Code that writes code

Exponential-looking growth in computing emerges when systems can improve their own code. Once a loop exists that can generate, execute, measure, and refine programs, capability compounds: each improvement enhances the next round of search over the program phase space. The driver is not just device density, but a feedback process that compresses errors, increases information flow, and reduces effective entropy in the distribution of behaviors.

In mathematical terms, let the capability at iteration \(t\) be \(C(t)\). A simple model of compounded self-improvement is \[ C(t+1) = (1 + r)\,C(t), \qquad r > 0, \] whose solution is \[ C(t) = C(0)\,(1 + r)^t \approx C(0)\,e^{rt}, \] capturing the "exponential-looking" growth. A more general continuous-time feedback model writes \[ \frac{dC}{dt} = f(I,C,E), \] where \(I\) measures information flow through the system and \(E\) is the average error. A negative dependence \(\partial f/\partial E < 0\) and positive \(\partial f/\partial I > 0\) encode that reducing errors and increasing information flow accelerates capability.

Let the program hypothesis space be a high-dimensional manifold \(\mathcal{H}\) with coordinates \(\theta \in \mathbb{R}^d\) parameterizing programs, and let a loss (or energy-like) function be \(L : \mathcal{H} \to \mathbb{R}_+\). Guided search corresponds to iterates \[ \theta_{t+1} = \theta_t - \eta_t \, \nabla_\theta L(\theta_t) + \xi_t, \] where \(\eta_t\) is a learning rate and \(\xi_t\) represents stochastic exploration. The associated probability density over programs, \(p_t(\theta)\), can be modeled as a Gibbs-like distribution \[ p_t(\theta) = \frac{1}{Z_t} \exp\big(-\beta_t L(\theta)\big), \] with partition function \(Z_t\) and inverse temperature \(\beta_t\) reflecting how strongly the system prefers lower-loss code. As \(t\) increases and \(\beta_t\) grows, the distribution contracts around high-quality programs, reducing effective entropy \[ S[p_t] = - \int_{\mathcal{H}} p_t(\theta)\, \log p_t(\theta)\, d\theta. \]

Mechanism of self-improvement

• Program synthesis as guided search over a high-dimensional landscape; gradients, heuristics, and discrete exploration move candidates toward lower loss (energy-like objective).

Formally, represent each candidate program by parameters \(\theta \in \mathbb{R}^d\) and define an objective (loss/energy) \(L(\theta)\). Gradient-based guidance is \[ \theta_{t+1} = \theta_t - \eta_t \, \nabla_\theta L(\theta_t), \] while purely heuristic or discrete search can be modeled by a Markov chain with transition kernel \(K(\theta'\mid\theta)\) that satisfies \[ p_{t+1}(\theta') = \int K(\theta'\mid\theta)\,p_t(\theta)\,d\theta, \] where \(p_t(\theta)\) is the probability density over program parameters at iteration \(t\). The stationary distribution \(p_*(\theta)\) often approximates a Boltzmann form \(p_*(\theta) \propto e^{-\beta L(\theta)}\), concentrating mass near minima of \(L\).

• Error-correcting feedback reduces uncertainty: observations act as measurements that collapse hypotheses, concentrating probability mass on higher-fidelity code.

Let \(\Theta\) be a discrete hypothesis set of programs, with prior \(P(\theta)\) for \(\theta \in \Theta\). Given observed data or test outcomes \(D\), Bayes' rule updates beliefs via \[ P(\theta \mid D) = \frac{P(D \mid \theta)P(\theta)}{P(D)}, \qquad P(D) = \sum_{\theta' \in \Theta} P(D \mid \theta')P(\theta'). \] The Shannon entropy of the hypothesis distribution is \[ H[P] = -\sum_{\theta \in \Theta} P(\theta)\,\log P(\theta). \] Error-correcting feedback corresponds to sequences of measurements \(D_1, D_2, \dots\) such that \[ H\big[P(\cdot \mid D_1,\dots,D_t)\big] \searrow H_* \quad \text{as} \quad t \to \infty, \] where \(H_*\) is low and the posterior probability mass is concentrated on a small set of high-fidelity programs.

• Thermodynamics of computation: every irreversible operation dissipates energy; efficient self-editing favors representations and compilers that minimize erasures while maximizing useful work per joule.

Landauer's principle states that erasing one bit of information incurs a minimum heat dissipation \[ Q_{\min} = k_B T \ln 2, \] where \(k_B\) is Boltzmann's constant and \(T\) is the ambient temperature. If a self-editing system performs \(N_\text{erase}\) bit erasures per update, the minimal thermodynamic cost per update is \[ Q_\text{update} \ge N_\text{erase}\,k_B T \ln 2. \] The computational efficiency can be expressed as useful information gain per unit energy, \[ \eta_\text{info} = \frac{\Delta I}{Q_\text{update}}, \] where \(\Delta I\) is the increase in mutual information between internal model parameters \(\Theta\) and task outcomes \(Y\): \[ I(\Theta;Y) = \sum_{\theta,y} p(\theta,y)\, \log \frac{p(\theta,y)}{p(\theta)\,p(y)}. \] Efficient self-editing corresponds to strategies that maximize \(\eta_\text{info}\) by minimizing unnecessary erasures and maximizing \(\Delta I\).

