möbius-path-filtering

Möbius Path Filtering

Core Concept

A Möbius filter is a non-orientable topological constraint that eliminates paths where traversal would require "doubling back" or revisiting a configuration from a different orientation.

In navigator terms: some paths appear valid locally but are invalid globally because they would require the navigator to traverse the same logical position in two different ways simultaneously—geometrically impossible on a Möbius surface.

This constraint is checked before path compilation, preventing invalid Navigators from ever being cached.

Why Möbius Filtering?

The Problem

Consider a nested structure:

data = Dict(
  "a" => [1, 2, 3],
  "b" => Dict(
    "c" => [4, 5, 6],
    "a" => [7, 8, 9]  # Same key name, different context!
  )
)

A naive path like [ALL_KEYS, ALL_VALUES, ALL] might:

1. Navigate to key "a" → [1, 2, 3]
2. Navigate to key "b" → {...}
3. Navigate to nested "a" → [7, 8, 9]

But what if we then try to transform all "a" keys? The path becomes ambiguous:

• Is "a" referring to the top-level key or the nested key?
• Can we transform both simultaneously?
• What if the nested "a" contains a reference back to the top-level "a"?

On a Möbius surface, this ambiguity is geometrically impossible—you can't have two distinct orientations of the same location.

The Solution

Möbius filtering rejects paths that would create "twists" in the navigation space. It asks:

"Does this path require the navigator to visit the same logical location from two different orientations?"

If yes → path is invalid and compilation is rejected.

Architecture

Möbius Invariant

For each path step, track:

1. Position - current location in structure
2. Orientation - the "direction" we're approaching from (forward/reverse)

A valid path maintains consistent orientation throughout:

• Cannot visit (position, orientation) pairs in contradictory ways
• Cannot have cycles where orientation flips