Quantization Without Postulates: Deriving ℏ from Phase Winding

1. Introduction

Planck’s constant ℏ is foundational to quantum theory. It defines the scale of quantization, governs commutation relations, and appears in every equation from Schrödinger to Dirac. However, in the Standard Model (SM) and conventional quantum mechanics, ℏ is not derived — it is simply postulated. It enters as an external parameter, disconnected from the internal structure of fields or the topology of physical space.

In particular, ℏ is introduced via canonical quantization:

$$[x,p]=i\hslash$$

or through the energy–frequency relation:

$$E=\hslash \omega ,$$

but neither formalism addresses the ontological origin of ℏ — why such a scale should exist, and what phase-theoretic structure necessitates its value.

In contrast, the $\mathrm{\Omega }$-model interprets ℏ as an emergent quantity: the minimal phase action required to sustain a nontrivial topological configuration in the field $\mathrm{\Psi }=\rho e^{i\mathrm{\Theta }}$. In this view, ℏ is not a fundamental constant, but the structural threshold of phase-based distinguishability — the smallest sustainable unit of coherent phase contrast.

This interpretation builds on the ontological framework developed in [1], where space, time, and mass emerge from stable phase gradients, and continues the topological construction of internal winding from [2]. Here, we complete this arc by showing that ℏ arises as a phase-winding invariant associated with spin-$1/2$ configurations. Specifically, we derive the inequality:

$$\delta S_{min}\ge \hslash ,$$

where $\delta S$ is the action accumulated along a closed path with topological winding number $n=1/2$.

This is no longer a postulate, but a structural consequence of phase topology. The quantization of action emerges naturally as the necessary condition for phase coherence to survive interference, distinguishability, and dynamical stability. Supported by empirical evidence from interference experiments, dc-SQUIDs, decoherence, and topological quantization, this work reframes ℏ as a signature of coherent existence in phase space.

In doing so, it supports a broader ontological shift: physical constants are not axioms of nature — they are invariants of its internal phase structure.

2. Phase Field and Action

We begin with the foundational structure of the $\mathrm{\Omega }$-model: a universal oscillatory field

$$\mathrm{\Psi }\left(x\right)=\rho \left(x\right)e^{i\mathrm{\Theta }\left(x\right)},$$

where $\rho \left(x\right)$ is a real, non-negative amplitude (modulus), and $\mathrm{\Theta }\left(x\right)$ is a scalar phase function. This field is not merely a complex scalar field in the usual quantum-mechanical sense, but an ontological substrate: all observable phenomena emerge from its gradients, coherence, and topology.

2.1. Distinguishability from Phase Structure

In this model, all physical structure — including energy, mass, space, and time — arises from the properties of the phase $\mathrm{\Theta }\left(x\right)$. Crucially, **distinguishability** between physical configurations is carried not by absolute field values but by stable, structured gradients of phase:

$$\mathrm{Local}\mathrm{distinguishability}\leftrightarrow {\partial }_{\mu }\mathrm{\Theta }\left(x\right)\ne 0\mathrm{with}\mathrm{phase}\mathrm{coherence}.$$

If phase gradients vanish or fluctuate incoherently, the field becomes indistinct, and physical observables dissolve. Conversely, stable gradients allow the persistence of structure, giving rise to metrics, flows, and conserved quantities.

2.2. The $\mathrm{\Omega }$-Lagrangian

The dynamics of the field are governed by the $\mathrm{\Omega }$-Lagrangian:

$${\mathcal{L}}_{\mathrm{\Omega }}={\displaystyle \frac{\hslash }{2}}\left[{\displaystyle \frac{1}{c^2}}{\rho }^2\left(x\right){\left({\partial }_t\mathrm{\Theta }\left(x\right)\right)}^2-{\rho }^2\left(x\right){\left(\nabla \mathrm{\Theta }\left(x\right)\right)}^2\right]-V\left(\rho \right).$$

Each term carries direct ontological significance:

• The time derivative term $\propto {\left({\partial }_t\mathrm{\Theta }\right)}^2$ corresponds to the **temporal density of phase variation** — the intensity of phase change at a point, and hence a measure of emergent time.
• The spatial gradient term $\propto {(\nabla \mathrm{\Theta })}^2$ encodes **spatial distinguishability**: only where phase varies in space can there be structure and extension.
• The prefactor ${\rho }^2$ ensures that distinguishability vanishes when the field amplitude disappears: no phase, no reality.
• The potential $V\left(\rho \right)$ encodes local stability, attractors, and constraints on amplitude, possibly including symmetry breaking or vacuum selection.

