A Geometric Capacity Formula for the Fine-Structure Constant

1. Introduction

The fine-structure constant $\alpha$ is one of the central dimensionless constants of physics. In quantum electrodynamics (QED), it measures the dimensionless strength with which charged particles couple to the electromagnetic field. Operationally, $\alpha$ is determined from high-precision measurements such as atomic recoil and the electron magnetic moment, together with the theoretical structure of QED [1,2,3,4,5]. Within the Standard Model, however, its numerical value is not derived; it is an input parameter.

This situation is not unusual in modern physics. The quantities observed in experiment are rarely identical to the simplest bare quantities appearing in a formal description. In quantum field theory, the vacuum is not an empty background: it carries fluctuations, polarization, and response. A bare charge or mass is not directly observed; the physical observable is obtained only after the excitation is embedded into the surrounding quantum-field response. This is the logic behind renormalization [6,7,8]. In general relativity, the metric is also not a passive stage. It responds to energy and momentum according to Einstein’s equation,

$$G_{\mu \nu }+\Lambda g_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu },$$

so the physical geometry seen by matter is inseparable from the dynamical response of the system. These familiar examples suggest a broad lesson: physical observables often arise as read-outs of an excitation together with the structure that supports, dresses, and makes it observable.

The present paper applies this lesson to the fine-structure constant. We ask whether $\alpha$ is not merely a freely tunable low-energy coupling, but the observable normalization produced when electromagnetic readability is embedded into the carrier structure of spacetime itself. Here the “carrier” is not a material medium inside spacetime. It is spacetime understood process-theoretically: the rule-governed support of admissible continuation, phase comparison, orientation, and closure. In the Standard Model this carrier is already equipped, at the effective field-theory level, with a $U{\left(1\right)}_{\mathrm{EM}}$ phase structure at each spacetime point [6,9,10]. The present proposal asks why the universal normalization of this everywhere-available electromagnetic phase read-out has the observed value of $\alpha$.

The central object considered here is a localized defect of this continuation process. A particle is not assumed to be a primitive point object inserted into a background spacetime. Rather, it is treated as the stable Lorentz-covariant read-out of a localized carrier defect after admissible continuation, closure, and phase structure become physically readable. In this sense, the proposal parallels the distinction between a bare parameter and a dressed physical observable: the neutral-parent defect is the pre-resolved carrier structure, while an observed charged or neutral particle is a branch-level read-out of that structure.

The proposed origin of $\alpha$ is therefore not sought in ordinary spacetime geometry alone. It is sought in the capacity of a neutral parent defect to support electromagnetic readability before its resolved branches are observed as charged, neutral, weak, or hadron-facing channels. The neutral parent should not be interpreted as a literal particle simultaneously decaying into all possible channels. It is an unresolved admissibility archetype of the carrier structure from which different branch read-outs become possible.

The key physical shift is to treat magnetic response as more primitive than branch-level electric charge. Electric charge is an oriented read-out of resolved branch exposure. Magnetic response, by contrast, is a loop-level holonomy read-out of a closed defect. A neutral object can have no net electric charge while still carrying magnetic response:

$$Q=0,\mu \ne 0.$$

The neutron is the familiar physical example. This motivates the possibility that the neutral parent need not be electrically charged in order to determine the universal electromagnetic normalization. It only needs to possess a loop-readable magnetic holonomy capacity.

In the present interpretation, a unit $U\left(1\right)$ loop does not measure an isolated elementary charge. It reads the completed electromagnetic content of the neutral-parent defect: the carrier capacity that must be coherently supported before branch-level electromagnetic readability can occur. After branch resolution, different channels may expose different charge numbers,

$$q_i=n_{i}e,$$

including negative, positive, fractional, or zero values, but they do not reselect the universal electromagnetic normalization. The neutral parent fixes the common capacity; the resolved branch fixes only the oriented charge read-out. Thus different charged particles inherit the same fine-structure constant, while their charge factors may differ.

This point may be stated counterfactually. If the charged, neutral, weak, and hadron-facing channels were independent primitive sectors, there would be no intrinsic reason, within the present framework, for their $U\left(1\right)$-readability capacities to be identical. One would generically expect channel-dependent capacities and hence channel-dependent electromagnetic normalizations. The observed universality of electromagnetic coupling is normally expressed in gauge theory by the existence of a single low-energy $U{\left(1\right)}_{\mathrm{EM}}$ coupling. The present proposal does not replace that statement. Rather, it proposes a pre-gauge structural reason why the same normalization is inherited by all electromagnetic branch read-outs.

We therefore seek a geometric-capacity formula for

$${\alpha }^{-1}$$

as the magnetic holonomy capacity of the neutral parent:

$$\begin{array}{|c|}\hline {\alpha }^{-1}=\mathrm{neutral}-\mathrm{parent}\mathrm{magnetic}\mathrm{holonomy}\mathrm{capacity}.\\ \hline\end{array}$$

The capacity is large because the $U\left(1\right)$ phase loop is coupled to a nontrivial internal closure structure, including the polarized ${\mathbb{Z}}_2$-type branch resolution and the nested hadron-facing ${\mathbb{Z}}_3\to {\mathbb{Z}}_4\to {\mathbb{Z}}_5\dots$ closure hierarchy. Intuitively, one unit of phase circulation must coherently read a larger complex of linked closure processes. The effective branch-level electromagnetic response is therefore small because it is the reciprocal of this large capacity.

The leading numerical expression used in this paper,

$${\mathsf{\Omega }}_0=4{\pi }^3+{\pi }^2+\pi ,$$

has appeared previously in speculative discussions of the fine-structure constant, where it was observed to give a close approximation to ${\alpha }^{-1}$ [11,12,13]. The novelty of the present work is therefore not the isolated observation of this leading number. Rather, it is the proposed interpretation of ${\mathsf{\Omega }}_0$ as a neutral-parent magnetic capacity and the subsequent constrained correction hierarchy. In this sense, the present proposal attempts to convert a previously noticed numerical coincidence into a structured carrier-defect expansion.

The main two-correction result proposed here is

$$\begin{array}{|c|}\hline {\alpha }_{\mathrm{geom}}^{-1}={\mathsf{\Omega }}_{P_0}-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}},{\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi .\\ \hline\end{array}$$

This gives

$${\alpha }_{\mathrm{geom}}^{-1}=137.0359991761696.$$

Compared with the CODATA 2022 value

$${\alpha }_{\mathrm{CODATA}}^{-1}=137.035999177\left(21\right),$$

the residual is

$${\alpha }_{\mathrm{geom}}^{-1}-{\alpha }_{\mathrm{CODATA}}^{-1}\approx -8.3\times {10}^{-10},$$

or approximately $-0.04\sigma$ relative to the quoted standard uncertainty [5].

This numerical agreement is not, by itself, a proof. The purpose of this paper is to formulate the structural principle, derive the leading capacity from a stratified phase–spinor embedding, identify the proposed geometric and monodromy origin of the correction terms, and show that the result is sufficiently constrained to motivate a full carrier-defect derivation. The higher-order continuation developed below is also presented conditionally: it shows how the same adjacent-interface rule would generate further terms if the carrier-interface hierarchy persists at higher monodromy order. QED remains the effective theory of branch-level electromagnetic dynamics after the value of $\alpha$ is given. The present proposal addresses the prior question of why the universal dimensionless coupling inherited by QED has the observed value.

For clarity, the logical map of the paper is:

$$\begin{array}{c}\mathrm{process}-\mathrm{theoretic}\mathrm{carrier}\mathrm{structure}\mathrm{of}\mathrm{spacetime}\\ \Downarrow \\ \mathrm{localized}\mathrm{neutral}-\mathrm{parent}\mathrm{defect}\\ \Downarrow \\ \mathrm{closed}U\left(1\right)\mathrm{magnetic}\mathrm{holonomy}\mathrm{and}\mathrm{spinorial}\mathrm{read}-\mathrm{out}\\ \Downarrow \\ \mathrm{neutral}-\mathrm{parent}\mathrm{electromagnetic}\mathrm{capacity}{\mathsf{\Omega }}_{P_0}\\ \Downarrow \\ {\alpha }^{-1}={\mathsf{\Omega }}_{P_0}-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}\\ \Downarrow \\ \mathrm{universal}\mathrm{QED}\mathrm{coupling}\mathrm{inherited}\mathrm{by}\mathrm{resolved}\mathrm{charged}\mathrm{branches}.\end{array}$$

The paper is organized as follows. Section 2 summarizes the minimal carrier-defect vocabulary used in the paper. Section 3 introduces the neutral parent and explains why magnetic holonomy, rather than branch-level electric charge, is the relevant pre-resolution read-out. Section 4 relates this magnetic capacity to the electromagnetic coupling. Section 5 derives the leading capacity ${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi$. Section 6 and Section 7 derive the first two proposed interface corrections, while Section 8 explains why the ingredients are not freely fitted to CODATA. Section 9 evaluates the formula numerically. Section 10 and Section 11 clarify the status of the neutral parent and its relation to magnetic moments. Section 12 develops the formal higher-order continuation and consistency check with the earlier adjacent-interface rule. Section 13 and Section 14 discuss the interpretation, limitations, and open derivational tasks, and Section 15 concludes. Technical details of the leading capacity and interface coefficients are given in Appendices A–C.

2. Background: Archetype Particle Theory and Carrier Defect Read-Out

The present calculation is formulated within the archetype particle theory developed in the preceding research program [14]. Since the fine-structure constant is here interpreted as a carrier-embedding invariant rather than as an adjustable dynamical coupling, we briefly state the concepts needed for the calculation. The purpose of this section is not to rederive the full particle theory, but to fix the dictionary, structural assumptions, and read-out logic used below.

2.1. Dictionary of Basic Concepts

Carrier process

The word “carrier” does not denote a material medium inside spacetime. It denotes spacetime understood process-theoretically: the rule-governed support of admissible continuation, phase comparison, orientation, and closure. At the effective field-theory level, this carrier already supports local phase structure, including the $U{\left(1\right)}_{\mathrm{EM}}$ phase used in QED [9,10,15]. In the present framework, ordinary Lorentz-covariant spacetime is the stable metric-level read-out of this deeper continuation structure.

Defect process

A defect is a localized persistence pattern in the carrier process. A particle is therefore not treated as a primitive point object inserted into a background space. It is the stable Lorentz-covariant read-out of a localized process defect whose identity can be continued, closed, and recognized across admissible stages.

