Error Propagation Resistance Overview

• Error propagation resistance is the capacity to prevent local inaccuracies from accumulating into persistent failures in sequential or coupled systems.
• It employs techniques such as reliability flow reinforcement, adaptive windowing, and anchor resets to mitigate cascading errors in communication, diffusion, and quantum circuits.
• Practical implementations in channel decoding, generative models, video compression, and quantum processing demonstrate that operator-level adjustments are key to controlling cumulative error effects.

Error propagation resistance denotes the capacity of a system, algorithm, or decoder to prevent local inaccuracies from accumulating into persistent downstream failures. Across the literature, the phenomenon appears when a sequential or coupled process reuses its own imperfect outputs: sliding-window decoders forward unreliable log-likelihood ratios, diffusion samplers reuse quantized predictions over many denoising steps, autoregressive predictors condition on past residuals, inter-frame codecs predict from previously reconstructed frames, and quantum or classical circuits expose later computation to earlier faults. The shared technical issue is not merely the presence of error, but the existence of a propagation channel that converts local perturbations into structured, temporally extended degradation. Correspondingly, resistance is achieved either by strengthening reliability flow, attenuating propagation operators, inserting anchors or resets, or training the system on trajectories that explicitly include its own mistakes (Zhu et al., 2018, Zhu et al., 2020, Li et al., 2023, Liu et al., 16 Aug 2025).

1. Propagation as a structural phenomenon

In the coding and sequential-inference literature, error propagation is typically tied to explicit recurrence or coupling. In blockwise sparsely braided convolutional codes (SBCCs), each information block is protected by two recursive systematic convolutional component encoders, and parity outputs at time $t$ feed back as inputs at time $t+1$ after permutation; the coupling memory is therefore $m=1$, and decoder reliability at block $t+1$ depends on soft information from block $t$ (Zhu et al., 2018). In sliding-window decoding of spatially coupled LDPC codes, the final LLRs of previously decoded blocks remain embedded in parity-check constraints inside the current window; if those LLRs have large wrong-sign magnitude, they contaminate later check-node updates and may trigger a cascade of block errors (Zhu et al., 2020).

A closely related formulation appears in diffusion models, where the backward denoising process is a chain of $T$ modules. The paper on diffusion-model error propagation defines a modular error,

$$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$

and a cumulative error,

$$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$

then shows the propagation inequality

$$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$

which formalizes accumulation across timesteps (Li et al., 2023). Quantized diffusion sampling exhibits an analogous recurrence at the sampler level: for DDIM, the cumulative state perturbation obeys

$\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$

where $t+1$0 is the per-step quantization error and $t+1$1 are timestep-dependent propagation coefficients derived from the sampler update (Liu et al., 16 Aug 2025).

In recurrent forecasting and encoder–decoder quantization, the same structure appears through Jacobian-mediated recursions. For ASR post-training quantization, linearization yields

$t+1$2

so activation error may grow with $t+1$3 unless later layers compensate for upstream distortion (Wang et al., 5 Jan 2026). In spatio-temporal forecasting, residuals are modeled as jointly correlated across time and space specifically because early residuals perturb latent states and compound over long horizons (Moghadas et al., 18 May 2026). This suggests that error propagation resistance is best understood as a property of the full state-update operator rather than of isolated component accuracy.

2. Reliability flow, windowing, and resets in channel decoding

The most explicit engineering treatments of error propagation resistance appear in sliding-window decoding. For SBCCs, low-magnitude a posteriori LLRs in the oldest block of the window produce unreliable extrinsic messages for subsequent blocks; because the component encoders are recursive, this unreliable soft information can be reinforced through feedback, producing a stall (Zhu et al., 2018). The paper defines the average absolute reliability of block $t+1$4 after horizontal iteration $t+1$5 as

$t+1$6

and enlarges the decoding window whenever one of the first $t+1$7 blocks satisfies

$t+1$8

With $t+1$9, $m=1$0, and $m=1$1, the window extension algorithm yields roughly one order of magnitude BER/BLER/FER improvement over a fixed $m=1$2 baseline, while average window size remains nearly unchanged when stalls are rare; adding a resynchronization mechanism with $m=1$3 improves BER and BLER by up to three orders of magnitude and FER by about one order of magnitude (Zhu et al., 2018).

