Foundation Pre Logical Cost

The cost floor

In Recognition Science, recognition (the act of classifying an event into one of a fixed set of categories) carries a cost that reality is forced to pay. The foundational framework, PreLogicalCost, studies the simplest possible version of that cost before any logical structure appears. The setup is stark: a configuration is a single real number between 0 and 1, and the cost of being at that number is the product of the number and its complement, written as cost(s) = s × (1 − s). This is the pre-logical cost, the cost paid by a state before it has any meaning as true or false.

The classical shape of this cost is a parabola that opens downward. It is zero at the two ends of the interval, at 0 and at 1, and it reaches its maximum value of 1/4 exactly in the middle, at 1/2. The framework proves a theorem about this shape: a configuration is stable, meaning its cost is zero, if and only if its value is either 0 or 1. This is the theorem stable_iff_boundary, and it is the entire point of the framework. The cost function does not merely prefer the two boundary states; it forces them. Any state that is not exactly 0 or exactly 1 pays a positive cost, and the only way to pay nothing is to sit at one of the two ends.

The framework then shows what those two stable states can do. It packages the stable states as arithmetic bits, numbers that are provably either 0 or 1. On these bits it defines three operations: conjunction as multiplication, disjunction as a + b − ab, and negation as 1 − a. These are exactly the arithmetic formulas for Boolean logic on 0 and 1. The theorem stable_forms_boolean_algebra verifies that these three operations behave as a Boolean-style algebraic fragment: conjunction multiplies, disjunction combines by a + b − ab, and negation subtracts from 1. The two stable states, 0 and 1, are not just arbitrary endpoints; they are the seeds of logical truth and falsity, and the cost function is what selects them.

In Recognition Science, the pre-logical cost is the floor beneath all later structure. The broader forcing chain derives more elaborate costs and constants from five plain conditions, but this framework shows the simplest case: a cost that is cheapest at exactly two points, and those two points are the ones that arithmetic logic needs. The framework establishes that the step from continuous configuration to discrete bit is not an assumption but a consequence of minimizing a cost. The reader can now see that the Boolean fragment does not have to be put in by hand; it falls out of a parabola.

/-- 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 framework does not derive the full cost function J(x) or any later constants like phi or hbar. The pre-logical cost is not claimed to be the unique cost function; it is a specific simple model before the forcing chain's five conditions apply. No claim is made that the Boolean fragment is a complete Boolean algebra; the theorem only verifies the three listed operations.

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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