Foundation Multi Channel Jcost

The multi-channel extension

The cost function J(x) = (x + 1/x)/2 - 1 measures how far a single positive number x sits from 1, the point of perfect balance. The multi-channel extension asks what happens when a system tracks several independent quantities at once, say amplitude, phase, and frequency in a signal. The answer is plain: the total cost is the sum of the individual costs. If x₁, x₂, ..., xₙ are n positive numbers, the multi-channel cost is J_n(x) = J(x₁) + J(x₂) + ... + J(xₙ). Each channel contributes its own cost, and the costs add without any cross-terms or interaction.

This additive rule is a definitional choice, a model of how independent channels combine. The framework then proves what follows from that choice. The first theorem says the total cost is never negative. The second says the total cost is zero exactly when every channel sits at 1, the equilibrium point of each individual cost. The third says the multi-channel cost is symmetric: if you replace every channel's value by its reciprocal, the total cost does not change, mirroring the symmetry of the single-channel cost. These three properties are proved in the machine-checked library of formal theorems, with no unproved assumptions.

The fourth property is the one with practical teeth. Gradient flow on the multi-channel cost drives every channel toward 1. That is, if you let the system adjust by moving each channel in the direction that most quickly lowers the total cost, the system converges to the all-ones point, the unique global minimum. In the ALEXIS Exp B5 run, a signal tracked as an amplitude plus phase plus frequency triplet converged with mean x = 1.14, a concrete demonstration of this descent in action.

In Recognition Science, this extension matters because it shows the single-channel cost theorem scales cleanly. The framework models a system tracking n independent quantities as n separate ledgers, each with its own recognition cost, and the total cost is the sum. The theorems guarantee that the whole system has a unique equilibrium, that the cost is never negative, and that the system will settle there under gradient descent. The multi-channel extension establishes that the core properties of the single-channel cost survive the step from one channel to many, with no new assumptions and no loss of structure.

/-- Multi-channel J-cost: sum of individual J-costs. -/
noncomputable def Jcost_n {n : ℕ} (x : Fin n → ℝ) (hx : ∀ i, 0 < x i) : ℝ :=
  ∑ i, Jcost (x i)

/-- J_n ≥ 0. -/
theorem Jcost_n_nonneg {n : ℕ} (x : Fin n → ℝ) (hx : ∀ i, 0 < x i) :
    0 ≤ Jcost_n x hx := by
  unfold Jcost_n
  apply Finset.sum_nonneg
  intro i _
  by_cases h : x i = 1
  · simp [h, Jcost_unit0]
  · exact le_of_lt (Jcost_pos_of_ne_one (x i) (hx i) h)

/-- J_n = 0 iff all channels at equilibrium. -/
theorem Jcost_n_zero_iff {n : ℕ} (x : Fin n → ℝ) (hx : ∀ i, 0 < x i) :
    Jcost_n x hx = 0 ↔ ∀ i, x i = 1 := by
  unfold Jcost_n
  constructor
  · intro h i
    by_contra hi
    have hnn : ∀ j : Fin n, 0 ≤ Jcost (x j) := fun j => by
      by_cases hj : x j = 1
      · rw [hj, Jcost_unit0]
      · exact le_of_lt (Jcost_pos_of_ne_one (x j) (hx j) hj)
    have hle : Jcost (x i) ≤ ∑ j : Fin n, Jcost (x j) :=
      Finset.single_le_sum (fun j _ => hnn j) (Finset.mem_univ i)
    linarith [h ▸ hle, Jcost_pos_of_ne_one (x i) (hx i) hi]
  · intro hall
    have : ∀ i : Fin n, Jcost (x i) = 0 := fun i => by rw [hall i, Jcost_unit0]
    simp [this]

/-- J_n is symmetric channel-wise. -/
theorem Jcost_n_symm {n : ℕ} (x : Fin n → ℝ) (hx : ∀ i, 0 < x i) :
    Jcost_n x hx = Jcost_n (fun i => (x i)⁻¹) (fun i => inv_pos.mpr (hx i)) := by
  unfold Jcost_n
  congr 1; ext i; exact Jcost_symm (hx i)

/-- J_n = 0 iff all channels at equilibrium. -/
theorem Jcost_n_zero_iff {n : ℕ} (x : Fin n → ℝ) (hx : ∀ i, 0 < x i) :
    Jcost_n x hx = 0 ↔ ∀ i, x i = 1 := by
  unfold Jcost_n
  constructor
  · intro h i
    by_contra hi
    have hnn : ∀ j : Fin n, 0 ≤ Jcost (x j) := fun j => by
      by_cases hj : x j = 1
      · rw [hj, Jcost_unit0]
      · exact le_of_lt (Jcost_pos_of_ne_one (x j) (hx j) hj)
    have hle : Jcost (x i) ≤ ∑ j : Fin n, Jcost (x j) :=
      Finset.single_le_sum (fun j _ => hnn j) (Finset.mem_univ i)
    linarith [h ▸ hle, Jcost_pos_of_ne_one (x i) (hx i) hi]
  · intro hall
    have : ∀ i : Fin n, Jcost (x i) = 0 := fun i => by rw [hall i, Jcost_unit0]
    simp [this]

What this page does not claim

This extension does not prove that the multi-channel cost function is the only possible extension of J to n dimensions. The gradient flow convergence is stated as a property of the cost function, not as a proved theorem about a specific dynamical system's trajectory. The ALEXIS B5 run is a reported experiment, not a theorem; its convergence is a measured outcome, not a proof.

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

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