l0_pos
The lemma establishes that the fundamental length scale ℓ₀ is strictly positive. Researchers deriving energy scales or UV cutoffs from the Recognition constants would cite this result to justify subsequent inequalities. The proof is a one-line term reduction that unfolds the definition ℓ₀ = 1 and applies the standard fact 0 < 1 in the reals.
claim$0 < ℓ_0$ where $ℓ_0$ denotes the fundamental length scale.
background
The Compat.Constants module supplies project-wide constants and minimal structural lemmas. Its definition sets ℓ₀ := 1 in native units. Upstream results include the abstract Constants structure from CPM.LawOfExistence (with nonnegativity on Knet) and the QFT.UVCutoff redefinition ℓ₀ = c · τ₀ that ties the placeholder to physical scales such as E₀ = ℏ / τ₀.
proof idea
One-line term proof that unfolds the definition of l0 and invokes zero_lt_one on the reals.
why it matters in Recognition Science
This basic positivity lemma anchors the length scale that enters mass formulas, energy scales, and the phi-ladder constructions in the Recognition framework. It supports downstream use of the eight-tick octave and D = 3 by guaranteeing a positive base length. Although currently unreferenced, it closes the minimal interface required for the compat layer to feed constants into RSBridge and Fermion structures.
scope and limits
- Does not assign a numerical value to ℓ₀ beyond the unit placeholder.
- Does not connect ℓ₀ to the J-cost function or phi-ladder rungs.
- Does not address physical unit conversions or tau0 scaling.
formal statement (Lean)
30lemma l0_pos : 0 < l0 := by
proof body
Term-mode proof.
31 simpa [l0] using (zero_lt_one : 0 < (1 : ℝ))
32
33end
34
35end Constants
36
37namespace RSBridge
38namespace Fermion
39