Astrophysics Planck Mass From Phi Ladder

The Planck mass and its ladder

The Planck mass is a physical constant that defines a natural unit of mass, roughly 2.18 times 10 to the minus 8 kilograms. It is constructed from the speed of light, the gravitational constant, and the reduced Planck constant, and it marks the scale at which the effects of quantum gravity are expected to become as strong as those of ordinary gravity. It is an enormous mass by particle standards, about 10 to the 22 times heavier than an electron, yet it is tiny by everyday standards, roughly the mass of a grain of dust.

The ratio of the Planck mass to the electron mass is a pure number, approximately 2.39 times 10 to the 22. A natural question is whether this ratio has any simple structure. In Recognition Science, the framework models physical quantities as sitting on a ladder of steps, where each step multiplies a value by the golden ratio, phi, approximately 1.618. The framework proposes that the Planck mass sits on this ladder, and that the number of steps from the electron mass to the Planck mass is around 106, which is the logarithm of the ratio divided by the logarithm of phi.

In Recognition Science, the framework's machine-checked library of formal theorems contains a module named PlanckMassFromPhiLadder. The module defines a cost function that measures the distance between two masses, and it proves three general facts about that cost. The first is that the cost is zero when the two masses are equal. The second is that the cost is never negative for positive masses. The third is that a certain threshold value, phi minus 1.5, is positive. These three facts are packaged together in a structure called a Planck mass certificate.

The module proves nothing specific to the Planck mass itself. The cost function is defined as a general function of a mass ratio, without any reference to the actual values of the Planck mass or the electron mass. The module is a template, shared verbatim with thousands of sibling modules, and it would become a theorem about the Planck mass only if the mass ratio were defined in terms of the physical quantities themselves. The research note in the source records the intended direction, not a result.

What the module does establish is a small piece of the framework's machinery: a cost function with the right basic properties, and a certificate that those properties hold. The empirical comparison, that the Planck mass to electron mass ratio is about 10 to the 22 and that the phi-rung is about 106, is a measurement check, not a theorem. The framework's claim is that the ladder structure is the right way to think about the ratio, and the module provides the formal scaffolding for that claim.

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)

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

What this page does not claim

The module proves that the Planck mass actually sits on the phi-ladder. The phi-rung number 106 is derived from the framework, not measured. The Planck mass value is derived from the framework's constants.

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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