SCC Canonical Spec — Part 3: Energy Principle & Provisional Operators

Part 3 of 5 — Soft Cognitive Cohesion canonical specification, current release CV-1.5.2 (2026-05-02). See the index for the status note, change log, and links to the other parts.

8. Minimal Energy Principle

The theory admits a variational formulation in which cohesive formations are characterized as approximate minimizers of a canonical energy functional. The minimal energy principle provides a unified optimization target that encodes all four structural requirements — closure, separation, morphology, and transport — as energetic penalties.

8.0. Volume Constraint

The energy is minimized on the constraint manifold

Σ_{m} = {u \in [0, 1]^{n} : x \in X_{t} \sum u_{t} (x) = m}

where $m > 0$ is the cohesive budget and $n = ∣ X_{t} ∣$ . The volume fraction $c = m / n$ is a structural parameter of the theory.

Ontological justification. Any finite relational system has finite capacity to sustain cohesion. The volume constraint formalizes this: a formation must select which regions to cohere, because universal cohesion ( $u \equiv 1$ everywhere) is as featureless as universal non-cohesion ( $u \equiv 0$ ). The cohesive budget $m$ is the finite resource that forces selectivity — and selectivity is the precondition for formation.

Without the volume constraint, the trivial field $u \equiv 0$ is the global energy minimizer (since all energy terms are nonnegative and vanish at $u = 0$ ). The volume constraint is therefore mathematically necessary for the existence of non-trivial minimizers (T8-Core) and ontologically natural as a commitment to finite cohesive capacity.

Admissible range. The phase transition theorem (T8-Core) requires $c \in (\frac{3 - 3}{6}, \frac{3 + 3}{6}) \approx (0.211, 0.789)$ , which is the range where the double-well second derivative $W^{''} (c) < 0$ .

8.1. Energy Functional

The canonical energy over a temporal window $W$ is:

E (u) = λ_{cl} E_{cl} + λ_{sep} E_{sep} + λ_{bd} E_{bd} + λ_{tr} E_{tr}

where $λ_{cl}, λ_{sep}, λ_{bd}, λ_{tr} > 0$ are weighting coefficients that control the relative importance of each structural requirement. The four terms are defined as follows.

Schematic showing the canonical energy E(u) decomposing into four boxed terms: closure, separation, boundary/morphology, transport. — Four conceptually independent structural requirements assembled into the canonical energy E(u). Closure (self-support under relational completion), separation (asymmetry vs exterior field), boundary/morphology (smooth interface + double-well), and transport (temporal inheritance) — each penalty is structurally distinct, though they interact mathematically (CN5).

8.2. Closure Term

E_{cl} (u_{t}) = x \in X_{t} \sum (u_{t} (x) - Cl_{t} (u_{t}) (x))^{2}

This term penalizes deviation of the cohesion field from its relationally completed form. Minimizing $E_{cl}$ drives the field toward self-support: a state in which the cohesion at every site is consistent with the relational support it receives from its neighborhood. A field that minimizes this term is approximately closed — it sustains itself under its own relational structure.

8.3. Separation Term

E_{sep} (u_{t}) = x \in X_{t} \sum u_{t} (x) (1 - D_{t} (x; 1 - u_{t}))

This term penalizes cohesive sites that are not structurally distinguished from the exterior. Minimizing $E_{sep}$ drives the field toward a state in which every site that participates in the formation is asymmetrically positioned relative to the non-cohesive surround. A field that minimizes this term exhibits clear structural contrast between interior and exterior.

The separation term is structurally distinctive: because $D_{t}$ depends on $u_{t}$ , the energy is self-referential — the field defines what "distinction" means, and then is penalized for failing to be distinctive. This self-referential loop has no analogue in standard Allen-Cahn or phase-field theories.

8.4. Boundary and Morphology Term

E_{bd} (u_{t}) = α x, y \in X_{t} \sum N_{t} (x, y) (u_{t} (x) - u_{t} (y))^{2} + β x \in X_{t} \sum u_{t} (x)^{2} (1 - u_{t} (x))^{2}

where $α, β > 0$ are morphological parameters and the sum $\sum_{x, y}$ ranges over ordered pairs (per Section 0).

