TUS OS Paper 4 — Technical Volume II: Orthogonality, Process Closure and Native Triadic Synthesis

TUS OS Paper 4 — Technical Volume II: Orthogonality, Process Closure and Native Triadic Synthesis completes the two-volume technical development of Paper 4. Technical Volume I fixed the 64-state carrier, complete terminal future corpus, parity-separated complement-reachability structure and invariant probability module. This volume closes the remaining questions of universally forced support orthogonality, native composition, representation closure, LANDING, GUIDANCE and downstream synthesis correspondence at their explicitly declared scopes. For every finite-dimensional exact common-effect representation, Atlas complement effects force complementary carrier states to have orthogonal supports. In every exact common-CPTP realisation, common-word complement reachability propagates this orthogonality backwards by trace-distance contractivity. The resulting provenance-certified graph is Gprov≅K32even⊔K32odd,ω(Gprov)=srk⊥(Gprov)=32. Hence every admitted common-CPTP predictive representation has persistent Hilbert dimension at least 32. An explicit 64-dimensional diagonal construction supplies an exact upper-bound realisation, establishing 32≤dCPTPmin≤64. The 1,024 cross-parity pairs remain unclassified by the certified forcing route. Accordingly, neither the complete universally forced support-orthogonality relation nor the exact common-CPTP minimum within this interval is determined. The paper then constructs a total, source-separated and non-circular conservative native whole-Triad object. Semantic equality erases custody identity, while a distinct provenance fibre preserves exact source, occurrence, relation and certificate custody. The construction is triadically governed at the input-incidence level, but its conservative semantic output is exactly decomposable and its generated relational layer is pairwise-generated. At this scope, native synthesis is therefore a relationally enriched constrained concatenation rather than an irreducible ternary semantic state. On the abstract canonical corpus, the structural quotient has exactly 64 singleton classes and is canonically isomorphic to the complete deterministic predictive carrier system. No predictive merger is possible: the minimum deterministic unifilar realisation has exactly 64 states. This carrier-faithful target exposes an exact static–dynamic separation. The inherited dimension-four static Atlas representation admits no affine or linear descent for the principal or continuation maps, whereas the 64-sector diagonal construction realises them by fixed CPTP channels. Exact LANDING state-only closure satisfies 32≤dL,statemin≤64, while recovery-complete selected-line instrument access forces and attains dL,instrmin=64. An explicit 32-state complement-orbit memory construction also gives an exact quantum-classical hybrid with the certified four-dimensional quantum component. LANDING further admits factorised two-step process-comb and exact relational representations, showing that neither non-Markovian process memory nor irreducibly relational form is necessary for existence at the canonical scope. GUIDANCE is classified separately. Its selected-line future language distinguishes every non-complement pair, and exact line and endpoint instruments attain their declared recovery contracts. Deterministic GUIDANCE line observation can be nondemolition and jointly compatible with LANDING. The GUIDANCE and LANDING endpoint transformations are sibling same-start updates and cannot be serialised on a single carrier without changing their meanings. An exact no-feedback sidecar instrument instead returns the authoritative LANDING successor together with the GUIDANCE receipt. Joint LANDING–GUIDANCE compatibility therefore closes at the canonical scope without adaptive feedback, hidden carrier lookup or source-role inversion. Finally, the native-to-emitted correspondence is formalised as a custody relation and a lawful status-constrained subrelation, rather than as an asserted deterministic projection. The active-v1 profile yields a total 106-class correspondence census with no material-loss, unjustified-field or unresolved class. One authenticated Pilot 2 package supplies an independently reconstructed unrestricted realised witness (ωP2,sP2)∈Clawful⊆Ccust. This establishes one admitted instance of custody-closed native-to-emitted synthesis. The results concern exact finite mathematical and operational models under frozen access, recovery and resource contracts. The certified same-parity graph is a proved universal subgraph of the full forced-orthogonality relation; cross-parity forcing remains unclassified, no universal 64-way support-orthogonality result is asserted, and the exact common-CPTP minimum remains open within 32−64. Likewise, the conservative native object is triadically governed, source-sovereign and non-circular, but no irreducible three-lane semantic degree of freedom is claimed at that scope. The paper makes no claim of global all-runs correspondence, truth of external domain claims, physical quantum execution, quantum advantage, production Domain Binding or activation of a production Technical Synthesis policy. TUS OS mathematics series Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic Witness Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine Architecture Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OS Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability Module Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic Synthesis Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource Separation The present publication is Technical Volume II of Paper 4. It takes the formal foundation and complete-future corpus established in Technical Volume I and closes the declared orthogonality, common-process, native-composition, LANDING, GUIDANCE and realised custody-correspondence questions at the scopes fixed by the Paper 4 technical programme. Files included TUS_OS_Paper_4_Technical_Volume_II_Orthogonality_Process_Closure_and_Native_Triadic_Synthesis_v1.0.pdfMain public v1.0 paper and public release authority. TUS_OS_Paper_4_Technical_Volume_II_v1.0_Public_Verification_Package.zipSupporting public verification package containing the Research Control and Detailed Verification Archive, the Volume II verification package, including its Pilot 2 verification materials, together with the outer release manifest and SHA-256 integrity controls. The main paper PDF and manuscript Markdown are intentionally not duplicated inside this package. TUS_OS_Paper_4_Technical_Volume_II_v1.0_SHA256.txtExternal integrity record binding the main public PDF and public verification package. SHA-256 — main PDF84f2defdc54fc0c50c1ac08ae6eade2e602f307e554900791f28a712c046c56b SHA-256 — public verification package94b7d79ab6ee946c4b0895da41f46cc8ef6bed1bb9762074263a161abea11a19 TUS OS® is a trademark of Mark Whitlock.