Finite-Order Port-Response Models and Chain Realizations
Node role: Concept.
Reader task: Specify a finite-order response target or derive its physical port-response model, then determine whether its coordinates have physical ownership and whether it admits a valid port-anchored chain realization.
What this node owns
This node distinguishes a finite-order response target from its physical port-response model, defines the port anchor, and states the extra conditions required to call one realization a Chain Model. The separate Hybridized-Circuit Subsystem Reduction and Design procedure owns the physical design, reduction, optimization, winner comparison, and Layout/EM handoffs.
It joins the physical analysis direction in Circuit Models to Bare Coordinates, Open EOM, and Normal Modes, which starts from a physical circuit and derives its finite coordinates and response.
For the derivation, read the Node/Line-Flux Input–Output Bridge first. This page owns realization names and physical ownership; it does not repeat the traveling-wave prefactors.
| Name | Meaning in this Knowledge Base |
|---|---|
| Finite-order port-response target | A desired finite-dimensional port response and its declared model class, without physical ownership of hidden coordinates. |
| Physical finite-order port-response model | A finite linear realization derived from a circuit, with declared representation, coordinate provenance, drive map, and observation map. Finite order is not a claim that an infinite distributed field has only \(N\) degrees of freedom. |
| Port-anchored basis | A coordinate basis whose first vector, or first block, is fixed by the subspace through which the declared ports drive and observe the internal system. |
| Chain model | A nearest-neighbor realization whose internal operator is tridiagonal, or block tridiagonal for a multiport anchor. |
| Jacobi model | A real symmetric tridiagonal chain with positive off-diagonal entries after allowed coordinate-sign choices. |
| Lanczos basis | The orthonormal Krylov basis that tridiagonalizes a Hermitian internal operator while retaining a declared anchor. |
| Minimal realization | A realization with no state that is unreachable from or invisible to the declared port experiment. Minimality is relative to the complete input/output boundary. |
These names describe a response realization and its coordinates. They do not by themselves assign capacitors, inductors, transmission-line sections, chip placement, or geometry.
Two valid origins, one physical-ownership rule
This is an engineering distinction made here: a response requirement and a circuit-derived model may share mathematical form without sharing provenance. Linear realization theory supplies related response-equivalence and minimality background (Gough and Zhang 2013). It does not assign chip geometry to hidden states.
The finite-order port-response layer may be entered in either direction:
- Circuit-derived. Start from physical node fluxes, form the Lagrangian, perform every declared reduction, take the Legendre transform, normalize the retained canonical coordinates, attach the ports, and derive the open EOM and \(S/Y/Z\) response.
- Target-specified. Start from desired passive response quantities, bounds, or an admissible response family for inverse design, before a physical circuit has been found.
Only the first route gives physical ownership to internal coordinates. A target-specified contract never acquires that ownership through fitting or response closure. A physical hybridized circuit acquires ownership from its topology and persisted coordinate and port maps. Target satisfaction qualifies that model as a feasible witness; it does not transfer internal meaning to an abstract response realization.
A target-specified contract does not need to assign every matrix coefficient or prescribe one complete numerical trace. It must state which quantities are fixed, targeted, bounded, intentionally free, derived, or diagnostic. The Design Target owns that quantity-role contract; the physical numerical model is then derived from the declared hybridized circuit.
Use finite-order port-response target for the independently specified observable requirement. Reserve physical finite-order port-response model for the circuit-derived realization with persisted coordinate and port maps. The canonical internal parameter basis is the physically anchored retained canonical basis derived from the node-flux circuit; it is not an untracked abstract realization and not a later Lanczos chain basis.
The circuit-derived route is
\[ \boxed{ \vec\Phi_{\rm node} \longrightarrow \mathcal L(\vec\Phi,\dot{\vec\Phi}) \longrightarrow (\vec\Phi_{\rm bare},\vec Q_{\rm bare}) \longrightarrow H_{\rm bare} \longrightarrow \vec a_{\rm bare} \longrightarrow \mathbf M(\omega) \longrightarrow \mathbf S(\omega). } \]
This does not mean that all objects use the raw node-flux basis. The artifact must persist the transforms and reductions between three state-form levels:
| Representation | State and order | Use |
|---|---|---|
| Second-order flux | \(N\) retained generalized fluxes in \(\mathbf K-\omega^2\mathbf C+\boldsymbol\Sigma_{\Phi,\rm env}\) | Exact finite-circuit \(S/Y/Z\) before oscillator or Markov reduction. |
| First-order canonical | \(2N\) flux/charge or doubled Nambu states | Exact Hamiltonian evolution when pairing terms are retained. |
| Number-conserving oscillator | \(N\) amplitudes governed by \(\mathbf h\) after a declared RWA drops \(\boldsymbol\Delta\) | Narrow-band input–output and Chain/Lanczos realization. |
This table classifies dynamical order, not physical basis identity. The separate node-flux \(\to\) anchored-local \(\to\) closed-normal ladder is defined by Circuit Models to Bare Coordinates, Open EOM, and Normal Modes. A finite-order artifact must state both axes: which state form it uses and which physical basis its internal coordinates represent.
