The Action Cost of Distinction: A Phase-Topological Foundation for Multi-Scale Complexity

1. Introduction: From the Cost of Erasure to the Cost of Distinction

1.1. Conceptual Anchor: Maxwell’s Demon

The deep link between information and physics was crystallized by Maxwell’s Demon and its resolution via Landauer’s principle [1]: erasing one bit of information has an irreducible thermodynamic cost of $k_{B}Tln2$. Subsequent developments in the thermodynamics of computation [2,3] established a rigorous association between erasure and energetic expenditure. Independent lines of work—from Bekenstein’s geometric entropy bounds [4] to Zurek’s decoherence program [5]—further reinforced that information is constrained by physical law.

Here we propose to interpret Planck’s constant ℏ as the minimal quantum of action required for a physical distinction to be sustainably realized,1 in contrast to the statistical cost of erasure. This lower bound is structural and originates in phase-topological consistency conditions. In geometric quantization, these conditions appear as integrality of the symplectic flux,

$${\displaystyle \frac{1}{2\pi \hslash }}{\int }_{\Sigma }\omega \in \mathbb{Z}\mathrm{for}\mathrm{every}\mathrm{closed}\mathrm{two}–\mathrm{surface}\Sigma ,$$

with $\omega$ the symplectic form on phase space [6,7,8]. Equivalently, the cohomology class $[\omega /2\pi \hslash ]$ is integral, ensuring globally consistent $U\left(1\right)$ phase holonomies. We note that the canonical flux quantum is $2\pi \hslash$; throughout this paper we adopt ℏ as the operative unit for sustainable distinction. In this sense, the unit that protects coherent distinctions is anchored in topology rather than in statistical mechanics.

1.2. The Problem of Complexity and a Proposed Framework

Recent advances in complexity science identify robust multi–scale regularities. The framework of Siegenfeld and Bar–Yam [9,10] formalizes a complexity profile, a multi–scale law of requisite variety, and a sum rule. These are presented as phenomenological regularities, leaving their physical origin unspecified.

To address this gap, we develop a two–level architecture that links phenomenology to structural constraints:

• Phenomenological level: information–theoretic descriptors (complexity profile, multi–scale requisite variety, sum rule) [11];
• Structural level: a symplectic–topological formulation in which constraints follow from a minimal–action requirement

  $$\delta S\left[\gamma \right]\ge \hslash ,\delta S\left[\gamma \right]:={\oint }_{\gamma }\Lambda ,$$

  where $\Lambda =pdq-Hdt$ is the Poincaré–Cartan one–form and $\gamma$ is a closed cycle in phase space.

This architecture parallels the relation between Thermodynamics and Statistical Mechanics: macroscopic laws emerge from microscopic structure [12]. In our formulation, the inequality $\delta S\ge \hslash$ is taken as an operational formulation of the underlying quantization dictated by geometric integrality.

2. Formalism of the Phase-Topological Framework: Foundations

2.1. Coherence Principle and Minimal Units

Definition 1

(Coherence Principle). For any physically distinguishable closed cycle γ in phase space,

$$\delta S\left[\gamma \right]={\oint }_{\gamma }\Lambda \ge \hslash ,$$

where $\Lambda =pdq-Hdt$ is the canonical Poincaré–Cartan one-form of the action. A minimal coherent unit is any configuration saturating this inequality, $\delta S=\hslash$.

Note. Here $\delta S\left[\gamma \right]$ denotes the line integral of $\Lambda$ over a closed trajectory, not a variational derivative in the Euler–Lagrange sense.

This condition is adopted here as a working structural postulate of the framework. The interpretation is operational: every sustainably realized distinction requires at least one quantum of action, meaning that it preserves phase coherence sufficiently to correspond to an (approximately) orthogonal support in Hilbert space.

The structural origin of this postulate lies in geometric quantization. The holonomy of the $U\left(1\right)$ prequantum bundle, together with the Maslov index correction, imposes the Bohr–Sommerfeld quantization rule,

$${\oint }_{\gamma }pdq=2\pi \hslash \left(n+\frac{\mu \left(\gamma \right)}{4}\right),n\in \mathbb{Z},$$

for any closed curve $\gamma$ on a Lagrangian submanifold of phase space, where $\mu \left(\gamma \right)$ is the Maslov index [8,13,14]. This quantization condition ensures the global consistency of phase holonomies. In our reformulation, it is interpreted as a minimal-action requirement for the sustainability of distinct physical configurations.

The role of the dynamical component $-Hdt$ in the Poincaré–Cartan form will be analyzed in later sections and appendices, where explicit Hamiltonian evolutions and quantum speed-limit bounds are discussed (Appendix A). For now, the principle is established at the level of structural consistency.

2.2. Measure of Distinction ${\mu }_{\Omega }$

To quantify the effective number of reliably distinguishable configurations, we introduce the structural measure:

• For a finite set of pure states U:

  $${\mu }_{\Omega }\left(U\right):=dim\mathrm{span}\left(U\right),$$

  i.e. the dimension of the subspace spanned by U. For non-orthogonal sets this counts the linear span, not the number of reliably distinguishable states.
• For a mixed state with density operator $\rho$:

  $${\mu }_{\Omega }\left(\rho \right):=exp\left(S_{\mathrm{vN}}\left(\rho \right)\right),$$

  where $S_{\mathrm{vN}}\left(\rho \right)=-(\rho log\rho )$ is the von Neumann entropy. This definition, sometimes referred to as the exponential of the entropy, interprets $S_{\mathrm{vN}}$ as an effective Hilbert space dimension, consistent with approaches to typical entanglement and effective dimension in quantum statistical mechanics [15,16].

Throughout we use the natural logarithm, so ${\mu }_{\Omega }$ is expressed in nats; using ${log}_2$ would instead yield bits. A fundamental property of this measure is subadditivity under composition of subsystems: for two subsystems A and B within a fixed context,

$${\mu }_{\Omega }\left({\rho }_{AB}\right)\le {\mu }_{\Omega }\left({\rho }_A\right){\mu }_{\Omega }\left({\rho }_B\right).$$

This inequality follows directly from the subadditivity of the von Neumann entropy:

$$S_{\mathrm{vN}}\left({\rho }_{AB}\right)\le S_{\mathrm{vN}}\left({\rho }_A\right)+S_{\mathrm{vN}}\left({\rho }_B\right),$$

and therefore

$$exp\left(S_{\mathrm{vN}}\left({\rho }_{AB}\right)\right)\le exp\left(S_{\mathrm{vN}}\left({\rho }_A\right)\right)exp\left(S_{\mathrm{vN}}\left({\rho }_B\right)\right).$$

Equality holds for product states, while strict inequality arises in the presence of correlations or entanglement. This expresses the structural limitation that the number of jointly sustainable distinctions cannot exceed the available resource budget. In later sections we show how this information-theoretic property is connected to the coherence bound $\delta S\ge \hslash$; the two are not independent assumptions but complementary expressions of the same structural limitation, linked explicitly in subsequent theorems.

3. Results: Structural Derivation of Complexity Laws

3.1. Multi-Scale Law of Requisite Variety

Theorem 1

(Capacity Bound). The number of mutually orthogonal coherent units that can be sustained in a region R is bounded above by the available action budget:

$$M_{max}\left(R\right)\le ⌊{\displaystyle \frac{\delta S\left[R\right]}{\hslash }}⌋.$$

Here $\delta S\left[R\right]$ is defined as the supremum of the action integral over all relevant closed cycles $\gamma$ contained within the spacetime region R,

$$\delta S\left[R\right]:=\underset{\gamma \subset R}{sup}{\oint }_{\gamma }\Lambda .$$

This definition ensures that $\delta S\left[R\right]$ is well-posed and consistent with the geometric setting. The inequality follows directly from the coherence principle: each coherent unit requires at least one quantum of action to remain stable. Hence the number of reliably distinguishable configurations cannot exceed the ratio $\delta S\left[R\right]/\hslash$.

Clarification. In geometric quantization the elementary flux quantum is $2\pi \hslash$; here we adopt ℏ as the operational threshold for sustainable distinction. This should be understood as the minimal action cost of distinguishability, not as a literal Bohr–Sommerfeld spacing. The bound of Theorem 1 may be further restricted by the finite Hilbert-space dimension of the system, as discussed in Appendix A.

Connection to Phenomenological Laws.