The first sub-term is a smoothness penalty: it discourages arbitrary high-frequency oscillation in the cohesion field by penalizing large differences between adjacent sites. For symmetric $N_{t}$ , this equals $2 α v^{T} Lv$ where $L$ is the graph Laplacian and $v = u_{t} - c 1$ , giving a Hessian contribution of $4 α L$ .

The second sub-term is a double-well penalty: it penalizes intermediate values of cohesion, favoring fields that are either close to $0$ (exterior) or close to $1$ (interior). This ensures that the field develops a structured morphology with discernible inside and outside regions rather than remaining at a featureless intermediate level.

Together, the two sub-terms drive the field toward a state with smooth, well-defined interior regions, coherent boundary bands, and clear exterior — the morphological signature of a genuine cohesive formation.

8.5. Transport Term

E_{tr} (u_{t}, u_{s}) = x \in X_{t} \sum y \in X_{s} \sum M_{t \to s} (x, y) ω (u_{t} (x), u_{s} (y)) (u_{s} (y) - u_{t} (x))^{2}

where $ω (a, b) : [0, 1]^{2} \to [0, \infty)$ is a weighting function that emphasizes inheritance at high-cohesion sites (for example, $ω (a, b) = a \cdot b$ or $ω (a, b) = min (a, b)$ ). This term penalizes discontinuities in cohesive content under temporal transport. Minimizing $E_{tr}$ drives the field toward a state in which the structurally significant part of the formation is smoothly inherited from one time to the next.

8.6. Why Four Distinct Terms

The four energy terms correspond to four conceptually independent structural requirements:

Closure ensures internal relational self-support.
Separation ensures structural contrast with the exterior.
Boundary/morphology ensures coherent spatial articulation.
Transport ensures temporal inheritance of cohesive organization.

These requirements are logically independent: a field may be closed but undistinguished, distinguished but morphologically incoherent, morphologically coherent but temporally discontinuous, and so forth.

Honesty Note. "Conceptual independence" means each term addresses a logically separable structural requirement. It does not mean the terms are mathematically decoupled — in practice, the smoothness penalty in $E_{bd}$ and the distinction requirement in $E_{sep}$ interact strongly, and the Allen-Cahn substrate is shared between the morphology term and the closure term. The assertion is structural (four independent reasons for penalization), not mathematical (four uncorrelated terms). Collapsing the four terms into fewer would obscure these independent structural motivations and reduce the theory's capacity to diagnose which condition a given formation fails to satisfy.

8.7. Natural Geometry

The constraint manifold $Σ_{m}$ carries a natural information-geometric structure. The Shahshahani metric, defined by $g_{ij} = δ_{ij} / u_{i}$ , induces a Riemannian structure under which the natural gradient differs from the Euclidean gradient. The replicator dynamics connection — whereby the gradient flow of $E_{bd}$ under the Shahshahani metric yields a replicator equation — is noted as commentary. Whether the natural gradient improves convergence properties is an implementation question, not a theoretical commitment.

Energy contours on a 2-simplex with five gradient flow trajectories descending to corner attractors. — Projected gradient flow on the constraint manifold Σ_m (here n = 3 simplex with mass m = 1). Trajectories from various initial points (open circles) descend to critical points (filled squares). The uniform field u ≡ c at the centroid is a saddle (star); corner-region attractors are non-trivial minimizers.

9. Provisional Concrete Operator Forms

This section presents currently favored functional realizations of the canonical operators. These are adopted because they jointly support self-sustaining cohesion, exterior asymmetry, and non-deterministic structural inheritance in a computationally tractable manner. They are presented explicitly as provisional: they are the best current candidates, not permanently fixed definitions. Future work may refine, replace, or generalize these forms while remaining within the canonical framework.

9.1. Local Relation Kernel and Aggregation

The adjacency structure is instantiated through a local relation kernel $K_{t} : X_{t} \times X_{t} \to [0, \infty)$ satisfying

K_{t} (x, y) \geq 0, K_{t} (x, y) = K_{t} (y, x) .

The associated aggregation operator is defined as

(P_{t} f) (x) = \frac{\sum _{y} K _{t} ( x , y ) f ( y )}{\sum _{y} K _{t} ( x , y ) + ε}

where $ε > 0$ is a small stabilization constant that prevents division by zero at isolated sites. The operator $P_{t}$ computes a locally weighted average of any field $f$ with respect to the relational kernel $K_{t}$ . It is the basic building block from which closure, distinction, and other operators are constructed.

