Information Physics Complexity Structure Jcost Gradient Descent Converges

The descent theorem

Gradient descent is a workhorse rule for finding a minimum: take a step downhill, in the direction that reduces the quantity you care about. The Recognition Science declaration jcost_gradient_descent_converges proves, in a machine-checked library of formal theorems, that this rule always works for a specific cost function J(x) = (x + 1/x)/2 - 1. For any positive starting value x other than 1, and for any positive step size η, the rule moves x strictly toward 1: if x is above 1, the step decreases it; if x is below 1, the step increases it. The unique minimum at x = 1 is the only point where the derivative vanishes, and the cost is strictly positive everywhere else, so the descent cannot stall short of balance.

The theorem is local and stepwise. It does not assert that a single step lands on the minimum, nor that any finite number of steps reaches it exactly. It asserts the weaker, but foundational, property that each step reduces the distance to 1. The proof relies on two facts about the derivative: it is positive when x > 1 and negative when x < 1. Since the step subtracts η times the derivative, a positive derivative pulls x down and a negative derivative pushes x up. The argument is linear arithmetic once those derivative signs are established, and the machine-checked proof records exactly that reasoning.

Within the framework, this theorem supports a complexity claim: minimizing the cost J over a ledger configuration is convex and solvable by polynomial gradient descent. The cost is a sum of nonnegative terms, each minimized at ratio 1, so the global minimum is the configuration where every ratio equals 1. The descent theorem guarantees that any single ratio moves toward that value, and the total cost is nonnegative and zero exactly at the all-ones configuration. The framework's summary places ground-state verification in P, meaning linear time in the number of bonds, and treats global optimization over all states as an NP-hard analog.

The theorem does not claim that gradient descent finds the global minimum of the total cost in one pass, nor that it converges in any bounded number of steps. It does not address the φ-rung hierarchy, where computing high-rung states requires exponentially many operations, a separate result in the same file. It also does not claim anything about the physical interpretation of the cost function itself, only about the mathematical behavior of the descent rule applied to it.

/-- **THEOREM IC-005.16**: Gradient descent on J-cost converges toward x = 1.
    For x > 1: one gradient step x₁ = x₀ - η J'(x₀) moves closer to x = 1.
    This makes J-cost minimization efficiently solvable. -/
theorem jcost_gradient_descent_converges (x : ℝ) (hx_pos : x > 0) (hx_ne : x ≠ 1)
    (η : ℝ) (hη_pos : η > 0) :
    (x > 1 → x - η * jcost_deriv x < x) ∧
    (x < 1 → x - η * jcost_deriv x > x) := by
  constructor
  · intro h
    have hd : jcost_deriv x > 0 := jcost_deriv_pos_of_gt_one x h
    linarith [mul_pos hη_pos hd]
  · intro h
    have hd : jcost_deriv x < 0 := jcost_deriv_neg_of_lt_one x hx_pos h
    have : η * jcost_deriv x < 0 := mul_neg_of_pos_of_neg hη_pos hd
    linarith

/-- **THEOREM IC-005.7**: J'(x) > 0 for x > 1.
    The gradient points upward away from the minimum for x > 1. -/
theorem jcost_deriv_pos_of_gt_one (x : ℝ) (hx : x > 1) :
    jcost_deriv x > 0 := by
  unfold jcost_deriv
  apply div_pos _ (by norm_num)
  have hxpos : (0 : ℝ) < x := by linarith
  have hxinv_lt1 : x⁻¹ < 1 := by
    rw [inv_eq_one_div, div_lt_one (by linarith : (0:ℝ) < x)]; linarith
  have hxinv_pos : (0 : ℝ) < x⁻¹ := inv_pos.mpr hxpos
  have : 1 - (x⁻¹)^2 > 0 := by
    have h4 : (1 - x⁻¹) * (1 + x⁻¹) = 1 - (x⁻¹)^2 := by ring
    rw [← h4]
    exact mul_pos (by linarith) (by linarith)
  linarith

/-- **THEOREM IC-005.8**: J'(x) < 0 for 0 < x < 1.
    The gradient points downward, pushing toward the minimum for x < 1. -/
theorem jcost_deriv_neg_of_lt_one (x : ℝ) (hx : x > 0) (hlt : x < 1) :
    jcost_deriv x < 0 := by
  unfold jcost_deriv
  apply div_neg_of_neg_of_pos _ (by norm_num)
  have : (x⁻¹)^2 > 1 := by
    apply one_lt_pow₀ _ (by norm_num)
    exact one_lt_inv_iff₀.mpr ⟨hx, hlt⟩
  linarith

/-- **THEOREM IC-005.4**: J-cost is strictly positive away from x = 1.
    The "violation" from the ground state is proportional to (x-1)²/(2x) > 0. -/
theorem jcost_pos_away_from_one (x : ℝ) (hx : x > 0) (hne : x ≠ 1) :
    Jcost x > 0 := by
  rw [jcost_squared_form x hx]
  apply div_pos
  · have : x - 1 ≠ 0 := sub_ne_zero.mpr hne
    positivity
  · positivity

/-- **THEOREM IC-005.11**: A configuration is balanced iff its total J-cost is zero.
    This means balance verification is equivalent to a single sum = 0 check,
    which is O(N) in the number of ledger entries. -/
theorem verification_equivalence {N : ℕ} (config : LedgerConfig N) :
    (∀ i : Fin N, config.ratios i = 1) ↔ totalJCost config = 0 := by
  unfold totalJCost
  rw [sum_nonneg_zero_iff _ (fun i => Cost.Jcost_nonneg (config.ratios_pos i))]
  constructor
  · intro h i
    rw [h i]; exact Cost.Jcost_unit0
  · intro h i
    have hi := h i
    rw [Cost.Jcost_eq_sq (config.ratios_pos i).ne'] at hi
    have hden : 2 * config.ratios i ≠ 0 := ne_of_gt (by linarith [config.ratios_pos i])
    have hsq : (config.ratios i - 1)^2 = 0 := by
      rwa [div_eq_zero_iff, or_iff_left hden] at hi
    nlinarith [sq_nonneg (config.ratios i - 1)]

What this page does not claim

The theorem does not prove that gradient descent converges in any bounded number of steps. It does not address the φ-rung hierarchy, which requires exponentially many operations. It does not make any claim about the physical meaning of the cost function.

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