The polar phase calculation and balanced tensor construction are exact conditional starting points. The open problem is to make their compatibility, observable algebra and completion rule consequences of one source dynamics.
From local phase to a coherent composite
The target is a source law whose admissible histories determine both phase-bearing carriers and their joint operation algebra. It should identify independently controllable subsystems, explain how their phase actions combine and specify which states and measurements are physically available.
This is a precise extension of SMM’s common-origin idea. Source unity does not itself choose a tensor product or every correlation between outputs. DTT adds a completion question: when a joint history becomes a record, does the interface retain coherent amplitudes, only classical labels, or additional memory? The answer must predict subsequent joint tests using the same retained state.
What real dynamics already supplies
For an invertible real skew-symmetric generator A, the positive operator Ω=(−A²)1/2 commutes with A. Its polar factor J=AΩ⁻¹ satisfies J²=−I and Jᵀ=−J. It therefore defines a compatible ordinary complex structure on the dynamic sector.
This selects phase for a supplied oriented flow. A stationary kernel requires separate treatment, and reversing A reverses J. The split-complex unit j²=+1 used for hyperbolic temporal pairing is a different structure. A source model must not exchange those units when moving from temporal interpretation to coherent quantum operations.
The balanced composition rule
Let real carriers V1,V2 have selected phases J1,J2. The ordinary complex tensor product identifies moving i between the two factors:
Imposing this balancing relation on the real tensor product recovers complex composition with the appropriate induced inner product. For complex dimensions m and n, the balanced space has real dimension 2mn. The raw real tensor product instead has real dimension 4mn. Local phase existence alone does not impose the quotient. The source needs an operational reason why the identified joint histories are indistinguishable under every admitted continuation.
Two phase planes and a joint test
Take each carrier to be one complex line, with real bases x,J1x and y,J2y. Balancing identifies the two singly rotated products. Applying the relation twice gives J1x⊗J2y=−x⊗y. Only two real basis directions remain, exactly one complex line.
With two complex two-level systems, the balanced rule permits the coherent state (|00⟩+|11⟩)/√2. Its expectation of X⊗X is one, whereas the equal incoherent mixture of |00⟩ and |11⟩ gives zero. Both yield matching labels in a Z test. A completion rule preserving those labels alone therefore fails to preserve every joint prediction. This supplies a simple discriminator for candidate source instruments.
A candidate reconstruction through admitted operations
Begin with a real source carrier and an oriented generator, then identify two commuting algebras of local operations through independently controllable preparations and tests. Derive their compatible complex structures from the shared dynamics. Define observational equivalence of joint histories using the full continuation language, not only an instantaneous readout.
Investigate whether this equivalence forces the balancing relation and whether repeated composition is associative. The six nested inner roles can implement preparation, routing, interaction and retention as dependent operations. Their number is not the number of tensor factors. An actual source model must show which records define subsystems and how joint operations act on their retained memory.
States, probabilities and completion need their own theorem
The polar calculation does not supply a probability rule. If the candidate model adopts ordinary positive density operators and completely positive instruments, say so and derive the consequences within that framework. A stronger reconstruction would explain why the admissible preparations and tests have those properties.
Completion must retain all outcomes in an unconditional description. A selected outcome can change conditional joint predictions, while a trace-preserving local channel cannot change the distant reduced state under the supplied tensor rule. A proposed inner common-source mechanism should preserve that distinction and identify the actual resources used to compare phase or export a record. Hidden classical correlations cannot substitute for the coherent joint test above.
The composition result that would move the programme forward
A successful theorem would derive one compatible phase and composition rule from declared source operations, classify any residual orientation or kernel choices, and predict product and entangled preparations with the same completion instrument. It should remain consistent when three or more subsystems are grouped in different orders.
A finite model that proves the balancing relation for an independently specified family of joint tests would already improve the current construction. A countermodel retaining local phase but violating that relation would locate the missing physical assumption. The composition guide supplies the exact algebraic examples; the open task is to explain their operational selection.
Continue with the articles
- Test languages and observable quotients
A test language determines which records an observer can distinguish. Enlarging that language refines the observable quotient and reveals when apparently different instruments retain the same information.
- Retained observations versus predictive states
A readable record need not contain enough information to predict its next update. An exact leakage criterion identifies when a linear completion is autonomous and measures the error when hidden information returns.
- A quantum source for generation and completion
A positive phase–dilation Hamiltonian ties one phase encoder to a powered coordinate and controlled work. It gives an explicit quantum realization of generation and completion while retaining its constitutive interfaces as inputs.
- Completion without biasing an unknown input
A factorized accepted channel gives the same success probability for every admitted unknown input, including inputs entangled with a reference. The failure channel and the amplification of errors by conditioning require separate checks.
- A phase-sensitive registration experiment
Registering a completion flag before or after further source evolution changes later phase-sensitive acceptance statistics. The proposed protocol measures that order effect and compares it with an ordinary qubit baseline.
- Completion, commutants, and positive frequency
This guide follows completion, compatible complex structures and positive-frequency selection through their separate mathematical roles. Finite examples connect the framework to SMM/DTT while keeping causal readout and physical calibration as additional constructions.
- Real-commutant phase selection: a paper guide
An oriented real orthogonal flow can select an ordinary complex structure on its dynamic sector. Planar examples explain residual zero-mode freedom, time-reversal antilinearity and the extra balancing rule needed for composite systems.