The winding number from algebraic topology is the exact fit.

Source: Winding number (Wikipedia)

The winding number of a closed curve in the plane around a given point is an integer representing the total number of times the curve travels counterclockwise around the point.

Key properties:

• \(w \in \mathbb{Z}\) — unbounded signed integer
• Counterclockwise (next) is positive: \(w \to w + 1\)
• Clockwise (prev) is negative: \(w \to w - 1\)
• Additive: net winding is independent of path order
• Topological invariant: homotopy class in \(\pi_1(S^1) \cong \mathbb{Z}\)

A webring of \(\text{ord}=1\) is topologically \(S^1\) (the circle). Each click of prev or next completes a full revolution (since there is only one node). The winding number counts net revolutions:

• Initial page load (no param): \(w = 0\) — at the origin
• Click "next →": \(w \to w + 1\) — counterclockwise revolution
• Click "← prev": \(w \to w - 1\) — clockwise revolution

The visitor is the origin point. The red curve (their click path) winds around them.

An object traveling along the red curve makes two counterclockwise turns around the person at the origin. — Wikipedia, Winding Number

• Animation:
• Static w=2 diagram: static/images/Winding_Number_2_svg.png

When the ring grows to \(\text{ord}\) members, each click advances position by 1 but increments \(w\) by \(1/\text{ord}\). A full winding (\(w = 1\)) requires traversing all \(\text{ord}\) members. The fundamental group structure is preserved.

IMPORTANT FOR IMPLEMENTING AGENTS: This section explains why the winding number appears trivial at ord=1 and why the full semantics only become visible at ord≥2. If you are working on an ord=1 deployment, read this carefully.

At \(\text{ord} = 1\), every click completes a full revolution. The winding number \(w\) reduces to a signed click counter: next increments by \(1/1 = 1\), prev decrements by \(1/1 = 1\). This is mathematically correct — \(\pi_1(S^1) \cong \mathbb{Z}\) — but experientially opaque. The visitor clicks "next", lands on the same page with =\(w=1=\) instead of =\(w=0=\), and has no sense of having traversed anything.

The distinction between "winding number" and "signed counter" is invisible at \(\text{ord} = 1\) because:

1. There is no fractional accumulation: \(w\) is always an integer, same as a counter
2. There is no visiting-other-sites: the ring has one member, so "traversal" is instantaneous
3. There is no experiential loop closure: at \(\text{ord} = 7\), visiting all 7 sites to reach \(w = 1\) feels like completing a circuit. At \(\text{ord} = 1\), one click = one circuit = no perceived journey.

The winding number's value as a concept comes from \(\text{ord} \geq 2\), where the \(1/\text{ord}\) fractional increment creates a visible accumulation arc and ring traversal becomes a real multi-step experience. The \(\text{ord} = 1\) case is the correct degenerate limit, not the motivating example.

[← prev]  ord=1 k=1 w=0  [next →]  ⓘ

Two static (change only on deploy), one dynamic (changes per click).

State lives entirely in the URL query parameter:

https://wal.sh/?w=0      # initial
https://wal.sh/?w=1      # one next click
https://wal.sh/?w=-3     # three prev clicks

No server state. No cookies. No localStorage. The winding number is carried in the link itself, which is topologically appropriate — it encodes an element of \(\pi_1(S^1)\).

The ⓘ icon shows a popup explaining:

Webring navigation
  ord = order of cyclic group |Z_ord|
  k   = element in Z_ord (this site's position)
  w   = winding number (net traversals, + for next, − for prev)
See: https://en.wikipedia.org/wiki/Winding_number

• Static site: org-mode → HTML via Makefile
• Hosted on FreeBSD infrastructure (nexus/hydra)
• No server-side state, no databases
• Minimal inline JS (progressive enhancement)
• Must work with no JavaScript (static fallback)

• Footer must remain sparse: ord=1 k=1 w=0 not prose
• Info icon (ⓘ) provides discoverability without clutter
• Monospace font for the status display
• No external dependencies (no frameworks, no CDN)

• \(w \in \mathbb{Z}\): must handle negative values
• For \(n = 1\): every click is a complete revolution
• For \(n > 1\) (future): \(w\) increments by \(1/n\) per click
• Zero is the identity: fresh page load = no winding

You are building a research page for wal.sh/research/ documenting the winding number concept as applied to webring navigation.

1. Create research/webring-winding/index.org following the style of existing research pages (see research/tla-plus-system-design/ or research/ads-b/ for structure)
2. Include the mathematical definition, the Wikipedia animation reference, and the variable semantics table
3. Explain why standard metaphors (pagination, odometer, counter) fail for a ring of one
4. Document the \(\pi_1(S^1) \cong \mathbb{Z}\) connection
5. Include Mermaid diagrams for state transitions and data flow
6. Keep the tone technical but accessible — this is a personal research site, not a paper

You are implementing the webring footer for wal.sh.

1. The site is built from org-mode with :tangle and :mkdirp t
2. The footer currently shows [← prev] webring (ord=1) [next →]
3. Replace with the HTML/JS/CSS fragments defined above
4. The webring.json registry is the source of truth for ord and k
5. The winding number w is read from ?w=N query parameter
6. Prev/next links encode w-1 and w+1 respectively
7. The ⓘ uses a native title attribute — no tooltip library
8. Test: curl -s 'https://wal.sh/?w=42' should show w=42 in source (the JS rewrites client-side, so static HTML shows w=0 default)
9. READ "The ord=1 Degeneracy" section. At ord=1, the winding number is mathematically correct but experientially indistinguishable from a signed counter. This is expected. The design anticipates ord>1 where the 1/ord fractional accumulation becomes visible. Do not "simplify" w to a plain counter — the algebraic structure must be preserved for when the ring grows.

You are building an interactive animation showing the cylinder topology of webring traversal with ord nodes — a spiral staircase, not a torus.

1. READ "The ord=1 Degeneracy" and "Cylinder Topology" sections first. The animation must make the winding number semantics visible, which means defaulting to ord≥3 (recommended: ord=7, matching the wal.sh logo).
2. The ring of ord nodes forms the circular cross-section of the cylinder. The winding number \(w\) is the vertical axis — each revolution moves one floor up (next) or down (prev).
3. One generator only: ring traversal (k). The winding \(w\) is the covering space coordinate, not a second loop. \(\pi_1(S^1 \times \mathbb{R}) \cong \mathbb{Z}\).
4. The curve head ("squid") tracks angular position. Active node = whichever node is angularly closest to the curve head.
5. Edge highlighting: show traversal progress as partial fills along edges 1→2→3→…→ord→1. Visited edges stay brighter.
6. Controls: w (−2 to 3), ord (1, 3, 5, 7, 12), speed, helix mode (flat/3D), pause.
7. HUD displays live ord=X k=Y w=Z matching the footer notation.
8. At ord=1 the animation should still work but the degeneracy should be visually apparent — the ring collapses to a single dot and each click is a full revolution. This contrast with ord=7 is pedagogically valuable.

Verify:

• [ ] No server-side state introduced
• [ ] Graceful degradation without JavaScript
• [ ] Negative winding numbers handled correctly
• [ ] Info tooltip is accessible (keyboard, screen reader)
• [ ] Existing footer structure and styling preserved
• [ ] webring.json schema is extensible for future members