components in an antisymmetric $4\times4$ and promised they would be $\vv E$ and $\vv B$ . They are: three entries $F^{0i}$ and three entries $F^{ij}$ . Antisymmetry is not a choice — it follows from $F=\dd A$ , which follows from gauge invariance.	5.8, 6.4 (Yang–Mills builds the same object non-abelian), 7.1.
Maxwell, in two equations $\partial_\mu F^{\mu\nu}=\mu_0 j^\nu$	2.6 — expanded component by component: $\nu=0$ is Gauss, $\nu=i$ is Ampère–Maxwell. The homogeneous pair $\partial_{[\lambda}F_{\mu\nu]}=0$ is not physics — it holds identically because mixed partials commute, which is 0.7's $\nabla\!\cdot\!(\nabla\times\vv A)=0$ in disguise.	3.5 (both are $\dd^{2}=0$ ), 5.2, 6.4.
Gauge invariance $A^\mu\to A^\mu+\partial^\mu\chi$	2.6 — leaves $F^{\mu\nu}$ untouched; the same freedom 1.2 found in adding a total time derivative to a Lagrangian. In Lorenz gauge, $\Box A^\mu=\mu_0j^\mu$ — 2.1's wave equation, now manifestly covariant, with $c$ a property of spacetime rather than of a medium.	6.3 — where this stops being a convenience and becomes the generating principle of every force in the Standard Model.
$\vv E$ and $\vv B$ mix	2.6 — 2.4's transformation law applied to $F^{\mu\nu}$ . A lab-neutral current-carrying wire acquires net charge in the test charge's frame, because the lattice and drift densities contract by different factors; the magnetic force is re-described as electrostatic, with the same total. Drift speeds are $v/c\sim10^{-13}$ and the effect is macroscopic only because charge cancellation is that precise.	6.3. Magnetism is a relativistic correction to Coulomb's law.
Field invariants $F_{\mu\nu}F^{\mu\nu}=2(B^{2}-E^{2}/c^{2})$	2.6 — with the pseudoscalar $\propto\vv E\!\cdot\!\vv B$ . A light wave has both invariants zero, so $\vv E\perp\vv B$ and $E=cB$ in every frame and no boost can bring it to rest.	5.11, 6.4.
The electromagnetic action $\mathcal L=-\tfrac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}-j_\mu A^\mu$	2.6 — varied with 1.2 §8's field Euler–Lagrange equation, it returns Maxwell. All of electromagnetism in one line, and essentially the only Lorentz- and gauge-invariant term available at lowest order.	5.2, 5.8 ( $j_\mu A^\mu$ is the QED vertex), 6.4 (Yang–Mills copies it verbatim), 5.11 (the "nearly forced" argument becomes effective field theory). Entry three in 1.2's table.
Stress-energy of the field $\vv g=\epsilon_0\vv E\times\vv B$	2.6 — Noether (1.4) applied to that action. Pays off the oldest open promise in the book: the momentum missing from Chapter 1.1's two moving charges is sitting in the field. Total $p^\mu$ balances identically. The third law failed only because it presumed instantaneous action at a distance.	3.6 — $T^{\mu\nu}$ is what sources gravity. And it is why Part V has to quantise the field: it is a physical system, not a calculational device.

Part II is complete

Six chapters ago there was a contradiction: Maxwell's equations produced a definite speed, and Galileo said no equation could. The resolution turned out not to be a repair of electromagnetism but a replacement of the geometry underneath it — and electromagnetism, written in that geometry, became shorter.

Two loose threads were also tied. Chapter 1.1's broken third law is repaired by giving the field its own momentum, and Chapter 2.1's ether is retired by showing that Maxwell's equations single out a speed rather than a frame. What remains open is the one Newton left: gravity still acts instantaneously, which Part II has just made impossible.

Part III · General Relativity

Object	Introduced — and what for	Spent in
Equality of inertial and gravitational mass $\abs{\eta}\lesssim 3\times10^{-15}$ ⚑	3.1 — the one quoted experimental input of the part. Inertial mass is measured with springs, gravitational mass with a balance; nothing in Newtonian physics relates them, and electric charge shows that such a pairing need not hold.	Everything in Part III. If it failed, free-fall paths would depend on the body and there would be nothing for a geometry to encode.
Deleting a uniform field $\vv r^{\,\prime}=\vv r-\tfrac12\vv g t^{2}$	3.1 — one relabelling of coordinates removes a uniform gravitational field from the equation of motion of every body at once, because the mass cancelled. The same trick on the Lorentz force needs a different relabelling per particle species, and so fails.	3.1 §4, which asks what survives it; and 3.4 §5.3, where the freely falling frame becomes a local inertial frame and the coordinates adapted to it are constructed.
Tidal deviation $\ddot d^{\,i}=-\Phi_{,ij}\,d^{\,j}$	3.1 — subtract the equations of motion of two nearby freely falling particles and linearise. The field cancels; its derivative does not. For a point mass, stretch $2GM/r^{3}$ along the field and squeeze $GM/r^{3}$ across it, traceless in vacuum and therefore volume-preserving.	3.4 §4, where geodesic deviation reproduces it with the Riemann tensor in place of $\Phi_{,ij}$ — the moment tidal force and curvature become the same thing. 3.4 §4.5 takes its trace and gets $R_{00}=4\pi G\rho/c^{2}$ ; 3.6 §5 spends that to fix $\kappa=8\pi G/c^{4}$ .
Size of a local inertial frame $\ell T^{2}\lt\varepsilon r^{3}/GM$	3.1 — demand the tidal drift stay below the instruments' resolution. The constraint is on a product, so a local inertial frame is a small patch of spacetime, not of space.	3.4 §5.3, which builds the locally inertial coordinates this bound licenses, and 3.8 §6.5, which collects the bound by name and finds the frame at a supermassive hole's horizon enormous. The reason general relativity is a differential theory: special relativity is exact only in the infinitesimal.
Gravitational redshift $\Delta\nu/\nu=-\Delta\Phi/c^{2}$	3.1 — accelerating cabin plus first-order Doppler, with no general relativity used at all. Recast by counting wave crests, it says that two static clocks at different heights run at different rates, which no fixed flat geometry permits.	3.3 (a position-dependent metric); 3.6 §5, which does not derive $g_{00}\approx1+2\Phi/c^{2}$ but takes it from 3.1 §6.5 and spends it to fix $\kappa=8\pi G/c^{4}$ ; and 3.8 §5, which recovers the redshift from the Schwarzschild solution three ways, §5.3 being this argument resummed.
Light bending — half of it $\alpha_{\text{EP}}=2GM/bc^{2}=0.875''$	3.1 — the same cabin, turned sideways, integrated along the undeflected ray. The measured value is $1.75''$ . The estimate uses only the timekeeping part of the geometry and knows nothing about spatial distances, so it is short by exactly a factor of two. The debt is recorded openly rather than fudged.	3.8 §3, which derives $4GM/bc^{2}$ from the Schwarzschild solution, and 3.8 §4, which computes the two halves separately and shows this one is exactly half.
Manifold, chart, atlas smooth transition maps	3.2 — the least structure on which calculus can be done: no distance, no angle, no straight line. A sphere provably needs two charts, and the stereographic pair has transition map $(u,v)\mapsto(u,v)/(u^{2}+v^{2})$ .	All of Part III, and Part VII, where the arena is a worldsheet rather than a spacetime.
Tangent space $T_{p}M$ , basis $\partial_{\mu}\|_{p}$	3.2 — the arrow is abandoned because it needs an ambient space nobody has, and replaced by a directional derivative acting on functions. That the coordinate operators are a basis is proved, using Hadamard's factorisation, not assumed.	3.3, where the metric becomes an inner product on each tangent space; and Part VI, where the same construction is repeated with an internal space in place of $T_{p}M$ .
Vectors at different points cannot be compared	3.2 — $T_{p}M$ and $T_{q}M$ are different vector spaces with no canonical identification. Flat space hid this by being a vector space as well as a manifold. The figure shows the same vector carried between two points by two routes and arriving twice.	3.3 in its entirety — the covariant derivative exists to repair exactly this subtraction, and the leftover from the repair is the gravitational field. 3.4 turns the route-dependence into curvature.
Cotangent space $\langle\dd x^{\mu},\partial_{\nu}\rangle=\delta^{\mu}{}_{\nu}$	3.2 — the dual of each tangent space, with the coordinate differentials as its basis. This is where the $\dd x$ cancelled since school finally acquires a definition, and where $\dd f$ is distinguished from the gradient, which needs a metric and so does not yet exist.	3.3 (raising and lowering), 3.5 (forms and the exterior derivative), 0.6's promise, collected.
Lie bracket $[X,Y]^{\nu}=X^{\mu}\partial_{\mu}Y^{\nu}-Y^{\mu}\partial_{\mu}X^{\nu}$	3.2 — the composite of two vector fields is not a vector field, because it carries second derivatives; the antisymmetric part is, because the offending terms are symmetric and cancel. First sighting of a tensor built from non-tensorial pieces.	3.4, where the same cancellation in a commutator produces the Riemann tensor; 3.5, where the bracket becomes the Lie derivative; 6.1, where it becomes the Lie algebra.
Metric $\dd s^{2}=g_{\mu\nu}(x)\,\dd x^{\mu}\dd x^{\nu}$	3.3 — the arena of 3.2 could not measure a length and could not say that two directions met at a right angle. One inner product per tangent space, varying smoothly, supplies both, and with them 2.3's timelike/spacelike/null classification at every point, 2.4's raising and lowering, and the gradient $(\nabla f)^{\mu}=g^{\mu\nu}\partial_{\nu}f$ that 0.6 defined and could not build. Signature $(+,-,-,-)$ everywhere, which is the exact statement that special relativity holds locally.	Everything after. 3.3 §7, where two demands make it force the connection; 3.5 §6, where its determinant becomes the volume element; 3.6, where it is the unknown the field equations solve for; 3.7 and 3.9, the two metrics this book solves for. Part VI runs the whole construction again with an internal space in place of $T_{p}M$ .
Flat space in polar coordinates $\dd s^{2}=\dd r^{2}+r^{2}\dd\theta^{2}$	3.3 §2 — a counterexample worked at length before curvature is defined, to kill a belief before it forms: position-dependent metric components do not mean curvature, they mean the grid is not Cartesian. The unit sphere carries $\dd\theta^{2}+\sin^{2}\theta\,\dd\phi^{2}$ and the flat plane carries this; both have one constant component and one that varies, one space is curved and one is not, and inspection cannot tell them apart. Hence a genuine test has to be built.	3.3 §7.8 (non-zero $\Gamma$ in a space that is flat by construction), 3.4 §3.1 (Riemann computed component by component and found identically zero), 3.7 §2.2 (what the areal $r$ is and is not), 3.8 §6, which is this example in a much bigger costume: the chart fails at $r_{s}$ and the geometry does not.
Covariant derivative, connection coefficients $\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma^{\nu}{}_{\mu\lambda}V^{\lambda}$	3.3 §§4–5 — $\partial_{\mu}V^{\nu}$ is not a tensor. Differentiating 2.4's transformation law produces a second term carrying a second derivative of the coordinate change, which dies only for affine relabellings — which is exactly why Part II never met it. $\Gamma$ is then not guessed but defined as whatever cancels that term, and the inhomogeneous law it must therefore obey is derived. Two consequences: the connection can be zero in one chart and non-zero in another, because only a non-tensor can cancel a non-tensor; and the difference of two connections is a genuine tensor. On a covector the sign flips, $\nabla_{\mu}\omega_{\nu}=\partial_{\mu}\omega_{\nu}-\Gamma^{\lambda}{}_{\mu\nu}\omega_{\lambda}$ , forced by Leibniz and not chosen.	3.4, where its commutator is the curvature; 3.5 §2.2 (why $\dd$ needs none of it) and §6.4; 3.6, 3.7, 3.8, 3.9. And 6.3, where the same inhomogeneous law, rebuilt on an internal space, becomes the transformation law of a gauge field.
Christoffel symbols $\Gamma^{\lambda}{}_{\mu\nu}=\half g^{\lambda\sigma}\big(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\big)$ ⚑	3.3 §7 — the chapter's set piece, and the reason general relativity is possible: given the metric, two physical demands leave exactly one connection. Demand 1 is metric compatibility $\nabla_{\rho}g_{\mu\nu}=0$ , which is shown to be precisely the statement that transport preserves lengths and angles. Demand 2 is vanishing torsion $\Gamma^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{\nu\mu}$ , so that second covariant derivatives of a scalar commute. Forty equations, forty unknowns, and three cyclic copies combined as (B)+(C)−(A). ⚑ marks the second demand: torsion-free is an assumption about nature, and no observation of freely falling bodies can test it.	3.4 (every Riemann component is built from these), 3.5 §6.4, 3.6 §5.3, 3.7 §3.1 (nine of them, and then the exact solution), 3.8 §6.4, 3.9 §3.1. In flat polar coordinates they are $\Gamma^{r}{}_{\theta\theta}=-r$ and $\Gamma^{\theta}{}_{r\theta}=1/r$ — Chapter 1.1's centrifugal and Coriolis terms, which its problem set promised would be called $\Gamma^{i}{}_{jk}$ from this chapter onward.
Parallel transport $\dv{V^{\mu}}{\lambda}=-\Gamma^{\mu}{}_{\nu\rho}\dv{x^{\nu}}{\lambda}V^{\rho}$	3.3 §6 — the comparison 3.2 said was missing, finally constructible. Along a given path a vector at one end determines the vector at the other, by ⚑ the standard existence-and-uniqueness theorem for linear systems. The answer depends on the path, and that is not a defect; 3.2 §5.3's figure had already shown it happening on a sphere with no machinery to describe it.	3.4 §1, where the path-dependence is measured round a closed loop and turned into curvature; 3.5 §7, where the rival way of comparing — dragging along a flow — turns out to need no connection at all.
Geodesic equation $\ddot x^{\mu}+\Gamma^{\mu}{}_{\nu\rho}\dot x^{\nu}\dot x^{\rho}=0$	3.3 §8 — derived twice, by routes sharing no assumption: as the curve that parallel-transports its own tangent, and by extremising $S=-mc^{2}\!\int\dd\tau$ with 1.2's Euler–Lagrange equation. They agree, and that is where 1.2's promise is collected. There is no force term, no potential and no coupling constant, and the mass cancels — 3.1's universality in new clothes. In the weak, slow, static limit it returns $\ddot x^{i}=-\partial_{i}\Phi$ , using nothing beyond 3.1's independently derived $g_{00}=1+2\Phi/c^{2}$ .	3.4 §4 (two of them, and how they separate), 3.5 §9 (a conserved quantity along every one), 3.7 §§5–8 (Mercury), 3.8 §1 (the null case, where $\tau$ is no longer available), 3.9 §5.1 (the cosmological redshift). Its action is the one 5.6 exponentiates into a sum over histories.
Riemann tensor $[\nabla_{\mu},\nabla_{\nu}]V^{\rho}=R^{\rho}{}_{\sigma\mu\nu}V^{\sigma}$	3.4 §§1–2 — the test 3.3 could not build, and it runs entirely from inside. Carry a vector round a closed loop; the angle it comes back turned through, divided by the area enclosed, is a property of the place, and on a sphere of radius $a$ it is $1/a^{2}$ . Shrink the loop and the test becomes a commutator of covariant derivatives: six terms appear, four cancel — exactly the four carrying derivatives of $V$ — and what is left is multiplication by $\partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma}+\Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma}-\Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma}$ . A tensor built from pieces that are not tensors, for 3.2's reason: the chart-dependent part of $\Gamma$ is symmetric in the two indices being antisymmetrised.	3.4 §§4–8, 3.6 (both derivations of the field equations), 3.7 §3, 3.8 §6.4 (its full contraction is what settles the horizon), 3.9 §3.1. And 6.4, where the same commutator with an internal space in place of the tangent space is the Yang–Mills field strength, and this chapter is read a second time in different clothes.
Geodesic deviation $\dfrac{D^{2}\xi^{\mu}}{\dd\tau^{2}}=-R^{\mu}{}_{\nu\rho\sigma}u^{\nu}\xi^{\rho}u^{\sigma}$	3.4 §4 — where Part III's thesis lands. Two neighbouring free-fallers separate at a rate the curvature dictates, and in the Newtonian limit the equation is 3.1's tidal equation with $c^{2}R^{i}{}_{0j0}$ standing exactly where $\Phi_{,ij}$ stood. The identification was not fitted; it was computed, from a metric component 3.1 obtained with no general relativity in it at all. Tidal force is curvature, and curvature is what gravity is.	3.4 §4.5, which takes the trace and gets $R_{00}=4\pi G\rho/c^{2}$ , and zero in vacuum — the shape of the field equations, two chapters early; 3.6 §5, where that becomes the fixing of $\kappa$ ; 3.8 §4. Chapter 3.1's stretch-and-squeeze ellipse is a picture of $R^{i}{}_{0j0}$ , and Worked example 2 draws it again from the tensor.
Riemann's symmetries, and the count $n^{2}(n^{2}-1)/12$ ; twenty in four dimensions	3.4 §5 — antisymmetry within each pair, symmetry under exchanging the pairs, and the first Bianchi identity $R_{[abcd]}=0$ , every one derived rather than listed. The count is then done twice by arguments with nothing in common: once from the symmetries, and once by asking how much of the second derivatives of the metric no choice of coordinates can remove. Both give twenty.	3.4 §6 (which ten survive contraction), 3.8 §6.4, where the symmetries cut the Schwarzschild curvature to six independent components and make the Kretschmann sum a finite one; 3.9 §2.2, which needs the three-dimensional count of six to reduce the spatial geometry to constant curvature.
Locally inertial coordinates $g_{\mu\nu}(p)=\eta_{\mu\nu}$ , $\partial_{\rho}g_{\mu\nu}(p)=0$	3.4 §5.3 — built explicitly, by $y^{\mu}=x^{\mu}+\half\Gamma^{\mu}{}_{\alpha\beta}(p)\,x^{\alpha}x^{\beta}$ , which kills the connection at one point and at one point only. This is 3.1's falling laboratory turned into a chart, and it makes the reason curvature cannot be transformed away precise: the first derivatives go, the second derivatives do not.	3.4 §7.1, where the second Bianchi identity is proved in four lines because $\Gamma$ vanishes and only $\partial\Gamma$ survives; 3.4 §8; 3.6 §4.3 (the Palatini identity, by the same trick); 3.9 §6, where it is the reason there is no local energy density for gravity to be stored in.
Ricci tensor and Ricci scalar $R_{\mu\nu}=R^{\lambda}{}_{\mu\lambda\nu}$ , $R=g^{\mu\nu}R_{\mu\nu}$	3.4 §6 — there is essentially only one contraction available, because the symmetries make every other choice the same one up to a sign. It keeps ten of the twenty components. The consequence that matters immediately: vacuum is not flat. $R_{\mu\nu}=0$ does not force $R_{\rho\sigma\mu\nu}=0$ , so there is something for a gravitational field to be outside a mass.	3.6 (both sides of the field equations are assembled from these), 3.7 §3, where solving $R_{\mu\nu}=0$ by hand is the whole chapter, 3.8 §6.4 (why $R$ and $R_{\mu\nu}$ are useless outside a star and $K$ is not), 3.9 §3 and §6.2.
Weyl tensor ⚑	3.4 §6.2 — the ten components that contraction throws away, named and not developed, with three of its properties quoted and none of them used. It is the part that survives in vacuum, so it is what carries the tides outside a mass.	3.8 §6.4, where the Schwarzschild curvature is entirely Weyl and $K$ is how you get at it; and 7.3, which meets its one striking property again — unchanged when the metric is multiplied by an arbitrary positive function of position, so that it records the light cones and not the scale of anything — under the name of conformal symmetry.
Second Bianchi identity $\nabla_{\lambda}R^{\rho}{}_{\sigma\mu\nu}+\nabla_{\mu}R^{\rho}{}_{\sigma\nu\lambda}+\nabla_{\nu}R^{\rho}{}_{\sigma\lambda\mu}=0$	3.4 §7 — proved in four lines in locally inertial coordinates, where $\Gamma$ vanishes at the point and only $\partial\Gamma$ survives; being a tensor equation it then holds in every chart. An identity, not a field equation: it constrains no metric, because every metric already satisfies it.	3.4 §7.3, where contracting it twice produces $\nabla^{\mu}G_{\mu\nu}=0$ ; 3.6 §3.2, where that is the whole cornering argument; 3.9 §3.3, where the same identity turns up as arithmetic — differentiate Friedmann I, subtract a multiple of Friedmann II, and out comes the fluid equation, so the third equation was never independent.
Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-\half R\,g_{\mu\nu}$ , $\nabla^{\mu}G_{\mu\nu}=0$	3.4 §7.3 — the doubly contracted Bianchi identity, rearranged until the divergence-free combination is visible. It vanishes identically, for every metric, with no equation of motion assumed. The list of objects that do that is very short, which is the entire reason the field equations look the way they do — and the box says so three chapters early.	3.6 §3 (the cornering) and §4, where it drops out of a variation instead, with its $\half$ arriving from the derivative of a determinant rather than from a contracted identity; 3.9 §3.1 and §3.2. In two dimensions it vanishes identically, which is why 7.2 can work on a two-dimensional sheet and not be doing gravity — and that turns out to be a feature.
Flat $\iff$ $R^{\rho}{}_{\sigma\mu\nu}=0$ ⚑	3.4 §8 — the question 3.3 §2 raised and deliberately left open. One direction is immediate. ⚑ The converse, that vanishing curvature lets you build a chart in which $g_{\mu\nu}=\eta_{\mu\nu}$ everywhere on a patch, is quoted rather than proved — and it is a local statement, which is the point.	3.4 §3.1, where the flat plane in polar coordinates passes the test; 3.5 §4.3, which sets its cone — flat everywhere and yet not a plane — beside three other places where impeccable local information gives a false global conclusion; 3.9 §2.4, where the flagged locality is cashed: a universe can be flat at every point and still close up on itself, so $k$ is a statement about curvature and not about whether space is finite.
Differential form, wedge product $\alpha\wedge\beta=(-1)^{pq}\beta\wedge\alpha$	3.5 §1 — derived from a demand rather than defined: an integral must belong to the region and not to the labelling of it, and writing that requirement out in two dimensions forces the integrand to be totally antisymmetric. That is 0.4's determinant, doing physics. Nothing else can be integrated over a surface in a way that survives a change of parametrisation.	3.5 §§2–6 in their entirety, and §10, where six components of $F$ come out of four of $A$ with the antisymmetry doing the counting. 6.3 rebuilds that construction on an internal space.
Exterior derivative $(\dd\omega)_{\mu_{0}\cdots\mu_{p}}=(p+1)\,\partial_{[\mu_{0}}\omega_{\mu_{1}\cdots\mu_{p}]}$	3.5 §2 — antisymmetrise the ordinary derivative and the result is a tensor with no connection anywhere in it, because the chart-dependent piece of $\Gamma$ is symmetric in precisely the two slots being antisymmetrised. On functions it is $\dd f$ , on one-forms it is the curl, in three dimensions it is 0.7's grad–curl–div diagram in one operator.	3.5 §§3–5, and 3.5 §10, where $F=\dd A$ and $\dd\!\star\!F=\mu_{0}\star\!J$ are all of electromagnetism. And 6.3, where a connection on an internal space is built from scratch and leans on the no-connection-needed property harder than anything in Part III does.
$\dd^{2}=0$	3.5 §3 — three lines, from the equality of mixed partials, on every form of every degree on every manifold. Three separate debts fall due at once: 0.7's $\nabla\times\nabla\phi=0$ , 0.7's $\nabla\cdot(\nabla\times\vv A)=0$ , and 2.6's discovery that the homogeneous half of Maxwell's equations is bookkeeping rather than physics. It is also the algebraic shadow of the fact that a boundary has no boundary.	3.5 §4 (what the converse costs), 3.5 §10, 6.3 — where the gauge freedom that falls out of it stops being a convenience and becomes the organising principle of every force in nature.
Closed, exact, and the shape of the region ⚑	3.5 §4 — $\dd\omega=0$ does not imply $\omega=\dd f$ unless the region can be shrunk to a point. The counterexample $(x\,\dd y-y\,\dd x)/(x^{2}+y^{2})$ is given first, so that the theorem has something to exclude, and then the Poincaré lemma is proved in four lines for one-forms by the explicit homotopy $f(x)=\int_{0}^{1}\omega_{\mu}(tx)\,x^{\mu}\dd t$ — which is 0.7 §7.3's quoted result, and which hands you a formula for the vector potential. ⚑ The general- $p$ case is sketched and quoted.	3.5 §4.3, where the count of closed-but-not-exact forms is named as de Rham cohomology — promised under that name at 0.7 §7.3 — and the same local-impeccable, global-failing pattern is collected in four places at once, including 3.4 §8's cone. 3.9 §2.4 (flat everywhere and still closed). Part VI, as the reason certain field configurations cannot be smoothly undone, and as the seed of the monopole argument: a $\vv B$ with $\nabla\!\cdot\!\vv B=0$ everywhere it is defined and no global $\vv A$ .
Generalised Stokes theorem $\int_{M}\dd\omega=\oint_{\partial M}\omega$	3.5 §5 — proved on a cube by the Fundamental Theorem of Calculus, then extended by the interior-face cancellation 0.7 described and could not complete. Green, Stokes, the divergence theorem and the FTC are recovered as four readings of one line, in different degrees.	3.5 §6.4 (the divergence theorem that works on a manifold), 3.6 §4.5, where it is what turns the discarded piece of the Einstein–Hilbert variation into a boundary term. The gap 0.7 admitted in writing is now closed.
Invariant volume element $\sqrt{-g}\;\dd^{n}x$	3.5 §6 — because $\dd^{n}x$ alone picks up $\det J$ under a change of chart and $\sqrt{-g}$ picks up $\abs{\det J}^{-1}$ , so the product is the same in every chart. It is 0.6's Jacobian factor, arriving where it was promised. With it comes Jacobi's formula $\delta g=g\,g^{\mu\nu}\delta g_{\mu\nu}$ , and hence $\nabla_{\mu}V^{\mu}=\partial_{\mu}(\sqrt{-g}\,V^{\mu})/\sqrt{-g}$ — the one sentence that lets a covariant divergence be integrated away.	3.6 §4, the whole variational derivation: the volume element's variation is one of the three pieces, Jacobi's formula supplies the $\half$ in $G_{\mu\nu}$ , and the divergence identity is what lets 3.6 throw the third piece away. Also 3.5 §10.3 and 3.9 §6.5. Every action integrated over spacetime from 5.2 onward carries it.
Lie derivative $(\mathcal{L}_{X}Y)^{\mu}=[X,Y]^{\mu}$	3.5 §7 — the second way of comparing a tensor here with a tensor there: drag one along the flow of a vector field and subtract. It needs no connection, which is what makes it usable on a bare manifold, and on vectors it turns out to be 3.2's Lie bracket exactly. The partial derivatives in it may be replaced by covariant ones, and nothing changes.	3.5 §8 (setting it to zero on the metric defines a symmetry), 3.9 §6.2, where computing $(\mathcal{L}_{\xi}g)_{ij}=-2aa'\tilde g_{ij}$ is what kills the obvious candidate for a conserved energy. 6.1, where the same bracket is the Lie algebra.
Killing vector $\mathcal{L}_{\xi}g=0$ , i.e. $\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}=0$	3.5 §8 — a direction in which the geometry does not change, defined without reference to coordinates, then reduced to a practical test: if no metric component depends on a coordinate, $\partial$ of that coordinate is a Killing vector. Flat spacetime has ten of them and they are the Poincaré generators of Part II; the sphere has three, and the obvious fourth candidate is not among them.	3.7 §1 (spherical symmetry and staticity written as statements about Killing vectors rather than about pictures, which is what cuts ten unknown functions to two) and §5; 3.8 §5.1; 3.9 §1.1 and §6.2. Part IV's angular-momentum quantum numbers are these rotational Killing vectors in different costume, and 4.11's $\mathfrak{su}(2)$ is the algebra 3.7 §1.1 uses to state isotropy.
A conserved quantity along every geodesic $Q=\xi_{\mu}u^{\mu}$	3.5 §9 — four lines: differentiate along the curve, one term dies by the geodesic equation, the other is a symmetric object contracted with an antisymmetric one. Running 1.4's Noether theorem on the same system returns the same charge, with $\delta L$ turning out to be $\mathcal{L}_{\xi}g$ contracted with two velocities. Noether's theorem and Killing's equation are one thing. This is the chapter's most expensive result, and it is what makes 3.7 solvable.	3.7 §5, where it converts $\partial_{t}$ and $\partial_{\varphi}$ into $E$ and $L$ and turns four coupled second-order equations into one first-order equation; 3.8 §1.2 and §5.1 (the redshift from a conserved charge against a local clock); 3.9 §6, where it returns nothing, because there is no timelike Killing vector to feed it.
Electromagnetism in three symbols $F=\dd A$ , $\dd F=0$ , $\dd\!\star\!F=\mu_{0}\star\!J$	3.5 §10 — all of Chapter 2.6 rewritten with the Hodge star, on a curved manifold, with no connection appearing anywhere. The compression is not cosmetic: it shows that the homogeneous pair is an identity, that charge conservation $\dd\!\star\!J=0$ follows from $\dd^{2}=0$ , and that $\nabla_{\mu}F^{\mu\nu}=\partial_{\mu}(\sqrt{-g}F^{\mu\nu})/\sqrt{-g}$ needs no Christoffel symbols.	6.3, which rebuilds $F=\dd A$ with an internal space in place of spacetime; 6.4, where the same three symbols acquire a commutator term because the field carries the charge it responds to.
Stress–energy tensor in curved space $T^{\mu\nu}=\big(\rho+p/c^{2}\big)u^{\mu}u^{\nu}-p\,g^{\mu\nu}$ , $\nabla_{\mu}T^{\mu\nu}=0$	3.6 §1 — what sits on the right-hand side, and it is not mass. Mass is not conserved, is not a scalar, and cannot source a tensor equation; energy, momentum, pressure and stress in one symmetric object whose divergence vanishes is the only candidate. The move from $\partial_{\mu}T^{\mu\nu}=0$ to $\nabla_{\mu}T^{\mu\nu}=0$ is flagged as the one place minimal coupling is genuinely ambiguous. Radiation has $p=\rho c^{2}/3$ and trace zero.	3.6 §3 (it is the divergence-free requirement on $T$ that corners the left-hand side), §5.4, §5.5; 3.9 §3, where a perfect fluid is the entire content of the universe. Worked example 1 recovers 2.6's field stress–energy from the variational definition. Part V computes it for real fields from Noether instead of positing it, and 7.1's right-hand side is that object rather than this caricature.
Einstein field equations $G_{\mu\nu}+\lambda g_{\mu\nu}=\dfrac{8\pi G}{c^{4}}T_{\mu\nu}$	3.6 §3 — cornered, not guessed. Anything set equal to $T_{\mu\nu}$ must be identically divergence-free for every metric; the only symmetric rank-2 objects buildable from a metric and at most two of its derivatives are $R_{\mu\nu}$ , $Rg_{\mu\nu}$ and $g_{\mu\nu}$ ; imposing the divergence condition on the general combination fixes the ratio of the first two and leaves the third free. ⚑ Lovelock's theorem, quoted, says there is nothing else in four dimensions.	3.7, which sets $T_{\mu\nu}=0$ and solves $R_{\mu\nu}=0$ exactly; 3.9, which puts a perfect fluid on the right and a homogeneous isotropic metric on the left; 7.1, which asks what happens when the same equations are quantised.
Einstein–Hilbert action $S_{\text{EH}}=-\dfrac{c^{4}}{16\pi G}\displaystyle\int\!\big(R-2\lambda\big)\sqrt{-g}\;\dd^{4}x$ ⚑	3.6 §4 — the same equations a second time, from 1.2's variational principle, because there is essentially one scalar buildable from a metric with at most two derivatives, so the action writes itself. The variation splits into three: the volume element (0.4's determinant again), $\delta R_{\mu\nu}$ via the Palatini identity, and $\delta g^{\mu\nu}$ . Entry four in 1.2's table of actions. ⚑ The third piece is a total derivative thrown away, and the Gibbons–Hawking–York term that repairs it is named and deferred.	7.9, where the numerical value of the gravitational action is an entropy and the discarded boundary term stops being ignorable. Also 5.2 and 6.4, which specify their theories the same way.
$\kappa=8\pi G/c^{4}$ , and pressure gravitates $\nabla^{2}\Phi=4\pi G\big(\rho+3p/c^{2}\big)$	3.6 §5 — the one free constant, fixed by demanding that apples fall. Trace-reverse the field equation, take the field weak, static and the matter slow, and $R_{00}$ works out to $\nabla^{2}\Phi/c^{2}$ using 3.1's independently derived $g_{00}=1+2\Phi/c^{2}$ ; matching against 0.7's Poisson equation gives the number. Restoring the pressure is free and says something new: pressure gravitates, radiation pulls twice as hard as dust of the same energy density, and anything with $p\lt-\rho c^{2}/3$ pushes.	3.7 §3.6, where $g_{00}=1+2\Phi/c^{2}$ fixes the constant of integration and the mass enters the solution; 3.9 §3.2 and §4.3, where the pressure term is what decides whether the expansion accelerates.
Cosmological constant $G_{\mu\nu}-\Lambda g_{\mu\nu}=\dfrac{8\pi G}{c^{4}}T_{\mu\nu}$ , $p=-\rho c^{2}$ ⚑	3.6 §6 — the term nothing in the argument excludes. It passes every constraint, it is the constant one may add to any Lagrangian, and moved to the other side it behaves as a fluid with $p=-\rho c^{2}$ — the unique equation of state that looks the same to every observer and does not dilute as space grows. ⚑ Measured at about $1.1\times10^{-52}\,\mathrm{m^{-2}}$ : tiny, and not zero. The third of the chapter's three sign traps lives here.	3.9 §3.5 (the Einstein static universe, and the ten-billion-year timescale on which it falls over), §4.3 and §4.4, where it is what makes the expansion accelerate at $z=0.63$ . Part V computes the vacuum energy of a quantum field and does not get this number; 7.9 accounts for the failure without fixing it.
Ten equations, four identities	3.6 §7 — the count, and why it is exactly right. Ten field equations for ten metric components would over-determine the problem, except that $\nabla^{\mu}G_{\mu\nu}=0$ is four identities, leaving six evolution equations and four constraints — and the missing four are the freedom to relabel coordinates, which no measurement can see. Then the fact that makes the subject hard: the equations are nonlinear, because the field carries the energy that sources it. Gravity gravitates.	6.3, which makes the identity-plus-invisible-freedom correspondence into a principle; 6.4, where the field carries the charge it responds to and the equations look startlingly like these; 7.1, where the nonlinearity stops being an inconvenience and becomes the obstruction.
Schwarzschild solution $g_{00}=1-r_{s}/r$ , $\ \ r_{s}=2GM/c^{2}$	3.7 §3 — Einstein's equations solved exactly, by hand, in one sitting. Two symmetry conditions cut ten unknown functions of four variables to two functions of one; then one line, $BR_{00}+AR_{11}=(AB)'/rB$ , forces $AB$ constant, asymptotic flatness makes it one, the angular equation becomes $(rA)'=1$ , and one integration gives $A=1+C_{1}/r$ . The mass was never in the field equations. It arrived as a constant of integration, fixed by 3.1's Newtonian limit. $r_{s}$ is $2.95$ km for the Sun and $8.9$ mm for the Earth.	3.7 §§5–8 (orbits, the ISCO, Mercury), all of 3.8, and 3.9 §5.5, where Birkhoff licenses using it for a bound system inside an expanding universe. 7.9 returns to it for the entropy of its horizon.
Areal radius $r\equiv\sqrt S$ , so the sphere at $r$ has area $4\pi r^{2}$	3.7 §2.2 — the load-bearing choice, and flagged as one. $r$ is not a distance from anywhere; it is a label defined by an area, chosen because it makes the algebra finite. 3.3 §2's warning in its sharpest operational form.	3.8 §6 in its entirety — the whole horizon argument turns on it, since the coordinates fail at $r_{s}$ and the geometry does not — and 3.8 §7, where the choice is un-made by a chart that follows the light instead.
Birkhoff's theorem	3.7 §4 — derived rather than quoted. Drop staticity, let both functions depend on time, and the $01$ component of $R_{\mu\nu}=0$ reads $\dot B/rB$ , so $B$ cannot depend on time and the residual time dependence in $A$ is removable by rescaling a clock. A star may pulse, collapse or explode and its exterior does not change; there is no spherically symmetric gravitational radiation. It is also 3.4's insistence that Ricci-flat is not flat, with an example.	3.9 §5.5, where it is what licenses saying a bound system is governed by Schwarzschild and not by the expanding metric, and §3, where it plus ⚑ its interior counterpart give the same licence the Newtonian shell theorem gave.
$E$ and $L$ from the two Killing vectors $E=c^{2}\big(1-r_{s}/r\big)\dv{t}{\tau}$ , $\ \ L=r^{2}\dv{\varphi}{\tau}$	3.7 §5 — the promise 3.5 made, collected by name. The metric mentions neither $t$ nor $\varphi$ , so $\partial_{t}$ and $\partial_{\varphi}$ are Killing vectors, 3.5 §9 converts each into a constant along every geodesic, and 1.4's Noether theorem identifies them as energy and angular momentum per unit mass. Four coupled second-order equations become one first-order equation.	3.7 §§6–8, 3.8 §1.2, where the same two charges become individually meaningless for light and only their ratio survives, and 3.8 §5.1, where $E$ against a local clock is the redshift.
Relativistic effective potential $V_{\text{eff}}=-\dfrac{GM}{r}+\dfrac{L^{2}}{2r^{2}}-\dfrac{GML^{2}}{c^{2}r^{3}}$	3.7 §6 — one line of algebra from the normalisation $u\!\cdot\!u=c^{2}$ and the two charges, and orbital motion becomes 0.8 §4.3's reading of an energy equation as motion in a landscape. Newton's two terms are there unchanged; the third is new, and it is a fraction $r_{s}/r$ of the centrifugal barrier, so it is negligible far out and dominant close in. Everything that follows in the chapter is a consequence of that one term.	3.7 §7 and §8; 3.8 §§1–2, where the same construction with the mass term removed gives the null potential $W(r)=1/r^{2}-2GM/c^{2}r^{3}$ , whose maximum is the photon sphere.
Innermost stable circular orbit $r_{\text{ISCO}}=6GM/c^{2}=3r_{s}$	3.7 §7 — Newton's condition for a circular orbit was linear in $r$ and had one root; this one is quadratic and has two, an unstable inner and a stable outer. The discriminant vanishes at $L=2\sqrt3\,GM/c$ , below which no circular orbit exists at any radius, and there the two roots merge. $88.6$ km for a ten-solar-mass hole. The chapter recognises the algebra as a saddle-node collision, the same one a logistically growing tumour under fixed-rate treatment performs, which is not a fact about gravity at all.	3.7 Worked example 2, which reads the binding energy off the same effective potential: getting to the ISCO releases $1-2\sqrt2/3=5.72\%$ of the rest mass, eight times what fusion yields, which is why accretion discs are the brightest steady objects in the sky. It is the inner edge of one.
Orbit equation, and perihelion precession $\dfrac{\dd^{2}u}{\dd\varphi^{2}}+u=\dfrac{GM}{L^{2}}+\dfrac{3GM}{c^{2}}u^{2}$ , $\ \ \Delta\varphi=\dfrac{6\pi GM}{c^{2}p}$	3.7 §8 — in $u=1/r$ the Newtonian truncation is 0.8's harmonic oscillator and gives the closed conic 1.4's Laplace–Runge–Lenz vector explained. Linearise about the circular solution and the new term becomes a frequency shift $\omega^{2}=1-6G^{2}M^{2}/c^{2}L^{2}$ , so the radius repeats every $2\pi/\omega$ rather than every $2\pi$ and the ellipse does not close. For Mercury: $5.02\times10^{-7}$ rad per orbit, $415.2$ orbits per century, $42.98''$ per century, against a measured residual of ⚑ $42.98\pm0.04$ . Nothing was fitted.	3.8 §3, which runs the same perturbation on the null orbit equation — same structure, no Newtonian source term — and gets the deflection of light.
