Citation: Zeng, X. (2026). Structural Determination Theory: A Single-Axiom Framework for Reality, Time, Irreversibility, and Space. PhilArchive. https://philpapers.org/rec/ZENSDT

Working version at this site. Published version at PhilArchive (use for academic citation).

Version 5.9 · Xiaozhou Zeng · Independent Researcher, Nancy, France · ORCID: 0009-0001-8244-7329 · April 2026

Abstract. This paper develops Structural Determination Theory (SDT) from a single ontological axiom: reality consists of distinctly differentiated structure. The argument proceeds in three stages. The first stage groups the axiom’s four constituents (reality, structure, consists of, distinctly differentiated) under existing philosophical traditions and establishes the undeniability of each through self-refutation arguments. The second stage unfolds the axiom’s ontological commitments into eleven formal concepts: centrally the realized-structure set $R$ , the global constraint $G$ , determination $D$ , and the determination dependency graph $G_{D}$ . The third stage derives three core arguments from this conceptual foundation: $R$ ‘s non-retraction and monotonic growth (Argument 1), $G$ ‘s constancy (Argument 2), and time’s emergence from the unfolding of $G_{D}$ ‘s partial-order structure (Argument 3); a structural corollary derives spatial distance as a joint derived quantity of the time structure and zero-carrier propagation. SDT borrows no specific physical theory, depends on no empirical data, and presupposes no concept of time or space; all arguments proceed by conceptual explication of the axiom. The theory’s principal contributions are: (i) a structural emergentism of time that jointly fulfills the core concerns of A-theory and B-theory rather than compromising between them; (ii) a fourth, independent kind of irreversibility, namely logical irreversibility, distinct from the statistical, thermodynamic, and initial-condition types (Boltzmann 1877; Prigogine 1980; Albert 2000; Price 1996), derived unconditionally from the modal structure of “realized” itself; (iii) the derived status of spatial distance, symmetric with the relationism of time and jointly supporting SDT’s via negativa strategy; (iv) a fourfold classification of determinations by $(∣ E ∣, ∣ S ∣)$ cardinality (Type I attribute transformation, Type II split, Type III merge, Type IV merge-split), providing an exhaustive meta-structural framework for composite phenomena; (v) the derivation of five structural conclusions (DAG structural-order, ontological direction, rule-background stability, time emergence, derived space) from a single axiom, providing a unified formal basis for traditionally independent metaphysical domains.

Keywords: ontological axiom; structural determination; logical irreversibility; emergence of time; derived space; via negativa; analytic metaphysics

Chapter 1. The Axiom

§1.1 Statement of the Axiom

Structural Determination Theory (henceforth SDT) rests on a single ontological axiom:

Reality consists of distinctly differentiated structure.

This section limits itself to the bare statement. The four terms in the axiom (reality, structure, distinctly differentiated, consists of) are technical terms; their precise meanings are explicated in §1.2 by being placed within established philosophical traditions. The argument that the axiom cannot be coherently denied is given in §1.3; the scope of the axiom and its self-limitation are stated in §1.4.

§1.1.1 Methodological coordination between statement and inference. To say that “this section limits itself to the bare statement” is to mark two methodological choices of SDT, which jointly determine the argumentative form of the theory.

First, the axiom as a fixed point. The axiom, as the ontological starting point of SDT, is not itself proved but exists as the fixed point of SDT’s reasoning system, with a methodological status analogous to that of the axioms of Euclidean geometry or of propositional calculus. The legitimacy of this status is secured by two considerations: (a) §1.3 demonstrates by self-refutation argument that the axiom cannot be coherently denied within SDT’s scope, so that the axiom as starting point is not an arbitrary choice but the weakest commitment that any consistent theory operating within this scope must already bear; (b) §1.4 explicitly delineates the scope, so that the boundary within which the axiom operates is open to scrutiny and the axiom is not applied to objects it cannot legitimately bear.

Second, the retrospective directionality of argumentation. SDT’s argumentative direction is opposite to the traditional forward explanation. Forward explanation derives consequences from initial states and laws; SDT does not adopt this direction. SDT begins from an irreducible fact, reality is the irreducible starting point of SDT’s argumentation, and retrospectively analyzes the structure that this fact necessarily presupposes. This directional choice is a logical consequence of the axiom itself: the axiom “reality consists of distinctly differentiated structure” states the ontological constitution of reality, not some dynamical process; the axiom’s undeniability is also retrospective (deriving “the axiom is true” from “the act of denial self-refutes within the scope”).

Accordingly, SDT does not attempt to answer questions of the form “how does possibility transform into actuality.” Such questions presuppose that possibility is a domain prior to reality, requiring some mechanism to “transmit” it into reality. SDT’s ontological commitment runs in the opposite direction: reality is the irreducible starting point, and possibility is a logical projection from reality. Possibility is the modal background that SDT necessarily presupposes for analyzing “realized structure,” not a parallel reality alongside it. This positioning runs through subsequent chapters: the possible-configuration set and admissible-configuration set introduced in Chapter 2 are defined from existing structure; the three core arguments of Chapter 3 all proceed retrospectively from realized structure.

The relation between the two methodological choices. Fixed-point status and retrospective directionality are not two independent methodological choices but two sides of the same methodological position: precisely because the axiom is an unproved starting point, the argumentation must proceed retrospectively from “existing reality”; conversely, precisely because the argumentation is retrospective, the axiom can stand as an unproved starting point. The axiom supplies the starting point of inference, and inference, on this basis, develops concepts and derives consequences; the axiom is not itself a consequence, but every consequence is fully traceable to commitments already established by the axiom. The formal unfolding of Chapter 2 and the arguments of Chapter 3 both follow this methodological framework.

§1.2 Explication of the Axiom

§1.2.1 Preliminary remark. At the meta-level, SDT adopts classical logic as the background for inference, and does not accept any tolerance of true contradictions; that is, SDT rejects dialetheism (Priest 1979). This is a working precondition of theoretical construction, not a second ontological axiom alongside §1.1: classical logic plays the role of inferential rules and occupies a different level from SDT’s ontological commitments.

The classical logic adopted by SDT refers to classical first-order predicate logic with its standard semantics (the principle of bivalence, the law of excluded middle, the law of non-contradiction, De Morgan’s laws), and does not commit to any specific choice among non-classical systems such as modal logic, tense logic, intuitionistic logic, intermediate logic, or paraconsistent logic. SDT rejects dialetheism because the latter’s tolerance of true contradictions would invalidate the self-refutation arguments of §1.3 (the act of asserting and the content of the assertion could simultaneously be true and false). SDT does not reject intuitionistic or intermediate logic, since these systems are more stringent than classical logic on bivalence and are compatible with SDT’s non-contradiction requirement.

The core arguments of SDT (self-refutation, non-retraction, $G$ constancy, time emergence) rely primarily on the law of non-contradiction and on formal analysis of self-refutation situations; they do not, at any critical step, invoke the strong form of the law of excluded middle (i.e., arguments using $φ \lor \neg φ$ as a premise for bivalent partitioning). SDT’s conclusions thus broadly hold within intuitionist and intermediate-logic readerships as well, though a rigorous demonstration is left for subsequent work. When concepts such as consistency, contradiction, and modal exclusion appear later in the paper, they are used under the standard semantics of classical logic.

§1.2.2 SDT’s methodological principle for the use of existing terms. SDT is a self-contained ontological theory: it develops all of its ontological commitments from within the axiom and the formalism. But SDT’s exposition must be expressed in language, and language has its limitations: discussion of ontological concepts such as reality, structure, possibility, realized, necessity, and identity must borrow from existing terms in natural language and philosophical traditions, since no fully independent conceptual system can be built from scratch. Moreover, if every use of a concept required redefinition from scratch, SDT’s exposition would become unworkably long and could not communicate its content effectively.

On the basis of the limitations of language and considerations of expository efficiency, SDT borrows certain expressions widely accepted in academic discussion (such as actuality, modal state, possible configuration, identity of indiscernibles, conceptual necessity, structural realism) as technical terms. SDT’s borrowing of these terms does not entail endorsement of: (a) any complete philosophical position to which a borrowed term belongs, for example, SDT’s borrowing of “possibility” does not commit it to Lewis’s modal realism (Lewis 1986), Stalnaker’s abstract-proposition view (Stalnaker 1976), Plantinga’s essentialism (Plantinga 1976), or Adams’s nominalism (Adams 1981); SDT’s borrowing of “reality” does not commit it to eternalism, presentism, or the growing-block view; SDT’s borrowing of “identity of indiscernibles” does not commit it to the full philosophical content of the Leibnizian PII tradition; (b) any linguistic interpretation that a borrowed term carries in its original tradition. For instance, the term realized admits a temporal-perfective reading in everyday usage, but SDT does not endorse this reading and uses realized instead as a modal-state marker; the term conceptual necessity carries different senses in different philosophical traditions (Kantian, Kripkean, Carnapian, Chalmersian), but SDT uses it instead in the technical sense of “directly analytically derivable from SDT’s internal conceptual definitions.”

§1.2.3 The technical meaning of borrowed terms. At first use, SDT gives each borrowed term its technical meaning within the SDT framework: what role the term plays inside SDT, what determines its content, and how it relates to other SDT concepts. This technical meaning may partially overlap with the term’s meaning in some classical tradition, but SDT does not commit to full overlap. A term’s final meaning is determined by SDT’s internal technical definition, not by the classical tradition to which the term belongs.

§1.2.4 Legitimacy of this principle. This principle is methodologically legitimate for two reasons. (a) The starting point of SDT (the axiom) is at the ontological level, where complete avoidance of natural-language vocabulary is impossible. Even an ostensibly “neutral” formal symbol must be associated with some natural-language concept upon first interpretation; otherwise the formal symbol cannot be understood. (b) Borrowing terms while declaring technical meaning preserves both the readability of exposition (readers can connect SDT’s concepts to their existing vocabulary) and the precision of the theory (technical meaning is constrained by SDT’s internal definitions, not by everyday usage). This dual function makes term-borrowing the optimal trade-off for SDT’s exposition.

§1.2.5 Costs and commitments of this principle. This principle has two acknowledged costs. (a) Readers may, on first reading, transfer the everyday or classical-philosophical meanings of a term into SDT, leading to misreading; SDT minimizes this risk by giving explicit technical definition at first use. (b) The principle carries a methodological commitment: when SDT later argues for some conclusion, the argument must rest on the term’s technical meaning, not on the term’s classical-philosophical content. This requires every step of SDT’s argumentation to be checkable for whether it relies on a classical philosophical thesis SDT has not endorsed. SDT explicitly accepts this commitment: every argument in this paper is tied to internal definitions and explicit derivations, not to external philosophical doctrines.

§1.2.6 Adjusting the reader’s expectations. Readers familiar with analytic philosophy may, at first reading, expect SDT to take a position within some classical debate (e.g., on possibility, on identity, on time). This expectation should be adjusted: SDT’s technical use of borrowed terms is not equivalent to taking such a position. The classical debates concern specific philosophical theses (e.g., whether possible worlds are concrete entities, whether identity is intrinsic), while SDT’s technical use is constrained only by what SDT’s internal concepts require. In most cases, SDT’s technical content is compatible with multiple positions in the classical debate, and SDT does not commit to any of them.

§1.2.7 The methodological status of §1.2: explication, not inference. §1.2 explicates the four constituents of the axiom; it is not itself an inference from the axiom. The function of §1.2 is to make the axiom readable and assessable: to translate its bare statement into a form that can be examined philosophically. The content of the axiom is fixed by §1.1; what §1.2 adds is interpretive precision. Inference proper begins with Chapter 2, where the axiom’s commitments are unfolded into formal concepts and derivations.

§1.2.8 Reality. The first constituent of the axiom is reality. Following SDT’s principle on borrowed terms, reality in SDT is taken as a technical term with the following content: the unique totality of states of affairs that have actually been realized, where “realized” is read in the modal sense (a state of affairs is realized iff it has crossed the boundary from possibility into actuality). On this reading, reality is what is, not what could be. Reality is uniquely identified (there is one reality, not many; the uniqueness is argued in §1.2.14) and contains all and only the states of affairs that have actually been realized.

This reading is the weakest actualism commitment compatible with SDT: SDT commits to the view that only one actual world exists, but does not commit to any further internal distinction among actualism positions (e.g., whether unrealized possibilities have any form of being, whether they exist as abstract propositions, etc.). SDT’s via negativa strategy works through this minimal commitment: SDT argues that certain widespread conceptions of reality (e.g., reality as a fixed totality of all possibilities, reality as a parameter-indexed family of worlds, reality as a continuous temporal manifold) make commitments beyond what the axiom requires, and SDT does not depend on them.

§1.2.9 Structure. The second constituent is structure. Structure in SDT is the technical term for a distinctly differentiated configuration of states: states existing as distinguishable parts standing in differentiated relations. This conception draws on, but is not equivalent to, structural realism (Worrall 1989; Ladyman 1998; French 2014) (Worrall 1989; Ladyman 1998; French 2014); SDT’s commitment is weaker than ontic structural realism (which holds that only structure exists) and does not depend on its rejection of object-level ontology.

For SDT, “structure” denotes the form-internal composition of any realized state of affairs: any realized object, situation, or arrangement is structured if and only if it consists of differentiated parts in differentiated relations. The axiom’s claim that “reality consists of structure” thus says that the unique totality of realized states of affairs is, at every level, internally differentiated: reality is not a featureless continuum.

§1.2.10 Consists of. The third constituent is the verb consists of. In SDT, consists of is read as a conjunctive existential claim: reality both exists (the realized totality is non-empty) and is composed of differentiated structure (there are realized parts standing in differentiated relations). The two conjuncts are inseparable: existence without internal differentiation would yield a featureless realized totality, contradicting “structure”; differentiation without existence would yield modal possibilities without actualization, contradicting “reality.”

This conjunctive reading is what allows the axiom to do work in subsequent chapters. The existence conjunct supports the existence claims of $R$ (the realized-structure set is non-empty) and of $G$ (the global constraint exists, §2.2). The composition conjunct supports the formalization of $R$ as a set of structured members rather than as an unstructured aggregate.

§1.2.11 Distinctly differentiated. The fourth constituent is distinctly differentiated. This phrase carries SDT’s strongest commitment: the parts that compose a realized state of affairs are not merely different but identifiably distinct under the identity of indiscernibles in its strong reading (henceforth PII strong reading): if two members of $R$ agree on every aspect of their realized structural content, they are the same member. PII strong reading thus functions as the identity criterion on $R$ .

PII strong reading is borrowed from the Leibnizian tradition (Leibniz’s Monadology §9; modern treatments in Black 1952; Forrest 2020). SDT commits to the strong reading specifically: the agreement at issue is full structural agreement (agreement on layout content (element identity, relational configuration, attribute specification) as defined in Chapter 2), not merely qualitative agreement. SDT does not commit to PII applied beyond $R$ : traces ( $T$ ) and dependency-graph nodes ( $G_{D}$ ) operate under their own identity criteria (event identity for traces, topological position for nodes), not under PII (see §2.3.13).

§1.2.12 Compatibility of PII with quantum indistinguishable particles. A common objection to PII concerns quantum indistinguishable particles (e.g., two electrons in a singlet state): standard quantum mechanics treats such particles as having identical intrinsic properties (Saunders 2006) yet as numerically distinct, apparently violating PII. SDT handles this objection through the categorical division between $R$ , $T$ , and $G_{D}$ (developed fully in §2.3.13). An entangled or coupled joint state (e.g., a singlet, a bound state, an uncollapsed multi-particle superposition) is a single $R$ -member whose layout content jointly specifies multiple particle-roles; these particle-roles are internal to the $R$ -member’s layout content and do not constitute separate $R$ -members at the SDT level. PII strong reading therefore does not encounter a “two-member comparison” within such joint states; PII concerns identity among $R$ -members, and an entangled joint state is a single $R$ -member to which PII applies trivially. When measurement, decoherence, or other interaction subsequently produces factorized constituent structures, these enter $R$ as new independent $R$ -members differentiated in their element-identity components — each new structure carries its own determination-chain ancestry, with identity borne jointly by $R$ ‘s relational-configuration component, $T$ ‘s event identities, and $G_{D}$ ‘s topological positions (see “Resolution of two factorized identical electrons” below). PII strong reading on $R$ is satisfied without conflict with quantum mechanics.

Resolution of “two factorized identical electrons” in SDT: the physical scenario “two identical electrons existing as factorized constituent states (e.g., produced by independent determination events)” necessarily resolves in SDT formalization to two distinct realized structures $S_{1} \neq = S_{2}$ in $R$ . Two grounds. First, by §2.4 usage exclusivity, the same specific $e$ cannot participate in two different $L$ ; therefore “two electrons,” if they are participating structures of two independent determination events, cannot be the same $e$ in $R$ . Second, $S_{1}$ and $S_{2}$ as products of their respective determinations have necessarily different element-identity components in their layout-content three-component sense: each electron’s $E$ contains the prior determination products that produced it; different electrons have different prior production chains; hence $E$ as a set of specific realized structures differs. By PII strong reading, $S_{1} \neq = S_{2}$ as members of $R$ , and PII is not violated in this scenario.

Resolution of “two measurements on the same electron” in SDT: the physical scenario “two measurements performed on the same electron” resolves in SDT to a determination chain $S_{0} \to S_{1} \to S_{2}$ : the original electron state $S_{0}$ participates in $D_{1}$ as the sole member of $L_{1}$ ‘s $E$ , producing $S_{1}$ (the electron’s state after the first measurement); $S_{1}$ participates in $D_{2}$ as the sole member of $L_{2}$ ‘s $E$ , producing $S_{2}$ . At each step, $E$ , $S$ , $τ$ , $G_{D}$ node are independent; adjacent $S$ on the determination chain differ in the layout-content three-component sense (at least in attribute specification, guaranteed by the input ≠ output criterion, see §2.3 core legitimacy criterion of determination). “The same electron” in SDT is not a same $e$ in $R$ spanning multiple determinations but a physical-object evolution trajectory depicted by consecutive $S$ on the determination chain; the physical intuition of “the electron’s continuity” is borne by the determination-chain structure, not by requiring that the same $e$ in $R$ span multiple determinations as input.

Strict applicable range of PII strong reading: PII applies only to $R$ , with the criterion being layout-content three components. $R$ , $T$ , $G_{D}$ are in strict one-to-one correspondence in SDT formalization (each determination $D$ adds one corresponding object to each), but categorically independent: $R$ judges structural identity by PII; $T$ bears trace identity through the determination event itself; $G_{D}$ bears node identity through topological position in the DAG. Equivalent determinations’ products $S$ and $S^{'}$ , by the joint application of usage exclusivity, necessarily differ in the element-identity component (as shown in the two scenarios above); therefore at the $R$ level, no situation arises where “two structures completely agree on layout-content three components yet count as distinct members.” PII strong reading holds strictly within SDT formalization, undiminished by equivalent determinations or by the existence of indistinguishable particles in quantum mechanics.