9.2. Closure Candidate

The currently adopted closure operator is

Cl_{t} (u) (x) = σ (a_{cl} ((1 - η_{cl}) u (x) + η_{cl} (P_{t} u) (x) - τ_{cl}))

where:

$σ$ is the logistic sigmoid $σ (z) = 1/ (1 + e^{- z})$ ,
$a_{cl} > 0$ controls the steepness of the closure response,
$η_{cl} \in [0, 1]$ controls the balance between self-retention and neighborhood aggregation,
$τ_{cl}$ is a threshold that determines the level of combined support required for cohesion to be maintained.

Parameter constraint. $a_{cl} < 4$ is required for A3 (contraction). The Lipschitz constant of $Cl_{t}$ is bounded by $max σ^{'} \cdot a_{cl} \cdot max (1 - η_{cl}, η_{cl}) \leq a_{cl} /4$ , which is $< 1$ when $a_{cl} < 4$ . At $a_{cl} \geq 4$ , multiple fixed points may exist but A3 is violated.

9.3. Distinction Candidate

The currently adopted distinction operator is

D_{t} (x; 1 - u) = σ (a_{D} ((P_{t} u) (x) - λ_{D} (P_{t} (1 - u)) (x)) - τ_{D})

where:

$a_{D} > 0$ controls the sensitivity of the asymmetry comparison,
$λ_{D} > 0$ scales the exterior aggregation relative to the interior aggregation,
$τ_{D}$ is a threshold.

Change from v1.0: The gradient term $b_{D} \cdot g_{t} (x; u)$ is removed ( $b_{D} = 0$ ). This is required for energy analyticity: the absolute value $∣ \cdot ∣$ in $g_{t}$ breaks the analyticity needed for the Łojasiewicz gradient inequality (T14). Boundary sensitivity (D-Ax3) is preserved through the spatial structure of $P_{t} (1 - u)$ , which implicitly encodes boundary configuration. The smoothness penalty in $E_{bd}$ already enforces spatial coherence of the field, making the explicit gradient term redundant for the variational theory.

9.4. Co-belonging Candidate

The currently adopted co-belonging operator is the resolvent form

C_{t} = (I - α W_{sym})^{- 1}

where $W_{sym}$ is the symmetrized cohesion-weighted adjacency matrix. The resolvent can be expanded as a Neumann series:

C_{t} = k = 0 \sum \infty α^{k} W_{sym}^{k}

converging when $α ρ (W_{sym}) < 1$ . The $k$ -th term captures the contribution of paths of length $k$ through the cohesion-weighted adjacency graph, providing non-local structural integration that is distinct from (and richer than) adjacency alone.

Parameter constraint. $α ρ (W_{sym}) < 1$ for convergence. The spectral radius $ρ (W_{sym})$ depends on both the adjacency structure and the cohesion field.

9.5. Transport Candidate

The currently adopted temporal transport kernel is

M_{t \to s} (x, y) = \frac{exp ( - \frac{∥ y - Ψ _{t \to s} ( x ) ∥ ^{2}}{2 σ _{M}^{2}} - γ _{M} ∥ φ _{t} ( x ) - φ _{s} ( y ) ∥ ^{2} )}{\sum _{y^{'}} exp ( - \frac{∥ y ^{'} - Ψ _{t \to s} ( x ) ∥ ^{2}}{2 σ _{M}^{2}} - γ _{M} ∥ φ _{t} ( x ) - φ _{s} ( y ^{'} ) ∥ ^{2} ) + ε}

where:

$Ψ_{t \to s} : X_{t} \to X_{s}$ is a predicted spatial correspondence (e.g., from motion estimation or optical flow),
$σ_{M} > 0$ controls the spatial tolerance of the transport,
$φ_{t} (x)$ is a feature representation of site $x$ at time $t$ (capturing qualitative or structural characteristics),
$γ_{M} > 0$ controls the importance of feature similarity relative to spatial proximity,
$ε > 0$ prevents degenerate normalization.

