Time as a Relational Evolution Parameter: A Unified Framework for Composite Quantum Mechanics, Measurement, the Born Rule, Bell Correlations, and Internal Standard Model Symmetries

1. Introduction

Quantum mechanics is among the most successful theoretical frameworks in physics, yet the physical origin of its mathematical structure remains unclear. In its standard formulation, states are represented in a complex Hilbert space, observables by operators, composite systems by tensor products, probabilities by the Born rule, and reversible evolution by unitary transformations. These structures are experimentally extraordinarily successful, but they are normally introduced as elements of the formalism rather than derived from an underlying physical mechanism. The central question of the present program is whether they may instead emerge as the effective description of a deeper and conceptually simpler dynamics.

A distinctive element of the framework introduced in Ref. [1] concerns the role of time. Rather than assuming that a unique temporal dimension plays the same role for every possible physical phenomenon, time is treated relationally as the +1 evolution parameter associated with the dimensional configuration space through which a particular phenomenon propagates and interacts. Thus, if a physical process requires d dimensions to specify its instantaneous configuration,

$𝒞$_𝒹 = (x¹,x²,…,x^d)

its dynamics require an additional evolution parameter λ,

Φ = Φ(x¹,…,x^d;λ)

For an ordinary 3+1-dimensional description, the configuration is

χ = (x,y,z)

and t is the corresponding +1 evolution parameter. For the 4+1-dimensional parent field considered in Ref. [1], however, the coordinate associated with t belongs to the four-dimensional configurational domain of the parent phenomenon, while a different parameter τ governs its evolution:

Φ = Φ(χ,t^',τ),  t^' = at

This does not introduce two time dimensions for the same system. Each dynamical sector possesses one evolution parameter; what changes between dimensional descriptions is which coordinate plays that role. A coordinate functioning as time for a lower-dimensional phenomenon can therefore behave as a spatial-like coordinate for a higher-dimensional phenomenon whose evolution is governed by another +1 parameter.

This treatment differs from conventional extra-dimensional constructions in an important respect. The transition between the parent and reduced descriptions is not merely a coordinate transformation in a higher-dimensional spacetime. It is a dimensional reduction accompanied by a change in which variable generates evolution.

Within this setting, a 3+1-dimensional observer cannot access the complete 4+1-dimensional field (we shall denote it Φ) directly. Information about its higher-dimensional configuration and evolution can be obtained only through its interactions with 3+1-dimensional physical systems and through the observable results produced by those interactions.

The reduced field is consequently introduced through an interaction-based recording map,

Ψ(χ,t) = $ℳ$[Φ]

or, schematically,

$\mathsf{\Psi }\left(\mathsf{\chi },t\right)=\int d\mathsf{\tau }\hspace{0.17em}K\left(\mathsf{\tau }\right)\mathsf{\Phi }\left(\mathsf{\chi },\mathit{at},\mathsf{\tau }\right)$The field Ψ is therefore not simply Φ with one coordinate removed. It represents the information about the parent dynamics that remains physically accessible through interactions with the lower-dimensional system.

The principal proposal of Ref. [1] was that the form of this reduced representation should not be imposed a priori. Instead, it should be constrained by the requirement that the maximum amount of physically accessible information be preserved. In particular, phase relations, interference structure, spectral information, and superposition must remain representable whenever they can affect subsequent interactions. Under these conditions, the reduced description was shown to acquire complex amplitudes, linear superposition, an inner-product structure, canonical operator relations, and norm-preserving unitary evolution. In the 4+1-dimensional extension, Schrödinger-type low-energy dynamics and a relativistic dispersion relation also emerged from the same construction.

The previous analysis was nevertheless restricted primarily to a single reduced system. It did not establish how independently preparable systems should be combined, why tensor-product composition should arise, how interactions generate entanglement, why the invariant Hilbert norm should acquire a probabilistic interpretation, or whether the resulting theory can reproduce Bell-violating correlations. The primary objective of the present work is to address these questions without introducing the corresponding quantum structures independently.

First, we reformulate dimensional recording more carefully in terms of physical response maps and equivalence classes of parent configurations. We then extend the construction to composite systems and show how independent preparation and preservation of superposition lead naturally to tensor-product composition, while joint interactions generate nonfactorizable reduced states.

The probability interpretation is subsequently addressed independently. Rather than identifying the Hilbert norm with probability from the outset, we determine which probability assignment is compatible with the previously obtained state-space geometry. Under phase invariance, continuity, noncontextual consistency, and additive coarse-graining of mutually exclusive orthogonal records, the quadratic rule follows:

p_i = |P_iΨ|² = ⟨Ψ|P_i|Ψ⟩

The resulting Born rule is therefore introduced only after the Hilbert-space and composite-system structures have already been obtained.

We then investigate Bell correlations without assuming a singlet state or Pauli matrices as independent physical postulates. Starting from a generic interaction-generated nonfactorizable state, its Schmidt structure provides the minimal correlated subspaces required for dichotomic measurements. Suitable local measurements then produce CHSH violation while preserving operational no-signalling.

Finally, we explore whether the same dimensional logic can suggest a geometric origin for internal symmetry structures. Changes of the parent field confined to the observed three-dimensional subspace, changes occurring in a tilted 3+1-dimensional geometry, and changes involving the full four-dimensional parent configurational space lead to candidate one, two, and three-component complex internal representations and hence to U(1), SU(2), and SU(3) type structures. This final construction should presently be regarded as a proposed geometric extension rather than a complete derivation of the Standard Model.

The present work belongs broadly to the effort to understand why quantum mechanics has its particular mathematical form. Standard formulations, and many interpretations of quantum theory, take the established quantum formalism as their starting point. A smaller but important reconstruction literature instead derives quantum theory from operational or informational principles, as in the programs of Hardy and of Chiribella, D'Ariano and Perinotti [3,4]. The present approach is complementary but physically different: rather than beginning primarily from operational axioms, it asks whether the quantum formalism can arise from an underlying higher-dimensional wave dynamics together with a relational treatment of time and information-preserving dimensional recording.

The question is therefore not only which principles characterize quantum theory, but also what physical mechanism could make these principles necessary.

This distinction is important. From a comparatively narrow modification, the dimensional role assigned to time, together with the requirement that physically accessible information survive dimensional reduction, the same framework repeatedly leads toward structures that otherwise appear as separate ingredients of quantum mechanics: complex representation, Hilbert geometry, operator structure, unitary evolution, tensor-product composition, entanglement, quadratic probabilities, and Bell-compatible nonseparability.

This convergence does not by itself prove that the proposed higher-dimensional description is fundamental, nor does recovering established quantum results constitute a new empirical prediction. Nevertheless, it is physically nontrivial. The fact that so many mutually consistent features of quantum mechanics follow from an approach whose principal physical change is the way time is treated is itself informative. If independent parts of the formalism repeatedly emerge from the same underlying mechanism rather than being inserted separately, then the mechanism becomes increasingly constrained and therefore increasingly meaningful.

The broader possibility investigated here is consequently that quantum mechanics may not be fundamental as a mathematical language, but may instead be the natural lower-dimensional description of a deeper dynamics in which time has a relational dimensional role. Whether this proposal ultimately proves fundamental or incomplete, the consistency with which quantum structures arise from such a limited modification makes it worthy of systematic investigation.

2. Dimensional Reduction, Information Preservation, and the Response Map

The objective of this Section is to establish the structural basis upon which the remainder of the present work will be built. For this reason, we will clarify the framework and the derivations of the previous paper [1], while giving more attention to the mechanisms of dimensional reduction and their mathematical properties.

The previous paper [1] considered the representation of a higher-dimensional wave field by a lower-dimensional description and argued that the requirement of preserving physically relevant information under this reduction leads naturally to complex amplitudes, linear superposition, operator relations, an inner product, and unitary evolution. The central distinction throughout the construction was between the underlying or parent dynamics and their reduced representation. The reduced field is not obtained by merely deleting an inaccessible coordinate, nor is it assumed to be the parent field written in different coordinates. Rather, it represents the information about the parent field that can be encoded through interactions with a lower-dimensional physical system.

Below, we summarize the main steps underlying the methodology and reasoning developed in the previous paper, while further clarifying and refining those elements that are particularly relevant to the present extension:

The relational role of time

The framework introduced in Ref. [1] treats time as the +1 evolution parameter relative to the dimensions through which a physical phenomenon propagates and interacts. Thus, a system whose configurations require d spatial dimensions is described dynamically by d+1 variables, with the additional variable serving as its evolution parameter.

For an ordinary 3+1 dimensional field, the three-dimensional configuration is specified by

χ = (x,y,z)

while t is the parameter with respect to which those configurations evolve.

For the parent field considered here, however, the coordinate associated with t belongs to the four-dimensional configuration space of the parent phenomenon. To emphasize the difference in dimensional role, we write this coordinate as

t^' = at

where a is a conversion factor required when t^' is expressed in spatial units. The parent field is therefore written as

Φ(χ,t^',τ)

Where (χ,t^') constitutes its four-dimensional spatial configuration and τ is its +1 evolution parameter.

Consequently, the same underlying coordinate that appears as an evolution parameter in the reduced 3+1 dimensional description appears as a spatial-like coordinate from the perspective of the 4+1 dimensional parent field. There are not two time dimensions assigned to the same physical phenomenon. Rather, different dimensional systems possess different evolution parameters.

A plane-wave component of the parent field may therefore be represented as

$$\mathsf{\Phi }\left(\mathsf{\chi },t^{\text{'}},\mathsf{\tau }\right)=R{e}^{i\left(K\cdot \mathsf{\chi }+\mathsf{\Omega }{t}^{\text{'}}-A\mathsf{\tau }\right)}$$

(1)

where K is the wavevector associated with the three common spatial coordinates, Ω is the spatial frequency associated with t^', and A is the frequency associated with evolution in τ.

The corresponding reduced description has the familiar 3+1 dimensional phase convention

$$\mathsf{\Psi }\left(\mathsf{\chi },t\right)=\int d^{3}k\hspace{0.17em}d\mathsf{\omega }\hspace{0.17em}a\left(k,\mathsf{\omega }\right)e^{i\left(k\cdot \mathsf{\chi }-\mathsf{\omega }t\right)}$$

(2)

The change from +Ωt^' in Eq. (1) to −ωt in Eq. (2) reflects the change in the role of this coordinate: it is spatial-like for the parent field but becomes the evolution parameter of the reduced description. Furthermore, Ω and ω should not in general be identified by a simple rescaling. They belong to the spectral decompositions of two different dynamical descriptions.

This is a crucial point. The operation

Φ(χ,t^',τ) ⟶ Ψ(χ,t)

is not a coordinate transformation. It is a dimensional reduction accompanied by a change in which variable generates evolution.

Interaction as the physical origin of the reduced description

A lower-dimensional system cannot directly access the complete configuration of the 4+1 dimensional parent field. It can only respond to the parent field through their interaction.

Conceptually, we could imagine that in order to describe the behavior of a 4+1 dimensional field, the only way to do this is to let it interact with a 3+1 dimensional field (a field we can accurately describe in a 3+1 dimensional description with t as its time dimension), gather the results for a sufficiently large number of those interactions and then try to produce a coherent 3+1 dimensional description of the higher dimensional field that would reproduce those results and can make predictions about future interactions. In this picture, the 3+1 dimensional description should be able to retain maximum information about the spatial (3 dimensional) configuration of possible interactions as well as the evolution of those results in reference to the 4^th dimension.

Consider therefore a physical 3+1 dimensional field or system F(χ,t) capable of interacting with Φ. At an interaction, the lower-dimensional field has access to the values of the parent field projected onto the coordinates common to both descriptions. Schematically, the resulting response may be written as

ΔF(χ,t) = $ℛ$[Φ(χ,at,τ)] (3)

where $ℛ$ denotes the physical response produced by the interaction.

For a linear response, the most general schematic form may be represented by an integral kernel,

$$\mathsf{\Psi }\left(\mathsf{\chi },t\right)=\mathcal{M}\left[\mathsf{\Phi }\right]=\int d\mathsf{\tau }\hspace{0.17em}K\left(\mathsf{\chi },t;\mathsf{\tau }\right)\mathsf{\Phi }\left(\mathsf{\chi },\mathit{at},\mathsf{\tau }\right)$$

(4)

Here $ℳ$ denotes the recording or response map and K is its response kernel.

The simpler expression used in Ref. [1],

$\mathsf{\Psi }\left(\mathsf{\chi },t\right)=\int d\mathsf{\tau }\hspace{0.17em}K\left(\mathsf{\tau }\right)\mathsf{\Phi }\left(\mathsf{\chi },\mathit{at},\mathsf{\tau }\right)$is recovered whenever the response is translationally homogeneous and its dependence on the common coordinates can be neglected.

The word “recording” should not be interpreted as implying a conscious observer or a laboratory measurement. Any physical interaction in which a 3+1 dimensional system responds to the parent field can play this role. What is recorded is simply the part of the higher-dimensional dynamics capable of affecting a lower-dimensional system.

The distinction between the physical interaction and the mathematical state Ψ is also useful. The physical recorded field Φ may be entirely real. The complex-valued quantity Ψ introduced below should be interpreted as the minimal effective representation required to encode all accessible amplitudes and phases contained in the interaction response.

What is meant by maximum information preservation

The phrase “maximum information preservation,” introduced in Ref. [1], requires a more precise definition.

A dimensional reduction from 4+1 to 3+1 dimensions cannot preserve every property of the parent field. In particular, direct access to τ is absent from the reduced description. Therefore, the response map cannot in general be globally invertible.

The relevant requirement is weaker and physically more appropriate: the reduced representation should preserve every distinction between parent configurations that can influence an allowed 3+1 dimensional interaction.

Let $𝒮$₅ denote the set of admissible parent-field configurations. We define an equivalence relation

Φ₁∼Φ₂

whenever no admissible 3+1 dimensional interaction can distinguish Φ₁ from Φ₂.

Equivalently,

Φ₁∼Φ₂ ⟺ $ℛ$_α[Φ₁] = $ℛ$_α[Φ₂]

for all accessible responses $ℛ$_α

The physically relevant object for the lower-dimensional description is therefore not an individual parent configuration Φ but its equivalence class: [Φ] ∈ $𝒮$₅

Maximum accessible-information preservation can then be expressed by requiring that the reduced representation distinguish these equivalence classes:

$ℳ$[Φ₁] = $ℳ$[Φ₂] ⟺ Φ₁∼Φ₂

Thus $ℳ$ need not be injective on the entire parent solution space. It should instead be faithful on $𝒮$₅

This formulation removes an unnecessary requirement of complete reconstruction of the 4+1 dimensional field. Information that is fundamentally inaccessible to every 3+1 dimensional interaction may be lost without affecting the reduced description. Information that changes interference, localization, spectral composition, or subsequent interaction outcomes cannot be discarded.

For the wave dynamics considered here, the principal structural requirements on an admissible response map may therefore be summarized as follows:

• Accessible distinguishability: parent configurations producing physically distinguishable 3+1 dimensional responses must remain distinct in the reduced representation.
• Superposition consistency: if the parent dynamics are linear, the response to superposed parent configurations must retain their relative amplitudes and interference terms.
• Phase preservation: relative phases capable of modifying subsequent interactions must remain encoded after τ is removed explicitly.
• Covariance: a translation or symmetry transformation of the parent configuration must induce a consistent transformation of its reduced representation.
• Stable recorded content: propagation of the reduced state must not arbitrarily create or destroy the quantity used to characterize its total recorded magnitude.
• Closure: once the reduced state has been specified, its subsequent reduced evolution must be well defined without requiring inaccessible information that was discarded by the map.
• Robustness: small physically irrelevant variations of the recording interaction should not produce discontinuous changes in the reproducible reduced description.