Convergence with physical fact: the standard treatment of “identical particles” in quantum mechanics (through exchange symmetry) is consistent with SDT’s analysis above. Exchange symmetry’s formal content — that swapping particle labels yields no physically distinct state — is naturally captured under SDT’s reading: in entangled or symmetrized joint states (single $R$ -members under SDT’s analysis), the “particle labels” are internal formal roles within the $R$ -member’s layout content, not references to distinct $R$ -members; relabeling internal roles within a single $R$ -member yields the same $R$ -member. For factorized constituent states (post-measurement, post-decoherence), each particle’s formal identity is borne by its structural position on the determination chain, its trace, and its $G_{D}$ node, without committing to “particle entities” as independent ontological units. SDT therefore does not respond to the identical-particle challenge by diluting PII strong reading; by clarifying the $R$ -membership status of entangled versus factorized states, it naturally aligns with quantum mechanics’ formal structure.

§1.2.13 Consistency. A further commitment implicit in distinctly differentiated is consistency: the realized totality cannot contain contradictory structures. If two structures $S_{1}, S_{2} \in R$ stand in relations whose joint instantiation would entail a contradiction (in the classical-logic sense), they cannot both belong to $R$ . Consistency is needed for $R$ to be coherently characterizable as a set: a set with contradictory members admits no consistent description, and SDT’s exposition would be undefined.

Consistency is implicit because SDT’s adoption of classical logic (§1.2.1) and PII strong reading (§1.2.11) jointly entail it: classical logic forbids contradictions, and PII strong reading requires that any two members of $R$ , which they cannot be if their joint instantiation is contradictory (a contradiction being indistinguishable from any other contradiction). Consistency thus does not need to be added as an independent commitment; it is a consequence of commitments already made.

§1.2.14 Uniqueness. Reality is unique: there are not two or more independent “realities.” This claim is not an additional definitional choice of SDT but an ontological necessity derived from the preceding explication.

Argumentative premise. This argument relies on the PII strong reading adopted in the distinctly differentiated clause (two structures whose layout content fully agrees in every respect are the same).

The precise meaning of “independent.” “Two independent realities $R_{1}$ and $R_{2}$ ” specifies a conjunction of three conditions: (a) $R_{1}$ and $R_{2}$ are not ontologically reducible to each other (they are not two descriptions of the same reality); (b) there are no shared structural relations (causal, co-existential, referential, etc.) between $R_{1}$ and $R_{2}$ ; (c) no structure in $R_{1}$ or $R_{2}$ holds simultaneously across the two. These three conditions jointly characterize the strong independence under which “the two realities do not touch at any level.” The uniqueness argument refutes precisely this strong independence; weaker independence (e.g., merely ontological irreducibility while permitting shared reference or description) lies outside the scope of this argument, since the introduction of any shared relation makes the two realities different regions of one reality.

The argument. Suppose two independent realities $R_{1}$ and $R_{2}$ exist. Calling them “two” presupposes some difference between them. By the distinctly differentiated clause, this difference must be borne by at least one point of difference in their layout content (the application from the structural level to the reality level rests on consists of: reality consists of structure, so the difference between two realities can only come from structural difference, and the PII criterion at the structural level applies at the reality level). But the layout content bearing this difference itself belongs to the structural category, and by the conjunctive claim of consists of, any structure belongs to reality and can belong only to $R_{1}$ or to $R_{2}$ . Once that layout content belongs to one of them, it brings the other into the referential scope of the same reality (since it carries a structural description of “the difference between the two” and must place both within its reference), and $R_{1}$ and $R_{2}$ are no longer two independent realities. The supposition of “two independent realities” self-refutes in every case. This argument is an early application of the §1.3 self-refutation arguments to the specific commitment of uniqueness: uniqueness is not an additional assumption beyond the axiom but a logical consequence of commitments the axiom already bears.

Response to “two wholly incomparable realities.” A possible evasion is to claim that two wholly incomparable realities exist (no description can simultaneously refer to both). This claim itself simultaneously refers to both (through the phrase “two realities” or any analogous expression), thereby establishing descriptive access to both within the act of asserting, in violation of the “wholly incomparable” assumption. This response parallels Branch three (referential acts) of §1.3: any assertion about “two realities” necessarily carries a joint reference to both, placing them within the same referential scope and thereby within a single reality.

Technical content. When SDT speaks of “reality,” it speaks of the unique totality of realized states of affairs, indexed by no observer or perspective. SDT does not commit to a privileged frame (the uniqueness is not Newtonian-absolute); it commits only to the fact that the totality is one and the same regardless of perspective. This minimal commitment is what allows SDT to refer to a single $R$ without indexing (the realized-structure set is the set, not a frame-relative or observer-relative set). The commitment is consistent with relativistic physics (where different frames give different coordinate descriptions of the same reality) and with quantum-mechanical observer-dependence (where different measurement contexts yield different outcomes within the same reality); both are interpreted in SDT as different descriptions of the unique reality, not as different realities.

§1.3 Self-Refutation Arguments

§1.3.1 Methodological note. §1.3 establishes that the axiom of §1.1 cannot be coherently denied within SDT’s scope. The argument is structured as four independent self-refutation arguments, one for each of the axiom’s four constituents (reality, structure, consists of, distinctly differentiated). Each argument shows that denying the constituent leads to an internal contradiction within the act of denial itself; thus the constituent must be already presupposed by the denier.

The four arguments are independent: each constituent is defended on its own terms, and the arguments do not depend on one another. This independence makes the axiom: an objector who denies any single constituent must answer the corresponding self-refutation argument, not merely point to an alternative constituent.

§1.3.2 Branch one: denying reality. Suppose someone asserts: “Reality does not exist.” This assertion is itself an act, a real act of asserting something. If the act exists, then at least the act is real; the assertion thus contradicts itself by performing an act it claims cannot occur. The denial is self-refuting because the denial is itself a counter-instance to what is denied.

A weaker version of the denial holds that “reality” as a meta-concept; the assertion holds that there is no totality of realized states. This weaker denial faces a similar problem: if the assertion is meaningful (asserts something rather than nothing), then the meaning-bearing act and its content together constitute realized states of affairs (the act and its content exist in the realized totality). If the assertion is meaningless, it cannot serve as a denial. Either way, the denial fails: it either contradicts itself or fails to be a denial.

§1.3.3 Branch two: denying structure. Suppose someone asserts: “Realized states can be without any differentiation: existing as a single integrated whole with no internal differences and no differences from other possible states.” The self-refutation of this claim does not depend on the nature of the assertion act itself (unlike Branch one), but on the inherent content of the concept “realized.” By §1.2 Reality, “realized” entails determinacy: a realized state is determinately in some particular state, not floating undecided among multiple possibilities. “Determinately in some particular state” already conceptually includes a minimal differentiation: it distinguishes “this state” from “other logically possible states,” and this distinction itself constitutes a differentiation of the state, namely it is this one rather than those others (“those others” refers to other logically possible configurations, without presupposing them as actual states existing simultaneously). Therefore a state with no differentiation whatsoever cannot be “determinately in some particular state”: the absence of differentiation means there is no content distinguishing it from other logical possibilities, and losing this distinction loses the content required by “determinately in some particular state.” “Realized but wholly without differentiation” is conceptually self-contradictory.

Response to suspicion of circular argument. A reader may object that arguing from “differentiation” to “structure” is circular. The response: this argument involves two distinct concepts of differentiation that must be strictly separated. Ontological differentiation: a state’s distinguishability from other logically possible states, that is, “this state is not other logically possible states.” This is the inherent content of the concept “realized,” directly entailed by determinacy (a state floating undecided among multiple possibilities does not satisfy determinacy). Structural differentiation: differences among distinguishable parts within a state, that is, “the state is composed of distinguishable parts standing in differentiated relations.” This is the content required by the concept of structure (§1.2 Structure). The argument’s chain is: ontological differentiation (from determinacy) entails structural differentiation (if a state’s ontological differentiation can bear the distinction “it is this one rather than those others,” this distinction necessarily requires the state to have identifiable content, and “identifiable content” within any consistent descriptive framework requires the state to have distinguishable internal parts). The two concepts of differentiation respectively bear two distinct conceptual functions (“distinguishing the state from other possibilities” vs. “differentiated structure within the state”), and the argument derives the latter from the former, which is a cross-level derivation rather than a tautology.

This argumentative mechanism belongs to conceptual contradiction: the content of the assertion (“realized can be without differentiation”) directly conflicts with the inherent content of the concept “realized” (“determinacy entails minimal ontological differentiation, which entails structural differentiation”); no appeal to the nature of the assertion act itself is needed to expose the contradiction.

A more sophisticated version of the denial. A reader may attempt a more sophisticated version: “fundamental” reality is unstructured, while only “appearance” is structured. This version faces the same problem at a different level: the distinction between fundamental and appearance is itself a structural distinction (two differentiated levels standing in differentiated relations). The denier cannot articulate the distinction without already presupposing structure. The two-level retreat does not escape the constituent structure; it only relocates it. The constituent structure therefore cannot be coherently denied.

§1.3.4 Branch three: denying “consists of”. Suppose someone asserts: “Reality does not: reality and structure are unrelated, or one of them is illusory.” This denial faces a dilemma. (a) If reality and structure are unrelated, then the denier’s assertion (which is a structured act in reality) creates a counter-example: the assertion as structure exists in reality, hence reality and structure are related at least in the case of this assertion. (b) If one of them is illusory, then either reality is illusory (but the denier’s assertion exists, so reality is not illusory) or structure is illusory (but the denier’s assertion is structured, so structure is not illusory). Either disjunct fails.

A more careful version of the denial holds that reality contains some structure but: structure is an external description applied to reality, not an internal feature of reality. This version still fails: if structure is an external description, then the description (as a structured medium) exists somewhere; that somewhere is part of reality (or, if it is outside reality, then “reality” is being used in a more restrictive sense than the axiom’s). On the broader sense of reality, the denial recreates the original contradiction.

§1.3.5 Branch four: denying distinctly differentiated. Suppose someone asserts: “Reality can contain two structures whose layout content is wholly the same (the layout content agrees in every respect) yet they are still ‘two’ rather than the same one.” The self-refutation of this assertion does not depend on the nature of the assertion act itself (unlike Branches one and three), nor on the inherent content of “realized” (unlike Branch two), but on the assertion’s internal use of concepts that presuppose what the assertion denies. The concept “two” used in the claim itself presupposes that the referenced objects are distinguishable: if the two are wholly the same in layout content (under the Leibniz PII strong reading adopted in §1.2 distinctly differentiated), they are the same structure, and the count “two” has no substantive bearer. The assertion attempts to claim simultaneously “they are two” and “there is no distinction between them,” but the former’s use of “two” already presupposes the latter’s denied content of “distinction.” The denial cannot form a meaningful proposition: it must either give up “two” (regressing to “there is one structure,” which is no longer a denial of distinctly differentiated) or give up “no distinction whatsoever” (admitting that there is a distinction between them, which is admitting distinctly differentiated). This argumentative mechanism belongs to the failure form of conceptual presupposition: the concept used by the denial already bears, at the level of its conditions of use, the content that it denies, so the denial cannot coherently form on the conceptual level.

Brief note on Black’s 1952 mirror-universe challenge. Black’s 1952 The Identity of Indiscernibles mirror-universe counterexample ostensibly challenges the PII strong reading. SDT’s response rests on the categorical division of $R$ / $T$ / $G_{D}$ (§2.3) and usage exclusivity (§2.4): if the two spheres correspond to two independent determination events, by usage exclusivity their $E$ membership identities must differ, and by PII strong reading they are different realized structures, not constituting a counterexample; if the two spheres correspond to no determination event whatsoever, they fall into the “dangling structures” not discussed by SDT (§2.3), and Black’s challenge does not enter SDT’s scope. The detailed mechanism is given in §1.2 in the parallel discussion of “PII compatibility with quantum indistinguishable particles.” The constituent distinctly differentiated therefore cannot be coherently denied.

§1.3.6 Concluding remark. The four branches respectively target the four constituents of the axiom with self-refutation arguments whose mechanisms differ: Branch one (reality) relies on the contradiction between the assertion act’s “realized” character and the assertion’s content; Branch two (structure) relies on the contradiction between the inherent content of “realized” and the assertion’s content; Branch three (consists of) relies on the contradiction between the differentiating treatment of the referenced object by the act of reference and the assertion’s content; Branch four (distinctly differentiated) relies on the assertion’s internal use of concepts that presuppose the very content denied.

Independence of the four branches. Although Branches one and three both involve “contradiction between the assertion act and the assertion’s content,” their argumentative mechanisms are independent at the critical points. Branch one’s contradiction occurs between “the existence of the assertion act itself” and “the realized status denied by the assertion’s content”; even if someone could make an abstract assertion without any specific reference, the making of that assertion still constitutes a counter-instance to “realized.” Branch three’s contradiction occurs between “the differentiating treatment of an object by the referential act” and “the absence of differentiation denied by the assertion’s content”; even if an assertion does not question that “the assertion act itself is realized,” so long as it must refer to some undifferentiated component, the referential act itself structures that component. The difference is: Branch one targets direct denial of reality, while Branch three targets acknowledging reality but denying the exhaustiveness of structure. The two are not two applications of an isomorphic argument but independent arguments aimed at two distinct denial targets. Similarly, Branch two (conceptual contradiction) and Branch four (conceptual presupposition) are independent in mechanism: the former depends on the inherent content of the concept “realized,” the latter on the presupposition by the concept used by the denial (“two”) of the very content denied. The four mechanisms differ in form but share the same logical structure: the act or concept used to deny some constituent of the axiom already bears, in its own operation, the content denied, so that the denial cannot be coherently completed.

The four constituents of the axiom therefore cannot be coherently denied within the range of consistent description, and the axiom as a whole cannot be coherently denied. It must be reiterated that the validity range of these arguments is strictly limited to reality that can be coherently described, a limit set out in detail in §1.4. The arguments do not assert that the axiom remains true for objects “transcending all descriptive frameworks”; SDT makes no commitment regarding such objects. This self-limitation is at once the boundary of §1.3’s argumentative strength and the source of its caution: within the scope it governs, SDT makes the strongest claim (the axiom is undeniable); outside that scope, it makes no claim.

§1.4 The Scope of the Axiom

§1.4.1 Methodological framing. §1.4 explicitly demarcates the scope within which the axiom of §1.1 holds. SDT does not claim universal applicability; it claims applicability within a specifically delimited scope, within which the self-refutation arguments of §1.3 are valid. Demarcation is necessary because every theoretical framework operates with implicit boundary assumptions, and SDT’s commitment is to make these assumptions explicit.

§1.4.2 Acknowledgment of the mechanistic limitation of transcendental arguments. The self-refutation arguments of §1.3 are transcendental in form: they show that the axiom is presupposed by any act of coherent assertion within SDT’s scope. As transcendental arguments, they share the well-known limitation noted by Stroud (1968) and Stern (2000): transcendental arguments establish that certain conceptual commitments are required for certain modes of inquiry to be possible, but they do not, by themselves, establish that those commitments correspond to mind-independent reality.

SDT explicitly accepts this limitation. The self-refutation arguments establish that the axiom is the weakest commitment any consistent description of reality within SDT’s scope must already bear; not that the axiom corresponds to mind-independent reality in some stronger metaphysical sense. SDT’s claim is conditional: if one accepts the project of consistent description of reality, then the axiom is already implicit; whether one should accept the project is a question SDT does not address.

§1.4.3 Positive demarcation of the scope. The scope of SDT comprises all states of affairs that can be coherently described. By “coherently described” SDT means: describable in a manner consistent with classical logic (no contradictions in the description) and meaningful in some natural or formal language (the description carries determinate content). This positive demarcation is broad: it includes all of physical reality, all of mathematics, all of formal sciences, all of empirical sciences, all of structured aspects of human and non-human experience that admit of articulation.

SDT does not commit to the specific content of these areas: physics is filled in by physical theory, mathematics by mathematical theory, etc. SDT commits only that whatever the specific content, it falls under the scope of the axiom: consists of distinctly differentiated structure.

§1.4.4 Negative demarcation of the scope. The scope of SDT excludes states of affairs that cannot be coherently described. Two examples: (a) hypothetical “objects” that defy any consistent description (e.g., the largest natural number, a square circle); (b) hypothetical totalities that exhibit internal contradictions (e.g., the totality of all sets that do not contain themselves, leading to Russell’s paradox).

SDT does not deny the existence of such objects in the strong sense; it merely declines to discuss them. The relation between SDT and such objects is one of methodological silence: SDT does not claim that they do not exist (such a claim would itself require describing them), only that SDT cannot describe them and thus does not extend to them.

§1.4.5 Methodological handling of recursivity. A recursive question arises: does the act of demarcating the scope (§1.4.3 and §1.4.4) itself fall under the scope? If yes, then the demarcation must be coherently describable, and meta-discourse about SDT itself becomes part of, introducing the possibility of self-reference paradoxes. If no, then the demarcation operates outside SDT, and SDT’s self-limitation is itself unconstrained.

SDT handles this recursivity by adopting a tiered approach: object-level discourse (about reality) is fully within SDT’s scope; meta-level discourse (about SDT’s own commitments) is conducted in classical logic with explicit awareness of the meta-status. The meta-level statements (such as the demarcation in this section) are not themselves theorems of SDT; they are methodological commitments about how SDT operates. This tiering avoids self-reference paradoxes while making the meta-commitments explicit and inspectable.

§1.4.6 Methodological legitimacy of self-limitation. Some readers may object that self-limitation weakens SDT’s claims: if SDT does not extend to all of reality, what does it tell us about reality as a whole? SDT’s response is that self-limitation is the price of precision. A theory claiming universal applicability without demarcation must either be vacuous (committing to nothing in particular about anything) or inconsistent (extending its commitments to objects it cannot coherently describe). SDT chooses precision and accepts the corresponding demarcation: SDT is a theory of the structurally describable, and what it says about this domain is the strongest claim it can responsibly make.

§1.4.7 Response to tool self-application. §1.4.5’s tiered approach handles self-reference paradoxes (meta-level discourse is not theorematized within SDT). But there is a separate phenomenon that requires direct treatment. SDT articulates a theory of structure using classical logic and language, and these articulation tools are themselves structured by §1.2’s account of structure: distinguishable parts (symbols, words) standing in differentiated relations (syntax, inference rules). The theory’s universal claim about structure therefore applies to its own articulation medium. This phenomenon is tool self-application.

Tool self-application is real, and SDT acknowledges it directly.

Tool self-application contains a form of circularity, but it is important to distinguish two senses of “circular”:

Narrow circularity (argument flaw): using P as a premise to prove P. SDT’s axiom defense (§1.3 self-refutation argument) is not narrowly circular. Its form is reductio: denying the axiom leads to an internal contradiction within the act of denial itself. The structural character of articulation tools is not used as a premise.

Broad circularity (structural fact): a theory cannot ground itself from outside its own scope, since any argument uses tools that fall within what the theory characterizes. SDT’s arguments are within broad circularity. This is a direct consequence of tool self-application.