This realization is sub-stochastic (satisfying E1) and structurally sensitive (satisfying E4), but it depends on external features $φ$ and external correspondence $Ψ$ , which are not generated by the cohesion field. A fully self-referential transport realization — where cost depends only on $u_{t}$ , $u_{s}$ , and $N$ — is an open problem for strong-regime analysis (Part 4 · Open Problems). A weak-regime implementation now exists using the cohesion fingerprint $φ (x) = (u (x), Cl (u) (x), D (x; 1 - u), C (x, x))$ as a self-referential feature vector, with entropy-regularized partial OT and fixed-point iteration for mutual consistency.

Continue to Part 4 · Structural Interpretation, Commitments & Open Problems →

Back to Canonical Spec index · ← Part 2 · Axioms

Part 3 of 5 — Soft Cognitive Cohesion canonical specification, current release CV-1.5.2 (2026-05-02). See the index for the status note, change log, and links to the other parts.

8. Minimal Energy Principle

8.0. Volume Constraint

The energy is minimized on the constraint manifold

Σ_{m} = {u \in [0, 1]^{n} : x \in X_{t} \sum u_{t} (x) = m}

where $m > 0$ is the cohesive budget and $n = ∣ X_{t} ∣$ . The volume fraction $c = m / n$ is a structural parameter of the theory.

8.1. Energy Functional

The canonical energy over a temporal window $W$ is:

E (u) = λ_{cl} E_{cl} + λ_{sep} E_{sep} + λ_{bd} E_{bd} + λ_{tr} E_{tr}

where $λ_{cl}, λ_{sep}, λ_{bd}, λ_{tr} > 0$ are weighting coefficients that control the relative importance of each structural requirement. The four terms are defined as follows.

8.2. Closure Term

E_{cl} (u_{t}) = x \in X_{t} \sum (u_{t} (x) - Cl_{t} (u_{t}) (x))^{2}

8.3. Separation Term

E_{sep} (u_{t}) = x \in X_{t} \sum u_{t} (x) (1 - D_{t} (x; 1 - u_{t}))

8.4. Boundary and Morphology Term

E_{bd} (u_{t}) = α x, y \in X_{t} \sum N_{t} (x, y) (u_{t} (x) - u_{t} (y))^{2} + β x \in X_{t} \sum u_{t} (x)^{2} (1 - u_{t} (x))^{2}

where $α, β > 0$ are morphological parameters and the sum $\sum_{x, y}$ ranges over ordered pairs (per Section 0).

8.5. Transport Term

E_{tr} (u_{t}, u_{s}) = x \in X_{t} \sum y \in X_{s} \sum M_{t \to s} (x, y) ω (u_{t} (x), u_{s} (y)) (u_{s} (y) - u_{t} (x))^{2}

8.6. Why Four Distinct Terms

The four energy terms correspond to four conceptually independent structural requirements:

Closure ensures internal relational self-support.
Separation ensures structural contrast with the exterior.
Boundary/morphology ensures coherent spatial articulation.
Transport ensures temporal inheritance of cohesive organization.

8.7. Natural Geometry

9. Provisional Concrete Operator Forms

9.1. Local Relation Kernel and Aggregation

The adjacency structure is instantiated through a local relation kernel $K_{t} : X_{t} \times X_{t} \to [0, \infty)$ satisfying

K_{t} (x, y) \geq 0, K_{t} (x, y) = K_{t} (y, x) .

The associated aggregation operator is defined as

(P_{t} f) (x) = \frac{\sum _{y} K _{t} ( x , y ) f ( y )}{\sum _{y} K _{t} ( x , y ) + ε}

9.2. Closure Candidate

The currently adopted closure operator is

Cl_{t} (u) (x) = σ (a_{cl} ((1 - η_{cl}) u (x) + η_{cl} (P_{t} u) (x) - τ_{cl}))

where:

$σ$ is the logistic sigmoid $σ (z) = 1/ (1 + e^{- z})$ ,
$a_{cl} > 0$ controls the steepness of the closure response,
$η_{cl} \in [0, 1]$ controls the balance between self-retention and neighborhood aggregation,
$τ_{cl}$ is a threshold that determines the level of combined support required for cohesion to be maintained.