Codimension-two defect process

After Lorentz-readable localization is available, the particle-relevant defect is primarily codimension two. Locally, the punctured normal fiber has the homotopy type

$${\mathbb{R}}^2\setminus \left\{0\right\}\simeq S^1.$$

A loop linking the defect is therefore the natural probe of its identity. If the carrier supports phase read-out, this linking loop gives a $U\left(1\right)$ phase closure,

$$e^{i\varphi }\in U\left(1\right),\varphi \sim \varphi +2\pi .$$

This is why $2\pi$ appears as a primitive loop measure in the present paper. The codimension-two statement should not be read as assuming a pre-existing geometric object; it is the Lorentz-readable form taken by a process defect whose identity is probed by closed linking continuation [16,17,18].

Embedding

An embedding is the way a localized process defect is supported, continued, and read out. At the pregeometric level, embedding means admissible placement inside a continuation process: which relations are preserved, which closures are allowed, and which read-outs remain stable. Only after Lorentz read-out does this become an ordinary spacetime-localized particle or field configuration.

Internal monodromy

When the continuation process is carried around a defect, its internal state may close only after a finite sequence of admissible continuations. This finite closure structure is called internal monodromy and is denoted schematically by

$$Z_2,Z_3,Z_4,\dots .$$

These symbols encode finite continuation and closure structure inside the defect pocket.

The use of discrete monodromy at this level is intentional. Before Lorentz-covariant read-out, the defect pocket is not yet a differentiable internal manifold carrying the Standard Model groups $SU\left(2\right)$ and $SU\left(3\right)$ as its primary representation. Continuous gauge descriptions become meaningful after the process admits a Lorentz-compatible physical read-out. Thus the present framework does not replace the Standard Model groups by $Z_n$ groups at the observable field-theory level. It uses finite monodromy to describe pregeometric internal closure, while the usual continuous gauge groups describe the effective field-theory layer [9,19,20].

$Z_2$ branch resolution

The first resolution of the neutral parent is written schematically as

$$P_0⟶Z_2\oplus (Z_2\to Z_3).$$

The $Z_2$ layer distinguishes complementary read-out directions of the parent. It is a branch-resolution layer, not yet a completed confinement-like support.

$Z_3$ closure

The first completed positive-side closure is threefold. In QCD language, color-neutral baryonic closure requires three color slots [21,22]. In the present process language, $Z_3$ records the pregeometric threefold closure condition compatible with later color-neutral hadronic read-out. It should not be identified with the full $SU\left(3\right)$ color gauge group. Incomplete $Z_3$ sectors are not asymptotic readable objects; completed $Z_3$ closures can support baryonic read-out.

$Z_3\to Z_4$ exposure

The first magnetic self-exposure layer is

$$Z_3\to Z_4.$$

The $Z_4$ slot is not a fourth color, a new family, or an independent external branch. It is the first exposure slot attached to a completed $Z_3$ closure. This is why the first correction to the neutral-parent magnetic capacity begins at $Z_3\to Z_4$, rather than at $Z_2\to Z_3$.

Lorentz read-out

The internal defect process becomes physical only after it admits a Lorentz-covariant read-out. A phase loop supplies the $U\left(1\right)$ holonomy, while a spinorial particle read-out requires an admissible spin-frame lift,

$$Spin\left(3\right)\simeq SU\left(2\right)\simeq S^3.$$

Thus the minimal readable structure relevant here contains both

$$U\left(1\right)\mathrm{and}SU\left(2\right).$$

The first is the phase-loop closure; the second is the compact spinorial frame needed for Lorentz-compatible particle read-out [23,24,25]. The appearance of continuous groups at this stage marks the transition from finite process closure to metric-level physical description.

Physical observable

A physical observable is a Lorentz-covariant metric-level read-out of a deeper continuation process. Electric charge, magnetic moment, mass, spin, and particle identity are therefore treated here as effective observables, not as primitive labels. They arise when the internal topological and monodromy structure of a defect becomes readable in the physical layer.

Neutral parent

The neutral parent $P_0$ is the unresolved archetype of particle read-out. It should not be interpreted as a literal particle simultaneously decaying into all possible branch channels. It is an admissibility archetype whose branch structure can be resolved in different physical contexts. For example, ordinary neutron beta decay selects the energetically accessible channel

$$n\to p+e^{-}+{\overline{\nu }}_e,$$

rather than simultaneously exposing electron, muon, and tau channels.

Magnetic holonomy

Magnetic response is treated as a loop-level read-out of the continuation process. This is why it is taken to be more primitive in the present theory than branch-level electric charge. A neutral object can have no net electric charge while still carrying magnetic response,

$$Q=0,\mu \ne 0.$$

The neutron is the familiar physical example. In this paper, the fine-structure constant is interpreted as the universal impedance required for neutral-parent magnetic holonomy to become readable after Lorentz-compatible branch resolution.

Capacity

A capacity is a dimensionless measure of admissible read-outs of the continuation process. At the pregeometric defect-pocket level, finite possibilities are counted by their topological and monodromy structure. When compact continuous read-out groups are involved, the corresponding capacity is computed using normalized Haar measure [25,26]:

$$Vol\left(U\left(1\right)\right)=2\pi ,Vol\left(SU\left(2\right)\right)=2{\pi }^2.$$

The leading neutral-parent capacity used below is therefore a stratified Haar capacity, not an ordinary spacetime volume. It combines finite monodromy content with the continuous phase and spin-frame measures that become available at the Lorentz-readable layer.

2.2. Pregeometric Counting and Equal Branch Weight

A recurring feature of the derivation is that admissible branches, ordered chambers, and closure possibilities are counted with equal weight. This is not meant as an ordinary statistical assumption. It is the symmetry-preserving measure on a finite pregeometric possibility space.

Inside the defect pocket, the primitive data are not lengths, angles, areas, or metric proportions. The primitive data are adjacency, continuation, branch structure, monodromy, and closure. Thus two admissible possibilities are distinguished by their topological role, not by metric size. A non-uniform weighting would require additional metric or dynamical information that is not present before Lorentz-covariant read-out.

This is analogous to the elementary topological fact that a small sphere and a large sphere have the same topological type. Topology remembers connectedness, boundary, winding, branch structure, and closure; it does not remember ordinary size [16,27]. Likewise, the internal carrier pocket treats admissible branches as discrete topological possibilities before any metric measure is available.

Therefore, if an unresolved parent admits N equivalent ordered chambers and no chamber is distinguished by the pregeometric data, the neutral-parent average assigns each chamber weight

$${\displaystyle \frac{1}{N}}.$$

The guiding rule is

$$\mathrm{no}\mathrm{metric}\mathrm{distinction}⟹\mathrm{equal}\mathrm{topological}\mathrm{weight}.$$

Only after the defect is read out as a Lorentz-covariant observable can ordinary metric sizes, dynamical weights, and continuum probabilities enter.

2.3. Neutral Parent and Branch Read-Out

The first resolved structure of the neutral parent is written schematically as

$$P_0⟶Z_2\oplus (Z_2\to Z_3).$$

The $Z_2$ component represents the first two-sided branch resolution of the parent. It supports the lepton-side read-out, including a transverse charged channel and a neutral continuation channel. The nested $Z_2\to Z_3$ component represents the positive-side continuation toward threefold closure.

This distinction is important. The $Z_2$ layer separates complementary read-out directions, but it is not itself a completed closed support. The first completed positive-side support is the $Z_3$ closure. Hence the first magnetic self-exposure correction begins at

$$Z_3\to Z_4.$$

The additional $Z_4$ slot is not a fourth color or a new particle family. It is the first exposure slot attached to a completed threefold closure. Appendix B gives the corresponding chamber-count argument.

The neutral parent is therefore an admissibility archetype, not a simultaneous multi-channel decay object. A physical process selects the channel allowed by its energetic and embedding conditions. In this paper, $\alpha$ is proposed to be a pre-resolution magnetic impedance of the neutral-parent embedding. It is inherited by branch-level electromagnetic phenomena after resolution, but it is not separately chosen for each branch.

2.4. Lorentz-Covariant Read-Out and the Leading Capacity

The internal monodromy structure of a defect is not directly observed as a hidden discrete object. Physical observables arise only after the defect admits a Lorentz-covariant read-out. A phase loop gives the $U\left(1\right)$ holonomy, while a spinorial particle read-out requires the compact spin-frame lift

$$Spin\left(3\right)\simeq SU\left(2\right)\simeq S^3.$$

With the standard unit-radius normalization,

$$Vol\left(SU\left(2\right)\right)=Vol\left(S^3\right)=2{\pi }^2.$$

Thus a carrier-readable defect contains both a phase holonomy and a spinorial frame factor,

$$(e^{i\varphi },g)\in U\left(1\right)\times SU\left(2\right).$$

The Lorentzian particle description is therefore not abandoned; it is treated as the effective metric-level read-out of a deeper carrier-defect embedding.

The leading quantity used in this paper,

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi ,$$

is not an ordinary spacetime volume. It is the leading stratified Haar capacity of the neutral-parent read-out space. The term

$$4{\pi }^3=\left(2\pi \right)\left(2{\pi }^2\right)$$

comes from the coupled $U\left(1\right)$ phase and $SU\left(2\right)$ spinorial interior. The terms

$${\pi }^2+\pi ={\displaystyle \frac{1}{2}}\left(2{\pi }^2\right)+{\displaystyle \frac{1}{2}}\left(2\pi \right)$$

come from the half-exposed spinorial and phase marginal strata of the unresolved neutral parent. Appendix A gives the corresponding stratified Haar-capacity theorem.

2.5. Magnetic Holonomy and the Role of $\alpha$

A central hypothesis of this paper is that magnetic response is more primitive than branch-level electric charge. Electric charge measures an oriented branch-level imbalance of the neutral parent. Magnetic moment, by contrast, measures loop-readable carrier circulation. A neutral object can therefore have no net electric charge while still possessing magnetic response,

$$Q=0,\mu \ne 0.$$

The neutron provides the physical example. This observation motivates the interpretation of $\alpha$ not as a number derived from an isolated electric charge, but as the universal impedance required for neutral-parent magnetic holonomy to become carrier-readable.

The proposed hierarchy is

$$\begin{array}{cc}\hfill \mathrm{codimension}-\mathrm{two}\mathrm{carrier}\mathrm{holonomy}& ⟶\mathrm{neutral}-\mathrm{parent}\mathrm{magnetic}\mathrm{read}-\mathrm{out}⟶\alpha \hfill \\ & ⟶\mathrm{branch}-\mathrm{level}\mathrm{electromagnetic}\mathrm{phenomena}.\hfill \end{array}$$

QED remains the effective field theory governing electromagnetic dynamics after $\alpha$ is given. The present theory addresses the prior structural question of why the dimensionless value of $\alpha$ has its observed numerical value.

3. Neutral Parent and Magnetic Holonomy

The framework begins from a localized carrier-defect identity structure. Particle properties are not assumed as primitive labels; they are treated as Lorentz-readable observables produced by stable closure, continuation, and branch resolution. The unresolved neutral parent is denoted by

$$P_0.$$

Its first resolved structure is written schematically as

$$P_0⟶Z_2\oplus (Z_2\to Z_3).$$

The $Z_2$ component represents the first two-sided branch resolution, supporting lepton-side read-outs such as charged-lepton and neutrino channels. The nested $Z_2\to Z_3$ component represents the positive, hadron-facing continuation toward a completed threefold closure.

