Constants Rsnative Units Phi Rung Zero

The zero rung

The golden ratio, usually written φ, is the number that solves r² = r + 1, so φ = (1 + √5)/2 ≈ 1.618. It appears throughout classical mathematics: in a regular pentagon's diagonals, in the Fibonacci sequence, and as the limit of ratios of successive Fibonacci numbers. Euclid called it the extreme and mean ratio around 300 BCE; later writers gave it the name golden ratio in the 19th century. Its continued fraction is the simplest possible, all ones, which is why it is the hardest irrational to approximate by fractions.

Recognition Science (RS) builds a system of units from the golden ratio. The framework's library defines a ledger, a discrete record of recognition events, and sets its base units so that the speed of light equals 1 and all dimensionless ratios are powers of φ. On this ladder, the integer n picks a scale: φ^n for masses, energies, times, and lengths. The declaration phiRung_zero states that the zeroth rung, φ^0, equals 1. The proof is immediate from the definition of a power with exponent zero, and the library records it as a lemma.

Why does a trivial fact matter? Because it anchors the ladder. Every other rung is defined relative to this one. The rung φ^1 is φ itself, and φ^−5 gives the fundamental energy quantum in these units. Without the zero rung pinned to 1, the entire scaling system would lack a fixed reference point. The declaration is the hinge that lets the framework say, in its own units, that a mass at rung zero is one unit and an energy at rung −5 is φ^−5 units.

In Recognition Science, the framework models physical quantities as real numbers multiplied by these rungs. The declaration phiRung_zero does not claim that this unit system matches SI measurements; conversion to SI is a separate, explicit calibration step. It does not claim that the golden ratio itself is derived from the ledger; the ladder assumes φ as its base. It only fixes the meaning of the zeroth exponent on that ladder.

lemma phiRung_zero : phiRung 0 = 1 := by simp [phiRung]

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

lemma phiRung_one : phiRung 1 = phi := by simp [phiRung]

What this page does not claim

This declaration does not claim that the unit system matches SI measurements. It does not claim that the golden ratio is derived from the ledger. It does not claim that any physical quantity at rung zero has a measured value.

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)

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