Broad circularity is logically inevitable for any sufficiently strong theoretical system; it is not a defect specific to SDT. Two layers of support:

The philosophical layer. Any totalizing theory of reality must be articulated through some medium; that medium is part of reality; if the theory makes a universal claim about reality, the claim applies to its own articulation medium. Materialism (articulating that reality is material through material media), idealism (articulating that reality is mental through mental acts), and structuralism (articulating that reality is structured through structured discourse) all face the same situation. Any theory claiming to escape this would itself be a totalizing claim about reality and would face the same broad circularity in its own articulation.

The formal layer. Gödel’s second incompleteness theorem (1931) establishes that any sufficiently strong formal system containing arithmetic cannot prove its own consistency from within itself. This is not directly the same problem as SDT’s tool self-application; Gödel concerns formal-system consistency proofs, while tool self-application concerns the relation between a theory and its articulation medium. But both are instances of the broader phenomenon that sufficiently strong theoretical systems cannot fully ground themselves. If pure formal logic faces this kind of self-limitation, no philosophical theory can claim escape.

SDT does not pretend to escape broad circularity; such an escape is logically impossible. SDT’s position is to acknowledge broad circularity as a logical inevitability and to conduct substantive arguments within this constraint.

The substantive arguments of Chapter 3 ( $R$ non-retraction, $G$ constancy, time emergence) operate within the structurally describable domain. They do not claim to transcend this domain; they acknowledge the constraint and remain rigorous within it.

§1.4.8 Brief response to the anti-realist challenge. Dummett-style anti-realism (Dummett 1991 The Logical Basis of Metaphysics) may object that SDT’s neutrality regarding “trans-descriptive objects” already presupposes some realist commitment, namely that “such objects might exist” is itself a realist commitment. Similarly, Putnam’s internal realism (Putnam 1981 Reason, Truth and History) may object that the criterion “coherently describable” implicitly treats epistemic access as an ontological standard. A more general anti-realist challenge may further claim: whatever can be described is structured, but this tells us nothing about reality independent of description.

SDT’s response to such challenges is twofold.

First, SDT claims only the conditional thesis. SDT does not claim more than the conditional thesis (anything coherently describable is structured), so the anti-realist challenge does not refute SDT. The conditional thesis is sufficient for SDT’s substantive arguments in Chapter 3 ( $R$ non-retraction, $G$ constancy, time emergence), because these arguments operate entirely within the structurally describable domain. SDT’s via negativa strategy is consistent with anti-realism about anything beyond this domain.

Second, SDT remains neutral on the metaphilosophical dispute. SDT’s “neutrality” is not an adoption of some metaphilosophical position in the realism/anti-realism dispute; it is a natural boundary statement of SDT’s scope. SDT only declares that it does not address such objects; it does not claim that “they exist or do not exist” or that “they can or cannot be known.” This self-limitation remains entirely neutral on the realism/anti-realism dispute: realists may interpret “whether non-coherently-describable objects exist” according to their own position; anti-realists may interpret “the relation between coherent describability and ontology” according to theirs. SDT bears no specific position in these metaphilosophical disagreements. SDT’s commitments are confined to its scope; within that scope the axiom cannot be coherently denied, and the formalization and arguments developed therein are legitimate.

§1.4.9 Categorial diagnosis of the dynamics question. A common form of criticism takes this shape: SDT does not explain why determinations occur. What “drives” the transition from possibility to actuality? What makes one $p \in A$ realized rather than another? This criticism demands that SDT provide a more fundamental physical or metaphysical mechanism to explain the occurrence of determination itself.

This demand constitutes a category mistake within SDT’s methodological framework. SDT’s ontological commitment is: reality exists, and possibility is mapped from reality. The two are not endpoints of a dynamical chain (possibility first, actuality second, some mechanism mediating between them), but two analytic aspects of the same ontological fact. Reality, as the irreducible starting point of SDT’s argumentation (§1.1.1), already contains possibility as its logical projection; possibility’s existence is not independent of reality but is the modal background analyzed retrospectively from existing reality. Determination $D$ is not an intermediate mechanism that “transforms possibility into actuality” but SDT’s formal designation of “realized structure” as a given fact.

Accordingly, the picture presupposed by the demand for an “explanation of dynamics”, in which possibility is a domain prior to reality requiring some mechanism to transmit it into reality, has been repositioned by the retrospective directional choice of §1.1.1. Within SDT’s methodological framework, this picture does not constitute a problem SDT must answer; it belongs to the internal problems of different methodological positions.

This diagnosis is not an evasion of the dynamics question. SDT does not claim that “dynamics does not exist” or “dynamics is unimportant”; it only claims that, with respect to SDT’s argumentative goal as a meta-structural theory, dynamics lies outside SDT’s scope. Specific physical dynamics (how exactly a given $p \in A$ comes to be realized, determination rates, probability distributions, and so on) are borne by entity theories (cf. §1.2’s methodological principle on the division of labor between SDT and specific philosophical disputes); SDT provides a framework at the meta-structural level and does not specify dynamics at that level.

§1.4.10 Concluding remark. The scope of SDT is set by the axiom together with the principle of coherent describability. Within this scope, the axiom is undeniable (§1.3) and the formal apparatus of Chapter 2 unfolds rigorously. Outside this scope, SDT is silent. This dual orientation (rigorous within scope, silent beyond it) is what gives SDT its theoretical economy: SDT commits only to what its scope demands, and what it commits to is unconditionally undeniable within that scope.

Chapter 2. Core Concepts

Set-theoretic strength. All formal constructions in Chapter 2 take place against the background of ZFC set theory. SDT does not require any commitment beyond standard ZFC; specific applications (e.g., to physical theories with continuum cardinalities) are filled in by entity theories on this set-theoretic foundation. The choice of ZFC reflects expository convenience rather than ontological commitment: SDT’s structural claims would translate to weaker set theories with appropriate modifications, but ZFC provides the most familiar framework.

§2.1 Elements and Possible Configurations

§2.1.1 Section overview. This section introduces the static components of SDT’s formal framework: elements ( $E$ ), possible configurations ( $P$ ), and a preview of compatibility conditions ( $C$ ). The constraint system ( $G$ , $R$ , $A$ , the full $C$ ) is treated in §2.2; the dynamical concepts (structural layer $L$ , determination $D$ , trace $τ$ ) are treated in §2.3. The order of presentation reflects the axiom’s logical structure: static components first, constraints next, dynamics last.

§2.1.2 Element $E$ (definition and ontological anchor). An element, denoted $e$ , is a member of a realized structure: that is, $e \in R$ for some realized structure (the realized-structure set $R$ is introduced in §2.2.5). The collection of elements relevant to a particular structural layer is denoted $E$ , a non-empty set: $∣ E ∣ \geq 1$ . The ontological anchor of $E$ is the consists of clause of §1.2.10: any realized state of affairs is composed of differentiated parts, and $E$ is the formalization of the part-collection for a specified structural context. Elements are not abstract labels; each $e \in E$ has a corresponding member in $R$ , namely the realized structure $S_{e}$ to which $e$ refers as a role within the structural layer.

§2.1.3 Element $E$ (scale-relativity, legitimacy criterion, boundary). The members of $E$ may be of any scale: fundamental particles, composite atoms, molecules, biological cells, organisms, social institutions, or any other realized structure that can serve as a part within a larger structural context. SDT does not commit to any privileged scale; the choice of scale is dictated by the structural layer under analysis. Legitimacy criterion: a candidate $E$ is legitimate iff each candidate $e$ corresponds to some realized structure in $R$ (i.e., the role $e$ has a corresponding $S_{e} \in R$ ). Candidates whose putative members do not correspond to any $R$ -member are illegitimate (they would refer to non-realized possibilities, which $E$ does not capture). Boundary: SDT does not specify which entities qualify as legitimate elements at any given scale: this is filled in by entity theories applied to the particular structural context. SDT commits only to the formal status of $E$ as a non-empty set of role-references whose targets are members of $R$ .

§2.1.4 Possible configurations $P$ (definition and ontological anchor). Given $E$ , the elements may be combined in various distinct, differentiated ways. A possible configuration, denoted $p$ , is a complete specification of $E$ : it simultaneously specifies (i) the relational configuration of $E$ (how the members of $E$ stand in relations to each other) and (ii) the attribute specification of $E$ (the determinate values of attributes for each member of $E$ , including those attributes that have not been specified in prior determinations). The set of all possible configurations is denoted $P$ . Here “attribute” refers to the structural features carried by $e$ as a realized structure: features that determine $e$ ‘s compatibility with other configurations (the formal definition of attributes is given in §2.3.9 Type I internal argument); “relational configuration” refers to a specific way of combining the members of $E$ (the formal definition is given in §2.3.9 Type III internal argument).

§2.1.5 The foundational principle that $P$ is the set of unrealized possible configurations. The members of $P$ are unrealized possible configurations (modal markers of unrealized possibilities); already-realized structures, as members of $R$ , fall under the category of “actuality” and are not members of $P$ . This principle is the internal requirement of $P$ ‘s ontological status (see “ontological status” below). A key consequence: when $∣ E ∣ = 1$ , $P$ does not contain $e$ ‘s current state. Reason: $e$ , as an input element of $L$ , is already in $R$ ; its current realization belongs to the actualized category, not to “unrealized possible configurations.” $P$ , as the set of possible configurations of $E$ , can in the $∣ E ∣ = 1$ case only contain the possible specifications of $e$ ‘s unspecified attributes: these are the unrealized modal objects.

