Notes

The absorption wavelength follows from the excitation energy by ${\lambda }_{\max }=1240/E$ with $E$ in eV; the bright band of a conjugated molecule is its HOMO$\to$LUMO $\pi \to {\pi }^{\ast }$ transition.

Every furnace rate in fabrication — oxidation, diffusion — is Arrhenius, which is why temperature is the master processing knob and thermal budget is the currency.

A thermally activated rate is exponential in the barrier height over temperature.

Because the dependence is exponential in $1/T$, a plot of $\ln k$ against $1/T$ is a straight line whose slope is $-E_a/k_B$ — how activation energies are measured.

Greedy decoding commits to the highest-probability token at each step and often paints itself into a corner; beam search keeps several candidate skeletons alive and fits constants to each. Widening the beam moved recovery from 3/20 to 11/20 in the from-scratch SymGPT model.

The Mott gap is a many-body quantity, the cost to move charge; it opens with $U$ and grows linearly, approaching $U-W$ in the atomic limit.

Predicting color factors as colorimetry $\circ$ excited-state QM: the colorimetry is solved, but cheap TDDFT is the bottleneck. A $\sim$1 eV error in an excitation energy flips a hue to its complement (B3LYP puts indigo's band near 400 nm, predicting yellow, when it is really 605 nm and blue).

The perceived color of a molecule is deterministic given its spectrum: broaden the excitations into an absorption curve, attenuate daylight by Beer--Lambert, convolve with the CIE 1931 color-matching functions to XYZ, then map to sRGB. This works for any molecule.

A pigment's perceived color is roughly the complement of the wavelength it absorbs: absorb blue ($\sim$450 nm) and it looks orange; absorb only in the UV and it is colorless.

Carotenoids ($\beta$-carotene, lycopene) are long polyenes ($\sim$11 conjugated C=C bonds) whose $\pi \to {\pi }^{\ast }$ band has red-shifted into the visible — the reason carrots are orange and tomatoes red.

Longer conjugation red-shifts absorption: more delocalized $\pi$ electrons shrink the HOMO--LUMO gap, so ${\lambda }_{\max }$ grows with chain length. The free-electron (particle-in-a-box) model captures the trend.

The free-electron model wrongly predicts a vanishing gap for the infinite conjugated chain; Peierls bond-length alternation keeps polyacetylene's gap finite ($\sim$1.5--2 eV), so ${\lambda }_{\max }$ saturates rather than diverging.

The density of states $g(E)$ counts states per unit energy: the number of states between $E$ and $E+dE$ is $g(E)dE$.

Most thermodynamic quantities are integrals of $g(E)$ against an occupation factor — it is the bridge that turns a spectrum into observables.

For free particles $g(E)\propto E^{d/2-1}$: diverging as $E^{-1/2}$ in 1D, constant in 2D, growing as $\sqrt{E}$ in 3D. Dimensionality reshapes the whole spectrum.

Van Hove singularities: $g(E)$ spikes or kinks wherever the dispersion is flat ($dE/dk=0$) — band edges and saddle points.

Description length is negative log posterior: the model-encoding cost is a prior, the data-encoding cost is the likelihood — the MDL–Bayes correspondence.

Minimum description length scores a model by the total bits to encode the model plus the data given the model — the best model is the one that compresses best.

It is Occam's razor made quantitative: extra parameters cost description bits, so a more complex model must earn them back in a tighter data fit.

Symbolic regression uses it to choose between candidate formulas: the winner minimizes description length on a Pareto front, not raw training error.

This is the scoring behind the site's AI-Feynman / symbolic-regression work.

Single-site DMFT's local–self-energy approximation fails in 2D, where the pseudogap and $d$-wave pairing live in the momentum dependence $\Sigma (\mathbf{k},\omega )$ it discards — the reason cluster extensions (DCA, cellular DMFT) exist.

Unlike static mean field, DMFT keeps the full frequency dependence $\Sigma (\omega )$ — which is exactly why it captures a Mott gap that Hartree--Fock cannot.

DMFT replaces a correlated lattice with a single impurity in a self-consistent bath; the mapping is exact in infinite dimensions, where the self-energy becomes local (momentum-independent).

Two-site DMFT (Potthoff) truncates the bath to one orbital: the impurity is a 16-state Anderson model solved by exact diagonalization, with self-consistency $V^2=Z{M}_2$.

Two-site DMFT yields the closed form $Z=1-(U/U_c{)}^2$ with $U_c=3D$ (Bethe lattice) — within about 4% of the full-DMFT (NRG) value $\approx 2.9D$.

DMFT maps the lattice onto a self-consistent quantum impurity, exact in infinite dimensions; it keeps the frequency dependence a Mott gap lives in and captures the transition static mean field misses.

Electron correlation is the energy a mean-field description misses by replacing the instantaneous electron–electron repulsion with an average field.

Correlation is a tiny fraction of the total electronic energy but comparable to chemical bond energies — which is why quantitative chemistry is a correlation problem.

The correlation energy is always negative: Hartree–Fock is variational, so the exact energy always lies below it.

It splits into dynamic correlation (electrons dodging each other moment to moment, short-range) and static correlation (near-degenerate configurations, as in bond dissociation) that no single determinant can capture.

Entanglement entropy is the von Neumann entropy of a subsystem's reduced density matrix.

It measures quantum correlation between a region and its complement, and is exactly zero for an unentangled product state.

For a pure global state a region and its complement carry equal entanglement, $S_A=S_B$ — a consequence of the Schmidt decomposition.

It is the resource that makes many-body quantum states hard to simulate classically — and the quantity the area law bounds.

Sure Independence Screening ranks every candidate feature by the magnitude of its correlation with the target and keeps only the top few dozen; it is dimensionality reduction with a high-probability guarantee that the truly relevant features survive.

The 2D Hubbard model — the minimal model of the cuprate superconductors — remains unsolved: quantum Monte Carlo hits the fermion sign problem away from half filling, where weights go negative and noise explodes.

The sign problem is basis/representation dependent, not physical: a clever basis can remove it (sign-free models), but finding one is itself hard. It is what leaves the doped 2D Hubbard model unsolved.

The average sign is a ratio of partition functions, $⟨\mathrm{sgn}⟩=Z/Z_{|W|}=e^{-\beta N\Delta f}$ with $\Delta f\ge 0$, so it decays exponentially in $\beta N$; the error bar $\propto 1/⟨\mathrm{sgn}⟩$ makes the cost exponential.

Frustration (a non-bipartite lattice) turns the sign problem on even at half filling — the reason frustrated magnets and the doped 2D Hubbard model resist quantum Monte Carlo.

The repulsive Hubbard model is sign-free at half filling on a bipartite lattice: particle--hole symmetry makes $\det M_{\uparrow }\det M_{\downarrow }\ge 0$. Doping or frustration breaks the protection.

The sign problem is NP-hard in general (Troyer--Wiese, 2005): a universal polynomial-time cure would solve NP-complete problems in polynomial time, so no such cure is expected to exist.

In fermion quantum Monte Carlo, antisymmetry makes some configuration weights $W(c)$ negative; you sample $|W(c)|$ and carry the sign as an observable, so $⟨A⟩=⟨A\mathrm{sgn}⟩/⟨\mathrm{sgn}⟩$.

The Jordan–Wigner transformation maps fermionic creation/annihilation operators onto qubit (Pauli) operators, so a fermionic Hamiltonian can run on qubits.

That string keeps a hopping term cheap in 1D but makes it nonlocal in 2D — which is why Bravyi–Kitaev and other encodings exist.

It enforces fermionic antisymmetry with a nonlocal string of $Z$ operators: the sign a fermion picks up under exchange becomes a Pauli-$Z$ string.

It is the standard front end for simulating quantum chemistry and the Hubbard model on quantum computers, e.g. inside VQE.

A finite box rounds the transition: below a crossover $\Delta {\phi }^{\ast }\sim 1/N$ the power law is masked, so the asymptotic exponent must be read from an intermediate offset window — not from the offsets closest to ${\phi }_J$, which are the most contaminated.

FIRE finds the energy minimum by running molecular dynamics and zeroing the velocity whenever it points against the force — momentum with a restart rule, reaching a minimum without ever forming a Hessian.

Frustration (a non-bipartite lattice) turns the sign problem on even at half filling — the reason frustrated magnets and the doped 2D Hubbard model resist quantum Monte Carlo.

Genetic programming evolves a population of expression trees by tournament selection, subtree crossover, and mutation, scoring each candidate by its fit to the data.

Geometric frustration (e.g. a triangular lattice) prevents all antiferromagnetic bonds from being satisfied at once, leaving a degenerate classical manifold — the doorway to quantum spin liquids.

Hamilton's equations are first-order and symmetric in $q$ and $p$.

The flow preserves phase-space volume (Liouville's theorem) and conserves $H$ whenever $H$ has no explicit time dependence.

Hamiltonian mechanics recasts dynamics in phase space — positions $q$ and conjugate momenta $p$ — governed by a single function $H(q,p)$, the energy.

This geometric structure is why symplectic integrators and Hamiltonian Monte Carlo exist: they preserve the phase-space geometry the equations respect.

The energy HF misses is the correlation energy, defined as exactly what is left over from the exact result.

Only a percent or two of the total energy, but it dominates the energy differences chemistry lives on — bond strengths, barriers.

Exchange ($\hat{K}$) has no classical analog and acts only between same-spin electrons. It is the energy of the Fermi hole — the avoidance antisymmetry forces on parallel spins.

Opposite-spin electrons feel only Coulomb, no exchange — which is why HF misses opposite-spin correlation entirely.

The Fock operator is the one-electron core ($\hat{h}$) plus the averaged Coulomb minus exchange of every occupied orbital.

Its eigenfunctions are the molecular orbitals; its eigenvalues are the orbital energies ${\epsilon }_i$.

Koopmans' theorem: minus an occupied orbital energy approximates the ionization potential, assuming the remaining orbitals don't relax.

It works better than it should by error cancellation — neglected orbital relaxation and neglected correlation pull in opposite directions.

Each electron moves in the average field of all the others, not their instantaneous positions. The coupled $N$-electron problem becomes $N$ one-electron problems solved self-consistently.

Replacing the instantaneous repulsion with an average is what makes HF solvable — and exactly what it gets wrong (correlation).

Restricted HF forces both electrons of a pair into the same spatial orbital. This breaks at dissociation: RHF cannot separate H${}_2$ into two neutral atoms, and the energy sails high as the bond stretches.

The fix (UHF) lets the spins localize on different atoms, but contaminates the spin state — broken symmetry, $⟨S^2⟩\ne 0$.

HF is solved by iteration, not in one shot: the Fock operator is built from the very orbitals you are solving for. Guess the density, build the field, solve, repeat until the orbitals stop changing.

It is a fixed-point iteration — "self-consistent" means the field reproduces the orbitals that produced it.

Hartree–Fock approximates the full many-electron wavefunction by a single Slater determinant — one antisymmetrized configuration. Every other choice in the method is just picking the best orbitals to fill it.

This one-determinant ansatz is the source of everything HF gets wrong: the true wavefunction is a sum of many determinants.

The 1D spin-$\frac{1}{2}$ Heisenberg antiferromagnet ground-state energy per site (Bethe ansatz, 1931):

The 1D chain is critical (gapless): fractionalized spin-$\frac{1}{2}$ spinon excitations, power-law correlations, and logarithmic entanglement growth (central charge $c=1$).

Its logarithmic entanglement is exactly why DMRG/MPS handle it so well.

At jamming the packing is isostatic: exactly $Z_{iso}=2d$ contacts per particle — the Maxwell minimum that constrains every degree of freedom, with none to spare.

