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
Work in ℝ³ with P=diag(1,1,0), Q=diag(0,0,1), Ω=(√p,0,√(1−p)), v=(√(1−p),0,−√p) and w=(0,1,0). Rotate the two excitation vectors by angle α to obtain u=cosα v+sinα w and z=−sinα v+cosα w. The transfer matrix is T=|Ω⟩⟨Ω|+t₁|u⟩⟨u|+t₂|z⟩⟨z|, where 0<t₁,t₂<r<1.
The corrected threshold form is G=rI−T+(1−r)|Ω⟩⟨Ω|. It vanishes on Ω and has positive excitation eigenvalues a=r−t₁ and b=r−t₂. Its low Schur form and punctured low projection are
P°=diag(0,1,0).
The model extends the vacuum-completion obstruction discussed in Vacuum obstruction and form Feshbach reduction, v2. The derivation is in the nontrivial-puncture guide. The earlier two-state explorer remains useful for seeing why the puncture is necessary.
Run the calculation
A positive coefficient on a nonzero punctured window
A rank-two low window contains part of the vacuum and one further direction. Complete the vacuum in the high window, then inspect the positive Schur coefficient on the remaining direction.
p, transfer eigenvalues, threshold r, matrices and Schur coefficients are dimensionless. α is radians. Energies are inverse declared transfer-time steps, with one step set to one.
Punctured low-space rank one. Positive Schur coefficient 0.4; toy excitation gap 0.6931472 per transfer step.
- Punctured low-space rank
- 1
- Positive Schur coefficient c
- 0.4
- Corrected high block D
- 0.225
- Norm of high-block inverse
- 4.44444444
- Toy excitation gap / step⁻¹
- 0.693147181
- Threshold energy / step⁻¹
- 0.223143551
The exact low Schur form is diag(0, 0.4). Its zero direction is the projected vacuum. The remaining coefficient lies between 0.3 and 0.6, the two positive excitation margins of G.
| Row | 1 | 2 |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0 | 0.4 |
| Quantity | Value |
|---|---|
| High vacuum component | 0.707106781 |
| Reconstructed high component | 0.707106781 |
| Direct high transfer compression QTQ | 0.675 |
| QTQ < r? | Yes |
| Logarithmic energy margin / step⁻¹ | 0.470003629 |
Inspect the transfer matrix and corrected blocks
| Row | 1 | 2 | 3 |
|---|---|---|---|
| 1 | 0.675 | -0.10606602 | 0.325 |
| 2 | -0.10606602 | 0.35 | 0.10606602 |
| 3 | 0.325 | 0.10606602 | 0.675 |
| Row | 1 | 2 | 3 |
|---|---|---|---|
| 1 | 0.225 | 0.10606602 | -0.225 |
| 2 | 0.10606602 | 0.45 | -0.10606602 |
| 3 | -0.225 | -0.10606602 | 0.225 |
| Row | 1 | 2 |
|---|---|---|
| 1 | 0.225 | 0.10606602 |
| 2 | 0.10606602 | 0.45 |
| Entry | Value |
|---|---|
| B₁ | -0.225 |
| B₂ | -0.106066017 |
| D | 0.225 |
| Row | 1 | 2 |
|---|---|---|
| 1 | 0 | 1.39e-17 |
| 2 | 1.39e-17 | 0.4 |
Read the rotation samples
| α / radians | Schur c | High block D | Cross-block norm |
|---|---|---|---|
| 0 | 0.3 | 0.3 | 0.3 |
| 0.19634954 | 0.305819786 | 0.294290965 | 0.29707692 |
| 0.39269908 | 0.323702572 | 0.278033009 | 0.287971099 |
| 0.58904862 | 0.354747946 | 0.253701257 | 0.271968387 |
| 0.78539816 | 0.4 | 0.225 | 0.248746859 |
| 0.9817477 | 0.458484852 | 0.196298743 | 0.219398432 |
| 1.1780972 | 0.523356251 | 0.171966991 | 0.187610357 |
| 1.3744468 | 0.578001141 | 0.155709035 | 0.160912485 |
| 1.5707963 | 0.6 | 0.15 | 0.15 |
Inspect numerical identity residuals
| Check | Residual |
|---|---|
| basisOrthogonality | 4.00e-16 |
| transferVacuum | 3.14e-16 |
| correctedVacuum | 3.93e-17 |
| transferDefinition | 2.15e-16 |
| schur | 1.96e-17 |
| schurVacuum | 9.81e-18 |
| vacuumReconstruction | 2.22e-16 |
| determinantIdentity | 0 |
The abstract transfer operator has positive eigenvalues, but its coordinate matrix is not assumed to be a positivity-improving Markov kernel. A positive finite Schur coefficient and a logarithmic excitation gap are different quantities; physical time calibration and limiting estimates remain separate tasks.
Equations and conventions used
D = p(a cos²α + b sin²α); FP(G) = diag(0,c).
c = ab/(a cos²α+b sin²α); min(a,b) ≤ c ≤ max(a,b).
P° = diag(0,1,0); toy gap = −log(max(t₁,t₂)); threshold energy = −log r.
A supplied three-state transfer operator with positive eigenvalues. Its abstract coordinate matrix need not be entrywise positive or a Markov kernel. The rank-one punctured low result does not imply a continuum or thermodynamic-uniform physical mass gap.
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
- Use the default parameters p=1/2, t₁=0.2, t₂=0.5, r=0.8 and α=π/4. Check the high block D=0.225 and punctured coefficient c=0.4.
- Choose a small vacuum weight p=0.01. The high block becomes smaller, but the exact punctured coefficient stays 0.4. Inspect the high-block inverse and the raw matrix residuals.
- Choose α=0 and then α=π/2. The coefficient moves between b and a, while the transfer eigenvalues remain 1,t₁,t₂.
- Set t₁=t₂. The coefficient becomes r−t₁ and is independent of the excitation rotation.
All valid controls keep the two excited transfer eigenvalues below r. If that inequality is violated, the explorer pauses and identifies that the visible results use the last valid settings.
Understand the calculated quantities
At the default point, the exact low Schur form is diag(0,0.4). Its first direction is the projected vacuum. Removing that direction leaves rank one, so the coefficient 0.4 is a nonvacuous positivity statement within this finite model.
The toy excitation energy is −log(max(t₁,t₂)) per declared transfer-time step. The threshold energy is −log r. These are logarithmic energies; the Schur coefficient is instead a dimensionless margin for G. At the default point the energies are approximately 0.69314718 and 0.22314355, respectively. Their difference is approximately 0.47000363, which is not the same quantity as c=0.4.
Scope and interpretation
At small p the direct high-window compression QTQ can exceed r even while the corrected high form QGQ remains positive. The added vacuum term matters because the vacuum is not contained entirely in either window. This is why the block calculation starts from the fully corrected form.
The exact coefficient is independent of p within 0<p<1, while the norm of the high-block inverse depends on p. A finite identity and a uniform inverse estimate are therefore separate claims. The threshold counterexamples and proof roadmap explain the further estimates needed in a regulated field theory. No continuum or thermodynamic-uniform mass gap is inferred from this matrix example.
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.