Physics Quantum Dot Exciton From Jcost

The quantum dot cost

A quantum dot is a semiconductor crystal only a few nanometers across, small enough that electrons and holes are confined in all three directions. That confinement raises the energy of the exciton, the bound electron-hole pair, and the energy gap grows as the dot shrinks. The practical result is size-tunable emission: a larger dot glows red, a smaller one glows blue, and the same material can cover the visible spectrum by changing only the radius.

The standard picture is a particle-in-a-box problem. For a spherical dot of radius R, the confinement energy scales roughly as 1/R², so the emission wavelength moves continuously with size. The Recognition Science account instead places the dot on a discrete ladder. It defines a cost function J(x) = (x + 1/x)/2 - 1, the unique function forced by five plain conditions on recognition cost, and applies it to the ratio of two masses m/e. The framework's library proves three general facts about that cost: it vanishes when m equals e, it is never negative for positive inputs, and the golden ratio phi exceeds 3/2.

In Recognition Science, the module models a quantum dot exciton by setting the cost to J(m/e), where m and e are masses to be identified. The radius is proposed to follow R = R_0 × phi^(-n) for integer n, giving confinement energies E_n = E_0 × phi^(2n). That is a model, not a theorem: the Lean file proves the three cost properties, but it proves nothing specific to dots, because the definition of m and e in the dot's own terms is absent. The research note records the intended direction, and the same template body is shared verbatim with 2383 sibling modules.

What the module actually establishes is a certificate: a packaged proof that the cost function is zero at equality, nonnegative on positive inputs, and that the golden ratio threshold is positive. That certificate is a small, machine-checked fact about the cost function. It does not derive the size-energy ladder, and it does not predict a specific emission spectrum. The honest summary is that the framework provides a forced cost function and a candidate scaling rule, but the bridge from the cost to the dot's physical masses remains open.

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)

def canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

The module does not prove the size-energy ladder for quantum dots. The module does not identify the physical masses m and e. The golden ratio threshold positivity is a fact about the cost function, not a prediction of dot emission.

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

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