U_c
The lemma establishes that the speed of light equals unity in the Recognition Science native gauge U. Researchers normalizing velocities or working with dimensionless ratios on the phi-ladder would cite this when setting the causal limit to one voxel per tick. The proof is a direct reflexivity on the unit definition that explicitly assigns c to 1.
claimIn the RS-native gauge $U$ with base time interval one tick and base length one voxel, the speed of light satisfies $c = 1$.
background
The module defines an RS-native measurement system whose base units are the tick (atomic time quantum) and voxel (causal spatial step). Derived quanta are the coherence energy quantum equal to phi to the minus five and the action quantum equal to the same value since the base time interval is unity. The structure U packages these into a gauge whose fields are tau0 equal to one tick, ell0 equal to one voxel, and c equal to one, with the coherence condition proved by simplification on the base definitions.
proof idea
The proof is a one-line term proof that applies reflexivity directly to the definition of U, which sets the c field to the constant already known to equal 1 in the native system.
why it matters in Recognition Science
This lemma anchors the native units so that all velocities are expressed relative to the causal limit of one voxel per tick. It implements the key property c equals 1 stated in the module documentation and supports downstream calculations of dimensionless constants such as alpha in the interval (137.030, 137.039) together with mass formulas on the phi-ladder. No parent theorems are listed among the used-by edges.
scope and limits
- Does not derive the numerical value of c from any forcing chain step.
- Does not supply conversion factors between native units and SI.
- Does not constrain phi-ladder scalings for mass or energy.
- Does not address empirical calibration or external anchoring.
formal statement (Lean)
87@[simp] lemma U_c : U.c = 1 := rfl
proof body
Term-mode proof.
88
89/-! ## Coherence and action quanta
90
91`Constants.E_coh = φ⁻⁵` is a **dimensionless** RS-derived number.
92In the RS-native system:
93- **1 coh** (energy quantum) := E_coh
94- **1 act** (action quantum) := ħ = E_coh · τ₀ = E_coh (since τ₀ = 1)
95-/
96
97/-- Coherence energy quantum: φ⁻⁵ ≈ 0.0902. -/