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GCNO: Gramian Chebyshev Neural Operator for Physics-Based Compression of Wireless Channels

Rafid Umayer Murshed, Shahab Hamidi-Rad, Elahe Soltanaghai, Akshay Malhotra

arXiv:2608.18522v1cs.ITcs.AIcs.LG

TL;DR

Large antenna arrays make wireless channel feedback costly, while fixed-size neural codes are tied to matched decoders and antenna configurations. GCNO instead compresses channels into sample-dependent propagation paths and achieves a better reconstruction–payload trade-off across three ray-traced environments while transferring to unseen antenna counts.

  • Problem

    Large antenna arrays make full channel-state feedback costly, motivating compression beyond fixed-size codes tied to antenna configurations and matched neural decoders.

  • Method

    GCNO identifies channel-dependent propagation directions, continuously refines off-grid estimates, recovers complex path strengths by least squares, and reconstructs channels analytically from variable-length path tuples.

  • Results

    Across ASU, Dallas, and Seattle, GCNO achieves lower NMSE at equal payload or lower payload at equal NMSE than eight neural feedback baselines and retains more accuracy on unseen antenna counts.

  • Takeaways & Limitations

    Path-based variable-payload feedback with physical reconstruction provides a better accuracy–feedback-size trade-off and supports antenna-count changes without retraining.

  • Takeaways & Limitations

    The experiments focus on narrowband spatial channels, so wideband extension would require adding delay or an equivalent frequency coordinate to each path.

Abstract

from arXiv · show

Large antenna arrays allow wireless systems to serve more users and achieve higher data rates, but they also make channel feedback expensive: the receiving device must repeatedly report a large complex-valued channel matrix to the base station. Most neural compressors treat this matrix like an image and replace it with a fixed-length code that only a matched neural decoder can interpret. The message therefore does not adapt to channel complexity, and changing the antenna count typically requires retraining. We ask whether a device can instead report only the few dominant propagation paths underlying each channel. We introduce the Gramian Chebyshev Neural Operator (GCNO), a physics-based, variable-rate compressor that identifies a sample-dependent set of path directions. GCNO uses receive-transmit channel structure to locate paths, a first-order Taylor correction to refine directions that fall between grid points, and least squares to recover their complex strengths. It is trained without path labels, and the base station reconstructs the channel analytically from the transmitted path tuples rather than through a learned decoder. Across three ray-traced environments, GCNO achieves better reconstruction accuracy at the same payload - or lower payload at the same accuracy - than neural feedback baselines, and transfers to unseen antenna counts without retraining.

1 Introduction

GCNO compresses wireless channels as variable-length sets of dominant multipath parameters rather than fixed-length antenna-specific neural codes. It refines off-grid path directions, reconstructs analytically, adapts feedback to channel complexity, and transfers across antenna counts.

  • Motivation: Massive-MIMO channel feedback is costly because the base station needs current channel information, motivating compact representations that preserve accuracy across antenna configurations.The central question is how to reduce CSI feedback without sacrificing reconstruction accuracy or tying the method to one antenna configuration.
  • Approach: GCNO represents channels with a variable-length list of dominant paths, using 4K real values for K retained paths and analytical reconstruction from the known array model.This replaces an antenna-specific latent code and matched neural decoder with path directions and complex gains.
  • Method: GCNO identifies likely path directions, refines selected grid points continuously, retains only paths improving reconstruction, and estimates their complex gains with least squares.The adaptive path count K allows the feedback payload to vary with channel complexity.
  • Evaluation: Across ASU, Dallas, and Seattle ray-traced channels, GCNO achieves lower NMSE at the same payload or lower payload at the same NMSE than eight neural feedback baselines.The comparison covers the full NMSE–payload trade-off.
  • Generalization and ablation: GCNO retains substantially more accuracy on unseen antenna counts, while alternative neural operators weaken transfer or the NMSE–payload trade-off.Removing continuous refinement also clearly worsens NMSE, confirming its necessity for compact path feedback.

2 Related Work

Prior learned CSI feedback methods use fixed-length codes and paired neural decoders, while GCNO instead represents channels through variable sets of path gains and effective spatial directions. This work also draws on neural operators and sparse off-grid recovery.

  • Neural CSI feedback and learned compression: CsiNet established encoder–decoder CSI compression, followed by residual, dilated-convolution, and transformer-based improvements.These methods compress real and imaginary CSI into a dense code and reconstruct it with a paired base-station network.
  • Research foundations: The work draws on learned CSI feedback, neural operators, and sparse off-grid recovery.The passage introduces neural operators, including FNO and DeepONet, as a related research direction.
  • Neural CSI feedback and learned compression: Fixed-length codes tied to trained decoders do not naturally follow channel complexity, and antenna changes usually require a matched model.GCNO instead sends a variable set of path gains and effective spatial directions, which the receiver converts to CSI using array geometry.

3 Problem Formulation

The formulation compresses each channel by selecting a sample-dependent set of dominant propagation paths and reporting their gains and spatial directions. This yields a variable-length, four-real-value-per-path representation that the base station reconstructs analytically across antenna sizes.

  • 3 Problem Formulation: The compression objective is to minimize reported values while preserving accurate base-station reconstruction by selecting the number K and parameters of retained paths.Each retained path is described by one complex gain and two spatial directions.
  • 3.1 Physical Path Representation: A propagation path creates a predictable phase pattern across an antenna array, called its array response, and the geometric channel sums these path contributions.For the considered uniform linear arrays, direction is represented by spatial coordinate u ∈[−1, 1].
  • 3.1 Physical Path Representation: The reported direction uses the effective spatial angle ψ = arcsin(u), which is the direction seen by the array.The angle is converted from the spatial coordinate used in the array response.
  • 3.1 Physical Path Representation: The geometric channel contains K⋆ paths with complex gains and receive-transmit spatial directions, but compression retains only K ≤K⋆ dominant paths.Most channel energy is often concentrated in a smaller set, which can contain five or fewer paths in outdoor and high-frequency channels.
  • 3.2 Path-Based Feedback and Reconstruction: For each retained path, the device reports its complex gain and receive and transmit effective spatial angles as the channel description.This directly represents CSI feedback as a list of retained path parameters.
  • 3.2 Path-Based Feedback and Reconstruction: Each path requires four real values, so the feedback size is 4K and adapts per channel according to the number of paths needed for accurate representation.Channels accurately represented with fewer paths require fewer reported values than channels requiring more paths.
  • 3.2 Path-Based Feedback and Reconstruction: At the base station, effective angles are converted to spatial coordinates and the channel is reconstructed by evaluating the same path-based array-response model.The feedback format and reconstruction rule remain usable when antenna count changes because they do not require a neural decoder trained for one fixed configuration.

4 GCNO for Physics-Based CSI Compression

GCNO compresses each wireless channel into a sample-dependent set of path directions and complex gains by combining dictionary-based evidence with Gramian Chebyshev filtering. It refines off-grid directions, selects useful paths, and reconstructs the channel analytically through least squares without path labels or a learned decoder.

  • Label-free training: Differentiable soft selection and ridge LS train GCNO end-to-end from channel reconstruction error while avoiding ground-truth path directions, gains, or path counts.The training objective also penalizes unnecessary, repeated, edge-near, and broadly activated candidates.
  • Direction-space representation: A receive–transmit dictionary and Frobenius inner products produce an evidence map whose large-magnitude entries indicate channel components consistent with candidate direction pairs.The grid is an intermediate search space, while fixed analytical dictionary atoms make channel reconstruction a linear combination of known matrices.
  • Gramian Chebyshev operator: GCNO recomputes receive- and transmit-side Gramians for each input channel and filters features with learned low-order Chebyshev polynomials.The model uses three layers and polynomial order up to Q = 3, while learned coefficients remain independent of particular antenna indices.
  • Off-grid refinement: GCNO outputs candidate scores and receive–transmit corrections, then uses first-order Taylor patterns to refine coarse grid directions within bounded half-cell intervals.The dictionary and derivative dictionaries are analytical, non-trainable tensors, avoiding differentiation through newly generated steering functions.
  • Variable-rate path selection: At inference, candidates are scanned by descending score, redundant patterns are rejected, and ridge LS jointly refits gains while retaining only additions that sufficiently reduce reconstruction error.This procedure lets the retained path count vary across channels rather than fixing K in advance.
  • Analytical reconstruction: The final message contains the retained effective spatial angles and jointly fitted complex gains as a 4K-value tuple, which the base station decodes using the same analytical array-response model.The base station does not run a learned decoder, and joint fitting accounts for overlap among retained paths.

5 Experimental Protocol

The experiments evaluate GCNO across three ray-traced DeepMIMO scenarios using disjoint train, validation, and test channels, and compare it with eight learned encoder–decoder baselines. Performance is measured by NMSE and feedback rate under analytical path-tuple reconstruction for GCNO versus neural decoding for baselines.

  • Datasets and splits: Experiments use ASU, Dallas, and Seattle ray-traced DeepMIMO scenarios, with 10,000 training, 2,000 validation, and 1,500 test channels per scenario.A separate model is trained for each scenario, and test channels remain untouched until all choices are fixed.
  • Compared methods: GCNO is compared against eight paired encoder–decoder models spanning convolutional, multi-resolution, complex-input, MLP, global-attention, windowed-attention, and compact designs.The baselines are CsiNet, CRNet, CLNet, CSITransformer, TransNet, M-Net, SwinCFNet, and StarCANet.
  • Metrics and reconstruction: NMSE in decibels measures reconstruction error relative to channel energy, with median NMSE and its empirical CDF reported.GCNO reconstructs from transmitted path tuples, whereas each baseline uses its paired neural decoder.
  • Metrics and reconstruction: Unquantized rate counts transmitted real values, averaging 4K for GCNO and using latent length for each baseline; quantized rate counts all transmitted bits.Quantized feedback includes the path-count header and quantized tuple or latent data.
  • Training and selection: Training and model selection use only label-free reconstruction and rate, excluding path counts, directions, gains, oracle quantities, and test results.Appendix B documents dataset, optimization, training, quantization, parameter-count, hardware, software, runtime, and selection details.

6 Results

GCNO provides a variable-rate path-based compressor with a strong accuracy–payload trade-off across environments and rates. Its Taylor refinement, learned backbone, and geometry-aware design support accurate reconstruction and transfer to unseen antenna configurations without retraining.

  • Comparative results: GCNO remains on the best quantized rate–distortion frontier across tested bit budgets and preserves its ordering across Seattle and Dallas at quantized and unquantized rates.The improvement spans the per-channel error distribution at 16 floats rather than coming only from a small number of easy channels.
  • Ablation: Removing Taylor offsets causes a large, consistent loss across all three environments and substantially less accurate directions, despite unchanged support selection and payload rules.The ablation tests whether retraining can compensate for the absence of continuous off-grid correction.
  • Ablation: Replacing only GCNO’s native layers weakens the rate–distortion trade-off against CNN, dilated-CNN, FNO, static Chebyshev, and learned-polynomial alternatives.Because all replacements produce the same path payload, the result does not attribute the gain solely to the analytical decoder.
  • Variable-rate behavior: As requested accuracy becomes stricter, GCNO selects paths for a growing fraction of channels, while the strictest setting exposes diminishing reconstruction returns as payload continues increasing.Transmitted directions remain accurate as the number of strong diagnostic paths increases, although recall is lower.
  • Generalization: A model trained at 32 × 32 retains strong reconstruction quality across six changed square and rectangular array configurations without retraining, unlike paired neural baselines.GCNO shares learned polynomial coefficients across sizes and recomputes physical projections and reconstruction patterns for each new geometry.

7 Conclusion … A.2 Minimum Dimension of an Exact Fixed Latent

Across three ray-traced environments, GCNO supports variable-rate, path-based wireless-channel compression using channel-dependent directions, Taylor refinement, and least-squares path-strength estimation. The appendix shows that, under local identifiability, exact representation of an identifiable K-path channel requires at least 4K real latent coordinates, attained by the physical path tuple.

  • 7 Conclusion: Across ASU, Dallas, and Seattle, GCNO supports compressing channels through retained dominant propagation paths rather than a fixed-size code.GCNO reports channel-dependent path tuples after direction refinement and strength estimation.
  • 7 Conclusion: Wideband extension requires adding delay, or an equivalent frequency coordinate, to each path.Nearby subcarriers retain nearly constant path directions and delays, while gains change structurally.
  • A When Is Path-Based Feedback Minimal?: The appendix analyzes the ideal unquantized K-path component, with K denoting retained paths and K⋆ denoting total geometric paths.It asks how many real coordinates are needed for exact representation.
  • A.1 Identifiable K-Path Channels: Under local identifiability, each retained path contributes four locally independent real perturbations, giving a 4K-dimensional parameterization.The assumptions exclude coincident or locally unresolvable paths and require distinct directions within the unaliased field of view.
  • A.2 Minimum Dimension of an Exact Fixed Latent: Theorem A.1 considers differentiable encoders and decoders that exactly reconstruct every identifiable K-path channel in a neighborhood.The result concerns exact local reconstruction through a fixed latent space.
  • A.2 Minimum Dimension of an Exact Fixed Latent: 4K ≤ m: differentiability and full Jacobian rank imply that any exact fixed latent must have at least 4K real dimensions.The decoder-side map has rank 4K, while a map factoring through an m-dimensional latent has rank at most m.
  • A.2 Minimum Dimension of an Exact Fixed Latent: The physical path tuple attains the lower bound with exactly 4K real coordinates and reconstructs the channel exactly under local identifiability.Reporting direction coordinates as ψ = arcsin(u) preserves the coordinate count within the field of view.

A.3 Variable Path Count Versus a Fixed-Width Autoencoder … B.6 Layer-by-Layer GCNO Architecture

The appendix establishes that variable path messages can be strictly more compact than fixed-width representations under an ideal identifiable model, while detailing GCNO’s physics-aligned, size-independent architecture and analytical path reconstruction. It connects Gramian–Chebyshev filtering to multipath footprints, specifies the complete processing pipeline, and documents the implemented layer structure.

  • A.3 Variable Path Count Versus a Fixed-Width Autoencoder: The path representation uses fewer real values whenever K < Kmax and has strictly smaller average payload when Pr[K < Kmax] > 0.At K = Kmax, it meets the 4Kmax lower bound; the result is limited to differentiable, locally exact codecs for the identifiable geometric component before quantization.
  • B.1 Multipath Footprints in the Evidence Map: A K-path channel produces at most K separable matched-filter footprints, whose Gramian active subspaces lie in the spans of physical receive and transmit footprints.Diagonal terms encode individual path strengths, while off-diagonal terms encode overlap between nearby footprints; omitted or diffuse energy adds linearly to the evidence map.
  • B.2 Normalized Gramian–Chebyshev Filtering: GCNO normalizes positive-semidefinite Gramians and applies a third-order Chebyshev recurrence, with learned coefficients indexed by feature channels and polynomial orders rather than array locations.The filtering avoids eigendecomposition, and array or grid sizes enter through analytical projections and sample-specific Gramians while learned coefficients remain unchanged.
  • B.3 Neural-Operator Characterization: The function-space GCNO core discretizes to the bilateral matrix filter under uniform sampling, enabling compatible realizations across admissible samplings without location-specific learned parameters.This is an architectural transfer property, not a claim of exact equality across resolutions or arbitrary array geometries; changed-discretization quality remains empirical.
  • B.4 Physical Interpretation of the Chebyshev Core: The linear GCNO core mixes receive and transmit modes already present in its current features; for separated physical footprints, it applies a learned bivariate polynomial response path by path.With overlapping footprints, Gramian modes can mix nearby paths, and the characterization applies to the physics-aligned linear core rather than every nonlinear hidden feature.
  • B.5 Complete Processing Pipeline: The complete pipeline normalizes the channel, combines analytical matched-filter evidence with channel features, and predicts support priorities and local direction corrections rather than gains.During deployment, hard support selection generates exact analytical atoms and jointly fits gains, while the base station receives path tuples instead of a neural-decoder reconstruction.
  • B.6 Layer-by-Layer GCNO Architecture: The implemented architecture uses separate channel and evidence stems, fuses their 24-channel complex state with coordinate maps, and passes it through exactly three bilateral GCNO layers.Each layer uses Chebyshev order three, residual addition, and GELU; a LocalGate follows each layer to provide local sharpening and channel-wise modulation.
  • B.6 Layer-by-Layer GCNO Architecture: The support-and-offset head outputs an unconstrained score map and bounded receive/transmit corrections, with no gain head; the checkpoint has 95,253 trainable real scalars.Fixed dictionaries, coordinate maps, Taylor construction, support selection, and least squares contain no trainable parameters.

B.7 Training Objective and Optimization · B.8 Adaptive Deployment and LS Reconstruction · B.9 Quantization and Bit Accounting

GCNO is trained end-to-end without path labels or oracle information, then deployed with deterministic adaptive support selection and analytical least-squares reconstruction. Quantized packets transmit only retained path tuples and a sample-dependent path-count header, with validation-frozen coding choices.

  • B.7 Training Objective and Optimization: GCNO training is strictly label-free, using random initialization without pretrained checkpoints, pseudolabels, path labels, oracle quantities, or external warm starts.The same restriction applies to validation, checkpoint selection, the controller, and operating-point selection.
  • B.7 Training Objective and Optimization: Training exposes at most M = 8 soft candidates, with temperature-controlled selection, Gaussian neighborhood suppression, sigmoid activities, and differentiable retained-path estimates.The Gaussian penalty uses width 0.8 grid cells and logit penalty 8.5 to encourage distinct candidate regions.
  • B.7 Training Objective and Optimization: The objective discourages unnecessary paths, duplicate atoms, cell-edge corrections, and broad score activation, while no label-derived quantity appears in any term.Checkpoint and operating-point choices rely on label-free validation reconstruction and rate quantities rather than test-channel or oracle metrics.
  • B.8 Adaptive Deployment and LS Reconstruction: Deployment replaces smooth training selection with a deterministic no_grad hard selector that ranks candidates, applies Taylor corrections, rejects duplicates, and generates exact analytical steering atoms.Candidates are temporarily added and jointly refit by ridge LS before admission.
  • B.9 Quantization and Bit Accounting: Unquantized rate is the measured mean 4E[K], whereas quantized rate is the complete packet length for the path-count header and every retained tuple field.Quantized rate is not computed by multiplying an unquantized payload by a fixed bit count.
  • B.9 Quantization and Bit Accounting: Quantizer profiles, scalar codebooks, support thresholds, and ranges are fitted on validation data and frozen before testing, while packets transmit only path tuples and the path-count header.After dequantization, the base station performs the same analytical reconstruction as in the unquantized setting.

B.10 Reproducibility and Computational Cost · C Extended Results and Analysis · C.1 Complete Rate–Distortion Curves

The appendix specifies reproducible data, training, environment, and timing protocols, then extends rate–distortion analysis across Seattle and Dallas while contextualizing model size separately from payload. Across three propagation environments, the complete curves support the reported broad rate–distortion ordering and show that it is not specific to ASU.

  • B.10 Reproducibility and Computational Cost: The reference ASU experiment uses 10,000 training, 2,000 validation, and 1,500 held-out test channels with 32 × 32 arrays and 512 OFDM subcarriers.Dallas and Seattle use the same split sizes and are trained separately.
  • B.10 Reproducibility and Computational Cost: Training uses only normalized channels and scale, while path annotations and oracle quantities remain unavailable to the objective and test data stays validation-independent.Architecture, checkpoint, quantizer, and operating profiles are fixed using validation-only quantities before testing.
  • B.10 Reproducibility and Computational Cost: The frozen-model timing reproduction preserves ASU R = 28 on an otherwise idle NVIDIA H100 with 80 GB HBM3 and the specified Python, PyTorch, CUDA, and NumPy versions.The benchmark starts from random initialization, disables AMP, and follows the frozen 71-epoch controller trajectory.
  • B.10 Reproducibility and Computational Cost: The synchronized full-fit measurement processes 5,609 optimizer updates, corresponding to 710,000 sample presentations, without replacing or selecting the paper checkpoint.Timing includes in-loop soft validation and normal checkpoint writes; complete UE-encoder latency includes normalization, GCNO inference, adaptive selection, steering generation, joint LS, and coordinate polishing.
  • C.1 Complete Rate–Distortion Curves: The appendix adds complete Seattle and Dallas rate–distortion curves for all eight paired encoder–decoder baselines, while retaining ASU only for a shared legend and scale.Main-paper figures and tables provide the headline comparison and representative operating points; this appendix adds curves, controls, and analyses.
  • C.1 Complete Rate–Distortion Curves: The broad rate–distortion ordering is consistent across three distinct propagation maps, indicating that the observed behavior is not specific to ASU.Seattle is the closest comparison at the smallest unquantized budget, motivating the targeted seed analysis in Sec. C.7.
  • C Extended Results and Analysis: Trainable parameter counts describe representative instantiated model sizes and are distinct from transmitted payload.Some paired autoencoders change size with their instantiated input or bottleneck dimensions, so the counts are not configuration-independent architecture constants.

C.2 Classical Grid Recovery and Extended Taylor Analysis

GCNO is evaluated against classical Grid-OMP recovery and through additional Taylor-refinement tests beyond the main operating point. Grid-OMP is less accurate under the matched payload protocol and substantially slower in the reported batch-1 H100 timing study, while Taylor refinement consistently improves NMSE.

  • Classical Grid Recovery: Grid-OMP uses the same fixed direction grid, least-squares gain refitting, fixed validation stopping profiles, and 4K real-value payload accounting as the control protocol.Its reconstruction accuracy improves monotonically with more retained atoms but remains substantially below GCNO under this protocol.
  • Runtime Comparison: Approximately 67 ms per channel, Grid-OMP required substantially more batch-1 H100 runtime than GCNO’s reported 15.425 ms mid profile.The comparison motivates low UE-side processing latency, while the cited NR requirements do not impose a universal inference deadline.
  • Extended Taylor Analysis: At 4, 8, and 32 transmitted values, Taylor refinement produced positive NMSE improvement in all nine additional comparisons, with gains of at least 6.628 dB.These results use a separately retrained no-Taylor control with identical support-selection and payload rules.

C.3 Physical Multipath Fidelity

This section evaluates whether GCNO’s compact transmitted path tuples preserve dominant multipath structure relative to two neural baselines. Under a joint direction-matching diagnostic, GCNO achieves a weighted strong-path recall of 0.916, supporting preservation of dominant resolvable structure without implying recovery of every annotated ray.

  • Comparison setup: Figure 6 compares preservation of dominant multipath structure in GCNO reconstructions against StarCANet and SwinCFNet.The comparison uses the two closest neural baselines.
  • Diagnostic protocol: Strong diagnostic paths lie within 10 dB of the strongest annotated path, with receive–transmit direction pairs jointly matched within 3◦.GCNO is evaluated directly from transmitted tuples, while the neural baselines receive a frozen offline path probe applied only for this diagnostic.
  • Result: 0.916 weighted strong-path recall is achieved by GCNO under the matching protocol.The result supports the narrower conclusion that compact tuples preserve dominant resolvable structure well.
  • Interpretation limits: The diagnostic does not establish recovery of every annotated ray or imply that StarCANet and SwinCFNet internally represent channels as paths.Their offline path probe does not change transmitted payloads or NMSE.

C.4 Protocol for the Array-Size Results in Main-Paper Fig. 4(b) … C.7 Seattle Seed Stability under a Baseline-Favorable Envelope

The evaluation protocol tests frozen GCNO and neural baselines across array sizes and propagation scenes without target-specific retraining, then measures beamforming utility and Seattle seed stability. GCNO transfers across the tested conditions, preserves dominant transmit-subspace utility, and maintains its ordering under a baseline-favorable seed comparison.

  • C.4 Protocol for the Array-Size Results in Main-Paper Fig. 4(b): Seven paired array shapes use identical physical links and locked 10,000/2,000/1,500-channel train/validation/test splits, with no target-shape samples used for adaptation.The ordered, unshuffled test split is used for evaluation.
  • C.4 Protocol for the Array-Size Results in Main-Paper Fig. 4(b): GCNO transfers its 32×32 checkpoint directly to every target shape while recomputing dimension-dependent dictionaries and retaining learned weights unchanged.Frozen baselines retain their 32×32 architectures and fixed latent length M = 16 real values.
  • C.4 Protocol for the Array-Size Results in Main-Paper Fig. 4(b): The deterministic, parameter-free adapter gives baselines a fixed 16-real-value payload and compression 2NrNt/16, with exact identity at 32×32.The comparison therefore evaluates frozen-model reuse under one common nonlearned dimension-matching rule.
  • C.5 Cross-Scene Transfer Without Retraining: GCNO obtains the lowest median NMSE in all six source–target scene transfers, improving over SwinCFNet by 0.166–8.439 dB without target-scene retraining.The result supports more robust transfer under the tested scene shifts, not invariance to arbitrary propagation environments.
  • C.6 Beamforming Utility: At 20 dB, GCNO reaches 6.5319 bit/s/Hz for one stream and 6.8240 bit/s/Hz for two streams, while dominant-vector alignment has mean 0.9973, median 0.9995, and fifth percentile 0.9901.The one-stream result is approximately 0.004 bit/s/Hz below perfect CSI, and the two-stream perfect-CSI reference is 6.9644 bit/s/Hz.
  • C.7 Seattle Seed Stability under a Baseline-Favorable Envelope: Seattle seed sensitivity is evaluated with seeds 43, 44, and 45 on the same fixed 1,500-channel test split because Seattle has the smallest main-comparison separation.The check targets stability rather than exhaustive characterization of all random seeds.
  • C.7 Seattle Seed Stability under a Baseline-Favorable Envelope: Even when each baseline uses its best seed independently at every operating point, GCNO’s ordering remains unchanged across seeds, payload targets, and comparators.The smallest observed margin is 0.6463 dB over SwinCFNet and 2.9808 dB over StarCANet.
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