Chemistry Micelle Cmc From Jcost

The CMC scaling law

The critical micelle concentration (CMC) is the concentration above which surfactant molecules cluster into spheres called micelles. In water, a surfactant such as sodium dodecyl sulfate has a CMC around 8 millimolar. A classical empirical rule, known since the 1950s, states that the logarithm of the CMC falls linearly as the alkyl chain length grows: log(CMC) = A − B×n, where n is the number of carbon atoms. For alkyl sulfates, the slope B is measured near 0.3 per carbon atom.

Recognition Science proposes a specific value for this slope. The framework models recognition events as entries in a discrete ledger, a record of events with a forced cost. The cost function J(x) = (x + 1/x)/2 − 1 is proved in the framework's machine-checked library of formal theorems to be the unique function satisfying five plain conditions. The proposed CMC law sets B ≈ J(phi)/2, where phi is the golden ratio, giving B ≈ 0.059 per carbon. This is an order of magnitude smaller than the measured 0.3, a discrepancy the framework's own research notes acknowledge.

The module named MicelleCMC_FromJCost establishes three general facts about the cost function, not about micelles. It proves that the cost vanishes when its two inputs are equal, that it is nonnegative for positive inputs, and that phi − 3/2 is positive. The module defines a quantity called domainCost as J(m/e), where m and e are real numbers, but it never defines what m and e mean for a surfactant. The research note attached to the module states plainly that this body of proofs is shared verbatim with 2383 sibling modules and that the module proves nothing specific to this subject.

What the module does establish is a template. It shows that the cost function has the formal properties a CMC law would need: zero cost at the reference point, nonnegativity away from it, and a positive threshold constant. What would turn this template into a theorem about micelles is a definition of m and e in the surfactant's own terms, for example a ratio of chain lengths or concentrations. That definition does not exist in the module. The CMC scaling law remains a research note, not a proved result.

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 CMC scaling law for any real surfactant. The value B ≈ 0.059 is not measured; it is a proposed slope from a research note. The framework's cost function is not claimed to be the only possible basis for a CMC law.

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

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