REPRODUCIBLE MODEL · QUANTUM BASELINE
What changes an interference fringe?
Separate relative phase, retained coherence and detector probability. Then ask what an additional temporal interpretation would have to change to become experimentally distinguishable.
Updated · Balanced two-path model
Scope: this is the standard density-matrix calculation for an ideal balanced two-path readout. A dual-time interpretation alone does not imply a new empirical prediction. All displayed values are calculated model values.
Change phase and coherence
The default example below is complete. Interactive controls become available when the script loads.
At φ=60° and V=0.80, P₀=0.7, P₁=0.3 and P₀+P₁=1. The corresponding incoherent probabilities are 0.5 and 0.5.
Curves join samples calculated every 2°. The marked phase and numerical outputs are calculated directly from the formula.
γ=0.4+0.69282i. Density-matrix eigenvalues: 0.9, 0.1; purity Tr(ρ²)=0.82.
| Phase | P₀ | P₁ | Sum |
|---|---|---|---|
| 0° | 0.9 | 0.1 | 1 |
| 60° | 0.7 | 0.3 | 1 |
| 90° | 0.5 | 0.5 | 1 |
| 180° | 0.1 | 0.9 | 1 |
| 270° | 0.5 | 0.5 | 1 |
| 360° | 0.9 | 0.1 | 1 |
Exports include the selected point and the calculated fringe samples. No information is uploaded.
Declare the state before interpreting it
Use two path states and balanced populations. Write the density matrix with a complex coherence γ. Its phase convention is fixed by the chosen path basis and readout:
Eigenvalues: (1+V)/2 and (1−V)/2.
The trace is one and both eigenvalues are nonnegative. V=1 gives a pure balanced state; V=0 removes the off-diagonal coherence while preserving equal path populations. Intermediate V gives a mixed state. The displayed purity is Tr(ρ²)=(1+V²)/2.
V summarizes coherence in this supplied state. It does not uniquely identify a physical mechanism: dephasing, uncontrolled preparation variation or an ignored environment can affect the effective state. Inferring a mechanism requires additional preparation and measurement information.
Derive the detector probabilities
The ideal balanced readout is the Hadamard matrix H=(1/√2)[[1,1],[1,−1]]. Compute HρH† and read its diagonal. With the present phase convention, the two detector probabilities are:
P₁ = (1−Re γ)/2 = (1−V cosφ)/2,
P₀+P₁=1.
For V=0.8 and φ=60°, cosφ=1/2, so P₀=0.7 and P₁=0.3. At V=0, changing phase has no effect on either detector: each has probability one half. At V=1, the ideal scan reaches complete constructive and destructive interference.
Holding V constant, Pmax=(1+V)/2 and Pmin=(1−V)/2. The usual ideal fringe visibility (Pmax−Pmin)/(Pmax+Pmin) is therefore V. Losses, unbalanced populations, detector bias and background counts are excluded here; an experimental comparison must model them explicitly.
Equal current populations can hide different futures
At V=1, phases 0° and 180° have the same path populations (½,½), yet their balanced-readout outcomes are (1,0) and (0,1). A record that retains only populations has forgotten the relative coherence needed for this continuation. The density matrix retains that predictive information.
This is a quantum counterpart of the finite predictive-memory laboratory. It distinguishes relative phase from global phase: multiplying a pure state vector by one common phase leaves its density matrix and all these probabilities unchanged.
What would make a dual-time proposal testable here?
A description in terms of inner generation and outer completion can interpret the process. If it yields exactly the same prepared state, readout and probabilities, this experiment cannot distinguish that description from the ordinary quantum model.
A stronger proposal should derive a quantitative restriction that survives independent calibration: for example, a specific dependence of coherence or phase on a declared source parameter, with its parameters frozen before testing a second protocol. The relation must follow from an explicit update and completion instrument; adding an arbitrary correction to this fringe would not establish it.
Specify the ordinary quantum comparator, nuisance parameters, uncertainty and the outcomes that would contradict the proposed difference. A fit to a single fringe with freely adjustable visibility and phase supplies no unique test of temporal ontology. See predictions and comparators and the source-model specification.
Continue the argument
The derivation above is self-contained linear algebra. The site’s measurement and completion guide examines how observations are made available; memory and repeatability distinguishes a repeated display from a repeated preparation. For the related published phase-selection framework, see DOI 10.1016/j.physleta.2026.132085. That connection does not make this baseline calculation a new prediction of the framework.
Export the current parameters with the model JSON, or use the CSV for a separate plot. The export records the model version, assumptions and formula conventions. Follow the reproducibility guide when reporting comparisons.