One Axiom : Nau Victoria

ONE AXIOM: Nau Victoria — A Navigation Manual for Exploration Domains FORBIDDEN States, the Navigator's Safety Code, and the Victoria Theorem This paper is the fifth installment in the ONE AXIOM Series M. It addresses the operational gap left by 4M (Exploration Theory): given that an Explorer operates in a structured domain M₄ GF (2) ⁶, which states must be avoided, and how does a coherent agent navigate around them? The central result — Theorem NEG-DEF ("The Axiom Knows the Map") — proves that the single structural condition (x) > 0 is exactly equivalent to the conjunction of 13 independently derived Navigator's Safety Conditions C (1) –C (13), each corresponding to one of the 13 FORBIDDEN states in M₄. The axiom is not merely a generator; it is a complete navigational map. No lookup table is required: > 0 is both necessary and sufficient. Main results (32 formal results, all O∩FR PROVEN) Lemma FORB-GEN: The generator set G^- = \7, 18, 36\ closes under XOR and Q₆-orbit operations to produce all 13 FORBIDDEN states in 4 rounds. States \28, 35, 49\ are Q₆-isolated and reachable only via XOR. Theorem FORB-COMPLETE: (x) = 0 and x Fix (M) if and only if x FORBIDDEN₁₃. The catalog is exhaustive. Theorem NEG-DEF ("The Axiom Knows the Map"): (x) > 0 C (k) (x) x ALLOWED. Constructive and non-circular in both directions. Navigator's Safety Code (NSC): 13 structural conditions partitioned into three failure classes by attractor orbit: O₇: Logical Collapse C (1) –C (7) O₄: Measure Collapse C (8) –C (11) O₂: Perspective Error C (12) –C (13) Includes a 13-row Recovery Map. Theorem NAV: Every ALLOWED trajectory maintains FreedomE _ throughout and carries coherence measure: () = ^T () /_ where = 63/64 (derived from 0A). Operational Protocols Definition VNA (Victory Navigation Algorithm): A 4-step viability-preserving protocol — not an optimizer. Maintains FreedomE _ as a structural invariant. Complexity bound (Cor. VNA-BOUND): Tₕ₍₀ ₂₎ₕ|DE|/_ |DE| 192. Theorem VICTORIA: VNA terminates with a trajectory ^* satisfying all 5 Victoria conditions on any connected ALLOWED domain. The Victoria set V is structurally rare and non-empty: 0 0 acting as a coherence attractor structurally violates the recipient Explorer's CSP. The -source (= 0) is the unique admissible coherence attractor. Structural Properties D₁₀ reachable from any ALLOWED state in 3 steps (BFS-certified). Prop. STRUCTURE-CAPACITY: N₂ₑ₈ₓ = (FreedomE - _) /_; N₂ₑ₈ₓ = 1 at Deep ALLOWED minimum. Prop. OVERLOAD-RELEASE: SC N₂ₑ₈ₓ + 5 sequential 5-step shedding releases 5_, enabling meta-D1 recovery. Remark STAG-IMPULSE A stagnation loop (periodic trajectory with 0) is structurally dominated by any allowed exit via -monotonicity (₄ₗ₈ₓ > ₋₎₎₏ for any period p 2). This is the structural source of any decision impulse — not a psychological postulate. 12 Falsifiable Predictions (Examples) P2: kₒₐ \12 1\ for LLM token sequences. P4: FORB-GEN requires 4 rounds of closure. P7: C (13) violation via autocorrelation of ₒ₄₋₅ outputs. P9: FEP emerges as the ₅₄₏-shadow of ALLOWED dynamics. P10: N₂ₑ₈ₓ = 1 at the point of minimum agency. P11: 3-step re-anchoring (BFS) after a domain change. P12: Attractor orbit (O₇/O₄/O₂) predicts the specific NSC failure class. Appendices A: Cross-system structural mappings (Viability Theory, Control Barrier Functions, FEP, Rough Sets). B: FORB-GEN 4-round computation table + NSC derivation (> 0 C (k) ) + WITNESS verification. C: NSC full independence table (78 pairs; 73 independent, 4 structural relations). D: -model (Optional; open verification task V5, incorporating previous /5 and /6 layer findings). E: BFS Certificate — Deep ALLOWED geometry: D₁₀ = \0, 1, 8, 9, 25, 40, 41, 42, 45, 56\; A₁₅ = \6, 19, 20, 22, 26, 30, 38, 39, 51, 52, 53, 54, 55, 59, 62\. Series position and dependencies: Preceded by: 4M — Exploration Theory (DOI: 10. 5281/zenodo. 19933072) Depends on: 0M, 000, ABC, 0A, 0B PSP (DOI: 10. 5281/zenodo. 18233261), 0C, 3M, 4M. Connects to: 4B v5. 0 (Prediction Watch) and 3C (P vs NP structural bridge via State 36).