Astrophysics Stellar Mass Function From Phi Ladder

The stellar mass function

The stellar mass function is a census of star formation. It counts, for each mass M, how many stars are born with that mass, and it is usually written as N(M) ∝ M^-α. Edwin Salpeter's 1955 survey of nearby stars found α ≈ 2.35, a value that has held up across many later observations and is now a standard benchmark in astrophysics. The slope describes a steep decline: low-mass stars are far more common than high-mass ones.

Recognition Science enters with a candidate for that slope. The framework's phi ladder, built from the golden ratio φ ≈ 1.618, suggests the exponent might be -(1 + 3/φ²) ≈ -2.146. That is close to Salpeter's 2.35, but the match is not exact, and the framework does not claim it is. The proposed formula is a research note, not a derived result.

In Recognition Science, the relevant module defines a cost function, a measure of the price of recognition, as J(m/e), where J(x) = (x + 1/x)/2 - 1 and m and e are masses. The module proves three general facts: the cost vanishes when m equals e, it is nonnegative for positive masses, and φ - 3/2 is positive. These are true of the cost function itself, not of any particular stellar population.

What the module does not do is connect that cost to a specific definition of m and e in stellar terms. Without such a definition, the Salpeter slope remains an open target, not a theorem. The framework's machine-checked library of formal theorems proves only the general properties of the cost, and the leap to a mass function slope is a hypothesis awaiting a falsifier.

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

This answer does not claim the Salpeter slope is derived from the phi ladder. This answer does not claim the framework's cost function applies to stars without a definition of m and e.

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

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