Cross-Step Consistency Control (CSCC)

• Cross-Step Consistency Control (CSCC) is a framework that maintains targeted feature coherence across consecutive steps in both generative modeling and robotic control.
• It uses token fusion and reachability analysis to couple local step data with global context, ensuring ingredient fidelity in recipes and state boundedness in robotics.
• Empirical studies show that proper parameter tuning in CSCC, such as optimal lambda settings, significantly improves cross-step consistency and step faithfulness metrics.

Cross-Step Consistency Control (CSCC) refers to a class of methods explicitly designed to maintain, across discrete steps of a process, the consistency of targeted features—semantic content, ingredients, or state variables—in structured multi-step generative or control settings. In contemporary literature, CSCC arises in two distinct but conceptually related domains: (1) sequential text-to-image generative modeling of instructional content, where it enforces ingredient and visual semantic consistency across a multi-image recipe sequence, and (2) symbolic hybrid control of legged robots, where it certifies bounded recurrence of dynamical states across hybrid impacts. In all cases, CSCC leverages knowledge of the multi-step structure to couple local step representations with global or temporally extended information, utilizing mechanisms ranging from text-token fusion to symbolic reachability arguments (Zhang et al., 3 Dec 2025, Coënt et al., 2019).

1. Cross-Step Consistency Challenges in Sequential Generation

In multi-step text-to-image generation, such as cooking recipe illustration, a significant challenge is ensuring that semantic content about ingredients and their transformations is preserved across all generated images. Standard diffusion-based frameworks tend to lose track of fine-grained ingredient states—tiny objects may vanish, reappear, or undergo spurious visual changes from step to step (the "Tiny Ingredient Continuity Problem"). In robot locomotion control, the analogous challenge is to ensure that the state of the biped is reliably driven back to an acceptable region at the completion of each step, regardless of the hybrid system's nonlinearities and discrete transitions, thereby guaranteeing stable repeated locomotion.

2. Mathematical Foundations of CSCC in CookAnything

In "CookAnything" (Zhang et al., 3 Dec 2025), CSCC is formulated purely at the text-token level. Let the entire recipe $\mathcal{R}$ be represented as a sequence of tokens $$\smash{C^{\mathrm{recipe}}\in\mathbb{R}^{M\times d}}$$ by a T5 encoder, and let each step $n$ be individually encoded as $\smash{C^{(n)}\in\mathbb{R}^{t^{(n)}\times d}}$. The $n$th step occupies positions $[b^{(n)}, \ldots, b^{(n)}+t^{(n)}-1]$ in the global recipe token stream. The fusion is given by: $$\dot C^{(n)}[0\!:\!t^{(n)}] = C^{(n)}[0\!:\!t^{(n)}] + \lambda\,C^{\mathrm{recipe}}[b^{(n)}\!:\!b^{(n)}\!+\!t^{(n)}]$$ with $\lambda\in[0,1]$ controlling the trade-off between step-local semantics and global ingredient continuity. The fused tokens $\dot C^{(n)}$ are concatenated and input to the diffusion model, with the remainder of the denoising objective and network unchanged: $$\mathcal{L}_{\mathrm{diff}} = \mathbb{E}_{t,\mathbf{z}_t,\epsilon} \left\| \epsilon - \epsilon_\theta(\mathbf{z}_t, t, C^{\mathrm{input}}) \right\|^2$$ where $$C^{\mathrm{input}} = [\dot C^{(1)}; \ldots; \dot C^{(N)}]$$.

3. Architectural Integration and Workflow

CSCC is integrated within the overall CookAnything architecture as follows:

• Text Processing and Tokenization: A Cooking Agent (GPT-4o) explicitly annotates all ingredient attributes across steps for maximal reference continuity. The T5 encoder then produces both per-step and global recipe token sequences.
• Cross-Step Token Fusion: Each step’s encoded tokens are fused with the corresponding segment from the global encoding, scaled by $\lambda$.
• Latent Initialization: Diffusion latents for all $N$ visual steps are tiled into vertical regions; positional encoding (Flexible RoPE) prevents cross-talk.
• Denoising and Step-wise Regional Control: The fused tokens are supplied to a DiT with attention masks confining each region’s context. The global summary from the whole recipe can be injected at each layer via interpolation ($\alpha$).
• Inference and Decoding: After denoising, the latent is decoded and segmented into step-wise output images.