In the exact retained flux representation,
\[ \boxed{ \begin{aligned} \mathcal D_\Phi(\omega) &=\mathbf K_\Phi-\omega^2\mathbf C_\Phi +\boldsymbol\Sigma_{\Phi,\rm env}(\omega),\\ \mathcal D_\Phi\vec\Phi &=\mathbf F_\Phi\vec s_{\rm in},\\ \vec s_{\rm out} &=\mathbf S_{\rm dir}\vec s_{\rm in} +\mathbf O_\Phi\vec\Phi, \end{aligned} } \]
Here \(\mathbf F_\Phi\in\mathbb C^{N\times P}\) maps \(P\) incident port waves to the \(N\) retained generalized-force coordinates, while \(\mathbf O_\Phi\in\mathbb C^{P\times N}\) maps the flux response back to outgoing waves. \(\mathbf S_{\rm dir}\) contains the declared direct path that bypasses those retained coordinates.
so the circuit-derived response is
\[ \boxed{ \mathbf S(\omega) =\mathbf S_{\rm dir}(\omega) +\mathbf O_\Phi(\omega) \mathcal D_\Phi^{-1}(\omega) \mathbf F_\Phi(\omega). } \]
An exact doubled \(2N\) canonical/Nambu form is another representation of this same solve after every operator and port map is transformed consistently. An \(N\)-amplitude number-conserving form additionally applies the declared RWA/Markov reduction and must close against the exact response. The Node/Line-Flux Input–Output Bridge owns the traveling-wave prefactors, the exact environment boundary, and the proof that \(S/Y/Z\) are invariant under a consistent invertible internal coordinate change.
In the physically anchored oscillator basis, the complete matrix is allowed to be dense. For example,
\[ \mathbf h_{\rm bare} = \begin{pmatrix} \omega_q&g_{qr}&g_{qp}\\ g_{qr}^*&\omega_r&g_{rp}\\ g_{qp}^*&g_{rp}^*&\omega_p \end{pmatrix}. \]
No off-diagonal entry is set to zero merely because the schematic was drawn as a chain. A zero requires a topology or symmetry proof, or a closure-qualified approximation recorded in the artifact.
When it is a Chain Model
A finite-order model is not automatically a Chain Model. The name is valid only when, in one declared port-anchored basis, its internal operator is tridiagonal,
\[ \boxed{ \mathbf H_{\mathrm{chain}} = \begin{pmatrix} \alpha_1&\beta_1&0&\cdots&0\\ \beta_1&\alpha_2&\beta_2&\ddots&\vdots\\ 0&\beta_2&\alpha_3&\ddots&0\\ \vdots&\ddots&\ddots&\ddots&\beta_{N-1}\\ 0&\cdots&0&\beta_{N-1}&\alpha_N \end{pmatrix}, } \]
so coordinate \(n\) couples directly only to \(n-1\) and \(n+1\). The drive map, output map, and initial/reference conditions must be transformed with the same basis. A multiport anchor generally produces a block-tridiagonal Chain Model, not a scalar chain.
This chain is a response-coordinate realization. The physical node-flux or physically anchored retained-coordinate basis is itself a Chain Model only if its declared operator already satisfies the tridiagonal or block-tridiagonal condition above. A Lanczos transform can otherwise construct an equivalent chain, but its coordinates are Krylov combinations and no longer individual physical nodes or anchored device coordinates. A physical cascade interpretation belongs to a later equivalent-circuit realization.
From a port anchor to a chain
The intuition is successive exploration from the driven subspace: apply the internal operator to the port anchor, remove already represented directions, and normalize what remains. This section reorganizes the Hermitian Lanczos construction (Paige 1976), with the port interpretation discussed in linear realization theory (Gough and Zhang 2013). The physical-coordinate ownership statements are this page’s engineering interpretation, not extra claims of either source.
Consider a passive number-conserving internal model with Hermitian frequency matrix \(\mathbf H\in\mathbb C^{N\times N}\) and one normalized port-coupling anchor \(\mathbf q_1\). Its Krylov sequence is
\[ \mathcal K_m(\mathbf H,\mathbf q_1) =\operatorname{span} \left\{ \mathbf q_1, \mathbf H\mathbf q_1, \ldots, \mathbf H^{m-1}\mathbf q_1 \right\}. \]
Lanczos orthogonalization produces \(\mathbf Q=(\mathbf q_1,\ldots,\mathbf q_N)\) and
\[ \boxed{ \mathbf T=\mathbf Q^\dagger\mathbf H\mathbf Q = \begin{pmatrix} \alpha_1&\beta_1&&0\\ \beta_1&\alpha_2&\ddots&\\ &\ddots&\ddots&\beta_{N-1}\\ 0&&\beta_{N-1}&\alpha_N \end{pmatrix}. } \]
The diagonal \(\alpha_n\) are local chain frequencies and the off-diagonal \(\beta_n\) are nearest-neighbor coherent couplings in this basis. Coordinate phases or signs may make nonzero \(\beta_n\) positive without changing the transfer function.
The scalar chain is minimal for this anchor only if
\[ \boxed{ \operatorname{rank} \begin{pmatrix} \mathbf q_1&\mathbf H\mathbf q_1&\cdots&\mathbf H^{N-1}\mathbf q_1 \end{pmatrix} =N. } \]
Early Lanczos breakdown exposes a smaller reachable/observable subspace. It must not be hidden by retaining disconnected chain coordinates. Conversely, a system with several independent port anchors generally requires block Lanczos and a block-tridiagonal realization. Calling that system a scalar Jacobi chain requires an additional cyclic-anchor proof.
A consistent invertible state-coordinate transformation preserves the declared transfer function when the internal operator, drive map, output map, metric, and initial/reference conditions are transformed together. It need not preserve the physical meaning of an individual internal coordinate. The transformed coefficients are therefore response-realization or chain-basis parameters. Even when the chain was derived from a physical circuit, they do not inherit local physical-node or element ownership unless a chain-structured equivalent circuit separately establishes that interpretation.
For canonically normalized annihilation operators, however, an ordinary number-conserving basis rotation must be unitary; the exact quadratic change is symplectic/Bogoliubov in doubled space. A general non-unitary similarity map is only an abstract response realization unless its energy metric is also carried.
Open-system locality is a separate question
Standard Hermitian Lanczos tridiagonalizes only the coherent matrix \(\mathbf H\). Under the same transform \(\mathbf Q\), every open-system object must also move:
\[ \boxed{ \begin{aligned} \mathbf H'&=\mathbf Q^\dagger\mathbf H\mathbf Q, & \boldsymbol\Gamma'&=\mathbf Q^\dagger\boldsymbol\Gamma\mathbf Q,\\ \boldsymbol\Sigma'_{a,\rm env}&=\mathbf Q^\dagger \boldsymbol\Sigma_{a,\rm env}\mathbf Q, & \mathbf K'_{\rm port}&=\mathbf Q^\dagger\mathbf K_{\rm port},\\ \mathbf D'_{\rm port}&=\mathbf D_{\rm port}\mathbf Q.&& \end{aligned} } \]
If only \(\mathbf H'\) is tridiagonal, call the result a coherent-H chain with its transformed bath and port maps. Call the complete port-response model a Chain Model only when the declared damping/self-energy and port maps also satisfy its stated local or block-local chain boundary.
Analytical scattering formula
Now read the chain through the same external boundary. Tridiagonalizing a matrix does not create a new experiment; the port maps must be transformed with it. This page uses the reorganized coupled-mode transfer relation (Suh et al. 2004) and the consistent-coordinate derivation in the bridge.
For a general linear open model, the internal EOM and input–output boundary give
\[ \boxed{ \mathbf S(\omega) =\mathbf S_{\mathrm{dir}}(\omega) +\mathbf D_{\mathrm{port}}(\omega) \mathbf M^{-1}(\omega) \mathbf K_{\mathrm{port}}(\omega). } \]
The Multimode Input–Output Scattering node owns this formula and its normalization. The finite-order claim means that \(\mathbf M\) retains all coordinates of the declared representation. Its basis, model order, drive map, and output map must be the ones carried from the physical derivation or declared by the target specification. The measured \(S_{21}\) is selected only after port order, reference impedance, reference planes, direct path, and calibration are fixed.
Limits before physical realization
The readiness checks below assess the finite-order model. Continue with Hybridized-Circuit Subsystem Reduction and Design to relate it to a physical distributed/lumped circuit and a fabrication-legal Layout. A local Equivalent Circuit is a mapped reduced representation, not an independent source of physical ownership.
Physical finite-order model readiness and acceptance contract
This retained engineering contract documents the earlier design workflow’s handoff questions and quantity ownership. It is not a universal physical criterion or a newly activated acceptance Gate. Publication of this explanation does not accept its design choices, and no numeric thresholds are introduced.
After a physical circuit and its coordinate map supply physical ownership, its finite-order port-response model may hand off to response comparison only when all applicable rows pass.
| Gate | Required evidence |
|---|---|
| Boundary identity | Model order, coordinate order, port order, wave convention, reference impedances, reference planes, calibration, frequency window, and fixed approximations are explicit. |
| Coordinate provenance | The artifact declares whether it is circuit-derived or target-specified. A circuit-derived artifact persists the node-flux-to-anchored-coordinate transform; a target-specified artifact does not claim physical anchored-bare open-EOM parameters before coordinate and reduction closure. |
| Representation order | The artifact distinguishes an \(N\)-coordinate second-order flux model, a \(2N\) first-order canonical/Nambu model, and an \(N\)-amplitude number-conserving reduction. |
| Target performance | Every cared quantity meets its declared target or bound over the complete review window. A complete calibrated complex \(S_{21}\) trace is required as a target only when the Design Target explicitly specifies it. |
| Stability and passivity | Open poles use the declared stable half-plane; the full available scattering or immittance evidence satisfies the applicable passive-network checks. |
| Chain scope | The artifact says whether only \(\mathbf H\) is tridiagonal or the complete open operator and port boundary are chain-local after transforming \(\boldsymbol\Gamma/\boldsymbol\Sigma_{a,\rm env}\), \(\mathbf K_{\rm port}\), and \(\mathbf D_{\rm port}\). |
| Minimality | Controllability/observability rank, Krylov rank, or an equivalent test excludes hidden unreachable or unobservable coordinates for the declared ports. |
| Witness identity | The topology, basis, fixed/free parameter map, ports, calibration, fit window, initializer, and bounds state whether the fitted coordinates are fabrication controls or a local finite-order response witness. A response fit is not a fabrication witness without a separate provenance-bearing realization map. Global uniqueness of every internal coefficient is not required; materially different eligible fits must agree on the promoted outputs or the extraction remains unresolved. |
| Feature resolution | Poles, zeros, residues, linewidths, and narrow features are stable under local grid refinement and branch continuation. |
| Reproducible handoff | Units, matrices, parameter values, target residuals, source identities, and candidate status are persisted without implying Human promotion. |
Passing these gates makes the analytical candidate eligible for Human review and handoff. It does not accept the model, equivalent circuit, distributed circuit, or Layout. Only an explicit Human decision accepts each semantic scope.
Connections
- Dynamics Conventions and the Flux-to-Mode Symbol Bridge fixes Fourier, phasor, and amplitude conventions.
- Circuit Models to Bare Coordinates, Open EOM, and Normal Modes owns the physical-circuit-to-Hamiltonian analysis direction and the three closure layers.
- From Node and Line Flux to Input–Output Scattering derives the physical port boundary used by an analytical realization.
- Network Trace Views owns observable and calibration semantics; Poles, Zeros, and Residues owns feature interpretation.
References
- J. E. Gough and G. Zhang, “On Realization Theory of Quantum Linear Systems,” arXiv:1311.1375, especially the minimal passive and chain-mode realizations.
- C. C. Paige, “Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix,” IMA Journal of Applied Mathematics 18, 341–349 (1976).
| Field | Value |
|---|---|
| Status | Seed |
| Review state | Candidate terminology and acceptance contract; pending Human review. |
| Human review gate | Can a reader distinguish a response target, a circuit-derived finite-order model, a port-anchored chain/Jacobi realization, an equivalent circuit, and a Layout, and explain why identical \(S_{21}\) does not make their internal parameters interchangeable? |