a node `n` represents a nascent thought or perceptual cue. resonance, `n1 -> n2`, signifies n1 evokes or tends towards n2. bundling, `n1 & n2`, indicates n1 and n2 are concurrently active or associated. emergence, `(n1 & n2) => c`, shows their co-activation crystallizing into a new node `c`. flux, `?n`, marks node `n` as partially formed or uncertain, while pulse, `n`, denotes that node `n` has fluctuating salience.

fleeting connection: `?p(warm_color) & ?p(round_shape) -> n(comfort_idea)`, where faint perceptual cues nudge towards an abstract idea.

pattern recognition: `n(melody_fragment_1) & n(rhythm_pattern_x) => n(familiar_song_id)`. fluctuating recall of musical elements suddenly triggers song identification.

interdependent ideas: `n(project_scope) -> ?n(required_effort)` and `n(available_time) -> ?n(required_effort)`. the `required_effort` crystallizes from multiple weaker influences.

system: directional arithmetic (`da`)

primitives:
  `q`: scalar magnitude (non-negative real).
  `δ`: a direction from a finite, structured set `δ ∈ {δ_0, δ_1, ..., δ_n-1}`.
  `d_val = (q, δ)`: a directed value.

structure on `δ`:
  existence of `inv(δ_i)` (inverse direction).
  existence of relation `comp(δ_i, δ_j) ∈ [ -1, 1 ]` (compatibility/alignment; `1` for identical, `-1` for inverse, `0` for orthogonal, intermediate for others).

rules for binary operation `ω(d_val_1, d_val_2) → d_val_r`:
  `d_val_1 = (q_1, δ_1)`, `d_val_2 = (q_2, δ_2)`
  `q_r = f(q_1, q_2, comp(δ_1, δ_2))`
  `δ_r = g(δ_1, δ_2, context)`

each thought fragment anchors in its specific moment of origin, including the concurrent cognitive atmosphere.

relationships between insights develop through explicit tagging, semantic understanding, and shared timing.

the system reintroduces past realizations when they resonate with current explorations.

it facilitates combination of disparate insights, charting their evolution into complete ideas.

when a new notion appears, it actively seeks prior related musings.

its interface flexes, providing structured organization during focused periods and encouraging serendipitous discovery when clarity wanes.

patterns in thought development and interconnection emerge through system observation.