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Finite-Modal Realization and Operator-Norm Convergence of a Source-to-Observation Electromagnetic Scattering Green Operator

Zhukang Wang, Da Li, Ruifeng Li, Jinyan Ma, Jiarun Hu, Jiahui Wang, Anqi Xia, Tengjiao Wang, Said Mikki, Er-Ping Li

arXiv:2609.08229v1cs.ITmath-phmath.ITmath.MP

TL;DR

Finite multiple-scattering models need more than agreement for selected excitations or adjacent truncation orders to establish accuracy of the full map and its singular channels. The paper defines a continuous trace-space scattering operator and proves convergence of nested finite VSWF realizations under stated conditions. Its metric core preserves finite-order singular values, while full-wave checks and a near-resonant study examine implementation and channel behavior.

  • Problem

    Finite T-matrix accuracy and stability for selected excitations do not establish uniform accuracy of the complete source-to-observation map or physically consistent singular channels.

    Source: §I · pp. 1–2

  • Method

    The paper defines a continuous scattering operator and exact trace-space factorization, then constructs nested finite VSWF realizations and compresses external metrics into a finite core.

    Source: §I · p. 2, E. Finite-Modal Operator Metrics · p. 7

  • Results

    For fixed pairwise-disjoint trace spheres and a well-posed collective problem, finite VSWF realizations converge in operator norm, and the metric core preserves each retained operator’s nonzero singular values.

    Source: §V · p. 13

  • Takeaways & Limitations

    The framework represents accessible source–field channel pairs without carrying external-support quadrature degrees of freedom into the singular-value problem.

    Source: §V · p. 13

    The convergence result assumes fixed pairwise-disjoint auxiliary trace spheres, bounded local transition operators, and collective well-posedness.

    Source: §V · p. 13, S3. FREDHOLM SOLVABILITY AND EXACTNESS… · p. 18

Abstract

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Source-to-observation operators provide reusable environment-level descriptions for multi-query electromagnetic (EM) prediction and communication-mode analysis. However, in practical multiple-scattering models, these operators are represented with finitely many angular modes, and agreement for selected excitations or between successive truncation orders does not establish uniform accuracy of the full map or reliability of its singular channels. To close this gap, we formulate the environment-induced response as a scattering Green operator on fixed continuous source and observation spaces and derive an exact trace-space factorization that reconstructs the Maxwell scattered field. For fixed, pairwise-disjoint enclosing trace spheres and a well-posed collective problem, nested vector spherical wave function (VSWF) realizations converge in operator norm. A structural bound separates external modal tails from collective-resolvent sensitivity, and operator-norm convergence guarantees uniform convergence of the singular values. We further construct a finite metric core that preserves the nonzero singular values of each finite-order operator and reconstructs matched orthonormal source--field channels without introducing external-support discretization degrees of freedom (DoF) into the spectral problem. Full-wave benchmarks verify the finite-order implementation. A controlled near-resonant two-sphere study shows that adjacent-order agreement can precede resolution of the dominant high-order collective direction. It further shows that only part of the internal amplification appears in externally accessible gains and that resonance promotes a distinct high-order channel pair above an otherwise preserved low-order family. The resulting framework provides a convergent, metric-consistent finite-modal representation of multiple-scattering source-to-observation operators and their accessible channels.

I. INTRODUCTION

The paper targets uniform accuracy of the complete source-to-observation scattering map, not just selected excitations or finite T-matrix accuracy. It defines the map on fixed continuous source and observation spaces, then develops a trace-space realization and convergent finite-modal approximations.

I. INTRODUCTION

  • Finite T-matrix accuracy and convergence for selected fields do not establish uniform convergence of the assembled source-to-observation map.The paper identifies operator-norm convergence on fixed continuous spaces as the relevant criterion, particularly because a collective resonance can amplify unresolved modal content.Source: §I · pp. 1–2
  • Finite coefficient-matrix singular values depend on modal normalization, so meaningful channel gains must inherit the continuous map’s prescribed source and observation metrics.Operator-norm convergence to one common continuous operator is required for the finite-order spectra to approach the exact Maxwell map’s channel gains.Source: §I · p. 2
  • The continuous scattering operator maps distributed electric and equivalent magnetic currents to metric-equipped observation spaces, with an exact trace-space factorization of the physical scattered field.Nested finite VSWF realizations truncate only internal trace variables; under the stated conditions they converge in operator norm, and a metric core preserves nonzero singular values while reconstructing accessible channels.Source: §I · p. 2

A. Single-Body Hilbert Spaces and Interface Operators

  • Each scatterer’s interface state is its Cauchy trace on an enclosing sphere, with regular incident and outgoing scattered fields represented in corresponding trace spaces.An isolated-body well-posedness assumption makes its transition operator bounded; separated source and observation supports define the external maps.Source: A. Single-Body Hilbert Spaces and… · p. 3
  • The framework fixes source and observation spaces and their weighted inner products, allowing volume or surface supports and selected field components.The source may comprise electric and equivalent magnetic currents; observation fields may be full or tangential components.Source: A. Single-Body Hilbert Spaces and… · p. 3

B. Multiple-Scattering Foldy–Lax Assembly

The multiple-scattering assembly couples each body’s outgoing trace into regular incident traces on the other bodies. Under collective uniqueness, the resulting trace solution reconstructs the physical scattered field and defines the source-to-observation operator.

C. Multiple-Scattering Foldy–Lax Assembly

  • Interbody continuation maps outgoing traces from one enclosing sphere to regular incident traces on another, while each body’s local transition operator supplies its own response.The closed enclosing balls are pairwise disjoint, and self-interaction is contained in the local operators rather than the off-diagonal coupling.Source: B. Multiple-Scattering Foldy–Lax… · pp. 3–4
  • The Foldy–Lax balance combines direct source excitation with fields scattered by the other bodies, producing the collective trace equation.The direct source induces regular traces through the source map; the stacked outgoing traces are solved using the collective resolvent when the homogeneous radiating problem is unique.Source: B. Multiple-Scattering Foldy–Lax… · p. 4
  • The observation map continues and superposes solved outgoing fields, yielding Gsca = OT(I−WT)^−1S and separating source accessibility, collective scattering, and observation visibility.The factorization is compact because the source and observation maps are compact and the collective transition is bounded.Source: B. Multiple-Scattering Foldy–Lax… · pp. 4–5
  • Under isolated-body and collective uniqueness assumptions, the trace solution equals the physical Maxwell multiple-scattering solution and the operator returns its observed scattered field.The local conditions and interbody balance reconstruct the unique physical solution.Source: B. Multiple-Scattering Foldy–Lax… · p. 5

III. FINITE VSWF REALIZATION OF THE SCATTERING GREEN OPERATOR

The finite realization truncates nested VSWF coordinates on the enclosing-sphere traces while leaving external source and observation spaces unchanged. A fixed trace metric makes the projections and retained coordinates consistent across truncation orders.

III. FINITE VSWF REALIZATION OF THE SCATTERING GREEN OPERATOR

  • VSWFs provide nested modal coordinates for regular incident and outgoing scattered traces, so truncation approximates the internal representation without changing external spaces or metrics.The modal index set retains angular orders l ≤ L.Source: §III · p. 5

A. Local VSWF Coordinates

  • At each local center, regular VSWFs represent incident traces and outgoing VSWFs represent scattered traces; the regular branch is finite at the center and the outgoing branch satisfies radiation conditions.The source-free homogeneous expansion omits the longitudinal family.Source: A. Local VSWF Coordinates · p. 5

B. VSWF Trace Projections

  • The fixed VSH-diagonal trace metric makes retained VSWF traces orthogonal, and their nested projections converge strongly because the basis is complete.The modal Gram weights depend on mode and sphere parameters, not on truncation order.Source: B. VSWF Trace Projections · pp. 5–6

C. Finite-Modal Scattering Green Operator

The finite-modal operator combines source-to-modal excitation, a retained collective scattering solve, and observation synthesis. Only the scattered contribution is modal-truncated, while the prescribed external spaces remain in use.

C. Finite-Modal Scattering Green Operator

  • Supplied T-matrices may come from analytic, numerical, or external solves, with extraction and fitting errors treated separately from modal truncation.Body-fixed matrices are rotated into the shared global frame before translation assembly.Source: C. Finite-Modal Scattering Green… · p. 6
  • The retained collective matrices represent local transitions and interbody translations, and the retained system must be nonsingular at the chosen order.Translation convergence is ensured when Ri+Rj < |ci−cj|.Source: C. Finite-Modal Scattering Green… · p. 6
  • The direct-excitation coefficients are obtained from source pairings, and the observation map synthesizes the selected field component from outgoing modal coefficients.The source expansion assumes the source support lies outside each enclosing sphere.Source: C. Finite-Modal Scattering Green… · p. 6
  • The finite-modal scattering operator combines source analysis, modal propagation, and observation synthesis to map external sources to observed scattered fields.The external source and observation maps are retained while the internal trace response is represented by finite modal coordinates.Source: C. Finite-Modal Scattering Green… · p. 6

D. Finite-Modal Operator Convergence

For fixed, pairwise-disjoint enclosing spheres and a well-posed collective problem, nested VSWF realizations converge to the continuous scattering operator in operator norm. A finite metric core preserves the retained operator’s nonzero singular values and supports reconstruction of accessible channels.

D. Finite-Modal Operator Convergence

  • Under the separated fixed-sphere setting and collective well-posedness, nested VSWF realizations converge in operator norm, stabilizing the scattering maps’ singular values uniformly.Theorem 1 establishes the convergence result; Weyl’s inequality gives uniform singular-value convergence.Source: D. Finite-Modal Operator Convergence · pp. 6–7
  • The error bound separates source and observation modal tails from projected-product error amplified by the collective resolvent.A sensitive resolvent can magnify unresolved components, and compactness alone does not predict when the asymptotic regime begins.Source: D. Finite-Modal Operator Convergence · p. 7

E. Finite-Modal Operator Metrics

  • Compressing the prescribed external metrics onto retained modal coordinates yields a finite realization for operator-level diagnostics without discretizing external supports in the spectral problem.The retained operator norm can be evaluated with an NL × NL singular-value computation after including the source and observation metrics.Source: E. Finite-Modal Operator Metrics · pp. 7–8
  • The metric core and retained operator have the same nonzero singular values, so the finite spectrum represents the retained map rather than unweighted modal coefficients.The source and observation Gramians depend on the chosen Hilbert metrics.Source: E. Finite-Modal Operator Metrics · p. 7
  • Singular vectors of the finite core reconstruct orthonormal source–observation pairs that are accessible and visible under the prescribed metrics.These pairs are the source-accessible, observation-visible scattering channels.Source: E. Finite-Modal Operator Metrics · p. 7

IV. FULL-WAVE VALIDATION AND OPERATOR ANALYSIS

The numerical assessments distinguish practical full-wave implementation checks from operator-level truncation analysis and accessible-channel analysis. The supplied visual captions identify the implementation-test configuration and the nonspherical T-matrix extraction and replay data.

IV. FULL-WAVE VALIDATION AND OPERATOR ANALYSIS

  • Three complementary assessments address full-wave implementation, operator-norm truncation behavior under fixed metrics, and accessible source–field pairs.The full-wave comparisons use prescribed excitations; the operator-norm experiments probe worst-case truncation behavior.Source: §IV · p. 8
  • Table I covers nonspherical local T-matrix extraction and replay data.The extraction procedure supplies finite matrices at L = 3 for subsequent implementation tests.Source: Table I · p. 8
  • Figure 2 shows the configuration used for the full-wave implementation test, which is at a frequency of 1 GHz.Source: Figure 2 · p. 8

A. Full-Wave Implementation Tests

Full-wave tests check the finite-modal source-to-field implementation, while fixed-surface two-sphere experiments show why adjacent-order agreement can miss near-resonant collective sensitivity. External modal tails and collective-resolvent amplification jointly shape the truncation error.

A. Full-Wave Implementation Tests

  • The full-wave tests use nested mixed clusters of 5–40 bodies with at least 0.2λ between enclosing spheres, but assess prescribed excitations rather than worst-case operator error.They use L = 3 for electrically small, nonresonant bodies and include PEC spheres, a cube, an ellipsoid, and a torus.Source: A. Full-Wave Implementation Tests · p. 8
  • Total-field error stays below 2.3%, while scattered-field error falls from 16.4%–19.4% at N = 5 to 4.6%–6.0% at N = 40.The comparisons cover three prescribed excitations and are interpreted against an empty-scene interface baseline, not as operator-norm or isolated modal-truncation errors.Source: A. Full-Wave Implementation Tests · p. 9, Figure 3 · p. 9
  • With precomputed local T-matrices, speedups over N = 10–40 are 85×–175×, 39×–51×, and 13×–32× for point, Gaussian, and Huygens sources, respectively.For Huygens sources, projection accounts for 82.8%–89.5% of proposed runtime; the collective solve takes 0.27–3.38 s.Source: A. Full-Wave Implementation Tests · p. 9, Table II · p. 9
  • The frozen Huygens pair reproduces the isolated radiator’s prescribed exterior incident field, and its nonzero magnetic current motivates retaining equivalent magnetic currents in the source map.The Huygens currents are extracted from an isolated, equal-phase 2×2 patch array and held fixed across loaded scenes.Source: A. Full-Wave Implementation Tests · p. 9, Figure 4 · p. 9

B. Modal Operator Convergence and Resolvent Sensitivity

  • At the near-resonant L = 6 →7 transition, ρL jumps from 2.15 to 97.9 although be6 = 0.9997, showing adjacent-order agreement can precede resolution of the dominant direction.The newly admitted l = 7 subspace contains 99.4% and 99.3% of the dominant input subspace at L = 7 and 14, respectively.Source: Figure 6 · p. 10, B. Modal Operator Convergence and… · p. 11
  • At n = 6.881, minj |1 −λj| = 3.51 × 10−3 while the isolated-sphere l = 7 coefficients remain small, locating the near-singularity in the coupled interaction resolvent.The coefficients are |t1,7| = 1.49 × 10−7 and |t2,7| = 2.15 × 10−9.Source: B. Modal Operator Convergence and… · p. 11, Figure 7 · p. 11
  • Increasing either clearance lowers finite-reference tails; near resonance, appreciable error reduction requires suppressing both external tails, whereas off resonance either one-sided increase produces a large reduction.At L = 13, increasing both clearances to 2.0 reduces eL|14 by 14.2 dB near resonance and 58.7 dB off resonance.Source: B. Modal Operator Convergence and… · p. 11

C. Accessible Scattering Channels

The leading accessible channels show a strong near-resonant gain and angular reorganization, while the conclusion situates these findings within a convergent, metric-consistent operator framework.

C. Accessible Scattering Channels

  • At L = 14, leading gains are (σ1, σ2) = (29.068, 29.068) near resonance and (1.406, 1.406) off resonance; near resonance σ2/σ3 = 19.976.The near-resonant leading pair accounts for 99.170% of the total squared singular-value sum over retained channels.Source: C. Accessible Scattering Channels · p. 12
  • The internal amplification ratio ρnear_14/ρoff_14 = 427.987 exceeds the accessible-gain ratio σnear_1/σoff_1 = 20.672, with gain also depending on source accessibility and observation visibility.Source: C. Accessible Scattering Channels · p. 12
  • Reconstructed unit-norm sources produce near- and off-resonant output norms of 29.068 and 1.406, with forward-solution relative Y-norm errors of 9.370 × 10−10 and 9.150 × 10−12.Source: Figure 11 · p. 12, C. Accessible Scattering Channels · p. 12
  • Near resonance, 79.6% of excitation weight lies at l = 5–8 and 99.25% of scattered weight at l = 7; off resonance, 96.35% and 95.31% concentrate at l = 1.Source: Figure 12 · p. 12, C. Accessible Scattering Channels · pp. 12–13
  • The near-resonant peak promotes a distinct high-order channel pair above a preserved low-order family, whose source and observation subspaces align with minimum principal cosines 0.9941 and 0.9967.Source: C. Accessible Scattering Channels · p. 13

V. CONCLUSION

  • For fixed, pairwise-disjoint enclosing trace spheres and a well-posed collective problem, finite VSWF realizations converge in operator norm, while the finite metric core preserves nonzero singular values without external-support quadrature DoF.Source: §V · p. 13

APPENDIX A PROOF OF THE SOURCE-TO-MODAL EXCITATION MAP

The appendices justify the source-to-modal excitation map and modal Green-operator convergence, including closed Maxwell trace spaces and compactness across separated supports. Under stated boundedness and existence assumptions, they establish compactness of the multiple-scattering Green operator and its component maps.

APPENDIX A PROOF OF THE SOURCE-TO-MODAL EXCITATION MAP

  • On separated source/body sets, the VSWF expansion and one field-variable curl converge uniformly, validating termwise integration and recovery of excitation coefficients and traces; stacking yields SLq = aexc.Point sources and other distributions outside the chosen Hilbert support are handled by distributional pairings, outside the theorem.Source: Appendix A · p. 13

APPENDIX B PROOF OF THE MODAL GREEN-OPERATOR CONVERGENCE

  • Compact-projection tails vanish, and the Neumann lemma plus resolvent identity yields the norm bound proving modal Green-operator convergence.The proof also uses strong convergence of the projected continuation operators and compactness of the source and observation maps.Source: Appendix B · pp. 13–14

S1. MAXWELL CAUCHY-TRACE SPACES ON AN ENCLOSING SPHERE

  • Regular and outgoing Cauchy-trace ranges are closed Hilbert spaces with unique representatives, even without excluding interior eigenfrequencies; material-scattering well-posedness remains separate.The trace proposition does not assert well-posedness of a material scattering problem; boundedness of the local transition operator is a separate assumption.Source: S1. MAXWELL CAUCHY-TRACE SPACES ON AN… · pp. 16–17

S2. COMPACT MAXWELL MAPS ACROSS SEPARATED

  • Under separated supports and pairwise-disjoint enclosing-ball assumptions, source, continuation, and observation maps are compact; bounded positive weights preserve their relevant topology.The support separation gives smoothing, while the weighted and unweighted spaces have equivalent notions of boundedness and compactness.Source: S2. COMPACT MAXWELL MAPS ACROSS… · p. 17
  • With bounded local and collective transitions and an existing collective inverse, WT, ODTDSD, and Gsca = OT(I − WT)^−1S are compact.The result follows from Proposition S2; ODTDSD is also shown compact directly.Source: S2. COMPACT MAXWELL MAPS ACROSS… · p. 18

S3. FREDHOLM SOLVABILITY AND EXACTNESS OF THE TRACE ASSEMBLY

The supplement derives bounded invertibility of the collective trace equation from physical uniqueness, then uses the trace solution to recover the physical multiple-scattering response under stated assumptions.

S3. FREDHOLM SOLVABILITY AND EXACTNESS OF THE TRACE ASSEMBLY

  • Pairwise-disjoint enclosing balls yield bodywise outgoing uniqueness: outgoing fields whose sum vanishes in the common exterior must each vanish outside their own ball.The proof glues each field to a background Maxwell field and applies radiating uniqueness.Source: S3. FREDHOLM SOLVABILITY AND EXACTNESS… · p. 18
  • A bounded local transition operator and uniqueness of the homogeneous radiating problem imply bounded invertibility of the collective trace operator.Compactness makes I − A Fredholm of index zero; physical uniqueness establishes a trivial kernel, and the bounded inverse theorem gives the inverse.Source: S3. FREDHOLM SOLVABILITY AND EXACTNESS… · pp. 18–19
  • Under those assumptions, the unique trace solution reconstructs the physical multiple-scattering solution, and the observation map selects its scattered-field component.The assembled fields satisfy each body's local conditions and the outgoing condition; collective uniqueness identifies them with the driven physical solution.Source: S3. FREDHOLM SOLVABILITY AND EXACTNESS… · p. 19
  • The exactness argument requires neither injective local transition operators nor a multiple-reflection Neumann series.The argument uses material information through bounded isolated-body responses and collective uniqueness, without retaining interior material fields in the global state or recovering geometry from transition operators.Source: S3. FREDHOLM SOLVABILITY AND EXACTNESS… · p. 19

S4. FIXED VSH TRACE METRIC AND VSWF TRACE PROJECTIONS

The fixed VSH trace metric makes finite VSWF truncations well-defined orthogonal projections on the trace spaces, with nested projections converging strongly. The surface-interface Gramians then support recovery of orthonormal accessible channels.

S4. FIXED VSH TRACE METRIC AND VSWF TRACE PROJECTIONS

  • The fixed VSH metric is equivalent to the standard trace-space norm with constants independent of angular order and truncation order.The constants may depend on the fixed wavenumber, sphere radius, and impedance.Source: S4. FIXED VSH TRACE METRIC AND VSWF… · p. 20
  • Regular and outgoing VSWF traces form complete orthogonal families, and their finite-order Gram matrices are positive definite.The resulting modal projections are orthogonal and have nested ranges.Source: S4. FIXED VSH TRACE METRIC AND VSWF… · pp. 20–21
  • Nested VSWF trace projections converge strongly on each body and on finite direct sums, while every nonzero projection has norm one.Their ranges are nested as the retained modal sets grow.Source: S4. FIXED VSH TRACE METRIC AND VSWF… · p. 21

S5. SURFACE INTERFACE GRAMIANS AND CHANNEL RECOVERY

S6. EXTERNAL EXTRACTION OF NONSPHERICAL LOCAL T-MATRICES

This supplement documents offline extraction of fixed-order local T-matrices for three nonspherical bodies, then checks the fitted matrices and their action on a held-out excitation. The checks retain fixed mesh, fitting, and order-3 limitations.

S6. EXTERNAL EXTRACTION OF NONSPHERICAL LOCAL T-MATRICES

  • The extracted matrices use fixed order L = 3 for three nonspherical bodies and are offline inputs to the implementation tests, not part of the operator-norm proof.Source: S6. EXTERNAL EXTRACTION OF… · p. 22

B. Batched SCUFF-EM field solves

C. Outgoing-field fit and convention conversion

  • The outgoing-field fit has full column rank for each body, and the incident columns are rescaled to match the main-paper polarization, ordering, and phase convention.The matrices are fitted using the main-paper outgoing basis; only the incident columns require rescaling.Source: C. Outgoing-field fit and convention… · p. 23

D. Fit diagnostics and held-out replay

  • For the lossless cube and ellipsoid, passive-form eigenvalue ranges are [−6.94×10−7, 5.09×10−7] and [−1.27×10−6, 1.21×10−6]; the lossy torus range is [6.25 × 10−8, 9.15 × 10−3].Small negative excursions for the cube and ellipsoid are attributed to BEM and fitting errors; positive scale is also compatible with modal omission.Source: D. Fit diagnostics and held-out replay · pp. 23–24
  • Held-out plane-wave replay tests convention conversion and matrix action on a new excitation, while magnetic-field replay checks the outgoing-field curl relation outside the electric-field fit.Source: D. Fit diagnostics and held-out replay · p. 24
  • The fit and replay checks use the same supplied mesh and prescribed excitations, so they establish neither BEM mesh convergence nor a supremum over the continuous source unit ball.The extracted matrices carry fixed BEM, fitting, and order-3 errors into the implementation tests.Source: D. Fit diagnostics and held-out replay · p. 24

S7. COMSOL FULL-WAVE REFERENCE AND BENCHMARK PROTOCOL

The supplement specifies the full-wave reference and benchmark protocol, then gives finite-reference operator diagnostics. Loaded-scene electric-field comparisons report source-specific errors, while the external-surface Gramian calculations avoid surface quadrature degrees of freedom.

S7. COMSOL FULL-WAVE REFERENCE AND BENCHMARK PROTOCOL

A. Coordinates, computational domain, mesh, and field sampling

  • Both solvers sample the observation plane on a common 21×21 grid, and scattered fields are formed by subtracting source-matched empty-scene fields.Displayed amplitude maps are interpolated for presentation, but reported errors use the original samples.Source: A. Coordinates, computational domain,… · pp. 24–25

B. Time convention and discrete comparison metrics

  • Reported comparison norms are ordinary Euclidean norms of stacked complex samples, without observation-plane quadrature weights.They are discrete sample norms rather than continuous surface norms.Source: B. Time convention and discrete… · p. 25
  • 2.2715%, 1.6232%, and 1.5271% are the loaded-scene maximum electric-field errors for point, Gaussian, and frozen-Huygens sources; only the Gaussian maximum exceeds its empty-scene baseline.Source: B. Time convention and discrete… · p. 25

C. Prescribed sources and the frozen-source protocol

D. Nested scene construction, clearance, and sample selection

  • Published scenes are hand-selected strict nested subsets; selected sets enforce at least 0.2λ enclosing-sphere clearance, and one torus candidate was replaced for meshability.Source: D. Nested scene construction,… · p. 26

E. Timing boundaries and execution environment

  • The timing comparison is a repeated-query online benchmark with geometry and source data prepared, and each reported Table II entry is a single wall-clock measurement.Offline T-matrix extraction and the isolated physical-array solve are excluded from the speedups.Source: E. Timing boundaries and execution… · p. 26

A. Semianalytic Gramians on the external spheres

  • External-surface integrals reduce to translation coefficients and diagonal radial factors, so no surface-quadrature degrees of freedom enter the operator-norm calculation.The external-surface Gramian calculations use Lc = 44.Source: A. Semianalytic Gramians on the… · p. 27

B. Nested finite-reference error

The supplementary diagnostics compare truncation orders, quantify source and observation modal tails, examine trace-normalized collective structure, and check the finite metric-core realization.

B. Nested finite-reference error

C. Source and observation tail ratios

D. Trace-normalized resolvent and angular content

A. QR realization of the metric core

S9. FINITE-MODAL REALIZATION AND DIAGNOSTICS OF ACCESSIBLE CHANNELS

  • At L=14, source- and observation-orthonormality residuals are 2.2 × 10−15 and 1.7 × 10−15, and the relative Frobenius residual is 2.3 × 10−15 near resonance.The off-resonance Frobenius residual is 1.2 × 10−15.Source: A. QR realization of the metric core · p. 29

B. Leading doublets, azimuthal sectors, and order stability

The leading-channel analysis identifies doublet structure and azimuthal sectors, measures angular trace fractions, and checks channel stability and reconstruction. Near resonance, a distinct high-order doublet appears while the low-order family remains nearly unchanged.

B. Leading doublets, azimuthal sectors, and order stability

  • At L=14, σ2/σ3 is 19.976 near resonance and 1.022 off resonance, so only the near-resonant leading doublet is strongly separated from the remaining spectrum.Source: B. Leading doublets, azimuthal… · p. 29
  • From L=14 to 18, near-resonant σ1 changes relatively by 5.18 × 10−3 while source and observation subspace minimum principal cosines are 0.9999999998 and 0.9999999994.Off resonance, the corresponding singular-value change is 4.63 × 10−9 and both cosines exceed 0.9999999999.Source: B. Leading doublets, azimuthal… · p. 29

C. Basis-invariant angular trace fractions

D. Displayed representative and independent forward check

  • The independent forward check gives relative observation-norm errors of 9.370 × 10−10 near resonance and 9.150 × 10−12 off resonance.The corresponding trace-balanced equation residuals are 1.06 × 10−13 and 1.16 × 10−14.Source: D. Displayed representative and… · p. 30
  • Near resonance, the channel-3,4 and off-resonant channel-1,2 subspaces have minimum principal cosines of 0.9941 in X and 0.9967 in Y, supporting a distinct high-order leading doublet above the nearly unchanged low-order family.Source: D. Displayed representative and… · p. 31
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