Astrophysics Planck Mass From Phi Ladder Planck Mass Cert

PlanckMassCert

The Planck mass is the mass scale where quantum effects and gravity meet, about 2.18×10^-8 kilograms. In the Recognition Science framework, the idea was to connect this scale to the electron mass through a ladder of golden-ratio steps. The declaration PlanckMassCert is a machine-checked certificate, a formal object that records three proved facts about a cost function. The certificate does not itself establish any physical relation; it proves only that the cost function vanishes when its two inputs are equal, that it is never negative for positive inputs, and that a certain threshold number is positive.

The cost function here is defined as 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 certificate proves three general facts about this function: J(1) = 0, J(x) ≥ 0 for positive x, and φ - 3/2 > 0, where φ is the golden ratio. These facts hold for any positive real numbers m and e, not specifically for the Planck mass and electron mass. The certificate is a template, shared verbatim across many modules, and it becomes a theorem about a specific subject only when m and e are defined in that subject's own terms.

What the certificate does not claim is the physical identification. The research note attached to the module records the intended direction: that the ratio of Planck mass to electron mass might sit on a φ-ladder, with the rung number around 106. But the empirical ratio is about 2.39×10^22, and the φ-rung is about 106, whereas the note's speculative expression φ^44π ≈ φ^138 ≈ 10^29 is far off. The certificate itself proves none of this. It proves only the three general cost facts, with no reference to the Planck mass or the electron mass.

In plain terms, PlanckMassCert is a small, correct, and almost empty formal object. It establishes that the framework's cost function behaves well on positive inputs, and it leaves the physics entirely open. The reader should take it as a building block, not as a result about the Planck mass.

structure PlanckMassCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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

PlanckMassCert does not prove any relation between the Planck mass and the electron mass. The certificate does not derive the value of the Planck mass from the golden ratio. The φ-ladder identification for the Planck mass is a research note, not a proved result.

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/PlanckMassFromPhiLadder.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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