Chemistry Viscosity3 From Phi Ladder

Viscosity and the phi ladder

Viscosity is a fluid's resistance to flow: honey pours slowly, water quickly, and pitch, over years, barely at all. The SI unit is the pascal-second (Pa·s). Measured values span a vast range, from helium at about 1.6e-5 Pa·s to pitch near 1e8 Pa·s, a spread of more than twelve orders of magnitude.

In Recognition Science, the framework models this spread with a recognition cost, a forced penalty for mismatch between two quantities. The cost function J(x) = (x + 1/x)/2 - 1 vanishes when its inputs are equal and grows as they diverge. The framework's library, a machine-checked collection of formal theorems, defines a domain cost as J applied to the ratio m/e for two quantities m and e, and proves three general facts: the cost is zero when m equals e, it is nonnegative for positive inputs, and the number phi - 3/2 is positive. These are true for any positive m and e; nothing in the proofs ties them to viscosity.

The phi ladder is the framework's proposed scaling rule: physical quantities are said to sit at powers of the golden ratio phi, about 1.618. A research note in the module suggests matching viscosity ratios to phi powers: pitch over helium, 6.25e12, is roughly phi^60, and water's viscosity is near phi^7 times helium's. These are numerical observations recorded as notes, not results the library proves.

What the module actually establishes, in plain language, is modest: the cost function behaves sensibly on positive inputs, and a threshold constant is positive. The viscosity ladder itself remains a research idea awaiting a definition of m and e in fluid terms. The library's theorems would become statements about viscosity only if such a definition were added.

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]

What this page does not claim

The module does not prove that any real liquid's viscosity follows a phi-power law. The phi ladder observations are research notes, not machine-checked results. No definition connects the abstract cost function to a specific fluid property.

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/Viscosity3_FromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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