The factor of two is because each contact is shared by two particles.

Rattlers — particles with fewer than $d+1$ contacts — bear no load and must be removed iteratively before counting the contact number, because removing one can strand another.

A finite box rounds the transition: below a crossover $\Delta {\phi }^{\ast }\sim 1/N$ the power law is masked, so the asymptotic exponent must be read from an intermediate offset window — not from the offsets closest to ${\phi }_J$, which are the most contaminated.

Above jamming the excess contact number grows as the square root of the distance to the transition — the signature power law that made jamming look critical.

At jamming the packing is isostatic: exactly $Z_{iso}=2d$ contacts per particle — the Maxwell minimum that constrains every degree of freedom, with none to spare.

The factor of two is because each contact is shared by two particles.

Jamming is the onset of rigidity: the packing fraction at which a random arrangement of soft particles can no longer relax to zero energy.

The order parameter is the minimized energy itself: zero below ${\phi }_J$ (a gap-free packing exists) and nonzero above it (every minimum has residual overlap).

For 2D 50:50 bidisperse disks at size ratio 1.4, the jamming point is ${\phi }_J\approx 0.84$. The size ratio exists to frustrate crystallization — equal disks would form a triangular lattice instead of jamming.

Rattlers — particles with fewer than $d+1$ contacts — bear no load and must be removed iteratively before counting the contact number, because removing one can strand another.

Given planetary periods and semi-major axes, symbolic regression recovers Kepler's third law $T\propto a^{3/2}$ ($T^2=a^3$) from the numbers alone — the elbow of the Pareto front at $R^2=1$.

They need $A$ only as a black box $\mathbf{x}\mapsto A\mathbf{x}$ — which is why they dominate large sparse and matrix-free problems.

Convergence is set by the eigenvalue distribution: clustered spectra converge fast, which is exactly what preconditioning engineers.

The order-$k$ Krylov subspace is everything reachable from $\mathbf{b}$ using only repeated multiplication by $A$.

Krylov methods (CG, GMRES, Lanczos, Arnoldi) find the best available solution inside this growing subspace, without ever forming $A^{-1}$ or factoring $A$.

The two master factorizations are the eigendecomposition $A=V\Lambda V^{-1}$ (a square map on its own space) and the SVD $A=U\Sigma V^{\top }$ (any map between two spaces).

Linear algebra is the study of vector spaces and linear maps between them; a matrix is just a linear map written down in a chosen basis.

Numerically you almost never invert a matrix — you factor it. Choosing the factorization (LU, QR, Cholesky, SVD) is choosing the algorithm.

Almost everything reduces to a matrix's four fundamental subspaces — column space, null space, row space, left null space — and how the map moves vectors among them.

The condition number bounds how much relative error in the data is amplified into the solution.

An ill-conditioned system can be solved exactly in arithmetic and still be useless in floating point.

Solve by factorization — LU in general, Cholesky for SPD — not by forming $A^{-1}$, which is slower and numerically worse.

Large sparse systems are solved iteratively (CG, GMRES), which touch $A$ only through matrix–vector products — never storing or factoring it.

$A\mathbf{x}=\mathbf{b}$ has a unique solution exactly when $A$ is invertible ($\det A\ne 0$); otherwise it has zero or infinitely many.

The bond dimension of those matrices caps the entanglement across any cut — which is why MPS efficiently represent low-entanglement, area-law 1D states.

A matrix product state writes the amplitude of each many-body configuration as a product of small matrices, one per site.

DMRG is variational optimization over MPS — the reason 1D quantum ground states are essentially a solved problem.

MPS is the 1D member of the tensor-network family; PEPS and MERA extend the idea to higher dimensions and to critical systems.

A Mott insulator is a half-filled band that band theory calls a metal but that insulates from electron repulsion — a metal–insulator transition driven by interaction, with no change in filling and no broken symmetry.