• Compression and generalization: shorter effective descriptions (lower algorithmic complexity) tend to transfer across tasks, enabling accelerated reuse and faster convergence on new problems.

Let the Kolmogorov complexity (algorithmic information content) of a solution for task \(T\) be \(K(T)\), defined as the length (in bits) of the shortest program that solves \(T\) on a fixed universal Turing machine \(U\). For two tasks \(T_1, T_2\), the shared structure can be quantified by an information-theoretic overlap \[ I(T_1;T_2) \approx K(T_1) + K(T_2) - K(T_1,T_2), \] where \(K(T_1,T_2)\) is the complexity of jointly solving both tasks. Reusable compressed representations correspond to internal codes \(z\) of small description length \(L(z)\) that minimize expected description length over tasks: \[ \min_{z} \, \mathbb{E}_{T \sim \mathcal{D}}\big[ L(z) + L(T \mid z) \big], \] with \(\mathcal{D}\) a task distribution. Lower \(L(z)\) and \(L(T\mid z)\) imply faster adaptation and thus accelerated convergence on new problems.

How this page is assembled

This page was assembled with automated assistance: generative tooling produced structure and text, which were reviewed and emitted as static HTML. In other words, the site itself serves as a small example of code that helps author more code. We always keep a human in the loop.

Abstractly, denote the human editor as \(H\) and the generative model as \(G\). Let \(x\) be an initial specification and \(y\) the final HTML. The interaction can be seen as an alternating minimization over drafts \(y_t\): \[ y_{t+1} = \operatorname*{arg\,min}_{y} \Big( \mathcal{L}_\text{spec}(y \mid x) + \mathcal{L}_\text{style}(y) + \mathcal{L}_\text{error}(y) \Big), \] with updates produced by either \(G\) or \(H\): \[ y_{t+1} = \begin{cases} G(y_t,x,\xi_t) & \text{with probability } p_G, \\ H(y_t,x) & \text{with probability } 1-p_G, \end{cases} \] where \(\xi_t\) captures model stochasticity. Convergence is reached when successive edits satisfy a small-difference condition, e.g. \[ d(y_{t+1},y_t) < \varepsilon, \] for a suitable distance metric \(d\) on documents.

# Pseudocode using a Qiskit-like API
from qiskit import QuantumCircuit

qc = QuantumCircuit(2, 2)  # 2 qubits, 2 classical bits

# 1. Put qubit 0 into a superposition
qc.h(0)

# 2. Use it as control for a CNOT on qubit 1
qc.cx(0, 1)

# 3. Measure both qubits
qc.measure(0, 0)
qc.measure(1, 1)

Minimal meta-programming sketch

// Pseudocode for a self-improving loop
population = initialize_programs()
while (time < budget):
  for prog in population:
    result = execute(prog, tests)
    score  = measure(result)
    log(result, score)
  models = fit_surrogates(log)          // learn to predict score from program features
  proposals = propose(models)           // synthesize or mutate new programs
  population = select(population, proposals, log) // keep better, diverse candidates
  if converged(population): break
deploy(best(population))

Let the population at iteration \(t\) be \(\mathcal{P}_t = \{\theta_t^{(1)},\dots,\theta_t^{(N)}\}\), and let each candidate have a score (fitness) \(F(\theta)\). The selection step can be modeled by a softmax (Boltzmann) sampling distribution \[ P_t(\theta) = \frac{\exp\big(\beta_t F(\theta)\big)}{\sum_{\theta' \in \mathcal{P}_t} \exp\big(\beta_t F(\theta')\big)}, \] with inverse temperature \(\beta_t\) controlling selection pressure. Proposals (mutations or synthesized programs) \(\tilde{\theta}\) are drawn from a proposal kernel \(q_t(\tilde{\theta} \mid \theta)\), so the expected update of the population distribution is \[ p_{t+1}(\tilde{\theta}) = \sum_{\theta} P_t(\theta)\, q_t(\tilde{\theta} \mid \theta). \] Convergence of the loop can be expressed via a stopping condition such as \[ \operatorname{Var}_{\theta \sim p_t}[F(\theta)] < \delta \quad \text{or} \quad \max_{\theta \in \mathcal{P}_t} F(\theta) - \max_{\theta \in \mathcal{P}_{t-k}} F(\theta) < \epsilon, \] for some window size \(k\) and tolerances \(\delta, \epsilon\).