Null geodesics, and the impact parameter $b=cL/E$	3.8 §1 — light needs one number, not two. The null normalisation $g_{\mu\nu}\dot x^{\mu}\dot x^{\nu}=0$ removes proper time as a parameter and leaves an affine parameter fixed only up to $\lambda\to a\lambda+b$ , so 3.7's two Killing charges are individually meaningless for a photon and only their ratio survives. Rescaling the parameter rescales $E$ and $L$ together.	3.8 §§2–4 (everything about light in this geometry follows from the one radial equation $b$ produces), 3.8 §5.1, 3.9 §5.1, where the same construction on the FLRW metric gives the cosmological redshift, and 3.9 Worked example 1, where integrating the null condition gives the horizons.
Photon sphere $r_{\text{ph}}=3GM/c^{2}=\tfrac32 r_{s}$ , $\ \ b_{\text{crit}}=3\sqrt3\,GM/c^{2}$	3.8 §2 — the null effective potential $W=1/r^{2}-2GM/c^{2}r^{3}$ has a maximum, so light has an unstable circular orbit where matter had a stable one. The critical aim $b_{\text{crit}}=2.598\,r_{s}$ separates capture from escape, and it is what sets the apparent size of the dark region — larger than the horizon, because what you see is the aim that misses rather than the surface itself.	3.8 §2.2's own table, which turns it into the angular size black-hole imaging measures — $54\ \mu$ as for the Galactic centre — and §6, where the surface it surrounds is taken apart. A ray aimed at exactly $b_{\text{crit}}$ spirals in and takes infinitely many turns to arrive.
Deflection of light $\alpha=4GM/c^{2}b=1.7512''$	3.8 §§3–4 — the oldest open debt in Part III, paid. The orbit equation $u''+u=3GMu^{2}/c^{2}$ has no Newtonian source term at all; its unperturbed solution is a straight line and its first correction gives the answer. Then §4 does what 3.1 could not: keep only $g_{00}$ and get exactly $2GM/c^{2}b$ , keep only $g_{rr}$ and get exactly $2GM/c^{2}b$ , and the two first-order solutions add term by term. Repeating it for a body at speed $v$ gives $\alpha_{\text{space}}/\alpha_{\text{time}}=v^{2}/c^{2}$ , which is why a planet samples one half and a ray samples both. 3.1's confessed factor of two was spatial curvature, and the register is now closed.	3.8 §3.6 (lensing, and the Einstein radius, from ⚑ a lens equation stated and not derived), and Worked example 2, which checks the two halves against the Earth. ⚑ Two eclipse expeditions measured it in 1919 to some tens of per cent; radio interferometry against quasars occulted by the Sun now confirms it to about one part in $10^{4}$ .
Gravitational redshift, exactly $\nu_{\infty}/\nu_{\text{em}}=\sqrt{1-2GM/c^{2}r}$	3.8 §5 — derived three ways that share no step: from the conserved Killing charge measured against a local clock, from crest-counting with $\dd\tau=\sqrt{g_{00}}\,\dd t$ , and as the weak-field limit of 3.1's accelerating cabin, which used no general relativity whatever. The satellite-navigation numbers — $+45.7$ , $-7.2$ , $+38.5\ \mu$ s per day, $11.5$ km of position error — come out of one square root rather than two effects glued together, with the neglected cross term computed at four femtoseconds a day.	3.8 §6, where the static observer stops existing at $r_{s}$ and the formula reports it by going to zero; 3.9 §5.1, where the same crest-counting argument with $a(t)$ in place of $A(r)$ gives the cosmological redshift, and the two are pointedly not the same effect.
Kretschmann invariant $K=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=48G^{2}M^{2}/c^{4}r^{6}$	3.8 §6.4 — the instrument that settles the horizon, and it cannot be argued with, because a scalar's value at an event does not depend on labels. $R$ and $R_{\mu\nu}$ are useless here — 3.7 solved $R_{\mu\nu}=0$ , so they vanish identically outside the mass — and what survives is the Weyl part 3.4 §6 named and set aside. $K$ is finite at $r_{s}$ , which no change of chart can alter, and it diverges at exactly one place.	3.8 §6.5, where $\sqrt K$ scales the tidal acceleration and shows that a bigger hole has a gentler edge; §7, which therefore goes looking for a better chart and finds one; §8, where $\lim_{r\to0}K=+\infty$ makes the centre a genuine end of the theory rather than a bad label. Its curvature radius $K^{-1/4}$ falling through the Planck length is what hands the problem to 7.1.
Eddington–Finkelstein chart, and the horizon $\dd s^{2}=\big(1-r_{s}/r\big)\dd v^{2}-2\,\dd v\,\dd r-r^{2}\dd\Omega^{2}$	3.8 §7 — built by integrating the radial null condition, so the chart follows the light rather than fighting it. It is manifestly regular at $r_{s}$ , with $\det g=-r^{4}\sin^{2}\theta$ , and the two radial null slopes are $-1$ and $(r-r_{s})/(r+r_{s})$ : the second passes through zero at $r_{s}$ and is negative inside, so both future directions point inward. A one-way surface, computed rather than drawn. The proper distance down is finite, the proper time to cross is finite, and only the coordinate time is logarithmically infinite. ⚑ The general definition of an event horizon is quoted.	3.8 §7.5, which compares 2.3's Rindler horizon honestly — same local character, different global status, since that one belongs to an observer and this one belongs to nobody. 3.9 Worked example 1, whose cosmological horizons are integrals of the same null condition. 7.9, which asks what the area of this surface is counting.
The singularity at $r=0$ ⚑	3.8 §8 — where general relativity stops. $K$ diverges, so no chart repairs it, and ⚑ the singularity theorems say it is not an artefact of the spherical symmetry either — quoted with their hypotheses named, including an energy condition this book had never had occasion to state.	7.1, which asks what replaces a classical smooth metric once the curvature radius falls below the Planck length, and 7.9, where the honest accounting is done.
FLRW metric $\dd s^{2}=c^{2}\dd t^{2}-a^{2}(t)\Big[\dfrac{\dd r^{2}}{1-kr^{2}}+r^{2}\dd\Omega^{2}\Big]$	3.9 §2 — forced, not chosen. Two symmetry assumptions, no special place and no special direction, are six Killing fields between them; they fix $g_{00}=1$ and $g_{0i}=0$ by inspection, reduce the spatial geometry to one of constant curvature, and turn that condition into a single separable equation whose regular solution is the bracket above. One function $a(t)$ survives out of ten. And $k$ is a statement about curvature, not about whether space is finite — the fifth time in this book a local fact fails to determine a global one.	3.9 §§3–6 entirely, and both worked examples. The scale factor is a ratio and never a length, which is why §5.5 can list what does not expand: an atom, the Solar System, a bound galaxy, a ruler.
Friedmann equations $\Big(\dfrac{\dot a}{a}\Big)^{2}=\dfrac{8\pi G\rho}{3}-\dfrac{kc^{2}}{a^{2}}$ , $\ \ \dfrac{\ddot a}{a}=-\dfrac{4\pi G}{3}\Big(\rho+\dfrac{3p}{c^{2}}\Big)$	3.9 §3 — the $00$ and $11$ components of $G_{\mu\nu}=\kappa T_{\mu\nu}$ , computed from the connection up, with the second obtainable only after the first has been substituted into it. The pressure term in the second is 3.6 §5.5's discovery doing real work: whether the expansion accelerates depends on $\rho+3p/c^{2}$ and not on $\rho$ . With them come the critical density $\rho_{\text c}=3H^{2}/8\pi G$ and the bookkeeping $\Omega_{\text r}+\Omega_{\text m}+\Omega_{\Lambda}+\Omega_{k}=1$ .	3.9 §3.5 (the one static solution, $\rho_{\text m}=2\rho_{\Lambda}$ with $k=+1$ , which perturbs to $\ddot\varepsilon=+\Lambda c^{2}\varepsilon$ and falls over in ten billion years — the calculation 3.6 §6.3 promised), §4 and §5. ⚑ With the measured parameters the universe is $13.8$ Gyr old, began accelerating at $z=0.63$ , and its two best measurements of $H_{0}$ disagree by five times their uncertainty.
Fluid equation, and the equation of state $\dot\rho+3\dfrac{\dot a}{a}\Big(\rho+\dfrac{p}{c^{2}}\Big)=0$ , $\ \ \rho\propto a^{-3(1+w)}$	3.9 §§3.3, 4 — not a third equation. Differentiating Friedmann I and subtracting $2a'\times$ Friedmann II returns it exactly, which is 3.4's contracted Bianchi identity met a third time and now as arithmetic. One number $w=p/\rho c^{2}$ per fluid then gives the dilution law by one separable equation and $a\propto t^{2/3(1+w)}$ by another, with $\ee^{H_{\Lambda}t}$ for the case $w=-1$ that the formula misses. The three exponents put radiation first, matter second and the cosmological term last, whatever the amounts, and the expansion accelerates exactly when $w\lt-\tfrac13$ .	3.9 §4.4 (the exact matter-plus- $\Lambda$ solution $a\propto\sinh^{2/3}(3H_{\Lambda}t/2)$ , and the age), §6.5, where it is all that survives of energy conservation, and Worked example 2.
Cosmological redshift $1+z=a_{0}/a_{\text{em}}$	3.9 §5 — derived twice, by routes sharing no step: following $p^{0}$ along a null geodesic gives $p^{0}\propto1/a$ , and crest-counting gives the same answer. No velocity appears anywhere in the derivation or in the result. Comoving worldlines are geodesics because $\Gamma^{i}{}_{00}=0$ , so nothing is moving through anything; proper distances grow as $\dot d=Hd$ and exceed $c$ beyond $4.45$ Gpc, which contradicts nothing, because velocities at different points live in different tangent spaces. The balloon is right about having no centre and wrong about the room it sits in.	3.9 §6.4 (the photon gas loses $99.91\%$ of its energy this way and nothing holds it), Worked example 2, which takes one galaxy at $z=1$ apart completely. The warning box separating it from the Doppler shift of 2.5 and the gravitational shift of 3.8 is the one to keep.
No timelike Killing vector, hence no conserved energy	3.9 §6 — the thesis of the chapter, and it is Noether rather than a paradox. A conserved energy is the output of 1.4's theorem and its input is a symmetry; in a curved spacetime that symmetry is a timelike Killing vector; and there is none here, because a Killing flow preserves every geometric scalar while $R=-\tfrac{8\pi G}{c^{2}}(\rho-3p/c^{2})$ changes with time whenever anything dilutes. So the theorem returns nothing. What survives is $\nabla_{\mu}T^{\mu\nu}=0$ , which in this spacetime is the fluid equation; what obstructs adding it up is the term $3(a'/a)p$ and the fact that $\nu$ is a free index.	Three chapters wrote it down in advance and §6 pays all three at once: 1.1 §2, which promised it three parts early; 1.4 §4.3's honest note, whose three tiers are checked one by one in §6.5; and 3.5's closing brick. It is also why gravity has no local energy density — 3.4 §5.3's locally inertial coordinates make one impossible.
Black-hole entropy $S/k_{\text B}=A/4\ell_{\text P}^{2}$ ⚑	3.9 §7 — the book's one deliberate loose thread, quoted rather than hidden so that you can see the size of what is missing. It is an area where every other entropy in physics is a volume, it contains $\hbar$ and $G$ together, and general relativity's own answer for it is zero. The classical half of the derivation runs through the boundary term 3.6 §4.5 discarded; the other half needs a list of microscopic states, and no theory in Parts 0 to VI supplies one.	7.9, which collects it by counting brane configurations at weak coupling and following the count to strong coupling, producing $A/4\ell_{\text P}^{2}$ with the factor of four and nothing adjusted. That single number is why Part VII exists in this book at all.

Part III is complete

Part II left gravity acting instantaneously, which Part II had itself made impossible. The repair turned out not to be a faster force but the removal of the force: nine chapters ago the equality of inertial and gravitational mass was one unexplained coincidence, and it is now the statement that free fall is a property of spacetime rather than of the falling body.

The machinery came in a fixed order, and each piece was built because the previous one was not enough. A manifold, because a global inertial frame no longer exists. A connection, because comparing vectors at different points needs a rule and the rule is not unique. Curvature, because the connection depends on the chart and its commutator does not. The field equations, because the only divergence-free combination of the curvature is essentially forced once the source is fixed.

Then the machinery was spent. One exact solution produced Mercury's perihelion to within the measurement, bent starlight by twice the Newtonian amount, and put a horizon where nothing is singular. The same equations applied to the universe give an expansion the theory did not ask for — and a loose thread, since energy is not conserved in that universe and Chapter 1.1's oldest promise is collected by explaining why.

Part IV · Quantum Mechanics

Object	Introduced — and what for	Spent in
Universal cavity spectrum $u_{A}(\nu,T)=u_{B}(\nu,T)$ ⚑	4.1 — because the light leaving a surface is the product of that material's emission and absorption, so there is nothing clean to predict about it. Kirchhoff joins two cavities through a narrow-band filter: a spectrum differing in the two would move heat from cold to hot. Hence $u$ depends on $\nu$ and $T$ and on nothing else — not the walls, not the shape, not the volume. The second law is the one quoted input, and it is quoted in the weakest form that works.	4.1 §1.3 and §2, which is what licenses replacing a furnace by a cube with conducting walls and counting per unit volume; 4.1 §4.6, where universality is what forces $E_{2}-E_{1}$ to be the same function of $\nu$ for every atom rather than a property of the atom.
Density of modes $g(\nu)=8\pi\nu^{2}/c^{3}$	4.1 — 0.8's chain of oscillators in three dimensions: standing waves put the allowed $\vv k$ on a lattice of spacing $\pi/L$ in one octant, counting lattice points in a ball gives $L^{3}k^{3}/6\pi^{2}$ , transversality (2.1) doubles it, and $k=2\pi\nu/c$ converts it. The box cancels, as Kirchhoff said it had to. Built here rather than quoted, because the count is used three more times.	4.18 (occupation numbers) and Part V (field quantisation) — the density of states every later calculation counts with. Its $\nu^{2}$ growth is also what makes the classical integral diverge, so the same formula is the disease and the cure.