Mott vs band insulator: a band insulator has completely filled bands (even electron count); a Mott insulator has a half-filled band and remains a magnet, its localized spins coupling as $J=4t^2/U$.

Two-site DMFT yields the closed form $Z=1-(U/U_c{)}^2$ with $U_c=3D$ (Bethe lattice) — within about 4% of the full-DMFT (NRG) value $\approx 2.9D$.

The 2D Hubbard model — the minimal model of the cuprate superconductors — remains unsolved: quantum Monte Carlo hits the fermion sign problem away from half filling, where weights go negative and noise explodes.

The Mott gap is a many-body quantity, the cost to move charge; it opens with $U$ and grows linearly, approaching $U-W$ in the atomic limit.

A Mott insulator is a half-filled band that band theory calls a metal but that insulates from electron repulsion — a metal–insulator transition driven by interaction, with no change in filling and no broken symmetry.

Double occupancy collapses from $\frac{1}{4}$ (uncorrelated) toward zero as $U$ grows: electrons pay kinetic energy to avoid sharing sites — correlation a single Slater determinant cannot represent.

In an orthonormal basis, a vector's coordinate along each basis vector is just an inner product, $c_i=⟨e_i,\mathbf{v}⟩$ — no linear system to solve.

Two vectors are orthogonal when their inner product is zero; an orthonormal set is orthogonal and unit-length.

Gram–Schmidt turns any basis into an orthonormal one by subtracting off projections; numerically its modified form, or a QR factorization, is preferred for stability.

The projection of a vector onto a subspace is the closest point in it, and the residual is orthogonal to the subspace — the geometric content of least squares.

'Best approximation' and 'orthogonal residual' are the same statement; the normal equations enforce it.

The input $(x_i,y_i)$ points are a set, not a sequence, so each is embedded by an MLP and fed as context with no positional encoding, leaving the model permutation-invariant over the samples. The generated expression tokens do get positional encodings and a causal mask.

Poisson's equation relates the electrostatic potential to the charge density that sources it.

With no charge it becomes Laplace's equation, whose solutions have no interior maxima or minima — the maximum principle.

The electric field is minus the gradient of the potential; field lines run downhill in $\phi$.

In a semiconductor the charge $\rho$ itself depends on $\phi$ through the Boltzmann carrier statistics, making it a nonlinear Poisson equation solved by Newton's method.

The classical Néel state is not the quantum ground state: the $S_i^{+}{S}_j^{-}+S_i^{-}{S}_j^{+}$ terms flip antiparallel pairs, so the true ground state is an entangled superposition lying below the classical energy.

The coupling is antiferromagnetic because virtual hopping to a neighbor is Pauli-allowed only for antiparallel spins, so antiparallel alignment lowers the energy.

The quasiparticle weight $Z$ sets the effective mass $m^{\ast }/m=1/Z$; at the Mott transition $Z\to 0$ and the mass diverges — the metal localizes without the carriers disappearing.

$n$ qubits live in a $2^n$-dimensional space — the exponential that makes them powerful and makes classical simulation hard.

Gates are unitary matrices — reversible and norm-preserving — so a computation is a product of unitaries applied to the state.

Measurement is not unitary: it projects the state onto a basis outcome with probability equal to the squared amplitude, collapsing the superposition.

A qubit is a normalized vector in ${\mathbb{C}}^2$: unlike a bit it can be any superposition $\alpha |0⟩+\beta |1⟩$ with $|\alpha {|}^2+|\beta {|}^2=1$.

The measured charge radius folds the finite proton size onto the point-proton distribution: $r_{ch}^2=⟨r^2{⟩}_p+r_{proton}^2$ (the point radius alone is not the observable).

For a zero-range (contact) interaction the pair energy $\iint \rho v\rho$ collapses to $\int {\rho }^2$: the energy density becomes a pointwise function of local densities.

The direct Coulomb energy carries the factor $\frac{1}{2}$ of classical electrostatics, $E=\frac{1}{2}\int V\rho$, so each proton pair is counted once.

