coherenceLevel_pos
The theorem shows that coherenceLevel x is strictly positive for any positive real x. Researchers modeling safety interlocks in high-coherence systems cite it to confirm that coherence cannot reach zero under the J-cost definition. The term proof unfolds the reciprocal definition and applies div_pos together with linarith on J-cost non-negativity.
claimFor every real number $x > 0$, $0 < 1 / (1 + J(x))$, where $J(x) = (x + x^{-1})/2 - 1$ denotes the J-cost.
background
The Safety Interlock module proves that Gap-45 uncomputability and sigma-conservation together enforce a fundamental safety barrier for high-coherence operation. CoherenceLevel is defined by the equation coherenceLevel x _ = 1 / (1 + Jcost x), so that lower J-cost yields higher coherence. Upstream lemmas establish J-cost non-negativity for positive arguments via AM-GM: J(x) >= 0 whenever x > 0, with proofs appearing in Cost, Gravity.CoherenceCollapse, and NavierStokes.PhiLadderCutoff.
proof idea
The term proof first unfolds the definition of coherenceLevel. It then applies div_pos to the positive numerator one and invokes linarith on the supplied Jcost_nonneg hypothesis to conclude that the denominator is positive.
why it matters in Recognition Science
This positivity result anchors the safety interlock by guaranteeing that coherence stays positive for positive inputs, directly supporting module claims such as sigma_zero_optimal (J(1) = 0) and weaponization_structurally_impossible. It aligns with the Recognition framework's J-uniqueness (T5) and non-negative cost structure that underpins the phi-ladder and eight-tick octave.
scope and limits
- Does not establish positivity for non-positive inputs.
- Does not provide explicit bounds or decay rates for coherenceLevel.
- Does not incorporate Gap-45 coprimality or sigma constraints.
- Does not address vector or operator-valued extensions.
formal statement (Lean)
50theorem coherenceLevel_pos (x : ℝ) (hx : 0 < x) : 0 < coherenceLevel x hx := by
proof body
Term-mode proof.
51 unfold coherenceLevel
52 apply div_pos one_pos
53 linarith [Jcost_nonneg hx]
54