Astrophysics Asteroids From Phi Ladder V2

The asteroid threshold

Asteroids are small rocky bodies that orbit the Sun, ranging from meters to hundreds of kilometers across. The classical definition of an asteroid is size-based but fuzzy: there is no sharp boundary where a rock becomes a planet. The Recognition Science module called asteroids from phi ladder v2 asks whether a single mathematical rule, derived from the framework's cost function (a formula that assigns a price to any ratio of two quantities), can draw that line cleanly.

The module defines a domain cost: for any two masses m and e, the cost is J(m/e), where J(x) = (x + 1/x)/2 - 1. This is the same cost function that the framework proves is forced by five plain conditions. The theorem domainCost_at_eq shows that when the two masses are equal, the cost is exactly zero: J(1) = 0. This is the baseline, the point where no recognition effort is needed because the two things are the same.

The module then defines a canonical threshold at phi - 3/2, where phi is the golden ratio, about 1.618. The threshold is about 0.118. The theorem canonicalThreshold_pos proves this number is positive, using the fact that phi is greater than 1.5. The threshold is a proposed boundary: masses whose ratio to some reference mass falls below this value would be classed as asteroids, while those above it would be something else.

In Recognition Science, the framework models this as a ledger (a discrete record of recognition events) where each mass ratio is a bookkeeping entry. The cost of recognizing one mass against another is forced, not chosen. The module packages these two facts, zero cost at equality and a positive threshold, into a certificate structure. The certificate is inhabited, which means the conditions are consistent and provable.

What this establishes in plain language: the framework can produce a specific, positive number that could serve as a size cutoff for asteroids, and it proves the two basic properties any such cutoff needs. It does not yet connect that threshold to any observed asteroid belt or planetary mass. The bridge from this abstract cost to actual astronomical data remains open.

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 canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty AsteroidsFromPhiLadder_v2Cert := ⟨cert⟩

What this page does not claim

The threshold has not been tested against any observed asteroid population. The module does not prove that real asteroids obey this cost function. No claim is made about the physical mechanism that would set the reference mass.

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/AsteroidsFromPhiLadder_v2.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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