OR5 Holomorphic Continuation
OR5 is the fifth of six original-rule narrowing principles (the Six Ontic Requirements, VII.D37). It states the holomorphic continuation requirement: any τ-categorical reality candidate must support a notion of analytic continuation in which local data determine global structure — so that lawfulness is encoded as a structural continuation property, not appended as extra postulates.
τ-Definition
OR5 is the fifth of six original-rule narrowing principles (the Six Ontic Requirements, VII.D37). It states the holomorphic continuation requirement: any τ-categorical reality candidate must support a notion of analytic continuation in which local data determine global structure — so that lawfulness is encoded as a structural continuation property, not appended as extra postulates.
Categorical invariant. Holomorphic-continuation narrowing: ∀ candidate ontic structure τ', τ' supports a continuation operator C such that local data on an open set U ⊆ τ' uniquely determine global data on the maximal continuation domain. Lawfulness = the continuation property of admissible operators.
Primary registry anchor:
VII.D37
τ-Derivation Chain
-
I.K0— Universe Postulate -
VII.D37— Six Ontic Requirements (OR1-OR6) — OR5 is holomorphic continuation -
VII.L33— OR5+OR6 Narrowing — analytic and spectral conditions linking combinatorial structure to functional analysis -
VII.P08— Each Requirement Independently Necessary — dropping OR5 admits purely combinatorial categories with no analytic structure
Phenomenological Correlate
OR5 is instantiated whenever lawful behavior is recovered as the analytic continuation of local data, rather than postulated as an extra axiom. Examples: complex-analytic continuation in mathematical physics; the Wightman axioms' analytic continuation between Euclidean and Lorentzian QFT; the τ-framework's recovery of physics from holomorphic structure (Book IV); operator realism (the Central Theorem 𝒪(τ³) ≅ A_spec(L), Book II).
Examples:
- Complex analysis: a holomorphic function on a connected domain is determined by its values on any open subset — laws emerge from local data
- Quantum field theory: Wick rotation / analytic continuation between Euclidean and Lorentzian signatures encodes Lorentz invariance structurally
- τ-framework: the Central Theorem 𝒪(τ³) ≅ A_spec(L) (Book II) — physics emerges from the holomorphic structure of τ
- Operator realism: laws of nature are not extra postulates but the continuation property of admissible τ-operators
- Determinism: classical mechanics' principle of least action expresses local-data-determines-global continuation
Register codomain: Proof (diagrammatic — holomorphic continuation is a structural property of admissible operators, lives in Reg_D's codomain; with downstream readout into Reg_E for empirical lawfulness)
Manuscript reference: manuscript-sources/book-07/part02/ch29.tex
Lean Coverage
Status: Planned