9.3. Distinction Candidate

The currently adopted distinction operator is

D_{t} (x; 1 - u) = σ (a_{D} ((P_{t} u) (x) - λ_{D} (P_{t} (1 - u)) (x)) - τ_{D})

where:

$a_{D} > 0$ controls the sensitivity of the asymmetry comparison,
$λ_{D} > 0$ scales the exterior aggregation relative to the interior aggregation,
$τ_{D}$ is a threshold.

9.4. Co-belonging Candidate

The currently adopted co-belonging operator is the resolvent form

C_{t} = (I - α W_{sym})^{- 1}

where $W_{sym}$ is the symmetrized cohesion-weighted adjacency matrix. The resolvent can be expanded as a Neumann series:

C_{t} = k = 0 \sum \infty α^{k} W_{sym}^{k}

Parameter constraint. $α ρ (W_{sym}) < 1$ for convergence. The spectral radius $ρ (W_{sym})$ depends on both the adjacency structure and the cohesion field.

9.5. Transport Candidate

The currently adopted temporal transport kernel is

M_{t \to s} (x, y) = \frac{exp ( - \frac{∥ y - Ψ _{t \to s} ( x ) ∥ ^{2}}{2 σ _{M}^{2}} - γ _{M} ∥ φ _{t} ( x ) - φ _{s} ( y ) ∥ ^{2} )}{\sum _{y^{'}} exp ( - \frac{∥ y ^{'} - Ψ _{t \to s} ( x ) ∥ ^{2}}{2 σ _{M}^{2}} - γ _{M} ∥ φ _{t} ( x ) - φ _{s} ( y ^{'} ) ∥ ^{2} ) + ε}

where:

$Ψ_{t \to s} : X_{t} \to X_{s}$ is a predicted spatial correspondence (e.g., from motion estimation or optical flow),
$σ_{M} > 0$ controls the spatial tolerance of the transport,
$φ_{t} (x)$ is a feature representation of site $x$ at time $t$ (capturing qualitative or structural characteristics),
$γ_{M} > 0$ controls the importance of feature similarity relative to spatial proximity,
$ε > 0$ prevents degenerate normalization.

Continue to Part 4 · Structural Interpretation, Commitments & Open Problems →

Back to Canonical Spec index · ← Part 2 · Axioms

8. Minimal Energy Principle#

8.0. Volume Constraint#

8.1. Energy Functional#

8.2. Closure Term#

8.3. Separation Term#

8.4. Boundary and Morphology Term#

8.5. Transport Term#

8.6. Why Four Distinct Terms#

8.7. Natural Geometry#

9. Provisional Concrete Operator Forms#

9.1. Local Relation Kernel and Aggregation#

9.2. Closure Candidate#

9.3. Distinction Candidate#

9.4. Co-belonging Candidate#

9.5. Transport Candidate#

8. Minimal Energy Principle#

8.0. Volume Constraint#

8.1. Energy Functional#

8.2. Closure Term#

8.3. Separation Term#

8.4. Boundary and Morphology Term#

8.5. Transport Term#

8.6. Why Four Distinct Terms#

8.7. Natural Geometry#

9. Provisional Concrete Operator Forms#

9.1. Local Relation Kernel and Aggregation#

9.2. Closure Candidate#

9.3. Distinction Candidate#

9.4. Co-belonging Candidate#

9.5. Transport Candidate#

8. Minimal Energy Principle

8.0. Volume Constraint

8.1. Energy Functional

8.2. Closure Term

8.3. Separation Term

8.4. Boundary and Morphology Term

8.5. Transport Term

8.6. Why Four Distinct Terms

8.7. Natural Geometry

9. Provisional Concrete Operator Forms

9.1. Local Relation Kernel and Aggregation

9.2. Closure Candidate

9.3. Distinction Candidate

9.4. Co-belonging Candidate

9.5. Transport Candidate

8. Minimal Energy Principle

8.0. Volume Constraint

8.1. Energy Functional

8.2. Closure Term

8.3. Separation Term

8.4. Boundary and Morphology Term

8.5. Transport Term

8.6. Why Four Distinct Terms

8.7. Natural Geometry

9. Provisional Concrete Operator Forms

9.1. Local Relation Kernel and Aggregation

9.2. Closure Candidate

9.3. Distinction Candidate

9.4. Co-belonging Candidate

9.5. Transport Candidate