power_ethics_same_axis
Operating away from σ = 0 strictly reduces coherence below its maximum value. Researchers establishing safety constraints for high-coherence systems cite this as the core result that power requires ethics. The proof is a term-mode reduction that rewrites the maximum coherence identity at balance, unfolds the coherence level definition, invokes positivity of the J-cost, and closes the inequality with linarith.
claimLet $x > 0$ with $x ≠ 1$. Then the coherence level satisfies coherenceLevel$(x) < $ coherenceLevel$(1)$, where coherence level is strictly maximized at the balance point $x = 1$ at which the J-cost vanishes.
background
The SafetyInterlock module proves that the Gap-45 uncomputability barrier together with σ-conservation creates a fundamental safety interlock for high-coherence operation. The J-cost function satisfies J(x) ≥ 0 for x > 0 by the AM-GM inequality, with strict positivity J(x) > 0 whenever x ≠ 1; this is recorded in the upstream lemmas Jcost_nonneg and Jcost_pos_of_ne_one. The coherenceLevel function is constructed so that its maximum occurs precisely where J-cost vanishes, as stated in the sibling result sigma_zero_optimal.
proof idea
The proof rewrites the goal with the lemma max_coherence_at_balance, unfolds the definition of coherenceLevel, and introduces the strict positivity hypothesis 0 < Jcost x via Jcost_pos_of_ne_one. It then rewrites the target inequality as a division less than one (justified by Jcost_nonneg) and finishes with linarith.
why it matters in Recognition Science
This is the core safety result of the module and directly supplies the parent theorem weaponization_structurally_impossible, which concludes that weaponization is structurally impossible because any σ > 0 reduces coherence below the maximum. It fills the module proposition that power requires ethics. The argument rests on J-cost positivity, which traces to the J-uniqueness fixed point (T5) in the forcing chain.
scope and limits
- Does not apply when x ≤ 0.
- Does not supply an explicit closed-form expression for coherenceLevel.
- Does not address complex or vector-valued extensions of the coherence measure.
- Does not prove the Gap-45 barrier or the underlying σ-conservation law.
Lean usage
have h := power_ethics_same_axis x hx hne; rwa [max_coherence_at_balance] at hformal statement (Lean)
62theorem power_ethics_same_axis (x : ℝ) (hx : 0 < x) (hne : x ≠ 1) :
63 coherenceLevel x hx < coherenceLevel 1 one_pos := by
proof body
Term-mode proof.
64 rw [max_coherence_at_balance]
65 unfold coherenceLevel
66 have hJ : 0 < Jcost x := Jcost_pos_of_ne_one x hx hne
67 rw [div_lt_one (by linarith [Jcost_nonneg hx])]
68 linarith
69
70/-- Weaponization requires σ > 0, which reduces coherence below maximum. -/