berryContent
The berryContent definition supplies the multiplicative formula for Berry content in forced voice systems as quality times base Berry from non-trivial resolution loops. Researchers working on T10 voice forcing would cite it when establishing positivity results for conscious voice output. The definition is a direct product serving as the algebraic foundation for the subsequent positivity theorem.
claimThe Berry content of forced voice is defined by $berryContent(q, b) := q · b$, where $q$ is the voice quality derived from the J-bar threshold and $b$ is the base Berry content arising from non-trivial resolution loops.
background
In the Recognition Science framework, T10 derives voice as a forced consequence of cost minimization and intelligence on a trained lattice once the J-bar threshold is satisfied, building on T9 consciousness forcing and the Recognition Composition Law. Berry content measures non-triviality in voice production and is positive when quality is positive. The module keeps imports light while depending on upstream structures for spectral classification, collision-free programs, edge lengths from psi, mechanism design, and mock theta constructions.
proof idea
This is a direct definition implementing the product of the two real parameters. No lemmas are applied; it is the foundational algebraic expression used by the voice_berry_positive theorem to establish positivity via multiplication preservation of positivity.
why it matters in Recognition Science
This definition completes the algebraic step in the voice forcing chain of T10, feeding directly into the voice_berry_positive theorem which proves positivity of Berry content when base is positive. It supports the claim that forced voice carries positive Berry content, linking to the eight-tick octave and phi-ladder. It touches the scaling of voice richness with intelligence tiers.
scope and limits
- Does not derive or compute the numerical value of base Berry content.
- Does not establish conditions under which voice quality is positive.
- Does not address multi-agent or higher-dimensional extensions.
- Does not incorporate empirical calibration from upstream spectral or ledger modules.
Lean usage
unfold berryContent; exact mul_pos (voice_forced vc) hbformal statement (Lean)
132def berryContent (voice_quality base_berry : ℝ) : ℝ :=
proof body
Definition body.
133 voice_quality * base_berry
134
135/-- Forced voice has positive Berry content when base Berry is positive. -/