Foundation Determinism Constrained Problem

The constrained problem

A constrained optimization problem asks for the best choice among a restricted set of options. The constraint set is the list of allowed options; the goal is to minimize some cost function over that set. The Recognition Science declaration ConstrainedProblem formalizes the minimal ingredients such a problem needs: a nonempty set of feasible points, all of which are positive real numbers. It is a definition, not a theorem: it states what counts as a well-formed problem, not that any particular problem has a solution.

The framework's cost function, written J(x), is strictly convex on positive reals: its second derivative is positive everywhere. A strictly convex function has at most one minimum on any convex set. The theorem unique_minimizer_principle proves exactly this: for any constrained minimization of J-cost over a convex set of positive reals, the minimizer is unique. This is the determinism core. It says that if a problem satisfies the conditions, then the next state is uniquely determined by the current state plus the constraint. There is no room for choice.

The declaration also introduces an observer with finite resolution, and a projection map that sends a real-valued state to one of finitely many observed outcomes. The theorem projection_lossy proves that distinct states can map to the same observation. This is the formal origin of apparent randomness: a deterministic process, seen through a coarse-grained lens, can look random even though the underlying dynamics are unique.

In Recognition Science, this structure resolves the determinism question. The universe is deterministic because each step has a unique minimizer. Apparent randomness is a feature of the observer's finite resolution, not of reality. The framework's ledger, a discrete record of events, updates by this unique minimization at each step.

What ConstrainedProblem does not claim is just as important. It does not prove that every optimization problem has a solution; it only guarantees uniqueness when a solution exists. It does not prove that the constraint set is convex; that is a separate hypothesis. It does not derive the Born rule or quantum mechanics; the docstring sketches that as a future step, but no theorem in the pack establishes it. The structure is a foundation, not a complete physical theory.

/-- **Theorem (Determinism core)**: For any constrained minimization of J-cost
    over a convex set of positive reals, the minimizer is unique.

    This means the next ledger state is uniquely determined by the current
    state plus the constraint. There is no "choice" — the dynamics are
    deterministic. -/
theorem unique_minimizer_principle (p : ConstrainedProblem)
    (h_convex : Convex ℝ p.feasible)
    (x_min : ℝ) (hx_feas : x_min ∈ p.feasible)
    (hx_min : ∀ y ∈ p.feasible, Jcost x_min ≤ Jcost y)
    (y_min : ℝ) (hy_feas : y_min ∈ p.feasible)
    (hy_min : ∀ z ∈ p.feasible, Jcost y_min ≤ Jcost z) :
    x_min = y_min := by
  by_contra h_ne
  have hx := hx_min y_min hy_feas
  have hy := hy_min x_min hx_feas
  have h_eq : Jcost x_min = Jcost y_min := le_antisymm hx hy
  -- Strict convexity: Jcost is strictly convex on (0,∞), so equal cost at two points forces equality.
  have hJ_pos : StrictConvexOn ℝ p.feasible Jcost :=
    StrictConvexOn.subset Jcost_strictConvexOn_pos
      (fun z hz => Set.mem_Ioi.mpr (p.positive z hz)) h_convex
  have h_mid_mem : (x_min + y_min) / 2 ∈ p.feasible := by
    have hsmul := h_convex hx_feas hy_feas (by norm_num : (0 : ℝ) ≤ 1/2) (by norm_num : (0 : ℝ) ≤ 1/2)
      (by norm_num : (1/2 : ℝ) + (1/2 : ℝ) = 1)
    simp only [smul_eq_mul] at hsmul
    convert hsmul using 1
    ring
  have h_strict : Jcost ((x_min + y_min) / 2) < (1/2) * Jcost x_min + (1/2) * Jcost y_min := by
    have heq : (1/2 : ℝ) • x_min + (1/2 : ℝ) • y_min = (x_min + y_min) / 2 := by
      simp only [smul_eq_mul]; ring
    rw [← heq]
    exact hJ_pos.2 hx_feas hy_feas h_ne (by norm_num : (0 : ℝ) < 1/2) (by norm_num : (0 : ℝ) < 1/2)
      (by norm_num : (1/2 : ℝ) + (1/2 : ℝ) = 1)
  -- RHS = Jcost y_min since Jcost x_min = Jcost y_min
  rw [h_eq, show (1/2 : ℝ) * Jcost y_min + (1/2) * Jcost y_min = Jcost y_min by ring] at h_strict
  have h_min := hx_min ((x_min + y_min) / 2) h_mid_mem
  rw [h_eq] at h_min
  linarith

/-- **Theorem**: Multiple distinct states map to the same observation.
    This is the origin of "apparent randomness." -/
theorem projection_lossy (obs : Observer) :
    ∃ x y : ℝ, x ≠ y ∧ project obs x = project obs y := by
  use 0, 1
  constructor
  · norm_num
  · simp [project]

/-- A constrained optimization problem on positive reals. -/
structure ConstrainedProblem where
  /-- The constraint set (e.g., log-sum = constant) -/
  feasible : Set ℝ
  /-- Feasible set is nonempty -/
  nonempty : feasible.Nonempty
  /-- All feasible points are positive -/
  positive : ∀ x ∈ feasible, 0 < x

What this page does not claim

The declaration does not prove that every constrained optimization problem has a solution. The declaration does not derive the Born rule or any quantum mechanical prediction. The uniqueness theorem requires the constraint set to be convex, which is a separate hypothesis.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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