blend_step
The blend_step definition supplies the single-step update rule for node activation under linear blending: new value equals (1-η) times current activation plus η times incoming signal. Researchers modeling discrete causal graphs in Recognition Science cite it when deriving hop decay and ordering preservation on directed chains. The definition is a direct algebraic expression with no lemmas or case analysis.
claimThe blended activation is given by $(1 - η) ψ_{current} + η ψ_{incoming}$, where η is the blend rate and ψ denotes node activation.
background
The Causal Propagation Ordering module examines whether SpMV propagation on directed graphs preserves causal ordering and whether the 8-tick octave suffices for multi-hop reach. The blend rate η is fixed at 1/φ² in RS-native units, matching the engineering constant ETA_BLEND. The fundamental time quantum is the tick τ₀ = 1, with one octave defined as 8 ticks.
proof idea
This is a direct definition that implements the convex combination formula in one algebraic line. No upstream lemmas are invoked; the expression is written verbatim from the propagation model described in the module doc-comment.
why it matters in Recognition Science
The definition supplies the core update rule underlying module results such as hop_ordering_preserved and effective_reach_bound. It operationalizes the propagation step inside the 8-tick octave framework (T7) and connects to the question of whether directed chains maintain ordering before shortcuts appear. The module doc-comment explicitly ties it to the blended = (1-η)·my_psi + η·prop model.
scope and limits
- Does not fix the numerical value of the blend rate η.
- Does not encode graph topology or multi-hop iteration.
- Does not reference the phi-ladder, J-cost, or spatial dimension D=3.
- Does not address shortcut edges or bidirectional links.
formal statement (Lean)
48def blend_step (current : ℝ) (incoming : ℝ) (η : ℝ) : ℝ :=
proof body
Definition body.
49 (1 - η) * current + η * incoming
50
51/-- Activation at hop distance d from source after t ticks of pure blend,
52 assuming the source has unit activation and directed chain topology.
53 At distance d, signal arrives at tick d and decays as η^d per hop. -/