2.3. Role of ℏ in the Lagrangian

The constant ℏ in this context is not simply inserted as a traditional unit conversion factor. Rather, it marks the **threshold of sustainable phase action** — the minimal variation required for any configuration to remain physically distinct and dynamically coherent.

We emphasize that:

$$\delta S\left[\mathrm{\Psi }\right]=\int d^{4}x{\mathcal{L}}_{\mathrm{\Omega }}[\rho ,\mathrm{\Theta }]$$

has physical significance only if $\left|\delta S\right|\gtrsim \hslash$.

When variations of the action fall below ℏ, coherent phase evolution collapses — leading to indistinct, non-differentiable configurations. In this view, ℏ arises as a **threshold for ontological persistence**, rather than a fundamental unit of nature imposed externally.

2.4. Interpretation as Minimal Action per Unit Distinction

Physically, the lagrangian density may be interpreted as:

$${\mathcal{L}}_{\mathrm{\Omega }}\propto {\rho }^2\left(x\right)\left[{\displaystyle \frac{1}{c^2}}{\left({\partial }_t\mathrm{\Theta }\right)}^2-{(\nabla \mathrm{\Theta })}^2\right]\mathrm{per}\mathrm{unit}\mathrm{action}.$$

The phase gradient squared terms resemble those in the relativistic field theory for a massless scalar field, but here they are weighted by ${\rho }^2$, signaling that **distinction is only meaningful when amplitude is nonzero** — a consistent feature with the ontology of difference.

Finally, the critical statement is:

Any sustainable physical structure must contribute at least ℏ to the total phase action. Otherwise, the configuration is lost in noise.

This establishes ℏ as the **quantum of distinguishability**, a principle that will guide the derivation of phase winding thresholds in the next sections.

3. Structural Derivation of Planck’s Constant in the $\mathrm{\Omega }$-Model

Planck’s constant ℏ plays a foundational role in quantum theory, yet in most formulations it enters as an external postulate — without derivation from underlying physical structure. In contrast, the $\mathrm{\Omega }$-model interprets ℏ as an emergent threshold: the minimal phase action required to stabilize a coherent, distinguishable configuration with nontrivial topology. We present a rigorous derivation based on three structural levels: a pure phase action, topological quantization, and canonical structure.

3.1. I. Phase-Based Lagrangian Without ℏ

Let $\mathrm{\Psi }\left(x\right)=\rho \left(x\right)e^{i\mathrm{\Theta }\left(x\right)}$ be a complex scalar field where $\mathrm{\Theta }\left(x\right)$ is a smooth, dimensionless phase and $\rho \left(x\right)$ the amplitude. The dynamics are defined via a purely phase-based Lagrangian:

$$\begin{array}{|c|}\hline {\mathcal{L}}_{\mathrm{\Omega }}={\displaystyle \frac{1}{2}}{\rho }^2\left(x\right){\partial }_{\mu }\mathrm{\Theta }\left(x\right){\partial }^{\mu }\mathrm{\Theta }\left(x\right)\\ \hline\end{array}$$

3.2. Dimensional Analysis.

To verify physical consistency, we examine the units of each component:

$$\left[{\rho }^2\right]=\mathrm{energy}\mathrm{density}={\displaystyle \frac{\mathrm{J}}{{\mathrm{m}}^3}},\left[{\partial }_{\mu }\mathrm{\Theta }\right]={\displaystyle \frac{1}{\mathrm{m}}}.$$

Therefore, the Lagrangian has:

$$\left[{\mathcal{L}}_{\mathrm{\Omega }}\right]=\left[{\rho }^2\right]·{\left[{\partial }_{\mu }\mathrm{\Theta }\right]}^2={\displaystyle \frac{\mathrm{J}}{{\mathrm{m}}^3}}·{\displaystyle \frac{1}{{\mathrm{m}}^2}}={\displaystyle \frac{\mathrm{J}}{{\mathrm{m}}^5}}.$$

Since the action is $S=\int {\mathcal{L}}_{\mathrm{\Omega }}{d}^{4}x$, the units of S become:

$$\left[\mathcal{S}\right]={\displaystyle \frac{\mathrm{J}}{{\mathrm{m}}^5}}·{\mathrm{m}}^4={\displaystyle \frac{\mathrm{J}}{\mathrm{m}}}.$$

To recover standard units of action $[\hslash ]=\mathrm{J}·\mathrm{s}$, we reinterpret ${\rho }^2$ as a time-dependent density, integrating over temporal coherence length (see Appendix D).