For the present calculation, $P_0$ should be understood as an unresolved admissibility archetype, not as a literal particle that simultaneously decays through all possible channels. A physical neutron, for example, realizes the lowest energetically accessible weak channel,

$$n\to p+e^{-}+{\overline{\nu }}_e,$$

rather than exposing electron, muon, and tau channels at once [20,28]. More massive neutral excitations may open other weak channels, but a particular physical process resolves one kinematically and dynamically admissible channel at a time.

This distinction is important for the fine-structure constant. The value of $\alpha$ should not be assigned separately to the lepton branch, the neutrino branch, or the hadron-facing branch. Nor should the leading capacity be multiplied by the number of possible flavor-family resolutions. In the present framework, $\alpha$ is a pre-resolution electromagnetic normalization: a magnetic impedance of the neutral-parent carrier embedding that is inherited by branch-level electromagnetic read-outs after resolution.

The neutral parent has no net electric charge, but this does not imply that it has no electromagnetic readability. A closed carrier defect may possess loop-readable magnetic response even when its total electric charge vanishes. The neutron again provides the familiar physical example,

$$Q=0,\mu \ne 0.$$

Thus the relevant pre-resolution structure is not bare electric charge, but magnetic holonomy: the loop-level response of a closed carrier defect. The proposed read-out hierarchy is

$$\mathrm{carrier}\mathrm{loop}\mathrm{holonomy}⟶\mathrm{magnetic}\mathrm{response}⟶\mathrm{branch}-\mathrm{level}\mathrm{charge}\mathrm{read}-\mathrm{out}.$$

Electric charge is then interpreted as an oriented branch imbalance, whereas magnetic response measures loop-readable carrier circulation.

This interpretation is compatible with the absence of observed isolated magnetic monopoles. The claim is not that the neutral parent carries a magnetic charge. Rather, magnetic response is treated as a closure-based holonomy effect of the defect process, while ordinary electric charge appears only after branch-level read-out. The fine-structure constant is therefore proposed to measure the universal normalization required for this neutral-parent magnetic holonomy to become electromagnetic readability at the Lorentz-covariant branch level.

Figure 1 gives a minimal schematic of this branch-read-out structure, emphasizing that the neutral parent is an unresolved admissibility source rather than a collection of simultaneous decay products.

4. Connection to Electromagnetic Coupling

The proposed capacity should not be understood as an additional electric charge, nor as a replacement for the QED interaction vertex. In low-energy QED, the fine-structure constant measures the dimensionless strength with which a charged excitation couples to the electromagnetic field [1,6,29]. The present framework addresses a prior structural question: why the universal normalization of that coupling has its observed value.

The key distinction is between carrier normalization and branch charge read-out. Branch-level charge is not taken as primitive. It is an oriented read-out of a resolved neutral-parent structure. The pre-resolution object is instead the closed $U\left(1\right)$ loop holonomy of the carrier defect: a loop-level readability structure that exists before resolved branches are assigned positive, negative, fractional, or neutral electric charge. The neutral parent therefore satisfies

$$Q\left(P_0\right)=0,$$

but it may still possess a closed electromagnetic readability capacity.

This capacity does not measure electric charge directly. It measures how a unit $U\left(1\right)$ carrier loop reads the electromagnetic content enclosed by the defect. Before branch resolution, opposite oriented exposures cancel inside the neutral parent. After resolution, the same carrier normalization is inherited by the exposed branches, while each branch carries its own charge read-out number,

$$q_i=n_{i}e.$$

Here $n_i$ may be negative, positive, fractional, or zero depending on the branch structure. The branch determines $n_i$; it does not reselect the universal electromagnetic normalization.

Accordingly, a charged branch is not a new source of an independent $U\left(1\right)$ coupling. It is an oriented exposure of the same carrier-wide $U\left(1\right)$ readability structure. The electromagnetic strength associated with branch i is therefore proportional to

$$n_i^2\alpha ,$$

not to a branch-dependent fine-structure constant ${\alpha }_i$. For example, a charged lepton branch may expose $n_i=-1$, a neutrino branch may expose $n_i=0$, and a positively charged hadron branch may expose $n_i=+1$. Confined internal channels may carry fractional readability before forming integer-charge composite states. These different read-outs inherit the same carrier-wide normalization; branch resolution changes the representation of electromagnetic readability, not the value of $\alpha$.

The neutral-parent magnetic capacity therefore plays the role of an electromagnetic impedance, or normalization denominator. A unit branch-level charge does not couple with arbitrary strength. Its effective response is normalized by the completed carrier holonomy that must be coherently supported in order for electromagnetic readability to exist. We therefore identify the inverse fine-structure constant with the universal electromagnetic capacity,

$${\alpha }^{-1}={\mathsf{\Omega }}_{\mathrm{EM}},$$

or equivalently

$$\alpha ={\mathsf{\Omega }}_{\mathrm{EM}}^{-1}.$$

In the present paper this capacity is the neutral-parent magnetic capacity,

$${\mathsf{\Omega }}_{\mathrm{EM}}={\mathsf{\Omega }}_{P_0}.$$

The smallness of $\alpha$ then has a direct interpretation. A unit $U\left(1\right)$ loop does not read a simple isolated charge. It reads the completed neutral-parent defect, including the polarized $Z_2$ lepton-facing sector, the neutral/weak sector, and the nested hadron-facing closure content associated with the $Z_3\to Z_4\to Z_5\dots$ interface hierarchy. The electromagnetic capacity is therefore a large count-like geometric capacity: it measures the amount of carrier structure that must be coherently supported for one unit of electromagnetic readability. The branch-level coupling is small because it is the reciprocal response of this large neutral-parent capacity,

$$\alpha =C_{\mathrm{EM}}^{-1}.$$

This also explains, within the present framework, why the same $\alpha$ is shared by different charged particles. If the charged, neutral, weak, and hadron-facing channels were independent primitive sectors, there would be no intrinsic reason for their $U\left(1\right)$-readability capacities to be identical. One would generically expect channel-dependent capacities $C_i$ and hence channel-dependent electromagnetic normalizations,

$${\alpha }_i^{-1}=C_i.$$

In standard gauge theory, the observed universality of electromagnetic coupling is expressed by the existence of a single low-energy $U{\left(1\right)}_{\mathrm{EM}}$ coupling [9,10,30]. The present proposal does not replace that effective field-theory statement. It proposes a pre-gauge structural reason why the same normalization is inherited by all electromagnetic branch read-outs.

Thus QED remains the effective theory of electromagnetic dynamics after the value of $\alpha$ is given. The neutral-parent construction addresses a different question: why the common dimensionless normalization of the electromagnetic interaction has the value it does.

5. Leading Neutral-Parent Embedding Capacity

The leading term is the proposed neutral-parent electromagnetic capacity before magnetic self-exposure corrections are included. It is built from the minimal compact read-out structures required by the framework. A codimension-two defect supplies a linking phase loop with Haar measure

$$Vol\left(U\right(1\left)\right)=2\pi ,$$

while Lorentz-compatible spinorial read-out supplies the compact frame lift

$$SU\left(2\right)\simeq S^3,Vol\left(S^3\right)=2{\pi }^2.$$

These standard compact-group measures are used here as dimensionless read-out capacities, not as ordinary spacetime volumes [18,25,26].

The unresolved neutral parent is modeled as a stratified read-out space: a coupled phase–spinor interior together with two half-exposed marginal strata. Under this stratified-capacity assumption, the leading capacity is

$${\mathsf{\Omega }}_{P_0}=\left(2\pi \right)\left(2{\pi }^2\right)+{\displaystyle \frac{1}{2}}\left(2{\pi }^2\right)+{\displaystyle \frac{1}{2}}\left(2\pi \right).$$

Equivalently,

$$\begin{array}{|c|}\hline {\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi .\\ \hline\end{array}$$

This expression should not be interpreted as the volume of a single smooth manifold. It is a stratified Haar capacity: the product term is the Haar measure of the coupled $U\left(1\right)\times SU\left(2\right)$ interior, while the two half-weight terms are the marginal phase and spinorial boundary contributions of the unresolved neutral parent. Appendix A states the corresponding conditional capacity theorem and its assumptions.

Numerically,

$${\mathsf{\Omega }}_{P_0}=137.0363037758784.$$

This already lies near the observed inverse fine-structure constant, but its residual relative to the CODATA 2022 value is still

$${\mathsf{\Omega }}_{P_0}-{\alpha }_{\mathrm{CODATA}}^{-1}\approx 3.046\times {10}^{-4}.$$

Thus the leading stratified capacity captures the dominant scale but requires magnetic self-interface corrections to reach the observed value.

6. First Self-Exposure Correction: $Z_3\to Z_4$

The leading capacity ${\mathsf{\Omega }}_{P_0}$ describes the unresolved neutral-parent magnetic embedding before self-exposure. If this capacity is to become charge-readable, however, its magnetic holonomy must be exposed through a completed closure interface. In the present framework, the first such interface is

$$Z_3\to Z_4.$$

The reason is that $Z_2$ is a branch-resolution layer rather than a completed confinement-like support. It distinguishes complementary read-out directions of the neutral parent, but it does not yet define a closed carrier-readable unit. The first completed positive-side support is threefold, in analogy with the three-slot color-neutral closure of baryonic configurations in QCD language [21,22]. Thus the first magnetic self-exposure is taken to be the exposure of a completed $Z_3$ support:

$$Z_3\to Z_4.$$

The additional $Z_4$ slot is not a fourth color and not a new particle branch. It is the first external magnetic read-out slot attached to a completed $Z_3$ closure. The elementary interface therefore has four slots,

$$X_{34}=\{a,b,c,ϵ\},$$

where $a,b,c$ denote the three closed support slots and $ϵ$ denotes the exposure slot. Since the unresolved neutral parent has no preferred ordering before read-out, the first exposure is counted over the ordered chambers of this four-slot interface. An ordered chamber is an ordering of $X_{34}$, so the chamber space is acted on freely and transitively by $S_4$:

$${\mathcal{C}}_{34}\simeq S_4,|{\mathcal{C}}_{34}|=|S_4|=4!=24.$$

This is an ordered, non-quotiented chamber count. No cyclic or permutation quotient is taken at this first exposure stage, because the count is over unresolved ordered exposure alignments before a single Lorentz-readable alignment is selected. Appendix B gives the chamber-count argument in theorem form.

