Astrophysics Neutron Star Mass From Phi Ladder

The mass ladder

A neutron star is the collapsed core left behind when a massive star explodes. Its mass typically falls in a narrow band, and the most common value is close to 1.33 times the Sun's mass. That number matters because it sits below the Chandrasekhar limit, the maximum mass a white dwarf can hold before gravity wins and it collapses further. The standard limit is about 1.44 solar masses, so the observed peak sits at roughly 92.4 percent of that ceiling.

The framework's research note proposes a specific relationship: the neutron star mass peak equals phi to some power times the Chandrasekhar mass, where phi is the golden ratio, about 1.618. The ratio 1.33 divided by 1.44 is approximately 0.924, which the note identifies with 1 minus the square root of the framework's cost function evaluated at phi. In plain terms, the idea is that neutron star masses cluster one rung below the Chandrasekhar limit on a ladder whose steps are powers of the golden ratio. This is a suggestion about where the masses should be, not a demonstrated fact.

What the machine-checked library of formal theorems actually proves is much narrower. It defines a cost function that takes a measured value and an expected value, and returns the framework's cost, a forced penalty for mismatch. The formal results show three things: the cost is zero when the measured value equals the expected value, the cost is never negative for positive inputs, and a certain threshold constant, phi minus 1.5, is positive. These are general properties of the cost function, not statements about neutron stars.

The module does not define what the measured mass or the expected mass is in astrophysical terms. It never connects the cost function to actual stellar data. The research note records where the idea was meant to go, and the formal file itself says it proves nothing specific to this subject. The gap is explicit: to make this a theorem about neutron stars, someone would need to define the measured and expected masses in the star's own terms, and that definition does not exist yet.

What a reader can take away is the shape of the intended argument. The framework's cost function has the formal properties needed to compare a measured mass against an expected one. The proposed expected mass is a golden-ratio-scaled Chandrasekhar limit. But the link between the formal cost and the astrophysical numbers remains a research aspiration, not a proved result.

theorem domainCost_at_equilibrium (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 proves no astrophysical fact about neutron star masses. The golden-ratio mass ladder is a research note, not a derived result. The cost function's formal properties do not by themselves connect to stellar data.

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

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