Masses Excitation Ordering Excitation Cost Strict Mono

The strict cost ordering

In the Recognition Science framework, particle generations are associated with a numerical quantity called torsion, a discrete record of structural complexity. The theorem excitationCost_strictMono establishes that the cost of exciting a generation, measured by the framework's cost function J, is strictly increasing as the torsion number increases. Specifically, for any two non-negative torsion values τ₁ and τ₂, if τ₁ is less than τ₂, then the excitation cost of τ₁ is strictly less than the excitation cost of τ₂. This is a formal, machine-checked theorem in the framework's library.

This strict monotonicity is not an assumption; it is derived. The cost function J is defined as J(x) = (x + 1/x)/2 - 1, and the excitation cost of a torsion τ is J(φ^τ), where φ is the golden ratio. The theorem builds on the fact that φ is greater than 1, so φ raised to a larger power is strictly larger. Since J is strictly increasing for arguments greater than 1, the ordering follows. The proof is a direct consequence of these definitions and the monotonicity of the power function and J.

The theorem is used to show that in the framework's model of the three-dimensional cube, the first excited generation (with torsion 11) has a strictly lower cost than the second excited generation (with torsion 17). This provides a structural reason for the ordering of the generations, which is otherwise an unexplained feature of the model. The ordering is not a numerical accident; it is a consequence of the geometric fact that edges (dimension 1) come before faces (dimension 2) in the cube's cell structure.

What the theorem does not claim is that this cost ordering is the physical mechanism by which generations are excited. The theorem only establishes a mathematical relationship between the cost function and the torsion numbers. The physical premise that excitations couple to the cube's subcells in order of their dimension is a separate assumption, called the filtration principle. The theorem does not prove this principle; it takes it as a given. The theorem also does not claim that the specific torsion numbers 11 and 17 are the only possible values; it only proves the ordering for the values used in the model.

/-- Excitation cost is strictly monotone for non-negative torsion:
    0 ≤ τ₁ < τ₂ implies J(φ^τ₁) < J(φ^τ₂). -/
theorem excitationCost_strictMono {τ₁ τ₂ : ℤ} (h1 : 0 ≤ τ₁) (h2 : τ₁ < τ₂) :
    excitationCost τ₁ < excitationCost τ₂ := by
  apply Jcost_strict_mono_pos (zpow_pos phi_pos τ₁) (zpow_pos phi_pos τ₂)
  · rcases eq_or_lt_of_le h1 with rfl | hpos
    · simp
    · exact le_of_lt (one_lt_zpow₀ one_lt_phi hpos)
  · exact zpow_lt_zpow_right₀ one_lt_phi h2

/-- J-cost is strictly increasing on [1, ∞).

    Proof: write `J(x) = (x + 1/x)/2 - 1` and show `x + 1/x` is strictly
    increasing for `x ≥ 1` via the identity
      `(y + 1/y) - (x + 1/x) = (y - x)(xy - 1)/(xy)`,
    which is positive when `1 ≤ x < y`. -/
theorem Jcost_strict_mono_pos {x y : ℝ} (hx : 0 < x) (hy : 0 < y)
    (hx1 : 1 ≤ x) (hxy : x < y) :
    Jcost x < Jcost y := by
  have hx0 : x ≠ 0 := ne_of_gt hx
  have hy0 : y ≠ 0 := ne_of_gt hy
  simp only [Jcost]
  suffices h : x + x⁻¹ < y + y⁻¹ by linarith
  have hxy_pos : 0 < x * y := mul_pos hx hy
  have hyx : 0 < y - x := sub_pos.mpr hxy
  have hxy1 : 0 < x * y - 1 := by nlinarith
  have key : y + y⁻¹ - (x + x⁻¹) = (y - x) * (x * y - 1) / (x * y) := by
    field_simp
    ring
  linarith [div_pos (mul_pos hyx hxy1) hxy_pos]

/-- 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 the filtration principle that excitations couple to subcells in order of dimension. The theorem does not claim that the torsion values 11 and 17 are the only possible values for the generations. The theorem does not establish that the cost ordering is the physical cause of the generation ordering.

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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