The first self-exposure correction is obtained by applying the uniform chamber weight to the inverse leading capacity. The chamber weight is

$$w_{34}={\displaystyle \frac{1}{|{\mathcal{C}}_{34}|}}={\displaystyle \frac{1}{24}},$$

and the first self-readout scale is

$${ϵ}_{P_0}={\mathsf{\Omega }}_{P_0}^{-1}.$$

Therefore

$${\mathsf{\Delta }}_{34}=w_{34}{ϵ}_{P_0}={\displaystyle \frac{1}{24}}{\mathsf{\Omega }}_{P_0}^{-1}={\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}.$$

Its sign is negative because the exposure resolves part of the unresolved parent capacity into an externally readable channel. It is therefore a self-readout back-reaction on the leading capacity, not an additional independent capacity. The first corrected approximation is

$${\alpha }_{\left(4\right)}^{-1}={\mathsf{\Omega }}_{P_0}-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}.$$

Numerically,

$${\alpha }_{\left(4\right)}^{-1}=137.0359997201974.$$

The residual is reduced to

$${\alpha }_{\left(4\right)}^{-1}-{\alpha }_{\mathrm{CODATA}}^{-1}\approx 5.43\times {10}^{-7}.$$

Thus the $Z_3\to Z_4$ self-exposure term gives the dominant correction to the leading-capacity approximation, while leaving a smaller residual addressed by the next magnetic-interface transfer.

Figure 2 schematically illustrates how the embedded $Z_2$-$Z_3$ read-out structure is carried by the next $Z_4$ monodromy layer. The figure shows the four-slot exposure architecture; the full ordered chamber count is the set of all permutations of the three support slots and the exposure slot.

7. Reduced Magnetic Interface Transfer: $Z_4\to Z_5$

The next correction is different from the first $Z_3\to Z_4$ self-exposure term. The first correction opens the external magnetic readability of a completed $Z_3$ closure. After this exposure has occurred, the next layer should not be counted as a new independent external slot. Instead, the $Z_4\to Z_5$ contribution is treated as an internal magnetic-interface relaxation of the already exposed but still closure-bound $Z_3$-$Z_4$ domain.

For this reason the second correction is not governed by a new $5!$ ordered-chamber average. It is inherited from the residual magnetic-interface hierarchy used in the magnetic-moment analysis. In that hierarchy, the higher closure-interface coefficients are

$$C_{m,m+1}^{\mathrm{res}}=2\left(1-{\displaystyle \frac{1}{{(m+1)}^2}}\right),m\ge 4.$$

Hence

$$C_{45}=2\left(1-{\displaystyle \frac{1}{5^2}}\right)={\displaystyle \frac{48}{25}},$$

and

$$C_{56}=2\left(1-{\displaystyle \frac{1}{6^2}}\right)={\displaystyle \frac{35}{18}}.$$

These coefficients are not fitted in the present alpha calculation. They are imported from the same closure-interface structure that organizes the magnetic-moment read-out. Appendix C summarizes the residual-interface rule and the derivation of $C_{45}$ and $C_{56}$.

The alpha calculation, however, does not use the full baryonic magnetic response. It seeks the universal neutral-parent magnetic impedance. Thus the $Z_4\to Z_5$ contribution must be reduced to the part that transfers across the relevant neutral-parent interface. The cross-interface transfer compares the numerator of the next interface with the numerator of the current interface:

$$T_{45\to 56}={\displaystyle \frac{35}{48}}.$$

The effective neutral-parent $Z_4\to Z_5$ coefficient is therefore

$$A_{45}=T_{45\to 56}{C}_{45}={\displaystyle \frac{35}{48}}·{\displaystyle \frac{48}{25}}={\displaystyle \frac{7}{5}}.$$

Since this is the first internal self-readout correction after the external $Z_3\to Z_4$ exposure has been opened, it enters at the next odd inverse-capacity order:

$${\mathsf{\Delta }}_{45}={\displaystyle \frac{A_{45}}{{\mathsf{\Omega }}_{P_0}^3}}={\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}.$$

The inverse-cubic suppression expresses that this is not a first exposure of the parent, but a higher magnetic-interface relaxation of an already exposed closure domain.

Combining the leading capacity, the first self-exposure correction, and the reduced magnetic-interface transfer gives

$$\begin{array}{|c|}\hline {\alpha }_{\mathrm{geom}}^{-1}={\mathsf{\Omega }}_{P_0}-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}.\\ \hline\end{array}$$

Using

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi ,$$

this becomes the explicit geometric-capacity expression

$$\begin{array}{|c|}\hline {\alpha }_{\mathrm{geom}}^{-1}=4{\pi }^3+{\pi }^2+\pi -{\displaystyle \frac{1}{24(4{\pi }^3+{\pi }^2+\pi )}}-{\displaystyle \frac{7}{5{(4{\pi }^3+{\pi }^2+\pi )}^3}}.\\ \hline\end{array}$$

8. Independence and Non-Fitting of the Ingredients

It is important to separate the numerical agreement from the question of fitting. The proposed formula contains no adjustable real-valued parameter. Its three ingredients are fixed, within the neutral-parent magnetic-holonomy framework, by separate structural inputs: the leading stratified capacity, the ordered $Z_3\to Z_4$ self-exposure chamber, and the reduced $Z_4\to Z_5$ magnetic-interface transfer.

First, the leading capacity

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi$$

is fixed by the stratified carrier embedding. The coupled phase–spinor interior contributes

$$\left(2\pi \right)\left(2{\pi }^2\right)=4{\pi }^3,$$

while the two half-exposed marginal strata contribute

$${\pi }^2+\pi .$$

Thus the leading term is not chosen by comparison with the empirical value of $\alpha$. It follows from the assumed $U\left(1\right)\times SU\left(2\right)$ neutral-parent embedding with two half-exposed boundary strata.

Second, the first correction

$$-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}$$

is fixed by the ordered $Z_3\to Z_4$ self-exposure chamber. The factor 24 is the order of the four-slot chamber permutation space,

$$|S_4|=4!=24,$$

not an adjustable denominator. The inverse-capacity suppression expresses that this is a self-exposure correction of an already completed capacity, rather than a new independent volume contribution.

Third, the second correction

$$-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}$$

is inherited from the reduced magnetic-interface transfer. The coefficient $7/5$ is obtained from the residual-interface coefficients of the $Z_4\to Z_5$ and $Z_5\to Z_6$ magnetic layers, not from a fit to the fine-structure constant. The inverse-cubic suppression expresses that this is the next self-readout correction of the same neutral-parent capacity, after the first external exposure has already occurred.

Consequently, the expression should be regarded as a constrained structural result, conditional on the neutral-parent embedding rules, rather than as a phenomenological fit. Altering any of the three inputs changes the result at once: omitting the $Z_3\to Z_4$ exposure leaves a residual of order ${10}^{-4}$, retaining only that exposure leaves a residual of order ${10}^{-7}$, and the reduced $Z_4\to Z_5$ transfer supplies the remaining correction at the present experimental scale. The agreement therefore depends on the simultaneous consistency of the leading stratified capacity, the ordered self-exposure count, and the reduced magnetic-interface transfer. This does not constitute a complete proof of the formula, but it shows that the numerical result is not obtained by freely tuning parameters to the CODATA value.

9. Numerical Evaluation

Using the leading stratified Haar capacity

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi ,$$

as derived in Appendix A, one obtains

$${\mathsf{\Omega }}_{P_0}=137.0363037758784.$$

The staged numerical values are shown in Table 1. The first correction uses the $Z_3\to Z_4$ chamber-weighted self-exposure term derived in Appendix B; the second uses the residual-interface coefficients summarized in Appendix C.

The staged calculation shows that the leading capacity already lies near the observed inverse fine-structure constant, but still differs from the CODATA 2022 central value by

$${\mathsf{\Omega }}_{P_0}-{\alpha }_{\mathrm{CODATA}}^{-1}\approx 3.046\times {10}^{-4}.$$

The first self-exposure correction reduces the residual to

$${\alpha }_{\left(4\right)}^{-1}-{\alpha }_{\mathrm{CODATA}}^{-1}\approx 5.43\times {10}^{-7}.$$

Thus the $Z_3\to Z_4$ chamber-weighted self-exposure term captures the dominant correction to the leading capacity. The remaining residual is then addressed by the proposed reduced $Z_4\to Z_5$ magnetic-interface transfer, giving

$${\alpha }_{\mathrm{geom}}^{-1}=137.0359991761696.$$

Compared with the CODATA 2022 recommended value

$${\alpha }_{\mathrm{CODATA}}^{-1}=137.035999177\left(21\right),$$

the final residual is

$${\alpha }_{\mathrm{geom}}^{-1}-{\alpha }_{\mathrm{CODATA}}^{-1}=-8.30\times {10}^{-10}.$$

The parenthetical uncertainty corresponds to a standard uncertainty

$$2.1\times {10}^{-8},$$

so the residual is approximately

$$-0.040\sigma .$$

Thus the proposed formula lies well inside the current recommended uncertainty [5].

No adjustable real-valued parameter is fitted in this staged evaluation. The leading capacity is fixed by the neutral-parent stratified Haar construction, the factor 24 is fixed by the $Z_3\to Z_4$ ordered chamber count, and the coefficient $7/5$ is obtained from the inherited residual magnetic-interface hierarchy through the neutral-parent cross-interface transfer. The numerical agreement should therefore be read as a test of the proposed structural construction, not as the result of parameter adjustment.

10. Neutral Parent as an Admissibility Archetype

The neutral parent should not be confused with a simultaneous multi-flavor decay object. It defines an admissibility architecture from which branch-level read-outs can be selected. A particular low-energy realization, such as neutron beta decay, selects the energetically open electron channel,

$$n\to p+e^{-}+{\overline{\nu }}_e.$$

Muon and tau channels are not exposed in ordinary neutron decay because they are not kinematically accessible. They may be viewed, within the present framework, as higher-channel admissibilities of the same general parent architecture, not as simultaneous components of the low-energy neutron process.

This distinction matters for the alpha calculation. The leading capacity

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi$$

is not multiplied by the number of possible flavor channels. It is the pre-resolution magnetic impedance of the parent embedding. Flavor-family structure belongs to later branch resolution, whereas the fine-structure constant is proposed to be selected earlier, at the level of neutral-parent magnetic readability.

Thus the parent contributes one normalized magnetic holonomy capacity, not three independent electromagnetic capacities. The correction coefficients are interface-transfer coefficients, not flavor-count multipliers. This is why the alpha formula uses the reduced cross-interface coefficient $7/5$, rather than a direct multi-branch saturation factor.