In the framework of Ref. [9], the multi-scale law of requisite variety requires that the system complexity $C_X\left(n\right)$ must not fall below that of the environment, $C_Y\left(n\right)$, for all scales n. The capacity bound provides the corresponding structural resource constraint: if system X has a smaller action budget than needed to encode the states of its environment Y, then necessarily $C_X\left(n\right)<C_Y\left(n\right)$ for some scales. This is best interpreted as a structural limitation underlying Ashby’s law, not a literal proof of its full phenomenological formulation.

3.2. Sum Rule

Principle 2

(Balance of the Action Budget). The total action budget $\delta S\left[R\right]$ associated with a region R provides a fixed upper bound on the system’s informational capacity. This budget remains constant under redistribution of coherent structures, provided the region and its boundary conditions are fixed. While the distribution of complexity may shift between microscopic and macroscopic scales, the overall budget $\delta S\left[R\right]$ does not change.

This expresses that action functions as a finite resource: a system may allocate its budget to many small-scale distinctions or to fewer large-scale correlated structures, but the total capacity remains constrained by $\delta S\left[R\right]$.

Connection to Phenomenological Laws.

The sum rule of Ref. [9],

$$\sum _n{C}_X\left(n\right)=\sum _{x\in X}H\left(x\right),$$

is interpreted here as the phenomenological projection of this balance principle. The right-hand side, ${\sum }_{x\in X}H\left(x\right)$, represents the total information capacity of the system’s components, structurally bounded by the fixed action budget $\delta S\left[R\right]$.

Illustrative Example.

In a spin chain, the same budget $\delta S$ can be allocated in two limiting ways: (i) into N independent coherent units (a microscopic peak of complexity), or (ii) into a single highly entangled GHZ-type configuration (a macroscopic peak). Both respect the same resource bound, though the distribution of $C_X\left(n\right)$ across scales differs. In both cases one finds ${\sum }_n{C}_X\left(n\right)=N$, consistent with the budget constraint.

3.3. Complexity Profile and the Distinction Measure ${\mu }_{\Omega }$

Theorem 3

(Projection of Distinction). For a coarse-grained system, the phenomenological complexity profile $C_X\left(n\right)$ can be expressed as the per-scale increment of the structural distinction measure:

$$C_X\left(n\right)=ln{\mu }_{\Omega }\left({\Pi }_{n+1}\rho \right)-ln{\mu }_{\Omega }\left({\Pi }_n\rho \right),$$

where ${\Pi }_n$ denotes the coarse-graining operator at resolution scale n, modeled as an entropy non-decreasing channel such as a conditional expectation (pinching) onto the subalgebra corresponding to scale n. Here ${\Pi }_{max}$ refers to the coarsest-graining operator, corresponding to the maximal available scale.

This definition guarantees $C_X\left(n\right)\ge 0$ whenever each coarse-graining step is entropy non-decreasing. In typical systems the profile decays with scale, consistent with phenomenological observations, though local fluctuations may occur. The telescoping sum

$$\sum _n{C}_X\left(n\right)=ln{\mu }_{\Omega }\left({\Pi }_{max}\rho \right)-ln{\mu }_{\Omega }\left({\Pi }_1\rho \right)$$

ensures consistency with the action budget constraint.

We emphasize that the entropic formulation given above is taken as the primary definition of the complexity profile. The combinatorial model of Appendix B serves as an illustrative discrete analogue; their equivalence holds under canonical coarse-graining, as discussed in Appendix C.

3.4. Conclusion

Within this formulation, the phenomenological laws of complexity—the multi-scale law of requisite variety, the sum rule, and the scale-dependent complexity profile—can be interpreted as emergent consequences of a single structural action principle, rather than as independent postulates.

Schematic Correspondence of Structural Results

Figure 1. Correspondence between structural principles (left/middle) and phenomenological laws of multi-scale complexity (right). Dashed arrows indicate the projection of structural results into phenomenological formulations.

Figure 1. Correspondence between structural principles (left/middle) and phenomenological laws of multi-scale complexity (right). Dashed arrows indicate the projection of structural results into phenomenological formulations.

This diagram highlights the logical structure: the Coherence Principle generates both the Capacity Bound and the Balance principle. Together with the distinction measure, they underlie the projection theorem, from which the phenomenological complexity laws follow.

