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
A covariant difference around a discrete circle can be put into an axial form with a single residual holonomy. In an eigendirection of that holonomy, the boundary condition carries a phase exp(iθ). With N sites, length L and spacing aτ = L/N, the corresponding finite Laplacian eigenvalues are 4aτ−2sin²[(2πk + θ)/(2N)].
The explorer compares one twisted fiber with an untwisted fiber on the same circle. In the adjoint setting, centralizer directions can be untwisted even while other directions carry a twist. The zero shown for the untwisted k = 0 mode is a statement about this auxiliary fiber operator. It is not the identification of the physical vacuum, nor the spectrum of a physical translation-neutral Hamiltonian. [Vacuum obstruction, v2]
Run the calculation
Twisted auxiliary-cycle spectrum
Compare a holonomy-twisted fiber with the untwisted fiber on the same discrete circle. Let the twist approach zero to see why compactness alone does not supply a uniform physical gap.
L is in a chosen length unit; aτ = L/N. Eigenvalues are inverse square length in that unit. N counts cycle sites in this model.
Twisted fiber minimum 0.3597796; untwisted fiber minimum zero, in inverse square chosen length units.
- Twisted fiber minimum λ
- 0.3597796
- Untwisted fiber minimum λ
- 0
- Cycle spacing aτ
- 0.1428571
- Continuum fiber comparison θ²/L²
- 0.36
| Mode k | Twisted λ | Untwisted λ |
|---|---|---|
| 0 | 0.3597796 | 0 |
| 1 | 43.68166 | 36.898 |
| 2 | 127.9064 | 119.8071 |
| 3 | 189.611 | 186.2949 |
| 4 | 182.3306 | 186.2949 |
| 5 | 111.5476 | 119.8071 |
| 6 | 30.56297 | 36.898 |
| Fraction of chosen twist | Twist / rad | Twisted minimum λ |
|---|---|---|
| 1 | 0.6 | 0.3597796 |
| 0.5 | 0.3 | 0.08998623 |
| 0.1 | 0.06 | 0.003599978 |
| 0.01 | 0.006 | 0.000036 |
| 0 | 0 | 0 |
The twisted minimum tends to zero as the holonomy twist does. The untwisted k = 0 fiber mode is exactly zero throughout. These labels do not identify charged and neutral character sectors of the quantum Hamiltonian, and this fiber zero is not the physical vacuum. A lower bound on selected twisted fibers needs an explicitly restricted holonomy window; physical coercivity also requires a common-space comparison, the vacuum puncture and control of sector mixing.
Equations and conventions used
aτ = L/N; λmin(θ) → θ²/L² as N → ∞ for |θ| ≤ π; λmin(θ) → 0 as θ → 0.
These are eigenvalues of a finite auxiliary fiber Laplacian. The untwisted zero mode is not identified with the physical vacuum or the spectrum of the translation-neutral quantum Hamiltonian. N counts regulator sites, not additional observable time axes. In JSON results, the retained field names charged and neutral mean the twisted and untwisted fibers respectively; they do not identify physical charge sectors.
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 fixed N and L, reduce θ toward zero. Compare the exact minimum with θ²/L².
- Choose zero twist and identify the exactly zero k = 0 mode. Other modes need not vanish.
- Keep L fixed and increase N. The low twisted mode approaches its continuum-circle value while the high discrete modes continue to depend on the regulator.
- Double L while keeping N and θ fixed. Every eigenvalue scales down by a factor of four.
The mode label k follows the discrete Fourier branch and is not an ordering by energy. The table retains that label, so the spectrum is not silently sorted when θ changes sign. The lowest value is calculated separately from all modes.
Interpret the result
For |θ| ≤ π, the lowest branch has λmin ≈ θ²/L² at fine spacing. This tends to zero as θ tends to zero. A bound valid only away from identity holonomy therefore cannot become a global mass floor without an argument restricting and controlling the relevant configurations.
Physical states live in a larger Hilbert space with a Hamiltonian, vacuum and interactions. Multiplication by a pointwise auxiliary eigenvalue, an isolated finite fiber matrix and the physical transfer generator are different operators. The cycle derivation, sector guide and mass-gap article explain what further estimates would be needed.
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.
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Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.