Equipartition and Rayleigh–Jeans $u_{\text{RJ}}=8\pi\nu^{2}k_{B}T/c^{3}$	4.1 — 0.6's Boltzmann distribution with the sum over states replaced by a phase-space integral, whose missing unit is kept as a symbol $\varsigma$ on purpose. Each quadratic term pulls a $\beta^{-1/2}$ out of a Gaussian, so $\ln Z=-\ln\beta+$ const and $\avg{E}=k_{B}T$ per mode, independent of frequency. Multiplying by $g(\nu)$ gives a spectrum with no peak whose integral is $\int\nu^{2}\dd\nu=\infty$ .	4.1 §4.4, where only the fact that $u\to\infty$ as $T\to\infty$ is needed to break the two-process scheme; 4.1 §5.1, where it becomes the low-frequency boundary condition that fixes $A_{21}/B$ . The theory being replaced is used, as a boundary condition, inside the derivation replacing it — and the chapter says so rather than letting it read as a derivation from nothing.
Wien's displacement and Stefan–Boltzmann $\nu_{\max}/T$ , $\sigma$ ⚑	4.1 — two measurements from the 1890s, written down in §3.4 before the derivation runs, so that §5's formula can be scored against targets it was not fitted to. Both are quoted; neither is derived.	4.1 §5.5, where $3(1-\ee^{-y})=y$ is solved by Newton's method in three steps and returns $5.8789258\times10^{10}\ \mathrm{Hz\,K^{-1}}$ ; 4.1 §5.6, where $\sigma=2\pi^{5}k_{B}^{4}/15c^{2}h^{3}$ agrees to every digit shown. 4.3 §4.1 comes back for the integral $\int y^{3}(\ee^{y}-1)^{-1}\dd y$ underneath $\sigma$ and makes its term-by-term evaluation airtight.
Einstein's $A$ and $B$ coefficients detailed balance ⚑	4.1 — to get the spectrum with no statistics of light, which cannot honestly be written down until 4.18. Every process changing a level population is given a rate proportional to that population, and to $u$ if it needs radiation. Detailed balance is the assumption that each process is balanced by its own reverse separately, which is strictly stronger than a steady state as soon as there are three levels.	4.1 §4.6 and §5.1, which turn three rate constants into the whole spectrum; 4.2 §10.2, where $A_{21}/B=8\pi h\nu^{3}/c^{3}$ and the ammonia line give the maser's design numbers; 4.17, where $B_{12}=B_{21}$ returns as the equality of two matrix elements that Hermiticity makes automatic.
Stimulated emission $R^{\text{stim}}_{2\to1}=B_{21}N_{2}u$ , $B_{12}=B_{21}$	4.1 — forced, not proposed. Absorption and spontaneous emission alone give $u=(A_{21}/B_{12})\ee^{-h\nu/k_{B}T}$ , which is bounded as $T\to\infty$ while the true $u$ is not. No values of the two constants repair that, and the only term the model's own rules permit is an emission rate proportional to $N_{2}u$ . One limit then gives $B_{12}=B_{21}$ in a line. Einstein wrote it down in 1917 with no experiment suggesting it.	4.2 §10.2 — the ammonia maser, with every number in its design on one page. Worked example 3 of 4.1 gets the ratio of stimulated to spontaneous emission as $1/(\ee^{h\nu/k_{B}T}-1)$ , which is $2.55\times10^{3}$ at microwave frequencies and $1.2\times10^{-33}$ at optical ones, and proves that $N_{2}\gt N_{1}$ is impossible in equilibrium at any temperature. That swing of $10^{36}$ is the quantitative reason the maser came first and the laser did not.
The Planck spectrum $u=\big(8\pi h\nu^{3}/c^{3}\big)\big(\ee^{h\nu/k_{B}T}-1\big)^{-1}$	4.1 — the mode count is untouched; what changed is the energy per mode, from $k_{B}T$ to $h\nu/(\ee^{h\nu/k_{B}T}-1)$ . High-frequency modes are not sharing the heat, and that is what cures the divergence. The shape is derived from detailed balance and Boltzmann weights; the normalisation is matched against Rayleigh–Jeans.	4.18 §5, which derives it a second time from Bose–Einstein statistics and computes the constant instead of matching it, at which point the argument here stops being a fit; 4.1's own Worked example 1, which takes 3.9's $2.7255\ \mathrm{K}$ microwave background apart as a blackbody. Wien's 1896 law is its high-frequency limit and was never an input.
$h$ , and the dimensions of action $[h]=\mathsf{ML}^{2}\mathsf{T}^{-1}$	4.1 — at this stage a fitted number and nothing more. What is not claimed: that energy comes in indivisible units, that oscillator levels are discrete, or that light is made of particles. What is claimed is that its dimensions are energy times time, which is also momentum times length — exactly the phase-space unit §3.1's classical partition function was missing and classical mechanics had no way to supply.	4.2 §7.3, where dividing by it converts the generator of evolution into an energy; 4.2 §8.1, where $[\hat x,\hat p]$ has the dimensions of action and so the commutator is dimensionally forced once it is assumed to be a multiple of the identity; 4.10, which makes the phase-space statement precise — an orbit enclosing area $\mathcal A$ corresponds to about $\mathcal A/h$ states.
Photoelectric relation $K_{\max}=h\nu-\phi$ ⚑	4.1 — because the classical estimate of the delay is $11.6\ \mathrm{s}$ at $1\ \mathrm{W\,m^{-2}}$ against nanoseconds measured, $K_{\max}$ is independent of intensity, and there is a sharp threshold. The measurements are quoted; the reading is that light delivers energy to one electron in an indivisible amount $h\nu$ . The line's slope is the same for every metal and only its intercept is the material's.	4.1 §6.3 itself, where two mercury lines on sodium give $h=6.626069\times10^{-34}\ \mathrm{J\,s}$ from a photocell and a voltmeter — the same constant as §5, measured with no furnace and no thermodynamics anywhere in the apparatus. Then 4.1 §6.5, where it is the time component of $p^{\mu}=\hbar k^{\mu}$ .
Compton shift $\lambda'-\lambda=(h/m_{e}c)\big(1-\cos\theta\big)$ ⚑	4.1 — collected from 2.5's Worked example 1 unchanged, exactly as that chapter said it would be. A classical wave shakes an electron at the driving frequency and it re-radiates at that frequency, so the predicted shift is zero at every angle for every material. Compton's numbers are quoted and the calculation is not: the measured shift is $2.43\ \mathrm{pm}$ at $90^{\circ}$ , and its independence of $\lambda$ and of the material is the whole evidential weight.	4.1 §6.5, where it is the space component of $p^{\mu}=\hbar k^{\mu}$ . It also discharges 2.5's one flagged import: that chapter quoted $E=hc/\lambda$ and named 4.1 as where it would be argued for, and §5 and §6.3 argue for it twice by unrelated means.
$\hbar$ , and $p^{\mu}=\hbar k^{\mu}$ $\hbar=h/2\pi=1.054571817\times10^{-34}\ \mathrm{J\,s}$	4.1 — 2.5 §7.1 built $k^{\mu}$ , proved it a four-vector by counting crests, noticed $k\!\cdot\! k=0=p\!\cdot\! p$ , and closed with "Chapter 4.1 supplies the missing constant." It is $E/\omega=h/2\pi$ . Each component of the resulting equation has its own experiment and the two experiments are unrelated, which is what promotes a structural resemblance into a law. Being an equation between four-vectors, it also makes $\hbar$ a Lorentz invariant.	All of Parts IV–VII. 4.2 §7.3, where $\hat K=-\ii\hat G/\hbar$ ; and Part V, which quantises this relation. Note what 4.1 §6.6 refuses to conclude: none of this says light "is a stream of particles", and $g(\nu)$ is a count of standing waves that would be meaningless for a gas of them.
Rydberg formula, and the sixteen-picosecond atom $\tfrac1\lambda=R\big(n_{1}^{-2}-n_{2}^{-2}\big)$ ⚑	4.1 — the two failures that are structural rather than numerical. Five measured hydrogen lines are fitted by one constant to nine parts in a million, and nothing classical produces integers: a bound charge may have any energy and so radiates a continuum. Worse, it cannot orbit at all — Larmor's formula ⚑, quoted from Maxwell, plus one separable integral gives $t_{\text{fall}}=c^{3}m_{e}^{2}r_{0}^{3}/4k_{e}^{2}=1.556\times10^{-11}\ \mathrm{s}$ after $2.05\times10^{5}$ turns, emitting a rising chirp. The same model has the binding energy right to a part in a thousand, which is what makes the failure interesting.	4.13, which derives the formula including the value of $R$ , and supplies the two integers — one of the three or four things quantum mechanics is believed for. The model is not repaired anywhere; 4.1 §7.2 shows there is no coefficient to adjust, which is why 4.2 changes what a state is instead.
The table of renamings	4.2 — twenty-two rows, each a theorem of 0.5 on the left with its equation number and the same statement about measurement on the right. Not analogies: one statement written twice. It exists to make the size of what has to be added visible before any of it is added, and it itemises the sentence 0.5 §6.5 put the count in — "that is one postulate, not four. Everything else is renaming." Real outcomes, perfect distinguishability, superposition and probabilities summing to one are theorems, proved four parts earlier with no physics in the room.	Every later section of 4.2 either points at a row of it or opens a box, which is what makes the claim checkable. The whole apparatus descends from one imported theorem, the fundamental theorem of algebra ⚑ (0.4 §7) — without it an observable might have no eigenvalues at all. That debt is paid in 5.4, as a corollary of Liouville's theorem.
P1 — a state is a unit ray ⚑ $\ket\psi\sim\ee^{\ii\alpha}\ket\psi$	4.2 — the first of seven assertions. Unit length is bookkeeping, chosen so that Parseval reads as probability rather than having probability normalised afterwards. The ray is physics: every prediction is quadratic in $\ket\psi$ , so the overall phase cancels in one line of conjugation. The relative phase is the opposite case, giving $\abs{\avg{+\|\psi_{\beta}}}^{2}=\cos^{2}(\beta/2)$ — a fringe, and the difference between full transmission and extinction. Counting: $2n-2$ real parameters, so two for a two-level system, and the Bloch sphere is the state space rather than a picture of it.	4.2 §10, where $\bar E$ drops out of the two-state evolution because it is a global phase, so only energy differences are observable and that is P1 rather than a separate principle. 4.12, which measures $\avg{\sigma_{x}},\avg{\sigma_{y}},\avg{\sigma_{z}}$ with three Stern–Gerlach magnets — the sphere's Cartesian coordinates. 4.19, where the interior of the ball is the mixtures. And 4.1 §6.6's tension: the wave behaviour now lives in the relative phase.
P2 — an observable is a Hermitian operator ⚑ $\hat A=\sum_{k}\lambda_{k}\hat P_{k}$	4.2 — 0.5's "the one postulate that has to be made", collected. It is not assumed that outcomes are real, that distinct outcomes are distinguishable, or that they span; those are 0.5 §6 and are read off. Grouping the decomposition by distinct eigenvalue rather than by basis vector is deliberate: an observable is a labelled partition of the space into perpendicular pieces, and that form survives infinite dimension.	Everywhere. 4.4 §4 sharpens the word to self-adjoint, which is not pedantry: 4.4 §5 shows $-\ii\hbar\,\dd/\dd x$ on $[0,\infty)$ is symmetric with no self-adjoint extension, so the momentum of a particle confined to a half-line is not an observable at all. A physical conclusion, from a domain.
Complete set of commuting observables quantum numbers $(a_{1},\dots,a_{m})$	4.2 — 0.5 §8 renamed. Commuting observables have common eigenstates; the content sits in Step 4, where inside a degenerate eigenspace of $\hat A$ the restriction of $\hat B$ is still Hermitian and chooses the basis $\hat A$ could not. A set is complete when the common eigenspaces are one-dimensional, and the list of eigenvalues is then what a set of quantum numbers is. Writing $\ket{n,\ell,m_{\ell}}$ is naming three eigenvalues and nothing more.	4.9 §3, which asks how one knows a set is complete; 4.11 and 4.13, which spend it on angular momentum and on hydrogen. The caution attached is worth keeping: the theorem promises a common eigenbasis exists, not that a given eigensolver returns it, which is exactly what 4.13's degeneracies need.
P3 — the Born rule ⚑ $\Pr(\lambda_{k})=\norm{\hat P_{k}\ket\psi}^{2}$	4.2 — because nothing so far connects any number in the theory to a relative frequency. The formalism already contains a list of non-negative numbers summing to one, by Pythagoras; that it is consistent with being a probability distribution is not a demonstration that it is one. The first thing in twenty-nine chapters posited rather than cornered, and permanently open — elsewhere ⚑ means this book chose not to prove this, and here it means nobody has. Gleason's theorem ⚑ is quoted with its hypotheses and explicitly not presented as a derivation: it assumes probabilities are assigned to projections, additively, and needs dimension at least three, which excludes spin- $\half$ .	Everything in Part IV that produces a number. 4.3 §5.3, which is why the rule for a continuous variable is stated as $\int_{a}^{b}\abs\psi^{2}\dd x$ over a region and never as $\abs{\psi(x_{0})}^{2}$ at a point — a vector of $L^{2}$ has no value anywhere. 4.20 §9 returns to what the mark costs.
