With α′ⱼ=αⱼ+χⱼ₊₁−χⱼ, all frame angles cancel in the loop sum. The U(1) holonomy is W=exp(iΦ), whose real and imaginary trace components are cosΦ and sinΦ.
Try the demonstration
Loop phase Φ = 120° modulo 360°. W = exp(iΦ); Re tr W = -0.5, Im tr W = 0.866. Frame-change sum difference = 0°. Changing χ changes the links but leaves the closed-loop holonomy invariant.
| Link | Original α | After frame change α′ |
|---|---|---|
| 1 → 2 | 30° | 30° |
| 2 → 3 | 45° | 45° |
| 3 → 4 | -15° | -15° |
| 4 → 1 | 60° | 60° |
| Sum | 120° | 120° |
Exact Abelian link and frame arithmetic on a supplied loop. No gauge-field dynamics, coupling selection, non-Abelian evolution, or physical temporal holonomy is derived.
Separate the physical links from their description
The first four controls set oriented link phases around the displayed loop. Changing these values changes the loop itself. The next four controls change the local frame phases χ at its vertices. They change how each link is represented while preserving the closed-loop result.
The table shows original and transformed link angles before reduction modulo a full turn. The live summary displays the loop phase modulo 360°, both trace components, and the cancellation error. This makes an invariant quantity visible alongside the description-dependent data used to calculate it.
Derive the endpoint rule
Let a component at vertex j transform as ψ′ⱼ=exp(iχⱼ)ψⱼ. The link from j to j+1 is Uⱼ=exp(iαⱼ). Requiring the transported component to transform consistently gives:
α′ⱼ=αⱼ+χⱼ₊₁−χⱼ.
At the fourth link, the arrival frame is the first vertex again. Adding all four equations makes the χ contributions telescope to zero. This is exact endpoint cancellation, not a statistical average over frames.
What the trace means for U(1)
The loop product is W=U₄U₃U₂U₁=exp(iΦ), where Φ is the sum of link angles. A one-dimensional U(1) matrix has trace equal to itself. Therefore the full trace is complex:
The central plot labels the real part, while the summary retains both components. CosΦ alone cannot distinguish Φ from −Φ. A coherent two-arm comparison with a reference can yield probability (1+cosΦ)/2, but that additional instrument is not implemented by merely computing the loop.
Try a local frame shift
The initial link angles are 30°,45°,−15°,60°, summing to 120°. Their holonomy has real trace −1/2 and imaginary trace √3/2. Set only the first frame to χ₁=30°. The first link becomes 0° and the fourth becomes 90°, while the other two stay unchanged. The sum remains 120°.
Next change a link angle itself. The holonomy generally changes because this modifies the comparison data rather than just their local description. Reset restores all original links and zero frame angles, allowing the two operations to be compared repeatedly.
A retained comparison in SMM/DTT
The construction illustrates how completed records can support relational phase information without a preferred local basis. A source model would need to generate the carriers, links, and a physical comparison instrument. The six nested inner roles could organize those operations; outer completion would retain the measured loop result.
Gauge invariance does not by itself create curvature energy, select a gauge group, or fix a force strength. The four-link example contains no dynamical field equation. Its exact cancellation is a consistency test that a more complete construction should preserve. The default SVG and numerical table remain available without interactive scripting.
Continue with the articles
- Relational transport and gauge freedom
Explain why independently unframed record fibers require a rule for comparison. Introduce gauge transport as a conditional reconstruction idea, distinct from the draft’s exact Hamiltonian results.
- An exact singlet-loop response
Introduce the four-position mobile carrier and stationary dual used in the solvable model. Explain how the holonomy-dependent resolvent determines the ground-state response under the supplied Hamiltonian.
- Invisible holonomy in inaccessible sectors
Explain the draft’s classification of residual parallel monodromy invisible to the prepared interface. Nontrivial structure can survive outside the behavior an experiment can access.