Hierarchical graph of matter and interactions

Universe
└─ Quantum fields
   ├─ Fermion fields (matter)
   │  ├─ Quarks (color charge: red/green/blue)
   │  │  ├─ Flavors: up, down, charm, strange, top, bottom
   │  │  └─ Bound states (hadrons)
   │  │     ├─ Baryons (3 quarks)
   │  │     │  ├─ Proton: u u d
   │  │     │  ├─ Neutron: u d d
   │  │     │  └─ Antibaryons (3 antiquarks)
   │  │     │     └─ Antiproton: ū ū d̄
   │  │     └─ Mesons (quark + antiquark)
   │  └─ Leptons
   │     ├─ Electron, Muon, Tau
   │     └─ Neutrinos (three types) + corresponding antiparticles (e.g., positron)
   ├─ Boson fields (interaction carriers)
   │  ├─ Gluons (strong interaction, SU(3) color)
   │  ├─ Photon (electromagnetic, U(1))
   │  ├─ W± and Z⁰ (weak interaction)
   │  └─ Gravitational quantum (graviton, hypothetical)
   └─ Scalar field associated with electroweak symmetry breaking
      └─ Scalar boson (mass-generating excitation)

Composite structures
└─ Atomic nucleus: protons + neutrons (held by residual strong force)
   ├─ Atoms: nucleus + electron cloud (quantized orbitals)
   ├─ Molecules: atoms bound via electromagnetic interaction
   └─ Condensed phases: solids, liquids, gases, plasmas, exotic matter

“Matter” typically denotes fermions and their composites. Gauge bosons (such as gluons) mediate interactions and are included for completeness.

In the Standard Model, the fundamental fields can be written as a Lagrangian density \(\mathcal{L}_\text{SM}\) of the form \[ \mathcal{L}_\text{SM} = -\frac{1}{4} \sum_{a} F_{\mu\nu}^a F^{a\,\mu\nu} + \sum_{f} \bar{\psi}_f\big(i\gamma^\mu D_\mu - m_f\big)\psi_f + (D_\mu \phi)^\dagger(D^\mu \phi) - V(\phi) + \mathcal{L}_\text{Yukawa}, \] where \ F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c \\ encodes the gauge bosons (gluons, \(W^\pm, Z^0\), photon), \(\psi_f\) are fermion fields (quarks and leptons), \(\phi\) is the Higgs scalar field, and \(D_\mu = \partial_\mu - i g A_\mu^a T^a\) is the gauge-covariant derivative.

Color-charged quarks \(q\) interact via the SU(3) gauge field \(G_\mu^a\) with coupling constant \(g_s\), described by \[ \mathcal{L}_\text{QCD} = \bar{q}\big(i\gamma^\mu D_\mu - m_q\big)q - \frac{1}{4} G_{\mu\nu}^a G^{a\,\mu\nu}, \] where \(D_\mu = \partial_\mu - i g_s T^a G_\mu^a\) and \(G_{\mu\nu}^a\) has the same non-Abelian structure as \(F_{\mu\nu}^a\) above. Bound states such as protons and neutrons are color-singlet combinations of three quarks (baryons), while mesons are quark–antiquark pairs \(q \bar{q}\), consistent with overall color neutrality.

Leptons (electron \(e\), muon \(\mu\), tau \(\tau\) and their neutrinos \(\nu_e, \nu_\mu, \nu_\tau\)) couple to the electroweak SU(2)\(_L\)\timesU(1)\(_Y\) gauge fields. After electroweak symmetry breaking, the Higgs field acquires a vacuum expectation value \[ \langle \phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix}0 \\ v \end{pmatrix}, \qquad v \approx 246~\text{GeV}, \] which generates fermion and weak boson masses through Yukawa terms \(y_f \bar{\psi}_f \phi \psi_f\) and gauge interactions, while leaving the photon massless.

Composite structures are organized hierarchically. A nucleus with \(Z\) protons and \(A-Z\) neutrons has baryon number \(B = A\) and charge \(Q = Z e\). An atom adds \(Z\) electrons, leading to a many-body Hamiltonian \[ H = \sum_{i=1}^{Z} \bigg( -\frac{\hbar^2}{2m_e} \nabla_i^2 - \frac{Z e^2}{4\pi \varepsilon_0 r_i} \bigg) + \sum_{1 \le i < j \le Z} \frac{e^2}{4\pi \varepsilon_0 \lvert \mathbf{r}_i - \mathbf{r}_j \rvert} + H_\text{nucleus}, \] whose eigenstates \(\Psi_n\) correspond to quantized orbitals and energy levels \(E_n\) solving \[ H \Psi_n = E_n \Psi_n. \] Molecules and condensed phases arise when these atomic states combine via electromagnetic interactions into multi-atom bound states and extended many-body systems.

