voice_emerges_below_threshold
plain-language theorem explainer
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.
Claim. Let $0 ≤ J̄ < 0.5$. 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
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.
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