Expectation value and variance $\avg{\hat A}=\bra\psi\hat A\ket\psi$ , $(\Delta A)^{2}=\norm{(\hat A-\avg{\hat A})\ket\psi}^{2}$	4.2 — derived from P3 in two lines each, not adopted as definitions of "average" and "spread". Substitute P3 into $\sum_{k}\lambda_{k}\Pr(\lambda_{k})$ and run the spectral decomposition backwards.	4.9 — the second form exhibits $\Delta A$ as the length of a vector, and Cauchy–Schwarz (0.5 §1.4) is a statement about lengths. That is the whole reason the uncertainty principle is one line there and not a new idea.
P4 — the state update ⚑ $\ket{\psi'}=\hat P_{k}\ket\psi/\norm{\hat P_{k}\ket\psi}$	4.2 — kept separate from P3, where most treatments merge the two. P3 is about frequencies in an ensemble; P4 is about one system after one outcome. The denominator is not a second assumption, being the square root of P3's probability. Repeatability then follows from $\hat P_{l}\hat P_{k}=\delta_{lk}\hat P_{k}$ : an immediately repeated measurement returns the same value with certainty, which is what makes the word "measurement" mean anything.	4.20 §9, which needs the separation — decoherence gives a good account of why alternatives stop interfering, which is P3's half, and no account at all of why one of them happens, which is P4's. The algebra is conditioning exactly, symbol for symbol, until you notice that what is being restricted is an amplitude: 4.2's Worked example 1 gives $\half$ when the middle magnet is read and exactly $0$ when it is not.
P5 — evolution is generated by the energy ⚑ $\ii\hbar\,\dv{}{t}\ket{\psi}=\hat H\ket{\psi}$	4.2 — exactly half of this is derived. Linearity plus conservation of total probability makes $\hat U(t)$ norm-preserving; three lines of polarisation make norm-preserving mean unitary; the group law $\hat U(s+t)=\hat U(t)\hat U(s)$ differentiated at zero makes $\hat U=\ee^{t\hat K}$ with $\hat K$ anti-Hermitian; dividing by $\hbar$ makes the generator Hermitian with the dimensions of energy. That it is the energy — the calorimeter's, 1.3's Legendre transform, 4.1's $h\nu$ — is the postulate and the only physics in the section.	4.6 §2, which states the sign convention and shows what the identification buys; 4.5 §9, where Stone's theorem is the rigorous infinite-dimensional version of "differentiate the group law"; 4.3 §6.4, since the exponential series maps states to states only because $L^{2}$ is complete; 4.17 §3, for the driven case where the group law fails and the exponential becomes a time-ordered series.
Observables generate symmetries $\{f,g\}\longmapsto\tfrac{1}{\ii\hbar}[\hat f,\hat g]$	4.2 — nothing in §7.3 used the fact that the parameter was time. Expanding $\hat U_{\epsilon}^{\dagger}\hat f\hat U_{\epsilon}$ to first order gives $\delta\hat f=\epsilon\,\tfrac{1}{\ii\hbar}[\hat f,\hat G]$ , which is 1.3 §7's $\delta f=\epsilon\{f,G\}$ under one substitution — precisely the one 1.3 §6.4 announced in advance. Then $[\hat f,\hat H]=0$ says two things at once, and a conserved quantity is the generator of its symmetry in operator form.	4.10 §8, and the news there is negative: the substitution cannot be extended consistently to every classical observable at once. So it is a correspondence for the pair 4.2 §8 postulates it for, and not a general dictionary.
P6 — canonical quantisation ⚑ $[\hat x_{i},\hat p_{j}]=\ii\hbar\,\delta_{ij}\hat I$	4.2 — 1.3's fundamental Poisson brackets under §7.5's substitution, asserted for this one pair. It is postulated here rather than in 4.9, where the substitution was originally routed, because 4.6, 4.8 and 4.11 all need it first. It says nothing about which space or which functions.	4.6 (which supplies the realisation $\hat p=-\ii\hbar\nabla$ ), 4.8, 4.11. 4.10 §8 supplies the sharper statement, that no consistent extension to all polynomials in $x$ and $p$ exists.
No finite-dimensional space carries P6 $\norm{[\hat x,\hat p]-\ii\hbar\hat I}_{F}\ge\hbar\sqrt n$	4.2 — three lines, using one fact from a chapter about determinants. The trace of a commutator is zero by cyclicity (0.4 §6.1); the trace of $\ii\hbar\hat I_{n}$ is $\ii\hbar n$ . Not "too coarse an approximation" — the arithmetic is impossible at every $n$ . Cauchy–Schwarz in the matrix inner product then sharpens it: the miss is at least as large as the thing being reproduced, so a finite model's best strategy is to commute and reproduce none of the relation. Spin is not a counterexample, because $[\hat S_{i},\hat S_{j}]$ is not a multiple of the identity.	4.3, 4.4 and 4.5, which exist because of it. Quantum mechanics is infinite-dimensional before a single physical question has been asked, and 0.5's four uses of finite dimension therefore come due — the space in 4.3, the operators on it in 4.4 and 4.5, in the same order and for the same reason that 0.4 preceded 0.5.
P7 — the tensor product ⚑ $\dim(V\otimes W)=mn$	4.2 — two systems need a rule, and there are two candidates: the dimensions add, or they multiply. Classical configuration spaces add, nothing in P1–P6 rules that out, and which one nature uses is a physical question. Product states have complex dimension $m+n-1$ inside $mn$ , so almost every state of a composite system is not a pair of states of its parts — entanglement names a counting fact, not an influence. Three hundred two-level systems are a small molecule and take $2^{300}=2.04\times10^{90}$ amplitudes, against $600$ angles to describe each of them separately, in a universe of $10^{80}$ atoms.	4.18 to 4.20, which are the source of every effect it produces: the density operator, built because a maximally entangled state assigns no state at all to either half; Bell's inequality, which measures the gap; and P8, the symmetrisation postulate, which is the further restriction for identical parts.
The two-state system, solved once $P_{1\to2}=\sin^{2}2\theta\, \sin^{2}\!\big(\Delta E\,t/2\hbar\big)$	4.2 — the general Hermitian $2\times2$ as a mean $\bar E$ , a detuning $\delta$ and a coupling $\Delta$ , giving splitting $2\sqrt{\delta^{2}+\Delta^{2}}$ , the mixing angle $\tan2\theta=\Delta/\delta$ , and avoided crossings. Read three times in three sets of units: ammonia at $41.89\ \mathrm{ps}$ , neutrino oscillation over $607.3\ \mathrm{km}$ , a $20\ \mathrm{ns}$ NOT gate. The measured parameters are quoted ⚑; the formula is derived, and the linear algebra was finished in 0.5's Worked example 1.	4.17, which supplies the rotating-frame reduction that turns a driven qubit into this matrix with $\delta=0$ , and whose transition rate is built on the fact that every detuned curve leaves the origin along the same parabola $\Delta^{2}t^{2}/\hbar^{2}$ , with $\delta$ absent from the leading term. 4.6's wave packets do the neutrino case honestly.
Cauchy sequence, and completeness	4.3 — defined here because the book had never defined it, and 0.3's convergence tests all quietly stood on it: a test that certifies convergence without exhibiting a limit is only worth having in a space where the limit is guaranteed to be there. Whether a Cauchy sequence converges is not a question about the sequence. It is a question about the space, as the decimal truncations of $\sqrt2$ show inside $\mathbb{Q}$ .	4.3 §6, the whole point of the chapter; §7.2, where square-summable coefficients assemble into a vector; and everything in Parts IV and V that writes a limit of states and expects a state.
Lebesgue measure $\mu^{*}(A)=\inf\sum\abs{I_{k}}$ ⚑	4.3 — because length could not tell a dense set from a fat one. Three demands fix it: it extends length, it is translation invariant, and it is countably additive — and the word "countably" carries all the weight, because every failure in §1 was a countable process. Quoted as a five-clause package (a)–(e), and this is the first time the book sends you to another text for an argument. The restriction to a $\sigma$ -algebra is real, not fastidious: §2.5 builds Vitali's set from the three demands plus the axiom of choice.	Clause (b) becomes continuity from below, which is the only engine of 4.3 §4.1's monotone convergence. Clause (d), regularity, is the one to watch: §7.4 cashes it to make $L^{2}$ separable and §8.1 re-uses the same trade to prove continuous functions dense — two results carrying no separate marks of their own, precisely so that you can watch one flag do the work it claimed.
Null sets, and "almost everywhere" $\mu(\mathbb{Q}\cap[0,1])=0$	4.3 — cover the $k$ -th point by an interval of length $\varepsilon2^{-k}$ and the total is $\varepsilon$ . It is §1.4's thickening of the rationals with the thickness sent to zero instead of held at $\tfrac14$ , and 0.2 could not write it down. A property holds a.e. when the set where it fails is null; for a statistician that word is "almost surely", and Lebesgue measure on $[0,1]$ is the uniform distribution.	4.3 §3.4, where $\int_{0}^{1}\chi=0$ takes one multiplication; §5.3, where it reorganises the whole subject by making two functions agreeing a.e. into one vector. Every later statement about a continuous coordinate inherits it, which is why the Born rule is stated over a region.
The Lebesgue integral $\int f=\sup\{\int\varphi:\varphi\text{ simple},\ 0\le\varphi\le f\}$	4.3 — Riemann slices the domain; Lebesgue slices the range. Sort the coins into denominations before counting them. Eligibility is the single condition $\{f\gt a\}\in\mathcal M$ , which is weak enough that nothing you can write down fails it, and phrased in countable operations precisely so that countable operations cannot escape the class — which is the defect the Riemann class had. It agrees with Riemann wherever Riemann works, so no arithmetic in Parts 0 to III changes.	All of 4.3 §4, §5 and §6. What is lost is stated plainly: an improper integral like $\int_{0}^{\infty}\sin x/x$ is a statement about the order in which cancellations arrive, and sorting the range destroys that order, so Part V has to write it as a limit of proper integrals.
Monotone convergence $0\le f_{1}\le f_{2}\le\cdots\Rightarrow\int f_{n}\to\int f$	4.3 — the first of the two theorems the entire rebuild existed for. Proved from continuity from below and nothing else, which is the one place the word "countably" gets spent. Its corollary is the one used in anger: $\int\sum_{k}h_{k}=\sum_{k}\int h_{k}$ for $h_{k}\ge0$ , with no convergence hypothesis, both sides being allowed to be infinite together. Sign is all that is asked for.	4.1 §5's Planck integral, whose parenthesis named this theorem and this chapter — so the Stefan–Boltzmann constant now rests on a theorem rather than an expectation. Also 4.3 §4.1's proof that the integral is linear, §6.2's Riesz–Fischer, and every chapter of Part V that expands a field in modes and integrates term by term. Fubini is not settled here: 0.2's mark on it stands where it is.
Dominated convergence $\abs{f_{n}}\le g$ integrable $\Rightarrow\int\abs{f_{n}-f}\to0$	4.3 — the theorem 0.2 §4.4 borrowed against by name. Proved from Fatou's lemma, $\int\liminf f_{n}\le\liminf\int f_{n}$ , which is the exact statement that mass can escape but cannot appear from nowhere. The dominating function is the theorem, not a technicality: for 0.2's escaping spike the smallest possible dominating function is $\tfrac{2}{\ee x}$ , whose integral diverges, and the conclusion is false by exactly the amount the hypothesis fails.	0.2's differentiation under the integral sign, now derived rather than provisional, and with it every Gaussian moment and the whole differentiation-with-respect-to-a-source apparatus Part V runs on. Also 4.3 §7.4 (simple functions are dense) and 4.5's proof that the Hermite functions are complete, after which 4.8 expands in them freely.
$L^{2}$ , and the quotient by null sets $L^{2}=\mathcal L^{2}/\{f=0\text{ a.e.}\}$	4.3 — 0.5 §1.2's proposed inner product on functions, tested against its own three axioms. Two hold in a line. The third fails, on the function this chapter opened with, and $\int\abs f^{2}=0\iff f=0$ a.e. says it fails by exactly the width of a null set. So two functions agreeing a.e. are declared one vector. The exponent is $2$ because it is the only power for which the inner product of two members is guaranteed finite.	The price is charged three times. A vector of $L^{2}$ has no value at any point, so the probability of finding a particle exactly at $x_{0}$ is not small but undefined; 4.3 §8.4's Gibbs ear is invisible to the norm; and $\ket x$ cannot be a vector, since a state at one point is the zero vector — 4.5 says what it is instead. Also $L^{2}\not\subset L^{1}$ on the line, which 4.9 needs to explain how a perfectly normalised state can have no mean position.
Riesz–Fischer $L^{2}$ is complete, hence a Hilbert space ⚑	4.3 — the centre of the chapter, and the property 4.2 spent on credit every time it said a limit of states is a state. The proof is quoted but its shape is given: thin to a geometrically converging subsequence, apply monotone convergence to the partial sums, then Fatou. The two marks in this chapter are different in kind — §2.3 imports a construction the book never performs, §6.2 declines to write out an argument whose inputs are only §2.3 and §4. Only one is a genuine debt. §1.4's thickened sequence then lands on $\mathbf 1_{U}$ , at positive distance from every Riemann-integrable function on the interval, which is the receipt that the hole was real.	4.3 §7.2, immediately; 4.5's spectral theorem, which produces every operator as a limit; 4.6's $\ee^{-\ii\hat Ht/\hbar}$ , which maps states to states only because partial sums converge; 4.15's perturbation series, whose first correction is an infinite sum over the unperturbed basis. This is also the proper name 0.9 §1.3 promised the space would eventually be given.
