module module high

`IndisputableMonolith.Quantum.ClassicalEmergence`

This module introduces the J-cost for a product state of N particles along with supporting definitions for entangled states, pointer states, and decoherence scales. It equips the Recognition Science framework with cost metrics that enable the emergence of classical descriptions from quantum configurations. Quantum physicists exploring RS interpretations would cite these when deriving crossover scales or minimum-cost pointer states. The module consists entirely of definitions without proof obligations.

claimJ-cost for a product state of $N$ particles, with extensions to entangled states satisfying the Recognition Composition Law, pointer states for position and momentum, and the quantum-classical crossover scale.

background

Recognition Science derives all physics from the J functional equation and the forcing chain T0-T8 that fixes three spatial dimensions. This module operates in the quantum domain and imports the fundamental time quantum τ₀ = 1 tick from Constants together with the base cost machinery from the Cost module. It introduces J-cost for product states of N particles, higher costs for entangled states, einselection driven by J-cost, and the decoherence time scale. These rest on the Recognition Composition Law for combining costs across particles.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The definitions feed directly into QMInterpretationStructure, which states that the classical description emerges as a J-cost minimum. The module thereby supports the RS interpretation of quantum mechanics by supplying the cost-based bridge from the Cost module to classical emergence, consistent with the phi-ladder and eight-tick octave.

scope and limits

Does not derive the Schrödinger equation or other dynamical laws.
Does not provide explicit numerical predictions for decoherence rates in specific systems.
Does not address the measurement problem beyond cost minimization.
Does not incorporate gravitational or relativistic corrections.

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