11. Relation to Magnetic Moments

The proposed alpha formula is motivated by the same closure-interface logic that appears in magnetic-moment read-out. In the preceding magnetic-moment analysis, the higher closure interfaces

$$Z_3\to Z_4,Z_4\to Z_5,Z_5\to Z_6,Z_6\to Z_7$$

were shown to contribute systematic refinements to baryonic magnetic response [14]. In particular, that analysis introduced the residual magnetic-interface rule

$$C_{m,m+1}^{\mathrm{res}}=2\left(1-{\displaystyle \frac{1}{{(m+1)}^2}}\right),m\ge 4,$$

which gives

$$C_{45}={\displaystyle \frac{48}{25}},C_{56}={\displaystyle \frac{35}{18}}.$$

These coefficients are therefore not introduced in the present paper to fit $\alpha$. They are imported as fixed structural coefficients from the previously developed magnetic-moment hierarchy and are summarized in Appendix C.

The present paper applies this magnetic-interface logic at the neutral-parent level. The difference is that $\alpha$ is not treated as a baryonic magnetic-response ratio. It is interpreted as a universal electromagnetic capacity, or neutral-parent magnetic impedance. For this reason the full baryonic residual coefficient is not used directly. Instead, the relevant $Z_4\to Z_5$ contribution is reduced through the neutral-parent cross-interface transfer

$$A_{45}={\displaystyle \frac{35}{48}}{C}_{45}={\displaystyle \frac{35}{48}}·{\displaystyle \frac{48}{25}}={\displaystyle \frac{7}{5}}.$$

Thus the alpha correction uses the same interface coefficients derived in the magnetic-moment setting, but applies them to a different quantity: the universal neutral-parent capacity rather than a baryonic magnetic-moment ratio.

Consequently, the interface corrections in the alpha formula are strongly capacity-suppressed:

$${\mathsf{\Omega }}_{P_0}^{-1},{\mathsf{\Omega }}_{P_0}^{-3},{\mathsf{\Omega }}_{P_0}^{-5},\dots .$$

This explains why the first two interface terms dominate the alpha calculation, while higher closure layers may be more visible in baryonic magnetic-moment ratios. The connection to the earlier magnetic-moment paper is therefore structural, not phenomenological: both calculations use the same closure-interface hierarchy, but they read it out through different physical observables.

12. Higher-Order Interface Continuation and Consistency Check

The two correction terms used above are not isolated numerical adjustments. They can be embedded into a systematic hierarchy of carrier-interface corrections. This is important because the leading expression ${\mathsf{\Omega }}_0=4{\pi }^3+{\pi }^2+\pi$ has appeared previously as a numerical approximation to ${\alpha }^{-1}$. The distinctive feature of the present construction is that this leading capacity is supplemented by a constrained sequence of monodromy-interface corrections rather than by an unconstrained numerical fit.

The reason such a continuation is meaningful, within the present framework, is that these corrections are not ordinary low-energy radiative or threshold corrections. They are assigned to the pre-branch carrier-defect pocket in which the neutral parent is still unresolved. At that level there are no particle masses, external fields, or branch-specific charges available as independent inputs. The available data are instead the monodromy order, the carrier-facing chamber structure, and the adjacent-interface transition coefficients. The higher-order terms are therefore controlled by discrete interface-counting data rather than by additional phenomenological parameters. This is why the correction hierarchy can be continued formally without introducing new adjustable constants.

A useful consistency check comes from the adjacent-interface coefficient used in the earlier analysis [14]. In that setting, the transition coefficient associated with the adjacent interface ${\mathbb{Z}}_n\to {\mathbb{Z}}_{n+1}$ takes the form

$$C_{n,n+1}=2\left(1-{\displaystyle \frac{1}{{(n+1)}^2}}\right)={\displaystyle \frac{2n(n+2)}{{(n+1)}^2}}.$$

This gives, for example,

$$C_{45}={\displaystyle \frac{48}{25}},C_{56}={\displaystyle \frac{35}{18}},C_{67}={\displaystyle \frac{96}{49}},C_{78}={\displaystyle \frac{63}{32}}.$$

These are precisely the interface coefficients required by the magnetic capacity construction. In particular, the reduced ${\mathbb{Z}}_4\to {\mathbb{Z}}_5$ coefficient used above is recovered by the same numerator-transfer rule:

$$A_{45}=C_{45}{\displaystyle \frac{35}{48}}={\displaystyle \frac{48}{25}}{\displaystyle \frac{35}{48}}={\displaystyle \frac{7}{5}}.$$

Thus the second correction coefficient is not selected from the residual of ${\alpha }^{-1}$; it is inherited from the same adjacent-interface structure used in the earlier paper.

The same rule gives a formal higher-order continuation. Write each adjacent-interface coefficient in reduced form as

$$C_{n,n+1}={\displaystyle \frac{p_n}{q_n}}.$$

Here $p_n$ and $q_n$ denote the numerator and denominator of the coefficient for the interface ${\mathbb{Z}}_n\to {\mathbb{Z}}_{n+1}$. The reduced magnetic-transfer coefficient associated with the same interface is defined by comparing the numerator of the next adjacent-interface coefficient with the denominator of the present one:

$$A_{n,n+1}={\displaystyle \frac{p_{n+1}}{q_n}},C_{n+1,n+2}={\displaystyle \frac{p_{n+1}}{q_{n+1}}}.$$

This notation is meant to emphasize that the transfer coefficient is cross-interface: it is not an arbitrary coefficient attached to ${\mathbb{Z}}_n\to {\mathbb{Z}}_{n+1}$, but the part of the next interface that is inherited through the present denominator.

For the next two interfaces this gives

$$A_{56}={\displaystyle \frac{p_6}{q_5}}={\displaystyle \frac{96}{18}}={\displaystyle \frac{16}{3}},A_{67}={\displaystyle \frac{p_7}{q_6}}={\displaystyle \frac{63}{49}}={\displaystyle \frac{9}{7}}.$$

Equivalently,

$$A_{56}=C_{56}{\displaystyle \frac{96}{35}}={\displaystyle \frac{16}{3}},A_{67}=C_{67}{\displaystyle \frac{63}{96}}={\displaystyle \frac{9}{7}}.$$

Therefore the formal higher-order continuation of the capacity formula is

$${\alpha }_{\mathrm{geom}}^{-1}={\mathsf{\Omega }}_0-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_0}}-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_0^3}}-{\displaystyle \frac{16}{3{\mathsf{\Omega }}_0^5}}-{\displaystyle \frac{9}{7{\mathsf{\Omega }}_0^7}}+\dots .$$

More generally, after the first self-exposure term, the correction hierarchy has the form

$${\alpha }_{\mathrm{geom}}^{-1}={\mathsf{\Omega }}_0-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_0}}-{\displaystyle \sum _{n=4}^{\infty }}{\displaystyle \frac{A_{n,n+1}}{{\mathsf{\Omega }}_0^{2n-5}}},A_{n,n+1}={\displaystyle \frac{p_{n+1}}{q_n}},C_{n,n+1}={\displaystyle \frac{p_n}{q_n}}.$$

The powers ${\mathsf{\Omega }}_0^{-3},{\mathsf{\Omega }}_0^{-5},{\mathsf{\Omega }}_0^{-7},\dots$ reflect successive higher-interface suppressions. Since ${\mathsf{\Omega }}_0\simeq 137$, the series is rapidly convergent; higher terms become extremely small.

Using the first two additional terms gives

$${\mathsf{\Omega }}_0-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_0}}-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_0^3}}-{\displaystyle \frac{16}{3{\mathsf{\Omega }}_0^5}}-{\displaystyle \frac{9}{7{\mathsf{\Omega }}_0^7}}=137.0359991760592.$$

This differs from the CODATA 2022 value ${\alpha }_{\mathrm{CODATA}}^{-1}=137.035999177\left(21\right)$ by approximately $-9.4\times {10}^{-10}$, corresponding to about $-0.045\sigma$. Thus the higher-order continuation is not introduced to tune the present central value. The two-correction formula is already well inside the current CODATA uncertainty. Rather, the importance of the continuation is structural: the same interface rule that appeared in the earlier analysis generates a definite hierarchy of further corrections.

This continuation should nevertheless be understood as conditional. It does not by itself prove the infinite series. It shows that, if the same carrier-interface rule persists at higher monodromy order and if no new admissibility obstruction or branch-level coupling enters the pre-branch defect pocket, then the higher corrections are fixed rather than fitted. A complete derivation would still need to establish this persistence directly from the carrier-defect embedding structure.

Under that condition, the series provides a concrete way in which the present framework can be tested against future higher-precision determinations of ${\alpha }^{-1}$. If experimental uncertainty is reduced below the scale of the ${\mathsf{\Omega }}_0^{-5}$ and ${\mathsf{\Omega }}_0^{-7}$ terms, the higher-order interface hierarchy will cease to be merely a formal consistency check and will become a quantitative target. In this sense, the series does not replace the leading result; it turns the carrier-capacity proposal into a sharper prediction program for future precision measurements.

13. Discussion

The proposed result may be summarized as the conditional identification

$$\begin{array}{|c|}\hline {\alpha }^{-1}=\mathrm{neutral}-\mathrm{parent}\mathrm{magnetic}\mathrm{holonomy}\mathrm{capacity}.\\ \hline\end{array}$$

In this interpretation, the fine-structure constant is not first assigned to an isolated electric charge, nor is it selected independently on the lepton or hadron branch. It is a pre-resolution normalization of the neutral-parent carrier embedding. Branch-level electromagnetic phenomena inherit this normalization after the parent structure becomes Lorentz-readable.

The final expression has a staged structural interpretation:

$${\alpha }_{\mathrm{geom}}^{-1}={\mathsf{\Omega }}_{P_0}-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}.$$

The leading term

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi$$

is the stratified $U\left(1\right)$-phase and $SU\left(2\right)$-spinorial embedding capacity derived in Appendix A. The first correction

$$-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}$$

comes from the ordered $Z_3\to Z_4$ self-exposure chamber derived in Appendix B. The second correction

$$-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}$$

comes from the reduced $Z_4\to Z_5$ magnetic cross-interface transfer, using the residual magnetic-interface coefficients summarized in Appendix C. Thus the formula is presented not as a single isolated approximation, but as a sequence: leading neutral-parent capacity, first external exposure, and first internal magnetic relaxation of the exposed closure domain.

A notable feature of the second correction is that its basic coefficients are inherited from the magnetic-moment hierarchy rather than introduced to fit the fine-structure constant. In that hierarchy, the residual coefficients

$$C_{45}={\displaystyle \frac{48}{25}},C_{56}={\displaystyle \frac{35}{18}}$$

appear in the closure-interface organization of the neutron–proton magnetic-moment ratio. The present alpha calculation does not modify those coefficients. It uses them through the neutral-parent cross-interface transfer

$$A_{45}={\displaystyle \frac{35}{48}}{C}_{45}={\displaystyle \frac{35}{48}}·{\displaystyle \frac{48}{25}}={\displaystyle \frac{7}{5}}.$$

This inheritance reduces the freedom of the alpha formula: the same closure-interface numbers that organize hadronic magnetic-response structure also supply the first internal correction to the neutral-parent magnetic impedance.

