Astrophysics Circumstellar3 From Jcost

Circumstellar zones and the cost function

A circumstellar habitable zone is the ring of distances from a star where a planet's surface could hold liquid water. In our own solar system, rough empirical bounds place this zone between about 0.75 and 1.7 astronomical units (AU), where one AU is the Earth-Sun distance. The module circumstellar3 from the Recognition Science framework takes up this idea, but its name is a research note, not a description of what the machine-checked code proves.

The module defines a cost function on two real numbers, written as domainCost m e = Jcost (m / e). Here Jcost is the framework's cost function, a forced measure of recognition cost that equals (x + 1/x)/2 - 1. The module proves three general facts about this cost: it is zero when the two inputs are equal, it is never negative for positive inputs, and the number phi - 3/2 is positive. These are facts about the cost function itself, not about stars or planets.

The library's own docstring is explicit: the module proves nothing specific to astrophysics, because the cost is defined without reference to any astronomical quantity. What would make this a theorem about habitable zones is a definition of the two inputs in terms of a star and a planet, for example a ratio of luminosities or temperatures. That definition is absent. The module is one of 2383 siblings sharing the same universal template, and the content is stated once in a central file.

The research note records the intended direction: inner zone edge from Venus at 0.72 AU is close to the golden ratio inverse, phi^(-1) about 0.618, and the outer edge at 1.77 AU is close to phi about 1.618. The note claims the framework's zone spans [phi^(-1), phi] in AU, which is consistent with the empirical [0.75, 1.7] AU range. This is a stated hope, not a proved result. The three proved facts about the cost function do not depend on these astronomical numbers.

What the module does establish is a small, clean certificate: the cost function vanishes at equality, stays nonnegative, and has a positive threshold. These properties hold for any positive inputs, so they are universal facts about the cost structure. The habitable zone idea remains a research note awaiting a real physical definition. Until that definition appears, the module is a template instance, not an astrophysics result.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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 module proves the habitable zone spans [phi^(-1), phi] in AU. The empirical bounds 0.75 and 1.7 AU are measured by the framework. The cost function Jcost is derived from astrophysical principles.

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/Circumstellar3_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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: