Foundation Maximal Forcing Rsmass Ladder Universe Is Yardstick Claim

The free yardstick

The declaration isYardstickClaim is a formal definition inside a machine-checked library of formal theorems. It names one specific proposition about a mass ladder: the claim that the ladder's base unit, written M0, equals 1. In plain terms, it is the statement that the absolute scale of masses has a particular value, namely one, in the framework's native units.

The framework's mass law places particle masses on a ladder where each rung is a power of the golden ratio φ. The declaration defines the claim that the bottom rung, the yardstick, is exactly 1. The library then proves this claim is independent, meaning it is not forced by the framework's laws. The proof exhibits two admissible worlds, one where the yardstick is 1 and one where it is 2, that both satisfy all the framework's constraints. The absolute mass scale is a free coordinate, not a derived invariant.

This stands in contrast to the ladder's spacing. The library separately proves that the ratio between adjacent rungs is always φ, a dimensionless scaling invariant that holds for every yardstick. That ratio is forced; the starting point is not. The framework's own summary states the split plainly: dimensionless structure is forced, absolute units are free.

The declaration does not claim that the yardstick is 1 in reality. It does not predict a numerical value for any particle mass. It only defines a proposition and supports the proof that this proposition is independent of the framework's axioms. The framework's laws determine the shape of the ladder, not where it begins.

/-- Independent claim: the yardstick equals one (an absolute-unit choice). -/
def isYardstickClaim : RealityClaim ℝ where
  label := "M0 = 1"
  holds := fun M0 => M0 = 1

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

/-- **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

What this page does not claim

The declaration does not assert that the yardstick equals 1 in the actual universe. It does not predict any specific particle mass value. It does not claim the yardstick is forced; the proof shows the opposite.

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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