Foundation Pre Logical Cost Pre State

The Pre-Logical State

A pre-logical state is the simplest possible configuration in a recognition system: a single real number that is constrained to lie between 0 and 1, inclusive. Think of it as a dial that can be set anywhere from fully off (0) to fully on (1), with every fractional position in between allowed. The framework defines a cost for each setting, cost (the price the system pays for being in that configuration), using the formula cost = value × (1 − value). This is a parabola that opens downward: it is zero at the two endpoints, 0 and 1, and reaches its maximum of 0.25 at the middle value 0.5.

The central theorem about this setup is that a configuration is stable exactly when its cost is zero, which happens only at the two boundary values. In other words, the only settings that cost nothing are the fully off and fully on positions. This is proved in the machine-checked library of formal theorems, and it means that the framework's most basic notion of a stable state is a binary choice: a bit. These two stable states, 0 and 1, can then be combined using arithmetic operations that mirror Boolean logic. Multiplication of two bits acts as logical AND, the formula a + b − ab acts as logical OR, and 1 − a acts as logical NOT. The framework proves that these operations on the two stable states form a Boolean-style algebraic fragment, meaning they obey the same algebraic laws as classical logic.

In Recognition Science, this pre-logical layer is the foundation on which more complex structures are built. The framework models the transition from a continuous range of possible values to a discrete set of stable, cost-free states. This is the first step in a chain that the framework's library shows leads to more elaborate structures, but this particular declaration does not itself claim anything about those later stages. It establishes only the behavior of this single number and its two stable states, along with the Boolean operations that those states support.

What this declaration does not claim is just as important as what it proves. It does not claim that the cost function is unique or that it is forced by deeper principles; that is a separate theorem about a different cost function. It does not claim that these Boolean operations are the only way to combine stable states, nor that they constitute a complete logic. It does not claim that real-world systems actually have pre-logical states; it simply defines a mathematical object and proves its properties. The declaration is a precise, limited building block, and its value lies in being exactly that: a small, verified piece of a larger framework.

/-- Stability is equivalent to the two boundary values `0` and `1`. -/
theorem stable_iff_boundary (s : PreState) :
    IsStable s ↔ s.val = 0 ∨ s.val = 1 := by
  unfold IsStable preCost
  constructor
  · intro h
    have hfact : s.val * (1 - s.val) = 0 := h
    rcases mul_eq_zero.mp hfact with h0 | h1
    · exact Or.inl h0
    · right
      linarith
  · intro h
    rcases h with h0 | h1
    · simp [h0]
    · simp [h1]

/-- The stable arithmetic states form a Boolean-style algebraic fragment. -/
theorem stable_forms_boolean_algebra :
    (∀ a b : StableState, (band a b).bit = a.bit * b.bit) ∧
    (∀ a b : StableState, (bor a b).bit = a.bit + b.bit - a.bit * b.bit) ∧
    (∀ a : StableState, (bnot a).bit = 1 - a.bit) := by
  constructor
  · intro a b
    rfl
  constructor
  · intro a b
    rfl
  · intro a
    rfl

What this page does not claim

This declaration does not claim that the cost function preCost is unique or forced by any principle. It does not claim that the Boolean operations defined here are the only possible ones on stable states. It does not claim that pre-logical states exist in any physical system; they are a mathematical definition.

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/PreLogicalCost.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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