Spatially coupled LDPC decoding uses a conceptually parallel strategy. Error propagation in SWD arises when previously decoded blocks remain inside the current parity-check constraints but are no longer updated, so incorrect high-confidence LLRs at the window boundary contaminate subsequent belief propagation (Zhu et al., 2020). The paper introduces two forms of internal anchoring. Check-node doping inserts occasional reduced-degree CNs that locally emulate a termination boundary, while variable-node doping fixes occasional VN blocks to known values and maintains their LLRs at a saturated constant, thereby forcing neighboring CNs to emit strong messages without modifying the decoder schedule (Zhu et al., 2020). For the $m=1$4 SC-LDPC code with $m=1$5 and $m=1$6, both CN and VN doping improve BER by up to approximately two orders of magnitude and BLER by approximately one order of magnitude relative to undoped chains; the VN-doped design performs essentially equivalently to CN doping while avoiding window-shape modification (Zhu et al., 2020).

The same paper also proposes an LLR-based window extension rule. If any of the first $m=1$7 blocks in the window has average LLR magnitude below a threshold $m=1$8, the decoder accepts two additional blocks and restarts BP, up to $m=1$9 (Zhu et al., 2020). With $t+1$0 and $t+1$1, this dynamic windowing yields more than approximately two orders of magnitude BER and BLER improvement, while the average window size—and hence average latency—drops by about one-third relative to fixed $t+1$2 decoding with comparable performance (Zhu et al., 2020). Across both SBCCs and SC-LDPC codes, the recurring design pattern is that resistance is improved by restoring a reliable left boundary, either adaptively by enlarging context or structurally by introducing internal anchors.

3. Training and compensation against accumulation in generative, compression, and quantized models

In diffusion models, resistance has been formulated directly as cumulative-error control. The MMD-based regularizer introduced for denoising models penalizes

$t+1$3

with weights $t+1$4, thereby placing greater emphasis near the output end of the chain (Li et al., 2023). Using $t+1$5, $t+1$6, $t+1$7, $t+1$8, and bootstrap length $t+1$9, the reported FID improves from $t$0 to $t$1 on CIFAR-10, from $t$2 to $t$3 on ImageNet, and from $t$4 to $t$5 on CelebA relative to vanilla DDPM (Li et al., 2023). Quantized diffusion then extends the analysis from model error to quantization error along the sampling trajectory. The timestep-aware cumulative error compensation scheme reconstructs $t$6 as

$t$7

and applies a local correction

$t$8

after showing that, under a contraction argument, choosing $t$9 suffices to obtain a strictly smaller upper bound on the corrected cumulative error (Liu et al., 16 Aug 2025). On SDXL W4A4, this raises sDCI PSNR from $T$0 to $T$1 and reduces MJHQ FID from $T$2 to $T$3 relative to SVDQuant; the reported end-to-end latency overhead is at most $T$4 (Liu et al., 16 Aug 2025).

Content-adaptive video compression addresses the same issue in inter predictive coding, where current frames are predicted from previously reconstructed frames. The proposed multi-frame training objective

$T$5

makes gradients flow through the reference chain, so the encoder learns reconstructions that are not only accurate for the current frame but also strong references for future frames (Lu et al., 2020). With EPA training, the PSNR at the $T$6th frame improves by $T$7 dB over DVC, and bitrate savings relative to DVC grow with GoP size, reaching $T$8 at GoP $T$9 (Lu et al., 2020). The same system adds online encoder updating, optimizing encoder parameters at inference under a fixed decoder, which reduces domain gap and further improves the quality of reconstructed references (Lu et al., 2020).

Quantization-aware resistance in heterogeneous encoder–decoder ASR models is addressed by FADE, which replaces QEP’s single $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$0 with a layerwise coefficient

$$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$1

where $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$2 combines intrinsic vulnerability, calibration gain, and a stability penalty (Wang et al., 5 Jan 2026). With defaults $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$3, $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$4, and $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$5, FADE improves both mean WER and run-to-run stability. On Whisper-Tiny at 3-bit quantization, GPTQ+QEP gives $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$6, whereas FADE yields $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$7; on Whisper-Tiny at 4-bit, GPTQ+QEP gives $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$8 and FADE $$\mathcal{E}_t^{\mathrm{mod}}= \mathbb{E}_{\mathbf{x}_{t}\sim p_\theta(\mathbf{x}_{t})} \Big[ D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1}\mid \mathbf{x}_t)\,\|\,q(\mathbf{x}_{t-1}\mid \mathbf{x}_t) \big) \Big],$$9 (Wang et al., 5 Jan 2026). A plausible implication is that resistance in quantized sequential models depends as much on variance control as on bias correction.