A nucleus is doubly-magic when both its proton number $Z$ and neutron number $N$ are magic (2, 8, 20, 28, 50, 82, 126) — both shells exactly filled.

The Hartree-Fock energy is not the sum of occupied single-particle energies: each ${\epsilon }_i$ contains the interaction with every other particle, so the sum counts each pair twice.

The Skyrme mean fields are functional derivatives of the energy: $U_q=\delta E/\delta {\rho }_q$, $B_q=\delta E/\delta {\tau }_q={\hslash }^2/2m_q^{\ast }$, $W_q=\delta E/\delta J_q$.

Gershgorin: every eigenvalue of a matrix lies within ${\sum }_{j\ne i}|H_{ij}|$ of some diagonal entry $H_{ii}$ — so ${\min }_i(H_{ii}-{\sum }_{j\ne i}|H_{ij}|)$ bounds the spectrum from below.

A symmetric discretization of $-\nabla \cdot B(r)\nabla$ on a uniform mesh requires the same half-point value $B_{i+1/2}$ in the diagonal and the off-diagonal it couples; asymmetric discretizations can return complex eigenvalues.

If both spin-orbit partners $j=\ell \pm \frac{1}{2}$ are filled with identical radial functions, their contributions to the spin-orbit density cancel exactly: $\ell (2\ell +2)-(\ell +1)(2\ell )=0$.

Inverse iteration with shift $\sigma$ converges to the eigenvector whose eigenvalue is closest to $\sigma$; a shift below the whole spectrum therefore yields the ground state.

The spin-orbit splitting between the $j=\ell \pm \frac{1}{2}$ partners of one $\ell$-level is proportional to $2\ell +1$: s-levels never split, and the split grows with $\ell$.

The number density integrates to the particle count: $\int {\rho }_q{d}^{3}r=N_q$; in spherical symmetry ${\rho }_q={\sum }_{occ}\frac{2j+1}{4\pi }{R}^2$.

In any central potential the 3-D single-particle equation separates as $\phi =[u(r)/r]Y_{\ell m}\chi$; $u$ obeys a 1-D radial equation with centrifugal term $\ell (\ell +1)/r^2$ and boundary $u(0)=0$.

$⟨\sigma \cdot l⟩=j(j+1)-\ell (\ell +1)-\frac{3}{4}$, which equals $\ell$ for $j=\ell +\frac{1}{2}$ and $-(\ell +1)$ for $j=\ell -\frac{1}{2}$.

The kinetic density $\tau ={\sum }_{occ}|\nabla {\phi }_i{|}^2$ integrates to the total kinetic energy: $E_{kin}=({\hslash }^2/2m)\int \tau d^{3}r$.

The Sparsifying Operator is ${\ell }_0$ regularization: for each descriptor dimension $D$ it finds the $D$-feature subset whose linear least-squares fit is best, reporting the smallest $D$ that suffices. The jump from a near-perfect 1D fit to an exact 2D fit signals two independent physical contributions.

The quadratic form ${\mathbf{x}}^{\top }A\mathbf{x}$ is an upward-opening bowl exactly when $A$ is SPD — which is why an SPD system has a unique minimum, not a saddle.

A symmetric matrix is positive definite iff ${\mathbf{x}}^{\top }A\mathbf{x}>0$ for all $\mathbf{x}\ne 0$, iff every eigenvalue is positive, iff it has a Cholesky factorization $A=L{L}^{\top }$ — three equivalent tests.

Minimizing $\frac{1}{2}{\mathbf{x}}^{\top }A\mathbf{x}-{\mathbf{b}}^{\top }\mathbf{x}$ is the same problem as solving $A\mathbf{x}=\mathbf{b}$ when $A$ is SPD — the bridge conjugate gradient walks across.

The Hutchinson estimator computes a trace as the average of ${\mathbf{z}}^{\top }A\mathbf{z}$ over random probe vectors with mean-zero, unit-variance entries.

