Foundation Relational Qm3 From Jcost

Relational quantum mechanics from cost

Relational quantum mechanics, proposed by Carlo Rovelli in the 1990s, holds that the state of a quantum system is not an absolute property but a description relative to a particular observer. Two observers can assign different states to the same system, and there is no observer-independent state of affairs underneath. What matters is how observers relate to each other and to what they measure.

In Recognition Science, this relational picture takes a specific form. The framework models each observer's recognition system as defining a cost frame: a way of assigning a cost to the act of recognizing one quantity relative to another. The central object is a cost function, a number that measures how expensive it is to recognize a ratio between two quantities. For a ratio of two positive numbers m and e, the cost is written as J(m/e), where J(x) = (x + 1/x)/2 - 1. This function has three proved properties: it is zero when m equals e, it is never negative for positive inputs, and a canonical threshold phi - 3/2 is positive, where phi is the golden ratio, about 1.618.

These three facts are assembled into a machine-checked certificate, a formal object that packages the properties together. The certificate proves that the cost function vanishes at agreement, stays nonnegative, and has a positive threshold. What it does not do is define what m and e are in the context of relational quantum mechanics. The cost is defined as J(m/e) without reference to any specific physical quantities. The paragraph in the source file describing Rovelli's observer-dependent states is a research note, not a theorem. The established facts are general properties of the cost function, nothing specific to quantum mechanics.

In plain language, the result establishes a scaffold. It shows that a cost function with the right shape can encode agreement and disagreement between observers: agreement costs zero, disagreement costs something positive, and the golden ratio sets a natural boundary. The step from this scaffold to a full relational quantum mechanics would require defining m and e in the theory's own terms, for example as measurement outcomes or reference values. That step remains open. The value of the result is that it makes the scaffold explicit and machine-checked, so the missing piece is visible and named.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

These results prove nothing specific to relational quantum mechanics. The paragraph about Rovelli's observer-dependent states is a research note, not a theorem. No definition of m and e in quantum terms is provided here.

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/RelationalQM3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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