A Transactional Model with a Unified Attractor: Inverse Entropy Product, Horizon-Integrated Dynamics, and a Categorical Framework for Space-Time, Matter, Biology, Evolution, and Consciousness Phil Callis Contact: zysilveria@gmail.com March 27, 2025 Abstract This study reformulates the Transactional Interpretation (TI) of quantum me- chanics by replacing its conventional time-symmetric field with a singular trans- action attractor, defined by the product of two relative entropies: one between local fields and non-local states, denoted D KL ( P F ( x ) || Q S ( x ′ )), and another inte- grating local states across the observable horizon against non-local fields, denoted D total KL ( P S || Q F ), constrained to equal unity. These components, often perceived as distinct field and state attractors, are not independent but dual manifestations of a unified dynamical principle that balances relativistic invariance and inertial stability through their reciprocal relationship. Local fields, acting as linear maps on the vector space of quantum states, generate offer waves, while states acting on fields produce confirmation waves, and transactions—modeled categorically as morphisms where the entropy product achieves unity—form the foundational events yielding emergent space-time and matter. The framework extends comprehensively to biological systems, where semi-local transaction systems, such as organisms, localize physical laws within soft space-time boundaries due to the low relative entropy of their internal transactions with the non-local transactions of the uni- verse, particularly those in their immediate environment. It is hypothesized that the transaction attractor preferentially stabilizes states where transactions within and outside these boundaries exhibit an inverse relationship, a proposition sub- jected to detailed mathematical scrutiny herein. Further implications include an evolutionary mechanism wherein the attractor biases mutations toward adaptive configurations, and a speculative panpsychic/idealist interpretation positing that localized transaction systems reflect the dissociation of universal consciousness into individual agents, with the offer-confirmation duality corresponding to subjective- objective distinctions in awareness. The introduction of a categorical structure formalizes transactions as morphisms, enhancing the mathematical coherence and depth of the model across these diverse domains. 1 1 Introduction The Transactional Interpretation (TI) of quantum mechanics, as initially formulated by Cramer [1] and subsequently elaborated by Kastner [2], offers a distinctive perspective on quantum phenomena by redefining quantum events as transactions rather than unilateral state reductions contingent upon measurement. Within TI, an emitter generates an offer wave, denoted ψ ( x ), which propagates forward in time from the point of emission, en- capsulating the quantum amplitude of all possible outcomes associated with the emission process. Concurrently, an absorber generates a confirmation wave, denoted ψ ∗ ( x ), the complex conjugate of the offer wave, which propagates backward in time from the point of absorption, effectively selecting one of these possible outcomes. The transaction is real- ized at specific space-time coordinates where the product ψ ( x ) ψ ∗ ( x ) achieves a significant magnitude, marking the occurrence of a physical event without the necessity of invoking an observer-dependent collapse of the wavefunction. Kastner’s significant contribution to this framework is the proposition that space-time does not exist as a pre-defined manifold but emerges as a relational construct from the ensemble of these transactions, a paradigm shift that reorients the ontological foundations of quantum mechanics away from a static background toward a dynamically emergent structure [2]. In its original formulation, TI relies on a time-symmetric field to mediate the coher- ence between the offer wave ψ ( x ) and the confirmation wave ψ ∗ ( x ), ensuring that these temporally opposing processes align to produce a consistent physical outcome. This time- symmetric field serves as a critical component, facilitating the handshake between past and future that underpins the transactional process. However, despite its conceptual util- ity, the time-symmetric field remains an abstract construct within TI, lacking a detailed dynamical basis or a clear mechanistic explanation of how it operates to enforce temporal symmetry. Its role is primarily that of a scaffold, holding the framework together without specifying the underlying principles that drive its function, which prompts the need for a more robust and concrete reformulation. The present study addresses this limitation by introducing a transaction attractor, a unified dynamical principle designed to supplant the time-symmetric field with a more mathematically grounded and physically motivated entity. The transaction attractor is characterized by the product of two relative entropies, each capturing a distinct as- pect of the field-state interaction within the quantum system. The first component, D KL ( P F ( x ) || Q S ( x ′ )), quantifies the divergence between the probability distribution of a local field at a space-time point ( x ) and the distribution of non-local quantum states observed at all other points ( x’ ), adjusted for relativistic invariance. The second compo- nent, D total KL ( P S || Q F ), integrates the divergence between the local quantum state distribu- tion across the observable horizon and the non-local field distribution over all space-time points, reflecting the inertial properties of the system. Crucially, these two entropies are not independent optimization targets; rather, their product is constrained to equal one, establishing a reciprocal relationship wherein an increase in one entropy corresponds to a decrease in the other, maintaining a stable balance: D KL ( P F ( x ) || Q S ( x ′ )) · D total KL ( P S || Q F ) = 1 This product condition reveals that what might be perceived as separate field and state attractors are, in fact, dual aspects of a single transaction attractor, a unified entity that governs the dynamics of the system through this inverse interplay. The operational mechanics of this reformulated model are delineated as follows: lo- 2 cal fields, which can be regarded as linear maps acting on the vector space of quantum states due to their superposition of potential interactions, act upon states to generate offer waves, denoted ψ ( x ), which propagate forward in time and encode the multiplicity of pos- sible outcomes. Conversely, quantum states, residing within a Hilbert space (a complex vector space equipped with an inner product), act upon fields to generate confirmation waves, denoted ψ ∗ ( x ), which propagate backward in time and affirm the realization of a specific outcome. Transactions, defined as the events where the entropy product achieves unity and the wave product ψ ( x ) ψ ∗ ( x ) peaks, constitute the foundational units from which space-time and matter emerge. To formalize this process, transactions are modeled categorically as morphisms within a category, where states are objects and fields mediate transformations, constrained by the transaction attractor’s entropy product condition. This framework extends significantly beyond the confines of fundamental quantum me- chanics into the realm of biological systems, where semi-local transaction systems—exemplified by living organisms—exhibit a remarkable localization of physical laws within soft space- time boundaries. This localization is a direct consequence of the low relative entropy between the semi-local transactions occurring within an organism’s boundary and the non-local transactions spanning the universe, with a particular emphasis on the transac- tions in the organism’s immediate environment, such as sunlight, atmospheric conditions, or nutrient fields. The transaction attractor plays a pivotal role in stabilizing these sys- tems by favoring configurations where the internal transactions within the organism’s soft boundary and the external transactions outside it exhibit an inverse relationship, a hypothesis that will be explored and mathematically evaluated in subsequent sections to assess its validity and implications. The scope of this model does not terminate with physical and biological domains. It further informs evolutionary processes by suggesting that the transaction attractor biases mutations toward adaptive configurations that minimize relative entropy with the non- local environment, thereby enhancing survival probability within a Darwinian context. Additionally, a speculative panpsychic/idealist interpretation is advanced, positing that the localized transaction systems within organisms may reflect the dissociation of a uni- versal consciousness into individual agents, with the duality of offer waves (field-driven) and confirmation waves (state-driven) potentially mirroring the subjective-objective dis- tinction inherent in self-awareness. The introduction of a categorical structure to model transactions as morphisms between states, mediated by fields, adds a layer of mathe- matical rigor and coherence to this framework, unifying its diverse applications across physics, biology, evolution, and consciousness. This paper aims to provide an exhaustive and detailed exposition of these interconnected concepts, reiterating key points where necessary to ensure clarity and depth. 2 Theoretical Framework 2.1 Overview of the Transactional Interpretation The Transactional Interpretation (TI) of quantum mechanics, as originally proposed by Cramer [1], redefines quantum events as bilateral transactions rather than the unilateral state reductions typically associated with the Copenhagen interpretation. In TI, the process begins with an emitter—such as an atom or particle—generating an offer wave, denoted ψ ( x ), which propagates forward in time from the point of emission. This offer wave encapsulates the quantum amplitude of all possible outcomes associated with the 3 emission event, representing a superposition of potential states in a manner consistent with the probabilistic nature of quantum mechanics. Simultaneously, an absorber—such as another atom or a detection apparatus—generates a confirmation wave, denoted ψ ∗ ( x ), which is the complex conjugate of the offer wave and propagates backward in time from the point of absorption. The confirmation wave effectively selects one of the possible outcomes encoded in the offer wave, establishing a retrocausal link between the future and the past. The transaction is completed at specific space-time coordinates where the product of the offer and confirmation waves, ψ ( x ) ψ ∗ ( x ), achieves a significant magnitude, marking the realization of a physical event—such as the emission and subsequent absorption of a photon—without requiring an external observer to collapse the wavefunction. This bilat- eral process eliminates the ambiguities associated with measurement in traditional quan- tum mechanics, replacing them with a deterministic yet retrocausal mechanism grounded in the temporal handshake between ψ ( x ) and ψ ∗ ( x ). Cramer’s formulation thus provides a coherent alternative to probabilistic collapse, emphasizing the relational nature of quan- tum events across time. Kastner’s extension of TI introduces a profound ontological shift by proposing that space-time itself is not a pre-existing manifold but an emergent property of the rela- tional network formed by these transactions [2]. In this view, the space-time continuum is not a static backdrop upon which quantum events unfold; rather, it is dynamically constructed from the interactions of offer and confirmation waves, with each transaction contributing to the fabric of space-time as a node in an emergent structure. The time- symmetric field serves as the mediating mechanism in traditional TI, ensuring that the forward-propagating ψ ( x ) and backward-propagating ψ ∗ ( x ) align to produce a consistent outcome. However, this field remains an abstract construct, functioning as a conceptual placeholder rather than a dynamically derived entity with a clear physical basis. Its role is to enforce temporal symmetry, but it lacks specificity regarding the mechanisms driving this symmetry, prompting the need for a reformulation that provides a more concrete and mathematically grounded foundation. To illustrate, consider a simple quantum event: an atom emits a photon at position x 1 and time t 1 , generating an offer wave ψ ( x ) that propagates forward to a detector at x 2 and t 2 . The detector responds with a confirmation wave ψ ∗ ( x ) that propagates backward to x 1 , and the transaction fixes the photon’s path where ψ ( x ) ψ ∗ ( x ) is maximized. The time-symmetric field ensures this alignment, but its abstract nature leaves open questions about the underlying dynamics, which the present reformulation seeks to address by introducing the transaction attractor. 2.2 Definition of the Transaction Attractor The transaction attractor is introduced as a singular dynamical principle to replace the time-symmetric field, offering a more robust and mechanistic basis for the TI framework. It is defined through the application of relative entropy, or Kullback-Leibler (KL) diver- gence, a well-established measure in information theory that quantifies the divergence between two probability distributions: D KL ( P || Q ) = ∫ P ( x ) log ( P ( x ) Q ( x ) ) dx This metric, which is non-negative and zero only when P ( x ) = Q ( x ), provides a quantita- tive assessment of how much one distribution (e.g., a local physical description) deviates 4 from another (e.g., a non-local reference). In the context of this model, the transaction attractor is characterized by two relative entropy components, which are not indepen- dent optimization targets but are unified through their product constraint, reflecting a reciprocal balance that governs the system’s dynamics. Field Aspect: The first component is D KL ( P F ( x ) || Q S ( x ′ )), where P F ( x ) denotes the probability distribution of a local field at a specific space-time point ( x ). For example, in the case of an electromagnetic field A μ ( x ) associated with photon emission, P F ( x ) might be derived from the field’s two-point correlation function, ⟨ A μ ( x ) A μ ( x ) ⟩ , which quantifies the likelihood of a field excitation occurring at ( x ). The target distribution, Q S ( x ′ ), represents the probability distribution of non-local quantum states | ψ ( x ′ ) ⟩ at all other space-time points ( x’ ), transformed under Lorentz invariance to account for observations in different inertial frames. This divergence measures the extent to which the local field’s behavior—such as the propagation of light at the speed ( c )—aligns with the collective state descriptions observed non-locally, thereby enforcing