bound_from_phi
plain-language theorem explainer
Recognition Science replaces the classical Bremermann limit with a tighter bound based on the eight-tick recognition cycle. The product of the maximum resolution rate and the energy quantum per resolution equals φ^5 over 8. Information theorists studying fundamental computation limits would cite this equality. The proof reduces directly to the definitions of the bound and the energy term after substituting the octave length.
Claim. The maximum resolution rate (one per eight ticks) multiplied by the energy quantum per resolution equals $φ^5 / 8$.
background
The module develops a recognition-theoretic version of Bremermann's limit on computation rate. In this setting the fundamental time unit is the tick τ₀, and the minimum cycle for one complete debt resolution is the eight-tick octave. The Bremermann bound is therefore defined as one resolution per eight ticks, or 1/8 resolutions per tick in units where τ₀ = 1. Each resolution carries an energy quantum φ^5 because ħ = φ^{-5} in RS-native units.
proof idea
The proof unfolds the definitions of bremermannBound and energyPerResolution, rewrites using the lemma octave_is_eight, and simplifies the resulting arithmetic expression with the ring tactic.
why it matters
This result confirms that the RS power bound incorporates φ^5, directly tying the information limit to the eight-tick octave (T7) and the φ-ladder energy scale. It fills the explicit claim in the module documentation that the bound involves φ^5 and supports the broader statement that no physical process resolves debt faster than eight ticks. No downstream theorems yet reference it, leaving open its integration with mass formulas or the alpha band.
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