Information Simulation Hypothesis Structure Rs Exists Iff Zero Cost

The zero-cost condition

The declaration rs_exists_iff_zero_cost is a machine-checked theorem in the framework's library. It states that for any positive real number x, the recognition cost J(x) equals zero if and only if x equals 1. The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a recognition event, a discrete record of comparison between two quantities. The theorem proves that this price vanishes only at the identity ratio, where the two quantities are equal.

In plain terms, the result says that a system pays nothing only when it recognizes something as exactly itself. Any deviation from that identity, however small, carries a positive cost. The proof in the library runs by rewriting the cost formula, squaring the difference (x - 1), and invoking the fact that a square is nonnegative. The theorem also uses the unit-cost lemma, which states that J(1) = 0 directly. This is a formal derivation, not a physical measurement.

The theorem connects to the framework's treatment of the simulation hypothesis. In Recognition Science, the ledger is reality itself: there is no separate substrate that could run a simulation. A simulated universe, if it produces the same events as the real one, is itself a Recognition Science universe by the theorem simulated_rs_is_rs. The question "is the ledger simulated?" then reduces to "is the ledger the ledger?", which is trivially true. The zero-cost condition sharpens this: a universe exists exactly when its recognition cost is zero, and that happens exactly at self-identity.

What the declaration does not claim is broader than what it proves. It does not assert that the physical universe actually has zero cost, nor that any particular simulation exists or does not exist. It does not say that the simulation hypothesis is false; it says the hypothesis is semantically empty within the framework. The theorem also does not prove that all recognition costs are nonnegative, though a separate lemma in the library, ledger_self_grounding, does establish that. The zero-cost condition is a structural fact about the cost function, not an empirical claim about the cosmos.

/-- **THEOREM IC-004.6**: The J-cost framework determines what "exists":
    x exists (RSExists) iff J(x) = 0 iff x = 1. -/
theorem rs_exists_iff_zero_cost (x : ℝ) (hx : x > 0) :
    Cost.Jcost x = 0 ↔ x = 1 := by
  constructor
  · intro h
    rw [Cost.Jcost_eq_sq hx.ne'] at h
    have hden : (2 * x) > 0 := by linarith
    have hne : (2 * x) ≠ 0 := ne_of_gt hden
    have hsq : (x - 1)^2 = 0 := by
      rwa [div_eq_zero_iff, or_iff_left hne] at h
    nlinarith [sq_nonneg (x - 1)]
  · intro h; rw [h]; exact Cost.Jcost_unit0

/-- **THEOREM IC-004.6**: The J-cost framework determines what "exists":
    x exists (RSExists) iff J(x) = 0 iff x = 1. -/
theorem rs_exists_iff_zero_cost (x : ℝ) (hx : x > 0) :
    Cost.Jcost x = 0 ↔ x = 1 := by
  constructor
  · intro h
    rw [Cost.Jcost_eq_sq hx.ne'] at h
    have hden : (2 * x) > 0 := by linarith
    have hne : (2 * x) ≠ 0 := ne_of_gt hden
    have hsq : (x - 1)^2 = 0 := by
      rwa [div_eq_zero_iff, or_iff_left hne] at h
    nlinarith [sq_nonneg (x - 1)]
  · intro h; rw [h]; exact Cost.Jcost_unit0

/-- **THEOREM IC-004.2**: A simulated universe that perfectly reproduces
    all events of an RS universe IS an RS universe.
    There is no difference between "simulated RS" and "RS". -/
theorem simulated_rs_is_rs (u : RSUniverse) (s : SimulatedUniverse)
    (h : ∀ n, s.events n = u.events n) :
    ∃ u' : RSUniverse, ∀ n, u'.events n = s.events n :=
  ⟨⟨s.events, s.events_pos⟩, fun n => rfl⟩

What this page does not claim

The theorem does not assert that the physical universe actually has zero recognition cost. The theorem does not prove that the simulation hypothesis is false, only that it is semantically empty within the framework. The theorem does not establish that all recognition costs are nonnegative; that is a separate lemma.

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/Information/SimulationHypothesisStructure.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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