Foundation Multi Channel Jcost Jcost N Zero Iff

The zero-cost condition

In the Recognition Science framework, a ledger (a discrete record of events) assigns a cost to each possible state of a system. The single-channel cost J(x) = (x + 1/x)/2 - 1 measures how far a single positive number x sits from its equilibrium value 1. The multi-channel extension J_n sums these individual costs across n independent channels: J_n(x) = Σᵢ J(xᵢ) for a vector x of n positive entries. This additive form treats each channel as an independent contributor to the total cost, with no interaction terms.

The theorem Jcost_n_zero_iff establishes the unique global minimum of this multi-channel cost. It proves, in the framework's machine-checked library of formal theorems, that J_n(x) = 0 if and only if every channel xᵢ equals 1. In plain language: the total cost vanishes precisely when every independent part sits at its own equilibrium. No other configuration achieves zero cost. This is the multi-channel analogue of the single-channel fact that J(x) = 0 exactly when x = 1, and it extends the uniqueness property from one dimension to any finite number of dimensions.

The theorem also carries two supporting properties proved in the same library. First, J_n is always non-negative: no configuration produces a negative cost, so zero is a true floor. Second, J_n is symmetric under componentwise inversion: swapping each xᵢ for its reciprocal 1/xᵢ leaves the total cost unchanged. This symmetry mirrors the single-channel identity J(x) = J(1/x) and means the cost treats a channel and its reciprocal as equally expensive deviations from equilibrium.

What the theorem does not claim is equally important. It does not say that the multi-channel system reaches this zero-cost state dynamically; the theorem is a statement about the cost function's values, not about how a system evolves over time. It does not assign physical meaning to the channels themselves, which could represent amplitudes, phases, frequencies, or any other positive quantities in a given application. And it does not extend to channels that are not positive: the theorem's hypothesis requires every xᵢ to be strictly greater than zero, so the result says nothing about configurations with zero or negative entries.

The practical consequence is a clean diagnostic. If a multi-channel system in the framework reports a total cost of zero, every channel must be at its equilibrium value 1. Conversely, if any channel deviates from 1, the total cost is strictly positive. This gives a precise, checkable condition for when a multi-part system sits at its unique rest point, and it is the foundation for the framework's descent result, which shows that gradient flow on J_n drives any positive configuration toward the all-ones vector.

/-- 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 ≥ 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 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)

What this page does not claim

The theorem does not describe how a multi-channel system evolves toward the zero-cost state over time. The theorem does not assign physical meaning to the channels themselves. The theorem does not apply to channels with zero or negative entries.

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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