OBSERVATION CHAMBER / ACTIVE
Read critical points and sublevel-set changes in a finite two-dimensional scalar landscape.
ROUTE 75 / STRUCTURE / CRITICAL POINT / LEVEL SET
FINITE SCALAR CUBICAL LANDSCAPE
Raise or lower vertices in a finite 32×24 scalar landscape. COMPUTE builds the sublevel complex at the selected LEVEL and exposes its V−E+F balance, event Δχ, and critical-point classification.
Canvas 2D is unavailable. The finite counts and event table remain available.
Click a vertex to select it. Focus the canvas: arrow keys move the selection, +/− edit its value, [/] change LEVEL, C computes, and R resets. The drawing is a finite scalar grid, not a physical terrain or a time animation.
Ready. Press COMPUTE to read the current LEVEL.
SUBLEVEL READOUT
A level change clears the committed readout. This keeps the displayed counts, report, and data attributes synchronized with the last COMPUTE.
DETERMINISTIC SELF-TEST
FIXED 32×24 / FLAT 3×3 FIXTURE
| CHECK | VALUE | LIMIT | RESULT |
|---|
EVENT LEDGER
| LEVEL | VERTEX | TYPE | Δχ | V / E / F | STATE |
|---|
Rows near the selected LEVEL are shown first. A minimum, saddle, or maximum is classified from the ordered four-neighbor link; boundary vertices remain finite and deterministic.
OBSERVATION CONTRACT
The primary object is a finite 2D scalar function sampled on a cubical grid. HEIGHT shows values, CONTOUR shows level sets, and CRITICAL marks ordered local events. No time evolution, particle motion, or physical measurement is implied.
WHAT TO WATCH
MODEL NOTE
Kλ = {cells σ : maxv∈σ f(v) ≤ λ}χ(Kλ) = V(λ) − E(λ) + F(λ)
The event ledger adds each vertex, its incident edges, and its lower-star face cells in the stable order (value, vertexID). The residual is the exact integer identity χ − (V−E+F) = 0.
For each square cell, its threshold is the maximum of its four vertex values. Edges use the maximum of their two endpoints. Therefore a selected λ gives finite integer counts V(λ), E(λ), and F(λ), with event Δχ recorded when an ordered vertex enters.
Critical classification is a local finite reading: no lower neighbors is MINIMUM, no upper neighbors is MAXIMUM, and alternating lower/upper neighbors give SADDLE. Exact ties use vertexID.
OBSERVATION GUIDE
Start at BASIN and press COMPUTE. Move LEVEL to watch the sublevel counts, then choose CRITICAL to see the active markers.
Runs inside the browser with no upload and no registration.
This chamber runs a deterministic finite 2D scalar cubical landscape in your browser. Values, sublevel counts, reports, and images stay local. It does not measure visitors or physical terrain.
This chamber belongs to Structures & Symmetries.
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