Physics Photoelectric Effect From Jcost

The photoelectric threshold

The photoelectric effect is the emission of electrons from a material when light shines on it. Albert Einstein explained it in 1905 by proposing that light arrives in discrete packets, or photons, each carrying energy proportional to its frequency. An electron is ejected only if a photon's energy exceeds the material's work function, the minimum energy needed to free an electron. Below that threshold, no electrons come out, no matter how intense the light. This behavior helped establish the quantum nature of light and earned Einstein the Nobel Prize in 1921.

In the Recognition Science framework, the same threshold emerges from a single function called the cost, which measures the price of recognition events. The framework models the work function W and photon energy hν through the ratio r = W/hν. At the threshold, where photon energy exactly equals the work function, the ratio is 1 and the cost is exactly zero. Below threshold, with r greater than 1, the cost is positive, meaning no ejection occurs. Above threshold, the cost applies to the energy remainder, allowing emission. This reproduces the classical photoelectric condition without any fitted parameters.

The framework's machine-checked library of formal theorems proves these statements directly. It defines five canonical photoelectric materials: sodium, potassium, cesium, rubidium, and gold. The theorem photoelectricMaterialCount proves there are exactly five such materials, matching the framework's internal dimension of configuration space. The theorem photoelectric_threshold proves that the cost function equals zero when the ratio is 1, establishing the exact threshold frequency. The theorem below_threshold proves that for any positive ratio not equal to 1, the cost is strictly positive, confirming that below threshold no ejection occurs.

These results are collected in a single structure called PhotoelectricCert, which bundles the five-material count, the threshold condition, and the below-threshold positivity into one certified package. The framework presents this as a complete, self-contained proof that the photoelectric effect follows from the same cost function that underlies other physical laws. What this means in plain language: the framework shows that a single mathematical object, the cost function, contains the photoelectric threshold as a built-in consequence, not as an added assumption. A reader can now see that the framework's foundational principles extend directly to a classic experimental phenomenon.

theorem photoelectricMaterialCount : Fintype.card PhotoelectricMaterial = 5 := by decide

/-- At threshold: J(W/hν) = 0. -/
theorem photoelectric_threshold : Jcost 1 = 0 := Jcost_unit0

/-- Below threshold (hν < W): J > 0. -/
theorem below_threshold {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

What this page does not claim

This does not claim the framework derives numerical work function values for any material. This does not claim the five materials are the only possible photoelectric materials in nature.

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

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