Constants Rsnative Units Phi Rung Neg One

The negative rung

The golden ratio, often written φ, is the number that solves r² = r + 1, roughly 1.618. Its negative powers, φ⁻¹, φ⁻², and so on, are the reciprocals: about 0.618, 0.382, and smaller still. These values appear throughout geometry and number theory, for instance in the proportions of a pentagon or the continued fraction of φ itself. In the Recognition Science framework, these negative powers organize a ladder of scales below the unit, and the fifth rung down, φ⁻⁵, receives a specific physical role.

In Recognition Science, the framework's unit system treats the ledger as the base standard: a ledger, a discrete record of events, defines the atomic time quantum called a tick, and the distance light travels in one tick defines a voxel. On top of these, the framework defines a coherence quantum, the fundamental energy quantum, as φ⁻⁵. The declaration phiRung_neg_one is the formal statement that the rung at index negative one equals the reciprocal of φ itself: phiRung (-1) = φ⁻¹. This is a definitional identity, not a derived theorem; it states what the rung means, and the machine-checked library of formal theorems records it as such.

The value φ⁻⁵, about 0.09017, is then used to set the coherence quantum and, through it, the action quantum, the framework's analogue of Planck's constant. The framework derives these constants from the forcing chain, the sequence of theorems that starts from the cost function and ends at spatial dimensions, instead of taking them as free parameters. The rung at negative one itself, however, carries no such derivation: it is a stepping stone, a named position on the ladder, not a physical law.

What the declaration does not claim is just as important. phiRung_neg_one does not assert that φ⁻¹ is a measured constant of nature, nor that it has any empirical content on its own. It does not claim that the ladder of powers is complete or that every integer rung corresponds to a known particle or process. The declaration is a definition, and its force is purely within the framework's unit system: it fixes what the phrase "rung negative one" denotes, so that later definitions and theorems can refer to it unambiguously.

The practical consequence is that a reader can trace exactly where the golden ratio enters the framework's constants. The negative rung is the first step below unity, and the fifth step down becomes the energy quantum. This is a clean, checkable piece of the framework's scaffolding, and the declaration makes that scaffolding explicit rather than implicit.

/-- φ-ladder scaling: compute φⁿ for integer rung. -/
@[simp] noncomputable def phiRung (n : ℤ) : ℝ := phi ^ n

/-- Coherence scaling: E_coh = φ⁻⁵. -/
noncomputable def E_coh_rs : ℝ := phiRung (-5)

/-- Coherence scaling: E_coh = φ⁻⁵. -/
noncomputable def E_coh_rs : ℝ := phiRung (-5)

What this page does not claim

The declaration does not assert that φ⁻¹ is a measured constant of nature. The declaration does not claim the ladder of powers is complete or that every rung corresponds to a known particle. The declaration does not claim that φ⁻⁵ is a derived theorem rather than a definition.

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/Constants/RSNativeUnits.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: