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.
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 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.
We study how much of the Hilbert-space and Born-rule structure of quantum theory can be obtained from operational primitives that involve records but not probabilities. The setting is a regime in which coherent alternatives recombine without leaving a stable record. We obtain three results of different kinds. Geometry. From composable contributions, record-inert reversible controls and amplitude scalings, under named conditions, we obtain a real inner-product event geometry. The existence of the inner product is equivalent to bounded mixing and is assumed in that form. In a worked interferometer with one mode per channel, the metric is not an input: metric-free port swaps and phase flips fix the form up to scale, and boundedness only forces the other controls to preserve it. Its uniqueness up to scale, the orthogonality of distinct outcomes, and the representation of events by orthogonal projections are derived. Obstruction. This geometry, even with full symmetry about the preparation and exact dyadic standards, does not select the quadratic weight: the preparation mixed with the maximally mixed state survives. The structural operations exhibited here certify only values on which the quadratic weight equals the counting weight. On restricted event families where Gleason’s theorem is unavailable, a calibration order oriented by the quadratic magnitude selects it: given exact standards, a state is the quadratic one exactly when it respects all such brackets. Whether such an orientation can be earned is the open problem. Given such brackets and refinement only to a finite depth K, every admissible state lies within 2^-K of the quadratic weight, without the Archimedean step. On the full projection lattice, Gleason’s theorem with null support, or sphere-transitive symmetry with null support, gives other conditional routes. Bridge. A predictive probability kernel is introduced separately. Its identification with the real quadratic weight ||P_A ψ||² / ||ψ||² is conditional on kernel-admission hypotheses, one of which is equivalent to the Born value; the complex form needs phase invariance in addition. The hypotheses select a class of contexts, and none is shown to be forced. For each we state the physical feature it expresses and a sufficient condition where one is known, and we work out one apparatus in which the geometry-stage hypotheses hold together. For all but one we give a context in which it fails (for the founding conditions, one that leaves the regime); for the remaining one, TCC-1, we give an illustration. This paper is the first in a series built on a common framework, Structural Ledger Consistency (in preparation).
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 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.
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.
Complementarity can organize a relational starting point for quantum reconstruction, but a completion relation does not yet supply preparations, effects, probabilities, continuous trans- formations, composites, or a scalar field. We formulate the missing passage as an assumption-control problem. Beginning from the typed primitive C= [a⊣b], we separate incidence, operationalization, continuity, scalar coherence, transition probability, effects, instruments, composition, purification, and process identification. Every negative reconstruction bridge is tied to an exact premise package and a typed target; geometric or pre-operational witnesses are not promoted into countermod- els of a larger theory they do not instantiate. At exact complex-closure resolution, let AOPT denote the finite-dimensional causal operational framework of an external reconstruction theo- rem, L local equivalence, and E equivalent-system purification. Finite classical theory witnesses AOPT + L without E, while finite-dimensional real quantum theory witnesses AOPT + E without L; the external theorem closes only AOPT + L+ E. Neither selector is derived from primitive complementarity here. The real-quantum stress test is sharpened by the two-rebit decomposition Sym(4) = Sym(2) ⊗Sym(2) ⊕R(J ⊗J): product effects see a nine-dimensional shadow of a ten- dimensional carrier. The identity setting together with one additional algebraically transverse joint-interaction setting gives complete terminal-product state exposure. This later observability remains distinct from original local tomography, universal process tomography, and complex-field selection. The contribution is a source-generated dependency architecture, exact finite witnesses, and a precise selector-origin problem: complex closure is formally conditional, while the physical origin of local equivalence and equivalent-system purification remains open.