Foundation Maximal Forcing Rsmass Ladder Universe Mass Scaling Forced Yardstick

The mass ladder

The mass ladder is a way of organizing particle masses in the Recognition Science framework. The framework models masses as sitting on a ladder of rungs, where the mass at rung r is the yardstick M0 multiplied by the golden ratio φ raised to the rth power. The yardstick is the base mass, the unit from which all other masses are measured, and the rung is the index counting steps up or down the ladder.

A central question is which parts of this ladder are forced by the framework's logic and which are free choices. The framework's library, a machine-checked collection of formal theorems, proves two things. First, the ratio between adjacent rungs is always φ, regardless of the yardstick. This is a structural identity of the ladder, so it holds for every possible base mass. Second, the yardstick itself is independent: the framework does not determine whether the base mass equals one, two, or any other number. Two different yardsticks, 1 and 2, both satisfy the framework's conditions, but they disagree on whether the yardstick equals one.

The declaration mass_scaling_forced_yardstick_free bundles these two results into a single theorem. It states that the scaling invariant is forced and the yardstick is independent. This is a formal distinction between a dimensionless structure, the ratio, and an absolute coordinate, the unit. The framework forces the shape of the ladder but leaves its position free.

This result is notable because it is the first instance in the library where the classification machinery uses both of its branches. The framework's closure logic can label a claim as forced, meaning it holds in every admissible model, or independent, meaning there are admissible models where it holds and admissible models where it fails. The mass ladder exercises both labels, showing the machinery is not trivially always-forced. It genuinely distinguishes invariants from coordinates.

What this does not claim is that the framework predicts the actual value of the yardstick, and therefore the actual masses of particles. The yardstick is a free coordinate, so the framework does not derive the absolute mass scale from its principles. It also does not claim that the golden ratio is the only possible ratio for a self-similar mass ladder, only that within this framework's definition, the ratio is forced to be φ.

/-- **The phi-ladder scaling invariant is forced over every yardstick.** No gate
is required: the recurrence is a structural identity of the ladder. -/
theorem forced_ladderRatio : Forced Lmass0.admissible isLadderRatioClaim := by
  intro M0 _ r
  show ladderMass M0 (r + 1) = φ * ladderMass M0 r
  unfold ladderMass
  rw [pow_succ]
  ring

/-- The yardstick claim is independent (Prop-level), via the witness. -/
theorem yardstick_independent : Independent Lmass0.admissible isYardstickClaim :=
  independent_of_witness yardstickIndepWitness

/-- The honest mass-layer summary: the scaling invariant is forced, the yardstick
is independent. Dimensionless structure is forced; absolute units are free. -/
theorem mass_scaling_forced_yardstick_free :
    Forced Lmass0.admissible isLadderRatioClaim ∧
    Independent Lmass0.admissible isYardstickClaim :=
  ⟨forced_ladderRatio, yardstick_independent⟩

What this page does not claim

The framework predicts the actual value of the yardstick or the absolute mass scale. The golden ratio is the only possible ratio for a self-similar mass ladder in general mathematics.

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

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