OBSERVATION CHAMBER / MVP
Riemann Sheet Navigator
Trace a closed path through analytic branches and observe how returning to the same z can land on a different sheet.
01 / DRAW THE LOOP
z-plane
Drag A outside the loop, then trace again.
COMPUTED SURFACE
Riemann surface and lifted path
Advanced paths, sheets, sampling, and diagnostics
Path and sheet settings
DiagnosticsNot traced
- Path status
- closed
- Sampling quality
- not evaluated
- Source type
- parametric-preset
- Active branches
- A, B
- Requested / effective
- 512 / —
- Start z
- —
- Current z
- —
- Current sheet
- —
- Closed path
- —
- Samples
- —
- Max angular step
- —
- Min branch clearance
- —
- Candidate ambiguity
- —
- Max jump
- —
- Finite
- —
- Previous valid trace
- none
- Safe-stop reason
- none
Observation report
SYSTEM NOTE
At each sampled z_n, the apparatus evaluates every finite branch candidate and selects w_n = argmin_j |w_j(z_n) - w_(n-1)|. Winding around a branch point p is accumulated segment by segment as (1 / 2pi) sum atan2(cross(z_i-p, z_(i+1)-p), dot(z_i-p, z_(i+1)-p)). Parametric paths are certified by independent N and 2N evaluations before a final sheet or monodromy is reported.
OBSERVATION GUIDE
Touchpoints for Observation
Observe analytic continuation as a lifted path: the z-loop closes while the followed value can arrive on another sheet.
- TRACE the default Pair sqrt loop, drag A outside the loop to obtain identity, restore it, then compare single, product, and ratio logarithms.
- Branch cuts are display conventions; changing their angle does not change closed-loop monodromy.
- Open, branch-contact, non-finite, or uncertified paths stop without assigning a final sheet.
- The 3D surface uses a finite mesh and masks the immediate neighborhood of singularities.
This is finite-range browser computation of analytic continuation and monodromy, not a mathematical proof. Sampling certification can refuse a result, and the rendered surface is not a complete infinite Riemann surface.
Runs inside the browser with no upload and no registration.
OBSERVATION POLICY
This lab runs in your browser. No image, pointer trace, or input is uploaded by this prototype.
Route
This chamber belongs to Shape & Space.
GEOMETRY Back to Shape & Space Route.