phi_pos
The theorem establishes that the golden ratio φ is strictly positive. Researchers working on the φ-ladder hypothesis cite this fact when assigning rungs to mass scales or timescales in the Recognition framework. The proof is a short term-mode reduction that unfolds the definition of φ, invokes positivity of the square root of five, and closes by linear arithmetic.
claimLet φ = (1 + √5)/2. Then 0 < φ.
background
The module states the φ-ladder hypothesis: privileged physical scales satisfy X = X₀ · φⁿ for integer n. Examples include particle masses m = B · E_coh · φʳ and timescales τ = τ₀ · φⁿ. The declaration supplies the elementary positivity fact presupposed by all rung calculations on the ladder.
proof idea
The proof unfolds the definition of φ, applies Real.sqrt_pos to obtain 0 < √5, and closes with linarith.
why it matters in Recognition Science
This positivity result underpins the φ-ladder hypothesis in the RRF module and supports derivations of mass scales via the phi-ladder. It connects to the self-similar fixed point φ forced in the T6 step of the unified forcing chain. No open questions are directly addressed.
scope and limits
- Does not establish uniqueness of φ as the positive root of x² - x - 1 = 0.
- Does not connect φ to physical constants such as the fine-structure constant.
- Does not prove any specific rung assignment for particles or timescales.
formal statement (Lean)
39theorem phi_pos : 0 < phi := by
proof body
Term-mode proof.
40 unfold phi
41 have h : 0 < Real.sqrt 5 := Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 5)
42 linarith
43
44/-- φ > 1. (√5 > 1, so (1 + √5)/2 > 1) -/