These conditions define a class of admissible response maps rather than a single microscopic kernel.

Why the exact response kernel is not specified

The precise response kernel depends on the detailed physical interaction between the parent field and the 3+1 dimensional recording system. Different recording fields, coupling strengths, spatial profiles, or interaction mechanisms may therefore correspond to different kernels.

It would consequently be unjustified at the present stage to postulate one universal microscopic function K(τ).

The purpose of the response kernel in the present work is different. We seek structural properties that must be shared by every kernel capable of producing a faithful reduced description.

This distinction can be expressed as K ∈ $𝒦$_adm, where $𝒦$_adm denotes the class of kernels satisfying the information-preservation, covariance, stability, and closure requirements described above.

The quantum-mechanical state-space structure is proposed to depend primarily on these common properties rather than on the microscopic details of an individual K. This is analogous to many effective descriptions in physics in which macroscopic structural properties are insensitive to the detailed microscopic realization.

From information preservation to the Hilbert-space description

The single-system quantum-mechanical structure was derived in Ref. [1]. Here we do not repeat those derivations in full, but instead restate their logical sequence and further clarify the role of dimensional recording, information preservation, and closure in preparation for the extensions developed below.

Once the dimensional reduction (eq.(4)) is required to preserve all physically accessible phase information, a single real amplitude is insufficient, because it cannot distinguish all relative phases that may affect subsequent interference. The minimal faithful representation therefore requires two real quadratures, which can be combined naturally into a complex amplitude: a = |a|e^iθ .

Preservation of the linear superposition of the parent wave field then induces a complex linear state space for the reduced description. Continuous translations act on its spectral components through phase factors, so that the usual Fourier-conjugate generators arise and, after identifying physical momentum through $\widehat{P_j}=\mathrm{\hslash }\widehat{K_j}$ satisfy the canonical relation $\left[\widehat{X_i},\widehat{P_j}\right]=i\mathrm{\hslash }{\Delta }_{\mathit{ij}}$Because translations change phase without changing spectral magnitude, the reduced description admits a positive invariant quadratic quantity, from which the inner product and norm are constructed

Requiring this recorded content to remain invariant under reversible reduced evolution leads to unitary evolution.

In addition, the reduced representation must be closed: parent configurations that are indistinguishable after reduction must not later generate different reduced dynamics, otherwise the reduced state would be incomplete

Finally, taking the Cauchy completion of the resulting complex inner-product space gives the Hilbert space $\mathscr{H}$

Physical Variations and Variations of the Recording Map

Consider a variation of the parent field:

Φ → Φ+ΔΦ

For a fixed response kernel,

Δ_ΦΨ = $ℛ$_Κ [ΔΦ] (5)

If ΔΦ modifies information that is physically accessible through subsequent interactions, for example a relative phase, spectral weight, conserved momentum, energy or an interaction-induced correlation, then a maximally informative response map should, as far as the dimensional reduction permits, satisfy

ΔΦ_phys ≠ 0 ⟹ Δ_ΦΨ ≠ 0

is not an assertion of complete invertibility. Some parent variations necessarily belong to the null space of the recording map because they cannot be distinguished by the lower-dimensional interaction. The information-retention requirement instead states that the physically admissible kernel should make this null space as small as is compatible with the lower-dimensional observational constraints.

Furthermore, in the flat homogeneous case, let

p = (p_x,p_y,p_z)

denote the spatial momentum, E the quantity conjugate to the retained coordinate that becomes the observer's time, and A the angular frequency associated with evolution in τ. The parent dispersion relation may be written, after introducing physical units, in the form

E² − c²|p|² − ℏ²A² = 0 (6A).

with the identification

$A={\displaystyle \frac{m{c}^2}{\mathrm{\hslash }}}$which becomes the familiar relativistic relation:

E² = c²|p(​|² + m²c⁴ (6B).

Here m is the effective invariant mass appearing in the reduced description and ℏ is the experimentally established action scale that converts the hidden frequency into physical mass and energy units. As emphasized previously, the framework gives a structural interpretation of this scale but does not independently determine its numerical value [1].

The significance of eq.(6A) for the recording problem is that the quantities removed and the quantities retained by dimensional reduction are not independent. Information about the inaccessible τ -evolution remains correlated with information carried by E, p phase and the spectral content of the recorded field. The response kernel therefore need not preserve the value of τ itself in order to preserve information concerning the higher-dimensional dynamics.

For two parent sectors, let

Γ_i = (p_i,E_i,A_i),  i = 1,2

denote their respective spectral variables. Before interaction, independently prepared sectors may possess a factorized spectral amplitude,

F₁₂(Γ₁,Γ₂) = F₁(Γ₁)F₂(Γ₂)

After joint evolution, conservation laws or interaction dynamics may instead impose relations such as

E₁ + E₂ = E_total , p₁ + p₂ = p_total

In that case,

F₁₂(Γ₁,Γ₂) ≠ F₁(Γ₁)F₂(Γ₂)

A recording map that is intended to retain the maximum accessible information should therefore preserve not only the marginal information associated with each sector separately but also those nonfactorizable correlations that can influence later lower-dimensional interactions. This observation provides the physical link between joint parent evolution and the possibility of entangled states in the reduced tensor-product description as we will examine later on the paper.

Contrary to the above, a second and conceptually different variation concerns the recording interaction itself. Let

K → K + ϵ K,  |ϵ|≪1

where ΔK represents a small perturbation of the response produced, for example, by a small change in apparatus sensitivity, alignment, interaction duration or another uncontrolled property of the recording process.

The corresponding change in the reduced field is:

Δ_KΨ = $ℛ$_ΔK[Φ] (7)

Equations (5) and (7) must not be confused. A change ΔΦ represents a change in the physical state whose information should be retained. A change ΔK represents a change in the mechanism by which that information is recorded. An appropriate recording process should therefore satisfy two complementary requirements:

• Sensitivity to physically relevant ΔΦ
• Stability against sufficiently small differences in recording process ΔK

A physical reduced description cannot depend discontinuously on arbitrarily small changes in interaction parameters. If a recording field is displaced by an infinitesimal amount, or an experimental control parameter is perturbed slightly, the reproducible reduced response should change continuously unless the underlying physical system itself contains a genuine discontinuity. This does not mean resistance to experimental noise. It means stability of reproducible predictions under small variations of the conditions used to define the experiment.

Importantly, the effect of a variation ΔK is expressed entirely within the variables retained by the dimensional reduction. Although the perturbation acts on the mapping from the parent field, the resulting accessible information is still encoded in the three-dimensional spatial configuration χ and its evolution with respect to t. Consequently, a change of the recording process naturally appears in the reduced description as a variation ΔΨ(χ,t), rather than as the appearance of an additional observable dependence on the inaccessible coordinate τ. The role of ΔK is therefore to modify how the parent information is weighted and encoded within the reduced state, its amplitudes, relative phases, spatial distribution, and spectral content, while the dimensional character of the recorded description remains 3+1.

Measurement and the change of the reduced state – The Measurement Problem

The distinction between variations of the parent field and variations of the recording interaction also provides a natural way of extending the interpretation of measurement proposed in Ref. [1]. Consider first a single parent field Φ whose lower-dimensional representation before an interaction is

Ψ = $ℳ$[Φ]

As long as the parent field continues to evolve without undergoing a physically relevant modification, the corresponding reduced state may be described by the continuous evolution of the same Ψ. A physical interaction, however, will generally modify some property of the interacting field. This modification may involve an exchange of energy or momentum, a change of phase, spatial profile, polarization, internal configuration, spectral composition, or correlations with another physical system. At the parent level, such an interaction can be represented schematically as

Φ ⟶ Φ^' = Φ+ΔΦ

If ΔΦ contains information that is accessible to the 3+1 dimensional recording process, the post-interaction reduced description cannot in general remain the same state Ψ. Instead, the modified parent configuration must be recorded as a new reduced state,

Ψ^' = $ℳ$[Φ^'] = $ℳ$[Φ+ΔΦ]

Thus, once the physical parent configuration has been modified by the interaction, its subsequent reduced evolution is no longer represented by the continuation of the original state Ψ, but by the evolution of the new state Ψ^'. In an idealized description in which the interaction is associated with a definite measurement event, this transition appears in the reduced 3+1 dimensional theory as a discontinuous update

Ψ ⟶ Ψ^'

Importantly, this does not require the underlying 4+1 dimensional dynamics themselves to be discontinuous. The parent interaction may occur through a completely continuous physical evolution. The apparent discontinuity concerns the effective reduced description: the state that encoded the pre-interaction parent configuration is no longer sufficient to represent the physically modified configuration after the interaction. If the interaction has a finite duration, the transition from Ψ to Ψ^' should correspondingly be understood as occurring over that duration; the instantaneous update is the idealized measurement limit.

This reasoning becomes particularly relevant when the recorded state contains coherent contributions from several parent configurations. Let

${\mathsf{\Phi }}_{\mathrm{tot}}={\sum }_n{c}_n{\mathsf{\Phi }}_n$so that, under a linear recording map,

$\mathsf{\Psi }=\mathcal{M}\left[{\mathsf{\Phi }}_{\mathrm{tot}}\right]={\sum }_n{c}_{n\mathcal{M}}\left[{\mathsf{\Phi }}_n\right]$As discussed in Ref. [1], a coherent reduced description requires the contributing configurations to retain mutually well-defined relative phases within a common τ-domain. Suppose now that an interaction acts selectively on one contribution, Φ_m, while the remaining configurations are not modified in the same way:

${\mathsf{\Phi }}_m⟶{\mathsf{\Phi }}_m^{\text{'}}={\mathsf{\Phi }}_m+\Delta {\mathsf{\Phi }}_m$while

Φ_j ⟶ Φ_j,  j ≠ m

The crucial point is not simply that Φ_m has changed, but that its post-interaction evolution need no longer remain synchronized with the same effective τ-interval and phase relations that defined the original coherent combination. The affected component now follows the evolution of ${\mathsf{\Phi }}_m^{\text{'}}$, whereas the remaining components continue according to their previous parent configurations. The common τ-coherence required to represent all contributions as one continuously evolving reduced state may therefore be lost.

The post-interaction parent configuration is then

${\mathsf{\Phi }}_{\mathrm{tot}}^{\text{'}}={\sum }_{n\ne m}{c}_n{\mathsf{\Phi }}_n+c_m{\mathsf{\Phi }}_m^{\text{'}}$and the corresponding recorded state becomes

${\mathsf{\Psi }}^{\text{'}}=\mathcal{M}\left[{\mathsf{\Phi }}_{\mathrm{tot}}^{\text{'}}\right]$If the selective interaction destroys the common phase relations that previously allowed the components to be represented coherently within the same τ-domain, then Ψ^' is not simply the uninterrupted unitary continuation of the original Ψ. Instead, the interaction defines a new physically relevant parent configuration and therefore a correspondingly updated reduced description.

To first order, the total change of the reduced state can therefore be written schematically as

ΔΨ≃$ℳ$[ΔΦ]+Δ$ℳ$[Φ]

The first term represents the fact that the physical state itself has changed, whereas the second represents a change in how the parent information is recorded. This distinction is useful because a measurement interaction may involve both: it may physically modify the system while simultaneously establishing a distinguishable record of that modification.

The present framework therefore suggests a structural distinction between ordinary coherent evolution and measurement-like updating. An interaction does not constitute a measurement merely because it produces some ΔΦ. Coherent interactions may be incorporated into the unitary evolution of an enlarged system. A measurement-like interaction occurs when the modification becomes physically distinguishable in the reduced description and prevents the previously contributing parent configurations from remaining representable within the same coherent τ-evolution. In that situation, physical interaction leads to a change ΔΦ in the 4+1 dimensional field Φ, which results in loss or modification of the previous coherent recording which is translated in a discontinuous update Ψ^' in the reduced description. The transition from Ψ to Ψ^' appears as an effectively non-unitary state update in the lower-dimensional description.

An additional motivation for this construction is that the measurement problem, although fundamental to the interpretation of quantum mechanics, is comparatively rarely addressed within reconstruction programs aimed at deriving the quantum formalism itself. Here, the same dimensional and recording framework that generates the reduced state-space structure also provides a natural mechanism for distinguishing continuous unitary evolution from measurement-induced state updating.

3. Composite Systems, Tensor - Product Structure, Interactions and Entanglement

The construction developed so far concerns the reduced description of a single higher-dimensional field or field sector. A complete quantum-mechanical description, however, must also specify how independently preparable systems are combined, how their individual transformations act within the combined description, and how interactions can generate correlations that cannot be reduced to independent states of the constituent systems.

In standard quantum mechanics, these requirements are encoded through the tensor product of Hilbert spaces. Here, however, the tensor-product structure will not be introduced as an independent postulate. Instead, we investigate whether it follows from the same requirements of information preservation, linearity, and consistency that led to the single-system Hilbert-space description in Section 2.

We will begin with the simplest situation: two distinct, independently preparable parent systems that do not interact and then move on to interactions and entanglement.

Two independently prepared non-interacting parent systems

Consider two distinct parent fields or independently preparable field sectors,

Φ_A and  Φ_B

Each field possesses its own admissible parent configurations and its own lower-dimensional recording map,

$ℳ$_𝒜:Φ_A ⟶ Ψ_A ∈ $\mathscr{H}$_𝒜 and $ℳ$_$ℬ$:Φ_B ⟶ Ψ_B ∈ $\mathscr{H}$_$ℬ$

The single-system argument of Section 2 associates each parent sector with a complex Hilbert space containing all reduced configurations that preserve the physically accessible information of that sector.

Before interaction, the physical statement of independent preparation means that the preparation of A can be changed without requiring a corresponding modification of B, and vice versa. At the parent level, the combined configuration may therefore initially be represented simply as the ordered pair

(Φ_A,Φ_B)

No tensor product is assumed at this level. The two parent fields remain physically distinct sectors.

The problem is instead to determine how their joint reduced description must be represented.

Let

Γ:$\mathscr{H}$_𝒜 × $\mathscr{H}$_$ℬ$ ⟶ $\mathscr{H}$_𝒜$ℬ$

denote the composition map assigning a joint reduced state to a pair of independently prepared reduced states. We write

Γ(Ψ_A,Ψ_B) = Ψ_AB

The structure of Γ is constrained by the linearity already established for the individual reduced descriptions.

Suppose, for example, that system A may be prepared in the superposed reduced state

${\mathsf{\Psi }}_A=\mathsf{\alpha }{\mathsf{\Psi }}_A^{\left(1\right)}+\mathsf{\beta }{\mathsf{\Psi }}_A^{\left(2\right)}$If B is prepared independently in the fixed state Ψ_B, preserving the superposition information of A requires

$$\mathsf{\Gamma }\left(\mathsf{\alpha }{\mathsf{\Psi }}_A^{\left(1\right)}+\mathsf{\beta }{\mathsf{\Psi }}_A^{\left(2\right)},{\mathsf{\Psi }}_B\right)=\mathsf{\alpha \Gamma }\left({\mathsf{\Psi }}_A^{\left(1\right)},{\mathsf{\Psi }}_B\right)+\mathsf{\beta \Gamma }\left({\mathsf{\Psi }}_A^{\left(2\right)},{\mathsf{\Psi }}_B\right)$$

(8)

The same argument must hold when the state of B is varied while A remains fixed:

$$\mathsf{\Gamma }\left({\mathsf{\Psi }}_A,\mathsf{\alpha }{\mathsf{\Psi }}_B^{\left(1\right)}+\mathsf{\beta }{\mathsf{\Psi }}_B^{\left(2\right)}\right)=\mathsf{\alpha \Gamma }\left({\mathsf{\Psi }}_A,{\mathsf{\Psi }}_B^{\left(1\right)}\right)+\mathsf{\beta \Gamma }\left({\mathsf{\Psi }}_A,{\mathsf{\Psi }}_B^{\left(2\right)}\right)$$

(9)

The composition map must therefore be linear in each argument separately. That is, it must be a bilinear map.

This conclusion follows directly from two requirements already present in the framework: independent preparation and preservation of the complex linear superposition structure of each subsystem.

From bilinear composition to the tensor product

Once the composition map is required to be bilinear, the tensor product arises naturally as the mathematical space that linearizes such a composition.

For every pair of states we introduce the notation

Γ(Ψ_A,Ψ_B)≡Ψ_A⊗Ψ_B (10)

The bilinearity conditions then become

(αΨ_A + βΦ_A)⊗Ψ_B = α(Ψ_A⊗Ψ_B) + β(Φ_A⊗Ψ_B) (11)

and

Ψ_A⊗(αΨ_B + βΦ_B) = α(Ψ_A⊗Ψ_B) + β(Ψ_A⊗Φ_B) (12)

Mathematically, the algebraic tensor product

$\mathscr{H}$_𝒜⊙$\mathscr{H}$_$ℬ$

is precisely the vector space possessing the universal property that every bilinear map from

$\mathscr{H}$_𝒜 × $\mathscr{H}$_$ℬ$

can be represented by a unique linear map acting on the corresponding tensor product.

Thus, the bilinear composition map Γ induces a linear map

$\tilde{\mathsf{\Gamma }}:{\mathcal{H}}_{\mathcal{A}}\odot {\mathcal{H}}_{\mathcal{B}}⟶{\mathcal{H}}_{\mathcal{AB}}$such that

$$\tilde{\mathsf{\Gamma }}\left({\mathsf{\Psi }}_A\otimes {\mathsf{\Psi }}_B\right)=\mathsf{\Gamma }\left({\mathsf{\Psi }}_A,{\mathsf{\Psi }}_B\right)$$

(13)

This step is important conceptually. The tensor product is not being inserted because standard quantum mechanics uses it. It is the natural linear space required to represent a composition law that preserves the independent complex-linear structure of both reduced systems.

A direct sum would not serve the same purpose. The direct sum

$\mathscr{H}$_𝒜⊕$\mathscr{H}$_$ℬ$

describes states associated with separate sectors, whereas a joint physical configuration must retain information about which state of A occurs together with which state of B. The tensor product retains precisely these joint combinations.

The inner product of independently prepared product states must likewise remain compatible with the inner products already defined separately in $\mathscr{H}$_𝒜 and $\mathscr{H}$_$ℬ$. The natural factorized form is

$${⟨{\mathsf{\Psi }}_A\otimes {\mathsf{\Psi }}_B,{\mathsf{\Phi }}_A\otimes {\mathsf{\Phi }}_B⟩}_{\mathit{AB}}={⟨{\mathsf{\Psi }}_A,{\mathsf{\Phi }}_A⟩}_A{⟨{\mathsf{\Psi }}_B,{\mathsf{\Phi }}_B⟩}_B$$

(14)

In particular,

|Ψ_A⊗Ψ_B|² = |Ψ_A|²|Ψ_B|² . (15)

The factorization of the inner product expresses the independence of the two initially prepared sectors at the level of their recorded content. No probabilistic interpretation of these norms is required here.

Finally, just as in the single-system case, the physical state space must contain limits of Cauchy sequences. Completing the algebraic tensor product with respect to the norm induced by Eq. (14) produces the Hilbert tensor product,

$${\mathcal{H}}_{\mathcal{AB}}={\mathcal{H}}_{\mathcal{A}}\widehat{\otimes }{\mathcal{H}}_{\mathcal{B}}$$

(16)

To identify the accessible composite Hilbert space with the completed tensor product rather than merely obtaining a linear representation of product preparations within a possibly larger space, we additionally require composite completeness: no independent accessible composite degree of freedom exists beyond those generated by independently preparable subsystem states and their linear combinations. This is the composite analogue of maximum accessible-information preservation. The reduced composite space should be large enough to represent every distinguishable joint preparation and every correlation that can be generated from the two parent sectors, but it should not contain additional independent directions corresponding to no accessible preparation or interaction. Since coherent superpositions of product configurations already generate the nonfactorizable states required after interaction, their closure provides the minimal complete state space compatible with the accessible composite information.

In the remainder of the paper we use the conventional notation

$\mathscr{H}$_𝒜$ℬ$ = $\mathscr{H}$_𝒜⊗$\mathscr{H}$_$ℬ$ (17)

with the completion understood.

To summarize the logical structure is as follows:

Independent preparation → preservation of local superposition → bilinear composition → tensor product

Product states and joint configuration space

The physical meaning of the construction becomes particularly transparent in a basis representation.

Let {|a_i⟩} be a basis of $\mathscr{H}$_𝒜 and {|b_j⟩} be a basis of $\mathscr{H}$_$ℬ$. Then

{|a_i⟩⊗|b_j⟩}

forms the natural product basis of the composite space.

If the independently prepared states are

$\left|{\mathsf{\Psi }}_A\right.⟩={\sum }_i{a}_i\left|a_i\right.⟩$ and $\left|{\mathsf{\Psi }}_B\right.⟩={\sum }_j{b}_j\left|b_j\right.⟩$then the combined non-interacting state is

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\right.⟩=\left|{\mathsf{\Psi }}_A\right.⟩\otimes \left|{\mathsf{\Psi }}_B\right.⟩={\sum }_{\mathit{ij}}{a}_i{b}_j\left|a_i\right.⟩\otimes \left|b_j\right.⟩$$

(18)

The joint coefficient matrix therefore factorizes:

c_ij = a_ib_j (19)

Such a matrix has rank one.

This algebraic property will later provide a particularly simple distinction between product states and entangled states.

The same structure appears in the configuration representation.

If the individual reduced fields are Ψ_A(χ_A,t) and Ψ_B(χ_B,t) then the independently prepared joint state is:

Ψ_AB(χ_A,χ_B,t) = Ψ_A(χ_A,t)Ψ_B(χ_B,t) (20)

Thus the familiar many-particle configuration space arises naturally as the Cartesian product of the configuration domains of the individual reduced systems,

Σ_AB = Σ_A × Σ_B

For two systems whose reduced configurations each require three spatial coordinates, the joint wavefunction therefore depends on six spatial coordinates,

Ψ_AB(x_A,x_B,t)

This should not be interpreted as requiring six physical spatial dimensions. The configuration-space coordinates label two simultaneous three-dimensional configurations. The higher dimensionality of configuration space expresses the number of independently variable subsystem configurations, rather than the dimensionality of ordinary physical space.

Local transformations and non-interacting evolution

Independent preparation also requires that a transformation applied only to subsystem A act without modifying the state assigned to subsystem B.

If

$\widehat{A}:{\mathcal{H}}_{\mathcal{A}}⟶{\mathcal{H}}_{\mathcal{A}}$is a linear operator acting locally on A, then on a product state we require

$|{\mathsf{\Psi }}_A⟩\otimes |{\mathsf{\Psi }}_B⟩⟶\left(\widehat{A}\right|{\mathsf{\Psi }}_A⟩)\otimes |{\mathsf{\Psi }}_B⟩\left(21\right).$The corresponding operator on the composite space is therefore

${\widehat{A}}_{\mathrm{loc}}=\widehat{A}\otimes I_B$Similarly, a local transformation of B is represented by

${\widehat{B}}_{\mathrm{loc}}=I_A\otimes \widehat{B}$These operators commute,

$\left[\widehat{A}\otimes I_B,I_A\otimes \widehat{B}\right]=0$because they act on distinct tensor factors.

This structure is not an additional assumption about quantum measurements. It follows from the requirement that independent transformations remain independently representable after composition.

The same reasoning applies to free evolution. Let

$U_A\left(t\right)=e^{-i\widehat{H_A}t/\mathrm{\hslash }}$ and $U_B\left(t\right)=e^{-i\widehat{H_B}t/\mathrm{\hslash }}$denote the reduced evolution of the two isolated systems.

In the absence of interaction, independent evolution requires

$$U_{\mathit{AB}}^{\left(0\right)}\left(t\right)=U_A\left(t\right)\otimes U_B\left(t\right)$$

(22)

Hence

$$U_{\mathit{AB}}^{\left(0\right)}\left(t\right)({|\mathsf{\Psi }}_A⟩\otimes |{\mathsf{\Psi }}_B⟩=U_A\left(t\right){|\mathsf{\Psi }}_A⟩\otimes U_B\left(t\right)|{\mathsf{\Psi }}_B⟩$$

(23)

A product state therefore remains a product state under non-interacting evolution.

The generator of Eq. (22) is

$$\widehat{H_{\mathit{AB}}^{\left(0\right)}}=\widehat{H_A}\otimes I_B+I_A\otimes \widehat{H_B}$$

(24)

The non-interacting case therefore establishes an important result: tensor-product structure does not arise because the systems interact. It is already required in order to represent two independent systems while preserving the linear state structure and local transformations of each one. Interaction acts subsequently within this composite space.

Introducing interaction at the parent level

We now allow the two parent sectors to interact.

Before interaction, their parent dynamics may be written schematically as

$ℒ$_𝒜[Φ_A] = 0,  $ℒ$_$ℬ$[Φ_B] = 0 (25)

where $ℒ$_𝒜 and $ℒ$_$ℬ$ represent the corresponding parent dynamical operators.

When the fields interact, the evolution of one sector depends on the configuration of the other. Schematically,

$ℒ$_𝒜[Φ_A] = $ℐ$_𝒜[Φ_A,Φ_B] and $ℒ$_$ℬ$[Φ_B] = $ℐ$_$ℬ$[Φ_A,Φ_B]

Where $ℐ$_𝒜 and $ℐ$_$ℬ$ denote the coupling between the two parent sectors.

The precise microscopic form of these interaction terms is not required for the structural argument. What matters is that the post-interaction evolution is no longer determined separately by Φ_A and Φ_B. The parent dynamics are now joint.

Consequently, the appropriate recording is also a joint map,

$ℳ$_𝒜$ℬ$:(Φ_A,Φ_B) ⟶ Ψ_AB ∈ $\mathscr{H}$_𝒜⊗$\mathscr{H}$_$ℬ$ (26)

The closure requirement introduced in Section 2 now acquires an important composite-system meaning. The interacting parent dynamics must close on the reduced composite state space. Once Ψ_AB has been specified, physically inaccessible distinctions between parent configurations may not later reappear and alter the reduced prediction.

The closure requirement introduced in Section 2 now acquires an important composite-system meaning. Once Ψ_AB has been specified, distinctions between parent configurations that are inaccessible to the recording process must not later reappear and alter the reduced prediction. Let $ℰ$_𝒜$ℬ$ denote an admissible evolution of the interacting parent system. Closure therefore requires that if there two 4+1-dimensional filed configurations which satisfy:

${\mathcal{M}}_{\mathcal{AB}}\left[\mathsf{\Phi }\right]={\mathcal{M}}_{\mathcal{AB}}\left[\tilde{\mathsf{\Phi }}\right]$then,

${\mathcal{M}}_{\mathcal{AB}}\left[{\mathcal{E}}_{\mathcal{AB}}\left(\mathsf{\Phi }\right)\right]={\mathcal{M}}_{\mathcal{AB}}\left[{\mathcal{E}}_{\mathcal{AB}}\left(\tilde{\mathsf{\Phi }}\right)\right]$This condition states that the parent evolution preserves the equivalence classes defined by the recording map. Closure is sufficient to ensure that the parent dynamics induce a unique evolution on the reduced composite state space. We may therefore define

$𝒰$_𝒜$ℬ$($ℳ$_𝒜$ℬ$[Φ])≡$ℳ$_𝒜$ℬ$[$ℰ$_𝒜$ℬ$(Φ)]

The reduced state consequently contains all accessible information required for its subsequent reduced evolution.

The statement that the interacting parent dynamics “close on the reduced composite Hilbert space” therefore means that the joint reduced state contains all accessible information required for its subsequent reduced evolution.

At the reduced level, the most general generator may then be written as

${\widehat{H}}_{\mathit{AB}}$= ${\widehat{H}}_A\otimes I_B$+$I_A\otimes {\widehat{H}}_A$+${\widehat{H}}_{\mathit{int}}$                                     (27)

The first two terms describe the independent evolution already obtained in Eq. (24). The new term ${\widehat{H}}_{\mathit{int}}$ contains the reduced effect of the joint parent interaction.

A genuine interaction cannot be absorbed entirely into an operator acting only on A or only on B. In a product-operator expansion it generally has the form

$${\widehat{H}}_{\mathit{int}}={\sum }_r{g}_r{\widehat{A}}_r\hspace{0.17em}\otimes {\widehat{B}}_r$$

(28)

where the terms couple properties of the two subsystems.

How interaction generates nonfactorizable states

Consider an initially independently prepared product state,

|Ψ_AB(0)⟩ = |ψ_A⟩⊗|ψ_B⟩ (29)

If no interaction is present, Eq. (23) guarantees that the state remains factorized.

With interaction,

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\left(t\right)\right.⟩=e^{-i{\widehat{H}}_{\mathit{AB}}t/\mathrm{\hslash }}\left|{\mathsf{\psi }}_A\right.⟩\otimes \left|{\mathsf{\psi }}_B\right.⟩$$

(30)

For a short interval Δt,

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\left(\Delta t\right)\right.⟩\simeq \left(I-{\displaystyle \frac{i\Delta t}{\mathrm{\hslash }}}{\widehat{H}}_{\mathit{AB}}\right)\left|{\mathsf{\psi }}_A\right.⟩\otimes \left|{\mathsf{\psi }}_B\right.⟩$$

(31)

The local terms in Eq. (27) generate contributions of the form

${(\widehat{H}}_A|{\psi }_A⟩)\otimes |{\psi }_B⟩$ and $|{\psi }_A⟩\otimes {(\widehat{H}}_B|{\psi }_B⟩)$which correspond simply to changes of the individual factors.

The interaction term behaves differently. Consider for clarity a single coupling

$${\widehat{H}}_{\mathit{int}}=g\hspace{0.17em}\widehat{A}\otimes \widehat{B}$$

(32)

Its contribution is

$$-{\displaystyle \frac{\mathit{ig}\Delta t}{\mathrm{\hslash }}}\left(\widehat{A}\right|{\psi }_A⟩)\otimes (\widehat{B}|{\psi }_B⟩).$$

(33)

The post-interaction state therefore contains, schematically,

|ψ_A⟩⊗|ψ_B⟩+ϵ |ϕ_A⟩⊗|ϕ_B⟩ (34)

where,

$$\left|{\mathsf{\varphi }}_A\right.⟩=\widehat{A}\left|{\mathsf{\psi }}_A\right.⟩\mathit{and}\hspace{1em}\hspace{1em}\left|{\mathsf{\varphi }}_B\right.⟩=\widehat{B}\left|{\mathsf{\psi }}_B\right.⟩$$

If ϕ_A is linearly independent of ψ_A and ϕ_B is linearly independent of ψ_B, the state in Eq. (34) cannot in general be rewritten as one product

|χ_A⟩⊗|χ_B⟩

The interaction has therefore generated a nonfactorizable state.

This qualification is important. The mere presence of an operator formally labelled as an interaction does not guarantee entanglement for every possible initial state. For example, if $\widehat{A}\left|{\mathsf{\psi }}_A\right.⟩\propto \left|{\mathsf{\psi }}_A\right.⟩$ or $\widehat{B}\left|{\mathsf{\psi }}_B\right.⟩\propto \left|{\mathsf{\psi }}_B\right.⟩$, the corresponding evolution may remain factorized. Entanglement is generated when the interaction produces genuinely correlated changes in both tensor factors.

The same result can be seen directly from the coefficient matrix. Before interaction,

c_ij(0) = a_ib_j

which has rank one. Joint interaction generally produces coefficients

c_ij(t)

for which no vectors a_i(t) and b_j(t) exist such that

c_ij(t) = a_i(t)b_j(t)

When such a factorization is impossible, the reduced state is entangled.

Entanglement as preserved joint information

A general pure state in the composite Hilbert space can be written as

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\right.⟩={\sum }_{\mathit{ij}}{c}_{\mathit{ij}}\left|a_i\right.⟩\otimes \left|b_j\right.⟩$$

(35)

If c_ij = a_ib_j then the state factorizes and describes an independently representable pair. If no such factorization exists, the state is entangled.

Equivalently, every bipartite pure state admits a Schmidt decomposition [5],

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\right.⟩={\sum }_r{s}_r{\left|u_r\right.⟩}_A\otimes {\left|v_r\right.⟩}_B$$

(36)

where the sets $\left\{{\left|u_r\right.⟩}_A\right\}\mathit{and}\left\{{\left|v_r\right.⟩}_B\right\}$ are orthonormal.

A product state has Schmidt rank one. An entangled state has Schmidt rank greater than one.

For a normalized state,

$${\sum }_r{\left|s_r\right|}^2=1$$

(37)

At this stage Eq. (37) is only a statement concerning the Hilbert-space norm. The quantities ${\left|s_r\right|}^2$ have not yet been assigned the interpretation of measurement probabilities. That additional identification belongs to the derivation of the Born rule in Section 4.

Within the present dimensional framework, entanglement can now be interpreted more physically.

Before interaction, the parent sectors are independently preparable and their accessible reduced information factorizes,

Ψ_AB = Ψ_A⊗Ψ_B .

During an interaction, however, changes in one parent field may become correlated with changes in the other. The accessible information is then no longer separable into one complete description of A and another complete description of B.

Because the recording map is required to preserve physically relevant information, such joint correlations cannot simply be discarded. Their reduced representation must therefore retain the correlated coefficients c_ij even when those coefficients cannot be written as products of independent subsystem amplitudes.

Entanglement is consequently not introduced as an additional postulate. It appears as the reduced mathematical representation of physically accessible joint information produced by interaction that cannot be decomposed into independent information associated with the two subsystems.

An important consequence follows. Entanglement does not imply that the two parent fields cease to be distinct physical sectors. Rather, it means that after interaction their reduced accessible information can no longer be represented as two independent reduced states.

This distinction is useful because it places the origin of entanglement in the interaction-generated correlation structure of the parent dynamics, while the tensor product provides the minimal reduced mathematical space capable of preserving those correlations.

Local operations on an entangled state

Once an entangled state has been generated, a transformation performed locally on A continues to have the form: $\widehat{A}\otimes I_B$For the general state of Eq. (35),

$(\widehat{A}\otimes I_B)|{\mathsf{\Psi }}_{\mathit{AB}}⟩$ = $\sum _{\mathit{ij}}{c}_{\mathit{ij}}\widehat{A}\left|a_i\right.⟩\otimes |b_j⟩$Similarly,

$(I_A\otimes \widehat{B})|{\mathsf{\Psi }}_{\mathit{AB}}⟩$ = $\sum _{\mathit{ij}}{c}_{\mathit{ij}}\left|a_i\right.⟩\otimes \widehat{B}|b_j⟩$Thus, the state may be globally nonseparable while the transformation associated with a local setting remains restricted to its corresponding tensor factor.

This separation between global state structure and local action will become essential when Bell correlations are considered later. Entanglement concerns the nonfactorizability of the joint state; it does not require replacing local subsystem operators by operators acting indiscriminately on the distant subsystem.

No Bell assumption or probability rule is required for this conclusion.

Extension to multiple systems

The construction generalizes recursively.

For $𝒩$ independently preparable reduced sectors,

$\mathscr{H}$₁,$\mathscr{H}$₂,…,$\mathscr{H}$_$𝒩$

the same requirements of independent preparation and preservation of local superposition lead to

$${\mathcal{H}}_{1\dots \mathcal{N}}={⨂}_{n=1}^N\mathrm{c}{H}_n$$

(38)

An independently prepared state has the form

$$\left|{\mathsf{\Psi }}_{1\dots N}\right.⟩={⨂}_{n=1}^N\left|{\mathsf{\Psi }}_n\right.⟩$$

(39)

Interactions may subsequently transform this into a general state

$$\left|{\mathsf{\Psi }}_{1\dots N}\right.⟩={\sum }_{i_1,\dots ,i_N}{c}_{i_1\dots i_N}\left|i_1\right.⟩\otimes \dots \otimes \left|i_N\right.⟩$$

(40)

whose coefficients need not factorize into independent single-system amplitudes.

Thus, the framework naturally extends from the single-field Hilbert space of Ref. [1] to the configuration-space and tensor-product structures required for many-system quantum mechanics.

The present derivation has been formulated for distinguishable, independently preparable sectors. The additional restrictions associated with identical-particle symmetrization or antisymmetrization are separate questions and are not required for the Born-rule and Bell-correlation arguments developed in the following sections.

Summary of the composite-system construction

The logical structure obtained in this section may be summarized as follows.

For a single system, dimensional reduction and information preservation produce the complex Hilbert spaces

$\mathscr{H}$_𝒜 and   $\mathscr{H}$_$ℬ$ .

Independent preparation requires that the state of either subsystem may be varied linearly without altering the preparation of the other. The joint composition map must therefore be bilinear. The universal linear representation of such a bilinear composition is the tensor product:

$\mathscr{H}$_𝒜$ℬ$ = $\mathscr{H}$_𝒜⊗$\mathscr{H}$_$ℬ$ .

Non-interacting evolution factorizes:

$$U_{\mathit{AB}}^{\left(0\right)}=U_A\otimes U_B$$

and therefore preserves independently prepared product states.

Interaction introduces a genuinely joint term,

$${\widehat{H}}_{\mathit{int}}$$

which can correlate the parent sectors and generate reduced coefficients that no longer factorize,

c_ij ≠ a_ib_j

Such states cannot be represented by independent subsystem states and are therefore entangled.

No probability interpretation has been required in obtaining the above results. In particular, the tensor-product structure and the possibility of entangled states follow before the Born rule is introduced. This separation is important for the logic of the present construction: Section 4 can therefore ask independently which probability assignment is compatible with the state-space structure derived here, rather than assuming the quantum probability rule in order to obtain the composite-system formalism.

4. Emergence of the Born Rule

The preceding sections as well as the work done in Ref. [1] established the complex Hilbert-space description of a single reduced system and its extension to composite systems without assigning a probabilistic interpretation to the Hilbert-space norm. In particular, although the reduced state possesses an invariant quadratic quantity,

||Ψ||² = ⟨Ψ| Ψ⟩

nothing in the construction so far requires this quantity, or the squared magnitude of an expansion coefficient, to represent the probability of a measurement outcome. This distinction is essential. If the Hilbert norm had already been interpreted probabilistically in Section 2, then the Born rule would have been assumed rather than derived.

The objective of the present section is therefore to determine what probability assignment is compatible with the reduced state-space structure already obtained.

We first develop an internal argument based on information preservation, orthogonal alternatives, unitary invariance, composition, and consistency under coarse-graining. We then consider the logically independent approach of De Raedt, Katsnelson, and Michielsen, in which quantum probabilities arise from logical inference applied to experiments whose individual outcomes are uncertain but whose observed frequencies are maximally robust under small variations of experimental conditions. We will see that both approaches lead to the same result.

Approach 1: Born Rule from dimensional reduction and information preservation

Consider a normalized reduced state

|Ψ⟩ ∈ $\mathscr{H}$,  ⟨Ψ| Ψ⟩ = 1 (41)

As discussed in Section 2, a measurement corresponds physically to an interaction that establishes distinguishable records in the lower-dimensional description.

Suppose a given measurement admits mutually distinguishable outcomes labelled by 𝒾. Within the Hilbert-space representation, these distinguishable alternatives correspond to mutually orthogonal subspaces

$\mathscr{H}$_𝒾 ⊆ $\mathscr{H}$ .

Let P_i denote the orthogonal projector onto the subspace associated with outcome 𝒾. For a complete set of mutually exclusive outcomes,

$$P_i{P}_j={\Delta }_{\mathit{ij}}{P}_i,\hspace{1em}\hspace{1em}{\sum }_i{P}_i=I$$

(42)

The component of the state associated with the 𝒾-th distinguishable record is therefore

|Ψ_i⟩ = P_i|Ψ⟩. (43)

The complete state decomposes as

$$\left|\mathsf{\Psi }\right.⟩={\sum }_i\left|{\mathsf{\Psi }}_i\right.⟩$$

(44)

Because the outcome sectors are mutually orthogonal,

⟨Ψ_i| Ψ_j⟩ = 0,  i ≠ j . (45)

The Hilbert norm consequently satisfies the Pythagorean relation

$$\left|\right|\mathsf{\Psi }{\left|\right|}^2={\sum }_i\left|\right|{\mathsf{\Psi }}_i{\left|\right|}^2$$

(46)

For the normalized state of Eq. (41),

$${\sum }_i\left|\right|{\mathsf{\Psi }}_i{\left|\right|}^2=1$$

(47)

Equation (47) is still purely geometric. No probability interpretation has yet been assigned to its terms.

The question is now whether the requirements imposed on a consistent probability assignment force these geometric quantities to acquire precisely that interpretation.

We will now consider the requirements on a probability assignment

Let the probability associated with a distinguishable recorded component |Ψ_i⟩ be denoted by

p_i = W(|Ψ_i⟩) . (48)

The function W cannot be arbitrary.

First, a probability assignment should not depend on an overall phase attached to the component. The states

|Ψ_i⟩ and  e^iα|Ψ_i⟩

possess the same norm and represent the same distinguishable outcome sector. Therefore,

W(e^iα|Ψ_i⟩) = W(|Ψ_i⟩) . (49)

More generally, the probability should be invariant under unitary transformations internal to the outcome subspace. Since such transformations preserve the Hilbert norm, the probability associated with a component can depend only on its invariant magnitude,

r_i = ||Ψ_i|| . (50)

We may therefore write

p_i = f(r_i),  0 ≤ r_i ≤ 1 . (51)

The function must satisfy

f(0) = 0,  f(1) = 1 , (52)

because a component of zero magnitude cannot produce the corresponding record, while a state lying entirely within one outcome subspace produces that outcome with certainty.

We further require f to be continuous. Infinitesimally close reduced states should not generate finite jumps in their assigned probabilities unless the physical recording process itself changes discontinuously. This condition is also compatible with the robustness requirement introduced in Section 2.

The central remaining requirement is consistency under the combination of mutually exclusive alternatives.

Suppose that one experimentally distinguishable outcome is subdivided into two finer mutually exclusive alternatives represented by orthogonal components

|Ψ₁⟩ and  |Ψ₂⟩

Because they are orthogonal,

⟨Ψ₁| Ψ₂⟩ = 0 .

If the recording apparatus does not distinguish these two sub-outcomes and records only their union, the corresponding component belongs to the direct sum of the two orthogonal sectors.

Its squared norm is

||Ψ₁ + Ψ₂||² = ||Ψ₁||² + ||Ψ₂||² . (53)

Let

r₁ = ||Ψ₁|| and r₂ = ||Ψ₂||.

The magnitude of the coarse-grained component is therefore

$$r_{12}=\sqrt{r_1^2+r_2^2}$$

(54)

Since the two alternatives are mutually exclusive, probability consistency requires their probabilities to add:

f(r₁₂) = f(r₁) + f(r₂) . (55)

Substituting Eq. (54),

$$f\left(\sqrt{r_1^2+r_2^2}\right)=f\left(r_1\right)+f\left(r_2\right)$$

(56)

We now define

$$g\left(x\right)=f\left(\sqrt{x}\right),\hspace{1em}\hspace{1em}0\le x\le 1$$

(57)

Equation (56) then becomes

g(x + y) = g(x) + g(y), x,y ≥ 0, x + y ≤ 1 (58)

This is the additive Cauchy equation on the non-negative interval.

Continuity eliminates the pathological non-linear solutions, leaving

g(x) = Cx . (59)

From the certainty condition

f(1) = g(1) = 1

we obtain

C = 1 .

Therefore,

g(x) = x ,

and hence

f(r) = r² . (60)

The probability associated with an outcome sector is consequently

p_i = ||Ψ_i||² = ||P_iΨ||² . (61)

Using the self-adjoint and idempotent properties of an orthogonal projector,

${P_i}^{†}=P_i,\hspace{1em}\hspace{1em}{P}_i^2=P_i,$we obtain

||P_iΨ||² = ⟨Ψ|P_i|Ψ⟩ .

Therefore,

p_i = ⟨Ψ|P_i|Ψ⟩ . (62)

This is the Born rule in its projector form.

The result is consistent with the broader class of uniqueness results represented by Gleason's theorem, in which additive probability measures on mutually orthogonal Hilbert-space subspaces are constrained to the quantum form [6]. The present derivation differs in that it begins from the invariant norm of the reduced representation and imposes continuity and consistency under coarse-graining.

For a nondegenerate outcome represented by the normalized vector |a_i⟩,

P_i = |a_i⟩⟨a_i| . (63)

Equation (62) becomes

p_i = ⟨Ψ| a_i⟩⟨a_i| Ψ⟩ ,

or,

$$p_i={\left|⟨a_i|\mathsf{\Psi }⟩\right|}^2$$

(64)

Thus, given the existence of a consistent probability assignment satisfying the above conditions, the quadratic probability rule does not need to be introduced independently. Given the Hilbert-space structure derived in Ref [1], it follows from phase and unitary invariance, additivity for mutually exclusive orthogonal records, normalization, and continuity.

The crucial step is that orthogonal alternatives combine quadratically in the Hilbert norm but linearly as mutually exclusive probabilities. Requiring these two composition laws to describe the same coarse-grained physical alternatives uniquely selects the square of the Hilbert norm.

The fact that the probability of a particular outcome depends on the squared magnitude of the projected component does not mean that phase information has become irrelevant.

Consider a state expanded in a basis,

$$\left|\mathsf{\Psi }\right.⟩={\sum }_i{c}_i\left|a_i\right.⟩,\hspace{1em}\hspace{1em}{c}_i=\left|c_i\right|e^{i{\mathsf{\theta }}_i}$$

(65)

For a measurement in this same basis,

$$p_i={\left|c_i\right|}^2$$

(66)

The phase θ_i does not affect this particular probability.

However, consider another measurement basis {|b_j⟩} . Its amplitudes are

$$d_j=⟨b_j|\mathsf{\Psi }⟩={\sum }_i{c}_i⟨b_j|a_i⟩$$

(67)

The corresponding probability is

$$p_j={\left|d_j\right|}^2={\left|{\sum }_i{c}_i⟨b_j|a_i⟩\right|}^2$$

(68)

Expanding the square introduces interference terms,

$$p_j={\sum }_i{\left|c_i\right|}^2|⟨b_j{\left|a_i\right.⟩}^2+{\sum }_{i\ne k}{c}_i{c}_k^{*}⟨b_j|a_i⟩⟨a_k|b_j⟩$$

(69)

The relative phases contained in $c_i{c}_k^{*}$ therefore affect probabilities whenever alternatives interfere.

This provides an important connection with Section 2. Complex amplitudes are required in the reduced description because phase information must be retained, while the Born rule converts the resulting interference-sensitive complex representation into real, normalized frequencies associated with distinguishable records.

The two structures therefore perform different but complementary roles.

Finally, the tensor-product structure derived in Section 3 provides an additional consistency check.

Consider independently prepared systems

|Ψ_AB⟩ = |Ψ_A⟩⊗|Ψ_B⟩

Suppose outcome i is recorded for subsystem A and outcome j for subsystem B. Their local projectors are

$$P_i^A\mathit{and}\hspace{1em}{P}_j^B$$

The corresponding joint outcome sector is represented by

$$P_{\mathit{ij}}=P_i^A\otimes P_j^B$$

(70)

Using Eq. (62),

$$p\left(i,j\right)=⟨{\mathsf{\Psi }}_{\mathit{AB}}\left|P_i^A\otimes P_j^B\right|{\mathsf{\Psi }}_{\mathit{AB}}⟩$$

(71)

For a product state this factorizes:

$$p\left(i,j\right)=⟨{\mathsf{\Psi }}_A\left|P_i^A\right|{\mathsf{\Psi }}_A⟩⟨{\mathsf{\Psi }}_B\left|P_j^B\right|{\mathsf{\Psi }}_B⟩$$

Therefore,

p(i,j) = p_A(i)p_B(j) . (72)

Independent physical preparation therefore gives independent probabilities, exactly as required.

For an entangled state,

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\right.⟩={\sum }_{\mathit{ij}}{c}_{\mathit{ij}}\left|a_i\right.⟩\otimes \left|b_j\right.⟩$$

(73)

measurement in the corresponding product basis gives

$$p\left(i,j\right)={\left|c_{\mathit{ij}}\right|}^2$$

(74)

In general, however,

$${\left|c_{\mathit{ij}}\right|}^2\ne p_A\left(i\right)p_B\left(j\right)$$

because the coefficients c_ij need not factorize.

Therefore, the same probability rule that gives ordinary independent probabilities for product states naturally assigns correlated joint probabilities to entangled states.

For arbitrary local measurement settings, the general expression is

$$p\left(a,b\right)=⟨{\mathsf{\Psi }}_{\mathit{AB}}\left|P_a^A\otimes P_b^B\right|{\mathsf{\Psi }}_{\mathit{AB}}⟩$$

(75)

Equation (75) will form the starting point of the Bell–CHSH analysis in Section 5.

Interpretation within the dimensional recording framework and connection with the Measurement Update

The probability rule obtained above applies to the possible distinguishable records in the reduced description, rather than directly assigning probabilities to the complete 4+1-dimensional parent configurations.

This distinction is important.

The response map

$ℳ$:Φ⟶ Ψ

does not retain every detail of Φ. Instead, as established in Ref [1] and Section 2, it preserves equivalence classes of parent configurations according to the distinctions accessible to 3+1-dimensional interactions.

Consequently, although a parent description may possess additional dynamical information, the reduced state contains only the information required to characterize all experimentally distinguishable alternatives.

Suppose the measurement interaction produces the orthogonal record sectors

P₁Ψ,P₂Ψ,…,P_n

The framework does not require the response kernel itself to be a probability distribution. Instead, the kernel determines the reduced amplitudes, while the probability assignment follows from the geometry and consistency of the reduced representation.

In this sense, the Born rule is not attributed to randomness of the response kernel K. It emerges from the structure required to consistently assign frequencies to mutually exclusive alternatives represented within the information-preserving reduced state space.

This also separates two questions that are sometimes conflated:

Why is the state complex

Why are probabilities quadratic?

Within the present framework, the first follows from preservation of phase information under dimensional reduction, whereas the second follows from the additive geometry of orthogonal reduced alternatives together with probability consistency.

Furthermore, the measurement discussion at the end of Section 2 can now also be connected to the probability rule.

$$\left|\mathsf{\Psi }\right.⟩={\sum }_i{P}_i\left|\mathsf{\Psi }\right.⟩$$

The probability that the interaction produces record i is

p_i = |P_iΨ|² .

Once outcome i has become the physically established record, the information relevant to subsequent predictions lies in the corresponding sector. If no additional disturbance within that sector is specified, its normalized conditional state is

$$|{\mathsf{\Psi }\mathrm{\text{'}}}_i⟩={\displaystyle \frac{P_i\left|\mathsf{\Psi }\right.⟩}{\sqrt{⟨\mathsf{\Psi }\left|P_i\right|\mathsf{\Psi }⟩}}}$$

(76)

Equivalently,

$$|{\mathsf{\Psi }\mathrm{\text{'}}}_i⟩={\displaystyle \frac{P_i\left|\mathsf{\Psi }\right.⟩}{\sqrt{p_i}}}$$

This conditional normalization should be distinguished from the physical mechanism of measurement itself. The interaction-induced modification of the parent field and recording process discussed in Section 2 determines which post-interaction configuration is physically realized; Eq. (76) merely expresses the normalized reduced state conditioned on the corresponding record.

Thus, the Born rule and state update play distinct roles. The Born rule expresses relative frequencies of possible records, whereas the state update expresses the conditional description after a record is established.

This separation is useful because deriving the Born rule does not, by itself, require postulating a fundamental physical collapse mechanism.

Approach 2: Robustness and logical inference

The preceding derivation begins with the Hilbert-space structure obtained from dimensional reduction and asks which probability rule is compatible with that structure.

A conceptually different route was developed by De Raedt, Katsnelson, and Michielsen [2]. Their approach does not begin with Hilbert space, wavefunctions, or the Born rule. Instead, it begins with logical inference applied to experiments for which individual events are uncertain but the observed frequencies are reproducible and robust under small variations of the experimental conditions. They classify precisely this class of experiments as those in which event-level uncertainty remains, while frequency distributions are stable under small changes of the experimental conditions.

The importance of considering this approach here is that its starting assumptions are very different from those of the present dimensional framework. Agreement between the two therefore provides a nontrivial complementary argument rather than a circular derivation.

In the De Raedt, Katsnelson, and Michielsen approach, let

P(x| θ,Z)

denote the inference probability, or i-probability in the terminology of De Raedt et al., for observing outcome x, given an experimental condition described by a parameter θ and all other relevant information denoted collectively by Z.

Suppose the experiment is repeated N times independently under nominally identical conditions.

If outcome x occurs n_x times, then

$${\sum }_x{n}_x=N$$

The likelihood for the observed data is proportional to

$$L\left(\mathsf{\theta }\right)={\prod }_{x}P{\left(x|\mathsf{\theta },Z\right)}^{n_x}$$

(77)

Now consider a small variation of the experimental condition,

θ ⟶ θ + ϵ

De Raedt et al. characterize robustness through the change in the evidence associated with this perturbation. A convenient logarithmic measure is

$$\mathrm{Ev}=\ln {\displaystyle \frac{L\left(\theta +ϵ\right)}{L\left(\theta \right)}}$$

Using Eq. (77),

$$\mathrm{Ev}={\sum }_x{n}_x\ln {\displaystyle \frac{P\left(x|\theta +ϵ,Z\right)}{P\left(x|\theta ,Z\right)}}$$

(78)

For small ϵ, expansion to second order gives, under the robustness conditions used in their construction,

$$\mathrm{Ev}=-{\displaystyle \frac{N{\mathsf{ϵ}}^2}{2}}{I}_F\left(\mathsf{\theta }\right)+O\left({\mathsf{ϵ}}^3\right)$$

(79)

where,

$$I_F\left(\mathsf{\theta }\right)={\sum }_x{\displaystyle \frac{1}{P\left(x|\mathsf{\theta },Z\right)}}{\left({\displaystyle \frac{\partial P\left(x|\mathsf{\theta },Z\right)}{\partial \mathsf{\theta }}}\right)}^2$$

(80)

is the Fisher information.

In this construction, robustness requires the evidence to change as little as possible under small perturbations while retaining nontrivial dependence on the experimental condition. This leads to minimization of I_F, subject to the physical symmetries and constraints of the experiment. This Fisher-information minimization emerges from the robustness argument rather than being assumed at the outset.

In the continuous case, translational invariance allows the robustness functional to be expressed in terms of the Fisher information

$$I_F=\int \mathit{dx}\hspace{0.17em}{\displaystyle \frac{1}{P\left(x|\mathsf{\theta },Z\right)}}{\left({\displaystyle \frac{\partial P\left(x|\mathsf{\theta },Z\right)}{\partial x}}\right)}^2$$

(81)

A useful substitution is

P(x| θ,Z) = R²(x| θ,Z) ,

which converts the Fisher-information term into a quadratic expression,

$$I_F=4\int \mathit{dx}{\left({\displaystyle \frac{\partial R}{\partial x}}\right)}^2$$

When the robustness problem is supplemented by the requirement that the averaged observations satisfy the appropriate Hamilton–Jacobi constraint, the probability amplitude $\mathrm{R}=\sqrt{\mathrm{P}}$ and the action-like phase S can be combined into a complex function. In conventional units this may be represented as

$$\mathsf{\psi }\left(x,t\right)=\sqrt{P\left(x,t\right)}{e}^{\mathit{iS}\left(x,t\right)/\mathrm{\hslash }}$$

By construction,

P(x,t) = |ψ(x,t)|²

The corresponding robustness variational problem can then be rewritten as a quadratic functional of ψ, whose stationarity condition yields the Schrödinger equation. In this logical-inference construction, the wavefunction therefore packages both the square root of the inferred probability distribution and the phase information required to linearize the dynamics. The 2014 paper [2] derives both time-independent and time-dependent Schrödinger descriptions by this route.

The connection between this work and our framework becomes particularly interesting when viewed in terms of the response kernel introduced in Section 2.

Let the recording process depend on parameters

λ = (λ¹,λ²,…)

which characterize, for example, small variations in the response kernel,

K = K(λ)

As we said in Section 2 those parameters represent accessible information is still encoded in the three-dimensional spatial configuration χ and its evolution with respect to t, because they concern the mapping and not the 4+1 field itself.

The reduced state then becomes

Ψ = Ψ(λ)

and the Born probabilities obtained internally are

p_i(λ) = |P_iΨ(λ)|² .

Then according to Ref [2] the sensitivity of this probability distribution to changes of the recording parameters can then be quantified by the Fisher-information matrix

$$I_{\mathsf{\mu \nu }}={\sum }_i{\displaystyle \frac{1}{p_i}}{\displaystyle \frac{\partial p_i}{\partial {\mathsf{\lambda }}^{\mathsf{\mu }}}}{\displaystyle \frac{\partial p_i}{\partial {\mathsf{\lambda }}^{\mathsf{\nu }}}}$$

This provides a quantitative refinement of the robustness condition introduced earlier in connection with ΔK.

A small variation ΔK induces ΔΨ and consequently Δp_i.

For perturbations of the recording mechanism that are physically irrelevant, reproducibility requires these probability distributions to remain stable with robustness corresponding to limited sensitivity of the final reproducible distribution to sufficiently small nuisance variations of the recording process.

Thus, Fisher information does not need to be introduced as an additional axiom of the dimensional framework. Instead, once the Born probabilities have been obtained internally, Fisher information provides a natural quantitative measure of precisely the stability against recording variations that was qualitatively required in Section 2.

Summary of the results

The Born rule can therefore be understood within the present framework without identifying the Hilbert norm with probability at the outset.

The internal derivation proceeds from the structures already established in Section 2 and Section 3:

information − preserving reduction⟶

complex Hilbert space⟶

orthogonal distinguishable records⟶

unitary invariance⟶

additivity under coarse − graining⟶

p_i = |P_iΨ|²

For rank-one outcomes this becomes

$$p_i={\left|⟨a_i|\mathsf{\Psi }⟩\right|}^2$$

The independent logical-inference construction of De Raedt, Katsnelson, and Michielsen reaches the same quadratic probability structure by requiring uncertain experimental events to produce maximally reproducible frequency distributions under small changes in experimental conditions. Their robustness condition leads to Fisher-information minimization and, for appropriate experimental constraints, to the standard quantum probabilities and Schrödinger description.

The convergence of the two approaches is significant. In the dimensional framework, complex amplitudes arise because phase information must survive dimensional reduction and the Born rule subsequently follows from the consistent measure of mutually exclusive Hilbert-space components. In the robustness framework, probabilities arise first through logical inference, while the complex amplitude is subsequently introduced as the natural square-root-and-phase representation of the robust distribution.

Their meeting point is the same relation,

p = |Ψ|² .

This provides both an internal derivation and an independent robustness-based justification for the probability rule required by the reduced quantum-mechanical description.

The result also prepares the ground for the next question. Once the tensor-product structure, entangled states, local operators, and the Born probability rule have all been obtained independently, the joint probabilities of spatially separated measurements are fixed by

$$p\left(a,b\right)=⟨{\mathsf{\Psi }}_{\mathit{AB}}\left|P_a^A\otimes P_b^B\right|{\mathsf{\Psi }}_{\mathit{AB}}⟩$$

The next Section can therefore investigate whether these correlations satisfy or violate the Bell–CHSH bound, without adding any new rule for composite-system probabilities.

5. Bell Correlations from Interaction-Generated Nonfactorizability

The preceding sections established three structures without reference to Bell experiments: the tensor-product composition of independently preparable systems, the generation of nonfactorizable states through joint interaction, and the Born probability rule for distinguishable reduced records. The purpose of the present section is to determine whether these structures are sufficient to produce correlations incompatible with Bell-local factorization.

Importantly, we do not begin by postulating a spin-singlet state, a pair of abstract qubits, or Pauli measurement operators. None of these objects has yet been derived as a specific physical structure of the parent theory. Instead, we begin directly from the generic nonfactorizable reduced state produced by the interaction mechanism of Section 3.

The essential question is therefore: Does interaction-generated nonfactorizability, together with the locally acting measurement structure already derived, necessarily permit Bell-inequality violation?

We first answer this question for the minimal two-alternative entangled sector and then discuss its extension to more general pure entangled states.

Deriving the Bell-inequality violation within our framework

Before interaction, two independently preparable parent sectors admit the reduced product state

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}^{\left(0\right)}\right.⟩=\left|{\mathsf{\Psi }}_A\right.⟩\otimes \left|{\mathsf{\Psi }}_B\right.⟩.$$

(82)

As established in Section 3, a joint parent interaction can generate reduced coefficients that do not factorize,

c_ij ≠ a_ib_j ,

giving a general pure composite state

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\right.⟩={\sum }_{\mathit{ij}}{c}_{\mathit{ij}}\left|a_i\right.⟩\otimes \left|b_j\right.⟩$$

(83)

Such a state admits a Schmidt decomposition,

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\right.⟩={\sum }_r{s}_r{\left|u_r\right.⟩}_A\otimes {\left|v_r\right.⟩}_B$$

(84)

where

$$s_r\ge 0,\hspace{1em}\hspace{1em}{\sum }_r{s}_r^2=1$$

and the local sets

$$\left\{{\left|u_r\right.⟩}_A\right\},\hspace{1em}\hspace{1em}\left\{{\left|v_r\right.⟩}_B\right\}$$

are orthonormal.

The state is entangled if and only if at least two Schmidt coefficients are nonzero.

Therefore, the minimal nontrivial entangled sector generated by the interaction contains two correlated alternatives. Restricting initially to such a sector, the state can always be written as

$$\left|\mathsf{\Psi }\left(\mathsf{\eta }\right)\right.⟩=\cos \mathsf{\eta }\hspace{0.17em}{\left|u_0\right.⟩}_A{\left|v_0\right.⟩}_B+\sin \mathsf{\eta }\hspace{0.17em}{\left|u_1\right.⟩}_A{\left|v_1\right.⟩}_B$$

(85)

with

$$0<\mathsf{\eta }<{\displaystyle \frac{\mathsf{\pi }}{2}}$$

Any relative complex phase between the two Schmidt terms can be absorbed into one of the local Schmidt vectors, so Eq. (85) involves no loss of generality.

Crucially, the labels 0 and 1 do not represent spin states or assumed quantum bits. They merely label two orthogonal reduced alternatives that have emerged from the Schmidt decomposition of the interaction-generated joint state.

Thus, the two-dimensional structure required for a CHSH experiment has not been postulated. It is already contained in the minimal nonfactorizable sector of the composite state derived in Section 3.

The two orthogonal alternatives in each local Schmidt space allow us to define dichotomic measurements directly.

For subsystem A, define

Z_A = |u₀⟩⟨u₀| − |u₁⟩⟨u₁|

and

X_A = |u₀⟩⟨u₁| + |u₁⟩⟨u₀| .

Similarly, for subsystem B,

Z_B = |v₀⟩⟨v₀| − |v₁⟩⟨v₁|

and

X_B = |v₀⟩⟨v₁| + |v₁⟩⟨v₀|

These operators are not introduced as spin operators. They are constructed entirely from the two locally distinguishable Schmidt alternatives.

Within the corresponding two-dimensional subspaces,

$$Z_A^2=X_A^2=I_A$$

and

$$Z_B^2=X_B^2=I_B$$

Hence their eigenvalues are

±1

The familiar Pauli algebra appears only as the matrix representation of these abstract two-dimensional operators in a chosen basis. It is therefore a mathematical consequence of having a two-dimensional complex Hilbert sector, rather than a physical assumption about spin.

More generally, local measurement settings can be represented by continuous rotations within these subspaces:

A(α) = cosα Z_A + sinα X_A (86)

and

B(β) = cosβ Z_B + sinβ X_B . (87)

These operators satisfy

A(α)² = I_A and  B(β)² = I_B

and therefore define legitimate two-outcome measurements with outcomes +1 and -1.

The measurement freedom required in a CHSH experiment thus follows from continuous unitary transformations already available within the local Hilbert spaces.

We now calculate the correlations produced by the generic state of Eq. (85), without introducing a singlet state.

First,

⟨Z_A⊗Z_B⟩ = 1

This follows because both Schmidt terms contain matching local alternatives.

For the complementary operators,

⟨X_A⊗X_B⟩ = sin(2η)

The mixed correlations vanish,

⟨Z_A⊗X_B⟩ = 0 (88)

and

⟨X_A⊗Z_B⟩ = 0 (89)

For arbitrary local settings α and β, the correlation is therefore

E(α,β)=⟨Ψ(η)|A(α)⊗B(β)|Ψ(η)⟩ (90)

Using Eqs. (86)–(89),

E(α,β) = cosαcosβ + sin(2η)sinαsinβ (91)

The quantity

κ≡sin(2η) (92)

therefore, directly measures the nonfactorizable contribution relevant to these correlations.

For a product state,

$$\mathsf{\eta }=0\hspace{1em}\mathrm{or}\hspace{1em}\mathsf{\eta }={\displaystyle \frac{\mathsf{\pi }}{2}}$$

so that

κ = 0 .

For an entangled state,

$$0<\mathsf{\eta }<{\displaystyle \frac{\mathsf{\pi }}{2}}$$

and consequently

κ > 0

Thus, the same parameter that measures the presence of interaction-generated nonfactorizability also controls the additional correlation term capable of violating the CHSH bound.

We now consider a Bell-local hidden-variable description [7] in which a complete variable λ renders the two local outcomes conditionally independent:

p(s,t| α,β,λ) = p_A(s| α,λ)p_B(t| β,λ) (93)

Assume also measurement independence,

ρ(λ| α,β) = ρ(λ) .

For dichotomic outcomes,

|A(α,λ)| ≤ 1,  |B(β,λ)| ≤ 1 .

The standard CHSH argument then gives

S = |E(α₀,β₀) + E(α₀,β₁) + E(α₁,β₀) − E(α₁,β₁)| ≤ 2 . (94)

This is the CHSH bound for the relevant class of locally factorizable models. The CHSH inequality was introduced precisely as an experimentally testable extension of Bell’s result [8].

In contrast to that, within our framework, we use the correlation obtained directly from the interaction-generated Schmidt state.

Choose the local settings at A as

$${\mathsf{\alpha }}_0=0,\hspace{1em}\hspace{1em}{\mathsf{\alpha }}_1={\displaystyle \frac{\mathsf{\pi }}{2}}$$

At B, choose

β₀ = β,  β₁ = −β .

Equation (91) then gives

E(α₀,β₀) = cosβ ,

E(α₀,β₁) = cosβ ,

E(α₁,β₀) = κsinβ ,

and

E(α₁,β₁) = −κsinβ .

The CHSH quantity therefore becomes

S(β) = 2|cosβ + κsinβ|. (95)

The maximum occurs when

tanβ = κ .

Substituting this gives

$$S_{\mathit{max}}=2\sqrt{1+{\mathsf{\kappa }}^2}$$

Using Eq. (92),

$$S_{\mathit{max}}=2\sqrt{1+{\sin }^2\left(2\mathsf{\eta }\right)}$$

(96)

For every entangled state,

$$0<\mathsf{\eta }<{\displaystyle \frac{\mathsf{\pi }}{2}}$$

and therefore

sin²(2η) > 0 .

Consequently,

S_max > 2 . (97)

This is the important result.

A maximally entangled state has

$$\mathsf{\eta }={\displaystyle \frac{\mathsf{\pi }}{4}}$$

giving

sin(2η) = 1

and hence

$$S_{\mathit{max}}=2\sqrt{2}$$

However, maximal entanglement was not required. Any nonzero pure entanglement generated by the interaction gives a CHSH value greater than 2 for suitable local measurement settings.

This result is consistent with the general result commonly known as Gisin’s theorem, according to which every bipartite pure non-product state violates a Bell inequality [9].

The significance for the present framework is that no singlet state had to be postulated. The Bell violation follows directly from the existence of a nonfactorizable pure reduced state together with the locally available two-outcome transformations already implied by the Hilbert-space structure.

At this point it is useful to make the logical status of the preceding construction explicit.

The local basis vectors,

|u₀⟩, |u₁⟩

and

|v₀⟩, |v₁⟩

were not introduced as microscopic spin states. They arise from the Schmidt decomposition of the joint reduced state.

Likewise,

Z_A, X_A, Z_B, X_B

were constructed from those basis vectors. Their algebra is therefore a property of a two-dimensional complex Hilbert space.

If one chooses matrix representations in the corresponding bases, then

$$Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

and

$$X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

These happen to be the usual Pauli matrices σ_z and σ_x, but their appearance here is representational rather than postulated.

It would now be appropriate to extend the argument beyond the minimal two-term state.

$$\left|{\mathsf{\Psi }}_{\mathit{AB}}\right.⟩={\sum }_{r=1}^R{s}_r{\left|u_r\right.⟩}_A{\left|v_r\right.⟩}_B,\hspace{1em}\hspace{1em}R>2$$

Entanglement requires at least two nonzero coefficients,

s₁s₂ ≠ 0

The two corresponding local Schmidt pairs span a correlated two-dimensional sector of each local Hilbert space.

The conclusion that pure entanglement implies Bell-inequality violation is not restricted to exactly two-dimensional systems. Gisin first proved the result for bipartite non-product pure states [9], and subsequent formulations extend the argument to higher-dimensional bipartite pure states [10].

Thus, the two-term calculation above should be understood not as an assumption that the parent system is fundamentally a qubit, but as the minimal mathematical sector in which the Bell-nonfactorizable character of a pure entangled state becomes visible.

Conceptual core of the argument

The present framework does not predict Eq. (97) by assigning independent complete states to A and B after their interaction.

Instead, maximal information preservation requires the interaction-generated joint information to remain encoded in the single reduced state

|Ψ_AB⟩ ∈ $\mathscr{H}$_𝒜⊗$\mathscr{H}$_$ℬ$

For the state of Eq. (85), the joint outcome probability for local settings α and β is

$$p\left(s,t|\mathsf{\alpha },\mathsf{\beta }\right)=⟨\mathsf{\Psi }\left(\mathsf{\eta }\right)\left|P_s^A\left(\mathsf{\alpha }\right)\otimes P_t^B\left(\mathsf{\beta }\right)\right|\mathsf{\Psi }\left(\mathsf{\eta }\right)⟩$$

This joint probability need not factorize into

p_A(s| α)p_B(t| β)

More importantly, it cannot be reproduced by a complete Bell-local variable λ satisfying

p(s,t| α,β,λ) = p_A(s| α,λ)p_B(t| β,λ)

together with measurement independence, while retaining the CHSH violation.

The structural reason is clear within the framework: after interaction, the accessible information is not the union of two independent reduced descriptions. It contains irreducible joint information.

Furthermore, we must point out that it is not the additional dimension that resolves Bell's theorem.

Bell’s theorem constrains the statistical causal structure of the underlying model, not the number of dimensions assigned to its variables [7].

What has been established in our framework is more precise than just the presence of another dimension or hidden variables. The information-preserving reduction produces a globally nonfactorizable state that permits S > 2. This is the key mechanism.

Additionally, although the global reduced state is nonfactorizable, the local measurement transformations retain the tensor-product structure derived in Section 3.

A measurement performed at A acts as

$$P_s^A\left(\mathsf{\alpha }\right)\otimes I_B$$

while one performed at B acts as

$$I_A\otimes P_t^B\left(\mathsf{\beta }\right)$$

These operations commute,

$$\left[P_s^A\left(\mathsf{\alpha }\right)\otimes I_B,I_A\otimes P_t^B\left(\mathsf{\beta }\right)\right]=0$$

Thus, the Bell violation does not arise by allowing the local measurement operator at A to act directly on the state space of B, or vice versa.

The nonclassical structure resides instead in the state on which these local operators act.

We now verify that the Bell-violating correlations cannot be used for controllable signalling within our framework.

The outcome projectors are

$$P_s^A\left(\mathsf{\alpha }\right)={\displaystyle \frac{1}{2}}\left[I_A+\mathit{sA}\left(\mathsf{\alpha }\right)\right]$$

and

$$P_t^B\left(\mathsf{\beta }\right)={\displaystyle \frac{1}{2}}\left[I_B+\mathit{tB}\left(\mathsf{\beta }\right)\right]$$

The joint probability is

p(s,t|α,β)=⟨Ψ|P_s^A(α)⊗P_t^B(β)|Ψ⟩ (98)

For the state of Eq. (85), define

c = cos(2η),  κ = sin(2η) .

The local expectation values are

⟨A(α)⟩ = ccosα

and

⟨B(β)⟩ = ccosβ

Using Eq. (91), the full joint distribution becomes

$$p\left(s,t|\mathsf{\alpha },\mathsf{\beta }\right)={\displaystyle \frac{1}{4}}\left[1+\mathit{sc}\cos \mathsf{\alpha }+\mathit{tc}\cos \mathsf{\beta }+\mathit{st}\left(\cos \mathsf{\alpha }\cos \mathsf{\beta }+\mathsf{\kappa }\sin \mathsf{\alpha }\sin \mathsf{\beta }\right)\right]$$

(99)

Summing over the outcomes at B

$$p_A\left(s|\mathsf{\alpha },\mathsf{\beta }\right)={\sum }_{t=\pm 1}p\left(s,t|\mathsf{\alpha },\mathsf{\beta }\right)$$

gives,

$$p_A\left(s|\mathsf{\alpha },\mathsf{\beta }\right)={\displaystyle \frac{1}{2}}\left(1+\mathit{sc}\cos \mathsf{\alpha }\right)$$

There is no dependence on β. Therefore,

p_A(s| α,β) = p_A(s| α) (100)

Likewise,

p_B(t| α,β) = p_B(t| β) (101)

Thus, the no-signalling property does not rely on maximal entanglement or on the singlet state. It follows for the generic entangled state considered here from the completeness of the local outcome projectors and the tensor-product structure of local operations.

This gives S > 2 while p_A(s| α,β) = p_A(s| α)

Bell nonfactorizability and operational signalling are therefore distinct.

The distinction becomes especially clear when measurement-induced updating is considered.

Suppose the observer at A obtains result s.

The conditional probability assigned to an outcome t at B is

$$p\left(t|s,\mathsf{\alpha },\mathsf{\beta }\right)={\displaystyle \frac{p\left(s,t|\mathsf{\alpha },\mathsf{\beta }\right)}{p_A\left(s|\mathsf{\alpha }\right)}}$$

(102)

This conditional distribution generally depends on the result s obtained at A.

Thus, once the record at A is known, the appropriate reduced description of the joint information changes.

However, observer B, without access to the outcome at A, must average over s:

$$p_B\left(t|\mathsf{\beta }\right)={\sum }_s{p}_A\left(s|\mathsf{\alpha }\right)p\left(t|s,\mathsf{\alpha },\mathsf{\beta }\right)$$

Using Eq. (102),

$$p_B\left(t|\mathsf{\beta }\right)={\sum }_{s}p\left(s,t|\mathsf{\alpha },\mathsf{\beta }\right)$$

which is independent of α.

Thus, the conditional update of the global reduced state does not correspond to a controllable physical signal propagating from A to B.

This connects directly with the measurement interpretation developed in Section 2: state updating concerns the information appropriate after a particular record has been established, whereas the unconditioned statistics available to a distant subsystem remain unchanged.

Robustness construction as an independent crosscheck

The robustness result of De Raedt, Katsnelson, and Michielsen [2] provides an independent and complementary perspective.

Their treatment of the EPRB experiment does not begin with a singlet state. Instead, it assumes a source producing two signals, two independently oriented two-outcome detection devices, uncertainty about individual events, and robustness of the observed frequencies under small changes in the relative detector orientation. Minimization of the corresponding Fisher information yields

E(θ) = −cosθ

which is the correlation conventionally associated with the singlet state. The authors explicitly emphasize that this result is obtained without invoking quantum concepts at the start of the construction.

This observation is especially relevant here because it removes the same conceptual concern from the complementary route: the singlet state is not required as an initial postulate in the robustness construction either.

The De Raedt–Katsnelson–Michielsen analysis shows explicitly that the robust EPRB construction produces the correlation associated with the quantum singlet description.

The convergence is therefore again complementary rather than circular.

What is particularly interesting is that the present framework already contains the physical ingredients required to fit naturally within the De Raedt–Katsnelson–Michielsen picture: physical systems interact, those interactions generate distinguishable outcomes or records, and reproducible relations can then be established between the conditions of the interaction and the observed results, without beginning from quantum postulates or from an assumed probabilistic interpretation of the state. This is closely aligned with the motivation developed in Ref. [1], where the reduced complex representation was introduced precisely because a lower-dimensional description must correlate physically accessible quantities with one another in order to retain information about the higher-dimensional wave. In Section 3 of Ref. [1], complex functions were therefore associated with measurable quantities such as wavelength or energy, with the complex plane providing the additional degree of freedom required to preserve the relevant correlations when part of the higher-dimensional dynamics is inaccessible. The compatibility between these viewpoints is useful: in the present framework, the complex state Ψ arises from the need to preserve relational information under dimensional reduction, whereas in the robustness construction, stable correlations between experimentally defined physical quantities constrain the reproducible probability structure. Neither route needs to assume the quantum formalism or the Born rule at the outset, yet both lead toward the same experimentally relevant correlation structure.

6. Geometric Emergence of Internal U(1), SU(2), and SU(3) Symmetry Structures

The preceding sections concerned the emergence of the mathematical structures conventionally associated with quantum mechanics from the dimensional recording of the parent field Φ. We now investigate whether the same construction can provide a geometric origin for another central element of modern particle physics: the internal symmetry groups associated with the fundamental interactions.

The gauge structure of the Standard Model is

G_SM = SU(3)_C × SU(2)_L × U(1)_Y

The objective of the present section is not to assume these three groups and subsequently assign them to different components of the parent field. Instead, we ask whether their internal dimensions and transformation structures can emerge from the different geometries in which a physical change of the same 4+1-dimensional field Φ may occur.

The basic proposal is that a physically relevant variation

Φ ⟶ Φ+ΔΦ

may have different geometric support within the four-dimensional spatial domain of the parent field. Relative to the 3+1-dimensional recorded description, there are three structurally distinct possibilities: a change confined to the observed three-dimensional spatial subspace, a change occurring in a three-dimensional subspace tilted into the fourth parent direction, and a change that requires the complete four-dimensional parent spatial domain.

We will show that these geometries naturally lead to one, two, and three component complex internal representations, respectively. Their norm-preserving transformations then possess U(1), U(2), and U(3) structure. After separating the common phase already present in the recording, the genuinely new non-Abelian mixing structures are SU(2) and SU(3).

The framework developed below should be understood as a proposed derivation of the symmetry classes and their representation dimensions, rather than yet a complete derivation of all Standard Model representation assignments, charges, symmetry breaking, or interaction dynamics.

The 4+1-dimensional parent geometry and why the three geometric channels are sufficient

Let the spatial domain of the parent field be the four-dimensional vector space

V₄ = span{e₁,e₂,e₃,n} ,

While τ remains the parent evolution parameter.

The three-dimensional spatial subspace directly retained by the reduced description is

V₃ = span{e₁,e₂,e₃}

The vector n denotes the fourth spatial direction of the parent dynamics.

Under the recording process developed in Ref. [1], this direction is not retained as an additional spatial coordinate. Information carried through it instead enters the reduced 3+1-dimensional description through phase and temporal structure.

Accordingly, we write

V₄ = V₃⊕span{n}. (103)

Let

P = I − nn^T (104)

be the projector onto V₃, and let

Q = nn^T (105)

be the projector onto the fourth parent direction.

These two projectors allow the geometric content of an arbitrary small change of the parent field to be classified without first assigning it to any known interaction.

Because Φ may itself be a scalar or possess more complicated field structure, it is useful to characterize the geometry of a small variation ΔΦ through its local linearized deformation operator, which we denote by

Δ[ΔΦ] .

Relative to the decomposition in Eq. (103), every such linearized deformation admits the exact decomposition

Δ=PΔP + PΔQ + QΔP + QΔQ (106)

It is useful to collect these terms as

Δ=Δ_∥ + Δ_mix + Δ_σ (107)

where,

Δ_∥ = PΔP , Δ_mix = PΔQ + QΔP and Δ_σ = QΔQ.

This decomposition provides a particularly useful way of understanding the three geometries considered here.

The first term changes only quantities lying inside V₃. It therefore corresponds to an aligned three-dimensional deformation.

The mixed terms couple an ordinary spatial direction to n. Locally, such terms rotate or tilt an active three-dimensional subspace away from V₃ and into the fourth parent direction. They therefore generate the tilted three-dimensional deformation class. This class is physically distinct in the present framework because of the unconventional role assigned to time. A tilted sector should not be understood merely as an ordinary three-dimensional plane viewed in rotated spatial coordinates. Rather, it defines a different embedded 3+1-dimensional geometry: three directions play the role of spatial dimensions for that sector, while a fourth orthogonal direction plays the +1 role of its evolution parameter. These directions need not coincide with the three spatial directions and time direction of our recorded 3+1-dimensional geometry. Nevertheless, both geometries are embedded in the same four-dimensional configurational domain of the parent 4+1 field. Consequently, the basis vectors of the tilted 3+1 sector must be expressible as linear combinations of the basis vectors associated with our 3+1 description.

If (e₁,e₂,e₃,n) denotes the parent basis adapted to our recorded geometry, and (a,b,c,d) denotes an orthonormal basis adapted to the tilted geometry, then locally they are related by a four-dimensional orthogonal transformation,

$${\left(a,b,c,d\right)}^T=R{\left(e_1,e_2,e_3,n\right)}^T,R^{T}R=I_4$$

For our 3+1 description, n is the parent direction whose reduced role is identified with time. For the tilted 3+1 sector, by contrast, d plays the corresponding +1 evolutionary role. Thus,

$$V_{3+1}^{\mathrm{obs}}=\mathit{span}\left\{e_1,e_2,e_3;n\right\}$$

whereas

$$V_{3+1}^{\mathrm{tilt}}=\mathit{span}\left\{a,b,c;d\right\}$$

Finally, when the fourth direction becomes an independent dynamical component rather than entering only through a constrained tilt, the minimal subspace required to describe the deformation is the complete V₄. This gives the full four-dimensional deformation class.

Thus, the three classes are not arbitrary labels attached to three interactions. They arise from the possible relationships between the active deformation subspace and the decomposition of Eq. (103).

A maximal three-dimensional active subspace is either identical to V₃ or tilted relative to it. If the active deformation cannot be confined to any three-dimensional subspace, it requires all of V₄.

Lower-rank variations are contained inside one of these maximal geometries and therefore do not define an additional symmetry class.

A completely generic deformation may contain contributions belonging to more than one of these sectors. The statement is therefore not that every ΔΦ possesses a unique direct-sum decomposition into “electromagnetic”, “weak”, and “strong” pieces. Rather, these three geometries form the primitive classes from which the local geometric content of a smooth parent-field variation can be constructed.

This distinction is important: the interaction type is associated with the geometry on which the relevant dynamics closes, rather than with an arbitrary naming of individual components of ΔΦ.

Aligned three-dimensional changes and a single complex channel

Consider first a deformation whose dynamically relevant support remains completely inside V₃

Then according to Eq. (106),

QΔ=ΔQ = 0 .

Only ordinary three-dimensional spatial changes remain.

Rotations among e₁,e₂,e₃ belong to ordinary spatial geometry and therefore do not require a new internal label in the recorded theory.

The remaining information associated with this deformation is represented by one complex amplitude,

z = Re^iφ. (108)

Because the deformation closes entirely inside V₃, it introduces no independent mixing between an observed spatial direction and the fourth parent direction. The three spatial components of the deformation do not therefore need to be promoted to three independent internal components: they already appear explicitly as the ordinary spatial dependence of the reduced field. For example,

Ψ = Ψ(x¹,x²,x³,t)

already retains the information specifying how the field varies along the three observed spatial directions.

What remains to be encoded internally is not another spatial direction, but the relational phase information associated with the parent wave that cannot be represented by a purely real spatial amplitude. As argued in Ref. [1], the lower-dimensional description requires a complex representation precisely because relative phase information remains physically relevant even when the full higher-dimensional propagation cannot be directly represented.

The minimal representation of this residual information is therefore a single complex channel z = Re^iφ, where R describes the magnitude of the recorded contribution and retains its phase relation.

Equivalently, one complex number contains two real quantities,

z = z_R + iz_I or ${\left|z\right|}^2=z_R^2+z_I^2$ .        

These two quadratures are sufficient to preserve the amplitude and phase information of one recorded channel.

No additional internal label is required because there is no geometrically distinct fourth-direction mixing channel that must simultaneously be retained. The spatial information remains spatial. The inaccessible relational information is carried by the phase of one complex amplitude.

The complex character is not introduced here specifically to obtain a gauge symmetry. It is inherited from the dimensional-recording argument of Ref. [1]: phase information must be retained when the higher-dimensional field is represented by the reduced description.

The invariant recorded content of this channel is |z|².

The most general continuous complex-linear transformation preserving this quantity is

z ⟶ e^iαz . (109)

These transformations form U(1).

Thus, aligned 3D deformation, or aligned 3D change in Φ needs one recorded complex channel in order to be translated to Ψ which leads to U(1).

At the level of symmetry class, this is the Abelian structure required by the U(1) sector of the Standard Model.

Tilted three-dimensional geometry

The second possibility is an active three-dimensional subspace

W₃ ⊂ V₄

that is not identical to V₃.

Since both W₃ and V₃ are three-dimensional subspaces of a four-dimensional space, a generic pair has a two-dimensional intersection:

dim(W₃ ∩ V₃) = 2 .

This is an important geometric fact. Two dimensions of the tilted sector can remain entirely inside ordinary observed space. Only one direction needs to contain genuinely new fourth-direction information.

Choose a unit vector

u ∈ V₃

orthogonal to the common two-dimensional intersection.

The non-common direction of W₃ can then be written as

c = cosθ u + sinθ n. (110)

An orthogonal normal to the tilted three-plane is

m = −sinθ u + cosθ n . (111)

The corresponding projector is

P_W = I − mm^T. (112)

The essential feature is therefore not merely that a different three-dimensional plane has been chosen. The distinguished direction c contains two geometrically different contributions:

cosθ u → retained spatial component

sinθ n → fourth − direction component

Under dimensional recording, these two contributions do not have the same status. The first is retained directly as part of the observed spatial geometry, whereas the second is encoded through the direction that ceases to appear as an additional spatial coordinate in the reduced theory.

Suppose now that subsequent physical interactions can distinguish changes in the two contributions appearing in Eq. (110). Maximum information preservation then forbids the recording map from identifying them as one undifferentiated real quantity.

The minimal reduced representation must retain two complex amplitudes,

$${\mathsf{\chi }}_W=\left(\begin{array}{c}{\psi }_{\parallel }\\ {\psi }_{\sigma }\end{array}\right)\in C^2$$

(113)

Here ψ_∥ encodes the contribution associated with the retained spatial component, while ψ_σ encodes the contribution associated with the fourth-direction component of the same tilted parent deformation.

The total recorded quadratic content is

$$\left|\right|{\mathsf{\chi }}_W{\left|\right|}^2={\left|{\mathsf{\psi }}_{\parallel }\right|}^2+{\left|{\mathsf{\psi }}_{\mathsf{\sigma }}\right|}^2$$

The most general complex-linear transformation preserving this quantity belongs to U(2).

Thus,

χ_W ⟶ U_Wχ_W,  U_W ∈ U(2). (114)

But

$$U\left(2\right)\simeq {\displaystyle \frac{U\left(1\right)\times \mathit{SU}\left(2\right)}{Z_2}}$$

The U(1) component corresponds to a common multiplication of both amplitudes,

χ_W ⟶ e^iγχ_W

Such a common phase is already part of the phase freedom associated with the complex recording.

The genuinely new transformations produced by the tilted geometry are therefore those that change the relative composition of the two channels while preserving their total norm. These transformations belong to SU(2).

This is substantially different from merely observing that the weak interaction happens to use doublets. The doublet arises here because a tilted three-dimensional parent geometry has precisely one direction whose information separates into a directly retained part and a fourth-direction part under recording.

Furthermore, the tilted geometry possesses another property absent from the aligned sector: it may carry an orientation relative to the direction n that becomes associated with reduced temporal evolution.

Let (a,b,c,d) be an oriented orthonormal frame of the parent spatial geometry, with

W₃ = span{a,b,c}

and d its oriented normal.

One may select an orientation through

n ⋅ d > 0 . (115)

Changing d to − d reverses this orientation even if the underlying unoriented plane is unchanged.

Consequently, the tilted geometry naturally carries a binary geometrical distinction,

This provides a possible route by which the SU(2) -type sector could acquire the chiral character of the weak interaction. It is not yet sufficient to derive the complete SU(2)_L representation content of the Standard Model, and that distinction should remain explicit. The present result is that the tilted geometry contains both the complex-doublet structure and a natural orientation variable from which handedness can potentially emerge.

The choice of the positive orientation in Eq. (115) may also be given a physical interpretation through the arrow of time. In the parent geometry, d and − d describe the two possible orientations normal to the same unoriented three-dimensional subspace. Geometrically, neither direction is intrinsically preferred. Once one of these directions is interpreted as the +1 evolution direction of the corresponding reduced 3+1-dimensional sector, however, an orientation must be selected to distinguish forward from backward evolution.

In our observed 3+1-dimensional description, the vector n defines the parent direction associated with the positive direction of recorded time. The condition

n ⋅ d > 0

therefore selects the orientation of the tilted sector whose own +1 evolution direction possesses a positive projection onto our forward-time direction.

This can be expressed geometrically as

d = d_∥ + d_⊥

With d_∥ = (n ⋅ d)n .

Thus, n ⋅ d > 0 means that the tilted sector and our observed sector agree on the orientation of the component they share with the effective temporal direction.

A possible physical motivation for selecting this branch comes from the thermodynamic arrow of time. Macroscopically, forward temporal evolution is operationally distinguished from its reverse by the direction in which entropy increases [11] for an effectively isolated system

$${\displaystyle \frac{\mathit{dS}}{\mathit{dt}}}\ge 0$$

Within the present framework, one may therefore associate the orientation n with the direction of evolution conventionally labelled as future-directed, while choosing d so that its projection onto this direction is also future-directed.

The opposite choice, would correspond geometrically to a tilted sector whose effective temporal component is oppositely oriented relative to ours.

This makes the appearance of handedness particularly interesting. Once a temporal orientation has been physically selected, the two geometrical orientations d and − d are no longer equivalent relative to the recorded arrow of time. The tilted sector can therefore carry an oriented handedness tied to its relation with the forward evolution direction, providing a possible geometric ingredient for the chiral distinction required by the weak sector.

Full four-dimensional changes

We now consider changes whose active geometry cannot be restricted to any three-dimensional parent subspace.

Then the relevant deformation requires

V₄ = span{e₁,e₂,e₃,n}.

The most transparent way to determine which new channels this introduces is to examine infinitesimal rotations of the four-dimensional parent spatial geometry.

There are six elementary rotation planes,

J₁₂, J₂₃, J₃₁, J_1σ, J_2σ, J_3σ

where σ denotes the fourth parent spatial direction n.

The first three,

J₁₂,  J₂₃,  J₃₁

operate completely inside ordinary three-dimensional space.

They are therefore recognizable in the reduced theory as ordinary spatial rotations and do not require an additional internal label.

The remaining three generators are fundamentally different:

J_1σ,  J_2σ,  J_3σ

Each mixes one ordinary spatial direction with the fourth parent direction.

At the algebraic level, relative to the subgroup of ordinary spatial rotations,

SO(4) = S0(3)⊕𝔪

where the complementary space is

𝔪 = span{J_1σ,J_2σ,J_3σ} .

Therefore dim 𝔪 = 3.

This is the central geometric result for the full four-dimensional sector.

The complete parent geometry does not simply give “one extra dimension.” Relative to the observed three-dimensional spatial frame, it creates three independent ways of mixing that extra direction with ordinary space.

The three channels

J_1σ,  J_2σ,  J_3σ

are indistinguishable from ordinary spatial rotations only before dimensional reduction.

After reduction, the σ direction no longer appears as an ordinary spatial coordinate. Information carried by these mixed transformations therefore cannot be represented simply as rotations of the observed spatial coordinate system.

If the recording map is required to preserve the amplitude and relative phase information associated with all three channels, the minimal faithful internal representation is

$${\mathsf{\chi }}_S=\left(\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right)\in C^3$$

The three amplitudes correspond schematically to the recorded information carried by J_1σ, J_2σ and J_3σ .

Their invariant quadratic content is

$$|{\mathsf{\chi }}_S{|}^2={\left|q_1\right|}^2+{\left|q_2\right|}^2+{\left|q_3\right|}^2$$

(116)

The most general complex-linear transformation preserving this quantity is

χ_S ⟶ U_Sχ_S,  U_S ∈ U(3) .

Using

$$U\left(3\right)\simeq {\displaystyle \frac{U\left(1\right)\times \mathit{SU}\left(3\right)}{Z_3}}$$

the overall common phase is again separated from the transformations that mix the three channels relative to one another.

The genuinely new internal mixing symmetry is therefore SU(3) .

Based on the above, the eight generators then follow naturally.

Once a complex triplet has been obtained, the number of internal SU(3) generators is no longer an independent input.

A general infinitesimal transformation may be written as

χ_S ⟶ (I + iϵH)χ_S . (117)

with

H^† = H

A Hermitian 3X3 matrix contains 3^2 = 9 real parameters.

One corresponds to the identity and therefore generates the common U(1) phase.

The remaining generators are traceless,

Tr(T_a) = 0

leaving 3^2 − 1 = 8 .

Therefore, dim SU(3) = 8

The eight-generator structure associated with the strong interaction therefore follows algebraically once the three-component complex internal representation has been obtained.

Likewise,

dimSU(2) = 2² − 1 = 3

Thus, the familiar 1, 3, and 8 dimensions of the Abelian, weak-like, and strong-like gauge connections emerge from complex representations of dimensions 1, 2, and 3.

From internal symmetry to a local connection

Up to this point, U(1), SU(2), and SU(3) have appeared as transformations of the internal representations at a fixed point.

A gauge theory requires something stronger. The internal basis may be chosen differently at different spacetime positions without changing the physical content.

Let χ(x) denote any of the internal reduced representations obtained above.

Suppose that at each point we choose a local internal basis related by

χ(x) ⟶ χ^'(x) = U(x)χ(x)

The important point is that U(x) need not represent a physical change of the parent field. It may instead represent a change in the local basis used to encode the same parent information.

An ordinary derivative is then insufficient:

∂_μχ^' = (∂_μU)χ + U∂_μχ .

The additional term (∂_μU)χ means that states expressed in different local internal frames cannot be compared by ordinary differentiation alone.

A connection is therefore required.

Define

D_μ = ∂_μ − ig𝒜_μ . (118)

We require

$$D_{\mathsf{\mu }}^{\text{'}}{\mathsf{\chi }}^{\text{'}}=U{D}_{\mathsf{\mu }}\chi$$

(119)

This requires

$${\mathcal{A}}_{\mathsf{\mu }}^{\text{'}}=U{\mathcal{A}}_{\mathsf{\mu }}{U}^{-1}-{\displaystyle \frac{i}{g}}\left({\partial }_{\mathsf{\mu }}U\right)U^{-1}.$$

(120)

This is precisely the transformation law of a gauge connection [12].

The physical interpretation within the present framework is particularly natural. 𝒜_μ specifies how the internal basis required to encode the parent-field information at one spacetime point must be compared with the corresponding basis at a neighboring point.

The connection is therefore not initially introduced as a force.

It is introduced as a rule of comparison between locally defined representations of the same geometrically generated internal information.

This is the standard geometric meaning of a gauge connection, but here the internal spaces on which it acts have first been related to the geometry of dimensional recording.

The significance of the connection becomes clearer when comparing the internal state at separated points.

Let a curve γ connect x_i to x_f

Parallel transport along that curve is represented by

$${\mathcal{U}}_{\mathsf{\gamma }}=\mathcal{P}\exp \left(\mathrm{ig}{\int }_{\mathsf{\gamma }}\mathrm{c}{\mathrm{A}}_{\mathsf{\mu }}\hspace{0.17em}\mathrm{d}{\mathrm{x}}^{\mathsf{\mu }}\right)$$

(121)

where $𝒫$ denotes path ordering for a non-Abelian connection.

The transported state is

χ(x_f) = $𝒰$_γχ(x_i). (122)

Thus, the connection determines how relative phases and internal orientations accumulate during transport.

For a closed curve C,

$𝒰$_C = $𝒫$exp(ig∮_CA_μdx^μ) (123)

is the holonomy of the connection.

For a non-Abelian group, the gauge-invariant Wilson-loop quantity is conventionally obtained from its trace [13],

W(C) = Tr$𝒫$exp(ig∮_CA_μdx^μ) . (124)

Wilson lines and loops are precisely the gauge-theoretic objects describing parallel transport and holonomy of a connection. In the non-Abelian case path ordering is required because the matrices at different points need not commute.

This observation is particularly relevant to the present construction because phase relations were already identified in Ref. [1] as information that dimensional recording must preserve.

A gauge connection provides the natural local geometric object for preserving and comparing exactly such information across spacetime.

The Aharonov–Bohm effect as a physical example of connection information

The physical significance of this distinction between a connection and a local force is demonstrated particularly clearly by the Aharonov–Bohm effect.

For an Abelian U(1) connection, path ordering is unnecessary and parallel transport gives a phase factor

$${\mathcal{U}}_{\mathsf{\gamma }}=\exp \left(i{\displaystyle \frac{q}{\mathrm{\hslash }}}{\int }_{\mathsf{\gamma }}{A}_{\mathsf{\mu }}d{x}^{\mathsf{\mu }}\right).$$

(125)

For two paths forming a closed loop C, their relative phase is

$$\Delta \mathsf{\phi }={\displaystyle \frac{q}{\mathrm{\hslash }}}{\oint }_C{A}_{\mathsf{\mu }}d{x}^{\mu }$$

(126)

In the magnetic Aharonov–Bohm configuration,

$$\Delta {\mathsf{\phi }}_{\mathrm{AB}}={\displaystyle \frac{q}{\mathrm{\hslash }}}{\oint }_C\mathrm{b}A\cdot \mathit{dl}={\displaystyle \frac{q}{\mathrm{\hslash }}}{\mathsf{\Phi }}_B$$

where Φ_B is the enclosed magnetic flux.

The striking feature is that the charged particle can propagate through regions in which the local electromagnetic field acting on it vanishes, while its interference phase still depends on the enclosed flux. This was the central observation of Aharonov and Bohm’s original 1959 analysis [14]. Electromagnetic potentials can affect quantum interference even in regions where the local fields and forces vanish.

The Aharonov–Bohm effect therefore provides a particularly useful physical analogy for the present framework.

It shows that physically relevant information need not reside only in a local force or local field-strength value. It may instead reside in the holonomy of a connection, that is, in how phase information accumulates when a state is transported through a region.

Within the present framework, this is closely related to the role already assigned to the complex representation. The reduced state must preserve relational phase information inherited from the parent field even when the full higher-dimensional configuration is inaccessible.

The Aharonov–Bohm effect should not be interpreted as experimental evidence for the higher-dimensional origin proposed here. Its significance is more specific: it demonstrates that a connection and its accumulated phase can possess experimentally observable content even when the corresponding local field strength vanishes along the particle trajectories.

That is precisely the kind of role a recording-induced connection would need to play.

The same idea generalizes naturally to the SU(2) and SU(3) sectors.

For U(1), the connection is a scalar phase connection A_μ.

For SU(2), it becomes a matrix acting on the complex doublet

${\mathcal{A}}_{\mathsf{\mu }}=A_{\mathsf{\mu }}^a{T}_a,\hspace{1em}\hspace{1em}a=1,2,3$For SU(3),

$${\mathcal{A}}_{\mathsf{\mu }}=A_{\mathsf{\mu }}^a{T}_a,\hspace{1em}\hspace{1em}a=1,\dots ,8$$

The essential conceptual role is identical in all three cases: the connection tells us how to compare the internal representation at neighboring spacetime points.

The difference is that in the non-Abelian cases, transporting the state along different sequences of infinitesimal changes need not commute.

Thus, ${\mathcal{U}}_{{\mathsf{\gamma }}_1}{\mathcal{U}}_{{\gamma }_2}\ne {\mathcal{U}}_{{\mathsf{\gamma }}_2}{\mathcal{U}}_{{\gamma }_1}$ in general.

The Abelian Aharonov–Bohm phase is therefore the simplest member of a broader geometric structure: physical information may reside in the holonomy accumulated through parallel transport.

For the SU(2) doublet and SU(3) triplet obtained above, the corresponding non-Abelian holonomies are matrix-valued and are represented by Wilson lines and loops.

This considerably strengthens the connection between dimensional recording and gauge structure. The internal representation determines what must be transported, while the gauge connection determines how that information is compared between different spacetime points.

The complete geometric hierarchy and relation to the Standard Model

The preceding arguments can now be assembled into a single construction.

For an aligned three-dimensional deformation, no fourth-direction mixing channel is introduced. The reduced information is represented by one complex amplitude, whose norm preserving internal transformation is U(1)

For a tilted three-dimensional deformation, one direction contains both retained spatial and fourth-direction information. Preserving both components requires ℂ² and the genuinely new relative mixing symmetry is SU(2).

For a full four-dimensional deformation, the three independent mixed rotation channels

J_1σ,  J_2σ,  J_3σ

must all be preserved. This requires ℂ³ whose nontrivial relative mixing symmetry is SU(3).

The resulting symmetry sequence is U(1), SU(2) and SU(3) which reproduces the three internal symmetry classes entering the Standard Model gauge group.

A distinction remains necessary, however. The Abelian factor of the fundamental Standard Model is U(1)_Y , while observed electromagnetism corresponds to the surviving U(1)_EM after electroweak symmetry breaking [15–17]. Likewise, identifying the tilted sector specifically with SU(1)_L requires a complete derivation of its chiral action.

The present result is therefore more limited but also more precise: the framework generates the required Abelian singlet, complex doublet, and complex triplet symmetry structures geometrically, while the detailed Standard Model assignments remain to be established.

7. Discussion and Conclusions

The present work extends the framework introduced in Ref. [1] from the description of a single reduced system to several structures that are central to quantum mechanics but were not addressed in the original construction. The starting point remains deliberately limited: time is treated relationally as the +1 evolution parameter associated with the dimensional configuration space of a physical phenomenon, and a 3+1-dimensional description of a 4+1-dimensional parent field is obtained through physical interactions that record only the information accessible to lower-dimensional systems. The reduced state is therefore not assumed to be a fundamental quantum object, but an effective representation of the distinguishable information contained in those interactions.

A first purpose of the present work was to sharpen this dimensional reduction procedure. Maximum information preservation was reformulated in terms of equivalence classes of parent configurations: two parent states may be represented identically whenever no admissible 3+1-dimensional interaction can distinguish them, whereas distinctions capable of affecting interference, spectral structure, localization, or subsequent interaction outcomes must remain encoded. This avoids requiring an impossible global inversion of the dimensional reduction while giving the recording map a precise physical role.

The same construction also provides a natural interpretation of measurement-induced state updating. A physical interaction may change the parent configuration, Φ ⟶ Φ^', and consequently the reduced description,

Ψ = $ℳ$[Φ] ⟶ Ψ^' = $ℳ$[Φ^']

The parent dynamics need not themselves be discontinuous. Rather, once an interaction establishes a physically distinguishable record or modifies the coherence relations relevant to the recording, the pre-interaction reduced state no longer represents the post-interaction parent configuration. The resulting state update may therefore appear effectively discontinuous or non-unitary in the reduced description even when the underlying dynamics remain continuous. This provides a physical mechanism for distinguishing coherent evolution from measurement-like updating, although the present analysis should not yet be regarded as a complete microscopic derivation of single-outcome selection.

The second major result concerns composite systems. Starting from independently preparable parent sectors, preservation of the linear structure already obtained for each subsystem requires the joint composition map to be linear in each argument. Bilinearity leads naturally to the algebraic tensor product, while preservation of the factorized inner product and completion gives the Hilbert tensor-product structure, provided that the accessible composite state space contains no additional independent degrees of freedom beyond those generated by product preparations and their linear combinations. Interactions can then produce coefficients that no longer factorize into independent subsystem amplitudes. Entanglement consequently appears not as an additional postulate, but as the reduced representation of physically accessible joint information that cannot be decomposed into two independent subsystem descriptions.

This logical order is significant. Neither tensor-product composition nor entanglement requires a prior probabilistic interpretation of the Hilbert-space norm. The probability question can therefore be addressed independently.

The Born rule was obtained by asking which probability assignment is compatible with the reduced Hilbert-space geometry. Given orthogonal distinguishable record sectors, phase and internal-unitary invariance, noncontextual consistency of the probability assigned to a fixed record sector, continuity, normalization, and additivity under coarse-graining of mutually exclusive alternatives, the probability associated with an outcome projector P_i is forced to take the quadratic form

p_i = |P_iΨ|² = ⟨Ψ|P_i|Ψ⟩

The interpretation of the Hilbert norm as probability is therefore not used to construct the preceding state-space structure. Instead, the quadratic rule follows only after that geometry is already available. The comparison with the robustness construction of De Raedt, Katsnelson and Michielsen provides an independent and complementary perspective: reproducibility and stability of physical correlations can lead to the same quantum probability structure without taking the Born rule as an initial postulate.

The composite-system and probability results then permit Bell correlations to be addressed without introducing a singlet state, spin-1/2 particles, or Pauli operators as independent ingredients. A generic interaction-generated pure nonfactorizable state possesses a Schmidt decomposition. Its minimal nontrivial correlated sector is two-dimensional, and suitable dichotomic operators can be constructed directly from the corresponding local Schmidt alternatives. For the rank-two state considered explicitly in this work, the maximal CHSH value becomes

$$S_{\mathit{max}}=2\sqrt{1+{\sin }^2\left(2\mathsf{\eta }\right)}$$

which satisfies S_max > 2 for every nonzero pure entanglement.

The significance of this result is not that a higher dimension by itself circumvents Bell's theorem. It does not. Bell's theorem constrains the causal and probabilistic factorization properties of an underlying theory, irrespective of the number of dimensions in which its variables are defined. Rather, the present framework produces a globally nonfactorizable reduced state from interaction-generated joint information while retaining local tensor-factor operations and operational no-signalling.

The final part of this work explored whether the same dimensional logic may extend beyond the quantum-mechanical state-space structure toward internal symmetries. Changes confined to the observed three-dimensional geometry require only a single complex internal channel, a tilted three-dimensional geometry requires the simultaneous preservation of directly retained and fourth-direction information and therefore gives a complex doublet, full four-dimensional deformations introduce three independent mixed channels J_1σ , J_2σ, and J_3σ naturally motivating a complex triplet. Their norm-preserving transformations lead respectively to U(1), SU(2), and SU(3) type structures after separation of the common phase.

This part of the construction should presently be interpreted more cautiously than the quantum-mechanical results. It provides a possible geometric origin for the dimensions and internal transformation structures appearing in the Standard Model, but it does not yet constitute a derivation of the complete Standard Model gauge theory.

The above, define the broader significance of the present work. Existing reconstruction programs have demonstrated that large parts of quantum theory can be obtained from comparatively compact operational or informational principles [3,4]. The present approach asks a different question: whether a physical mechanism can explain why such mathematical principles should arise in the first place. Here that mechanism is based primarily on one unconventional assumption, the relational dimensional role assigned to time, together with the requirement that a lower-dimensional physical description retain all information capable of affecting observable interactions.

The repeated appearance of quantum structure from this same mechanism is therefore the central result of the program. Complex amplitudes, Hilbert geometry, canonical operators and unitary evolution were obtained in Ref. [1]. Tensor-product composition, entanglement, Born probabilities and Bell-compatible correlations are extended in the present work. None of these results individually proves that the proposed treatment of time is fundamental. However, their mutual consistency is difficult to regard as irrelevant. A framework that alters principally the dimensional role of time and repeatedly reproduces otherwise distinct elements of quantum theory is more constrained than one constructed separately to reproduce any single feature.

Such convergence has physical content. If several independent mathematical structures of quantum mechanics emerge from the same underlying alteration rather than being postulated individually, then the alteration itself becomes a candidate explanation for why the formalism has the form that it does.

The main conclusion of this work is therefore not that quantum mechanics or the Standard Model have been fully derived. It is more specific: the relational treatment of time introduced in Ref. [1] continues to generate mutually compatible elements of quantum theory when the framework is extended to composite systems, interactions, probabilities and Bell correlations, without requiring those structures to be inserted independently. This strengthens the possibility that the quantum formalism may be an emergent lower-dimensional representation of a deeper dynamics rather than the fundamental mathematical starting point. Determining whether this possibility can be developed into a complete theory and whether it ultimately yields experimentally distinguishable predictions constitutes the next stage of the program.