Materials Photoresist Resolution From Jcost

Resolution limits

Photoresist resolution is the smallest feature a lithography process can print, and the standard way to state it is the Rayleigh criterion: R = k1 × λ / NA, where λ is the wavelength of the exposing light, NA is the numerical aperture of the optics, and k1 is a dimensionless factor that captures how well the whole process uses the optical system. In production extreme ultraviolet (EUV) lithography, k1 sits in a band from about 0.3 to 0.5, limited by resist chemistry, optics, and mask effects. The classical formula is a definitional model: it organizes the contributors to resolution, but it does not by itself say what the smallest possible k1 is.

The Recognition Science framework derives a theoretical lower bound for k1 from its cost function. In this account, the recognition-limited lithography factor is k1 = J(φ)/2 ≈ 0.059, where J is the framework's cost function and φ is the golden ratio. That value is a floor: it is what the framework's recognition ledger would permit if practical constraints vanished. The gap between 0.059 and the empirical 0.3 to 0.5 is large, and the framework does not claim to close it; the gap is where real resist chemistry and optics live.

What the machine-checked library actually proves about this subject is deliberately modest. The module defines domainCost, a discrete record of cost, as J(m/e), the cost function applied to the ratio of two real numbers m and e. Three general facts are proved: the cost is zero when m equals e, the cost is nonnegative for positive inputs, and the threshold φ − 3/2 is positive. These are true for any positive real numbers; the module proves nothing specific to photoresists, because m and e are not defined in resist terms. The research note in the file says plainly that the subject-specific claim is where the idea was meant to go, not a result that has landed.

The honest takeaway is that the framework offers a candidate theoretical floor for k1, but the module itself is a template. What would turn it into a theorem about photoresists is a definition of m and e in resist terms, for example a model of exposure dose and threshold. Until that definition exists, the 0.059 value is a prediction with a named falsifier: a measured k1 below that floor in a real process would refute it. No such measurement exists.

noncomputable def cert : PhotoresistResCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

noncomputable def cert : PhotoresistResCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

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 framework proves a photoresist-specific resolution limit. The 0.059 value is an achieved or achievable process result. The Rayleigh criterion itself is derived from the framework.

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/PhotoresistResolution_FromJCost.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: