Symbol Conventions

The project-wide notation contract for topology, coupling state, parameter layer, basis, channel, processing state, and units.

Node role: Concept.

Reader task: Name a circuit quantity so that another reader can identify the physical object, circuit state, parameter layer, basis, channel, and units before using its value.

Reading map

For lookup, begin with the notation-pattern table. For comparisons, keep topology state and parameter layer separate, then identify basis, operator, processing state and units. The examples are vocabulary illustrations, not a required circuit topology.

What this page owns

Engineering choice. Qualifier placement, abbreviation and unqualified matrix defaults are this project’s CONVERGING notation candidate. They are not universal physics notation and do not create scientific Gates.

This page owns the placement and meaning of project-wide semantic qualifiers. It does not redefine the quantity itself. Detailed meanings remain with their physics owners:

  • Parameter Layers defines reference, anchored-bare open-EOM, downstream Hamiltonian, normal-mode, and hybridized open-pole quantities;
  • Dynamics Conventions fixes phasor, Fourier, rotating-frame, quantum-evolution, and \(\Phi/Q\leftrightarrow a\) conventions; and
  • Network Trace Views defines the project defaults for \(\mathbf S\), \(\mathbf Y\), and \(\mathbf Z\).

One notation pattern

Use the base symbol for the physical kind of quantity. Add only the qualifiers that can change its meaning:

\[ \boxed{ \widetilde X_{ \text{identity},\,\text{channel},\,\text{layer} }^{ \text{topology},\,\text{coupling state},\,\text{processing} } (\text{controlled input}) } \]

The positions answer different questions:

Position Question Examples
Base symbol What physical quantity is this? \(\Phi\), \(\omega\), \(\kappa\), \(g\), \(\mathbf C\), \(\mathbf Y\)
Subscript Which coordinate, pole, channel, layer, or basis? \(r\), \(\mu\), \(\mathrm{ext}\), \(\mathrm{AB}\), \(\mathrm H\)
Superscript Which topology, coupling state, or processing state owns it? \(\mathrm{QRP}\), \(\mathrm{on}\), \(\mathrm{off}\), \(\mathrm{raw}\)
Tilde Is this the complete complex pole? \(\widetilde\omega_{\mu,\mathrm H}\)
Parentheses Which controlled input is varied? \((L_J)\), \((C_{\mathrm{probe}})\), \((\omega)\)

Not every symbol needs every field. A field is mandatory when omitting it could make two different physical quantities look identical.

Prefer a complete physical name over a local alias

If the existing base symbol and its semantic qualifiers identify the quantity, use them directly and explain its engineering role in prose. Do not introduce a second label only to shorten one page:

\[ \kappa_{p,\mathrm{AB}}^{E\downarrow;\,\mathrm{QRP,ext:on}} \qquad \text{(mandatory report-only cross-check)}. \]

The prose is not part of the symbol. It may describe a quantity as a target, cost operand, initializer, diagnostic, cross-check, or gate without renaming the physics. Introduce a new symbol only when it represents a genuinely new mathematical object that is reused in later equations.

Topology and coupling state are mandatory when they matter

Let

\[ \mathcal T\in\{\mathrm{QR},\mathrm{RP},\mathrm{QRP},\ldots\}, \qquad s\in\{\mathrm{on},\mathrm{off}\}. \]

Write both in the superscript:

\[ X^{\mathcal T,s}. \]

For example,

\[ f_{r,\mathrm{LB}}^{\mathrm{QRP,on}} \not\equiv f_{r,\mathrm{LB}}^{\mathrm{QRP,off}}. \]

\(\mathrm{on}\) means the declared physical cross-coupling branch is retained, including its resulting off-diagonal matrix entries. \(\mathrm{off}\) means the topology owner has disabled that branch. The owner must also state whether its diagonal loading is retained, grounded, or removed. \(\mathrm{off}\) does not mean that every off-diagonal entry of a larger circuit vanishes.

Changing coupling state changes the circuit matrices before normalization:

\[ \mathbf C_{\mathrm{bare}}^{\mathcal T,\mathrm{on}} \ne \mathbf C_{\mathrm{bare}}^{\mathcal T,\mathrm{off}}. \]

Therefore setting one fitted Hamiltonian coefficient to zero is not generally the same operation:

\[ g_{xy}=0 \quad\not\Rightarrow\quad \mathbf C^{\mathcal T,\mathrm{on}} \longrightarrow \mathbf C^{\mathcal T,\mathrm{off}}. \]

The inverse capacitance, normalization, diagonal coefficients, and mediated couplings may all change. A within-model \(g_{xy}=0\) comparison is only a diagnostic unless circuit-level closure proves equivalence.

Parameter layer is independent of topology state

Internal definition. The anchored-bare diagonal-root label below names one selected response extraction. It is not a Hamiltonian coefficient derived by a literature citation; the physical-coordinate method defines its operator and elimination.

\(\mathrm{AB}\) identifies a diagonal root of the exact open operator reduced to declared anchored coordinates before their mutual hybridization:

\[ \boxed{ \left[\mathbf D_R^{E\downarrow} (\widetilde\omega_{x,\mathrm{AB}}^{\mathcal T})\right]_{xx}=0. } \]

It is dressed by the eliminated complement and attached environment but bare relative to mutual hybridization among retained coordinates. Downstream \(h_{xx}\), \(h_{xy}\), and \(\Delta_{xy}\) remain Hamiltonian/quantization coefficients and are not aliases unless a separate local, nondispersive map proves equality.

Use the layer qualifier that matches how the value is defined:

Qualifier Meaning
\(\mathrm B\) Isolated component or isolated-subnetwork bare reference
\(\mathrm{ref}\) Value from a declared artificial reference fixture when no explicit loaded-coordinate mapping is claimed
\(\mathrm{AB}\) Anchored-bare open-EOM diagonal root before mutual retained-coordinate hybridization
\(\mathrm{LB}\) Downstream Hamiltonian-coordinate coefficient when that legacy/local alias is explicitly qualified
\(\mathrm N\) Closed conservative normal mode
\(\mathrm H\) Hybridized open-system pole

Changing or disabling a physical branch defines a different topology. Its roots may be reference values, but they do not own the anchored-bare roots of the unchanged physical system.

Identity, basis, and matrix type stay visible

Use fixed coordinate labels for pre-diagonalization coordinates and a different index for hybridized modes or poles:

\[ x,y\in\{q,r,p,\ldots\}, \qquad \mu,\nu\in\{1,\ldots,N\}, \qquad i,j\in\{\text{declared ports}\}. \]

Thus \(q\), \(r\), and \(p\) name retained coordinate directions. The index \(\mu\) names a continued normal mode or open pole; a pole can exchange q-like, r-like, and p-like character through an avoided crossing.

Keep the physical-kind symbol when coordinates change, and qualify its basis:

\[ \vec\Phi_{\mathrm{node}}, \qquad \vec\Phi_{\mathrm{bare}}^{\mathcal T,s}, \qquad \vec\Phi_{\mathrm N}^{\mathcal T,s}. \]

Bold symbols are vectors or matrices. A physical branch element and a matrix entry are written differently:

\[ C_{12} \quad\text{is a branch capacitance}, \qquad \left[\mathbf C_{\mathrm{bare}}^{\mathcal T,s}\right]_{12} \quad\text{is a matrix entry}. \]

Matrix form and basis must also remain visible when ambiguity exists:

\[ \mathbf C_{\mathrm{Maxwell}}, \qquad \mathbf C_{\mathrm{bare}}^{\mathcal T,s}, \qquad \mathbf K_{\Phi,\mathrm{bare}}^{\mathcal T,s}, \qquad \mathbf h_{\mathrm{bare}}^{\mathcal T,s}. \]

Do not multiply or compare coordinate-dependent objects until their basis, ordering, and normalization agree.

Dynamic operators and scattering maps use different symbols

Keep the internal frequency-domain operator distinct from every port observable:

\[ \boxed{ \boldsymbol{\mathcal D}_{\mathrm{open}}(\omega) \ \text{is the exact flux-domain dynamic operator}, \qquad \mathbf S_{\mathrm{dir}}(\omega) \ \text{is the direct port-scattering map}, \qquad \mathbf S(\omega) \ \text{is the complete scattering matrix}. } \]

Keep the exact second-order flux susceptibility separate from the reduced first-order oscillator susceptibility:

\[ \boxed{ \boldsymbol\chi_\Phi(\omega) =\boldsymbol{\mathcal D}_{\rm open}^{-1}(\omega), \qquad \boldsymbol\chi_a(\omega)=\mathbf M^{-1}(\omega). } \]

They have different state variables, units, and transformation laws even when they describe the same response after a closure-qualified reduction.

Use \(\mathcal D_{xy}=[\boldsymbol{\mathcal D}_{\mathrm{open}}]_{xy}\) for scalar entries of the dynamic operator. Reserve \(\mathbf D_{\mathrm{port}}\) for the reduced input–output emission map defined by Multimode Input–Output Scattering. Do not rename a direct scattering map \(\mathbf D_{\mathrm{dir}}\) merely to match generic state-space ABCD notation; in this Knowledge Base it remains \(\mathbf S_{\mathrm{dir}}\).

“Direct” is relative to the retained dynamical state. If finite feedline coordinates are retained, through propagation may remain inside \(\boldsymbol{\mathcal D}_{\mathrm{open}}\) while \(\mathbf S_{\mathrm{dir}}=-\mathbf I\) is only the terminal feedthrough. After eliminating that feedline, the same physical through path can instead appear inside \(\mathbf S_{\mathrm{dir}}\). State the retained representation before comparing the two.

Maxwell projection processing state

Engineering choice. Named-ground-only is the project’s declared projection and notation policy; it omits an identified row-sum coefficient. It is not an electrostatic identity or the only physically possible boundary replay.

The physical distinction between Named-GND-only and direct-retained Maxwell projection is defined in Field-Extracted Capacitance and Inductance Matrices. The project notation convention is asymmetric:

\[ \boxed{ \mathbf C_{\mathrm{node}} \equiv \mathbf C_{\mathrm{node}}^{\mathrm{named\text{-}GND\text{-}only}}, \qquad \mathbf C_{\mathrm{node}}^{\mathrm{direct\text{-}retained}} \text{ must be qualified explicitly.} } \]

Thus an unqualified node capacitance matrix uses only branches to the explicitly named ground and named conductors. It does not silently retain the implicit outer-reference row sum. The raw field-solver matrix remains \(\mathbf C_{\mathrm{Maxwell},E}\) and is never renamed as the projected node matrix.

Frequencies, couplings, poles, and linewidths

Internal definitions. Unit conversions are algebraic; coupling aliases and linewidth qualifiers below follow the linked local definitions rather than being attributed wholesale to a review paper.

Use angular symbols for angular quantities:

\[ \omega=2\pi f. \]

If \(\kappa\), \(g\), or \(J\) is defined as an angular rate, report \(\kappa/2\pi\), \(g/2\pi\), or \(J/2\pi\) in hertz. Do not attach GHz or MHz directly to an angular-rate symbol without this conversion.

Coherent couplings need both coordinate indices and topology state, but those fields alone are not sufficient to identify a value. A coupling that owns a value must also state its extraction or representation qualifier. For the same readout–filter coordinates and topology, representative distinct quantities are

\[ J_{rp,\mathrm{ex}}^{\mathrm{QRP,on}} =J_{rp,\mathrm H}^{\mathrm{QRP,on}}, \qquad J_{rp,\mathrm{RWA}}^{\mathrm{QRP,on}}, \qquad J_{rp,\mathrm{res}}^{\mathrm{QRP,on}}(\omega_*), \qquad J_{\mathrm{split}}. \]

\(J_{\mathrm{ex}}\) (also \(J_{\mathrm H}\)) is the exact fixed-anchored-basis number-conserving exchange-block coefficient accompanied by pairing; \(J_{\mathrm{RWA}}\) is its coefficient inherited by an explicitly stated number-conserving/RWA reduction; \(J_{\mathrm{res}}(\omega_*)\) is the local residue-normalized open-EOM coupling at a selected response point; and \(J_{\mathrm{split}}\) is an observed hybridized-pole half-splitting. They are not interchangeable definitions. These concise labels assume the stated canonical normalization and fixed anchored basis; an RWA coefficient inherits its meaning only under the declared approximation. See Which coupling is this? for the coefficient-inheritance statement and response-side conditions, not a full block derivation. The complete quadratic exchange/pairing derivation is not developed in this selected material.

Use \(g_{qp}\) as the canonical name. A bare \(J\) is permitted only as a page-local alias immediately after its complete qualified definition; a short alias such as \(G\equiv g_{qp}\) is likewise local to the equation that defines it.

A complete open pole is tilded:

\[ \boxed{ \widetilde\omega_{\mu,\mathrm H}^{\mathcal T,s} = \omega_{\mu,\mathrm H}^{\mathcal T,s} -i\frac{\kappa_{\mu,\mathrm{tot,H}}^{\mathcal T,s}}{2}. } \]

Every published linewidth names its identity, channel set, layer, topology, and coupling state:

\[ \kappa_{xj,\mathrm{ext,LB}}^{\mathrm{QRP,on}}, \qquad \kappa_{\mu,\mathrm{tot,H}}^{\mathrm{QRP,on}}. \]

Do not publish a naked \(\kappa\). Left/right or port-resolved rates require evidence for those channels; a scalar trace does not create that decomposition.

Response processing and observation scaffolds

The site-wide network convention is

\[ \mathbf S \ \text{has no implied PTC}, \qquad \mathbf Y,\mathbf Z \ \text{are design-facing PTC results}. \]

Before compensation, use

\[ \mathbf Y^{\mathrm{raw}}, \qquad \mathbf Z^{\mathrm{raw}}. \]

When raw and processed matrices appear together, label the processed object explicitly, for example \(\mathbf Z^{\mathrm{QRP,on,PTC}}\). Reference impedance, reference planes, port order, and exact PTC evidence remain required artifact metadata; they do not all belong inside the symbol.

Observation scaffold and physical loading also use different names:

\[ C_{\mathrm{probe}} \not\equiv C_{\mathrm{ext}} \not\equiv \mathbf C_{\mathrm{IDC}}(\vec u). \]

\(C_{\mathrm{probe}}\) is an artificial observation hook. \(\mathbf C_{\mathrm{IDC}}(\vec u)\) is a geometry-linked physical capacitance block. A scalar \(C_{\mathrm{ext}}\) is valid only when the component model declares that special case.

When local shorthand is allowed

A derivation may omit repeated qualifiers after one equation fixes the topology, coupling state, basis, units, and scope. Restore the complete qualifiers in:

  • reusable definitions;
  • comparisons across topologies, coupling states, or parameter layers;
  • Design Target and Parameter tables;
  • promoted artifacts and optimization outputs; and
  • any statement that assigns physical ownership.

References source notes and History pages may preserve source notation. Add a local mapping when that notation enters the canonical Knowledge or Design Target language; do not rewrite a source merely to match this page.

Connections

Field Value
Status Seed
Semantic state CONVERGING; Human review has not accepted this candidate convention.
Open review question Can a reader distinguish two quantities from their symbols before consulting the extraction method or numerical value?