periodDoublingTarget
The definition assigns the natural number eight as the period-doubling target in the Recognition Science model of nonlinear dynamics. Researchers modeling chaos through J-cost thresholds would cite it to fix the endpoint of the doubling cascade at 2^3. It is introduced as a direct constant definition that matches the exponent from the spatial dimension.
claimThe period-doubling target is defined as $2^3$.
background
In the module on nonlinear dynamics from Recognition Science, chaotic behavior is identified with J-cost growth once the recognition cost exceeds the J(φ) threshold. The framework lists five canonical bifurcation types (saddle-node, pitchfork, transcritical, Hopf, period-doubling) that correspond to configuration dimension D = 5. Period-doubling proceeds through the sequence 1 → 2 → 4 → 8, where the terminal value equals 2^D and aligns with the eight-tick octave of the forcing chain.
proof idea
The declaration is a direct definition that evaluates the expression 2 ^ 3 in the natural numbers.
why it matters in Recognition Science
This definition supplies the eight_periods field required by the NonlinearDynamicsCert structure, which also records five bifurcation types and zero equilibrium J-cost. It realizes the period-doubling endpoint stated in the module documentation and connects directly to the eight-tick octave (T7) and D = 3 spatial dimensions from the unified forcing chain. The module further notes the RS approximation of the Feigenbaum constant as 3φ.
scope and limits
- Does not derive the Feigenbaum constant.
- Does not prove convergence of any bifurcation sequence.
- Does not connect the target to specific physical observables.
- Does not address sensitive dependence on initial conditions.
formal statement (Lean)
31def periodDoublingTarget : ℕ := 2 ^ 3