OctantLevel
OctantLevel aliases Nat to count octant subdivision depth in the Morton geometric XOR family. Researchers encoding spatial SAT constraints cite it when indexing parity systems over 8^k regions. The definition is a direct abbreviation with no computation or bounds.
claimLet $k$ be the octant level, a natural number where $k=0$ denotes the undivided space and level $k$ partitions into $8^k$ octants.
background
The module defines the geometric XOR family extending linear masks with Morton-curve-aligned octant parity constraints. OctantLevel supplies the subdivision parameter appearing in sibling definitions such as inOctant, octantConstraint, octantsAtLevel, and mortonLocality. No upstream lemmas are required; the type simply indexes the recursive octant splitting.
proof idea
One-line abbreviation that directly equates OctantLevel to Nat.
why it matters in Recognition Science
OctantLevel is the parameter type for inOctant, mortonLocality, octantConstraint, octantsAtLevel, octantSystems, octantVars, and mortonLocality_holds. These assemble the geometric family for SAT instances carrying spatial locality. The octant count 8^k aligns with the eight-tick octave structure in the forcing chain.
scope and limits
- Does not bound level values relative to instance size n.
- Does not compute Morton indices or locality distances.
- Does not assign or verify XOR parities.
- Does not extend to non-octant partitions or higher dimensions.
formal statement (Lean)
24abbrev OctantLevel := Nat
proof body
Definition body.
25
26/-- Octant index within a level (0 to 8^level - 1). -/