OBSERVATION CHAMBER / ACTIVE
Hodge Decomposition Workbench
Decompose a discrete edge field on a square-cell complex into gradient, rotational, and harmonic parts, and see how holes determine the harmonic dimension.
Decomposition controls
Synchronized field views
Advanced matrices and diagnostics
Cell-complex statistics
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Solver and residuals
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Harmonic basis diagnostics
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Lifecycle guard
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Model limits
Finite 2D square-cell complexes, double-precision coefficients, one connected component, and a scale-aware numerical acceptance test. This is not a formal proof or an engineering design tool.
Result output
OBSERVATION GUIDE
Touchpoints for Observation
A discrete edge field separates orthogonally into gradient, rotational, and harmonic parts while holes determine the harmonic degrees of freedom.
- Start with One-hole circulation, run DECOMPOSE, switch Original, Gradient, Rotational, and Harmonic, then edit the field or domain and compare the accepted result.
- The connected two-dimensional square-cell model accepts at most 120 active faces and uses double-precision coefficients with a scale-aware tolerance.
- Its fixed orientation and free-boundary projection do not cover arbitrary meshes, weighted inner products, general boundary conditions, or three dimensions.
- Equivalent harmonic bases need not be unique; this apparatus applies a deterministic edge order, basis order, and sign convention. Play changes display interpolation only, never the committed result.
This is a finite numerical observation, not a formal proof, continuous PDE solver, cryptographic tool, or engineering design system.
Runs inside the browser with no upload and no registration.
SYSTEM NOTE
On a connected oriented square-cell complex, C⁰ --D0→ C¹ --D1→ C² satisfies D1D0=0. Decompose the edge 1-cochain as F=D0φ+D1ᵀψ+h, fixing φ(v0)=0 and solving D0ᵀD0φ=D0ᵀF and D1D1ᵀψ=D1F. The harmonic remainder obeys D1h≈0 and D0ᵀh≈0. Reconstruction, pairwise orthogonality, and ||F||²≈||D0φ||²+||D1ᵀψ||²+||h||² are checked at the reported tolerance. Topology gives b1=b0−V+E−F=basis count=dim H, and an orthonormal basis q_k reconstructs h≈Σ<h,q_k>q_k with Gram matrix near identity.
OBSERVATION POLICY
This lab runs in your browser. Domain edits, edge-field values, calculations, reports, and images are not uploaded by this lab.
Route
This chamber belongs to Structures & Symmetries.
STRUCTURE Back to Structures & Symmetries Route.