Materials Nanoscale Friction From Jcost

Friction at the nanoscale

Friction between macroscopic objects follows a simple rule: the force resisting motion is proportional to how hard the surfaces are pressed together, with a constant of proportionality called the coefficient of friction. At the nanoscale, where contacts involve only a few atoms or a single molecular layer, that rule acquires a second term. The standard description adds an adhesion force, so the total friction force is F_f = μ × F_N + F_adhesion. The coefficient μ still measures how much the normal force F_N contributes, but now the surfaces can also stick to each other, and that stickiness contributes its own resistance.

For a single layer of molecules sliding over another identical layer, experiments with atomic force microscopes measure coefficients in a range from about 0.05 to 0.2. The Recognition Science framework, which derives physical structure from a forced cost function, offers a candidate value in that window. Its model defines a cost based on the ratio of two masses, m over e, and evaluates that cost at the golden ratio φ. The result is μ_nano = J(φ) ≈ 0.118, a number that sits comfortably inside the measured range. This is a model's prediction, not a derived law: the framework chooses what m and e mean for a monolayer contact, and the agreement with experiment is an empirical check.

What the machine-checked library actually proves is far more modest. The formal ledger, a discrete record of events, contains a definition of the domain cost as J(m/e) and three theorems about it. First, the cost vanishes when the two masses are equal. Second, the cost is never negative for positive inputs. Third, the threshold φ − 3/2 is positive. These are general facts about the cost function itself, and they hold for any choice of m and e, not just for friction. The library's own documentation says so plainly: the proofs establish nothing specific to nanoscale friction, because the definition of domainCost does not reference any physical property of a contact.

In Recognition Science, this is the difference between a template and a theorem. The template states the general properties any cost of this form must have. To turn it into a statement about friction, someone must define m and e in terms of the contact's own physics, such as the masses or interaction energies of the sliding layers. That definition does not exist in the current library. The framework's library proves the skeleton, and the flesh of a material-specific derivation remains a target for future work.

What the reader can take away is a clean separation. The measured range 0.05 to 0.2 is a fact of experiment. The value 0.118 is a model's suggestion, consistent with that range but not forced by it. The proved theorems guarantee the cost function behaves sensibly: zero at equality, nonnegative everywhere, with a positive threshold. Those guarantees are real, but they are about the cost function's shape, not about the sliding of molecules.

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 library proves that nanoscale friction equals 0.118; it proves only general properties of the cost function. The library derives the adhesion force from the cost function; adhesion is an input to the model. The measured range 0.05 to 0.2 validates the Recognition Science framework; it is an empirical check, not a proof.

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

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