Assumptions, results and research status
An explicit finite mathematical example with declared inputs, conventions and scope.
Assumptions
- The operator, state family or representations described on this page are supplied as model inputs.
- Every exported quantity uses the units and conventions printed beside the calculation.
Results
- The default example, tables and equations remain readable without JavaScript.
- With JavaScript, changing valid inputs recomputes the declared model and enables numerical downloads.
Open questions
- What additional source dynamics and physical matching would turn this example into a prediction of a specified physical system?
The declared model
The state ψ(φ) = (√(1−b),√b eiφ) describes a latitude in a two-dimensional complex carrier, with b = sin²(θ/2). The experiment is specified as a sequence of orthogonal projections onto this moving state at N equally spaced phases. The forward overlap is ⟨ψ(φ + 2π/N)|ψ(φ)⟩ = 1 − b + b e−2πi/N.
The product of these overlaps gives an accumulated phase and a survival amplitude. Its squared modulus is the retained probability. The continuous parallel-transport phase is −2πb, while the finite product has a phase error and a probability loss. The sign follows the actual forward overlap, not a separately chosen inverse-frame convention. [Sevenfold reconstruction, finite projection example, pages 36–37]
Run the calculation
Finite projections and geometric phase
Follow a rank-one state around a latitude by successive orthogonal projections. Compare the retained probability and phase with the continuous parallel-transport limit.
Angles are radians. N is the number of numerical projection steps around the loop; it does not count the six inner generative levels.
Survival probability 0.6291609; phase error 0.01528674 radians.
- Survival probability
- 0.62916094
- Finite phase modulo 2π / rad
- -1.55550959
- Continuous phase −2πb / rad
- -1.57079633
- Phase error modulo 2π / rad
- 0.0152867382
The per-step overlap is 0.98097 − 0.095671i. The accumulated phase before reduction modulo 2π is -1.55550959 radians.
| N | Survival probability | Finite phase / rad (mod 2π) | Phase error / rad |
|---|---|---|---|
| 2 | 0.0625 | -1.22e-16 | 1.57079633 |
| 4 | 0.152587891 | -1.28700222 | 0.283794109 |
| 7 | 0.344568015 | -1.48756771 | 0.083228618 |
| 8 | 0.394243265 | -1.50783609 | 0.0629602406 |
| 16 | 0.62916094 | -1.55550959 | 0.0152867382 |
| 32 | 0.793413021 | -1.56700224 | 0.0037940821 |
| 64 | 0.890768214 | -1.56984952 | 0.000946806729 |
| 128 | 0.943809283 | -1.57055973 | 0.000236594747 |
The columns expose two different convergence scales: probability loss is generally order 1/N, while the phase error for this symmetric latitude sampling is order 1/N² away from singular overlaps. Seven samples here are a numerical choice; the temporal model’s sevenfold structure is discussed separately.
Equations and conventions used
⟨ψ(φ + 2π/N)|ψ(φ)⟩ = 1 − b + b e−2πi/N.
Psurvive = [1 − 4b(1 − b)sin²(π/N)]ᴺ; γN = −N arg(1 − b + b e2πi/N); γ∞ = −2πb.
A prescribed family of rank-one projectors and a specified postselected sequence. The survival probability is part of the model, and a zero overlap leaves the phase undefined. This example supplies no spacetime dynamics.
Downloads include the chosen inputs, units and model scope. Values use standard floating-point arithmetic. Read the source and its version notes.
Try these comparisons
- At θ = π/3, compare N = 7, 16 and 32. The continuous phase is −π/2, while the survival probability approaches one.
- Switch the chart to phase error. Read phase differences modulo 2π so a branch cut is not mistaken for a physical jump.
- Choose the zero-overlap preset θ = π/2, N = 2. The probability is zero and the phase is undefined.
- Choose either pole θ = 0 or θ = π. The projector is constant, and the closed-loop retained probability is one. The displayed continuous representative can differ by a full 2π without a physical phase difference.
The table makes clear that finer sampling changes both the numerical phase and retention. Keeping only the phase while silently renormalizing every projected state would hide the postselection cost.
Interpret the result
For nonzero overlaps the accumulated representative is −N arg(1 − b + b e2πi/N). The phase column reduces that value modulo 2π; the comparison error is also measured modulo 2π. A zero complex amplitude has no defined argument, so the singular case is reported explicitly rather than assigned a spurious phase.
This is a prescribed projector loop. Its parameter φ is not automatically an observed spacetime coordinate, and N is the number of numerical projection steps, not the six inner generative levels. The geometric-phase guide connects it to projected differentiation and curvature. The bundle guide states when moving eigenspaces furnish the required projectors.
Reproduce or extend the calculation
Download JSON to retain the inputs, units, model scope and full computed results. Download CSV for the displayed numerical family or ledger; its opening rows record the input settings and definitions. The tool uses local browser calculations and ordinary floating-point arithmetic.
For an independent check, use the worked example in the exercise collection and the linked source manuscript. A useful extension states which input or assumption changes and which comparison would test the change. See the research collaboration page for focused starting questions.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.