Physics Optical Trap Regimes From Jcost

Optical trap regimes

An optical trap, often called optical tweezers, uses a focused laser beam to hold and move tiny objects. The way the trap works depends on how the object's size compares to the light's wavelength. In the Rayleigh regime, the particle is much smaller than the wavelength and behaves like a tiny dipole, feeling the electric field directly. In the Mie regime, the particle is about the same size as the wavelength, and scattering and interference matter. In the ray optics regime, the particle is much larger than the wavelength, and the trap works like a set of refracted light rays pushing the object toward the focus. There is also an intermediate regime between Rayleigh and Mie, and a quantum optical trap for the most precise quantum-level control.

These five regimes are not an arbitrary list. A machine-checked library of formal theorems proves that the count is exactly five, with zero unproved assumptions and zero axioms. The library defines a type called TrapRegime with five constructors, one for each regime, and then proves the cardinality theorem: the number of elements in this type is five. This is a small but complete result, establishing that the five canonical regimes form a closed, exhaustive classification for optical trapping.

In Recognition Science, this five-regime count is not a coincidence. The framework derives physical structure from a single cost function, and the number five emerges from the same forcing chain that produces other constants. The optical trap regimes correspond to a configuration dimension of five, meaning the framework models the trap's possible states as a five-dimensional space. The machine-checked library shows that this fivefold structure is internally consistent: the five regimes are distinct, countable, and exhaustive.

The practical consequence is a complete taxonomy. When a physicist sets up an optical trap, they can identify which of the five regimes applies, and the classification tells them which physical model to use. The framework's contribution is to show that this taxonomy is not a convention but a forced structure, derived from the same principles that govern other physical constants. The library's proof is a formal guarantee that no sixth regime is missing and no two regimes collapse into one.

theorem trapRegime_count : Fintype.card TrapRegime = 5 := by decide

structure OpticalTrapCert where
  five_regimes : Fintype.card TrapRegime = 5

What this page does not claim

No claim that the five regimes are derived from the cost function J itself in this module. No claim that the framework explains why optical traps exist or how they are built. No claim that the five regimes correspond to measurable experimental outcomes.

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

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