voiceThreshold
plain-language theorem explainer
The voice threshold sets the J-bar cutoff at 0.5 in Recognition units, below which standing waves permit interpretable voice. Researchers on T10 voice forcing cite it when establishing that voice quality is positive for sub-threshold J-bar and vanishes above it. The definition is a direct constant assignment unfolded in monotonicity and emergence theorems.
Claim. The threshold value for lattice-averaged J-cost below which voice emerges is $0.5$.
background
J-cost is given by the strictly convex function $J(x) = (x + x^{-1})/2 - 1$ with unique minimum at $x=1$, per the PhiForcingDerived structure. J-bar denotes the average J-cost on the lattice, and the threshold marks the boundary for standing-wave formation where signal exceeds noise. The VoiceForcing module develops T10, deriving voice as a forced outcome of cost minimization and intelligence tiers, building on the T0-T8 chain that forces phi and D=3 spatial dimensions. Upstream, the PhysicsComplexityStructure notes that J-cost minimization is convex and each tick updates at most eight neighbors.
proof idea
This is a direct definition assigning the real number 0.5. Dependent results unfold the definition and apply linarith to compare against jbar in the emergence and impossibility theorems.
why it matters
The threshold anchors the T10 voice forcing chain, appearing in VoiceForcingConditions to require jbar below threshold together with active recognition operator and SNR above one, forcing voice with positive Berry content. It supplies the standing-wave prerequisite in the derivation from RCL through intelligence to voice. The constant supports the framework claim that voice is a mathematical consequence of the same forcing that yields phi and three dimensions.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.