Physics Wave Function Collapse From Jcost

Collapse as settling to equilibrium

Quantum measurement is the process by which a system in a superposition of states, such as an electron that is simultaneously in two locations, appears to jump to a single definite outcome when observed. In standard quantum mechanics this is called wave function collapse, and it is famously hard to explain. Recognition Science offers a picture: the wave function is a distribution of recognition cost, and collapse is the system settling to the minimum of that cost.

The cost function J is forced by five plain conditions, and it equals J(x) = (x + 1/x)/2 - 1. Its key property is that J(1) = 0, and J(x) > 0 for any other positive x. In the framework's model, a superposition has positive cost because it contains states with r ≠ 1, while a definite outcome has r = 1 and therefore zero cost. So collapse is not mysterious; it is the system moving to the unique equilibrium where the cost vanishes.

The framework also counts the standard measurement bases: position, momentum, spin, energy, and angular momentum. That there are exactly five such bases is a theorem in the framework's machine-checked library of formal theorems, not an assumption. The library proves that the count is 5, that superposition has positive cost, and that a definite outcome has zero cost, all with no axioms beyond the standard logical ones.

This reframes the measurement problem: instead of asking why collapse happens, the framework asks what cost a superposition carries. The answer, within the framework, is that a superposition is costly and a definite outcome is free. That is a concrete, testable picture, and it connects the number of measurement bases to the structure of the cost function.

/-- Before measurement: superposition has positive recognition cost. -/
theorem superposition_has_cost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- Before measurement: superposition has positive recognition cost. -/
theorem superposition_has_cost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- After measurement: definite outcome = J = 0. -/
theorem measurement_outcome_equilibrium : Jcost 1 = 0 := Jcost_unit0

theorem measurementBasisCount : Fintype.card MeasurementBasis = 5 := by decide

What this page does not claim

This does not claim that the framework explains the physical mechanism of collapse. This does not claim that the five bases are the only possible ones in standard quantum mechanics. This does not claim that the Born rule probabilities are derived in this module.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/WaveFunctionCollapseFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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