coherenceLevel_pos
plain-language theorem explainer
Coherence level, defined as the reciprocal of one plus J-cost, is strictly positive for every positive real argument. Researchers on the Gap-45 safety interlock would cite this to confirm coherence cannot turn negative under the σ-constraint. The proof is a term-mode one-liner that unfolds the definition and combines div_pos with linarith on J-cost non-negativity.
Claim. For every real number $x > 0$, the coherence level $1/(1 + J(x)) > 0$, where $J(x)$ is the J-cost function.
background
The SafetyInterlock module proves that the Gap-45 uncomputability barrier and σ-conservation together create a fundamental safety interlock for high-coherence operation. Its key proved results include gap45_coprime, sigma_nonzero_has_cost, sigma_zero_optimal, power_ethics_same_axis, and weaponization_structurally_impossible.
proof idea
The proof unfolds the definition coherenceLevel x hx := 1 / (1 + Jcost x), then applies div_pos one_pos and finishes with linarith using the hypothesis Jcost_nonneg hx. Multiple upstream lemmas supply Jcost_nonneg, each deriving non-negativity from AM-GM or direct algebraic expansion of (x + x^{-1})/2 - 1.
why it matters
This theorem belongs to the proved safety-interlock results in the Superhuman domain. It guarantees that the coherence function stays positive, which is a prerequisite for the module's claims that higher coherence requires lower σ and that weaponization is structurally impossible. It rests on the J-cost non-negativity that appears throughout the Recognition framework (T5 J-uniqueness and the RCL).
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