Masses Mass Genesis T10 Channel Cost Irreducibility Channel Cost Rational Of Irr

The irreducible channel cost

The framework's cost, a measure of the price of a recognition event, is built on a specific function. The theorem in question concerns a channel, a distinct pathway for recognition, and what happens when the number of channels is irreducible, meaning it cannot be split into two smaller positive counts. The declaration channel_cost_rational_of_irreducible_pricing proves a conjunction: for any pricing with an irreducible rung-per-degree-of-freedom, the exponent applied to the channel distinction model is exactly 2, and the cost function evaluated at phi squared is a rational number, specifically one half.

This is not a statement about the physical world directly. It is a theorem about the framework's own definitions, proved in the machine-checked library of formal theorems. The first part, the exponent being 2, follows from the definition of irreducible pricing combined with the fact that the only irreducible positive integer is 1. The second part, the cost being rational, relies on a separate theorem that the cost of phi squared is one half. The theorem does not say that any physical process has an irreducible channel count; it says that if a pricing has that property, then the cost is forced to be rational.

The result is a piece of the framework's internal consistency. It shows that the cost function, which is central to the framework, takes a rational value at a specific point that is tied to the golden ratio. This is a derived fact, not an assumption. It is a consequence of the cost function's defining properties, which are themselves proved from five plain conditions. The theorem is a small but necessary step in the framework's larger chain of derivations, which aims to show that the structure of reality, including particle masses, follows from the single principle of forced recognition cost.

The declaration does not claim that the cost function is always rational, nor that the number of channels in any physical situation is always irreducible. It does not claim that the exponent of 2 applies to all channel models, only to those with an irreducible pricing. It also does not claim that the rational value one half has any direct physical meaning; it is a mathematical consequence of the definitions. The theorem is a precise, limited statement about the framework's own structure, and it is proved with complete rigor.

/-- **THEOREM (rationality is a consequence, not a premise).** Under the
irreducible per-axis pricing and the axis model, the channel exponent is `2`,
and the channel cost `J(φ²) = 1/2` is rational by the banked
`RungBaseBoundary.jcost_phi_sq`. The rationality clause of the old conditional
closure is discharged by the axis-level pricing. -/
theorem channel_cost_rational_of_irreducible_pricing (p : DofPricing)
    (hirr : PosIrreducible p.rungPerDof) (M : ChannelDistinctionModel) :
    p.exponentOf M.distinctionsPerChannel = 2
      ∧ ∃ q : ℚ, Cost.Jcost (phi ^ (2 : ℕ)) = (q : ℝ) := by
  have hu := dofPricing_unit_eq_one_of_irreducible p hirr
  refine ⟨?_, ⟨1 / 2, ?_⟩⟩
  · simp [DofPricing.exponentOf, hu, M.distinctionsPerChannel_eq_two]
  · rw [show ((1 / 2 : ℚ) : ℝ) = 1 / 2 by norm_num]
    exact jcost_phi_sq

/-- **THEOREM (rationality is a consequence, not a premise).** Under the
irreducible per-axis pricing and the axis model, the channel exponent is `2`,
and the channel cost `J(φ²) = 1/2` is rational by the banked
`RungBaseBoundary.jcost_phi_sq`. The rationality clause of the old conditional
closure is discharged by the axis-level pricing. -/
theorem channel_cost_rational_of_irreducible_pricing (p : DofPricing)
    (hirr : PosIrreducible p.rungPerDof) (M : ChannelDistinctionModel) :
    p.exponentOf M.distinctionsPerChannel = 2
      ∧ ∃ q : ℚ, Cost.Jcost (phi ^ (2 : ℕ)) = (q : ℝ) := by
  have hu := dofPricing_unit_eq_one_of_irreducible p hirr
  refine ⟨?_, ⟨1 / 2, ?_⟩⟩
  · simp [DofPricing.exponentOf, hu, M.distinctionsPerChannel_eq_two]
  · rw [show ((1 / 2 : ℚ) : ℝ) = 1 / 2 by norm_num]
    exact jcost_phi_sq

/-- **THEOREM (the cone's unique irreducible is `1`).** Every `k ≥ 2` splits as
`1 + (k − 1)`; `1` splits as nothing. -/
theorem posIrreducible_iff_eq_one (k : ℤ) : PosIrreducible k ↔ k = 1 := by
  constructor
  · rintro ⟨hk1, hnd⟩
    by_contra hne
    have hk2 : 2 ≤ k := by omega
    exact hnd ⟨1, k - 1, le_refl 1, by omega, by omega⟩
  · intro h
    subst h
    refine ⟨le_refl 1, ?_⟩
    rintro ⟨a, b, ha, hb, hab⟩
    omega

What this page does not claim

The theorem does not claim that the cost function is rational for all inputs. It does not claim that any physical process has an irreducible channel count. It does not assign a direct physical meaning to the rational value one half.

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/T10ChannelCostIrreducibility.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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