The multi-configuration source of $P$ (i.e., the source of $∣ P ∣ > 1$ ) divides strictly into four mutually exclusive cases in SDT, corresponding to the four determination types defined in §2.3.9 by the two-dimensional cardinality $(∣ E ∣, ∣ S ∣)$ . Specifically: (a) when $∣ E ∣ = 1$ and $∣ S ∣ = 1$ (Type I), $P$ ‘s multi-configuration source comes entirely from possible new compatibility patterns of $e$ ‘s attributes; (b) when $∣ E ∣ = 1$ and $∣ S ∣ > 1$ (Type II), it comes entirely from possible split-modes of $e$ together with attribute specifications of each product; (c) when $∣ E ∣ > 1$ and $∣ S ∣ = 1$ (Type III), it comes entirely from possible relational configurations among the $e_{i}$ ; (d) when $∣ E ∣ > 1$ and $∣ S ∣ > 1$ (Type IV), it comes from relational configurations together with multi-product attribute specifications. Each case bears the multi-configuration role for its determination type, with strict structural division of labor (see §2.3.9 fourfold types and strict division section).

Ontological anchor of $P$ . The ontological anchor of $P$ falls directly on §1.2 structure: that subsection characterizes structure as concrete, differentiated configurations of states. $P$ is the formalization of the question “in what concrete, differentiated ways can $E$ be configured under its intrinsic attribute permissions”: each $p \in P$ corresponds to one such concrete differentiated way of configuration. Members of $P$ are pairwise distinguishable by construction: if two members $p_{1}, p_{2} \in P$ agree on every specification, then by §1.2.11 PII strong reading they are the same configuration. For them to figure as different members of $P$ , they must differ on at least one element’s relational or attribute specification, where (per the division above) this difference is sourced entirely from one of the four type-specific channels and does not mix sources within a single $P$ .

§2.1.6 Possible configurations $P$ (mutual exclusivity, ontological status, boundary). Mutual exclusivity: distinct members of $P$ are mutually exclusive in the sense that at most one of them can be realized in any given determination event. This follows from §1.2.13 consistency: realizing two distinct configurations of the same $E$ simultaneously would entail contradictory specifications (each element having two distinct attribute values or relational positions), violating consistency. Mutual exclusivity is therefore not an additional commitment of $P$ but a consequence of consistency applied to the context of $E$ .

Ontological status: $P$ is a modal set: its members are formal markers of unrealized possibilities (modal markers), not actual entities. This modal status aligns with the weakest actualism commitment of §1.2.8 (only one actual world exists), without committing to any specific position within actualism (e.g., concerning the metaphysical status of unrealized possibilities). $P$ is constructed as a set within ZFC, but its individual members do not exist as actual entities until realized through determination.

Boundary: SDT does not specify the cardinality of $P$ for any particular $L$ . The cardinality may be finite, countably infinite, or uncountably infinite, depending on the entity-theoretic content of the structural layer; SDT commits only to $P$ being a well-defined set with at least one member.

§2.1.7 Determination and single-configuration selection (explicitation). The mutual-exclusivity argument above depends on a key clarification: a determination $D$ selects exactly one configuration $p$ from $A$ , not multiple configurations simultaneously. This statement is intrinsic to the concept of determination, not an independent commitment about product cardinality. The grounds are: (i) determination is by §2.3 the formal marker of “exactly one configuration in $A$ becoming a realized structure”; “exactly one” is built into the concept itself; allowing multiple mutually exclusive configurations in $A$ to be simultaneously realized would render $e$ ‘s specification multi-valued and mutually exclusive, violating §1.2.13 consistency. (ii) “Exactly one configuration” concerns configuration selection uniqueness, not product quantity uniqueness. The selected $p$ , as a complete specification of $E$ , may internally describe attribute specification of a single object (yielding a single $S$ ) or relational configuration / joint specification of multiple objects (yielding multiple $S$ ); the latter is not “ $A$ ‘s multiple configurations being simultaneously realized” but rather “the multi-object specification within a single $p$ being jointly realized,” structurally distinct from the former. (iii) Multiple realized structures arising simultaneously in physics (e.g., decoherence of an entangled pair, BSM events of entanglement swapping) are handled under the four-type classification of §2.3.9: if multi-product comes from joint specification within a single $p$ , this is the legitimate product of a single determination (Type II or Type IV); if multi-product comes from independent realization of different $p$ in different $L$ , this is a combination of multiple determinations.

A misconception to avoid: physical superposition states (e.g., the spin of an unmeasured electron) are not “simultaneously in mutually exclusive multiple states” but rather “in a determinate state where some attribute has not yet been specified by any determination.” A superposed-state electron, as a realized structure, is determinate (in $R$ ); its superposition reflects the non-specification of some attribute, not simultaneous specification to multiple values. Dual case (already-specified attributes becoming unspecified): a physical object’s state may oscillate bidirectionally between specified and unspecified attributes (steady state to superposition, superposition to steady state, etc.). SDT meta-level does not separately characterize this bidirectionality, since the §2.3.8 input ≠ output criterion automatically covers both directions: any legitimate determination producing $S^{'}$ that differs from input $e$ in any of the three layout-content components is licensed. The specific dynamics of attribute directionality is content for entity theories.

Ontological status of $P$ : members of $P$ are not realized structures; they are the formalization of $E$ ‘s possible configurations. Following SDT’s principle on borrowed terms (§1.2.2), SDT borrows “possibility” as a technical term and does not commit to any specific modal-philosophical position (Lewis 1986; Stalnaker 1976; Plantinga 1976; Adams 1981). The technical content of “possibility” in SDT: members of $P$ are the formal-level unfolding of the structural question “in what ways can $E$ be configured” raised by §1.2 structure; their specific content is jointly determined by $E$ ‘s attributes and $G$ ‘s meta-structural rules (§2.2), filled in by entity theory. $P$ as a whole is constructed as a set in ZFC, but each member of $P$ before being selected by a determination remains a possibility-level formal marker, not an actual entity; only after determination does it enter “actuality” as some $S$ . $P$ is therefore an ontologically modal set: its members are modal markers, not actual entities. This modal status aligns with the weakest actualism commitment of §1.2 reality (only one actual world exists), without further committing to any specific actualism internal position.

§2.1.8 Meta-level statement on the cardinality of $P$ . SDT meta-level does not commit to any specific cardinality of $P$ . $P$ may have cardinality $∣ P ∣ = 1$ (a single possible configuration: rare but not excluded), $∣ P ∣$ finite, $∣ P ∣$ countably infinite, or $∣ P ∣$ uncountably infinite. The cardinality is determined by the entity-theoretic content of the specific $L$ : what configurations $E$ admits under $G$ ‘s meta-structural rules. SDT only commits to $P$ being a non-empty well-defined set; specific cardinality is filled in by entity theories.

§2.1.9 Independence between the discreteness of $G_{D}$ and the cardinality of $∣ P ∣$ . A potential confusion concerns the relation between $∣ P ∣$ and the discreteness of the determination dependency graph $G_{D}$ (§2.4). $G_{D}$ is discrete because each determination is a discrete countable structural event; this discreteness has no implication for $∣ P ∣$ . Even if $∣ P ∣$ is uncountably infinite, $G_{D}$ remains discrete because each determination selects only one $p$ from $A \subseteq P$ , and the realization events form a discrete sequence regardless of $P$ ‘s cardinality. This independence allows SDT to handle physical theories with continuum-cardinality configuration spaces (e.g., quantum mechanics with continuous parameters) while maintaining the discrete determination structure.

§2.1.10 Compatibility conditions $C$ (weakest definition and preview). Compatibility conditions, denoted $C$ , formalize the constraints that filter $P$ to the admissible configurations $A \subseteq P$ : those configurations that pass the constraint filter and are eligible for realization. A weakest-definition preview: $C$ is a function $C : P \to {0, 1}$ assigning 0 (rejected) or 1 (admitted) to each $p \in P$ ; $A := {p \in P : C (p) = 1}$ . The full definition of $C$ requires $G$ (the global constraint, introduced in §2.2.2) and $R$ (the realized-structure set, introduced in §2.2.5), and is given in §2.2.7 after both $G$ and $R$ are in place. SDT’s commitment at this stage is only that $C$ exists as a filtering function; its specific content awaits the constraint system.

§2.2 The Constraint System

§2.2.1 Section overview. This section introduces the constraint system: the global constraint $G$ , the realized-structure set $R$ , the admissible-configuration set $A$ , and the full definition of $C$ . These four formal objects together constitute the constraint apparatus of SDT: the formal machinery by which possible configurations are filtered into admissible configurations and then into realized structures. The presentation proceeds: $G$ first (the meta-structural rule), $R$ next (the actual realized totality), then the full $C$ (which uses both $G$ and $R$ ), and finally $A$ (the filtered subset of $P$ ).

§2.2.2 Global constraint $G$ (definition, ontological anchor, existence argument). The global constraint, denoted $G$ , is the meta-structural rule governing what configurations are compatible with each other globally: across the entire realized totality, not just within any single $L$ . Formally, $G$ is the rule that, for any two structures, determines whether their joint realization is consistent. The ontological anchor of $G$ falls on §1.2.13 consistency: consistency requires that the realized totality not contain contradictory structures, and $G$ is the formalization of the global consistency rule.

Existence argument: $G$ must exist if the realized totality $R$ is to be coherently characterizable. Suppose $G$ did not exist; then there is no global rule determining structural compatibility, and any two realized structures might or might not be compatible without any rule governing the matter. This would entail either (i) that $R$ contains some pair of mutually incompatible structures (violating consistency, §1.2.13), or (ii) that $R$ contains no pair of mutually compatible structures (violating consists of, §1.2.10, since the realized totality could have no internal differentiation at all). Either disjunct contradicts the axiom; therefore $G$ exists.

Ontological status of $G$ (explicitation of the functional definition): in SDT formalization, $G$ is a functional mode, i.e., its content is exhaustively borne by its compatibility judgments on all $S \in R$ together with its compatibility judgments on all $p \in P$ co-existing with $R$ . $G$ has no “intrinsic identity,” “metaphysical essence,” or “rule form independent of action” beyond its action. This anchoring is the rule-level cashing-out of the principle established in §1.2 consists of (“reality consists of structure, structure is borne by content that can be coherently described”): $G$ , as a rule universally holding for all $S$ , has its coherently describable content in SDT formalization being only its action; any “intrinsic essence” beyond action lies outside §1.4 SDT’s scope. This ontological anchoring is the conceptual basis for §2.2.3’s uniqueness argument.

§2.2.3 Uniqueness of $G$ . $G$ is unique: there is exactly one global constraint, not multiple. Uniqueness follows directly from §2.2.2’s functional definition of $G$ , without borrowing any identity principle external to SDT. Suppose two distinct global constraints $G_{1}, G_{2}$ ( $G_{1} \neq = G_{2}$ ) both held universally for all realized structures; consider their relation. Case 1: the two are action-equivalent (for any $S$ , $G_{1}$ and $G_{2}$ give the same compatibility judgment; for any $p \in P$ co-existing with $R$ , the two give the same compatible/incompatible judgment). By §2.2.2’s ontological status of $G$ , $G$ ‘s content is exhaustively borne by action: if $G_{1}$ and $G_{2}$ have completely identical action, they have no distinguishable content in SDT formalization; " $G_{1}$ " and " $G_{2}$ " are two labels for the same $G$ . This conclusion does not depend on any external philosophical tradition (Leibnizian PII, Quinean identity criteria, etc.); it depends only on $G$ ‘s functional-definition entailment: an object defined as a “mode of action” is identical to another iff the actions are identical. The supposition " $G_{1} \neq = G_{2}$ " does not constitute a substantive distinction under the functional definition; the supposition fails. Case 2: the two are not action-equivalent (there exists some $S^{*}$ for which $G_{1}$ and $G_{2}$ give different compatibility judgments). But by hypothesis, both hold universally for all realized structures, including $S^{*}$ ; then $S^{*}$ both satisfies $G_{1}$ and fails to satisfy $G_{1}$ (per $G_{2}$ ‘s judgment), a contradiction. Case 2 is impossible. The two cases jointly establish that $G$ must be unique. This argument unfolds entirely within SDT formalization; uniqueness is the conceptual entailment of $G$ ‘s functional definition and does not depend on any external philosophical commitment.

This uniqueness is meta-structural: SDT does not commit to the specific content of $G$ (what specific compatibility rules $G$ contains: this is filled in by entity theory, e.g., physics specifies the conservation laws, symmetry rules, etc., that $G$ instantiates). SDT commits only that $G$ is the unique meta-structural rule governing global compatibility.

§2.2.4 Global constraint $G$ (preview of constancy, specific content, boundary). Preview of constancy: $G$ is constant: it does not change across determinations. The full argument for this is given in §3.2 (Argument 2). At this stage, $G$ is introduced as a formal object whose content is specified entity-theoretically; its constancy is established later.

Specific content: $G$ ‘s specific content (the specific compatibility rules it contains) is determined by entity theory. Examples: in physics, $G$ contains the laws of physics (energy conservation, momentum conservation, gauge invariance, etc.); in mathematics, $G$ contains the consistency requirements of the axiomatic system; in any other domain, $G$ contains the meta-rules governing structural compatibility in that domain. SDT meta-level does not regulate the content of $G$ ; it only commits to $G$ ‘s formal status as the unique global compatibility rule.

Boundary: SDT does not commit to whether $G$ is finitely or infinitely specifiable; whether it is uniformly applicable across all of reality or has different applicability conditions in different regions; whether it is fully accessible to inquiry or only partially accessible. These are entity-theoretic questions.

§2.2.5 Realized-structure set $R$ (definition, ontological anchor, $E$ - $R$ relation). The realized-structure set, denoted $R$ , is the set of all realized structures, with members denoted $S$ . One or more structures, through some determination, become realized, each entering $R$ as $S$ , and permanently belonging to $R$ (the full argument for permanence is given in §3.1; this section gives an existence preview). $R$ is constructed in standard ZFC (per the section-opening set-theoretic strength declaration); it does not constitute an “absolute totality”: $R$ contains only those structures that have actually been realized, a concrete set object, not some meta-set encompassing all possibilities. The ontological anchor of $R$ falls directly on §1.2 consists of: that subsection’s conjunctive claim requires that “reality consists of concrete, differentiated configurations existing,” treating all realized structures as a totality; $R$ is the formal correspondent of this totality in SDT.

A misconception to avoid: $R$ is not equivalent to “reality”; strictly speaking, $R$ is the formal-level correspondent of reality in SDT, the formal tool for treating all realized structures as a unified set object. “Reality” as an ontological category is characterized in §1.2 (the unique totality of realized states); $R$ is the formal-level concretization of this category, with different ontological status from “reality.”

Finally, the $E$ - $R$ relation: per the §2.1.2 element- $E$ first paragraph’s preview that ” $e$ has a correspondent in $R$ ,” this is now concretized as $E \subseteq R$ : any structural layer’s element set $E$ is a subset of $R$ , and each member $e$ of $E$ is some $S$ in $R$ . The difference between $E$ and $R$ lies only in perspective: $R$ is global (containing all realized structures); $E$ is the member list circled out from the perspective of some particular structural layer (containing the realized structures relevant to that layer).

Indecomposability of $R$ -members. Members $S$ of $R$ , as members of $R$ , are not decomposed within $R$ : $R$ has no internal hierarchical nesting or containment relations; all $S$ exist as independent members on a flat plane. SDT does not at the meta-level discuss whether some structure is decomposable: decomposition is discussed only within determination, where a determination $D$ establishes the relation “certain $e$ as inputs are connected to some $S$ as product.” This connection is a dependency relation in $G_{D}$ (§2.4), not a “nesting” or “containment” relation at the $R$ level. Treatment of composite structures: a composite structure (e.g., a molecule as the product of multi-atom relational-configuration merge) enters $R$ as an independent $R$ -member; its relation to its input $S_{e_{i}}$ ‘s is borne by the $G_{D}$ dependency relation established by the producing determination, not by a “nested hierarchy” within $R$ . The molecule $S_{molecule}$ and the atoms $S_{atom_{i}}$ in $R$ are each independent $R$ -members; $G_{D}$ contains dependency edges $D_{atom_{i}} \to D_{molecule}$ expressing their structural relationship. Ontological anchor of this principle: §1.2.11 PII strong reading treats each $S$ as a distinguishable independent individual in $R$ ; if $R$ permitted nesting, the nested member would simultaneously exist as “independent $R$ -member” and “internal part of another $R$ -member,” violating §1.2 consistency. Indecomposability of $R$ -members is the direct consequence of applying PII strong reading at the $R$ level.

§2.2.6 Realized-structure set $R$ (preview of non-retraction, boundary). Preview of non-retraction: $R$ does not retract: once an $S$ enters $R$ , it remains in $R$ permanently. The full argument is given in §3.1 (Argument 1). This is one of SDT’s three core arguments, and the present section only previews its existence.

Boundary: SDT does not commit to the specific cardinality of $R$ (finite, countably infinite, or uncountably infinite). At any meta-temporal stage of $G_{D}$ ‘s growth, $R$ has a definite (though unknown to SDT meta-level) cardinality determined by the realized determinations up to that stage. SDT’s only commitment is that $R$ is non-empty and grows monotonically.

§2.2.7 Compatibility conditions $C$ (retrospective completion: full definition). Now that $G$ and $R$ are both in place, the full definition of $C$ can be given (extending the weakest preview of §2.1.10). $C$ is the function

$C (p) = 1 ⟺ p satisfies G and p is jointly compatible with all S \in R under G .$

Two pathways jointly determine $C$ ‘s filtering: (i) Pathway one ( $G$ ‘s direct constraint): $G$ excludes any $p$ that violates $G$ ‘s meta-structural rules directly (e.g., a configuration violating energy conservation in physics is excluded by Pathway one if energy conservation is part of $G$ ). (ii) Pathway two ( $G$ ‘s indirect constraint via $R$ ): $G$ excludes any $p$ that, if realized, would result in $R \cup {p}$ violating $G$ ‘s consistency requirements, i.e., $p$ may be locally consistent with $G$ but jointly inconsistent with the existing members of $R$ .

Both pathways are filtering functions of $G$ ; the difference lies in whether the constraint is applied to $p$ alone (Pathway one) or to $p$ jointly with $R$ (Pathway two). $A := {p \in P : C (p) = 1}$ is the admissible-configuration set, the subset of $P$ that has passed both pathway filters.

§2.2.8 Formal precisification of “satisfies” in the formula. “Satisfies” in the formula above has a precise formal meaning. For $p$ to satisfy $G$ : $p$ does not violate any of the structural compatibility rules in $G$ ‘s content. For $p$ to be jointly compatible with $R$ under $G$ : the union $R \cup {p_{realized}}$ (where $p_{realized}$ is the realization of $p$ ) does not contain any pair of mutually incompatible structures under $G$ ‘s rules. The first is a unary check on $p$ alone; the second is a joint check on $p$ and existing $R$ -members. Both must succeed for $p$ to enter $A$ .

§2.2.9 Admissible configurations $A$ (definition, derivation relation, dynamic character, boundary). $A := {p \in P : C (p) = 1}$ . Derivation relation: $A$ is derived from $P$ via $C$ , where $C$ is jointly determined by $G$ and $R$ . Different $L$ may have different $A$ (different filterings), depending on the specific $E$ , $P$ , and $C$ of each $L$ .