Paired with Lanczos it yields $\mathrm{Tr}(f(A))$ — the engine behind log-determinants, spectral densities, and other spectral sums.

Leans on the same Lanczos/moment kernel that recurs across the site's spectral pages.

It needs $A$ only through matrix–vector products, so you can estimate the trace of a matrix you can apply but never form — like a matrix function $f(A)$.

The error falls as $1/\sqrt{N}$ in the number of probes; Rademacher ($\pm 1$) probes minimize the variance among simple choices.

In the large-$U$ Hubbard model, virtual hopping generates an antiferromagnetic spin coupling $J=4t^2/U$ — magnetism with no magnetic force, only repulsion and the Pauli principle.

The effective model is Heisenberg with a constant shift; its singlet energy $-J$ matches the exact large-$U$ two-site Hubbard ground state, constant included.

The coupling is antiferromagnetic because virtual hopping to a neighbor is Pauli-allowed only for antiparallel spins, so antiparallel alignment lowers the energy.

Separability is detected from data: fit a smooth surrogate, hold all variables but one at a base point, and read off each 1-D profile. If the profiles ADD back to the data the law is additively separable; if they MULTIPLY back it is multiplicatively separable. The profiles are the subproblems.

Multiplicative separability is additive separability of $\log f$, since $f=g(x)h(y)$ gives $\log f=\log g(x)+\log h(y)$. One additive test covers both structures -- run it on $f$, then on $\log |f|$ for strictly positive data.

At each one-variable leaf a portfolio runs genetic programming, SISSO, and the trained transformer, keeping the simplest formula within a tolerance of the best fit. No single engine wins every leaf, so competing them is more robust than committing to one; the transformer can only play because the decomposition produced its trained one-variable case.

AI-Feynman's central reduction is separability: if $f(x,y)=g(x)+h(y)$ or $f(x,y)=g(x)h(y)$, the variables never interact and finding $f$ is really two independent one-variable searches. A separable $n$-variable law is $n$ one-variable problems in disguise.

The separability tests run on the fitted surrogate, not the true function, so surrogate accuracy caps everything downstream. AI-Feynman trains a neural network for this; a radial-basis fit suffices for a few variables and $\sim$100 points. Non-separable targets like $\sin (xy)$ or $g(x+y)$ defeat the tests and force a fallback.

Symbolic regression searches for the form of a law (the algebraic expression), not just the parameters of a form you chose; the search space is expression trees over a chosen primitive operator set.

Genetic programming evolves a population of expression trees by tournament selection, subtree crossover, and mutation, scoring each candidate by its fit to the data.

Given planetary periods and semi-major axes, symbolic regression recovers Kepler's third law $T\propto a^{3/2}$ ($T^2=a^3$) from the numbers alone — the elbow of the Pareto front at $R^2=1$.

Without a complexity penalty, symbolic regression overfits — a large enough tree memorizes noise. The fix is the Pareto front (best accuracy at each complexity); the law sits at the elbow, where accuracy stops improving.

Symbolic regression can only find laws expressible in its primitives: Kepler's $T=a^{3/2}$ needs ​ in the operator set. Dimensional analysis and symmetry (AI-Feynman) shrink the search enormously.

A transformer amortizes symbolic regression: trained once on millions of synthetic (points, formula) pairs, it answers a new problem in one forward pass whose cost is flat, independent of formula difficulty. A search instead pays its full, difficulty-growing cost on every problem.

A formula is a tree, and a tree has a unique prefix (Polish) serialization: a preorder walk that is parenthesis-free and bijective with the tree. This turns equation generation into next-token prediction over a small vocabulary of operators, the variable, and a constant placeholder.

Neural symbolic regression predicts a skeleton: every numerical constant collapses to a single placeholder token, the network outputs only structure, and the actual constants are fit numerically afterward. This decouples the discrete combinatorial problem from the continuous one and makes training tractable.