4. Propagation resistance in quantum information processing

Quantum work on error propagation resistance splits into two distinct lines: one studies how faults spread through entangling gates in noisy hardware, and another exploits quantum norm preservation to argue that some forms of poisoning do not amplify as they do in classical pipelines. In NISQ variational circuits for classical optimization, single-qubit errors inserted near the circuit midpoint spread through random two-qubit gates. After averaging over Haar-random two-qubit layers, a one-qubit error propagates to its partner with probability $$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$0, and the expected fraction of depolarized qubits in deep regimes scales as

$$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$1

leading to the requirement $$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$2 for the possibility of quantum advantage (González-García et al., 2022). For 2D QAOA with $$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$3 and estimated depth $$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$4, the resulting single-qubit error-rate condition is approximately $$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$5, matching the paper’s statement that practical 2D QAOA would require an error rate lower than $$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$6 (González-García et al., 2022). This work therefore defines resistance as a shrinking tolerance budget under scaling.

Compilation-level mitigation adopts a different perspective. In Pauli error propagation-based gate rescheduling, the circuit depth and gate count are fixed, but commuting gates are reordered so that noisy gates lie later in dependency chains and have fewer downstream dependents (Saravanan et al., 2022). The central surrogate is the weighted estimated success probability,

$$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$7

with $$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$8 based on downstream reachability (Saravanan et al., 2022). On a 5-qubit BV case study, two schedules with identical ESP produce PST $$\mathcal{E}_t^{\mathrm{cumu}}= D_{\mathrm{KL}}\big( p_\theta(\mathbf{x}_{t-1})\,\|\,q(\mathbf{x}_{t-1}) \big),$$9 and $$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$0, while WESP correctly distinguishes them; on IBM Melbourne QAOA experiments, post-mapping rescheduling improves AR and reduces ARG without increasing depth (Saravanan et al., 2022). Here resistance is increased not by reducing raw gate error, but by shortening the paths along which those errors can spread.

A complementary analytic result bounds state-level propagation under common gate-error models. For quantum circuits with probabilistic or Kraus gate noise, the propagation error in Frobenius norm satisfies

$$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$1

where $$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$2 is independent of qubit number and circuit depth beyond the gate count $$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$3 (Yu et al., 2022). The bound is tight in simulator experiments, shows linear growth $$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$4 when $$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$5, and saturates to $$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$6 as $$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$7 (Yu et al., 2022). By contrast, a different quantum-information argument emphasizes that unitary layers do not amplify distinguishability: for density matrices,

$$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$8

and CPTP maps are contractive in trace distance (Sultanow et al., 2024). This suggests that “propagation” in quantum settings is model-dependent: entangling hardware can spread local faults globally, while norm-preserving state evolution still prevents the kind of operator-norm blow-up seen in deep classical models.

5. Graph structure, residual correlation, and other domain-specific manifestations

In probabilistic spatio-temporal forecasting, resistance is linked to whether the graph supports or suppresses the spread of residual uncertainty. Teger models batch residuals with covariance

$$\mathcal{E}_t^{\mathrm{cumu}} \ge \mathcal{E}_{t+1}^{\mathrm{cumu}}+\mathcal{E}_t^{\mathrm{mod}},$$9

and then strengthens negatively curved bottleneck edges by

$\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$0

where curvature is measured by Balanced Forman curvature (Moghadas et al., 18 May 2026). The paper proves that the resulting Laplacian update is positive semidefinite, increases spectral connectivity, and reduces the Kirchhoff index; empirically, CRPS improves consistently across LSTM, Transformer, and xLSTM backbones on four traffic datasets (Moghadas et al., 18 May 2026). This suggests that over-squashing and error propagation are closely related: when a graph bottleneck prevents uncertainty from being shared, local residuals are under-modeled and compound during rollout.

Mechanical and inverse-problem settings exhibit analogous operator-level effects. In velocimetry-based pressure reconstruction, the error Poisson equation

$\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$1

implies spectral amplification $\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$2, so low-frequency source-error modes are amplified most strongly (Faiella et al., 2016). The worst-case mode is the principal eigenfunction of a fourth-order problem equivalent to an Euler–Bernoulli beam in one dimension or a Kirchhoff–Love plate in two dimensions, and the analysis shows that Dirichlet boundaries suppress propagation while Neumann boundaries magnify nearby error (Faiella et al., 2016). In auxetic materials, a localized shear transformation generates a quadrupolar far-field redistribution

$\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$3

so the amplitude of stress propagation decreases linearly with Poisson ratio and vanishes in the limit $\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$4 (Fielding, 2023). The reported consequence is increased yield strain and reduced tendency for plastic damage to percolate into system-spanning clusters (Fielding, 2023). In both cases, resistance is controlled by the spectrum or Green’s function of the underlying propagation operator.

6. Metrics, interventions, and recurring design principles

Across domains, error propagation resistance is measured either by end-task performance under recurrence or by explicit propagation statistics. In SBCC decoding, the key internal statistics are $\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$5 and the soft-BER stopping estimate

$\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$6

used to terminate decoding when $\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$7 (Zhu et al., 2018). In SC-LDPC sliding-window decoding, average block LLR magnitude identifies imminent stalls; in diffusion models, cumulative KL or its MMD surrogate quantifies accumulation; in quantized diffusion, $\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$8 is explicitly reconstructed; in circuit SER analysis, the functional sensitization probability

$\delta_{t-1}=A_t\delta_t+B_t\varepsilon_t,$9

measures the chance that an upset at node $t+1$00 reaches some output (0710.4712). In cooperative communications, degradation is seen in the loss of the nominal $t+1$01 diversity order under DF relay error propagation (Sanli et al., 2018). In surface-code decoding, accurate modeling of propagated fault signatures doubles the number of faults a distance-$t+1$02 code can reliably correct, from $t+1$03 to $t+1$04, and yields a quadratic improvement in logical error rate at low physical error rates (Fowler et al., 2010).

The interventions themselves fall into a small number of repeated classes. One class inserts reliable anchors or resets: VN/CN doping in SC-LDPC codes, resynchronization in SBCCs, known-bit injection in VN doping, or chain restarts in streaming decoders (Zhu et al., 2018, Zhu et al., 2020). A second class adds context or redistributes computation: window extension in both SBCCs and SC-LDPC codes, multi-frame training in video compression, multi-step regularization in diffusion, and curvature-aware rewiring in spatio-temporal models (Zhu et al., 2018, Zhu et al., 2020, Lu et al., 2020, Li et al., 2023, Moghadas et al., 18 May 2026). A third class attenuates propagation by modifying the local operator: timestep-aware compensation in quantized diffusion, layerwise $t+1$05 control in ASR PTQ, or scheduling noisy quantum gates later in dependency chains (Liu et al., 16 Aug 2025, Wang et al., 5 Jan 2026, Saravanan et al., 2022). A fourth class improves diagnosis rather than suppression alone: EPSTG/RSG constructions model how errors traverse circuits in space-time, and the shift-of-distribution measure quantifies how much a circuit increases expected syndrome errors relative to an empty circuit (Ye, 2024).

A common misconception is that local accuracy alone guarantees resistance. The surveyed results repeatedly contradict that view. Increasing local iterations in a too-small BCC decoding window only delays stalls marginally (Zhu et al., 2018); fixed per-layer PTQ objectives can still destabilize encoder–decoder ASR because they ignore cross-layer heterogeneity (Wang et al., 5 Jan 2026); and a quantum circuit with unchanged depth and gate count can still vary substantially in success probability depending only on gate order (Saravanan et al., 2022). The more robust interpretation is that resistance is an emergent property of trajectory geometry, coupling structure, and recovery mechanisms. Where a system reuses its own outputs, one must analyze the induced propagation graph, recurrence, or spectrum rather than only the fidelity of isolated components.