The CSCC module modifies only the text-token interface between step-wise encoding and the main diffusion transformer, coexisting with Step-wise Regional Control (SRC) and Flexible RoPE (Zhang et al., 3 Dec 2025).

4. Hyperparameters, Training Regime, and Ablation

Key hyperparameters and training settings in CookAnything include:

• CSCC fusion weight: $\lambda=0.1$ (training-based), best at $\lambda=0.2$ (training-free)
• SRC interpolation: $\alpha=0.1$
• 20,000 training steps on a single NVIDIA A100, batch size 2, Flux.1-dev base (12B parameters), LoRA rank 16, Adam optimizer, learning rate $1\text{e}{-4}$

Ablation studies demonstrate:

• Without CSCC, measured Cross-Step Consistency (CSC) degrades (CSC rises from 0.19 → 0.29; lower is better)
• Step Faithfulness (CLIP/GPT-score) marginally decreases from (30.45 / 8.69) to (30.28 / 8.67) without CSCC
• Excessive fusion ($\lambda > 0.2$) rapidly worsens CSC (e.g., CSC=3.15 at $\lambda = 1.0$) (Zhang et al., 3 Dec 2025).

5. Cross-Step Consistency Control in Symbolic Hybrid Robotics

CSCC in bipedal hybrid locomotion (Coënt et al., 2019) centers on recurrence certification: ensuring that after each hybrid "step" (swing-collision-reset), the system’s state returns to a specified invariant region $R$. The method constructs:

• Hybrid Model: State $$x=(\dot\theta_1,\dot\theta_2,\dot\theta_3,\theta_1,\theta_2,\theta_3)$$, with nonlinear swing dynamics, guard condition at $\theta_1+\theta_2=0$, and impact/reset maps incorporating angular momentum and symmetry.
• PD Control Law: $$u(t) = K_p(\theta_{SP} - [\theta_3(t) - \theta_1(t)]) - K_d[\dot\theta_3(t)-\dot\theta_1(t)]$$
• Controllable Recurrence Region $R$: Explicitly certified subset of $\mathbb{R}^6$ such that for all post-impact states $x(0)\in R$, a control mode exists ensuring $x(\text{next impact})\in R$.
• Verification: Zonotope-based reachability, adaptive bisection, and set-inclusion validation certify $R$ as a controlled invariant. For a given $R$, multi-valued Poincaré maps and symbolic inclusion are used to verify cross-step consistency.

Simulation results confirm that states remain bounded and recurrent in $R$ over consecutive steps under the certified control schedule (Coënt et al., 2019).

6. Practical Impact and Illustrative Outcomes

In recipe image synthesis, CSCC maintains ingredient shape and presence across visual steps. For example, in “Stir-Fried Carrot with Dried Tofu,” carrot cubes remain cubes across all illustrated steps with CSCC, while without, their shape degrades to strips. In "Steamed Chicken Wings with Taro," the taro remains visually persistent across steps due to CSCC's token fusion, but vanishes without CSCC (Zhang et al., 3 Dec 2025).

In hybrid robot control, CSCC guarantees that the bipedal walker state remains safely within operational bounds after every hybrid transition. Simulations show both angles and angular velocities recurring within the set $R$ over multiple steps, even in the presence of model nonlinearities and impacts (Coënt et al., 2019).

7. Domain-Specific Formalizations and Theoretical Significance

The underlying thread across both applications is the formalization and enforcement of "step-to-step" coherence:

• In generative modeling, this is realized as a token-level operation coupling per-step and global context without explicit architectural alterations or loss functions.
• In hybrid systems, it is realized through set-inclusion fixed-point arguments ensuring controlled invariance across discrete transitions.

A plausible implication is that CSCC-style strategies provide architecturally efficient avenues for enforcing global coherence in other structured generation or hybrid decision problems, provided suitable token or state representations and fusion/inclusion principles can be defined.

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Domain	Main CSCC Mechanism	Consistency Target
Recipe Generation	Token fusion with global context	Ingredient visuals/semantics
Legged Robotics	Symbolic recurrence via reachability	State boundedness/recurrence