Unlike genetic programming, SISSO has no random seed: the feature space is fixed, the screen is a deterministic sort, and the selection is an exhaustive best-subset search, so it returns the same descriptor every run. This reproducibility is why materials scientists publish SISSO descriptors.

Non-symmetric operators ($÷$, $-$) must be applied in both orderings when constructing features, or half the space is never built. If $(q_1{q}_2)/r^2$ is never constructed, no screen can find Coulomb's law; the failure is silent because a plausible high-correlation feature is returned instead.

SISSO has three deterministic steps: construct a huge feature space by applying an operator set in rungs, Sure Independence Screening (SIS) to keep the top features by $|$correlation$|$ with the target, then a Sparsifying Operator (SO) that ${\ell }_0$-selects the smallest feature subset whose linear combination fits.

A Hermitian operator equals its own conjugate transpose; real-symmetric is the real special case.

Hermitian operators have real eigenvalues and orthogonal eigenvectors — which is exactly why quantum observables are required to be Hermitian.

The spectral theorem writes any Hermitian operator as a sum of real eigenvalues times orthogonal projectors.

Critical 1D systems violate it logarithmically, $S\sim \frac{c}{3}\log \ell$, with $c$ the central charge — the entanglement fingerprint of a conformal field theory.

This is deeply non-generic: a random state has volume-law entanglement, so physical ground states occupy a vanishing, low-entanglement corner of Hilbert space.

The area law says the entanglement entropy of a region scales with the size of its boundary, not its volume — for gapped ground states.

The area law is why tensor networks work: MPS and PEPS are built to capture exactly boundary-law entanglement and nothing more.

A fixed-value boundary source gives an erfc profile; a fixed total amount with a sealed boundary gives a spreading Gaussian — the two dopant-diffusion regimes.

The diffusion (heat) equation says each point relaxes toward the average of its neighbors, at a rate set by the diffusivity $D$.

Its fundamental solution is a Gaussian that spreads with width $\sqrt{Dt}$ — the universal ruler of every diffusive process.

It smooths: a spatial mode of wavenumber $k$ decays as $e^{-D{k}^{2}t}$, fastest for the sharpest features. Running it backward is unstable — diffusion destroys information.

The eigenvalues are the roots of the characteristic polynomial; an $n\times n$ matrix has $n$ of them over $\mathbb{C}$, counting multiplicity.

An eigenvector is a direction the matrix only stretches, never rotates; the eigenvalue is the stretch factor.

A matrix with a full set of independent eigenvectors factors as $A=V\Lambda V^{-1}$, so a matrix power acts one eigenvalue at a time: $A^k=V{\Lambda }^k{V}^{-1}$.

This is why eigenvalues govern the long-run behavior of anything iterated (Markov chains, power iteration, dynamical systems).

Symmetric (real) or Hermitian (complex) matrices have real eigenvalues and a full orthogonal eigenbasis — the spectral theorem.

Convolution in one domain is pointwise multiplication in the other — the property behind filtering, diffraction imaging, and Green's-function PDE solutions.

The Fourier transform re-expresses a function as a superposition of sinusoids, mapping between position (or time) and frequency.

Differentiation becomes multiplication by $ik$ in Fourier space, turning constant-coefficient differential equations into algebra.

The FFT computes the discrete transform in $O(N\log N)$ rather than $O(N^2)$ — the algorithmic reason Fourier methods are everywhere.

Linear-response TDDFT reduces to Casida's equation, a generalized eigenvalue problem whose eigenvalues are the excitation energies and whose eigenvectors give the transition densities (hence oscillator strengths).

The condition number is the ratio of largest to smallest singular value — the factor by which a matrix can amplify relative error.

An induced matrix norm measures the maximum stretch a matrix applies to any unit vector.

The spectral norm $\Vert A{\Vert }_2$ is the largest singular value; for a symmetric matrix it is the largest eigenvalue magnitude.

A norm measures size and obeys the triangle inequality; the $p$-norms ($p=1,2,\infty$) are the usual vector choices.