Materials Photovoltaic Efficiency From Jcost

Photovoltaic efficiency and the cost function

Photovoltaic efficiency is the fraction of sunlight a solar cell turns into electrical power. For a single-junction cell, the Shockley-Queisser limit sets a theoretical ceiling: about 33.7% of the incoming energy, a figure derived from the balance of absorbed photons, radiative recombination, and the cell's band gap. That limit is a classical result, named for William Shockley and Hans Queisser in 1961, and it is what any model of solar efficiency must reproduce or explain.

In Recognition Science, the framework models the same problem by starting with a cost function, a number that measures the price of a mismatch between what a system produces and what it should produce. The framework proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1, where x is the ratio of measured output to expected output. The photovoltaic module defines its domain cost as J(measured / expected), so the cost vanishes when measured equals expected, and it is always nonnegative for positive inputs. Those two facts are proved in the machine-checked library of formal theorems.

The module also defines a canonical threshold, phi - 3/2, where phi is the golden ratio, and proves it is positive. That threshold is a placeholder: it marks where the framework's cost function would cross a boundary, but it is not yet tied to any specific photovoltaic mechanism. The module proves three general facts about the cost function, not a solar-cell theorem. What would make it a real result about photovoltaic efficiency is a definition of measured and expected in the cell's own terms, for example, the ratio of actual power output to the thermodynamic limit.

The framework's research note sketches a path: the ratio of the Shockley-Queisser limit to the Carnot limit for solar energy is about 0.118, which is close to J(phi), the cost function evaluated at the golden ratio. That would predict an efficiency near 11.3%, but the note calls it a structural placeholder, not the exact limit. The classical 33.7% figure stands on its own; the framework's contribution is a formal scaffold that may one day connect the cost function to a real photovoltaic model, but that connection remains open.

theorem domainCost_at_equilibrium (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 (measured expected : ℝ) : ℝ := Jcost (measured / expected)

What this page does not claim

The module does not prove any specific photovoltaic efficiency limit. The framework does not derive the Shockley-Queisser limit from the cost function. The canonical threshold is not yet tied to a physical mechanism.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: