Examples exercise organized mathematical objects, their evolution, interaction and observation under explicit laws. Choose a capability and its assumptions before running an instrument. The theory index owns definitions and mathematical scope; the execution plan owns research priorities. File numbers are stable identifiers, not a required learning order or a sequence of research milestones.

Install the repository and run the introductory example:

python -m pip install -e .
python examples/01_foundations/01_hello_world.py

Example 01 prepares a five-node cyclic organization with nonuniform signed form, circular phase and held capacity. Under its explicitly selected native relational law, the SDK certifies a winding-one recovery class, executes one shared finite step and assesses the resulting state again. The report retains the full state and law: winding identifies a geometric class, not everything needed to predict the object's response.

The initial organization is supplied. Each capture certificate supports ideal continuous recovery from its own snapshot; the computed finite-step response and numerical defect are separate evidence. This provides a small working object whose geometry, dynamics and recovery conditions can be inspected. Continue with the recovery-report guide and native execution guide.

Choose the continuation that answers your question:

The first route extends the native introductory object. The other routes select different execution contracts; a theorem or certificate transfers only when its complete premises match. A study export is not a resumable checkpoint.

Use rg --files examples -g "*.py" for the current file inventory. The grammar automaton examples reuse _flat_grammar_model.py; arithmetic and physical examples reuse their package owners. A finite fixture may recur as a controlled comparison without constituting a second implementation of its governing law.

Optional dependencies vary by script. Plotting examples require viz-basic; examples 91 and 92 additionally require scikit-learn, and 92 downloads and caches UCI data. The Torch demonstration uses compute-torch. Read module docstrings and available --help before an explicit run; the directory is not an instruction to execute every file. Importing a demonstration does not start its command-line workflow. Some demonstrations configure imports or plotting defaults at module scope, so this is not a promise of side-effect-free imports. Package ownership belongs to Architecture, and verification requirements to Testing.

Example 04 is a supplementary grammar control: it compares flat word admission with an independent, illustrative pressure proxy. It executes no operators on an evolving object. When adding operator events, use the event contracts and their live-state prerequisites as well as grammar admission.

Example 10 surveys supplied topology builders, templates and configured SDK operations. Use it as a construction reference. Example 07 is a larger prepared-state ensemble.

The type-hygiene directory retains these scoped entry points:

Entropy thresholds and scalar-only fixtures do not prove that a richer state type is necessary or impossible. Use the domain and execution contracts for signed-form admission, circular U3 admission, field availability and grammar observations.

For declared diffusion use 99. Prescribed phase/form response belongs to 179. Source recovery for retired examples belongs to the recovery index; interface changes belong to the API migrations.

These supplementary controls reuse the diffusion stability owner. The table distinguishes an exact declared model from finite engine evidence. Detailed bounds and assumptions remain in that owner instead of being duplicated as a second theorem ledger here.

Run the native evolution continuation with:

python examples/08_emergent_geometry/180_relational_exchange.py

Its detached certificate comparison distinguishes exact-reflection, full-state local and sector hypotheses. An unavailable narrower certificate can coexist with an admitted broader one. Read the execution guide for law and storage premises.

The scale bridge owns effective geometry and observation closure. The tetrad is a required diagnostic interface, not a proven complete state; reflection equivalence and loss of hidden state are separate questions. Auxiliary symplectic/U(2) demonstrations retain their own equations and premises; a Hamiltonian interpretation of engine operators requires a separate argument.

The curvature safety illustration compares supplied WS-graph field observations before and after operator events. Its fit and safety labels are configured advisories, not a stability theorem.

The unified-field showcase compares seeded supplied states on a path, cycle and barbell using the shared snapshot readout. It executes no operators or evolution. Its optional plot reports descriptive field statistics and explicitly leaves temporal conservation unavailable.

Example 179 checks a prescribed rotating phase contrast and its derived EPI response on the six-node prism. It evaluates detached analytic snapshots rather than a native trajectory. The phase/form owner states the exact assumptions, moving mean and same-input contraction scope. The imposed phase clock is not a derived autonomous maintenance mechanism.

python examples/08_emergent_geometry/179_phase_form_driven_response.py --output-dir output/phase_form_driven_response

Optional plots require the viz-basic extra. Retained outputs are the figure, JSON and CSV. Their recorded residuals and refinements are finite evidence, not autonomous formation or a physical identification.

The offline interface example compares a planted spatial target and supplied sinusoidal blocks using shared observation adapters. It executes no coupling law or nodal trajectory. The signal adapter currently omits coherence-length fit/fallback provenance; the example exposes that limitation and treats operator suggestions as telemetry.