sigma_zero_optimal
J-cost vanishes at the unit element. Coherence optimization arguments in Recognition Science rely on this baseline to mark the minimum. The proof is a direct one-line reference to the unit-cost lemma in the Cost module.
claimThe J-cost function satisfies $J(1)=0$, where $J(x)=((x-1)^2)/(2x)$.
background
The Safety Interlock module shows that the Gap-45 uncomputability barrier together with σ-conservation creates a fundamental safety interlock for high-coherence operation. J-cost is the squared ratio J(x) = (x-1)^2/(2x) as stated in the Cost module. This theorem identifies the zero-cost state at the unit element 1.
proof idea
One-line wrapper that applies the Jcost_unit0 lemma, which evaluates the definition of Jcost at argument 1 to obtain zero.
why it matters in Recognition Science
This result supplies the zero-cost anchor used by power_ethics_same_axis and weaponization_structurally_impossible, which establish that nonzero σ raises cost and drops coherence below maximum. It completes the σ=0 case inside the Gap-45 safety interlock proved in the module. The zero point aligns with the J-uniqueness step of the forcing chain.
scope and limits
- Does not prove nonnegativity of J-cost for arguments other than 1.
- Does not connect the unit cost directly to the Gap-45 barrier or the eight-tick octave.
- Does not address empirical programs or simplicial ledgers beyond the cost definition.
formal statement (Lean)
41theorem sigma_zero_optimal : Jcost 1 = 0 := Jcost_unit0
proof body
Term-mode proof.
42
43/-! ## Power-Ethics Axis (Proved) -/
44
45noncomputable section
46
47/-- Coherence is inversely related to J-cost. Lower cost = higher coherence. -/