{
  "schema_version": 1,
  "subjects": [
    {
      "id": "optimisation",
      "title": "Optimisation",
      "intro": "Follow an optimization proof from its objective and access model to a quantitative guarantee. Browse mathematical foundations separately from formal libraries, AI tools and new papers.",
      "libraries": ["optimisation"],
      "route": [
        {"title": "Specify the problem", "text": "State the feasible set, geometry, convexity and smoothness assumptions. Fix the oracle and whether the guarantee is an objective gap, stationarity or feasibility."},
        {"title": "Choose the mechanism", "text": "Start with a one-step descent or potential inequality. Distinguish gradient, proximal, mirror, accelerated and block methods rather than treating all convergence proofs as interchangeable."},
        {"title": "Check the guarantee", "text": "Track constants, step sizes, initialization and the actual oracle count. Read lower bounds and counterexamples under the same access model."}
      ],
      "boundary": "External methods and theorem candidates are reference material. Existing local proofs and source coverage are reported only by their canonical publication pages."
    },
    {
      "id": "sampling",
      "title": "Sampling",
      "intro": "Begin with the target, state space and access model. Choose a sampling regime before a named algorithm; use the textbook for foundations and SampleWiki for research results.",
      "libraries": ["log-concave-sampling", "samplewiki-route", "discrete-sampling", "mcmc"],
      "route": [
        {"title": "Specify the experiment", "text": "Record the target law, continuous or discrete state, initialization and permitted oracle. A sampler must define an actual measurable random output or probability kernel."},
        {"title": "Separate the proof obligations", "text": "Construction, non-explosion, invariance, convergence and implementation bias are different results. Discrete sampling is not the numerical discretization of a continuous process."},
        {"title": "Compose errors and costs", "text": "Match TV, Wasserstein, KL or another discrepancy to the theorem. Track conditional errors and costs under the actual input law; TV proximity alone does not control an unbounded expected query count."}
      ],
      "boundary": "Optimization primitives can support a sampling proof but cannot replace kernel semantics, integrability, domain assumptions, a mixing estimate or a cost theorem."
    },
    {
      "id": "geometry",
      "title": "Geometry and transport",
      "intro": "Keep the ambient space, metric and geometric structure visible. Read manifold optimization and statistical transport together where useful, without identifying their different notions of gradient or curvature.",
      "libraries": ["riemannian-optimization", "statistical-optimal-transport"],
      "route": [
        {"title": "Choose the space", "text": "Specify manifold, tangent space, Riemannian metric, retraction or geodesic. For transport, state measures, moments, couplings and the transport cost."},
        {"title": "Identify the adapter", "text": "Ask which Euclidean ingredient survives and which metric-dependent inequality must be proved anew. A similar proof diagram is not a theorem that transports between spaces."},
        {"title": "Expose the boundary", "text": "Distinguish geodesic convexity, displacement convexity and analytic curvature assumptions. Check completeness, cut loci, optimizer existence and moment hypotheses where the argument uses them."}
      ],
      "boundary": "Title-based discovery links below suggest reading, not source equivalence or a certified Euclidean-to-Riemannian/Wasserstein transport."
    }
  ],
  "libraries": {
    "log-concave-sampling": {"title": "Log-Concave Sampling", "path": "textbook/index.html", "role": "Chewi's source-ordered textbook and official supplement"},
    "samplewiki-route": {"title": "SampleWiki", "path": "example-cases/samplewiki.html", "role": "Sampling regimes, research problems and new results"},
    "optimisation": {"title": "Optimisation", "path": "libraries/optimisation/index.html", "role": "Chewi's Lectures on Optimization; formal-upstream reuse"},
    "riemannian-optimization": {"title": "Riemannian Optimization", "path": "libraries/riemannian-optimization/index.html", "role": "Boumal's manifold geometry and optimization route"},
    "statistical-optimal-transport": {"title": "Statistical Optimal Transport", "path": "libraries/statistical-optimal-transport/index.html", "role": "Chewi, Niles-Weed and Rigollet; couplings, duality and statistics"},
    "discrete-sampling": {"title": "Discrete Sampling", "path": "libraries/discrete-sampling/index.html", "role": "Finite-state models, spectral independence and mixing"},
    "mcmc": {"title": "Markov Chain Monte Carlo", "path": "libraries/mcmc/index.html", "role": "General-state kernels, scalable methods and estimation"}
  },
  "optlib_areas": {
    "Differential": {"title": "Differential calculus", "purpose": "Gradients, chain/product rules, mean-value and Taylor arguments.", "chapter": "libraries/optimisation/chapter-01.html", "consumer": "Shared analytic floor for Euclidean energy estimates. Riemannian gradients require their own tangent-space and metric adapter."},
    "Convex": {"title": "Convex analysis", "purpose": "Convexity, strong convexity, subgradients, cones and separation.", "chapter": "libraries/optimisation/chapter-01.html", "consumer": "Potential curvature can feed log-concavity prerequisites. It does not by itself prove a functional inequality, mixing bound or invariant law."},
    "Function": {"title": "Smoothness and proximal structure", "purpose": "Smoothness models, lower semicontinuity, proximal maps and Kurdyka–Łojasiewicz structure.", "chapter": "libraries/optimisation/chapter-08.html", "consumer": "A proximal minimizer and a Gibbs/RGO draw are different objects. Also distinguish Kurdyka–Łojasiewicz properties from Kullback–Leibler divergence."},
    "Optimality": {"title": "Optimality and duality", "purpose": "Unconstrained stationarity, constraint qualifications, KKT and weak duality.", "chapter": "libraries/optimisation/chapter-09.html", "consumer": "Transport and constrained algorithms need their own primal/dual spaces, attainment and constraint qualifications; do not infer them from an analogy."},
    "Algorithm": {"title": "Algorithms and complexity", "purpose": "Gradient/subgradient methods, proximal gradient, acceleration, LASSO, BCD and ADMM.", "chapter": "libraries/optimisation/chapter-03.html", "consumer": "Compare actual updates, potential estimates and oracle models. No optimization convergence theorem is silently promoted into sampling complexity."}
  }
}