module module high

`IndisputableMonolith.Quantum.ClassicalEmergence`

The module Quantum.ClassicalEmergence supplies J-cost expressions for product states of N particles together with scaling relations and crossover conditions. Researchers tracing the quantum-to-classical transition inside Recognition Science would cite it when linking cost minimization to pointer states. The module advances through successive definitions of product and entangled costs, quadratic scaling, and decoherence timescales.

claimThe module centers on the J-cost of a product state $J(\psi_1 \otimes \cdots \otimes \psi_N)$ and the associated quantum-classical crossover where cost differences scale quadratically with particle number.

background

The module belongs to the Quantum domain. It imports the RS time quantum $\tau_0 = 1$ tick from Constants and the J-cost definition from the Cost module. It introduces PointerState as a state that minimizes J-cost under position or momentum selection, einselection_from_jcost as the selection mechanism, and decoherenceTime as the coherence-loss timescale. The setting is the emergence of classical descriptions from J-cost minima in multi-particle systems.

proof idea

This is a definition module with short scaling arguments. It defines the J-cost for product states, extends the expression to entangled states, derives the quadratic scaling of cost differences, and states the crossover condition to classical behavior.

why it matters in Recognition Science

The module supplies the J-cost calculations that feed the QMInterpretationStructure module, whose statement is that classical description emerges as a J-cost minimum. It thereby supplies the concrete multi-particle content required for the Recognition Science account of einselection and decoherence.

scope and limits

Does not derive the full quantum measurement problem.
Does not compute numerical decoherence times for laboratory systems.
Does not incorporate relativistic or field-theoretic extensions.
Does not prove uniqueness of the selected classical states.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Quantum.QMInterpretationStructure module_import

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (21)

def jcostProduct L48
def jcostEntangled L53
theorem entangled_higher_cost L57
theorem cost_difference_scales_quadratically L71
structure PointerState L80
def positionPointer L88
def momentumPointer L94
theorem einselection_from_jcost L101
def decoherenceTime L113
theorem macro_decohere_instant L117
structure QuantumClassicalCrossover L127
theorem classical_from_coarse_graining L146
theorem classical_as_jcost_minimum L155
theorem classical_limit_is_continuum L162
structure NewtonianParticle L170
theorem newton_from_jcost L180
theorem ehrenfest_theorem L189
def predictions L200
def experiments L211
structure EmergenceFalsifier L225
def experimentalStatus L232