Masses Excitation Ordering Edge Is Minimal Nontrivial Excitation

The edge as first excitation

A cube has three kinds of building blocks: 8 vertices, 12 edges, and 6 faces. Order those blocks by dimension, vertices as 0, edges as 1, faces as 2. The framework's declaration edge_is_minimal_nontrivial_excitation states that among all blocks that can carry a nonzero excitation, the edge is the one with the smallest dimension. In plain terms, if any block is excited at all, an edge is excited, and no vertex is ever the first nontrivial excitation.

The proof is a short case analysis. The three possible blocks are vertices, edges, and faces. A vertex has zero coupling by definition, so it cannot be a nontrivial excitation. An edge has positive coupling, and its dimension is 1. A face also has positive coupling, but its dimension is 2, which is strictly larger than the edge's dimension. Since the edge's dimension is at most that of any positively coupled block, the edge is the minimal nontrivial excitation. The theorem is machine-checked in the framework's library of formal theorems.

This result is part of a larger story about why the framework's particle generations have the masses they do. The framework models three generations of fermions as excitations coupled to the cube's subcells in order of dimension: the ground state to vertices, the first excitation to edges, the second to faces. The edge-before-face ordering is what produces the torsion schedule {0, 11, 17}, and the declaration is the precise statement that the edge comes first. The cost function J, which measures the cost of recognition, is strictly increasing in the exponent, so the ordering is not just geometric but also energetic: the edge excitation is cheaper than the face excitation.

What the declaration does not claim is just as important. It does not say that excitations actually couple to the cube's subcells in this way. That coupling is a separate premise, called the filtration principle, and it remains a structural assumption about the mechanism, not a consequence of the framework's core axioms. The declaration only says that if a block is excited, the edge is the minimal one. It does not say why edges are excited at all, nor does it say that the edge excitation is the first in time. It is a statement about dimension and minimality, not about dynamics or causation.

/-- 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

@[simp] theorem passiveCoupling_vertex : passiveCoupling D .vertex = 0 := rfl

/-- 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

/-- 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 declaration does not prove that excitations actually couple to the cube's subcells in order of dimension. The declaration does not claim that the edge excitation is the first in time or that it causes the face excitation. The declaration does not derive the numerical values 11 and 17 from the cube's geometry alone.

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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