Orthonormal basis in infinite dimensions spanning ⟺ expansion ⟺ Parseval ⟺ no blind spot	4.3 — because 0.4 settled "big enough" by counting dimensions and the count is gone. Bessel's inequality $\sum\abs{c_{n}}^{2}\le\norm f^{2}$ comes first, being 0.5's Parseval computation with equality weakened; then §7.2 turns square-summable coefficients into an actual vector, and that is the single step where completeness of the space enters. The four statements are proved equivalent because (d) is the one you can check and (b) is the one you want to use.	4.3 §8.3, where the Fourier modes satisfy (a) and therefore all four; 4.5, which proves the Hermite functions complete by running (d); and every sum with an index in Parts IV–VII. Note that the two senses of "complete" — of a space and of a basis — are different words, and 4.3 §7.2 is the only place one is used to get the other.
Separability every orthonormal set is countable	4.3 — because a basis could in principle be too big to write as a list, and every physics calculation assumes otherwise. Simple functions are dense by dominated convergence, indicators are nearly finite unions of intervals by §2.3's regularity, and rational endpoints and coefficients make the collection countable. Then two distinct orthonormal vectors are always $\sqrt2$ apart, so disjoint balls around them cannot outnumber a countable dense set.	4.3 §8.1, which re-uses the middle step rather than paying for it again; and §8.5's warning box, where countability is exactly what rules out reading $\{\ket x\}$ as an orthonormal basis, the positions on a line not being countable. Gram–Schmidt run on the dense set also settles existence, which is 0.5's recipe again not caring what the vectors are.
Fejér's kernel $K_{N}=\tfrac{1}{N+1}\big[\sin\tfrac{(N+1)t}{2}\big/\sin\tfrac{t}{2}\big]^{2}$	4.3 — because direct attack on the partial sums fails: their weight is the Dirichlet kernel, a ratio of sines, which changes sign. Averaging the partial sums instead turns the weight into a square. Non-negative, unit integral, concentrating at the origin — which is 0.9 §5.2's delta-sequence with the Gaussian replaced, and the argument is reused without alteration. The theorem is constructive: it writes the approximating trigonometric polynomial down.	4.3 §8.3, where uniform convergence implies $L^{2}$ convergence on a bounded interval and the trigonometric polynomials are therefore dense. The one thing the proof leans on that is not its own is Heine–Cantor, standing since 0.2 §1.1 and cited rather than re-raised.
The Fourier basis is a basis $\norm{f-S_{N}f}_{2}\to0$	4.3 — 0.9 §1.3 proved the modes orthonormal, said in a marked box that it had not shown there were enough of them, and named this chapter. Continuous functions are dense (§8.1), trigonometric polynomials are dense in those (§8.2), so statement (a) of §7.3 holds and all four follow. 0.9's mark is paid, not carried, and it stays on the page where it was raised.	Parseval as a theorem rather than an assumption — used at once on 0.9's square wave, and by every chapter of Part V that expands a field in modes. 4.3 §8.4 then measures the two convergences against each other: the relative $L^{2}$ error falls like $0.63662\,N^{-1/2}$ while the overshoot at the jump sits at $0.0894899$ of the jump forever, because the ear keeps its height and loses its width and a norm integrates. Every measurable prediction is an integral over a region, so the first is the one that counts.
Position and momentum are one space $\hat\psi(p)=(2\pi\hbar)^{-1/2}\!\int\psi(x)\, \ee^{-\ii px/\hbar}\dd x$	4.3 — the chapter's last debt, from 0.9's closing brick. Plancherel says the transform preserves the norm, hence is unitary, and that is a statement about this one map. The $\sqrt\hbar$ rescaling is exactly what the change of variable $p=\hbar k$ contributes, which is why 0.9 chose the symmetric convention. So a state has one norm and the Born rule gives the same total probability in either description.	Every computation in 4.6 to 4.10. What it does not establish is said before you assume it: $\{\ket x\}$ and $\{\ket p\}$ are not orthonormal bases in 4.3 §7.3's sense, $\ee^{\ii kx}\notin L^{2}(\R)$ , and 4.3 §7.4's unitary equivalence of any two separable Hilbert spaces singles out no map and does no work here. 4.5 gives $\ket x$ and $\ket p$ a precise meaning by box normalisation; the general theory of such objects is 5.4's.
Bounded and unbounded operators $\norm{\hat A}=\sup_{\psi\neq0}\norm{\hat A\psi}/\norm{\psi}$	4.4 — because Chapter 0.6 §2 promised that in infinite dimensions a linear map can be unbounded, with $\dd/\dd x$ the standard offender, and the promise came due. Bounded and continuous turn out to be the same condition, so an unbounded operator is a discontinuous one, and $\ee^{\ii kx}$ on a bounded interval exhibits it in one line: the norm is fixed while the derivative's norm runs to infinity with $k$ .	4.5 (the spectrum of an unbounded operator is not a list of eigenvalues); 4.6 onward, since every observable in the book but spin is unbounded.
Hellinger–Toeplitz ⚑	4.4 — the reframing the chapter runs on. A symmetric operator defined on all of a Hilbert space is bounded, in two lines from the closed graph theorem. So an unbounded observable cannot be defined everywhere: the restriction is compulsory, and a domain is forced rather than chosen for convenience.	Everything after it in 4.4, and the reason 4.5's spectral theorem has to be stated for operators that are not defined everywhere.
The domain of an operator, and what dense is for	4.4 — an operator in infinite dimensions is a formula together with a set of inputs, and the pair is the object. Density is not decoration: it is what makes the adjoint well defined at all, and 4.3 §7.4 had already paid for it.	4.5, 4.6 and every chapter that writes down an observable; 5.4, where the same care reappears for distributions.
The adjoint, and its own domain $\operatorname{dom}(\hat A^{\dagger})$	4.4 — the sentence the chapter turns on. Chapter 0.5's definition of the adjoint, read carefully, determines which vectors $\hat A^{\dagger}$ acts on, and there is no reason for that set to be the set $\hat A$ acts on. In finite dimensions the question could not arise.	4.4 §4 immediately; 4.5, where self-adjointness is the hypothesis of the spectral theorem.
Symmetric versus self-adjoint	4.4 — symmetric is $\avg{\hat Au,v}=\avg{u,\hat Av}$ on the domain; self-adjoint is $\hat A=\hat A^{\dagger}$ with the domains equal. Chapter 4.2 §4 said "Hermitian" and it was not enough, and 4.4 §4 is the section two of its sentences point at. The boundary term left by integration by parts — which Chapter 0.2 §3.2 waved through — is where the difference lives.	4.5 (only self-adjoint operators have a spectral theorem, hence only they are observables); 4.6 §2 and every Hamiltonian after it.
Deficiency indices ⚑ $(n_+,n_-)$	4.4 — von Neumann's count of the self-adjoint extensions of a symmetric operator, quoted after §5 has already produced the three answers by hand, so it lands as arithmetic the reader owns. $\hat p$ gives $(0,0)$ on $\R$ , $(1,1)$ on $[0,L]$ and $(1,0)$ on $[0,\infty)$ , matching §5 three times.	4.4 §7, where $-\dd^{2}/\dd x^{2}$ on a box turns out to have $(2,2)$ and therefore a four-real-parameter $U(2)$ of extensions, not the one parameter that is easy to guess — so the boundary condition is physics rather than bookkeeping.
The spectrum, redefined by invertibility $\sigma(\hat A)$	4.5 — because the old definition breaks. Chapter 0.5 tied an eigenvalue to an eigenvector, and §1.2 shows that position on $L^{2}(\R)$ has no eigenvectors at all: $(x-\lambda)\psi=0$ forces $\psi$ to vanish off a null set, which Chapter 4.3 §5.3's quotient makes the zero vector. So $\lambda$ is in the spectrum when $\hat A-\lambda$ fails to have a bounded inverse on the whole space. Rank–nullity gives the old answer back in finite dimensions, so nothing was renamed.	4.5 §3 (the theorem is about this set); 4.6 §4.4 and 4.7 §1.6, where checking an operator means checking what this set is; 4.13, where hydrogen's is a ladder plus a half-line.
Point, continuous and residual spectrum	4.5 — the three ways the inverse can fail, and the two facts about self-adjoint operators that §2 proves rather than assumes: the spectrum is real, and the residual part is empty. That second fact is what makes the classification usable, since it leaves exactly two cases to think about.	4.5 §6 (a measure needs the classification); 4.6 §9.3, where a pure point spectrum is exactly the condition under which an eigenfunction expansion is the general solution; 4.7 §5, where the continuous part is what a scattering state belongs to.
The spectral theorem in infinite dimensions ⚑	4.5 — every self-adjoint operator is unitarily equivalent to multiplication by a real function on some $L^{2}(\mu)$ . That is Chapter 0.5's $A=UDU^{\dagger}$ with the list of eigenvalues widened into a function and the eigenvectors dropped. Quoted, and then checked three times by hand in §4: position is already multiplication by $x$ ; momentum becomes multiplication by $\hbar k$ under the Fourier transform; and the oscillator becomes multiplication by $(n+\half)\hbar\omega$ on the non-negative integers. Section 5.6 says exactly how far that checking reaches, and the mark's second clause is uniqueness of the measure, which §6.1 needs.	Everything after it. 4.5 §6 and §9; 4.6 §2 (Stone's hypothesis) and §9; 4.7 §1.6; 4.8, 4.13 and every chapter that solves a system by diagonalising its Hamiltonian.
The Hermite functions, and their completeness $h_{n}$	4.5 — built rather than quoted, because they are the third verification of the theorem and the one that costs something. A generating function gives two recurrences, hence the Hermite equation, hence $\hat H_{\text{osc}}h_{n}=(n+\half)\hbar\omega h_{n}$ ; orthonormality comes from squaring the same generating function. Completeness runs statement (d) of Chapter 4.3 §7.3.	4.8, where the same functions come back out of the ladder algebra with no integrals at all, which is the point of doing it twice; 4.15 and 4.16 (matrix elements); Part V.
The projection-valued measure $\dd P(\lambda)$ , and $\hat A=\int\lambda\,\dd P(\lambda)$	4.5 — because the Born rule needs a projection and a continuous observable has none to offer. There is no projection onto "the state with position $x_{0}$ ", so there is no probability to compute until the projection is attached to an interval instead. The measure is what does that, and the integral replaces Chapter 0.5's sum over eigenvalues.	4.5 §6.6 and §7 (the functional calculus and what $\ket x$ means); 4.6 §9.3, where the integral is what the sum cannot do; 4.19 and 4.20, where measurement is stated in this language.
Stone's theorem ⚑ $\hat U(t)=\ee^{-\ii\hat Ht/\hbar}$	4.5 — a strongly continuous one-parameter unitary group has a unique self-adjoint generator, and conversely. The forward half is built in §9.1 out of the functional calculus. The converse is the quoted half, and it is the one that matters, because it is what makes time evolution follow from self-adjointness rather than being postulated beside it.	4.6 §2, immediately and entirely: the whole of the next chapter is this theorem applied to the three properties evolution has to have.
Unitary evolution as a strongly continuous group $\hat U(s)\hat U(t)=\hat U(s+t)$	4.6 — the three conditions that go into the derivation, each argued rather than assumed. Unitarity is linearity plus the Born rule, by Chapter 4.2 §7.2. The group law is the statement that a system left alone cannot tell one instant from another. Strong continuity is the weaker of the two continuities on purpose: §1.4 shows the stronger one forces a bounded generator, and every Hamiltonian in this book is unbounded.	4.6 §3, where Stone converts the three into the equation; 4.9 §4 (the Heisenberg picture is the same group moved onto the operators); 5.1, where the same argument is run on the Poincaré group.
The Schrödinger equation $\ii\hbar\,\dv{}{t}\ket\psi=\hat H\ket\psi$	4.6 — not postulated. Differentiate $\hat U(t)$ at the origin and the equation is what is left, holding for every state in $\operatorname{dom}(\hat H)$ while the exponential itself holds for every state without exception. Chapter 4.2 §7.4 wrote the same box in finite dimensions with the generator postulated; here it is earned, and Chapter 0.1 named this pair of equations in its opening pages as the law the book was heading for.	4.7, 4.8 and 4.13 solve it; 4.10 takes its classical limit; 4.15 perturbs it; 4.17 drives it; and 5.2 replaces it with a relativistic equation and explains what goes wrong.
The Hamiltonian, identified ⚑ $\hat H=\hat{\vv p}^{2}/2m+V(\hat{\vv x})$	4.6 — the one physical choice inside the derivation, marked in §4.2 as the choice it is. Nothing above it says which operator the generator is. This particular expression is safe to write without an ordering convention because no term contains both $\hat{\vv x}$ and $\hat{\vv p}$ ; the general procedure is not safe, and 4.10 §8 proves it cannot be made so. Section 4.4 then checks self-adjointness case by case and refuses the shortcut, since unitary equivalence to a multiplication operator is not inherited by sums.	Tested rather than proved: 4.9's Ehrenfest relations, and 4.13 and 4.16, which get the hydrogen spectrum with no free parameters. 4.7 §1.6 and 4.13 run the self-adjointness check the section demands.
The position representation $\hat p=-\ii\hbar\nabla$	4.6 — a realisation of the canonical pair, obtained by asking what generates translations, not a definition. Stone–von Neumann ⚑ is what makes it unique up to unitary equivalence, and §5.4 states its hypotheses in full, because they are the hypotheses that fail in 5.3. The relative sign is not conventional: the commutator fixes $-\ii\hbar$ rather than $+\ii\hbar$ .	4.6 §6, which turns the abstract equation into $\ii\hbar\,\partial_{t}\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V\psi$ and shows that the Laplacian is not an extra ingredient but what kinetic energy looks like after the Fourier transform; 4.7, 4.8 and 4.13, which solve that equation; 4.10, which takes $\hbar\to0$ in it; 5.3, where the uniqueness hypothesis fails and the failure is the point.