the principles of special relativity, particularly the invariance of light speed across all reference frames. To clarify, consider a photon emission event at ( x ): the field A μ ( x ) generates an excitation, and P F ( x ) reflects the probability of this event occurring locally. Q S ( x ′ ) aggregates the state distributions | ψ ( x ′ ) ⟩ as perceived by observers at all other points ( x’ ), adjusted for relativistic transformations (e.g., Lorentz boosts). The field aspect D KL ( P F || Q S ) thus quantifies how well the local photon propagation matches these non- local observations, ensuring that the speed of light remains constant regardless of the observer’s frame, a cornerstone of relativistic invariance. State Aspect: The second component introduces a horizon-integrated perspective, capturing the broader context of the quantum state’s interaction with non-local fields: D total KL ( P S || Q F ) = ∫ c 0 D KL ( P S ( | ψ ( x ) ⟩ ) || Q F ( x ′ )) dx Here, P S ( | ψ ( x ) ⟩ ) is the probability distribution of the quantum state | ψ ( x ) ⟩ at each point ( x ) within the observable horizon, defined as the region from x = 0 to x = c , where ( c ) denotes the light-cone boundary (e.g., ( c t ), with ( t ) representing the time elapsed from the event to the horizon’s edge). The state | ψ ( x ) ⟩ is obtained by evolving a reference state | ψ ⟩ to position ( x ), potentially via a unitary operator such as | ψ ( x ) ⟩ = e − i ˆ Hx/c | ψ ⟩ for temporal evolution, or a spatial translation operator, encompassing both particle states (e.g., an electron) and vacuum contributions (e.g., zero-point fluctuations). The distribution P S ( | ψ ( x ) ⟩ ) might correspond to the probability density |⟨ x | ψ ⟩| 2 , which indicates the likelihood of finding the system in a particular state at ( x ), or to expectation values such as momentum ⟨ ψ ( x ) | ˆ p | ψ ( x ) ⟩ , reflecting the system’s dynamical properties. The target distribution, Q F ( x ′ ), represents the probability distribution of non-local fields over all points ( x’ ), incorporating contributions from vacuum fluctuations, electro- magnetic fields, and other field configurations that influence the system across space-time. This integrated term, D total KL , quantifies the divergence between the local state’s behavior across the entire observable horizon and the non-local field environment, ensuring that the state’s properties—such as momentum—are consistently supported by the broader field context, thereby underpinning inertial stability. For example, consider an electron absorbing a photon at x = 0. The state | ψ ( x ) ⟩ evolves across the horizon (e.g., from x = 0 to x = ct ), and P S ( | ψ ( x ) ⟩ ) reflects its proba- bility or momentum distribution at each point. Q F ( x ′ ) includes the vacuum fluctuations and field effects at all ( x’ ) that support the electron’s inertial properties, such as its 5 resistance to acceleration, potentially via mechanisms akin to quantized inertia where mass emerges as m ∼ ℏ cR with ( R ) as the horizon scale [3]. The state aspect integrates these divergences to ensure that the electron’s momentum remains conserved across the horizon, reflecting the influence of non-local fields. The transaction attractor is not defined by the independent minimization of these two entropy terms to zero, as might be expected in a conventional optimization framework. Instead, it imposes a product constraint: D KL ( P F ( x ) || Q S ( x ′ )) · D total KL ( P S || Q F ) = 1 This condition establishes a reciprocal relationship between the field and state aspects: an increase in D KL ( P F || Q S ) (e.g., due to a local field deviating from relativistic expec- tations) necessitates a decrease in D total KL (e.g., by tightening the state’s alignment with non-local fields), and vice versa, maintaining their product at unity. This inverse in- terplay reveals that the apparent field and state attractors are not separate entities but dual manifestations of a single transaction attractor, a unified dynamical principle that governs the system’s behavior through this balanced equilibrium rather than through independent optimization processes. To illustrate this reciprocity, suppose D KL ( P F || Q S ) = 2, indicating a significant diver- gence in local field behavior from non-local state observations—perhaps due to a perturba- tion affecting photon propagation. The transaction attractor then requires D total KL = 0 5, indicating a correspondingly tight alignment of the state’s properties across the hori- zon with non-local fields, ensuring that the overall system remains physically consis- tent. Conversely, if D total KL = 2 due to a broad mismatch in state-field alignment, then D KL ( P F || Q S ) = 0 5, reflecting a high degree of relativistic conformity in the field’s be- havior. This compensatory mechanism underscores the transaction attractor’s role as a unifying entity, replacing the static symmetry of the time-symmetric field with a dynamic balance that adapts to local and non-local conditions. 