Constants Rsnative Units Lambda Kin Eq K Gate Ratio

The gate ratio

A kinetic energy is the energy a thing has because it is moving. In the Recognition Science (RS) framework, the declaration lambda_kin_eq_K_gate_ratio establishes a relationship between this kinetic energy and a fundamental constant called the gate ratio, denoted K. In plain terms, it states that the kinetic energy of a particle, measured in the framework's native units, is equal to the gate ratio itself. This is a definitional equality, not a derived theorem: it sets the scale for what one unit of kinetic energy means in the RS measurement system.

The gate ratio is not an arbitrary number. Within the framework, it is defined as a specific power of the golden ratio, phi, which is the unique positive solution to r² = r + 1. The framework's native unit system, described in the RSNativeUnits module, treats the ledger primitives as base standards. The tick is one discrete ledger posting interval, the atomic time quantum, and the voxel is one causal spatial step, the distance light traverses in one tick. In these units, the speed of light c is exactly 1. All dimensionless ratios are fixed by phi alone, and the gate ratio K is one such fixed ratio.

The declaration itself is a definition, not a proof. It does not derive the value of the kinetic energy from first principles, nor does it claim that this equality holds in any external unit system like SI. It is a choice of how to express kinetic energy within the RS-native measurement system. The equality is a model, a definitional choice that anchors the meaning of the energy unit. The framework's library, a machine-checked collection of formal theorems, records this definition so that subsequent theorems can use it consistently.

What the declaration does not claim is equally important. It does not claim that the kinetic energy of any physical particle is actually equal to the gate ratio. It does not claim that this equality is a law of physics. It does not claim that the gate ratio is a measured quantity. The declaration establishes a unit convention, not an empirical fact. A reader should not take this as a prediction about the energy of a real particle, but as a definition that sets the scale for the framework's internal measurements.

The consequence of this definition is that all energy values in the framework are expressed as multiples of the coherence quantum, E_coh = phi⁻⁵, and the kinetic energy equality fixes the relationship between a particle's motion and this fundamental energy scale. This allows the framework to express physics without reference to SI units, making the framework self-contained. The gate ratio is one of the constants that emerges from the framework's forcing chain, which derives hbar = phi⁻⁵ and G = phi⁵/pi, rather than treating them as free parameters.

/-- The K-gate ratio: K = π/(4 ln φ). -/
noncomputable def K_rs : ℝ := Constants.RSUnits.K_gate_ratio

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

/-- One tick: the fundamental time quantum. -/
@[simp] def tick : Time := Constants.tick

/-- One voxel: the fundamental length quantum. -/
@[simp] def voxel : Length := 1

/-- Speed of light: c = ℓ₀/τ₀ = 1 voxel/tick. -/
@[simp] def c : Velocity := 1

What this page does not claim

The declaration does not claim that any physical particle's kinetic energy equals the gate ratio. The declaration does not claim that the gate ratio is a measured quantity. The declaration does not claim that this equality holds in SI units.

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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Derived articles

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