Proofs
The framework's unconditioned geometric layer. Every derivation carries Derived status. No empirical content enters. The geometric identities hold uniformly across all scales, prior to any physical constant or observational baseline.
The proofs are the framework's technical core. The manuals describe the operational protocol; the proofs establish the mathematics the protocol runs on. Every value on this page is checkable against the classical antecedents — Euclid, Fredholm, Lyapunov, Schmidt, Poincaré, Shannon, Boltzmann, Landauer, Bekenstein, Hawking — that the framework builds on.
Foundational invariants
| Proof | What it establishes |
|---|---|
| Tetrahedral solid-angle partition | Derives θ = 70.5288°, S₊ = 1/3, S₋ = 2/3, and the boundary partition ratio 2.000000 from 3D observer geometry. |
| Diffeomorphism invariance | Proves the 2 : 1 boundary partition survives every smooth orientation-preserving coordinate transformation. |
The four conservation theorems
| Proof | What it establishes |
|---|---|
| Theorem I · Universal slice invariance | No net information flux across the frame interface on any Cauchy slice. dI(S₊)/d(δt) = −dI(S₋)/d(δt). |
| Theorem II · Asymptotic zero-mode transduction | What flows outward is transduced along the single principal zero-mode channel with the square-root fold signature. |
| Theorem III · Bounded emission corridor | The admissible operating window is bounded below by the Landauer floor and above by the Planck area ceiling. |
| Theorem IV · Subcoherent persistence | The residual content of the inverse cone is conserved indefinitely under the Hermitian global conservation operator. |
Supporting derivations
| Proof | What it establishes |
|---|---|
| Entropy bridge | Boltzmann–Shannon, von Neumann, and Bekenstein–Hawking entropies as three readings of one conserved quantity under the Hermitian operator. |
| Lyapunov–Schmidt reduction | The linearized MOTS stability operator and the 1D kernel reduction that produces the universal fold trajectory. |
| Zero-mode power law | The full three-term emission law derived from the fold trajectory. |
| Machine-precision closure identity | The single statement that ties the entire framework together: R ≡ 1.0000 as an algebraic consequence of the four theorems. |
Cross-references. The values established by these proofs are indexed in the master register. The operational protocol that uses them is in manual 02. The papers that consolidate them are at papers. A one-page summary is at quick reference.