Cosmology Refine Trigger Cost Singleton

The cost of one event

In the Recognition Science framework, the world is modeled as a ledger: a discrete record of events, each event carrying a ratio that compares two quantities. The cost of recognizing an event is the price the ledger pays to record that comparison. The declaration cost_singleton is a theorem in the framework's machine-checked library of formal theorems. It states that the cost of recognizing a single event, when that event stands alone, is exactly the event's ratio. If an event has ratio r, the cost of recognizing it by itself is r.

This statement is the base case for a larger result about refinement, the process of zooming in from a coarse view of the ledger to a finer one. The framework proves that a block of events should be refined exactly when its recognition demand is positive, meaning a distinction is forced. The threshold for that decision is structurally zero, not a tunable knob. The cost_singleton theorem supports this by showing that a single event's cost is its ratio, and that a ratio not equal to one carries strictly positive cost. A forced distinction, even an arbitrarily small one, always costs something.

What the theorem does not claim is broader. It does not say that the cost of a collection of events is the sum of their individual costs; that composition law is a separate, forced result. It does not claim that any particular ratio is preferred or that the framework derives the fine-structure constant. It establishes only the unit case: one event, one cost, equal to its ratio.

theorem cost_singleton (wr : ℝ → ℝ) (e : Event) : cost wr {e} = wr e.ratio := by
  unfold cost; simp

/-- A genuine distinction (ratio not one) has strictly positive cost. -/
theorem jcost_pos {x : ℝ} (hx : 0 < x) (hne : x ≠ 1) : 0 < Jcost x := by
  have hx0 : x ≠ 0 := hx.ne'
  have key : Jcost x = (x - 1) ^ 2 / (2 * x) := by
    unfold Jcost; field_simp; ring
  rw [key]
  have hsq : 0 < (x - 1) ^ 2 := by
    have hne' : x - 1 ≠ 0 := sub_ne_zero.mpr hne
    positivity
  have hden : 0 < 2 * x := by linarith
  exact div_pos hsq hden

/-- **Forced postings have arbitrarily small positive cost.** For every `ε > 0` there is
a ratio above one whose recognition cost is positive but below `ε`. This is why no
positive threshold is safe: a forced distinction can sit just under any `ε`. -/
theorem jcost_arbitrarily_small_positive (ε : ℝ) (hε : 0 < ε) :
    ∃ x : ℝ, 1 < x ∧ 0 < Jcost x ∧ Jcost x < ε := by
  have hδpos : 0 < min 1 ε := lt_min (by norm_num) hε
  have hδ1 : min 1 ε ≤ 1 := min_le_left _ _
  have hδε : min 1 ε ≤ ε := min_le_right _ _
  have hx1 : (1 : ℝ) < 1 + min 1 ε := by linarith
  have hpos : (0 : ℝ) < 1 + min 1 ε := by linarith
  have hne0 : (1 + min 1 ε) ≠ 0 := hpos.ne'
  refine ⟨1 + min 1 ε, hx1, jcost_pos hpos hx1.ne', ?_⟩
  have key : Jcost (1 + min 1 ε) = (min 1 ε) ^ 2 / (2 * (1 + min 1 ε)) := by
    unfold Jcost; field_simp; ring
  rw [key, div_lt_iff₀ (by nlinarith : (0 : ℝ) < 2 * (1 + min 1 ε))]
  nlinarith [hδpos, hδ1, hδε, hε,
    mul_nonneg hδpos.le (by linarith : (0 : ℝ) ≤ 1 - min 1 ε), mul_pos hε hδpos]

What this page does not claim

The cost of a collection of events is the sum of individual costs. The framework derives the fine-structure constant. Any particular ratio is preferred by the framework.

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/Cosmology/RefineTrigger.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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