Astrophysics Dust Grain Size3 From Phi Ladder

Dust grain sizes and the phi ladder

Interstellar dust grains, the tiny solid particles that dim and redden starlight, span a well-known size range. Observations put them roughly between 0.01 and 1 micrometer, with very small grains (VSGs) near the bottom and larger grains approaching the top. The phi ladder in Recognition Science starts with 0.01 micrometers and multiplies by the golden ratio φ ≈ 1.618 at each step. At step 5 the ladder gives 0.11 micrometers, and at step 9 it gives 0.76 micrometers, so the full range φ^0 to φ^9 lands at 0.01 to 0.76 micrometers, consistent with the observed span.

The module DustGrainSize3FromPhiLadder in the framework's machine-checked library of formal theorems does not prove that dust grains actually follow this ladder. Its cost function, a measure of recognition effort, is defined as J(m/e), where J(x) = (x + 1/x)/2 - 1. The formal proofs establish three general facts about this cost: it vanishes when m equals e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive. These are properties of the cost function itself, not of dust.

In plain language, the module certifies that the cost function behaves sensibly: equal values cost nothing, costs never go negative, and the golden-ratio threshold is a real positive number. What would make this a theorem about dust grains is a definition of m and e in the subject's own terms, for instance m as a grain size and e as a reference size. Without that link, the module remains a general statement about J, shared verbatim with 2383 sibling modules in the library.

The practical takeaway is that the phi ladder offers a compact parametrization of the dust size range, but the framework's formal proofs currently stop at the cost function's general properties. The astrophysical connection is a research note, not a derived result. A reader should treat the size mapping as a suggestive model awaiting a subject-specific definition.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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]

What this page does not claim

The module does not prove that dust grains follow the phi ladder. The size mapping is not a derived result, only a research note. The cost function's properties are general and not specific to dust.

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/Astrophysics/DustGrainSize3FromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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