singleStep_le_chain
Voice richness at the single-step intelligence tier is bounded above by the value at the chain-reasoning tier. Researchers tracing the T10 voice-forcing derivation would cite this ordering when establishing monotonic increase of voice quality with intelligence level. The proof is a one-line wrapper that unfolds the tier definition and normalizes the resulting numerical comparison.
claimLet $r$ map intelligence tiers to real numbers via the assignments $r(.singleStep)=0.3$ and $r(.chainReasoning)=0.6$. Then $r(.singleStep) ≤ r(.chainReasoning)$.
background
The module develops T10, showing that voice emerges as a forced consequence of cost minimization once intelligence reaches a trained lattice and the J-bar threshold is met. The upstream definition tierVoiceRichness supplies the concrete map: retrieval yields 0.1, singleStep yields 0.3, chainReasoning yields 0.6, creativity yields 0.85, and selfAware yields 1.0. These values encode the qualitative enrichment of voice output at each successive tier.
proof idea
One-line wrapper that unfolds tierVoiceRichness and applies norm_num to the resulting numerical inequality.
why it matters in Recognition Science
The result supplies one monotonicity link inside the voice-forcing chain of T10. It supports the broader claim that higher intelligence tiers produce richer voice, consistent with the Recognition Composition Law and the earlier forcing of phi and D=3. No downstream uses are recorded yet, but the ordering is required for any proof that voice richness grows across the full tier sequence.
scope and limits
- Does not derive the numerical values assigned to each tier.
- Does not address voice emergence or the J-bar threshold condition.
- Does not connect the ordering to physical constants or the eight-tick octave.
formal statement (Lean)
92theorem singleStep_le_chain :
93 tierVoiceRichness .singleStep ≤ tierVoiceRichness .chainReasoning := by
proof body
One-line wrapper that applies unfold.
94 unfold tierVoiceRichness; norm_num
95