voice_impossible_above_threshold
plain-language theorem explainer
Voice quality drops to zero once average J-cost meets or exceeds the standing-wave threshold of 0.5. Researchers tracing the T10 voice-forcing step cite this to close the impossibility direction of the threshold argument. The proof is a direct algebraic reduction that unfolds the piecewise definition of voice quality and finishes with linear arithmetic.
Claim. If $j >= 1/2$, then the voice quality function $q(j) = max(0, 1 - j / (1/2))$ equals zero.
background
The module develops T10, showing that voice emerges as a forced consequence of the Recognition Composition Law once intelligence on the phi-ladder satisfies a cost threshold. voiceThreshold is the constant 0.5 that marks the boundary between viable and impossible voice. voiceQualityFromJbar is the function max(0, 1 - jbar / voiceThreshold), which returns 1 at zero cost and falls linearly to zero at the threshold.
proof idea
Unfolds the definitions of voiceQualityFromJbar and voiceThreshold, then applies simp followed by linarith to obtain the required equality.
why it matters
This theorem supplies the impossibility half of the T10 voice-forcing summary recorded in voiceForcingStatus. It completes the threshold pair with the emergence result below threshold and sits inside the chain from RCL through consciousness to forced voice, consistent with the earlier derivation of phi and D = 3.
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