Assumptions, results and research status
An exact finite example with declared preparation, operators, parameter domain and numerical diagnostics.
Assumptions
- The state, projector or transfer operator specified on this page is a model input.
- Numerical results use ordinary floating-point arithmetic in the stated units.
Results
- The default matrices and tables are available without JavaScript.
- Interactive results and exports implement the stated finite formulas.
Open questions
- Which source dynamics and physical observable correspondence justify applying this example to a specified system?
The declared model
Supply two real coordinates x,y and a real embedding parameter β. Define Z=diag(x+iβy,x−iβy), s=1+x²+β²y², u=[I₂;Z]/√s and P=uu†. The frame has two orthonormal columns in ℂ⁴. With a positive energy scale Δ, the Hamiltonian H=Δ(I₄−P) has the exact spectrum 0,0,Δ,Δ throughout the displayed domain.
The projected connection is ∇=P d. In the displayed frame its potential A=u†du is diagonal, with entries ±iβ(x dy−y dx)/s. Its curvature is
−Tr(Fxy²)=8β²/s⁴.
The geometry is on the supplied parameter chart. This worked construction develops the protection-versus-reconstruction issue in Sevenfold reconstruction, v18, PDF pp. 24–27 and the projector-curvature derivation. It does not identify x and y with measured spacetime coordinates.
Run the calculation
A fixed spectrum with changing projector curvature
Change the rank-two graph embedding and inspect its geometry. The Hamiltonian spectrum remains 0, 0, Δ, Δ; changing Δ alone leaves the projector and curvature unchanged.
x, y and β are dimensionless embedding parameters. Δ uses a chosen model energy unit. Loop eigenphases are radians; curvature uses the displayed coordinate chart.
Spectrum 0, 0, 1, 1. Curvature density 8; normalized loop trace 1.
- Exact Hamiltonian spectrum
- 0, 0, 1, 1
- Coordinate curvature density −Tr(Fxy²)
- 8
- Signed coefficient 2β/s²
- 2
- Trace of curvature
- 0
- Circle radius R
- 0
- Normalized loop trace
- 1
The two diagonal curvature entries have equal and opposite imaginary parts. The counterclockwise coordinate circle has eigenphases 0 and 0 radians, before reduction modulo 2π.
| Object | First diagonal entry | Second diagonal entry |
|---|---|---|
| Aₓ | 0 + 0i | 0 + 0i |
| Aᵧ | 0 + 0i | 0 + 0i |
| Fxy | 0 + 2i | 0 − 2i |
| Loop W | 1 + 0i | 1 + 0i |
Inspect the frame, projector and Hamiltonian
| Row | 1 | 2 |
|---|---|---|
| 1 | 1 + 0i | 0 + 0i |
| 2 | 0 + 0i | 1 + 0i |
| 3 | 0 + 0i | 0 + 0i |
| 4 | 0 + 0i | 0 + 0i |
| Row | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 1 + 0i | 0 + 0i | 0 + 0i | 0 + 0i |
| 2 | 0 + 0i | 1 + 0i | 0 + 0i | 0 + 0i |
| 3 | 0 + 0i | 0 + 0i | 0 + 0i | 0 + 0i |
| 4 | 0 + 0i | 0 + 0i | 0 + 0i | 0 + 0i |
| Row | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 0 + 0i | 0 + 0i | 0 + 0i | 0 + 0i |
| 2 | 0 + 0i | 0 + 0i | 0 + 0i | 0 + 0i |
| 3 | 0 + 0i | 0 + 0i | 1 + 0i | 0 + 0i |
| 4 | 0 + 0i | 0 + 0i | 0 + 0i | 1 + 0i |
Read the curvature samples
| x | Signed coefficient | Coordinate density |
|---|---|---|
| -4 | 0.00692041522 | 0.0000957842938 |
| -3.2 | 0.0158305999 | 0.000501215787 |
| -2.4 | 0.0437659746 | 0.00383092106 |
| -1.6 | 0.157808358 | 0.0498069554 |
| -0.8 | 0.743604997 | 1.10589678 |
| 0 | 2 | 8 |
| 0.8 | 0.743604997 | 1.10589678 |
| 1.6 | 0.157808358 | 0.0498069554 |
| 2.4 | 0.0437659746 | 0.00383092106 |
| 3.2 | 0.0158305999 | 0.000501215787 |
| 4 | 0.00692041522 | 0.0000957842938 |
Inspect numerical identity residuals
| Check | Residual |
|---|---|
| frameOrthonormalityError | 0 |
| projectorIdempotenceError | 0 |
| projectorHermiticityError | 0 |
| protectedEnergyError | 0 |
| spectralPolynomialError | 0 |
| connectionXError | 0 |
| connectionYError | 0 |
| projectorCurvatureError | 0 |
Changing Δ rescales the Hamiltonian while preserving this model's frame, connection and curvature. The loop is a specified coordinate circle, and its radius is derived from the selected x,y point. The connection lies in a diagonal Abelian subgroup of SU(2).
Equations and conventions used
A = diag(iβ(x dy−y dx)/s,−iβ(x dy−y dx)/s).
Fxy = diag(2iβ/s²,−2iβ/s²); −Tr(Fxy²) = 8β²/s⁴.
For R=√(x²+y²), flux = 2πβR²/√[(1+R²)(1+β²R²)]; W = diag(exp(−i flux),exp(i flux)).
An exact diagonal connection on a supplied parameter chart, with Abelian holonomy inside SU(2). The parameters are not identified spacetime coordinates. The curvature density uses this chart and trace normalization.
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 x=y=0 and β=1, verify the curvature coefficient 2 and the density 8. Change only Δ: neither geometric value changes.
- Choose β=0. The graph still defines the same spectral multiplicities, but its curvature is zero everywhere.
- Reverse β. The oriented curvature and holonomy phases reverse, while the nonnegative curvature density stays the same.
- Choose the radius-one circle with β=1. Its integrated connection has flux π, and forward holonomy is −I₂.
The chart varies x from −4 to 4 at the current y and β. The local matrix results use the selected point. The loop is separately specified as the counterclockwise coordinate circle centered at the origin with radius √(x²+y²); it is not an arbitrary path through that point.
Understand the calculated quantities
For that coordinate circle, the exact flux is 2πβR²/√[(1+R²)(1+β²R²)]. Since the displayed connection is diagonal, its matrices commute along the loop and no nontrivial path ordering is needed. With D=d+A, the parallel holonomy is diag(exp(−i flux),exp(i flux)). The explorer reports the unwrapped eigenphases and the normalized trace cos(flux).
The curvature density is invariant under unitary changes of frame within the retained carrier. Its value also uses the chosen x,y coordinates and trace normalization; a change of coordinate scale changes a coordinate component of a two-form. It is not presented as a coordinate-independent physical energy density.
Scope and interpretation
The example has a diagonal, Abelian connection inside SU(2). It supplies a concrete counterexample to reconstructing curvature from an eigenvalue list alone; it does not derive general non-Abelian gauge dynamics. The gap is deliberately a separate input, so changing it cannot select the embedding in this model.
The exported matrix residuals check orthonormality, idempotence, H²=ΔH and agreement between the frame and projector expressions for curvature. Their small floating-point values support the implementation of the declared identities; the proof is given in the article. For transport along a general supplied projector path, continue to Kato transport.
Reproduce the result
Download JSON to retain the complete model inputs, matrices, units, analytic quantities and numerical diagnostics. Download CSV for the chart's numerical family; its opening rows record the inputs and definitions. The default example remains readable with JavaScript disabled.
The worked exercises provide hand-checkable values. The package also includes a verifier that can be run independently with Node.js. For a research contribution, use the research note template and state which part of the model you have checked or changed.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.