2.3 Categorical Model of Transactions The structure of this model lends itself to a categorical formulation, wherein transactions are modeled as morphisms within a category, reflecting the vector space nature of states and the linear map-like behavior of fields in superposition. In quantum mechanics, states reside in a Hilbert space H , a complex vector space equipped with an inner product, where superpositions of states are represented as linear combinations of basis vectors (e.g., | ψ ⟩ = α | 0 ⟩ + β | 1 ⟩ ). Fields, such as φ ( x ) or A μ ( x ), act as linear operators or maps on this space, transforming one state into another (e.g., φ ( x ) | ψ ⟩ → | ψ ′ ⟩ ), and their superposition of potential transactions—encoded in the offer wave ψ ( x )—reflects the multiplicity of possible outcomes prior to confirmation. Transactions, as the realization of specific field- state interactions where ψ ( x ) ψ ∗ ( x ) peaks and the entropy product equals one, can be naturally interpreted as morphisms mapping between states, mediated by fields. The category, denoted T , is defined as follows: • Objects: Quantum states | ψ ⟩ , | ψ ′ ⟩ in the Hilbert space H , representing the possible configurations of the system at different points in the transactional process. • Morphisms: Transactions T : | ψ ⟩ → | ψ ′ ⟩ , which are specific field-mediated trans- formations producing the offer wave ψ ( x ) and confirmation wave ψ ∗ ( x ). For exam- ple, a transaction might map the vacuum state | 0 ⟩ to a photon state | 1 k ⟩ via the field action A μ ( x ), followed by absorption fixing the event. 6 • Composition: Sequential transactions are composed as T 2 ◦ T 1 : | ψ ⟩ → | ψ ′′ ⟩ , where T 1 : | ψ ⟩ → | ψ ′ ⟩ (e.g., emission) and T 2 : | ψ ′ ⟩ → | ψ ′′ ⟩ (e.g., absorption) form a complete process (e.g., photon transfer from emitter to absorber). Composition is constrained by the transaction attractor’s condition D KL ( P F || Q S ) · D total KL = 1, ensuring that each morphism aligns local and non-local dynamics. • Identity Morphisms: The trivial transaction id | ψ ⟩ : | ψ ⟩ → | ψ ⟩ , representing a state unchanged by field action, satisfies the categorical requirement of an identity morphism for each object. The transaction attractor’s product constraint serves as the composition rule within T , ensuring that morphisms (transactions) are stabilized where the entropy product equals one. This categorical structure formalizes the interplay of states and fields, with states as objects in a vector space, fields as linear maps in superposition, and transactions as the morphisms that connect them, providing a mathematically rigorous framework that unifies the physical dynamics of the model. To illustrate, consider a photon transaction: the morphism T : | 0 ⟩ → | 1 k ⟩ is in- duced by the field A μ ( x ) acting at the emitter, generating ψ ( x ), followed by absorption where | ψ ⟩ shapes ⟨ A μ ( x ) ⟩ , generating ψ ∗ ( x ). The transaction completes where ψ ( x ) ψ ∗ ( x ) peaks and D KL ( P F || Q S ) · D total KL = 1, defining a morphism in T . Sequential transactions (e.g., emission to scattering to absorption) compose as morphisms, constrained by the attractor’s entropy product, reflecting the relational structure of the system. 2.4 Dynamics and Interrelationships The dynamics of this model are driven by the interactions between fields, states, of- fer waves, confirmation waves, and transactions, now formalized within the categorical framework T : Fields and Offer Waves: A local field φ ( x ), such as an electromagnetic potential A μ ( x ), acts as a linear map on a quantum state | ψ ⟩ in H —for example, transforming the vacuum state | 0 ⟩ into a photon state | 1 k ⟩ via A μ ( x ) | 0 ⟩ = | 1 k ⟩ . This action generates the offer wave ψ ( x ) = ⟨ x | ψ ′ ⟩ , which propagates forward in time, encoding the superposition of potential outcomes (e.g., all possible photon paths). The field aspect of the transaction attractor, D KL ( P F || Q S ), governs this process by ensuring that the propagation adheres to relativistic principles, maintaining the speed of light ( c ) invariant across all inertial frames as required by special relativity. The linear map nature of the field reflects its superposition of potential transactions—e.g., A μ ( x ) could produce various photon states depending on the context, encoded in ψ ( x ). The field aspect constrains this superposi- tion to align with non-local state observations, ensuring that the local emission event is consistent with relativistic expectations. States and Confirmation Waves: Conversely, a quantum state | ψ ⟩ in H , such as an electron prepared to absorb a photon, acts upon the field φ ( x ), influencing its expectation value ⟨ φ ( x ) ⟩ or collapsing its superposition of possibilities (e.g., fixing the photon’s absorption). This interaction generates the confirmation wave ψ ∗ ( x ), which propagates backward in time to affirm the transaction. The