# O-R Lattice Visualizer - Recent Updates

I've made several significant improvements to the [O-R Lattice Collatz Visualizer](https://wildducktheories.github.io/o-r-lattice-explorer/?x=27) that enhance navigation and exploration of Collatz sequence structure. Here's what's new:

# 🎯 Congruence Navigation System

The biggest addition is a complete **congruence navigation** system that lets you explore structurally similar Collatz sequences.

# What are Congruences?

Values that differ by **x ± 2\^k** share identical Collatz sequence structure for the first k even steps. This means:

* **x=27** and **x=19** (27-8=27-2³) share the same pattern for the first 3 even steps
* They're "congruent" - like siblings in the same structural family

# Two Types of Congruences

**1. OE Block Congruences**

For each maximal ((OE)+E+) block, you now see clickable navigation:

    x=27: OEOEE
      k=3: ← 19 | 27 | 35 →

Click the arrows or numbers to jump to congruent values that share this block structure.

**2. First Descent (x\_fd) Congruences**

The visualizer now tracks **first descent terms** \- the nearest point below your current position that satisfies:

* x < x\_search
* L < L\_search
* o < o\_search
* Maximizes o (closest approach from below)

Displayed in the X₀ info panel with congruence navigation:

    x_fd = 40
      k=5: ← 21 | 53 | 85 →

Where k = 2(o\_x - o\_fd) - (r\_x - r\_fd) represents the even steps between x and x\_fd.

# Anchor Navigation (Persistence)

When you click a congruence link, the **anchor\_k parameter** persists in the URL:

`?x=19&anchor_k=3`

The page then shows:

    anchor: OEOEE
      k=3: ← 11 | 19 | 27 →
    
    x=19: OEOEEE
      k=4: ← 3 | 19 | 35 →

The **anchor section** displays:

* The OE prefix pattern (up to k E's) that's shared across the congruence family
* Your navigation path - how you got here
* Lets you continue exploring the same k-thread

Below that, you see the maximal OE blocks for the current value.

**Browser integration:**

* Back/forward buttons work correctly
* History preserves the anchor context
* Click "Plot Sequence" or examples to clear the anchor and start fresh

# 🎨 First Descent Term Visualization

Points are now highlighted in **cyan** when they're the first descent term for a hovered point. This shows the nearest structural "ancestor" in the lattice.

The search uses a sorted (L, o) index with binary search for efficient lookup - O(log n) instead of O(n).

# ⌨️ Tooltip Keyboard Controls

Tooltips now have **smart positioning** and keyboard controls:

* **Default:** Automatically positioned in the opposite quadrant from the hovered point
* u/d **keys:** Manually move tooltip to avoid obscuring interesting regions
* Tooltip positioning persists across different points

# 📐 Theta-Line Correspondence

The λ\_x layer now displays **horizontal reference lines** matching the theta-slope lines from the o-r lattice:

* Three horizontal lines on the λ\_x (log₂(x)) axis
* Each corresponds to a theta-slope line from the main lattice
* Shows the geometric relationship: lines with slope θ=(2-log₂(3)) on o-r lattice ↔ horizontal lines on log₂(x) graph

# 🔧 Bug Fixes

* Lambda\_x layer now properly respects "Show even terms" toggle
* First descent terms always visible regardless of even/odd filtering
* Fixed descent term search to maximize **o** (not L) among candidates

# Why This Matters

These features reveal **structural invariants** in Collatz dynamics:

1. **Congruence families** show how patterns persist under x → x±2^(k) transformations
2. **Anchor navigation** lets you explore an entire structural family systematically
3. **First descent terms** identify the nearest "parent" structure in the lattice
4. Together, they expose the **periodic and hierarchical** nature of Collatz sequences

Try it yourself: Start at [x=27](https://wildducktheories.github.io/o-r-lattice-explorer/?x=27), click a congruence link, and watch the anchor track your path through the family!

**Technical Details:**

* Pure vanilla JavaScript, single HTML file
* All changes: +494 lines, -61 deletions across 4 commits
* Full browser history integration
* Responsive keyboard controls

**Live demo:** [https://wildducktheories.github.io/o-r-lattice-explorer/?x=27](https://wildducktheories.github.io/o-r-lattice-explorer/?x=27)

Feedback welcome!

o-r lattice visualiser: major update - support for congruence based navigation