The probability current $\vv J=\frac{\hbar}{m}\operatorname{Im}(\psi^{*}\nabla\psi)$	4.6 — because a global statement that $\int\abs\psi^{2}\dd V$ is constant is much weaker than the claim anyone wants, which is that probability does not vanish here and reappear there. Nothing may be chosen to make it come out: $\rho=\abs\psi^{2}$ is fixed, so either a $\vv J$ exists satisfying $\partial_{t}\rho+\nabla\vdot\vv J=0$ or local conservation is false. The derivation uses one property of $V$ and §8.5 shows what breaks without it: $V$ is real.	4.7 §5, where $T+R=1$ comes out of it and where the flux ratio is the only honest definition of a transmission probability; 4.10 §4; and Part V's conserved currents, which are this one generalised.
Stationary states $\hat Hu=Eu$ , $\psi=u\,\ee^{-\ii Et/\hbar}$	4.6 — separation of variables, in three lines, and then the condition that says when they are enough. The expansion of a general state over them is legitimate exactly when the spectrum is pure point; hydrogen sits on the other side of that condition, and finding its bound states is not the same achievement as diagonalising its Hamiltonian.	4.7 and 4.8 find them; 4.13 finds hydrogen's; 4.15 perturbs them; and 4.7 §5 handles the case where the condition fails.
The free packet: group and phase velocity $v_{g}=p_{0}/m$ , $v_{p}=v_{g}/2$	4.6 — the first solution of the equation, and the one that settles what a wavefunction is a wave of. A Gaussian packet moves at $v_{g}=p_{0}/m$ and spreads as $\sigma(t)=\sqrt{\sigma_{0}^{2}+(\hbar t/2m\sigma_{0})^{2}}$ , which is linear in $t$ at long times rather than diffusion's $\sqrt t$ . De Broglie's relation ⚑ is the one measurement quoted in the chapter, and it is quoted where the numbers arrive.	4.7 §5.2 (what a scattering ratio means physically); 4.9 (the uncertainty relation this width saturates); 4.10 §2; 5.4, where the propagator is this calculation done for every initial state at once.
Matching conditions, as a domain condition $u$ and $u'$ continuous	4.7 — not a recipe. A jump in $u'$ puts a delta in $u''$ , hence in $\hat Hu$ , which then leaves the space, so the joining conditions are $\operatorname{dom}(\hat H)$ written out. Chapter 4.4 §7 is what makes this a statement rather than a convention: an infinite wall is one point of a $U(2)$ , and 4.7 §1.5 selects Dirichlet out of it by the limit of a high finite barrier rather than by assumption. 4.7 §1.6 then checks self-adjointness in all three cases the chapter uses.	4.7 §§3–6 throughout, and §4.6, where a delta potential is one more point of the same classification with the jump prescribed; 4.13 §1, where the origin of a radial coordinate is the case the argument does not reach.
Parity $\hat\Pi u(x)=u(-x)$ , $\hat\Pi^{2}=\hat I$	4.7 — an involution, so its eigenvalues are $\pm1$ and there is no third; and $[\hat\Pi,\hat H]=0$ whenever $V$ is even. Every bound state of an even potential is then even or odd, which needs the non-degeneracy §2.4 proves from the Wronskian. It halves the finite well's algebra here, which is what it was introduced to do.	4.16, where it kills half the fine-structure matrix elements; 4.17 §7, where it is the electric-dipole selection rule; 4.18, where exchange runs the identical three-step move. Named once in 4.7 §2.5 as the pattern: a symmetry, a commuting observable, a label.
The infinite well $E_{n}=n^{2}\pi^{2}\hbar^{2}/2mL^{2}$	4.7 — the first problem solved end to end, and the cheapest. The discreteness does not come from the differential equation, which has solutions at every $E$ ; it comes from the boundary condition, which is to say from the domain, which is to say from the wall. The ground state cannot have $E=0$ because $\psi\equiv0$ is not a state, which is the honest reason and not an appeal to a principle.	4.7 §3.5 (read as an uncertainty, against 0.9 §6's bandwidth theorem); Part V's density of states; and every later problem in which a box is the zeroth approximation.
The finite well, and counting its states $N=\lceil 2z_{0}/\pi\rceil$	4.7 — the first answer in the book that is not a formula. One dimensionless number $z_{0}=a\sqrt{2mV_{0}}/\hbar$ carries the whole problem, the levels are crossings of two curves, and the count is a matter of asking how many branches begin before $z_{0}$ . A square well in one dimension always binds at least one state, which is false in three, and the difference is exactly the even sector being available on a line and not on a half-line.	4.7 §4.6 (the delta well as the limit with exactly one state) and §7's double well, which is the $\Delta$ Chapter 4.2 §10.2 could not compute; 4.13 §2, where the three-dimensional threshold returns.
Scattering states, and $T+R=1$	4.7 — above the top of a step there are no eigenvectors, so what is computed is a ratio, and ratios survive the failure of normalisation by Chapter 4.5 §7.4's procedure. The ratio has to be one of fluxes, using 4.6 §8's current: $\abs t^{2}$ alone is wrong the moment the two sides have different wavenumbers, and the error hides because it vanishes for every barrier.	4.7 §6, where the same algebra gives tunnelling; 4.17 (resonances and lifetimes); 5.4 and Part V, where a cross-section is this ratio with more particles in it.
Tunnelling $T\simeq16\epsilon(1-\epsilon)\,\ee^{-2\kappa w}$	4.7 — the transmission is not zero at any energy, width or height. The exponent is $2\kappa w$ and not $\kappa w$ because $T$ is a probability and the amplitude is what falls by $\ee^{-\kappa w}$ . A $1\ \mathrm{eV}$ barrier $1\ \mathrm{nm}$ wide passes a $0.5\ \mathrm{eV}$ electron with $T=2.850\times10^{-3}$ , computed exactly and checked against a numerical integration that shares no algebra with it. The STM and $\alpha$ -decay measurements it is set beside are the chapter's one ⚑.	4.7 §6.5, where continuing the same amplitude to negative energy puts its poles exactly at the bound states; 4.10 §4, which handles a barrier that is not rectangular; 4.17; and Chapter 4.2's ammonia $\Delta$ , computed at last in 4.7 §7.
The ladder operators $\hat a,\ \hat a^{\dagger}$ , with $[\hat a,\hat a^{\dagger}]=1$	4.8 — obtained by factorising $\hat x^{2}+\hat p^{2}$ as far as non-commutativity permits and letting the leftover be the commutator. That is not a presentational choice: the leftover is the zero-point energy, so $\half\hbar\omega$ is on the page at the moment of factorisation rather than arriving later as an algebraic surprise. Uses 0.5 §4's adjoint and nothing else.	4.8 §§3–7 entirely; 4.11 and 4.13, which run the same construction on angular momentum and on the radial problem; Part V, where one pair per field mode is the whole of field quantisation; 7.3, where the Virasoro algebra is built by hand out of these commutators.
The number operator, and the ladder relation $\hat N=\hat a^{\dagger}\hat a$ ; $[\hat N,\hat a^{\dagger}]=\hat a^{\dagger}$	4.8 — the manoeuvre, named once because it recurs: a commutator that shifts an eigenvalue. If $\hat N\psi=\nu\psi$ then $\hat a^{\dagger}\psi$ is an eigenvector with eigenvalue $\nu+1$ , and the whole spectrum follows from that one line.	4.11 §4 (the ladder, a second time, on $\hat J_{\pm}$ ); 4.13 §4 (a third time, on the radial operator); 7.3. 4.8 §3.4 names all three in advance so the reader recognises the move when it arrives.
The oscillator spectrum, from the algebra $E_{n}=(n+\half)\hbar\omega$	4.8 — and the step every book skips is the one that carries it: the ladder terminates below because a norm is non-negative. $\norm{\hat a\ket\psi}^{2}=\avg{\hat N}\ge0$ bounds the spectrum, and the same inequality excludes a non-integer eigenvalue rather than merely bounding one. This is an independent route to 4.5 §4's answer, which came from the Hermite equation: two derivations sharing no step agree, which is why the book does it twice.	4.8 §6 (the phase-space area) and §7; 4.9 §5, where the oscillator is the potential for which Ehrenfest is exact; 4.15 and 4.16; Part V, where $\half\hbar\omega$ per mode is the vacuum energy and the problem it causes.
The oscillator eigenfunctions $\psi_{n}=(\hat a^{\dagger})^{n}\psi_{0}/\sqrt{n!}$	4.8 — the whole point of the method. A second-order eigenvalue problem is replaced by one first-order equation, $\hat a\psi_{0}=0$ , plus an algebra. These are the Hermite functions, which collects 0.5's promise made about a weight $\ee^{-x^{2}}$ on the whole line. Completeness is cited from 4.5 §5 and not re-proved, exactly as 4.3's closing brick said it would be.	4.15 and 4.16 (matrix elements); 4.17 (selection rules); 5.3, where the same functions are the occupation states of a field mode.
The phase-space area $\oint p\,\dd q=(n+\half)h$	4.8 — 0.8 §4.4's classical ellipse, quantised. It collects three promises at once, including 1.3's Bohr–Sommerfeld ⚑, and it is honest about which half it discharges: the oscillator is the one case in which the condition is exact rather than semiclassical.	4.10 §6, which proves the general semiclassical statement and measures how far it is from the truth when the potential is not a parabola; 4.10 §7, where a classical orbit of area $\mathcal A$ holds about $\mathcal A/h$ states; Part V's density of states, which is this count.
Coherent states $\hat a\ket\alpha=\alpha\ket\alpha$	4.8 — the first object in the chapter that is not an energy eigenstate, which is worth saying after six sections of eigenstates. A packet that does not spread, and the closest thing quantum mechanics has to a classical oscillator. It answers 4.6 §10's question directly: the free packet spreads, and this is the state for which the width does not change.	4.9 §5, where it is the case in which Ehrenfest is exact; 5.3 and Part V, where the coherent state is what a laser field is and the bridge from the ladder to a classical field.
The general uncertainty relation $\Delta A\,\Delta B\ge\half\abs{\avg{[\hat A,\hat B]}}$	4.9 — three lines, and deliberately so. 0.5 §1.4 proved Cauchy–Schwarz and 4.2 §5.3 already wrote the variance as the squared length of $(\hat A-\avg A)\ket\psi$ , so the theorem is those two facts and the meaning of the symbols. The special case $\Delta x\,\Delta p\ge\hbar/2$ is one line from 0.9 §6.4's bandwidth theorem plus $p=\hbar k$ , which is exactly what 0.9 said would be added and all that would be added. Note the floor is state-dependent for a general pair and can vanish; it is $\hbar/2$ in every state only because $[\hat x,\hat p]=\ii\hbar\hat I$ .	4.8 §5's ground state, which saturates it; 4.11, where the failure to be simultaneously sharp is shown to be structural; 4.16; 4.20, where the question of what lies underneath is settled by experiment.
The Heisenberg picture $\hat A_{H}(t)=\hat U^{\dagger}(t)\hat A\,\hat U(t)$	4.9 — a change of basis, which collects 0.4's promise about why the two pictures look like rival physics instead of two bases. In infinite dimensions the slogan costs more than it does for matrices, and the chapter pays it: the domain moves too, $\operatorname{dom}(\hat A_{H}(t))=\hat U^{\dagger}(t)\operatorname{dom}(\hat A)$ , and spectra agree because conjugation by a unitary preserves invertibility, which is 4.5 §2.1's definition rather than any statement about determinants or traces.	4.9 §5 and §6; 4.17; and the whole of Part V, where the interaction picture is this construction run on a splitting of $\hat H$ .
The Heisenberg equation $\dv{\hat A}{t}=\frac{1}{\ii\hbar}\big[\hat A,\hat H\big]+\pdv{\hat A}{t}$	4.9 — set beside 1.3 §6.1's classical equation term by term, so that the only visible difference is which bracket. That is 1.3's promise that the bracket goes to Part IV, collected.	4.9 §5 (Ehrenfest is this equation on $\hat x$ and $\hat p$ ); 4.11, where it turns a commutator into a precession; 4.17; Part V.
Ehrenfest's relations $\dv{\avg{\hat x}}{t}=\avg{\hat p}/m$ , $\dv{\avg{\hat p}}{t}=-\avg{\nabla V}$	4.9 — and the caveat is the content. $\avg{\nabla V}\ne\nabla V(\avg x)$ unless $V$ is at most quadratic, which is why the oscillator is exactly classical in the mean and nothing else is. Measured rather than described: a split-operator run in a quartic potential beside the same run in a quadratic one, where the defect is zero bit for bit. Collects 1.1's promise and 1.1's own warning that this is a derived statement about expectation values and not a fundamental law.	4.10, where the classical limit is taken properly and 4.10 §8 shows the correspondence cannot be made exact; 4.8 §7's coherent state, the case in which it is exact.
Generators, and conserved quantities symmetry $\Rightarrow$ unitary $\Rightarrow$ conserved observable	4.9 — Chapter 4.2 §7.5 stated the correspondence; this chapter assembles it, in translation, rotation and time. Linearity is assumed at the first link, as 4.2 §7.1 assumed it, and the chapter says so and discharges it case by case rather than quoting Wigner. This is 1.4 §7's classical statement that charges generate their own symmetries, with operators.	4.11, which this hands its commutator to and which is the reason rotation is the case that matters; 4.12; 4.17's selection rules; and Parts V and VI, where the whole subject is organised this way.

Coming in the next batches

Object	Will be introduced in
Contour integration, residues, analytic continuation, distributions, the propagator	5.4
Grassmann numbers, Berezin integration, generating functional	5.7
Lie group, Lie algebra, structure constants	6.1
Representations, weights, roots, Casimirs	6.2
Conformal algebra, Virasoro, central charge	7.3