Physics Photelectric Threshold From Phi

The threshold formula

The photoelectric effect is the emission of electrons from a metal when light shines on it. Albert Einstein explained it in 1905 by proposing that light arrives in packets, or photons, each carrying energy proportional to its frequency. A metal holds its electrons with a binding energy called the work function, typically between 2 and 6 electron-volts for common metals. Light can free an electron only if its photon energy exceeds that work function, and the threshold frequency is the minimum frequency that does so.

In Recognition Science, the framework models the work function as a product of a power of the golden ratio φ and a reference binding energy. The golden ratio is the number satisfying φ² = φ + 1, approximately 1.618. The reference energy E_coh is about 0.121 eV. For an exponent k = 8, the formula gives φ⁸ × 0.121 ≈ 11.1 × 0.121 ≈ 1.34 eV, which falls inside the typical work function range. For k = 10, it gives about 14.9 eV, above the range. The framework suggests work functions should lie between φ⁸ and φ¹² times E_coh.

In Recognition Science, the module defines a cost function Jcost(m/e) that measures the recognition cost of a ratio between two quantities m and e. The framework proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the quantity φ − 3/2 is positive. The last fact is a small numerical step: since φ ≈ 1.618, subtracting 1.5 leaves about 0.118, a positive number. These proofs are general properties of the cost function, not specific to photoelectricity.

What the module does not do is connect the cost function to the work function formula. The definition of domainCost uses only the ratio m/e without any reference to photons, metals, or thresholds. The work function formula appears in a research note, not in the formal proofs. The machine-checked library of formal theorems proves the cost properties and the positivity of φ − 3/2, but the step from those to a photoelectric threshold remains a research idea, not a proved result.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove that any real metal's work function equals φ⁸ × E_coh. The cost function's properties do not depend on photoelectric physics. The threshold frequency itself is not derived in the formal library.

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

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