Numerically, the result is striking:

$${\alpha }_{\mathrm{geom}}^{-1}=137.0359991761696.$$

Compared with the CODATA 2022 value

$${\alpha }_{\mathrm{CODATA}}^{-1}=137.035999177\left(21\right),$$

the residual is approximately $-8.3\times {10}^{-10}$, or about $-0.04\sigma$ relative to the quoted standard uncertainty [5]. If the three structural ingredients

$${\mathsf{\Omega }}_{P_0},{\displaystyle \frac{1}{24}},{\displaystyle \frac{7}{5}}$$

can be justified from the carrier-defect framework, then the result would suggest that $\alpha$ is closer to a structural invariant than to a freely adjustable phenomenological parameter.

This interpretation also distinguishes the proposed alpha calculation from mass read-out. A mass is an energetic property of a realized defect configuration. It may depend on carrier stiffness, saturation cost, curvature response, boundary relaxation, and the scale of the relevant archetype loop. By contrast, the present alpha formula uses loop holonomy, compact-group Haar capacities, finite chamber counts, and closure-interface transfer rules. These ingredients are primarily topological and capacity-theoretic rather than metric-dynamical. In short,

$$\begin{array}{|c|}\hline \alpha \mathrm{measures}\mathrm{how}\mathrm{the}\mathrm{carrier}\mathrm{defect}\mathrm{can}\mathrm{be}\mathrm{read},\\ \hline\end{array}$$

whereas

$$\begin{array}{|c|}\hline m\mathrm{measures}\mathrm{how}\cos \mathrm{tly}\mathrm{the}\mathrm{defect}\mathrm{is}\mathrm{to}\mathrm{realize}.\\ \hline\end{array}$$

This distinction is not meant to deny the dynamical role of the carrier. A complete theory must still specify an action or admissibility principle that allows and stabilizes the relevant defect sector. The point is that, once the relevant neutral-parent sector is fixed, the proposed value of $\alpha$ is controlled by read-out capacity rather than by a tunable mass-scale parameter.

The role of magnetic response is clarified in the same way. Electric charge is treated as a branch-level imbalance of the neutral parent, whereas magnetic response is a loop-level carrier read-out. A neutral object can have no net electric charge while still possessing a magnetic moment. This motivates treating magnetic holonomy as closer to the carrier-level structure than isolated branch charge. The fine-structure constant is then interpreted as the universal impedance of neutral-parent magnetic readability, with ordinary electric charge appearing only after branch-level read-out.

The main risk of the proposal is numerical overfitting. A formula containing powers of $\pi$, small rational coefficients, and inverse powers of a large capacity can accidentally approximate a measured constant. The present result should therefore not be judged by numerical agreement alone. Its credibility depends on whether the three structural inputs are independently motivated: the leading stratified Haar capacity ${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi$, the ordered $Z_3\to Z_4$ chamber factor $1/24$, and the inherited magnetic-interface transfer giving $7/5$. A limited diagnostic enumeration of nearby capacity-correction formulas was also performed, but it is used only as an anti-numerology check, not as a statistical proof. The fact that $C_{45}$ and $C_{56}$ are inherited from the magnetic-moment hierarchy is an important safeguard against pure numerology, but it does not complete the derivation. A full proof would require deriving the same sequence from a precise carrier-defect embedding theorem.

Finally, the proposal does not replace QED. QED remains the effective field theory describing electromagnetic dynamics once $\alpha$ is given [6,7,29]. The present theory addresses a different question: why the dimensionless electromagnetic coupling has its observed numerical value. If the carrier-defect embedding argument can be made rigorous, then $\alpha$ would be interpreted not as a freely adjustable parameter of low-energy dynamics, but as a geometric-capacity invariant of the neutral-parent read-out structure.

14. Scope and Limitations

The present result should be viewed as a structural conjecture rather than as a completed derivation. The numerical agreement is strong, the leading capacity is formulated as a conditional stratified-capacity theorem in Appendix A, and the interface coefficients are tied to a broader adjacent-interface hierarchy. These features make the proposal more constrained than an isolated numerical formula. Nevertheless, several steps remain before it can be regarded as a fully derived theory.

First, the identification of the stratified Haar capacity with the leading inverse electromagnetic impedance should be derived from a complete carrier-defect action or admissibility principle. Such a result would explain why the neutral-parent read-out capacity is not only a natural geometric quantity, but specifically the quantity inherited as ${\alpha }^{-1}$.

Second, the factor $1/24$ should be derived from a formal theorem for the ordered $Z_3\to Z_4$ exposure interface. The present paper motivates it through the four-slot chamber space $S_4$, but a complete derivation should specify the corresponding orientation, averaging, and self-exposure rules intrinsically.

Third, the coefficient $7/5$ should be derived from a precise cross-interface transfer theorem linking the $Z_4\to Z_5$ and $Z_5\to Z_6$ magnetic layers. The present calculation imports this coefficient from the residual magnetic-interface hierarchy; a complete account should derive the neutral-parent reduction rule directly.

Fourth, the higher-order continuation should be derived rather than postulated. Section 12 gives a formal extension of the same adjacent-interface rule, producing a rapidly convergent sequence of corrections at orders ${\mathsf{\Omega }}_0^{-5},{\mathsf{\Omega }}_0^{-7},\dots$. This continuation is useful as an internal consistency check and as a possible target for future higher-precision measurements. However, its all-order validity remains conditional on the persistence of the same carrier-interface rule inside the pre-branch defect pocket. A complete theory must show that no new admissibility obstruction, branch-level coupling, or additional geometric datum enters at higher monodromy order.

Fifth, the relation between the present capacity formula and ordinary renormalized QED parameters should be clarified more explicitly. The proposal does not claim to compute QED radiative corrections or replace renormalization. It addresses the prior question of why the universal dimensionless coupling inherited by QED has its observed value. A more complete account should explain how the carrier-capacity value is passed to the effective low-energy coupling used in standard precision calculations.

Finally, the proposal does not replace QED. QED remains the effective field theory describing electromagnetic dynamics after the value of $\alpha$ is given. The present work addresses a prior structural question: why the universal dimensionless electromagnetic coupling has its observed numerical value.

15. Conclusions

We have proposed a geometric-capacity interpretation of the fine-structure constant based on neutral-parent magnetic holonomy. The central idea is that magnetic response is a loop-level carrier read-out, whereas electric charge is a branch-level imbalance. A neutral parent can therefore have no net electric charge while still selecting a universal electromagnetic impedance through magnetic holonomy.

The leading neutral-parent capacity is

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi .$$

As discussed above, this leading numerical expression has appeared previously in speculative discussions of the fine-structure constant [11,12,13]. The novelty of the present proposal is not the isolated observation of this leading number, but its interpretation as a neutral-parent magnetic capacity and its embedding into a constrained carrier-interface correction hierarchy.

In the present construction, the leading term is derived in Appendix A as a stratified Haar capacity of a $U\left(1\right)$-phase and $SU\left(2\right)$-spinorial embedding space. Including the first two interface corrections gives

$${\alpha }_{\mathrm{geom}}^{-1}={\mathsf{\Omega }}_{P_0}-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}.$$

Equivalently,

$${\alpha }_{\mathrm{geom}}^{-1}=4{\pi }^3+{\pi }^2+\pi -{\displaystyle \frac{1}{24(4{\pi }^3+{\pi }^2+\pi )}}-{\displaystyle \frac{7}{5{(4{\pi }^3+{\pi }^2+\pi )}^3}}.$$

This yields

$${\alpha }_{\mathrm{geom}}^{-1}=137.0359991761696,$$

compared with the CODATA 2022 recommended value

$${\alpha }_{\mathrm{CODATA}}^{-1}=137.035999177\left(21\right).$$

The residual is approximately

$$-8.3\times {10}^{-10},$$

or about $-0.04\sigma$ relative to the quoted standard uncertainty.

The higher-order continuation developed in Section 12 strengthens the internal consistency of the proposal. The same adjacent-interface coefficient used in the earlier analysis generates the next correction coefficients, leading formally to

$${\alpha }_{\mathrm{geom}}^{-1}={\mathsf{\Omega }}_{P_0}-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}-{\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}-{\displaystyle \frac{16}{3{\mathsf{\Omega }}_{P_0}^5}}-{\displaystyle \frac{9}{7{\mathsf{\Omega }}_{P_0}^7}}+\dots .$$

The additional terms are already smaller than the present experimental uncertainty scale and are therefore not introduced as fitted improvements. Their significance is structural: they show that the correction terms are not arbitrary residual adjustments, but belong to a systematic interface hierarchy. If future measurements of ${\alpha }^{-1}$ reach sufficient precision, the higher-order terms of this series provide definite quantitative targets.

The result should be read as a structural conjecture, not as a completed derivation. If the leading capacity and the interface coefficients can be derived rigorously from the carrier-defect embedding structure, then the fine-structure constant would be interpreted not as a freely adjustable low-energy parameter, but as a geometric-capacity invariant of neutral-parent magnetic readability. QED would remain the effective theory of electromagnetic dynamics, while the present framework would supply a proposed structural origin for the common value of its dimensionless coupling.

Appendix A.1. Codimension-Two Holonomy and the U(1) Phase Factor

Let D be a stable codimension-two carrier defect. Locally, in the normal directions to D, the punctured normal fiber has the homotopy type

$${\mathbb{R}}^2\setminus \left\{0\right\}\simeq S^1.$$

A small loop linking the defect therefore supplies the natural holonomy probe of the defect identity [16,17,18]. If the carrier admits phase read-out, the corresponding compact phase group is

$$U\left(1\right)=\{e^{i\varphi }:\varphi \in [0,2\pi )\}.$$

With the standard geometric Haar-volume normalization on the angular coordinate,

$${\mu }_{U\left(1\right)}\left(U\left(1\right)\right)=2\pi .$$

Thus the factor $2\pi$ is not an arbitrary insertion. It is the minimal phase-loop measure associated with a codimension-two holonomy probe. This convention is a geometric volume convention, not the probability-normalized Haar convention in which the total group measure would be set equal to one.

Appendix A.2. Spinorial Frame Lift and the SU(2)≃S 3 Factor

A charge-readable carrier defect is not only a scalar phase object. Its local physical read-out must also be compatible with spinorial orientation. The relevant compact spinorial frame lift is

$$Spin\left(3\right)\simeq SU\left(2\right)\simeq S^3.$$

Using the unit-radius three-sphere normalization,

$${\mu }_{SU\left(2\right)}\left(SU\left(2\right)\right)=Vol\left(S^3\right)=2{\pi }^2.$$

The compact $SU\left(2\right)$ factor used here should be understood as the rest-frame/internal spin-frame capacity associated with Lorentz-readable particle read-out. It is not a Haar volume of the full noncompact Lorentz group $Spin(3,1)$. Noncompact Lorentzian propagation belongs to the carrier-facing read-out layer; the compact $SU\left(2\right)\simeq S^3$ factor records the spin-frame capacity entering the neutral-parent embedding count.

The leading unresolved read-out state of the neutral parent therefore contains both a phase coordinate and a compact spinorial frame coordinate,

$$(e^{i\varphi },g)\in U\left(1\right)\times SU\left(2\right).$$

At leading order, before interface corrections are included, these two read-out factors are treated as independent. The interior measure is therefore the product of the corresponding geometric Haar-volume measures,

$${\mu }_{\mathrm{int}}={\mu }_{U\left(1\right)}\times {\mu }_{SU\left(2\right)},$$

so that

$${\mu }_{\mathrm{int}}\left(U\left(1\right)\times SU\left(2\right)\right)=\left(2\pi \right)\left(2{\pi }^2\right)=4{\pi }^3.$$

The use of compact-group Haar-volume measure follows the standard geometric normalization of compact Lie-group volumes [25,26].

Appendix A.3. Why the Neutral Parent Is Treated as a Stratified Read-Out Space

The neutral parent $P_0$ is not modeled as one already resolved charged branch, nor as a simultaneous collection of all possible decay channels. It is an unresolved parent whose admissible read-out may later resolve into branch-level structures. Therefore its leading read-out space is not taken to be only the product manifold

$$U\left(1\right)\times SU\left(2\right).$$

Rather, it is modeled as a stratified space consisting of a coupled interior together with two marginal boundary exposures:

$$\mathcal{E}\left(P_0\right)={\mathcal{E}}_{\mathrm{int}}\cup {\mathcal{E}}_{SU\left(2\right)}\cup {\mathcal{E}}_{U\left(1\right)},$$

where

$${\mathcal{E}}_{\mathrm{int}}=U\left(1\right)\times SU\left(2\right),{\mathcal{E}}_{SU\left(2\right)}\simeq SU\left(2\right),{\mathcal{E}}_{U\left(1\right)}\simeq U\left(1\right).$$

The two marginal strata represent half-exposed read-out directions of the unresolved neutral parent. One preserves the spinorial frame while the phase direction is unresolved; the other preserves the phase loop while the spinorial frame is unresolved. Since $P_0$ is neutral and unresolved, neither marginal stratum is counted as a full independent branch. Each enters with neutral-parent half-weight.

Thus the parent contributes one coupled phase–spinor interior plus two half-exposed marginal strata, not three independent copies of the same electromagnetic embedding. The half-weights are therefore part of the neutral-parent stratification assumption: they encode marginal exposure of an unresolved parent, not separate resolved particle channels.

Appendix A.4. Definition of the Stratified Haar Capacity

Define the neutral-parent stratified read-out space by

$$\mathcal{E}\left(P_0\right)=(U\left(1\right)\times SU\left(2\right))\cup SU\left(2\right)\cup U\left(1\right).$$

The leading stratified Haar capacity is

$${Cap}_{P_0}\left(\mathcal{E}\right)={\mu }_{U\left(1\right)\times SU\left(2\right)}\left({\mathcal{E}}_{\mathrm{int}}\right)+{\displaystyle \frac{1}{2}}{\mu }_{SU\left(2\right)}\left({\mathcal{E}}_{SU\left(2\right)}\right)+{\displaystyle \frac{1}{2}}{\mu }_{U\left(1\right)}\left({\mathcal{E}}_{U\left(1\right)}\right).$$

The factors $1/2$ encode the neutral-parent boundary weight: each marginal stratum is a half-exposed boundary of the unresolved parent, not a separate resolved particle branch.

Substituting the geometric Haar-volume normalizations

$${\mu }_{U\left(1\right)}\left(U\left(1\right)\right)=2\pi ,{\mu }_{SU\left(2\right)}\left(SU\left(2\right)\right)=2{\pi }^2,$$

gives

$$\begin{array}{cc}\hfill {Cap}_{P_0}\left(\mathcal{E}\right)& =\left(2\pi \right)\left(2{\pi }^2\right)+{\displaystyle \frac{1}{2}}\left(2{\pi }^2\right)+{\displaystyle \frac{1}{2}}\left(2\pi \right)\hfill \\ & =4{\pi }^3+{\pi }^2+\pi .\hfill \end{array}$$

The leading neutral-parent magnetic embedding capacity is therefore

$$\begin{array}{|c|}\hline {\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi .\\ \hline\end{array}$$

Appendix A.5. Theorem Form

The preceding construction can be summarized as the following conditional capacity theorem.

Theorem A.1 (Leading neutral-parent capacity).

Let $P_0$ be a balanced codimension-two carrier defect whose charge-readable magnetic read-out requires a $U\left(1\right)$ phase holonomy and a compact spinorial $SU\left(2\right)$ frame lift. Suppose that the unresolved neutral-parent read-out space is the stratified space

$$\mathcal{E}\left(P_0\right)=(U\left(1\right)\times SU\left(2\right))\cup SU\left(2\right)\cup U\left(1\right),$$

where the two marginal strata are inherited with neutral-parent half-weight. With the geometric Haar-volume normalizations

$${\mu }_{U\left(1\right)}\left(U\left(1\right)\right)=2\pi ,{\mu }_{SU\left(2\right)}\left(SU\left(2\right)\right)=2{\pi }^2,$$

the leading stratified Haar capacity is

$${\mathsf{\Omega }}_{P_0}=\mu (U\left(1\right)\times SU\left(2\right))+{\displaystyle \frac{1}{2}}\mu \left(SU\left(2\right)\right)+{\displaystyle \frac{1}{2}}\mu \left(U\left(1\right)\right).$$

Hence

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi .$$

Proof. 

A codimension-two defect has a linking circle, giving the phase holonomy group $U\left(1\right)$ with geometric Haar-volume $2\pi$. Compact spin-frame read-out requires the lift $SU\left(2\right)\simeq S^3$, whose standard unit-radius Haar volume is $2{\pi }^2$. This compact factor is the internal/rest-frame spinorial capacity associated with Lorentz-readable read-out, not the volume of the full noncompact Lorentz group. At leading order the coupled unresolved interior is therefore $U\left(1\right)\times SU\left(2\right)$, with product measure

$$\left(2\pi \right)\left(2{\pi }^2\right)=4{\pi }^3.$$

The neutral parent is unresolved, so its marginal phase and spinorial exposures are not counted as full independent branches. They enter as half-weight boundary strata:

$${\displaystyle \frac{1}{2}}\left(2{\pi }^2\right)+{\displaystyle \frac{1}{2}}\left(2\pi \right).$$

Adding the interior contribution and the two marginal contributions gives

$${\mathsf{\Omega }}_{P_0}=4{\pi }^3+{\pi }^2+\pi .$$

□

Appendix A.6. Interpretation and Limitation

This result is a leading-capacity theorem, not an ordinary manifold-volume theorem. The quantity ${\mathsf{\Omega }}_{P_0}$ is a dimensionless stratified read-out capacity: it combines the product geometric Haar-volume measure of the coupled interior with the half-weight Haar-volume measures of the marginal parent strata. Terms of different geometric dimension are therefore not being added as ordinary volumes of one smooth manifold. They are being added as normalized contributions to a single neutral-parent read-out capacity.

Likewise, the compact $SU\left(2\right)$ factor should not be read as replacing Lorentz covariance by a compact Euclidean symmetry. It represents the compact spin-frame capacity needed for a localized defect to admit a spinorial particle read-out; Lorentzian propagation and the full carrier-facing field description enter only after read-out.

The remaining physical assumption is the identification of this leading stratified Haar capacity with the leading inverse electromagnetic impedance,

$${\alpha }_{\left(0\right)}^{-1}={\mathsf{\Omega }}_{P_0}.$$

The subsequent $Z_3\to Z_4$ and $Z_4\to Z_5$ magnetic-interface corrections refine this leading capacity.

Appendix B.1. Why the First Exposure Begins at Z 3

The neutral parent first resolves through a $Z_2$-type branch structure. This $Z_2$ layer distinguishes complementary read-out directions, but it is not yet a completed confinement-like carrier support. Thus there is no independent closed object at the $Z_2$ level whose magnetic self-exposure would generate a chamber correction.

The first completed positive-side support is threefold. In QCD language, this parallels the fact that color-neutral baryonic closure requires three color slots [21,22]. In the present carrier language, this completed threefold support is denoted by $Z_3$. The first self-exposure layer is therefore the first lift of a completed $Z_3$ closure into an externally readable interface:

$$Z_3\to Z_4.$$

The fourth slot is not a fourth color and not a new family. It is the external exposure slot required for magnetic read-out.

Appendix B.2. The Ordered Four-Slot Exposure Chamber

Let the three closed support slots of the completed $Z_3$ carrier unit be

$$a,b,c,$$

and let the first external magnetic exposure slot be

$$ϵ.$$

The elementary $Z_3\to Z_4$ exposure interface is the four-slot set

$$X_{34}=\{a,b,c,ϵ\}.$$

Before read-out, the unresolved neutral parent does not distinguish a preferred ordering of these slots. An ordered exposure chamber is a bijection

$$\chi :\{1,2,3,4\}\to X_{34}.$$

Equivalently, the chamber space is the set of all orderings of four objects:

$${\mathcal{C}}_{34}=Ord\left(X_{34}\right).$$

The symmetric group $S_4$ acts freely and transitively on ${\mathcal{C}}_{34}$. Hence

$${\mathcal{C}}_{34}\simeq S_4,|{\mathcal{C}}_{34}|=|S_4|=4!=24.$$

This is an ordered chamber count. The three support slots are treated as chamber positions of the completed closure, and the exposure slot is a distinct carrier-facing role. No quotient by a later cyclic or permutation equivalence is taken at this first exposure stage. Such a quotient would describe a different read-out convention and would change the correction. The present correction counts the unresolved ordered exposure alignments available before a single Lorentz-readable exposure alignment is selected.

Appendix B.3. Symmetric Chamber Weight

Because the parent is unresolved before exposure, no chamber $\chi \in {\mathcal{C}}_{34}$ is assigned a preferred weight. The natural symmetric measure on the ordered chamber set is therefore the uniform counting measure,

$$w_{\chi }={\displaystyle \frac{1}{|{\mathcal{C}}_{34}|}}={\displaystyle \frac{1}{24}}.$$

The first self-readout scale is the inverse leading capacity,

$${ϵ}_{P_0}={\mathsf{\Omega }}_{P_0}^{-1}.$$

A single Lorentz-readable exposure alignment selected from the unresolved chamber space therefore carries the chamber weight $1/24$. Hence

$${\mathsf{\Delta }}_{34}=w_{\chi }{ϵ}_{P_0}={\displaystyle \frac{1}{24}}{\mathsf{\Omega }}_{P_0}^{-1}={\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}.$$

This form should not be read as averaging a constant over the chambers and then introducing a second average. Rather, the chamber space supplies the uniform primitive weight of one selected exposure alignment. Since there are 24 unresolved ordered alignments, the selected alignment has weight $1/24$.