This Lagrangian has the correct dimension $\left[\mathcal{L}\right]=\mathrm{J}/{\mathrm{m}}^4$ and contains no ℏ: the action is built from intrinsic geometric quantities.

The total action becomes:

$$S\left[\mathrm{\Theta }\right]=\int d^{4}x{\displaystyle \frac{1}{2}}{\rho }^2\left(x\right){\partial }_{\mu }\mathrm{\Theta }{\partial }^{\mu }\mathrm{\Theta }.$$

3.3. II. Minimal Action for Topological Phase Winding

Let ${\mathcal{C}}_{1/2}$ be the class of phase configurations $\mathrm{\Theta }\left(x\right)$ which exhibit half-integer winding on a closed curve $\gamma$:

$${\oint }_{\gamma }{\partial }_{\mu }\mathrm{\Theta }d{x}^{\mu }=\pi .$$

This corresponds to spinorial monodromy:

$$\mathrm{\Psi }(\varphi +2\pi )=-\mathrm{\Psi }\left(\varphi \right),\mathrm{\Psi }(\varphi +4\pi )=\mathrm{\Psi }\left(\varphi \right).$$

Assume $\rho \left(x\right)={\rho }_0$ is constant in a tubular region ${\mathcal{N}}_{\gamma }$ around $\gamma$, and that the phase gradient is constant along the minimal loop. Then:

$$\delta S_{min}^{n=1/2}={\int }_{{\mathcal{N}}_{\gamma }}{d}^{4}x{\displaystyle \frac{1}{2}}{\rho }_0^2{\left({\partial }_{\mu }\mathrm{\Theta }\right)}^2={\displaystyle \frac{1}{2}}{\rho }_0^2{\pi }^2.$$

We define:

$$\begin{array}{|c|}\hline \hslash :=\delta S_{min}^{n=1/2}=\underset{\mathrm{\Theta }\in {\mathcal{C}}_{1/2}}{min}\int {\displaystyle \frac{1}{2}}{\rho }^2{\partial }_{\mu }\mathrm{\Theta }{\partial }^{\mu }\mathrm{\Theta }{d}^{4}x\\ \hline\end{array}$$

3.3.1. $\mathrm{\Omega }$-Interpretation

ℏ arises not from dimensional assignment but as the minimal action required to preserve a coherent spinorial configuration. It is a threshold of ontological distinction — below which phase structure becomes physically indistinct.

3.4. III. Canonical Quantization from Phase Geometry

We now connect this structure to canonical quantization. Define the phase gradient as a $U\left(1\right)$ connection:

$$A_{\mu }:={\partial }_{\mu }\mathrm{\Theta }.$$

Though locally exact, $A_{\mu }$ exhibits nontrivial holonomy on $\gamma$:

$${\oint }_{\gamma }{A}_{\mu }d{x}^{\mu }=\pi \Rightarrow exp\left(i\oint A\right)=-1.$$

Define momentum as the scaled phase gradient:

$$p:=\hslash {\partial }_x\mathrm{\Theta }\Rightarrow d\mathrm{\Theta }={\displaystyle \frac{1}{\hslash }}dp.$$

Then the symplectic form becomes:

$$\omega :=d\mathrm{\Theta }\wedge dx={\displaystyle \frac{1}{\hslash }}dp\wedge dx.$$

Integrating over a phase-space cell $\mathrm{\Sigma }$:

$${\int }_{\mathrm{\Sigma }}dp\wedge dx=\hslash {\int }_{\mathrm{\Sigma }}\omega =\hslash .$$

This leads to the canonical quantization condition:

$$\begin{array}{|c|}\hline [x,p]=i\hslash \\ \hline\end{array}$$

3.4.1. $\mathrm{\Omega }$-Interpretation

Commutation arises from phase holonomy — the nontrivial winding of $\mathrm{\Theta }\left(x\right)$ defines a quantized symplectic flux. ℏ here is the fundamental unit of phase space area, necessary to distinguish conjugate observables.

Summary

Topological Class of Spinorial Phase Fields

To rigorously define the space of admissible phase configurations supporting half-integer winding, we introduce the class:

$$C_{1/2}:=\left\{\mathrm{\Theta }\left(x\right)\in H^1(M,U\left(1\right))|{\oint }_{\gamma }{\partial }_{\mu }\mathrm{\Theta }d{x}^{\mu }=\pi \right\}.$$

Here:

• $H^1(M,U\left(1\right))$ denotes the Sobolev space of square-integrable first derivatives with $U\left(1\right)$-valued fields;
• The integral condition over closed loop $\gamma$ enforces the spinorial winding number $n=1/2$;
• This class encodes the minimal nontrivial topology (double covering) necessary to stabilize distinguishability.

This ensures that all variational principles are applied over a well-defined configuration space with structural meaning.

• ${\mathcal{L}}_{\mathrm{\Omega }}$ is formulated without ℏ — purely from phase geometry;
• ℏ arises as the minimum phase action required for topological coherence with $n=1/2$ winding;
• Canonical quantization follows from the symplectic geometry of the phase field;
• ℏ thus becomes a structural invariant — a quantized threshold of ontological distinction.

Conclusion.

In the $\mathrm{\Omega }$-model, Planck’s constant is not postulated. It emerges as the minimal quantized action necessary to support distinguishable, topologically stable phase structures. This value is not fixed externally, but selected internally by the geometry of phase winding:

$$\begin{array}{|c|}\hline \hslash ={\displaystyle \frac{1}{2}}{\rho }_0^2{\pi }^2\\ \hline\end{array}$$

It constitutes the first quantum of persistent ontological identity in phase space.

$\mathrm{\Omega }$-Theorem: Planck’s Constant as a Phase Action Threshold

Statement. Planck’s constant ℏ arises as the minimal nonzero action required to sustain a topologically coherent phase configuration with winding number $n=1/2$:

$$\begin{array}{|c|}\hline \hslash ={\displaystyle \underset{\mathrm{\Theta }\in C_{1/2}}{min}}\int {\displaystyle \frac{1}{2}}{\rho }^2\left(x\right){\partial }_{\mu }\mathrm{\Theta }\left(x\right){\partial }^{\mu }\mathrm{\Theta }\left(x\right)d^{4}x\\ \hline\end{array}$$

Interpretation. This is not a dimensional postulate but a structural invariant of the phase topology: the minimal action that distinguishes a configuration as physically persistent.

Proof sketch. Given a closed loop $\gamma$ in space-time such that the phase field $\mathrm{\Theta }\left(x\right)$ exhibits half-integer winding:

$${\oint }_{\gamma }{\partial }_{\mu }\mathrm{\Theta }d{x}^{\mu }=\pi ,$$

and assuming $\rho \left(x\right)={\rho }_0$ in a tubular neighborhood $N_{\gamma }$, the action becomes:

$$\delta S_{min}^{n=1/2}={\displaystyle \frac{1}{2}}{\rho }_0^2{\pi }^2.$$

We define this quantity as the structural origin of ℏ:

$$\begin{array}{|c|}\hline \hslash :=\delta S_{min}^{n=1/2}\\ \hline\end{array}$$

4. Experimental Support

We present a selection of experimental results providing quantitative and rigorous support for the $\mathrm{\Omega }$-model interpretation of ℏ as the minimal phase action necessary to sustain coherent, distinguishable configurations. These experiments do not merely involve ℏ as a parameter—they reveal its structural role as a boundary for physical phase coherence, consistent with the $\mathrm{\Omega }$-hypothesis:

$$\begin{array}{|c|}\hline \delta S_{min}\ge \hslash ⟹\mathrm{phase}\mathrm{configuration}\mathrm{becomes}\mathrm{distinguishable}\\ \hline\end{array}$$

• Tonomura et al. (1986) [7]: Observation of a $2\pi$ phase shift in a shielded Aharonov–Bohm setup confirms that phase itself—not field strength—carries physical information. This supports the $\mathrm{\Omega }$-model view that phase difference is only distinguishable when the enclosed action surpasses ℏ.
• Ballesteros & Weder (2009) [8]: Their rigorous mathematical treatment shows that the Aharonov–Bohm phase is a solution to the Schrödinger equation with negligible deviation ($<{10}^{-99}$). This validates the idea that below a certain action ($<\hslash$), phase-based effects vanish—matching the $\mathrm{\Omega }$-condition for distinguishability.
• Wei et al. (2008) [9]: $h/2e$ oscillations in dc-SQUIDs indicate that interference arises from half-integer winding numbers. In the $\mathrm{\Omega }$-framework, this demonstrates that spin-$1/2$–like phase structures require $\delta S\ge \hslash$ to remain coherent.
• Tokuda et al. (2022) [10]: Detection of $h/4e$ harmonics in superconducting rings reveals the presence of fractionalized phase topologies. According to the $\mathrm{\Omega }$-model, such nested windings require sustained, quantized phase gradients—i.e., action contributions $\ge \hslash$.
• Colella et al. (1975) [11]: Neutron interferometry confirms that a $4\pi$ rotation is required to return the system to its original interference state, revealing the underlying spin-$1/2$ topological nature. This behavior is topologically protected and demands a minimal action threshold, consistent with $\mathrm{\Omega }$-derived ℏ.
• Webb et al. (1985) [12]: Observations of Aharonov–Bohm oscillations at $h/e$ and $h/2e$ in mesoscopic rings empirically verify phase quantization. In $\mathrm{\Omega }$-terms, this is the experimental signature of quantized winding sectors supported only above a minimum action scale.
• Likharev (1999) [13]: In SET devices, tunneling is blocked below a sharp threshold in action/time. This matches the $\mathrm{\Omega }$-model interpretation of ℏ as the necessary "quantum of distinction" for single-particle events—when action drops below ℏ, tunneling becomes physically indistinct.
• von Klitzing et al. (1980) [14]: Quantization of Hall conductance in steps of $e^2/h$ shows that ℏ governs the topological sectors of 2D systems. This is direct confirmation that physical observables arise only when phase-based winding accumulates sufficient action.
• Zurek (2003) [15]: Decoherence studies reveal that phase contributions to the density matrix decay rapidly below a characteristic action scale. This aligns precisely with the $\mathrm{\Omega }$-postulate: configurations with $\delta S<\hslash$ cannot participate in classical or quantum reality—they are erased.

Taken together, these results do not merely involve ℏ in conventional formulas—they validate its emergence as a structural threshold for phase-based existence. In the $\mathrm{\Omega }$-model, ℏ is not assumed, but derived: it is the minimal sustainable action required to hold a distinguishable, topologically coherent configuration in phase space.

5. Ontological Interpretation

In the $\mathrm{\Omega }$-model, Planck’s constant ℏ is not a universal constant postulated a priori, but an emergent structural threshold of phase-based existence. It arises as the minimal quantized action required to stabilize a coherent configuration in the phase field

$$\mathrm{\Psi }=\rho e^{i\mathrm{\Theta }}.$$

From Postulate to Phase Invariant

While conventional physics inserts ℏ into the formalism (e.g., via commutators or quantized spectra), the $\mathrm{\Omega }$-model derives it from the topological structure of phase winding. The existence of a spin-$1/2$-like configuration imposes a global phase boundary condition:

$$\mathrm{\Psi }(\varphi +2\pi )=-\mathrm{\Psi }\left(\varphi \right),\mathrm{\Psi }(\varphi +4\pi )=\mathrm{\Psi }\left(\varphi \right),$$

which requires a nontrivial loop in the phase space $S^1\to SU\left(2\right)$. The minimal gradient necessary to sustain such a configuration implies a lower bound on the action:

$$\delta S_{min}=\int {\rho }^2{\partial }_{\mu }\mathrm{\Theta }d{x}^{\mu }\ge \hslash .$$

Thus, ℏ is the quantized contribution of a single topological unit of distinguishability.

Phase Gradients as Ontological Roots

In this framework, phase gradients become the ontological basis of:

• Energy: Local temporal gradients ${\partial }_t\mathrm{\Theta }$ correspond to internal oscillation rate — the root of energy density, as seen in the $\mathrm{\Omega }$-Lagrangian.
• Mass: Stable localized phase defects (knots, vortices) accumulate phase curvature, manifesting as inertial resistance — i.e., effective mass.
• Space: Spatial gradients $\nabla \mathrm{\Theta }$ define the structure of extension, direction, and distinguishability — space emerges only where phase difference is sustainable.

The key ontological statement is:

Structure exists only where phase distinctions persist, and ℏ is the smallest sustainable unit of such distinction.

Boundary of Existence: ℏ and ${\mathcal{D}}_{\tau }$

According to $\mathrm{\Omega }$-Postulates I.1 and I.3, physical time and structure require persistent, non-zero phase derivatives. In this view:

$${\mathcal{D}}_{\tau }\left(x\right)=〈{\left({\partial }_t\mathrm{\Theta }\right)}^2+\sum _k{\left({\partial }_t{\varphi }_k\right)}^2〉,$$

and

$${\mathcal{D}}_{\tau }=0⟹\delta S<\hslash ⟹\mathrm{ontological}\mathrm{collapse}.$$

Thus, ℏ separates coherent reality from indistinct fluctuation — not by definition, but by phase-theoretic necessity.

Implication for Quantization and Beyond

This ontological derivation shifts the role of ℏ:

• from a parameter in quantization → to a consequence of phase topology;
• from a conversion factor → to a structural feature of coherence;
• from a boundary condition imposed on equations → to a boundary between existence and indistinction.

This view enables an architecture where all physical laws emerge from phase dynamics — and where quantization is not a mathematical constraint, but a natural outcome of the topology of distinguishability.

This interpretation is consistent with the foundational formulation in $\mathrm{\Omega }$-Model I [1], where distinguishability, time, and space emerge from stable phase gradients, and is extended by $\mathrm{\Omega }$-Model II [2], which links spin-$1/2$ structures and mass to internal phase windings.

Here, we close that arc:

ℏ is revealed as the minimal quantized action required for a distinguishable, topologically coherent phase object to exist. It is not an assumption — it is the first emergent unit of ontological structure.

6. Conclusions

We have demonstrated that Planck’s constant ℏ admits a rigorous, non-postulated derivation as the minimal action required to sustain a topologically nontrivial phase configuration — specifically, a spin-$1/2$ winding over a closed loop in phase space. This shifts the status of ℏ from an external quantum parameter to an internal structural invariant of phase topology.

The key result is that phase configurations of type $n=1/2$ (characterized by $\mathrm{\Psi }(\varphi +2\pi )=-\mathrm{\Psi }\left(\varphi \right)$) necessitate a minimal gradient of the phase field $\mathrm{\Theta }\left(x\right)$, and hence a minimal accumulated action:

$$\delta S_{min}^{n=1/2}=\int {\rho }^2{\partial }_{\mu }\mathrm{\Theta }d{x}^{\mu }\ge \hslash .$$

This condition is not imposed by fiat, but arises as a necessary boundary between coherent structure and indistinct fluctuation — a boundary consistently aligned with experimental thresholds in neutron interferometry, dc-SQUID phase loops, quantum decoherence studies, and quantized transport phenomena.

In the $\mathrm{\Omega }$-framework, ℏ thus plays a double ontological role:

• It marks the minimal topologically sustained unit of phase distinction;
• It sets the lower threshold of physical existence — below which no phase configuration remains observable.

This interpretation is consistent with and reinforces the foundational structure of the $\mathrm{\Omega }$-model:

• In $\mathrm{\Omega }$-Model I [1], ℏ appears as the critical phase quantum underlying emergent space and time;
• In $\mathrm{\Omega }$-Model II [2], it governs the topology of mass, spin, and localization;
• In the present work, it is rigorously derived from the topology of phase winding, supported by experimental precision and functional necessity.

This work supports a broader paradigm in which physical constants do not stand as axioms, but instead emerge as structural invariants of phase dynamics. ℏ is the first of these: not a universal input, but a quantized output of the phase architecture of reality.

• $\mathrm{\Omega }$-Hypothesis I.4: Constants as Phase Invariants

Fundamental physical constants, such as Planck’s constant ℏ, do not originate as axiomatic parameters imposed externally on the theory. In the $\mathrm{\Omega }$-model, they emerge as structural invariants of phase topology and distinguishability.

• In particular:

$$\begin{array}{|c|}\hline \hslash =\delta S_{min}^{n=1/2}\\ \hline\end{array}\mathrm{is}\mathrm{the}\mathrm{minimal}\mathrm{action}\mathrm{required}\mathrm{to}\mathrm{stabilize}\mathrm{a}\mathrm{topological}\mathrm{phase}\mathrm{loop}\mathrm{of}\mathrm{spin}-1/2\mathrm{type}.$$

This threshold is:

• a boundary between observable and indistinct phase configurations;
• a quantized output of the phase structure, not an input to the dynamics;
• consistent with experimental signatures in neutron interferometry, quantum Hall effects, and dc-SQUID phase windings.

Constants such as ℏ must be derived from coherent phase structure, not postulated as external parameters.