Masses Mass Genesis Topology To Sector Rung From Topology Eq Components

The rung as a sum

A rung is a number assigned to a particle in the Recognition Science framework, a value that participates in the mass ladder. The declaration rungFromTopology_eq_components proves that this number is not an independent, arbitrary label. Instead, it is exactly the sum of three features of the particle's underlying pattern: its spin class, its generation torsion, and the count of its active edges.

In symbols, for any pattern T, the theorem states that rungFromTopology T = T.spinClass + T.generationTorsion + (T.q3ActiveEdges : ℤ). The proof is immediate, a reflexivity check in the machine-checked library of formal theorems. The statement is a definitional identity: the rung is defined to be that sum. What matters is the consequence: the rung is derived from the pattern's geometry, not stored as a separate fact.

This is part of a larger move in the framework's Mass Genesis module. The module shows that the sector, the rung, and another number Z are all read off from the same primitive pattern topology. The stored label fields in the pattern are audit slots only, a check that old values agree with the derived ones. The theorem also shows that these labels are invariant under the pattern evolution function, meaning they do not change as the pattern evolves.

In Recognition Science, this establishes that the rung, a component of the mass-generation story, is a structural consequence of the pattern, not a free parameter. The framework models a particle's identity as a geometric pattern, and the rung is one of the numbers that falls out of that geometry. The theorem does not claim that this rung value alone determines a particle's mass, nor does it prove the final load-to-mass-law identity. It is a step in a larger derivation, a clean statement about how a label is extracted from topology.

theorem rungFromTopology_eq_components (T : PatternTopology) :
    rungFromTopology T =
      T.spinClass + T.generationTorsion + (T.q3ActiveEdges : ℤ) := rfl

/-- Audit predicate: the legacy stored labels agree with the canonical
derived topology labels. Downstream proof does not need this predicate, but it
is useful for checking old PatternTopology values. -/
def TopologyAuditLabelsAgree (T : PatternTopology) : Prop :=
  T.sector = sectorFromTopology T ∧
  T.rung = rungFromTopology T ∧
  T.Z = ZFromTopology T

theorem derived_labels_evolvePattern
    (ψ : LightPattern Λ) (k : ℕ) :
    sectorOf (evolvePattern k ψ) = sectorOf ψ ∧
    rungOf (evolvePattern k ψ) = rungOf ψ ∧
    ZOf (evolvePattern k ψ) = ZOf ψ := by
  exact ⟨rfl, rfl, rfl⟩

What this page does not claim

The theorem does not prove that the rung value alone determines a particle's mass. The theorem does not establish the final load-to-mass-law identity. The theorem does not assign physical meaning to the spin class or generation torsion values.

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

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