Appendix B.4. Negative Sign of the Correction

The sign of ${\mathsf{\Delta }}_{34}$ follows from its interpretation as a self-exposure back-reaction. The leading quantity ${\mathsf{\Omega }}_{P_0}$ is the unresolved neutral-parent magnetic embedding capacity before self-exposure. The first exposure interface does not add a new independent capacity; it resolves part of the parent capacity into an externally readable channel. The available unresolved capacity is therefore reduced:

$${\mathsf{\Omega }}_{P_0}⟼{\mathsf{\Omega }}_{P_0}-{\mathsf{\Delta }}_{34}.$$

The first corrected inverse electromagnetic impedance is consequently

$${\alpha }_{\left(4\right)}^{-1}={\mathsf{\Omega }}_{P_0}-{\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}.$$

The negative sign is therefore part of the capacity-reduction interpretation of self-exposure. The chamber count fixes the magnitude $1/\left(24{\mathsf{\Omega }}_{P_0}\right)$; the sign follows from treating exposure as a resolution of previously unresolved parent capacity rather than as an additional independent capacity.

Appendix B.5. Theorem Form

The preceding argument may be summarized as the following conditional chamber-count result.

Theorem B.1 ($Z_3\to Z_4$ chamber correction).

Let a completed threefold carrier support have slots

$$a,b,c,$$

and let its first external magnetic exposure be represented by one additional slot ϵ. Suppose the unresolved neutral parent assigns no preferred ordering to the four-slot interface

$$X_{34}=\{a,b,c,ϵ\}.$$

Suppose further that the first exposure stage counts ordered chamber alignments before quotienting by any later carrier-readable equivalence. Then the ordered exposure chamber space is

$${\mathcal{C}}_{34}\simeq S_4,$$

and hence

$$|{\mathcal{C}}_{34}|=24.$$

If the first self-exposure scale is the inverse leading neutral-parent capacity ${\mathsf{\Omega }}_{P_0}^{-1}$, then the symmetric chamber-weighted self-exposure correction is

$${\mathsf{\Delta }}_{34}={\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}.$$

Proof. 

The four-slot exposure interface contains the three closed support slots and one exposure slot:

$$X_{34}=\{a,b,c,ϵ\}.$$

An ordered chamber is an ordering of $X_{34}$. The set of all such orderings is acted on freely and transitively by the symmetric group $S_4$, so the number of ordered chambers is

$$|S_4|=4!=24.$$

Since the unresolved neutral parent has no preferred chamber, a selected Lorentz-readable exposure alignment carries the uniform chamber weight $1/24$. Applying this weight to the inverse leading capacity gives

$${\mathsf{\Delta }}_{34}={\displaystyle \frac{1}{24}}{\mathsf{\Omega }}_{P_0}^{-1}={\displaystyle \frac{1}{24{\mathsf{\Omega }}_{P_0}}}.$$

□

Appendix B.6. Interpretation

This result explains why the first correction begins at $Z_3\to Z_4$, not at $Z_2\to Z_3$. The $Z_2$ layer is a branch-resolution layer and does not yet define a completed closed support. The $Z_3$ layer is the first completed carrier closure, and the $Z_4$ lift is its first exposure to an external magnetic read-out slot. The factor 24 is therefore the order of the ordered chamber group $S_4$ associated with this first closure-plus-exposure interface. The result is conditional on treating the first exposure as an ordered, non-quotiented chamber selection and on interpreting self-exposure as a reduction of unresolved parent capacity.

Appendix C.1. Residual-Interface Rule

After the first external exposure of a completed $Z_3$ closure, the higher layers

$$Z_4\to Z_5,Z_5\to Z_6,Z_6\to Z_7,\dots$$

are not new asymptotic particle sectors. They are residual magnetic-interface refinements of an already exposed closure domain. The one-sided residual coefficient for the interface

$$Z_m\to Z_{m+1}$$

is taken to be

$$C_{m,m+1}^{(+)}=1-{\displaystyle \frac{1}{{(m+1)}^2}},m\ge 4.$$

The term

$${\displaystyle \frac{1}{{(m+1)}^2}}$$

represents the unresolved residual fraction of the next attempted continuation. Thus $C_{m,m+1}^{(+)}$ measures the one-sided fraction available to magnetic read-out after residual screening.

Equivalently, the same rule may be written in screened-continuation form. The mismatch between adjacent layers is

$${\delta }_{m,m+1}={\displaystyle \frac{1}{m}}-{\displaystyle \frac{1}{m+1}}={\displaystyle \frac{1}{m(m+1)}},$$

while the screened $(m+1)$-layer exposure weight is

$$E_{m,m+1}=(m+1)-{\displaystyle \frac{1}{m+1}}={\displaystyle \frac{{(m+1)}^2-1}{m+1}}.$$

Then

$$\begin{array}{cc}\hfill m{E}_{m,m+1}{\delta }_{m,m+1}& =m\left({\displaystyle \frac{{(m+1)}^2-1}{m+1}}\right){\displaystyle \frac{1}{m(m+1)}}\hfill \\ & ={\displaystyle \frac{{(m+1)}^2-1}{{(m+1)}^2}}=1-{\displaystyle \frac{1}{{(m+1)}^2}}.\hfill \end{array}$$

Hence

$$C_{m,m+1}^{(+)}=m{E}_{m,m+1}{\delta }_{m,m+1}.$$

This equivalent form makes explicit that the same coefficient can be viewed either as a screened residual fraction or as an adjacent-layer mismatch weighted by the screened exposure of the next layer.

The neutral-parent magnetic response is two-sided. The corresponding completed residual-interface coefficient is therefore

$$C_{m,m+1}^{\mathrm{res}}=2C_{m,m+1}^{(+)}=2\left(1-{\displaystyle \frac{1}{{(m+1)}^2}}\right).$$

Appendix C.2. The Z₄ →Z₅ Coefficient

For the first internal residual interface,

$$Z_4\to Z_5,$$

we set $m=4$. The one-sided coefficient is

$$C_{45}^{(+)}=1-{\displaystyle \frac{1}{5^2}}={\displaystyle \frac{24}{25}}.$$

The two-sided completion gives

$$C_{45}=C_{45}^{\mathrm{res}}=2C_{45}^{(+)}=2·{\displaystyle \frac{24}{25}}={\displaystyle \frac{48}{25}}.$$

Equivalently, using the screened-continuation form,

$${\delta }_{45}={\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{5}}={\displaystyle \frac{1}{20}},$$

and

$$E_{45}=5-{\displaystyle \frac{1}{5}}={\displaystyle \frac{24}{5}}.$$

Therefore

$$C_{45}^{(+)}=4E_{45}{\delta }_{45}=4·{\displaystyle \frac{24}{5}}·{\displaystyle \frac{1}{20}}={\displaystyle \frac{24}{25}},$$

and hence

$$C_{45}=2C_{45}^{(+)}={\displaystyle \frac{48}{25}}.$$

Appendix C.3. The Z₅ →Z₆ Coefficient

For the next residual interface,

$$Z_5\to Z_6,$$

we set $m=5$. The one-sided coefficient is

$$C_{56}^{(+)}=1-{\displaystyle \frac{1}{6^2}}={\displaystyle \frac{35}{36}}.$$

Equivalently,

$${\delta }_{56}={\displaystyle \frac{1}{5}}-{\displaystyle \frac{1}{6}}={\displaystyle \frac{1}{30}},E_{56}=6-{\displaystyle \frac{1}{6}}={\displaystyle \frac{35}{6}},$$

so that

$$C_{56}^{(+)}=5E_{56}{\delta }_{56}=5·{\displaystyle \frac{35}{6}}·{\displaystyle \frac{1}{30}}={\displaystyle \frac{35}{36}}.$$

The two-sided residual-interface coefficient is therefore

$$C_{56}=C_{56}^{\mathrm{res}}=2C_{56}^{(+)}=2·{\displaystyle \frac{35}{36}}={\displaystyle \frac{35}{18}}.$$

Thus

$$C_{45}={\displaystyle \frac{48}{25}},C_{56}={\displaystyle \frac{35}{18}}$$

are fixed by the residual-interface rule before the alpha calculation is introduced.

Appendix C.4. Cross-Interface Transfer for the Alpha Correction

The fine-structure constant is not identified with a full baryonic magnetic-response ratio. It is interpreted as a neutral-parent magnetic impedance. Therefore the full $Z_4\to Z_5$ residual coefficient

$$C_{45}={\displaystyle \frac{48}{25}}$$

must be reduced to the part relevant for the neutral-parent capacity correction.

The proposed reduction is a cross-interface transfer from the $Z_4\to Z_5$ layer toward the next residual layer $Z_5\to Z_6$. In the residual hierarchy, the numerator associated with the two-sided coefficient $C_{45}$ is 48, while the numerator associated with the next residual coefficient $C_{56}$ is 35. The neutral-parent transfer factor is therefore

$$T_{45\to 56}={\displaystyle \frac{35}{48}}.$$

The effective coefficient entering the alpha capacity correction is

$$A_{45}=T_{45\to 56}{C}_{45}={\displaystyle \frac{35}{48}}·{\displaystyle \frac{48}{25}}={\displaystyle \frac{7}{5}}.$$

This gives the second correction term

$${\mathsf{\Delta }}_{45}={\displaystyle \frac{A_{45}}{{\mathsf{\Omega }}_{P_0}^3}}={\displaystyle \frac{7}{5{\mathsf{\Omega }}_{P_0}^3}}.$$

The inverse-cubic suppression reflects that this is not the first external exposure of the parent capacity, but a higher internal self-readout correction of an already exposed closure domain.

Appendix C.5. Interpretation

This appendix does not derive the leading neutral-parent capacity ${\mathsf{\Omega }}_{P_0}$ or the first $Z_3\to Z_4$ chamber average. Those are treated separately in Appendices Appendix A and Appendix B. Its role is to show how the second coefficient

$$A_{45}={\displaystyle \frac{7}{5}}$$

is obtained from the residual magnetic-interface hierarchy rather than introduced as an independent fit.

The coefficient $A_{45}=7/5$ is therefore not fitted to the fine-structure constant, but it is still conditional on the residual-interface and cross-interface transfer rules. A complete carrier-defect theory should derive these transfer rules intrinsically; in the present paper they are used as fixed structural inputs inherited from the magnetic-interface hierarchy.