Masses Mass Genesis Topology To Sector Sector Of Eq Sector From Topology

The topology-to-sector bridge

In the Recognition Science account of particle masses, each particle is represented by a LightPattern, a structured object that carries both a stored sector label (electroweak, lepton, up quark, or down quark) and a deeper layer of data called its PatternTopology. The topology records primitive features such as charge skew, loop rank, spin class, and generation torsion. The declaration sectorOf_eq_sectorFromTopology is a theorem stating that the sector label of any LightPattern is exactly equal to the sector computed from that pattern's topology via the function sectorFromTopology. In plain terms: the sector is not an independent fact about the particle; it is a derived consequence of the topology. The proof is immediate, a reflexivity step in the machine-checked library, because the definition of sectorOf is precisely to read the topology through that function.

The same bridge is built for the other two labels. Companion theorems rungOf_eq_rungFromTopology and ZOf_eq_ZFromTopology establish that the rung (a sum of spin class, generation torsion, and active edges) and the Z value (a polynomial in charge skew, plus loop rank for colored sectors) are likewise computed from topology, not stored independently. A further theorem, derived_labels_evolvePattern, shows that all three derived labels are invariant under the pattern evolution function: evolving a pattern by any number of steps leaves its sector, rung, and Z unchanged. The stored label fields in PatternTopology are explicitly demoted to audit slots only, useful for checking old values against the canonical derived ones.

What the declaration does not claim is as important as what it establishes. It does not assert that the topology-to-sector map is physically correct, only that the formal definitions make the sector a function of topology. It does not establish the final load-to-mass-law identity, the step that would connect these labels to actual particle masses; the module docstring says so directly. It also does not claim that the topology itself is derived from anything more primitive; the construction is a compact topology model, a definitional choice within the framework, not a theorem about nature.

The practical consequence is a cleaner architecture. Downstream proofs about sectors, rungs, and Z values can rely on the topology as the single source of truth, and the audit predicate TopologyAuditLabelsAgree provides a check that legacy stored labels match the canonical derived ones. The bridge turns three label lookups into three computed functions, and the evolution invariance means those computations are stable under the pattern dynamics.

theorem sectorOf_eq_sectorFromTopology (ψ : LightPattern Λ) :
    sectorOf ψ = sectorFromTopology ψ.topology := rfl

theorem rungOf_eq_rungFromTopology (ψ : LightPattern Λ) :
    rungOf ψ = rungFromTopology ψ.topology := rfl

theorem ZOf_eq_ZFromTopology (ψ : LightPattern Λ) :
    ZOf ψ = ZFromTopology ψ.topology := rfl

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⟩

/-- 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

What this page does not claim

This theorem does not assert that the topology-to-sector map is physically correct. This theorem does not establish the final load-to-mass-law identity. This theorem does not derive the topology itself from anything more primitive.

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)

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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