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 explorer supplies a five-dimensional complex carrier and, when selected, a decomposition into ranks three and two. With only a combined Hermitian metric, U(5) can mix all five directions. Preserving the two blocks gives U(3) × U(2). A retained parallel product-volume tensor then gives S(U(3) × U(2)). Fixing the two volume tensors separately gives SU(3) × SU(2).
This example is deliberately explicit about its inputs. The group-reconstruction guide explains the determinant reference and why a global parallel tensor is stronger than zero trace curvature. The recurrence guide explains the additional assumptions behind the source carrier construction. The explorer calculates a stabilizer of supplied data; it does not derive those data by moving a control. [Sevenfold reconstruction, v18]
Run the calculation
Retained structure and the gauge group
Choose exactly which tensors are retained, inspect the stabilizer dimension, and test the global quotient condition for a representation.
Group dimensions and q = 6Y are integers. κ₃ and κ₂ are illustrative responses in a common chosen unit.
S(U(3) × U(2)), dimension 12. Product-volume quotient residue 0.
- Selected stabilizer
- S(U(3) × U(2))
- Real Lie-group dimension
- 12
- Tr Q² for the product-volume case
- 30
- 12κ₃ + 18κ₂ (chosen response unit)
- 30
The separate blocks and a parallel product-volume tensor are retained exactly.
| Retained choice | Group | Dimension | Selected |
|---|---|---|---|
| combined | U(5) | 25 | |
| blocks | U(3) × U(2) | 13 | |
| product | S(U(3) × U(2)) | 12 | Yes |
| separate | SU(3) × SU(2) | 11 |
Global representation test
For S(U(3) × U(2)), q + 2t₃ + 3t₂ = 6, with residue 0 modulo 6. This representation descends through the stated ℤ₆ quotient.
| Quantity | Value |
|---|---|
| q = 6Y | 1 |
| Color triality t₃ | 1 |
| Weak parity t₂ | 1 |
| q + 2t₃ + 3t₂ modulo 6 | 0 |
| Kernel action on this representation | 1 + 0i |
The center generator Q is fixed by the stated embedding. Its trace weights give the displayed linear combination of two supplied block responses. A physical identification still requires action normalization, matter conventions, scale matching and any quantum corrections.
Equations and conventions used
A representation descends through ℤ₆ iff q + 2t₃ + 3t₂ ≡ 0 (mod 6).
Q = diag(−2I₃,3I₂); Tr Q² = 30; κcentral = 12κ₃ + 18κ₂.
The rank-three and rank-two carriers are supplied inputs here. Their source-level reconstruction requires the assumptions in the articles. The response sliders do not determine measured gauge couplings.
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
- Compare all four retained-structure choices. Explain the sequence of dimensions 25, 13, 12 and 11.
- Select the product-volume group and test Q with q = 1, t₃ = 1, t₂ = 1. The total is six, so the residue vanishes.
- Choose the conjugate up-quark preset. Use triality two and verify −4 + 2(2) = 0.
- Change κ₃ or κ₂. The central response changes with weights twelve and eighteen, while the group dimension does not depend on these coefficients.
When a different group is selected, the tool still explains the product-volume quotient diagnostic but marks it as inapplicable to the current group. A quotient rule for one global group is not a universal restriction on every group with related factors.
Interpret the result
The quotient map sends (A,B,z) to (z⁻²A,z³B). Its kernel is generated by (e2πi/3I₃,−I₂,eiπ/3), so a representation descends exactly when q + 2t₃ + 3t₂ vanishes modulo six. The matter ledger uses q = 6Y; changing that normalization changes how the same condition is written.
The center generator Q = diag(−2I₃,3I₂) has Tr Q² = 30. Two independent block responses consequently give 12κ₃ + 18κ₂ on that direction. This trace calculation is exact for the stated convention. A physical coupling assignment requires the extra steps in couplings and mixing.
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.