phi_cost_pos
The strict positivity of the J-cost for the golden ratio φ follows directly from its closed-form expression and the bound φ > 1.6. Researchers deriving the ground-state instability in Recognition Science cite this to establish that departure from the zero-defect state x=1 carries positive cost. The proof is a one-line wrapper that rewrites via the equality J(φ) = φ - 3/2 then applies linear arithmetic on the upstream lower bound.
claim$0 < J(φ)$, where $J(x) = (x + x^{-1})/2 - 1$ is the recognition cost from the Law of Existence and $φ$ is the golden ratio satisfying $φ^2 = φ + 1$.
background
The StillnessGenerative module derives that the unique zero-defect state x=1 is unstable and must generate structure, using only the T0-T8 forcing chain. The J-cost function from LawOfExistence quantifies the recognition defect of any state x via $J(x) = (x + x^{-1})/2 - 1$. PhiForcing supplies the golden ratio φ as the self-similar fixed point forced by T6, together with the explicit identity J(φ) = φ - 3/2 and the numerical bound φ > 1.6.
proof idea
One-line wrapper that rewrites the goal with phi_cost_eq to obtain 0 < φ - 3/2, then closes the inequality by linarith using the upstream lemma phi_gt_onePointSix.
why it matters in Recognition Science
Supplies the positivity clause required by origin_question_resolved, which assembles every sub-question on the origin of structure from T0-T8. It also feeds stillness_is_creative to confirm that the initial unity configuration cannot remain static. The result closes the finite-barrier step in the module derivation: J(φ) is positive yet bounded above by 1, so creation is forced yet possible. It directly supports the T5 J-uniqueness and T6 φ-forcing landmarks.
scope and limits
- Does not derive the explicit form of J itself.
- Does not compute a numerical value for J(φ) beyond its sign.
- Does not address the upper bound J(φ) < 1.
- Does not extend positivity to other rungs on the phi-ladder.
formal statement (Lean)
69theorem phi_cost_pos : 0 < LawOfExistence.J PhiForcing.φ := by
proof body
One-line wrapper that applies rw.
70 rw [phi_cost_eq]; linarith [PhiForcing.phi_gt_onePointSix]
71