Information-Theoretic Structure
The information-theoretic reading of the geometric partition. This section shows how the classical results of Shannon, Boltzmann, Landauer, and Bekenstein–Hawking fit into a single conservation structure — and how the identity between information entropy and physical entropy (the Shannon–Boltzmann correspondence) provides the bridge.
Every value here is tagged as either Derived (pure geometry or algebra) or Frame-Conditioned (tied to a specific thermal or spatial boundary). The classical names are preserved with attribution.
Contents
| Page | What it establishes | Tag |
|---|---|---|
| Entropy bridge | Shannon, Boltzmann, and Bekenstein–Hawking entropies map to each other under the Global Conservation Operator. The unit conversions are explicit. | Derived |
| Machine-precision closure identity | Establishes R ≡ (I_known + I_unknown) / I_total = 1.0000 as an algebraic conservation constraint across all frame transformations. | Derived |
| Thermodynamic bridge | Full operational handbook. The entropy ladder, bounded operational corridor, Landauer floor, Planck ceiling, and squeezed-vacuum mechanics. | Derived · Frame-Conditioned |
| Subcoherent persistence (Theorem IV) | A Hermitian global conservation operator unifies Shannon entropy, von Neumann entropy, and Bekenstein–Hawking horizon entropy. | Derived |
| Thermal & holographic limits (register §03) | The operational corridor values: Landauer floor at 7.155 × 10⁻³⁹ J, Planck area ceiling at 2.159 × 10⁻⁷⁰ m²/bit, Planck luminosity at 3.628 × 10⁵² W. | Derived |
| Consistent quantities (canonical) | The values that agree across all sources — Landauer floor, Planck ceiling, closure ratio, and the entropy-related constants. | Derived |
The Shannon–Boltzmann correspondence
Shannon's information entropy H = −Σ pᵢ log₂ pᵢ and Boltzmann's statistical entropy S = −k_B Σ pᵢ ln pᵢ are the same expression in different units. Shannon's 1948 paper credits the parallel explicitly. The correspondence — sometimes written as S_thermo = k_B · ln(2) · H_bits — is what allows bits to become thermal entropy, which in turn becomes area quanta through the Bekenstein–Hawking relation.
This is the framework's foundational bridge. Every subsequent unification — quantum mechanics, relativity, cosmology — rests on it. See the entropy bridge proof for the full derivation.
Related. The pure geometric foundation is in Classical Maths. The application of the entropy bridge to quantum and relativistic regimes is in QM–Relativity Unification.