4. Discussion

The results presented above suggest that the phenomenological laws of multi-scale complexity can be interpreted as emergent quantum-informational consequences of a single structural principle of action. This section outlines broader implications of this finding, ranging from reinterpretations of foundational concepts in physics to potential applications in quantum information, cybernetics, and the philosophy of science.

4.1. The Phase-Topological Framework as the “Statistical Mechanics” of Complexity

The analogy between the two-level framework proposed here and the relationship between Statistical Mechanics and Thermodynamics, introduced in Section 1, can now be made more precise. The results in Section 3 provide a formal bridge between these two levels. The laws of multi-scale complexity, as formulated by Siegenfeld and Bar-Yam [9], function as macroscopic, phenomenological rules governing the distribution of information in complex systems. The framework developed here provides a microscopic, structural foundation for these rules.

The Capacity Bound and the Principle of Budget Conservation are not postulated information-theoretic laws but physical constraints grounded in the symplectic-topological properties of the underlying phase space. Thus, the universal laws of complexity may be viewed as emergent quantum-informational laws, in parallel with how the laws of thermodynamics arise from the microscopic mechanics of constituent systems [10,12,17,18].

Earlier formulations of multiscale complexity and requisite variety [19,20] introduced these ideas descriptively. The present framework provides a structural origin, suggesting that what has been observed phenomenologically as “laws of complexity” may be interpreted as consequences of a more primitive constraint: the minimal action bound $\delta S\ge \hslash$, where $\delta S\left[R\right]$ is defined as in Section 3.

4.2. Reinterpreting Maxwell’s Demon through the Action Cost

The classic thought experiment of Maxwell’s Demon illustrates the deep connection between information and thermodynamics. Landauer’s principle resolved the paradox by identifying the thermodynamic cost of erasing information [1,2,3]. Within the present framework, we interpret each act of establishing a physical distinction as requiring at least one quantum of action, ℏ. Therefore, the demon’s ability to decrease entropy is limited not only by the thermodynamics of its memory but, more fundamentally, by the available action budget required to perform distinctions. This limitation may be regarded as phase-topological in nature, representing the cost of establishing information rather than merely erasing it. From this perspective, Ashby’s law of requisite variety [21] can be situated in a broader physical context.

4.3. Broader Implications

Quantum Information and Holography.

The Capacity Bound

$$M_{max}\left(R\right)\le ⌊{\displaystyle \frac{\delta S\left[R\right]}{\hslash }}⌋$$

establishes a fundamental limit on the number of distinguishable states that can be supported within a region defined by its available action. This resonates with holographic principles, such as the Bekenstein bound [4], which constrain the information content of a region in terms of its geometric properties. The present work suggests a possible connection between the action budget of a spacetime region and its geometric information bounds, though this analogy remains speculative and requires further investigation.

Biology and Cybernetics.

Ashby’s Law of Requisite Variety, a cornerstone of cybernetics [21], states that an effective regulator must have at least as much variety as the system it controls. The multiscale version of this law, as articulated by Siegenfeld and Bar-Yam [9], may find a possible physical grounding in the action framework. The requirement for a system to possess sufficient variety corresponds to the requirement of having a sufficient action budget to sustain the needed distinctions. This aligns with recent studies of complexity matching and eco-evolutionary dynamics [22,23], suggesting that cybernetic principles may be physically constrained by available action resources.

Philosophy of Science.

The framework developed here is consistent with structural realist interpretations of scientific theories [24,25]. What appears fundamental, within this framework, are not individual objects (particles, fields) but the structural constraints—action principles, topological quantization rules—from which observable entities emerge as stable, distinguishable configurations. This perspective resonates with Anderson’s dictum that “more is different” [17] and with the view that emergent macroscopic laws require structural explanation rather than independent postulation.

Dynamics versus Constraints.

It is important to note that the present framework primarily defines the structural constraints and the space of possibilities for complexity profiles, as dictated by the action budget. The specific dynamics, governed by a system’s Hamiltonian, determine which particular profile is realized within these bounds. In other words, the constraints define the feasible set, while the dynamics select the actual trajectory. A detailed study of this interplay represents a promising direction for future research.

The structural principles derived above can be summarized in a compact chain of correspondences:

Table 1. Correspondence between the phenomenological laws of Siegenfeld–Bar-Yam (Entropy, 2025) and the structural phase-topological framework developed here.

Phenomenological Law	Structural Foundation (This Work)	Interpretation
Multi-scale law of requisite variety:$C_X\left(n\right)\ge C_Y\left(n\right)$ for all n.	Capacity Bound:$M_{max}\left(R\right)\le \lfloor \delta S\left[R\right]/\hslash \rfloor$.	The system cannot sustain more distinctions than allowed by its action budget. If $\delta S_X<\delta S_Y$, then $C_X\left(n\right)<C_Y\left(n\right)$ necessarily follows for some scales.
Sum rule:${\sum }_n{C}_X\left(n\right)={\sum }_{x\in X}H\left(x\right)$.	Principle of Budget Conservation:$\delta S\left[R\right]$ provides a fixed upper bound under redistribution across scales (with fixed boundary conditions).	Redistribution of complexity across scales reflects the balance of the total action budget, while the total informational capacity remains constrained by $\delta S\left[R\right]$.
Complexity profile $C_X\left(n\right)$.	Projection of Distinction:$C_X\left(n\right)=ln{\mu }_{\Omega }\left({\Pi }_{n+1}\rho \right)-ln{\mu }_{\Omega }\left({\Pi }_n\rho \right)$.	The profile can be interpreted as a finite-difference projection of the structural measure ${\mu }_{\Omega }$ under hierarchical coarse-graining, rather than as an independent fundamental law.

Table 1. Correspondence between the phenomenological laws of Siegenfeld–Bar-Yam (Entropy, 2025) and the structural phase-topological framework developed here.

Phenomenological Law	Structural Foundation (This Work)	Interpretation
Multi-scale law of requisite variety:$C_X\left(n\right)\ge C_Y\left(n\right)$ for all n.	Capacity Bound:$M_{max}\left(R\right)\le \lfloor \delta S\left[R\right]/\hslash \rfloor$.	The system cannot sustain more distinctions than allowed by its action budget. If $\delta S_X<\delta S_Y$, then $C_X\left(n\right)<C_Y\left(n\right)$ necessarily follows for some scales.
Sum rule:${\sum }_n{C}_X\left(n\right)={\sum }_{x\in X}H\left(x\right)$.	Principle of Budget Conservation:$\delta S\left[R\right]$ provides a fixed upper bound under redistribution across scales (with fixed boundary conditions).	Redistribution of complexity across scales reflects the balance of the total action budget, while the total informational capacity remains constrained by $\delta S\left[R\right]$.
Complexity profile $C_X\left(n\right)$.	Projection of Distinction:$C_X\left(n\right)=ln{\mu }_{\Omega }\left({\Pi }_{n+1}\rho \right)-ln{\mu }_{\Omega }\left({\Pi }_n\rho \right)$.	The profile can be interpreted as a finite-difference projection of the structural measure ${\mu }_{\Omega }$ under hierarchical coarse-graining, rather than as an independent fundamental law.

This correspondence emphasizes that the complexity laws described in Ref. [9] should not be treated as independent postulates. Instead, they can be interpreted as quantum-informational consequences of a deeper structural principle: the minimal action cost of distinction.

5. Conclusions

The results of this work can be summarized as follows: the minimal action bound $\delta S\ge \hslash$ defines a bounded capacity for distinctions; this bounded capacity can be interpreted as giving rise to structural contextuality in the relations between system and environment; and from this structural contextuality the universal laws of multi-scale complexity can be understood to emerge.

Within this framework, Planck’s constant may be understood as the minimal quantum of action required to sustain distinguishable configurations. The coherence bound enforces a finite capacity for distinctions, providing, within this framework, a structural account of contextuality in system–environment interactions and supporting the appearance of the phenomenological laws of multi-scale complexity as formulated by Siegenfeld and Bar-Yam.

The framework developed here suggests a foundation for complexity theory analogous to the role of statistical mechanics in thermodynamics. What Siegenfeld and Bar-Yam formulated phenomenologically — the multi-scale law of requisite variety, the sum rule, and the complexity profile — can be interpreted as emergent consequences of a single structural principle: the minimal action cost of distinction.

By grounding these laws in a phase-topological principle, the present formulation offers a possible structural route toward integrating complexity science with the broader physical framework of quantized action and finite informational capacity.