Claims
A ledger of small, citeable facts — one declarative sentence each — and what happened when I tested them. Each is a hypothesis with a verdict: supported (it held, by an external reference or a numerical experiment), falsified (a test showed it false — kept anyway, because the negative result is the knowledge), or uncertain (untested or inconclusive).
Why a ledger of atoms instead of just pages?
Think about how a dictionary works. It defines a word using other words, which are defined using still other words. Follow any definition far enough and you either loop back to where you started or you reach a word no other words can pin down — one you fix only by pointing at an instance and saying this is one, that is not. There's no bottom made of pure definition. Knowledge is a graph of references with its leaves nailed to reality.
Programming is the same shape. A function is written in terms of other functions; types are built from other types; the call graph bottoms out in primitives, which bottom out in machine instructions, which bottom out in physics. Nothing is self-defining, and the structure only means something because the leaves touch hardware. This is the symbol-grounding problem, and it's not a bug — it's how any system of meaning has to work.
So I'm trying to build my own understanding the way both of those
are built: as a web of small claims that reference each other, with
the leaves grounded in computation and observation. Each claim is an
atom you could cite mid-sentence — "…because [Hartree-Fock is a
fixed-point iteration], DIIS applies directly." The
neighbors on each claim are the edges:
instance-of, generalizes, contradicts,
supports. The pages on this site are the syntheses; the
claims are the atoms those syntheses are made of.
And every atom has to be falsifiable, or it isn't worth keeping. That's the Popper part. I don't get to mark a claim true — I only get to say it survived testing (supported), the way a corroborated conjecture has survived but is never finally proven. What I can do cleanly is mark one falsified: when a numerical experiment shows it's wrong — like the time I tried to use rational functions in place of Slater orbitals and watched the integrals refuse to cooperate — that result stays on the ledger as a falsified claim. The vocabulary is asymmetric on purpose. You can knock a claim down with a single experiment; you can never prop it up forever.
The full authoring protocol lives in
CLAIM_AUTHORING.md.
- Most quantum many-body methods are the same move: replace the interacting N-particle problem with a single-particle (mean-field) one, then fill its ladder of one-particle levels by an occupation rule.
- The spectral function of any operator is a sum of Lorentzians centered at its eigenvalues, weighted by the probe state's overlap with each eigenvector and broadened by η (or the mode's inverse lifetime).
- The Obara-Saika angular-momentum recurrence is the Hermite three-term recurrence — the same 'multiply-by-x is tridiagonal' structure that underlies orthogonal polynomials, Gauss quadrature, and Lanczos.
- A relaxation time is the inverse of an eigenvalue of the system's linear relaxation operator; the RC time constant τ=RC, the Drude scattering time, and gradient descent's 1/λ are instances of one object.
- A rational (Padé / AAA) approximation of Slater orbitals can replace Gaussians for molecular integrals, using pole-residue summation in place of the Gaussian product theorem.
- The cusp at the origin of the 1s wavefunction ψ(r⃗) ∝ exp(−Z|r⃗|) comes from the absolute value of the position vector that appears when the radial form R(r) ∝ exp(−Zr) is written as a function of 3D position.
- A Slater determinant vanishes when two fermions occupy the same spin-orbital — two identical columns make the determinant zero (Pauli exclusion).
- The Casida equation is the linear-response, frequency-domain, adiabatic-kernel limit of the time-dependent Kohn-Sham equation.
- Solving the Casida eigenvalue problem yields excitation energies as eigenvalues and oscillator strengths from the eigenvectors (via transition dipoles).
- The Casida equation is most often used with the Tamm-Dancoff approximation.
- For neutral H₂, the Coulson-Fischer wavefunction has variational energy ≤ that of closed-shell LCAO (RHF), with strict inequality above the Coulson-Fischer bond length R_CF and equality below.
- Interpolating at Chebyshev nodes avoids Runge's phenomenon, which afflicts polynomial interpolation at equispaced points.
- The Coulson-Fischer point is the bond length where the symmetry-broken UHF solution first drops below the RHF solution in energy.
- Fibonacci is a degenerate dynamic-programming problem: no decision, just a sum. More complicated DP problems look back further and involve comparison operations.
- Hartree-Fock is a fixed-point iteration.
- Hashmap lookup by key is O(1) on average under a well-distributed hash; worst case is O(n) when collisions degenerate.
- Hodgkin-Huxley is an RC circuit with voltage-gated variable conductances; the inductive (RLC-like) behavior emerges from linearizing the K⁺ gating dynamics around a fixed point (Mauro 1970).
- Lanczos produces an orthonormal basis in which a symmetric operator is tridiagonal.
- Classical orthogonal polynomials are calculated in numerical contexts with their three-term recurrence relation, not explicit formulas — the latter suffer catastrophic cancellation at high degree.
- Any orthogonal-basis expansion (Fourier, Legendre, Chebyshev, Hermite, spherical harmonics, …) is computable by numerical integration of the weighted inner product cₙ = ⟨f, φₙ⟩_w / ⟨φₙ, φₙ⟩_w. The Fourier/FFT case is one instance among many.
- Padé-via-Lanczos (shift-invert Lanczos around a chosen σ) is used instead of plain Lanczos for circuit model-order reduction because plain Lanczos on K wastes its Ritz values away from the operating band — to span many decades of frequency it needs a chain so long that the reduced model costs as much as a dense direct solve.
- Random Fourier features give an explicit randomized feature map z(x) ∈ ℝ^D whose inner product z(x)·z(y) is an unbiased Monte Carlo estimate of a shift-invariant kernel k(x, y); this lets linear methods on z(x) emulate kernel methods at O(ND) cost instead of O(N²).
- The top-left entry of a tridiagonal matrix's resolvent has a closed-form continued-fraction expression.