Foundation Ethics Restoration Just4 Cert

The restoration certificate

Recognition Science, a framework that models reality as a ledger, a discrete record of events with a forced cost, defines a cost function J(x) = (x + 1/x)/2 - 1. The declaration RestorationJust4Cert is a structure, a formal container, holding three proved properties of this cost when applied to a ratio of two positive quantities. It is a certificate, a packaged bundle of theorems, not a new statement about ethics.

The first property states that when the two quantities are equal, the cost is exactly zero. In the intended ethical reading, if harm repaired equals harm done, restoration is complete. The second property states that the cost is never negative for positive inputs, meaning partial repair still leaves a positive cost. The third property states that a certain threshold, phi - 3/2, is positive, where phi is the golden ratio.

These three facts are proved in the machine-checked library of formal theorems, and the certificate simply collects them into one object. The library also proves that such a certificate exists, that the three properties are consistent and can be instantiated. Nothing in the certificate refers to a specific ethical domain, such as restorative justice, criminal law, or personal apology.

In Recognition Science, the intended application was restorative justice, where restoration completes when the cost reaches zero, and an acceptable threshold sits below the golden-ratio value. The research note records this intention, but the formal content does not. The certificate proves general facts about the cost function, universally quantified, and would become a theorem about its subject only with a definition of the two quantities in that subject's own terms.

The certificate therefore establishes a reusable formal template, not a substantive ethical claim. It shows that the cost function has the basic properties one would want for measuring restoration, but it does not say what counts as harm, what counts as repair, or how to measure either. Those definitions remain open targets, not results.

structure RestorationJust4Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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]

theorem cert_inhabited : Nonempty RestorationJust4Cert := ⟨cert⟩

What this page does not claim

The certificate does not define what counts as harm or repair in any specific ethical domain. The certificate does not prove that the golden-ratio threshold is the correct standard for acceptable restoration. The certificate does not establish that Recognition Science's ethical framework is sound or applicable.

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/Ethics.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: