A partial-transpose image of the positive-semidefinite cone supplies an established alternative ordered presentation on a fixed composite carrier. Transporting both the cone and the processes preserves an abstract equivalence, whereas fixing the process representation can discriminate the two orders. We study the iteration of one specified right-argument transpose rule on supplied finite qubit carriers. With three standard leaves, its two bracketings have different cones, so no linear identification fixing every product operator pointwise can make them agree. Subset-orientation labels give an explicit repair. Relative to a separately supplied tensor-closed standard process category, we construct the raw tensor of arrows, associator, symmetry, asymmetric unit maps, and quasi-inverse tensorator, and prove symmetric monoidal equivalence by faithful transport. Complementary subsets define the same isolated cone, but their identification is not a congruence for the raw tensor product. At fixed rank, a precisely defined two-lift groupoid is a split central Z₂ extension; its odd lift is not an ordinary completely positive endomorphism at fixed representatives. A five-qubit example then separates matched reference transport, a fixed calibrated anchor, and an effect-based quotient. The latter loses a linear subspace rather than only a two-point transpose orbit and is preserved only by histories satisfying the standard kernel-invariance condition. A named real local unitary connects the anchor pair, while a real controlled-NOT (CNOT) gate exposes the limitation of an enlarged single-anchor history class. The contribution is an assumption-controlled analysis of one finite composition rule and its interfaces, not a new general coherence, orientation, or resource theorem, and not a derivation of quantum theory from primitive complementarity. Selected supporting statements and finite witnesses are accompanied by a partial Lean 4.19.0 formalization. A recorded local build and type/axiom audit accepted 117 theorem declarations and 57 definitions or abbreviations. The full symmetric monoidal category and equivalence in Theorems 4.5 and 4.6, and other explicitly listed manuscript links, remain incompletely formalized. The public source supplement also includes three preserved exact Python implementations, expected results, pinned dependencies and claim-to-code maps. Packaging and source/log audits are not another Lean compiler run or external peer review. The public evidence selection preserves the successful type/axiom output and the failure/success chronology. Machine-local account paths are omitted from derived run summaries, and one failed-build diagnostic has an explicitly normalized home-directory prefix. Accepted Lean proof sources remain unchanged; private review archives and original machine-local receipts are not public upload files.
A partial-transpose image of the positive-semidefinite cone supplies an established alternative ordered presentation on a fixed composite carrier. Transporting both the cone and the processes preserves an abstract equivalence, whereas fixing the process representation can discriminate the two orders. We study the iteration of one specified right-argument transpose rule on supplied finite qubit carriers. With three standard leaves, its two bracketings have different cones, so no linear identification fixing every product operator pointwise can make them agree. Subset-orientation labels give an explicit repair. Relative to a separately supplied tensor-closed standard process category, we construct the raw tensor of arrows, associator, symmetry, asymmetric unit maps, and quasi-inverse tensorator, and prove symmetric monoidal equivalence by faithful transport. Complementary subsets define the same isolated cone, but their identification is not a congruence for the raw tensor product. At fixed rank, a precisely defined two-lift groupoid is a split central Z₂ extension; its odd lift is not an ordinary completely positive endomorphism at fixed representatives. A five-qubit example then separates matched reference transport, a fixed calibrated anchor, and an effect-based quotient. The latter loses a linear subspace rather than only a two-point transpose orbit and is preserved only by histories satisfying the standard kernel-invariance condition. A named real local unitary connects the anchor pair, while a real controlled-NOT (CNOT) gate exposes the limitation of an enlarged single-anchor history class. The contribution is an assumption-controlled analysis of one finite composition rule and its interfaces, not a new general coherence, orientation, or resource theorem, and not a derivation of quantum theory from primitive complementarity. Selected supporting statements and finite witnesses are accompanied by a partial Lean 4.19.0 formalization. A recorded local build and type/axiom audit accepted 117 theorem declarations and 57 definitions or abbreviations. The full symmetric monoidal category and equivalence in Theorems 4.5 and 4.6, and other explicitly listed manuscript links, remain incompletely formalized. The public source supplement also includes three preserved exact Python implementations, expected results, pinned dependencies and claim-to-code maps. Packaging and source/log audits are not another Lean compiler run or external peer review. The public evidence selection preserves the successful type/axiom output and the failure/success chronology. Machine-local account paths are omitted from derived run summaries, and one failed-build diagnostic has an explicitly normalized home-directory prefix. Accepted Lean proof sources remain unchanged; private review archives and original machine-local receipts are not public upload files.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.
Quantum-reconstruction programmes normally begin after several structures have already stabilised: a system type, a tensor-product composition rule, an operational perspective, and a reference-frame regime. Quantum-reference-frame (QRF) research, by contrast, shows that precisely these structures may depend on physical perspective. We develop Relational-Perspectival Quantum Reconstruction (RPQR), an operational architecture that places perspective formation and subsystem composition upstream of a fixed-perspective quantum-reconstruction core. The primitive data are complete candidate situations, physically implementable protocols, outcome statistics, sequential composition, and admissible multi-fragment gluing. An informational perspective is the quotient induced by the protocols available in that perspective. We prove a quotient-factorisation theorem defining an information order, then separate information perspectives from stronger process perspectives carrying actual statistics-preserving simulator functors. Coherent simulator data yield an indexed operational theory and hence a fibration. Composition is not assumed primitive: a perspective-local tensor product is introduced only when admissible multi-input gluing is representable; preservation of the corresponding universal gluing is the criterion for strong monoidality of a perspective change. Finite reference, clock and control resources are compared only after transport to a common operational comparison theory, allowing ideal backgrounds to be represented as context-stable completion limits. A fixed, resource-stable, representable sector then conservatively recovers the operational signature required by a companion fixed-perspective reconstruction, without deriving its independent quantum-selection principles. The architecture is shown to be nontrivial by an explicit three-qubit Z2 QRF realisation based on the extra-particle framework. Two invariant global states are identical in a naive Alice-relative description but orthogonal in Bob’s relative description. The missing information is identified with the central charge algebra C ⊕ C, providing a finite transition-remainder witness. The chargeaware QRF transformation induces, after an explicit operational-saturation model specification, a typed equivalence of whole process fibres. Each perspective contains a preferred representable compositional sector, while the preferred subsystem decomposition is perspective dependent. The QRF is an exact finite-resource witness, and fixing a charge sector yields an ordinary 𝑀4 (C) quantum subsector in which the fixed-perspective reconstruction signature is recovered. Thus RPQR possesses an explicit finite nontrivial model realising the full T1–T5 architecture. We also state falsifiability diagnostics and delimit the claims: no universal information-to-process lifting theorem, no universal cross-perspective tensor covariance, and no new quantum dynamics are assumed.
A supplied rank-two Complementarity-First carrier has a positive duality h and a canonical complement #. The companion time paper used the local twist h(x,y^#) to obtain an index-one form. Here the same complement acts on one leg of the identity-pairing tensor, Γ_# = e ⊗ e - Σ_{i=1}^m e_i ⊗ e_i, with ω_# = (1/4)Γ_#. Relative to the supplied duality this construction is basis independent, product-positive, and normalized by the rank-two deterministic effect. It gives P(a,b | n,m) = (1/4)(1 - ab n·m). The m = 1 branch is separable; for m ≥ 2, physical admission with the sharp product interface gives CHSH magnitude 2√2. On the selected complex-qubit carrier, ω_# is the Bell singlet projector. The structural complement is positive but not completely positive. For a separately supplied exchange generator we derive G(X) = 2i[ω_#,X]. A partial-swap witness rejects one specific statically self-dual partial-transpose-image cone under the frozen process representation; the stronger carrier-relative PSD closure remains an inherited result. Entangled extensions of Euclidean-ball systems and canonical maximally entangled cone tensors are established prior art. The contribution claimed here is the typed local/tensor/process synthesis, not priority for singlet correlations, maximally entangled tensors, probability, or the physical composite.
How a finite real cycle acquires a complex structure and a rule for composition Quantum theory uses a common imaginary unit when individual systems become a composite. This framework constructs that compatibility from finite real source histories. On the complete contrast sector of a three-cycle, the forward and backward comparisons give J=3P−P2,J2=−I The complex structure appears after the real comparison is specified. At a qualified unmarked boundary, opposite reference shifts select QAB=2I−JA⊗JB The range is the balanced complex tensor product. Its many-source composition is associative. A finite controlled contact supplies a nonfactorizable complex-linear operation, and exact residual bounds quantify departure from the selected sector. A source model that produces the contact A reversible controller compares operation orders on 24 proper signed frames. Its ordered, arrow-insensitive axis detector produces the retained cyclic response. The contact algebra gives [[G,G],[G,G]][G,G]≅C3. The construction prints the controller, scratch cleanup, full contrast probes, admissible operations and refusal states. A triaxial real response tensor realizes the detector and provides an explicit map between material contrasts and event-count contrasts. A record must survive the test of future operations A present tensor can hide a material marker that a later operation reveals. The framework derives independent force-gradient calibration and a delayed odd-field release. With a nonzero release field, energy and vector torque reconstruct the entire marker change: DU=−Δp⋅E,Dτ=Δp×E.Δp=∣E∣2E×Dτ−DUE The test includes shared calibration errors, stray fields, finite release pulses, marker relaxation and a complete operation inventory. It distinguishes detection of a retained record from agreement with its frozen prediction and from discrimination between physical theories. From proved structure to existing experiments The paper adds a conditional probability-correspondence theorem for the balanced composite, examines independent-source quantum-network experiments, and reproduces all eight measured rows of a public torque dataset. These supply composition benchmarks and demonstrated apparatus capabilities. Their published significances are kept attached to the hypotheses they actually test. The calibrated material release law is also a conventional electrostatic prediction; it is not presented as a QTT-exclusive signal. One material coefficient, two readouts A new archival calculation transfers a published same-particle libration calibration to a separate precession measurement. In the qualified rotor regime, the cross-readout relation is Ξ=ωlib2EpωsΩElcosϕl=alibaprec. An unchanged dipole-to-inertia coefficient gives unity. The calculation yields a slope of 2.909 from the libration calibration, compared with the published precession slope of 2.950, a central difference of 1.4135%. This shared electrostatic consistency relation eliminates the separate dipole and inertia; unequal field gains, dynamical corrections and shared uncertainty remain explicit. Reversal and long-term dipole-stability observations identify complementary laboratory capabilities. The complete matched-tensor delayed-release test remains an experimental target. When do two descriptions differ in any future experiment? The finite-reference extension follows a controlled source contact through shared references, fresh references, inverse operations, intermediate measurements and feedback. Its complete order-three response is gn=∑k=02pkζkn,ζ=e2πi/3,g3=1,g2=g1. A finite observable-span algorithm tests two fully specified models across every permitted future word. It either certifies agreement or returns the preparation, operation sequence, terminal outcome and exact probability difference. In the populated compatible-instrument benchmark, the complete quantum representation gives the same records without an additional target fit. A source-selected physical coupling or preparation remains a distinct route to predictive value. The framework includes nine vector figures, the retained source proofs and countermodels, complete reference-history benchmarks, calibration and vector-reconstruction theorems, runnable exact and numerical checks, published-input receipts, a data assessment and an experimental packet specification. The introduction defines the QTT terms needed to follow the construction without prior familiarity with the corpus. The probability-interface theorem, finite-memory bounds and covariance-weighted witness specify which further laboratory comparisons can distinguish physical models. Source anchors QTT Main Book, especially pp. 56, 248-250, 352 and 827-829; Artian's Completed-Event Four-Capacity Theorem; Frame-cycle derivation map, completed-event lexicon, and Artian's Universe and laboratory readout. Stable citation: 10.5281/zenodo.22882084.
Quantum theory combines three structures that are often introduced separately: complex norm-preserving evolution, entanglement of composite systems, and quadratic weights for measurement outcomes. This paper asks whether a complementarity-first architecture can organize their relation without importing one structure under the name of another. The motivating image is the Taichi Diagram: two distinguishable sides and two internal “eyes,” with each side carrying a trace of its complement. The image formulates a question; it is neither a historical anticipation nor a physical proof. On a supplied two-branch positive composite, a fixed-eye state family separates diagonal reciprocal inclusion from an off-diagonal coherent link. Positivity bounds that link, and within the declared two-qubit support the state is entangled exactly when the link is nonzero. On a supplied real two-plane, complement exchange as reversal admits elliptic, hyperbolic, nilpotent, and frozen generators. A non-fixed two-sided bounded orbit selects the elliptic branch, a positive invariant quadratic capacity, and a one-frequency Schrödinger normal form. One such plane is only one complex line and is projectively trivial, so relative phase and an entangling sector require an explicit aligned two-block lift. On that four-real-dimensional carrier, the complex-linear branch swap S is distinguished from the conjugating reciprocal reversal Theta = S C. A supplied exchange/XY Hamiltonian is Theta-even; its Schrödinger generator is Theta-odd, its unitary flow is reciprocal under Theta, and it produces an exact orbit from separability through maximal entanglement and back. Separately, passive repeatable lossless separators become orthogonal projections, while a nonnegative modular readout axiom uniquely linearizes channel response and yields a finite pure-projective Born-form valuation. This result is conditional and does not establish the density-operator trace rule, general effects or instruments, one actual outcome, or objective chance. Environmental restriction suppresses accessible pair coherence by an overlap factor and explains how a larger entangled whole can cast a classical-looking local shadow. The contribution is an assumption-controlled synthesis and countermodel chain, including an explicit carrier bridge and a typed distinction between branch exchange and reciprocal reversal; the component two-qubit, planar-flow, exchange, projection, and additive-response formulas are not claimed as individually new.
A supplied bounded reciprocal plane supports compatible elliptic, metric and symplectic geometry, but it does not itself select a physical quantum system or an absolute action scale. This paper organizes an assumption-controlled finite architecture of cell size, convex polarity, composition, paired transport and operational access. It distinguishes amplitude and phase-space interpretations, vector complementation and polarity of regions, local cell projections and global body composition, and geometric holonomy and measured response. The mathematical analysis includes a conditional polarity selector, a forward projection/section identity for supplied symplectic ellipsoids, an explicit counterexample to a fixed-splitting converse for general ellipsoids, inverse-transpose paired transport, a common-connection criterion and reference-dependent response comparisons. No primitive derivation of a quantum state cone, tensor product, minimum action, physical instrument or probability law is claimed. The accompanying source-and-evidence supplement contains the manuscript source, three preserved Python exact-check programs, and a partial Lean 4.19.0 formalization. Audited records of a local run support acceptance of 99 theorem/lemma declarations and 33 definitions/abbreviations. The actual set-level geometry of the concrete ellipsoid counterexample is included; the full convex-lattice, capacity, projective, all-rank geometric and smooth-connection results are not all formalized. This is not full-paper formal verification, external human peer review or a new compiler run. All selected command streams and type/axiom output remain verbatim. One local-run receipt is provided as a clearly labelled path-normalized derivative, with its original hash retained. The full private lineage is preserved separately. The current reading edition is version 1.0; the unchanged version 0.3 materials are retained only as verification baselines.