Gestalt Completion via E3 Minimization

• The paper introduces a unified framework where E3 minimization balances reconstruction fidelity, representational simplicity, and update stability for Gestalt completion.
• It outlines a systematic workflow involving candidate generation, threshold dynamics, and Bayesian model selection to complete perceptual stimuli and form hierarchical representations.
• The approach generalizes classical Gestalt phenomena to both visual perception and axiomatic intelligence, ensuring adaptive processing within finite capacity constraints.

Gestalt completion under E3 minimization refers to the principled act of reconstructing, grouping, or completing perceived or internally represented stimuli by selecting or creating internal structures that most reduce a global energy functional $E_3$, subject to competition, stabilization, structural simplicity, and adaptive constraints. This framework generalizes classical Gestalt phenomena—such as amodal and modal completion, figure-ground assignment, and perceptual organization—by making explicit the trade-offs among reconstruction fidelity, representational complexity, and update stability. In both perceptual modeling (e.g., amodal completion in vision) and axiomatic intelligence systems, the selection and stabilization of units that comprise a "Gestalt" are governed by minimization of $E_3$, linking local continuation, global regularity, and energetic rationality.

1. Formalization of the E3 Energy Principle

In the context of SANC($E_3$) and advanced perceptual modeling, the energy functional central to Gestalt completion is defined as:

$$E_3 \;=\; \lambda_1\,L_{\mathrm{rec}} \;+\; \lambda_2\,C_{\mathrm{struct}} \;+\; \lambda_3\,C_{\mathrm{update}}$$

where:

• $L_{\mathrm{rec}}$ quantifies reconstruction loss, i.e., the error incurred when reconstructing an observed or hypothesized event using the current representational inventory.
• $C_{\mathrm{struct}}$ penalizes representational complexity, typically by counting stabilized units, description length, or entropy.
• $C_{\mathrm{update}}$ penalizes instability due to frequent or disruptive updates in the token inventory.
• $\lambda_1, \lambda_2, \lambda_3 > 0$ modulate the trade-off among these pressures.

Minimizing $E_3$ directs the system to select composites and structural reorganizations that (i) reduce reconstruction loss, (ii) do not unduly increase complexity, and (iii) maintain organizational stability. In classical amodal completion (Oliver et al., 2015), an analogous minimization drives the selection of object boundaries and depth orderings such that the completed scene is both perceptually plausible and energetically optimal.

2. Axiomatic and Perceptual Foundations

Gestalt completion in SANC($E_3$) is governed by five foundational axioms:

1. Finite active capacity: The set of active tokens is bounded, enforcing competition.
2. Similarity-based competition: Only one member of each similarity neighborhood may be selected at a time.
3. Confidence-based stabilization and deletion: Units are stabilized or evicted based on selection history and confidence thresholds.
4. Energetic minimization: System dynamics continuously descend $E_3$.
5. Co-occurrence-based candidate generation: New composite candidates are generated from co-occurring tokens.

In perceptual modeling, global cues such as relatability (continuity of tangent directions at endpoints) and convexity are enforced as hard constraints on boundary initialization. For curve completion in images, the classical Euler’s elastica functional

$$E[\gamma] = \int_\gamma \left(\kappa(s)^2 + \beta \right) ds$$

is minimized subject to these constraints, balancing straight-line and smooth curved completions according to the weighting constant $\beta$ (Oliver et al., 2015).

3. Mechanistic Workflow of Gestalt Completion

The process of Gestalt completion via $E_3$ minimization proceeds as follows:

1. Generation of candidate hypotheses: Multiple complete distal scenes or composite tokens compatible with input are hypothesized.
2. Mask initialization via Gestalt cues: Only relatable and convex initializations are permitted, ensuring global and local regularity.
3. Numerical minimization: Iterative procedures (e.g., threshold dynamics, Grzibovskis–Heintz, Merriman–Bence–Osher steps) minimize the energy functional governing boundary completion or internal token arrangement.
4. Bayesian model selection or energetic updating: Likelihoods and priors are computed based on elastica and shape-complexity measures; in intelligence systems, reconstruction, structural, and update costs are evaluated.
5. Selection of preferred completion: The hypothesis or internal token set that globally minimizes $E_3$ is stabilized and used in perception, prediction, or further cognitive processing.

This pipeline is rigorously implemented in both computational models of visual disocclusion (Oliver et al., 2015) and axiomatic intelligence frameworks (Kwon et al., 13 Jan 2026).

4. Energetic Condition and Variational Principle

A formal energetic condition for Gestalt completion is:

$$\Delta E_3(u) = \lambda_1 [L_{\mathrm{rec}}(e; \Sigma) - L_{\mathrm{rec}}(e; \Sigma \cup \{u\})] + \lambda_2 [C_{\mathrm{struct}}(\Sigma \cup \{u\}) - C_{\mathrm{struct}}(\Sigma)] + \lambda_3 [C_{\mathrm{update}}(\Sigma \cup \{u\}) - C_{\mathrm{update}}(\Sigma)]$$

A composite $u$ is accepted as a Gestalt completion only if $\Delta E_3(u) < 0$.

The selection is typically posed as either a variational problem,

$\delta E_3/\delta u = 0, \quad \delta^2 E_3 > 0$

or by global maximization of posterior likelihood in Bayesian variants. This variational principle underpins both perceptual completion and internal cognitive reorganization, ensuring that selected completions achieve optimal trade-offs among fidelity, simplicity, and stability.

5. Instantiation in Perceptual and Cognitive Systems

Gestalt completion under $E_3$ minimization manifests in several phenomena:

• Amodal object completion: Underspecified or occluded boundaries are completed by minimizing elastica energy under relatability/convexity constraints, followed by probabilistic scene selection (Oliver et al., 2015).
• Category and hierarchy formation: New categories (shared sub-tokens) and hierarchical structures are created when they offer net energetic advantage in reconstruction and organizational cost (Kwon et al., 13 Jan 2026).
• Partial-match retrieval and replay: Internally replayed Gestalts (pseudo-memory-mapped I/O) follow identical energetic descent pathways as externally triggered ones, unifying perception, imagination, prediction, and planning.
• Unsupervised learning: All forms of learning and generalization are interpreted as instances of $E_3$-driven token selection and stabilization.

These principles hold recursively at all organizational levels and for both external sensory events and internally generated representations.

6. Comparative Table: E³-Minimization in Vision and Intelligence

Domain	Energy Terms	Gestalt Constraints	Completion Mechanism
Visual Perception	Elastica, complexity	Relatability, convexity	Threshold dynamics, scene selection
SANC($E_3$)	Reconstruction, structure, update	Capacity, similarity	Energetic descent, replay, hierarchy formation

Both domains instantiate E3 minimization principles, differing only in the specifics of constraints and operative mechanics.

7. Context and Implications

Gestalt completion under $E_3$ minimization provides a unified formalism that subsumes classical Gestalt psychology, Bayesian perceptual inference, and contemporary intelligence system organization. All phenomena—figure-ground segmentation, object completion, hierarchical generalization, planning, and prediction—are rigorously cast as select-and-create operations that globally reduce the total energy under finite capacity and competition. A plausible implication is that both biological and artificial systems implementing these principles can robustly reorganize internal representations for perception and cognition within bounded resources, and can adaptively balance complexity against fidelity and stability.