Astrophysics Stellar Mass Function From Phi Ladder Salpeter Imfcert

The Salpeter certificate

The initial mass function (IMF) describes how many stars form at each mass in a population. Astronomer Edwin Salpeter proposed in 1955 that the number of stars with mass M follows a power law, N(M) proportional to M^(-2.35). The exponent -2.35 is a measured property of real star-forming regions, not a number derived from first principles. The Recognition Science framework contains a declaration called SalpeterIMFCert that sounds as if it might certify this astrophysical law, but the certificate establishes something far more modest.

Within the framework, a cost function measures the recognition effort between two quantities, here written as J(m/e) where m and e are positive real numbers. The certificate proves three general facts about this cost function. First, when the two quantities are equal, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the golden ratio phi is greater than 1.5, which is a simple numerical fact. These three statements are true, and the machine-checked library of formal theorems verifies them.

The certificate does not connect these facts to stellar masses. The cost function is defined as J(m/e) without any reference to stars, masses, or the Salpeter law. The declaration merely packages three general properties of a cost formula under a name borrowed from astrophysics. The framework's own documentation states this plainly: it proves nothing specific to the subject, because the cost function is defined without reference to one. The Salpeter exponent -2.35 does not appear in any theorem.

What the certificate does is demonstrate that a certain cost formula has the basic properties one would want from a recognition measure: it vanishes at equality and stays nonnegative. This is a small but solid result. The research note attached to the framework records an attempt to relate the Salpeter exponent to the golden ratio, suggesting an exponent around -2.146 instead of -2.35, but this is an unpublished idea, not a proved claim. The Salpeter exponent itself remains a measured astrophysical quantity, not a derived one.

Readers should take the certificate for what it is: a formal record that three elementary facts about a cost function hold. It is not a derivation of the IMF, not a prediction of the Salpeter exponent, and not evidence that the framework explains stellar populations. The gap between the certificate's name and its content is deliberate in the framework's honesty system: the name flags an intended research direction, while the theorems state only what is actually proved.

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 Salpeter exponent -2.35 is derived or predicted by the certificate. The cost function J(m/e) has any connection to stellar masses in the proved theorems. The research note's suggested exponent -2.146 is a proved result.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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