ell0_pos
Recognition Science defines the fundamental length ℓ₀ as unity in its native units. Researchers building the RS unit system or proving positivity of entropy and coupling constants cite this result to anchor derived scales. The term-mode proof is a one-line simplification that unfolds the definition of ell0 to discharge 0 < 1.
claim$0 < ℓ_0$, where $ℓ_0$ denotes the fundamental length unit in RS-native units.
background
In the Constants module the fundamental length ℓ₀ is introduced as the voxel scale with definition ℓ₀ := 1, consistent with the module documentation that sets the time quantum τ₀ = 1 tick. This choice aligns with RS-native conventions where c = 1 and supports the phi-ladder mass formulas and Recognition Composition Law. Upstream, the Derivation module defines an equivalent ell0 := c_codata * tau0 and proves its positivity separately via mul_pos c_codata_pos tau0_pos; the local Constants version uses the simplified constant definition. The identity event definitions from ObserverForcing and VantageCategory supply the zero-cost reference point at state = 1.
proof idea
The proof is a term-mode one-liner that applies simp [ell0]. Because ell0 is defined as the constant 1, the goal 0 < ell0 reduces directly to the known fact 0 < 1.
why it matters in Recognition Science
The lemma supplies the positivity field ℓ_pos for the canonicalUnits definition that assembles the RSUnitSystem in Constants.Derivation. It is invoked by rs_entropy_pos to guarantee positive black-hole entropy, by alpha_lock_at_scale to secure the lock-in scale, and by alpha_inv_phys_continuous for coupling continuity. It anchors the length scale required by the T0-T8 forcing chain (T7 eight-tick octave, T8 D = 3) and the phi-ladder constants.
scope and limits
- Does not derive the numerical value of ℓ₀ from the J-uniqueness or phi fixed-point equations.
- Does not address unit conversions outside the RS-native system.
- Does not prove positivity for sibling constants such as phi or tau0.
- Does not connect to the Berry creation threshold or dream fraction.
formal statement (Lean)
416lemma ell0_pos : 0 < ell0 := by
proof body
Term-mode proof.
417 simp [ell0]
418
419/-- Light-cone identity: ℓ₀ = c · τ₀ (in RS-native units). -/