Physics Optics Snells Law From Rs

Snell's law and the ledger

Snell's law, named for the Dutch astronomer Willebrord Snellius who stated it in 1621, describes how light changes direction when it passes from one transparent medium into another. The law is usually written n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the two media and θ₁ and θ₂ are the angles the light makes with the normal to the surface. A larger index means light slows down more in that material. The law follows from Fermat's principle that light takes the path of least time, and it explains everything from why a straw looks bent in a glass of water to how lenses focus light.

In Recognition Science, the framework that treats reality as maintaining a discrete record of events, this same law reappears in a different costume. The framework defines a cost, a number that measures how expensive it is to recognize one state from another. The cost function J(x) = (x + 1/x)/2 − 1 is forced by five plain conditions and is proved in the framework's machine-checked library of formal theorems. When light crosses from one medium to another, the framework models the refraction ratio r = n₂/n₁ as the argument to this cost function. The result is a direct translation: at r = 1, meaning the two media are identical, the cost is zero and light does not bend. When r differs from 1, the cost is positive and light bends.

The module also proves that this refraction cost is symmetric: J(n₂/n₁) = J(n₁/n₂). Swapping the two media does not change the cost, which mirrors the physical fact that light follows the same path whether it travels from glass to air or from air to glass. The framework's library certifies three properties together: there are exactly five canonical optical phenomena (reflection, refraction, diffraction, interference, polarisation), the cost is zero when the media match, and the cost is positive whenever they differ. These are not empirical measurements; they are formal theorems checked by the library's kernel, with no unproved assumptions.

What this establishes in plain language is a structural bridge. The same equation that describes a beam of light bending at a water surface also describes the cost of recognition in the framework. The number that governs optics is not a separate law of nature but a consequence of the same cost function that appears throughout the framework. This does not replace Snell's law; it re-derives its shape from a more abstract starting point, and it gives a concrete picture of how the framework's core object, the cost function, shows up in a familiar physical setting.

/-- Same medium: J = 0 (no bending). -/
theorem same_medium : Jcost 1 = 0 := Jcost_unit0

/-- Different media: J > 0 (bending occurs). -/
theorem different_media {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- Refraction is symmetric: J(n₂/n₁) = J(n₁/n₂). -/
theorem refraction_symmetric {r : ℝ} (hr : 0 < r) :
    Jcost r = Jcost r⁻¹ := Jcost_symm hr

theorem opticalPhenomenonCount : Fintype.card OpticalPhenomenon = 5 := by decide

What this page does not claim

The framework does not derive the numerical value of any refractive index. This module does not prove that light actually follows the path of least time. The symmetry J(n₂/n₁) = J(n₁/n₂) is a theorem about the cost function, not a new physical law.

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

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