// n assets, vector of random returns R
// mean vector μ, variance–covariance matrix Σ

E[R]   = μ               // n×1
Cov(R) = Σ               // n×n (symmetric, PSD)

// portfolio weights (deterministic)
vector[n] w;
real R_p = dot_product(w, R);     // portfolio return

real mean_p   = dot_product(w, μ);
real var_p    = quad_form(Σ, w);  // w^T Σ w

// amplitudes ψ (complex allowed), normalized
complex psi[n];

// density matrix ρ_ij = ψ_i ψ*_j
complex rho[n, n] = outer_product(psi, conj(psi));

// return operator R̂ (for illustration, take it diagonal)
real R_vals[n];                    // possible asset returns
complex R_op[n, n];

for (i in 1:n) {
  for (j in 1:n) {
    R_op[i,j] = (i == j) ? R_vals[i] : 0;
  }
}

// quantum expectation of portfolio return
complex mean_p_q = trace(rho * R_op);  // ≈ classical w^T μ

ρ = |ψ><ψ|
<R̂>_ψ = <ψ| R̂ |ψ> = Tr(ρ R̂)

// very schematic Stan pseudo-flow
for (m in 1:M) {
  θ[m] ~ posterior(· | data);      // parameters: μ[m], Σ[m], etc.
  R[m] ~ mvnormal(μ[m], Σ[m]);     // draw returns
  R_p[m] = dot_product(w, R[m]);
}
// compute empirical mean/var/VaR of R_p from samples

// time evolution of portfolio state ψ_t
// i ℏ d|ψ_t>/dt = Ĥ |ψ_t>

psi_t+Δt ≈ exp(-i Δt Ĥ / ℏ) * psi_t;

// at horizon T, density matrix ρ_T = |ψ_T><ψ_T|
// expected payoff under operator Π̂
price_0 ≈ discount * trace(ρ_T * Π̂);

dS_t = r S_t dt + σ S_t dW_t

∂V/∂t + (1/2) σ^2 S^2 ∂^2V/∂S^2 + r S ∂V/∂S - r V = 0

∂φ/∂τ = (1/2) σ^2 ∂^2φ/∂x^2 - V_eff(x) φ

i ℏ ∂ψ/∂t = Ĥ ψ

// physical picture
// --------------------------------------------------
// motherboard: connects CPU, RAM, storage, peripherals
// CPU: executes instructions sequentially (with some parallelism)
// RAM: holds bits (0 or 1) for active programs

machine ClassicalComputer {
  Motherboard board;
  CPU         cpu;
  RAM         ram;
  Storage     disk;
}

// run a program
function run_classical(program P, input bits_in[]):
  // 1. load code and data into RAM
  ram.load(P.code)
  ram.load(bits_in)

  // 2. CPU executes instructions one by one
  while cpu.instruction_pointer not at END(P):
    instr = ram.fetch(cpu.instruction_pointer)
    cpu.execute(instr, ram)
    cpu.instruction_pointer++

  // 3. read out output bits from RAM
  bits_out = ram.read(P.output_region)
  return bits_out

// physical picture
// --------------------------------------------------
// classical control computer: compiles code, sends pulses
// quantum processing unit (QPU): array of qubits on a chip
// cryostat: keeps QPU near absolute zero

machine QuantumComputer {
  ClassicalControl ctrl;     // compiler, schedulers
  QuantumProcessingUnit qpu; // qubits + control lines
}

// high-level quantum run
function run_quantum(circuit C, classical_input bits_in[]):
  // 1. classical pre-processing
  //    e.g., encode bits_in into initial qubit states
  compiled_pulses = ctrl.compile(C, bits_in)

  // 2. upload pulse schedule to QPU
  qpu.load(compiled_pulses)

  // 3. apply quantum operations (unitaries)
  qpu.apply_pulses()
  // internally, each gate is a unitary U on |ψ>:
  //    |ψ_new> = U |ψ_old>

  // 4. measure qubits
  measurement_record = qpu.measure_all() // collapses |ψ> → classical bits

  // 5. classical post-processing
  result = ctrl.post_process(measurement_record)
  return result

function parity_classical(bits[]):
  acc = 0
  for b in bits:
    acc = acc XOR b
  return acc  // 0 = even, 1 = odd

function parity_quantum(bits[]):
  // 1. encode bits into computational basis states
  //    |b_1 b_2 ... b_N>
  |ψ> = |b_1 b_2 ... b_N>

  // 2. use a circuit of CNOTs to copy global parity into an ancilla qubit
  //    |ψ, 0>  →  |ψ, parity(bits)>
  for i in 1..N:
    CNOT(control = qubit_i, target = ancilla)

  // 3. measure only the ancilla qubit
  result = measure(ancilla)
  return result  // 0 = even, 1 = odd