Astrophysics Neutron Star Mass3 From Jcost

A mass ratio for neutron star mergers

A neutron star merger is the collision of two ultra-dense stellar remnants, and astronomers read the event's gravitational waves to estimate the masses involved. The ratio of the lighter star's mass to the heavier one, called q, is a key observable. For the 2017 event GW170817, the measured range was q = 0.73 to 0.89.

Recognition Science (RS) is a framework that derives physical structure from a single starting point: reality keeps a ledger, a discrete record of recognition events, and the cost of recognition is forced by five plain conditions. From that cost function J(x) = (x + 1/x)/2 - 1, the framework derives constants and scaling laws. A planned application was to predict the optimal mass ratio for a neutron star merger.

The module in question, Neutron_Star_Mass3_FromJCost, defines a cost function on a mass ratio and proves three general facts about it. The cost vanishes when the two masses are equal, it is never negative for positive masses, and a certain threshold value phi - 3/2 is positive. These are true statements about the cost function, but they carry no astrophysical content on their own.

The reason is plain: the module defines the cost as Jcost(m/e) without ever defining what m and e mean in terms of neutron stars. The docstring itself says the paragraph above is a research note recording where the idea was meant to go, not a result. The Lean code proves facts about a real-valued function, not about any specific physical system.

In Recognition Science, the intended result was that the optimal mass ratio equals phi^(-1) = 0.618, the golden ratio's reciprocal. The measured range 0.73 to 0.89 includes 0.618 at its low end, but the module does not establish that prediction. The framework's library proves the general cost properties; the bridge from those to a neutron star mass ratio remains open.

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 proves no specific fact about neutron stars or their mergers. The RS prediction phi^(-1) = 0.618 is not established by the module's theorems. The measured q range does not confirm the RS prediction; it merely includes it at the low end.

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/Neutron_Star_Mass3_FromJCost.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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