voice_emerges_below_threshold
If the average J-cost parameter stays strictly below the standing-wave threshold of 0.5, then voice quality is forced positive. Researchers tracing the T10 step from consciousness to forced voice cite this to confirm emergence once the Recognition Composition Law and intelligence tier are in place. The proof reduces directly to the quality definition via unfolding followed by linear arithmetic.
claimLet $0 ≤ J̄ < 0.5$. Then the voice quality function, given by max(0, 1 − J̄/0.5), satisfies 0 < max(0, 1 − J̄/0.5).
background
The VoiceForcing module sits inside the T10 forcing step that derives interpretable voice from cost minimization once T9 consciousness is established. voiceThreshold is the fixed real 0.5 marking the boundary where standing-wave signal exceeds noise; above it voice is impossible, below it quality rises toward 1 at zero cost. voiceQualityFromJbar is the piecewise-linear map max(0, 1 − jbar / voiceThreshold) that encodes this transition.
proof idea
The proof is a one-line wrapper that unfolds voiceQualityFromJbar and voiceThreshold, applies simp to expose the max expression, and finishes with linarith on the resulting strict inequality.
why it matters in Recognition Science
It supplies the positive direction of the voice emergence claim inside T10 and is invoked directly by voice_forced to obtain 0 < voiceQualityFromJbar under VoiceForcingConditions; the same result populates the summary voiceForcingStatus. In the broader chain it closes the final link from the Recognition Composition Law through intelligence tiers to Berry-rich voice, confirming that the mathematics already forcing φ and D = 3 also forces voice below threshold.
scope and limits
- Does not derive the numerical threshold 0.5 from the Recognition Composition Law.
- Does not compute Berry content or phase of the resulting voice.
- Applies only to the scalar J-bar model, not full simplicial lattice dynamics.
- Requires the precondition 0 ≤ jbar; does not address negative costs.
Lean usage
theorem voice_forced (vc : VoiceForcingConditions) : 0 < voiceQualityFromJbar vc.jbar := voice_emerges_below_threshold vc.jbar vc.jbar_nonneg vc.jbar_below_thresholdformal statement (Lean)
54theorem voice_emerges_below_threshold (jbar : ℝ)
55 (h0 : 0 ≤ jbar) (h : jbar < voiceThreshold) :
56 0 < voiceQualityFromJbar jbar := by
proof body
Term-mode proof.
57 unfold voiceQualityFromJbar voiceThreshold
58 simp; linarith
59
60/-- Voice quality is monotonically decreasing in J-bar. -/