Global Geometry · How Local Rules Make Worlds

A finite-radius law becomes global only after local data are glued through an incidence structure, iterated or accumulated, and observed at a declared scale. Boundary conditions, topology and initial state complete the specification.

The plane, cylinder and flat torus illustrate the non-uniqueness warning: locally flat neighbourhoods do not determine one global topology.

Sources: [2] [6]

The laboratory state combines a finite graph or triangular complex, positive intrinsic edge lengths, an explicit counting measure for metric balls, transport conductances, vertex fields and a display projection. Stored edge lengths are authoritative; changing only the drawing cannot alter an intrinsic observable.

The normalized graph Laplacian supplies the finite transport operator, while boundary matrices over GF(2) supply exact combinatorial topology.

Sources: [2] [8]

Forward experiments iterate a local law and measure the trajectory of topology, curvature, spectrum and dimension profiles. Inverse experiments compile an obstruction-checked target before runtime, then use radius-one curvature-error feedback. Recognition experiments compute descriptors of a complete supplied finite graph and expose ambiguity rather than claiming unique manifold recovery.

The runtime labels strict local steps, disclosed pre-runtime compilation and globally mediated analysis separately.

Sources: [2] [5]

GF(2) boundary-map ranks determine beta_0, beta_1 and beta_2, and the implementation checks Euler-Poincare exactly. For a valid triangulated two-manifold, interior and boundary angle defects sum to 2 pi times the Euler characteristic.

The exact curvature badge fails closed on duplicate faces, dangling surface edges, invalid triangle metrics or invalid vertex links. A prescribed target whose sum conflicts with Gauss-Bonnet is rejected, while a compatible sum remains only a necessary condition.

Sources: [2] [6] [7]

The browser reports volume-growth, spectral and walk dimensions across independently normalized logarithmic scales. It displays the native radius, heat time and walk step instead of silently identifying their clocks.

Every value is a finite point estimate. A fixed finite graph returns to zero spectral dimension at extreme heat scales, so continuum dimension requires a supported refinement or ensemble window that this release does not claim.

Sources: [2] [9]

Nine deterministic presets cover all three research directions and six bounded application families. Visitors can run and pause local dynamics, apply declared interventions, inspect vertices, change scale, compare two matched seeds, import a closed JSON configuration and export a complete finite-run record.

The browser evaluates no user code, makes no network requests and requires no dependencies or build step.

Sources: [4] [5]

Network, manifold-learning, programmable-sheet, morphogenesis, componentwise-swarm and relational-graph physics presets reuse the same mathematics. Their domain names remain analogies rather than validation claims.

The release contains no constitutive material law, biological calibration, robot motion controller, operational network failure model, latent-manifold uniqueness theorem or Lorentzian spacetime dynamics.

Sources: [2]

The immutable public commit contains the theory contract, runtime, pure UMD/CommonJS core, test suite, machine-readable manifest, citation metadata and permissive reuse terms. Twenty-six focused checks, fifty-eight combined browser-laboratory checks and the 1,100-test repository suite passed, with one documented opt-in skip.

The next targets are compactness and scaling-limit theorems, inverse non-realizability criteria, conditions for agreement of dimension notions, stability under local defects, universality classification and lower bounds for recognition from finite observations.

• node --test website/global-geometry-lab/test-global-geometry-core.js
• python -m http.server 8000 --bind 127.0.0.1
• Open http://127.0.0.1:8000/website/global-geometry-lab/

Sources: [3] [4]