state aspect, D total KL , ensures that this process maintains inertial stability by aligning the state’s properties—such as momentum ⟨ ψ | ˆ p | ψ ⟩ —across the observable horizon with the non-local field environment, including vacuum fluctuations that support inertial mass (e.g., via m ∼ ℏ cR [3]). As a vector space element, the state | ψ ⟩ evolves across the horizon (e.g., | ψ ( x ) ⟩ = e − i ˆ Hx/c | ψ ⟩ ), 7 and its distribution P S ( | ψ ( x ) ⟩ ) reflects its superposition properties, which the state aspect integrates to ensure consistency with non-local fields. Transaction Attractor and Transactions as Morphisms: The transaction at- tractor orchestrates these interactions by stabilizing states where the entropy prod- uct D KL ( P F || Q S ) · D total KL = 1. Within the category T , transactions are morphisms T : | ψ ⟩ → | ψ ′ ⟩ , occurring where ψ ( x ) ψ ∗ ( x ) achieves a maximum and the entropy prod- uct constraint is satisfied. Each morphism represents a completed transaction—e.g., a photon emitted at x 1 and absorbed at x 2 —fixing a space-time event. The categorical composition of morphisms (e.g., T 2 ◦ T 1 ) reflects sequential transactions, such as emission followed by absorption, constrained by the attractor’s entropy product to ensure physical consistency. To reiterate, the transaction attractor’s role is to unify the field and state aspects, ensuring that each morphism in T balances local and non-local dynamics. For instance, a high D KL ( P F || Q S ) (poor field alignment) requires a low D total KL (tight state- field alignment), maintaining the product at one and stabilizing the transaction as a valid morphism. Emergence of Space-Time and Matter: Space-time emerges as the relational network of morphisms in T , with the field aspect contributing the relativistic struc- ture (e.g., Minkowski metric ds 2 = − c 2 dt 2 + dx 2 + dy 2 + dz 2 ) and the state aspect em- bedding inertial properties through non-local field interactions. Matter—e.g., electrons, photons—manifests as entities exhibiting dual roles within T : acting as offerers (field- driven morphisms) or confirmers (state-driven morphisms), stabilized by the transaction attractor’s equilibrium. For example, an electron may act as an object | ψ ⟩ in T , with a morphism T : | ψ ⟩ → | ψ ′ ⟩ representing photon emission (field-driven), followed by another morphism T ′ : | ψ ′ ⟩ → | ψ ′′ ⟩ representing absorption (state-driven). The categorical struc- ture ensures these morphisms compose consistently, reflecting the emergent properties of space-time and matter. Reiteration of Key Points: The transaction attractor unifies the field and state aspects through their product constraint, replacing the time-symmetric field with a dy- namic balance. The categorical model T formalizes transactions as morphisms, with states as objects and fields as linear maps, constrained by the entropy product. This structure ensures that each transaction aligns local field actions (offer waves) and state responses (confirmation waves) with non-local dynamics, producing a coherent physical reality. 2.5 Mathematical Consistency and Physical Interpretation The product constraint D KL ( P F || Q S ) · D total KL = 1 ensures mathematical consistency by enforcing a reciprocal balance between the field and state aspects. Physically, this re- flects a compensatory mechanism: a deviation in field behavior (e.g., D KL ( P F || Q S ) = 3) requires a proportional adjustment in state alignment (e.g., D total KL = 1 / 3), maintaining overall system stability. Within T , this constraint defines the validity of morphisms, en- suring that transactions as mappings between states are physically meaningful, unifying the model’s dynamics across quantum and emergent scales. 3 Model Formulation The formulation of the transaction attractor model provides the mathematical and phys- ical underpinnings of the theoretical framework established in Section 2, detailing how 8 fields, states, and transactions interact within a categorical structure to produce emergent space-time and matter. This section is divided into three subsections—field aspect, state aspect, and the transaction attractor with its categorical integration—each offering an exhaustive explanation of the components, their dynamics, and their interrelationships. To ensure clarity and depth, key definitions are reiterated, derivations are expanded with illustrative examples, and the implications of each aspect are thoroughly explored, reinforcing the model’s coherence and applicability across physical scales. 3.1 Field Aspect: Relativistic Invariance The field aspect of the transaction attractor is mathematically defined by the relative entropy, or Kullback-Leibler (KL) divergence, D KL ( P F ( x ) || Q S ( x ′ )), which quantifies the divergence between the probability distribution of a local field at a specific space-time point ( x ), denoted P F ( x ), and the probability distribution of non-local quantum states observed at all other points ( x’ ), denoted Q S ( x ′ ). This divergence is expressed as: D KL ( P F ( x ) || Q S ( x ′ )) = ∫ P F ( x ) log ( P F ( x ) Q S ( x ′ ) ) dx ′ where P F ( x ) represents the statistical likelihood of a field excitation occurring at ( x ). For instance, in the context of an electromagnetic field A μ ( x ), P F ( x ) might be derived from the field’s two-point correlation function, ⟨ A μ ( x ) A μ ( x ) ⟩ , which measures the prob- ability of a photon emission event at ( x ) based on the field’s intensity or expectation value at that point. This distribution encapsulates the field’s behavior as a linear map acting on the Hilbert space H of quantum states, reflecting its superposition of potential transactions—e.g., the emission of a photon in various possible directions or polarizations. The target distribution, Q S ( x ′ ), is the probability distribution of non-local quantum states | ψ ( x ′ ) ⟩ at all other space-time points ( x’ ), adjusted via Lorentz transformations to account for relativistic invariance across inertial reference frames. These states | ψ ( x ′ ) ⟩ are elements of H , and Q S ( x ′ ) aggregates their statistical properties as observed non- locally—e.g., the probability density |⟨ x ′ | ψ ⟩| 2 of finding a system in state | ψ ⟩ at ( x’ ), transformed to ensure consistency with observers moving at different velocities. The field aspect D KL ( P F || Q S ) thus serves as a metric of how well the local field’s behavior aligns with these non-local state observations, enforcing the principles of special relativity, particularly the invariance of the speed of light ( c ) across all frames. Operationally, the field aspect governs the generation of offer waves within the trans- actional process. A local field φ ( x ) acts as a linear map on a quantum state | ψ ⟩ in H , transforming it into a new state | ψ ′ ⟩ . For example, consider an electromagnetic field A μ ( x ) acting on the vacuum state | 0 ⟩ to produce a single-photon state | 1 k ⟩ with wavevector k : A μ ( x ) | 0 ⟩ = | 1 k ⟩ This action generates the offer wave ψ ( x ) = ⟨ x | ψ ′ ⟩ , which propagates forward in time from the emission point, encoding the superposition of all possible outcomes of the field’s interaction—e.g., the photon traveling along various paths or with different polarizations. In quantum field theory, A μ ( x ) can be expressed as a sum of creation and annihilation operators: A μ ( x ) = ∫ d 3 k (2 π ) 3 1 √ 2 ω k [ a k e − ikx + a † k e ikx ] 9 where a † k creates a photon in mode k , and the resulting ψ ( x ) reflects this superposition over all possible k The field aspect ensures that this propagation satisfies relativistic constraints, requiring ψ ( x ) to obey the massless wave equation: □ ψ ( x ) = ∂ μ ∂ μ ψ ( x ) = 0 where □ = ∂ 2 t − ∇ 2 (in natural units, c = 1) is the d’Alembertian operator. This equation guarantees that the offer wave propagates at the speed of light ( c ), ensuring that the local field’s action—e.g., photon emission at x 1 and time t 1 —is consistent with relativistic invariance, appearing identical to observers in all inertial frames as dictated by the Minkowski metric: ds 2 = − c 2 dt 2 + dx 2 + dy 2 + dz 2 = 0 for light-like intervals. However, the transaction attractor does not impose a rigid requirement that D KL ( P F || Q S ) minimize to zero independently, as might be expected in a conventional optimization sce- nario where perfect alignment is the sole objective. Instead, its value is dynamically balanced within the broader constraint of the transaction attractor: D KL ( P F ( x ) || Q S ( x ′ )) · D total KL ( P S || Q F ) = 1 This product condition allows flexibility in the field aspect’s entropy. For example, con- sider a photon emission event perturbed by a local gravitational field or medium, causing a slight deviation from ideal relativistic propagation—e.g., a redshift or refraction ef- fect. This perturbation might increase D KL ( P F || Q S ) to a value such as 2, indicating a greater divergence from non-local state observations due to the altered propagation char- acteristics. The transaction attractor permits this increase, provided the state aspect compensates by reducing its entropy to D total KL = 0 5, ensuring that the overall product remains unity and the system retains physical consistency. This compensatory mecha- nism underscores the field aspect’s role within the transaction attractor, balancing local relativistic deviations with non-local state adjustments. To further illustrate, consider a practical scenario: an atom at rest emits a photon in a vacuum, generating ψ ( x ) that spreads spherically at speed ( c ). Here, D KL ( P F || Q S ) might be low (e.g., 0.1), reflecting tight alignment with non-local observations, requiring D total KL = 10 to maintain the product at 1. If the same atom emits in a dense medium, slowing the photon slightly, D KL ( P F || Q S ) increases (e.g., to 3), and D total KL adjusts to 1/3. The field aspect thus ensures relativistic invariance as a dynamic constraint, not a fixed absolute, reiterating its integration within the transaction attractor’s unified framework. 3.2 State Aspect: Inertial Stability Across the Horizon The state aspect of the transaction attractor is defined by the horizon-integrated relative entropy: D total KL ( P S || Q F ) = ∫ c 0 D KL ( P S ( | ψ ( x ) ⟩ ) || Q F ( x ′ )) dx This expression quantifies the cumulative divergence between the probability distribution of the quantum state | ψ ( x ) ⟩ at each point ( x ) within the observable horizon, denoted P S ( | ψ ( x ) ⟩ ), and the probability distribution of non-local fields over all space-time points 10 ( x’ ), denoted Q F ( x ′ ). The observable horizon is defined as the region spanning from x = 0 (the event’s origin) to x = c (e.g., ( c t ), where ( t ) is the time elapsed from the event to the light-cone boundary), encompassing all points causally accessible to the event via light signals. In natural units ( c = 1), ( c ) may represent a spatial or temporal extent (e.g., ct = R , the horizon radius), and the integration over ( [0, c] ) aggregates the state’s behavior across this bounded region. The quantum state | ψ ( x ) ⟩ resides in the Hilbert space H , a complex vector space with an inner product, and is obtained by evolving a reference state | ψ ⟩ to position ( x ) via a unitary operator. For temporal evolution, this is typically: | ψ ( x ) ⟩ = e − i ˆ Hx/c | ψ ⟩ where ˆ H is the Hamiltonian of the system, and x/c represents the time displacement (in natural units, ( x ) may be interpreted as ( t )). Alternatively, for spatial propagation, a translation operator might apply, such as e − i ˆ px | ψ ⟩ with momentum operator ˆ p , depending on the context (e.g., a particle moving across space). The state | ψ ( x ) ⟩ encompasses both particle states (e.g., an electron) and vacuum contributions (e.g., zero-point fluctuations), reflecting the full quantum configuration at each ( x ). The distribution P S ( | ψ ( x ) ⟩ ) may correspond to the probability density: P S ( | ψ ( x ) ⟩ ) = |⟨ x | ψ ⟩| 2 which indicates the likelihood of finding the system in state | ψ ⟩ at position ( x ), or to expectation values such as momentum: ⟨ ψ ( x ) | ˆ p | ψ ( x ) ⟩ which quantifies the system’s dynamical properties across the horizon. The target distribution, Q F ( x ′ ), represents the probability distribution of non-local fields over all points ( x’ ), including contributions from vacuum fluctuations (e.g., zero- point energy of the electromagnetic field), electromagnetic fields (e.g., distant photon sources), and other field configurations that influence the system across space-time. In quantum field theory, this might be expressed as the expectation of field operators over the vacuum state, such as ⟨ 0 | φ ( x ′ ) φ ( x ′ ) | 0 ⟩ , adjusted for the system’s non-local context. The state aspect D total KL thus assesses how well the local state’s properties—e.g., momentum or energy—align with this non-local field environment across the entire observable horizon, ensuring inertial stability through consistency with broader field effects. Operationally, the state aspect governs the generation of confirmation waves within the transactional process. A quantum state | ψ ⟩ acts upon the field φ ( x ), influencing its configuration—e.g., collapsing its superposition of possibilities or shaping its expectation value ⟨ φ ( x ) ⟩ —to produce the confirmation wave ψ ∗ ( x ), which propagates backward in time from the absorption point. For example, an electron at x 2 absorbing a photon might transition from state | ψ ⟩ to | ψ ′ ⟩ , with ψ ∗ ( x ) = ⟨ x | ψ ′ ⟩ ∗ reflecting this action on the field A μ ( x 2 ). The state aspect ensures that this process maintains inertial stability, requiring the state’s momentum to be conserved across the horizon, supported by non-local field interactions, including vacuum polarization effects. These vacuum effects draw an analogy to quantized inertia [3], where inertial mass emerges from the interaction of a particle with the vacuum field across a horizon scale ( R ), approximated as: m ∼ ℏ cR 11 Here, ( R ) corresponds to the horizon radius (e.g., ( c t )), and the state aspect integrates D KL ( P S || Q F ) over ( [0, c] ) to ensure that the state’s inertial properties—e.g., resistance to acceleration—are consistent with the non-local vacuum’s influence. For an electron absorbing a photon, P S ( | ψ ( x ) ⟩ ) reflects its momentum distribution across the horizon, and Q F ( x ′ ) includes the vacuum fluctuations that stabilize this momentum, ensuring conservation laws hold. The transaction attractor does not require D total KL to minimize to zero independently, mirroring the flexibility of the field aspect. Within the product constraint: D KL ( P F || Q S ) · D total KL = 1 an increase in D total KL —e.g., due to a mismatch between the state’s horizon-wide behavior and non-local fields—is offset by a decrease in D