Masses Excitation Ordering First Increment Is Passive Edges

The first excitation

The declaration first_increment_is_passive_edges is a theorem in the framework's machine-checked library of formal theorems. It states a precise arithmetic fact about a cube and a sequence of numbers called a torsion schedule. In plain language: the difference between the second and the first entry in the schedule is exactly 11, the number of passive edges of the cube. The first entry is 0, so the first nontrivial increment is the count of passive edges.

The cube in question is the three-dimensional cube Q₃, which has 8 vertices, 12 edges, and 6 faces. The framework models the three generations of fermions as excitations that couple to subcells of this cube in order of their CW dimension: vertices first (dimension 0), then edges (dimension 1), then faces (dimension 2). The torsion schedule that emerges is {0, 11, 17}. The first excitation adds the 11 passive edges; the second adds the 6 faces, giving 11 + 6 = 17. The theorem first_increment_is_passive_edges proves the first step of this schedule.

The geometric ordering is what makes the result structural rather than numerical. The theorem edge_dim_lt_face_dim proves that an edge has CW dimension 1 and a face has CW dimension 2, so edges come before faces by the cube's own geometry. The theorem edge_is_minimal_nontrivial_excitation proves that the edge is the lowest-dimensional cell with a positive coupling. Together these establish that the first excitation is edge-supported because edges are dimensionally prior to faces, not because 11 happens to be smaller than 6.

The cost ordering is a separate theorem. The framework's cost function J, applied to powers of the golden ratio, gives J(φ⁰) = 0, J(φ¹¹) > 0, and J(φ¹¹) < J(φ¹⁷). The theorem excitation_cost_ordering proves this strict ordering. The cost monotonicity is proved in Jcost_strict_mono_pos and excitationCost_strictMono.

What the declaration does not claim is equally important. It does not prove that excitations actually couple to subcells in CW dimension order. That is the filtration principle, a structural premise about the coupling mechanism, not a consequence of the framework's core logic. The theorem assumes the coupling and derives the arithmetic. The physical bridge from recognition to linking, which would justify the coupling, remains open.

/-- The first excitation increment equals the passive edge count. -/
theorem first_increment_is_passive_edges :
    cwCumulativeTorsion D .second - cwCumulativeTorsion D .first =
      (passive_field_edges D : ℤ) := by
  simp [cwCumulativeTorsion, passiveCoupling]

/-- CW-dimensional ordering: edges are strictly lower-dimensional than faces. -/
theorem edge_dim_lt_face_dim :
    CubeCell.cwDim (.edge : CubeCell D) < CubeCell.cwDim (.face : CubeCell D) := by
  decide

/-- Among all subcell types with nonzero passive coupling, edges have the
    smallest CW dimension. The variational principle (selecting cheapest
    excitation) therefore selects edge modes first. -/
theorem edge_is_minimal_nontrivial_excitation :
    ∀ (cell : CubeCell D),
    0 < passiveCoupling D cell →
    CubeCell.cwDim (.edge : CubeCell D) ≤ CubeCell.cwDim cell := by
  intro cell hpos
  cases cell with
  | vertex => simp [passiveCoupling] at hpos
  | edge => exact le_refl _
  | face => exact Nat.le_of_lt edge_dim_lt_face_dim

/-- The three generation torsion values have strictly ordered J-costs. -/
theorem excitation_cost_ordering :
    excitationCost 0 = 0 ∧
    0 < excitationCost 11 ∧
    excitationCost 11 < excitationCost 17 :=
  ⟨excitationCost_ground,
   excitationCost_pos_of_ne_zero 11 (by omega),
   excitationCost_strictMono (by omega) (by omega)⟩

What this page does not claim

The theorem does not prove that excitations couple to subcells in CW dimension order; that is the filtration principle, a structural premise. The theorem does not derive the number 11 from the framework's core logic alone; it assumes the coupling and derives the arithmetic. The theorem does not connect the torsion schedule to any measured particle mass.

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

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