INPUT
Coefficient plane
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OBSERVATION CHAMBER / MVP
Polynomial Root Braid Comparator
Trace the same two coefficient loops in opposite orders and compare the computed root paths, three-dimensional braids, and final root arrangements.
monodromy
Prototype
ROOT CONTINUATION / ORDER COMPARISON
Follow the same loops in two orders
Both sides compute every root independently, then track the same colored r0/r1/r2 identities through time.
ORDER COMPARISON
Same loops, different order.
Both traces use loop A once and loop B once. Only the order changes.
A→BA first, then B
versus
B→AB first, then A
Current
Next
revision 0
job —
worker —
running false
LIVE TRACE
Complex root plane
No certified trajectory yet.
TIME ↑
Unframed root braid
WebGL is unavailable. No 3D result is certified.
ORDER 01
A→B
Start → after A → after B
TRACE BOTH ORDERS to compare.
start
boundary pending
finish
WebGL is unavailable.
ORDER 02
B→A
Start → after B → after A
TRACE BOTH ORDERS to compare.
start
boundary pending
finish
WebGL is unavailable.
COMPUTED RESULT
Permutation not determined
A closed, independently certified trace is required.
Advanced · sampling and certification values
- Samples used
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- Assignment margin
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- Minimum root gap
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- Discriminant distance
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- Maximum root jump
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- Maximum residual
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- Vieta error
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- Certification
- draft
COMPUTED COMPARISON
Trace both orders to compare the final root arrangement.
Different final arrangement: not determined
A→B
Start→after A→after B
B→A
Start→after B→after A
Details: why can order matter?
The two based loops act in sequence. Their actions can be non-commutative: in braid-group language, changing the order can change the final root arrangement.
SYSTEM NOTE
Quadratic mode uses p(z;c)=z²−c with discriminant point c=0. Cubic mode uses p(z;c)=z³−3z+c with discriminant points c=−2,+2 and Δ=27(4−c²). At each step the tracked labels use σ*=arg min over σ in S_n of Σ_i |r_i(t+Δt)−r_σ(i)(t)|. A certified closed path induces a monodromy permutation. For based loops, π(A→B)=π_B∘π_A and π(B→A)=π_A∘π_B, and these need not agree.
OBSERVATION GUIDE
Touchpoints for Observation
Observe the same based loops A and B, each used once, produce two real root trajectories and potentially different final root arrangements when their order is reversed.
- Run TRACE BOTH ORDERS, compare the colored r0/r1/r2 mappings after the first loop and at the finish, then use the Quadratic one-loop preset as a simpler warm-up.
- This is sampled numerical continuation, certified with independently generated N and 2N paths up to a 4096-point cap.
- Discriminant contact, root collision, ambiguous assignment, nonfinite data, or insufficient sampling safely stops without a confirmed permutation.
- r0/r1/r2 are tracking identities, not an intrinsic ordering of polynomial roots.
This browser apparatus is neither a mathematical proof nor a computer algebra system. Its accepted result is limited to the sampled paths, solver checks, and certification bounds shown in the current report.
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 Structures & Symmetries.
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