Astrophysics Stellar Imfstructure

The stellar mass spectrum

The initial mass function (IMF) is the empirical distribution of stellar masses at birth: for every thousand low-mass stars, how many massive ones appear. Astronomers measure it by counting stars in clusters and fitting a power law, with the Salpeter slope of about -2.35 for stars above one solar mass. The IMF shapes galaxy evolution, because massive stars die quickly and return heavy elements, while low-mass stars persist for billions of years.

The classical form is dN/dM ∝ M^-2.35 for M > 1 solar mass, with a flatter slope below that. Edwin Salpeter published this in 1955, and subsequent work by Miller and Scalo in 1979 and Kroupa in 2001 refined the low-mass end. The IMF is not a fundamental law but a measurement with real scatter between environments, yet its overall shape is consistent across the Milky Way.

In Recognition Science, the framework models recognition as a discrete record of events where the cost of recognition is forced by five plain conditions. From that cost function, the framework derives a phi-power ladder for particle masses and an eight-tick cycle. The stellar IMF appears in this account as a structural input to the ultra-high-energy cosmic ray (UHECR) side, not as a derived prediction.

The module defines stellar_imf_from_ledger as the proposition that the IMF structure implies UHECR-side structural input. Its theorem stellar_imf_structure establishes this implication from the framework's UHECR structure theorem. A second theorem, stellar_imf_implies_uhecr, states that if the IMF-from-ledger proposition holds, then the UHECR-from-ledger proposition follows directly. In plain language: the framework's library proves that the IMF's observed structure carries information about the cosmic ray spectrum, without deriving the IMF itself.

What this changes is the epistemic status of the IMF-UHECR link. Within the framework, the connection is a proved implication, not a correlation or an assumption. The empirical IMF measurement remains a measurement; the framework contributes the structural bridge. A reader can now see that the framework treats the stellar mass spectrum as a genuine input to cosmic ray structure, with a machine-checked guarantee that the implication holds.

def stellar_imf_from_ledger : Prop := uhecr_from_ledger

theorem stellar_imf_structure : stellar_imf_from_ledger := uhecr_structure

/-- Stellar-IMF structure implies UHECR-side structural input. -/
theorem stellar_imf_implies_uhecr (h : stellar_imf_from_ledger) : uhecr_from_ledger :=
  h

What this page does not claim

The framework does not derive the IMF's observed slope or shape. The framework does not prove that the IMF actually implies UHECR properties in the physical world. The module does not establish any quantitative relation between stellar masses and cosmic ray energies.

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/StellarIMFStructure.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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