preferredBasisExamples
Recognition Science resolves the preferred basis problem by mapping pointer states to neutral windows of minimal J-cost in the landscape defined by the Recognition Composition Law. This definition supplies five canonical examples: position basis for macroscopic objects, energy eigenstates for atoms, coherent states for oscillators, field-aligned states for spins, and charge states for quantum dots. Quantum foundation researchers cite it when connecting RS environment symmetries to standard decoherence outcomes. The content is a direct list from
claimThe preferred basis examples are the associations: macroscopic objects with position basis (localization), atoms in vacuum with energy eigenstates, harmonic oscillators with coherent states, spins in magnetic field with field-aligned states, and quantum dots with charge states.
background
Recognition Science derives all structure from the J-cost functional obeying the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The shifted cost H(x) = J(x) + 1 satisfies the d'Alembert equation H(xy) + H(x/y) = 2 H(x) H(y), as given in CostAlgebra.H and reparametrized in FunctionalEquation.H. Pointer states are defined as the stable configurations that minimize J-cost locally; these are the neutral windows. The module QF-003 shows that environment interactions drive systems to these windows on the decoherence timescale, with the 8-tick octave selecting the concrete basis. Upstream LedgerFactorization.of supplies the multiplicative structure on positive reals used to calibrate J.
proof idea
This is a direct definition that enumerates five illustrative cases. No lemmas or tactics are invoked; the list follows immediately from the module description of how the 8-tick structure plus environment symmetries select pointer states.
why it matters in Recognition Science
This definition grounds the QF-003 claim that pointer states emerge as neutral windows in the J-cost landscape. It supplies concrete illustrations for the eight-tick octave (T7) and environment symmetries that select bases such as position for macroscopic objects. No theorems depend on it directly, yet sibling results such as pointer_states_are_neutral_windows and predictability_sieve_selects_pointer_states rest on the same neutral-window mechanism. It leaves open the full derivation of Lindblad operators from RS principles.
scope and limits
- Does not prove dynamical selection of these bases by the Lindblad equation.
- Does not compute decoherence timescales for the listed examples.
- Does not link the examples to explicit phi-ladder rung assignments.
- Does not quantify J-cost differences between superpositions and pointer states.
formal statement (Lean)
102def preferredBasisExamples : List (String × String) := [
proof body
Definition body.
103 ("Macroscopic objects", "Position basis - localization"),
104 ("Atoms in vacuum", "Energy eigenstates"),
105 ("Harmonic oscillators", "Coherent states (classical-like)"),
106 ("Spin in magnetic field", "Field-aligned states"),
107 ("Quantum dots", "Charge states")
108]
109
110/-! ## Mathematical Framework -/
111
112/-- The Lindblad equation describes open system evolution.
113
114 dρ/dt = -i[H, ρ] + Σ_k (L_k ρ L_k† - ½{L_k† L_k, ρ})
115
116 The Lindblad operators L_k encode environment coupling.
117 Pointer states are eigenstates of the L_k operators. -/