Foundation Recognition Forcing Nontrivial Recognition Positive Cost

The cost of recognition

A ledger, a discrete record of events, is the central image of Recognition Science. In that framework, every event is a recognition: an act of comparing two things and registering whether they match. The framework assigns each such event a cost, a real number that measures how much work the comparison takes. The declaration nontrivial_recognition_positive_cost states a simple fact about that cost: if the two things being compared are not identical, the cost is strictly positive. If they are identical, the cost is zero.

This is a theorem in the framework's machine-checked library of formal theorems. It derives from the framework's cost function J(x) = (x + 1/x)/2 - 1, where x is the ratio between the two compared quantities. That function is always greater than or equal to zero, and it equals zero only when x = 1, meaning the two quantities are equal. The theorem is the formal statement of that fact in the language of recognition events: a ratio different from 1 forces a positive cost.

The theorem does not say anything about what the cost is, only that it is positive. It does not say how large the cost is, or how it depends on the specific ratio, beyond the fact that it is greater than zero. It does not claim that the cost function is the only possible one, or that recognition is the only way to structure events. It is a narrow, precise statement about a specific function and the events built from it.

In Recognition Science, this theorem is a small but load-bearing piece of a larger structure. It is part of a chain of results that moves from the cost function to the golden ratio, to an eight-tick cycle, to the number 2^3, and to three spatial dimensions. The positive-cost theorem is the step that says: once you have the cost function, non-trivial recognition is not free. That fact is what makes the framework's later claims about structure and dimension possible.

For a reader encountering the framework for the first time, the practical upshot is this: in this account, difference is not free. Any act of recognizing something as other than yourself carries a cost, and only the act of recognizing yourself costs nothing. This is the foundation on which the framework builds its more ambitious claims about the structure of reality.

/-- Non-trivial recognition has positive cost.
    Uses the fact that J(x) = (x + 1/x)/2 - 1 ≥ 0, with = 0 iff x = 1. -/
theorem nontrivial_recognition_positive_cost (e : LedgerForcing.RecognitionEvent)
    (h : e.ratio ≠ 1) : recognition_cost e > 0 := by
  simp only [recognition_cost, LedgerForcing.J]
  have hpos := e.ratio_pos
  have h0 : e.ratio ≠ 0 := hpos.ne'
  -- (x - 1)² > 0 when x ≠ 1
  have hne : (e.ratio - 1)^2 > 0 := by
    have hsq : (e.ratio - 1)^2 ≥ 0 := sq_nonneg _
    have hne2 : (e.ratio - 1)^2 ≠ 0 := by
      intro heq
      have heq2 : e.ratio - 1 = 0 := sq_eq_zero_iff.mp heq
      have : e.ratio = 1 := by linarith
      exact h this
    exact lt_of_le_of_ne hsq (Ne.symm hne2)
  -- Expand: x² - 2x + 1 > 0
  -- So: x² + 1 > 2x
  -- So: (x² + 1)/x > 2 (since x > 0)
  -- So: x + 1/x > 2
  have h2 : e.ratio^2 + 1 > 2*e.ratio := by nlinarith [sq_nonneg (e.ratio - 1)]
  have h3 : e.ratio + e.ratio⁻¹ > 2 := by
    have heq : e.ratio + e.ratio⁻¹ = (e.ratio^2 + 1) / e.ratio := by field_simp
    rw [heq, gt_iff_lt, lt_div_iff₀ hpos]
    linarith
  linarith

/-- Recognition events with ratio = 1 are cost-free. -/
theorem self_recognition_zero_cost (e : LedgerForcing.RecognitionEvent) :
    e.ratio = 1 → recognition_cost e = 0 := by
  intro h
  simp only [recognition_cost, h, LedgerForcing.J]
  norm_num

/-- Non-trivial recognition has positive cost.
    Uses the fact that J(x) = (x + 1/x)/2 - 1 ≥ 0, with = 0 iff x = 1. -/
theorem nontrivial_recognition_positive_cost (e : LedgerForcing.RecognitionEvent)
    (h : e.ratio ≠ 1) : recognition_cost e > 0 := by
  simp only [recognition_cost, LedgerForcing.J]
  have hpos := e.ratio_pos
  have h0 : e.ratio ≠ 0 := hpos.ne'
  -- (x - 1)² > 0 when x ≠ 1
  have hne : (e.ratio - 1)^2 > 0 := by
    have hsq : (e.ratio - 1)^2 ≥ 0 := sq_nonneg _
    have hne2 : (e.ratio - 1)^2 ≠ 0 := by
      intro heq
      have heq2 : e.ratio - 1 = 0 := sq_eq_zero_iff.mp heq
      have : e.ratio = 1 := by linarith
      exact h this
    exact lt_of_le_of_ne hsq (Ne.symm hne2)
  -- Expand: x² - 2x + 1 > 0
  -- So: x² + 1 > 2x
  -- So: (x² + 1)/x > 2 (since x > 0)
  -- So: x + 1/x > 2
  have h2 : e.ratio^2 + 1 > 2*e.ratio := by nlinarith [sq_nonneg (e.ratio - 1)]
  have h3 : e.ratio + e.ratio⁻¹ > 2 := by
    have heq : e.ratio + e.ratio⁻¹ = (e.ratio^2 + 1) / e.ratio := by field_simp
    rw [heq, gt_iff_lt, lt_div_iff₀ hpos]
    linarith
  linarith

What this page does not claim

The theorem does not specify the magnitude of the cost, only that it is positive. The theorem does not claim the cost function is the only possible one. The theorem does not claim that recognition is the only way to structure events.

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

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