Foundation Maximal Forcing Rsmass Ladder Universe

The mass ladder

A ladder is a discrete record of values arranged in steps. The mass ladder places every mass on a rung: m(rung) = yardstick · φ^rung, where φ is the golden ratio, about 1.618, and the yardstick is the unit that sets the absolute scale. The rung is just the index, a whole number counting steps up or down. This is a definitional choice, a model of how masses might be organized, not a claim about the physical world by itself.

The framework then asks a sharp question about this model: which parts of it are forced, meaning they hold for every possible yardstick, and which parts are free, meaning we can change them without breaking the structure? The answer is split. The scaling invariant, the fact that adjacent rungs differ by exactly the factor φ, is forced. It is a structural identity of the ladder, true for any yardstick you pick, with no extra conditions. The absolute yardstick itself, by contrast, is independent. The framework shows this by constructing two explicit models, one with yardstick 1 and one with yardstick 2, that both satisfy the ladder definition but disagree on whether the yardstick equals one. The dimensionless structure is forced; the absolute unit is free.

This split is the point of the framework. It is the first concrete universe in the framework whose closure uses both branches of a trichotomy: every claim in the mass-ladder universe is classified as forced, independent, or selected, and here the closure splits into one forced invariant and one independent coordinate. The machinery that performs this classification is a machine-checked library of formal theorems, and the result is a certificate that every claim in this universe is accounted for. The theorem mass_scaling_forced_yardstick_free states the conjunction directly: the scaling invariant is forced and the yardstick is independent.

What this changes is the epistemic status of the mass ladder. The framework does not merely assert that masses follow a φ-power ladder; it proves that the ratio between rungs is unavoidable once you accept the ladder model, while the overall scale is a genuine choice. For a reader, the lesson is portable: in any structured family of values, some features are intrinsic to the structure and some are conventions of measurement. The framework's contribution is to make that distinction precise and machine-checkable for this specific ladder.

/-- **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

/-- **Crown trichotomy on the mass-ladder universe.** Every closure claim is
`Forced`, `Independent`, or `Selected`. Here the closure splits into one forced
invariant and one independent coordinate, with `Selected` empty. -/
theorem massUniverse_trichotomy
    (C : RealityClaim massUniverse.Realization)
    (hC : InClosure Primitive.lawOfLogic massUniverse C) :
    Forced Lmass0.admissible C ∨ Independent Lmass0.admissible C ∨ Selected Lmass0.admissible C :=
  maximal_forcing_closure_trichotomy massUniverseCert C hC

What this page does not claim

The mass ladder is a model, not a derivation from first principles. The framework does not claim that the yardstick has a specific value. The trichotomy theorem does not classify claims outside the mass-ladder universe.

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)

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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