OR6 Spectral Completeness
OR6 is the sixth of six original-rule narrowing principles (the Six Ontic Requirements, VII.D37). It states the spectral completeness requirement: any τ-categorical reality candidate must support a spectral decomposition of its internal operators, with all spectral data recoverable from the kernel's internal structure — so the foundation is self-diagnosing and can analyze its own operators without external spectral theory.
τ-Definition
OR6 is the sixth of six original-rule narrowing principles (the Six Ontic Requirements, VII.D37). It states the spectral completeness requirement: any τ-categorical reality candidate must support a spectral decomposition of its internal operators, with all spectral data recoverable from the kernel's internal structure — so the foundation is self-diagnosing and can analyze its own operators without external spectral theory.
Categorical invariant. Spectral-completeness narrowing: ∀ candidate ontic structure τ', every admissible operator T ∈ τ' has a spectral decomposition Spec(T), and the spectral data are recoverable internally from τ' (no external spectral theory required).
Primary registry anchor:
VII.D37
τ-Derivation Chain
-
I.K0— Universe Postulate -
VII.D37— Six Ontic Requirements (OR1-OR6) — OR6 is spectral completeness -
VII.L33— OR5+OR6 Narrowing — analytic and spectral conditions linking combinatorial structure to functional analysis -
VII.P08— Each Requirement Independently Necessary — dropping OR6 admits analytic spaces without full spectral decomposition (e.g., non-self-adjoint operator algebras)
Phenomenological Correlate
OR6 is instantiated whenever a foundation is required to *analyze its own operators* — to decompose them into spectral components and recover all relevant data from internal structure alone. Examples: τ's eight spectral forces (Book III); quantum mechanics' demand that observables be self-adjoint with real spectrum; the distinction between bound states and scattering states (discrete vs continuous spectra) as physically meaningful only in spectrally complete frameworks.
Examples:
- τ-framework: the eight spectral forces (Book III) provide self-diagnostic capacity — τ analyses its own internal operators
- Quantum mechanics: observables are self-adjoint operators precisely so they admit a real spectral decomposition (von Neumann)
- Bound vs scattering states: the discrete/continuous spectrum distinction grounds qualitatively different physical regimes
- Atomic physics: hydrogen spectral lines = direct empirical readout of spectral completeness in the Coulomb operator
- Counterexample (excluded by OR6): non-self-adjoint operator algebras — they have analytic structure but no full spectral theorem
Register codomain: Proof (diagrammatic — spectral completeness is an internal-structural / self-diagnostic property of admissible operators, lives in Reg_D's codomain; with strong downstream readout into Reg_E for measurable spectra)
Manuscript reference: manuscript-sources/book-07/part02/ch29.tex
Lean Coverage
Status: Planned