Dynamic character: $A$ is dynamic in the sense that $A$ varies with $R$ . As $R$ grows (new $S$ entering $R$ via determinations), $C$ ‘s Pathway-two filtering changes (new joint-compatibility constraints), and $A$ may monotonically contract: configurations admissible at an earlier stage may become inadmissible at a later stage, but never vice versa (configurations inadmissible at an earlier stage cannot become admissible at a later stage, since $R$ only grows and never shrinks). This monotonic contraction is the formal precondition of $G_{D}$ ‘s temporal-order emergence (Argument 3, §3.3).

Boundary: SDT does not commit to $A$ ‘s specific cardinality. $∣ A ∣$ may be 0 (in which case $L$ is illegitimate, see §2.3.5), 1 (constrained (exactly one admissible configuration), or greater than 1, free (multiple admissible configurations). The cardinality reflects $C$ ‘s filtering strength on $P$ ; details are entity-theoretic.

§2.2.10 Formalization of the directional relation between $A$ and $R$ . $A$ depends on $R$ (via $C$ ‘s Pathway-two filtering), but $R$ does not depend on $A$ in any prior sense: $R$ at any stage is the union of all realizations from prior stages, and the present $A$ is what $R$ permits at this stage. Formally: $R (stage n) \to A (stage n)$ ; $R (stage n + 1) = R (stage n) \cup {S_{n + 1}}$ where $S_{n + 1}$ is the realization of some $p_{n + 1} \in A (stage n)$ . This directional relation is fundamental to SDT’s temporal structure: $R$ accumulates monotonically, and each accumulation reshapes the next $A$ .

§2.2.11 $C$ -neutrality of $S$ produced by equivalent determinations. Two determinations are equivalent (formally defined in §2.4.5) if their structural layers are equivalent, i.e., have the same structural pattern but different specific objects. The products of equivalent determinations, $S$ and $S^{'}$ , are structurally distinct in $R$ (different element-identity components in their layout content) but property-equivalent (their compatibility patterns under $G$ are identical). For $C$ ‘s filtering: when applying $C$ to a candidate $p$ for some new $L$ , both $S$ and $S^{'}$ count equally as members of $R$ for Pathway-two filtering: their structural distinctness in $R$ does not give them differential weight in $C$ ‘s filtering, because $C$ ‘s Pathway-two checks joint compatibility under $G$ , and equivalent products have identical compatibility patterns. This $C$ -neutrality of equivalent products is essential to the formalization of equivalence classes in §2.4.5.

§2.3 Structural Layers and Dynamical Concepts

§2.3.1 Section overview. §2.1 and §2.2 introduced all static components of SDT’s formal framework: $E$ , $P$ , $C$ , $G$ , $R$ , $A$ . This subsection treats three matters. First, integrating these components into SDT’s basic analytic unit, the structural layer $L = (E, P, C)$ , and clarifying $L$ ‘s ontological status and legitimate discussion range. Second, introducing SDT’s dynamical concept: determination (the formal marker of “exactly one configuration in $A$ becoming a realized structure”), the four types of determination (distinguished by the two-dimensional cardinality $(∣ E ∣, ∣ S ∣)$ into Type I attribute transformation, Type II split, Type III merge, Type IV merge-split, with internal arguments unfolded), and trace (the formal marker of the determination event itself, complementary to the realized structure). Third, stating the categorical division of labor among $T$ , $G_{D}$ , and $R$ , as the formal foundation for PII’s strict applicability range at the $R$ level and for the individuation of determination events. After this subsection, the conceptual foundation of SDT’s formal framework is fully in place; Chapter 3 unfolds the core arguments on this basis. This subsection also introduces a new set object, the trace set $T$ , parallel to $R$ in bearing the ontological division of labor between “determination event” and “determination product.”

§2.3.2 Structural layer $L$ (definition, integration, ontological status). $L := (E, P, C)$ . The structural layer integrates SDT’s three static components into a single analytic unit. Ontological status: $L$ is a retrospective analytic model, not an ontologically primitive object. SDT does not commit to $L$ as a fundamental entity in reality; rather, $L$ is the formal-level reconstruction of “the structural context of some determination event,” used to analyze the conditions under which some realized $S$ came into being.

This retrospective character is essential. SDT does not posit $L$ ‘s existing prior to determination; on the contrary, given an $S \in R$ , SDT can retrospectively analyze the $L$ corresponding to the determination that produced $S$ . The relation is realized $S$ $\to$ retrospective analysis of $L$ , not prior $L$ $\to$ realization of $S$ . This direction is fundamental to SDT’s via negativa strategy: SDT does not commit to a pre-existing space of all possible $L$ , only to the $L$ corresponding to actually realized determinations.

§2.3.3 Bijective correspondence between $L$ and $D$ . Every legitimate $L$ corresponds to exactly one determination $D$ , and every determination corresponds to exactly one $L$ . This bijection follows from (a) $L$ ‘s retrospective definition (each legitimate $L$ is the analytic reconstruction of some specific determination’s structural context, hence each $L$ corresponds to one specific $D$ ); and (b) determination’s definition as “exactly one $p \in A$ becoming realized” (each $D$ has exactly one $L$ from which $A$ is derived). The bijection $L \leftrightarrow D$ is therefore not an additional commitment but a consequence of the definitions.

§2.3.4 Reconciliation between retrospective analysis and counterfactual reasoning. A potential tension: if $L$ is purely retrospective (defined relative to actually realized determinations), how can SDT engage in counterfactual reasoning, e.g., “if some other $p \in A$ had been realized instead, then…“? The reconciliation: counterfactual reasoning at the SDT meta-level operates on the structure of $A$ within an actually realized $L$ . Given a realized determination with admissible set $A = {p_{1}, p_{2}, \dots}$ and the actually realized $p_{i}$ , SDT can analyze “if $p_{j}$ had been realized instead” by treating the alternative $p_{j} \in A$ as a counterfactual specification of $E$ ‘s configuration. This counterfactual remains tied to the actual $L$ (the same $E$ , $P$ , $C$ ), so it does not require positing a non-actual $L$ .

This reconciliation has a price: SDT does not engage in “deeper” counterfactuals (e.g., “if the laws of physics had been different”: which would require $G$ to vary, but $G$ is constant per Argument 2). Counterfactual reasoning in SDT is bounded by the actual $L$ and its $A$ .

§2.3.5 Structural layer $L$ (legitimate discussion range, boundary). The previous subsection established $L$ ‘s ontological status as a retrospective analytic model. This status directly produces $L$ ‘s legitimate discussion range: an $L = (E, P, C)$ is a legitimate object for SDT discussion iff there exists some $S \in R$ such that $L$ is the retrospective formalization of “the structural context of the determination that produced $S$ .” This criterion excludes four classes of illegitimate $L$ : (i) dangling $L$ corresponding to no actually realized determination: SDT does not permit constructing ” $E$ contains some virtual elements, $P$ gives some abstract configurations, $C$ gives some hypothetical rules” purely fictional $L$ ; (ii) $L$ with $∣ A ∣ = 0$ : if $C$ excludes all configurations in $P$ , no configuration can be realized, no $S$ can correspond; (iii) $L$ exceeding the range of consistent description: per §1.4 negative demarcation, if $L$ involves objects ungraspable by any description or whose every description is internally contradictory, it lies outside SDT’s purview; (iv) $L$ with $P = \emptyset$ : if $E$ admits no possible configuration for determination to unfold, no $p$ can enter $A$ .

The fourth class $P = \emptyset$ typically occurs when $∣ E ∣ = 1$ and all of $e$ ‘s attributes have been locked by prior determinations: per the §2.1.5 “unrealized possible configurations” principle, $e$ ‘s current state, as a realized structure, is not in $P$ , and $e$ has no unspecified attributes that could yield new possible configurations. Such an $L$ corresponds to “the case where all possible new specifications of $S$ are jointly closed off by $G$ and $R$ ”; whether the determination chain with $S$ as sole element extends further depends on whether $G$ permits any new $p$ in $S$ ‘s state, to be judged by entity theory. $P = \emptyset$ and $∣ A ∣ = 0$ (class ii) are formally related but have different grounds: class ii’s ground is ” $C$ excludes all configurations in $P$ ” ( $P \neq = \emptyset$ but $A = \emptyset$ ); class iv’s ground is ” $E$ bears no unrealized possible configurations” ( $P = \emptyset$ , hence automatically $A = \emptyset$ ). The four classes of illegitimate $L$ are not denied existence by SDT; rather, SDT does not discuss them: it remains silent on them, consistent with §1.4’s self-limitation. Boundary: $L$ ‘s specific content (specific $E$ , $P$ , $C$ instances) is determined by entity theory; SDT only commits to $L$ ‘s formal structure as a triple.

§2.3.6 Determination $D$ (definition, necessity, production of $S$ and $τ$ ). A determination is the formal marker, denoted $D$ , of “exactly one configuration in $A$ becoming a realized structure.” $D$ is not a process or action but the formal identification of some structural event, marking “at some moment in some $L$ , some $p$ in $A$ crossed the boundary from possibility to actuality.” This positioning is consistent with the retrospective direction of SDT’s argumentation in §1.1.1: $D$ is SDT’s formal identification of the existing fact that “some $p$ in $A$ has been realized,” not a description of the dynamical event of “transforming possibility into actuality.” “Exactly one configuration” concerns configuration selection, not product quantity; the internal structure of $p$ (single-object attribute specification or multi-object joint specification) is determined by $L$ ‘s determination type (§2.3.9 details; §2.1.7 already explains this distinction).

Necessity: in SDT’s framework, this is a definitional consequence rather than an independent axiom. Per §2.3.5’s criterion for $L$ ‘s legitimate discussion range, every legitimate $L$ must correspond to at least one $S \in R$ ; the existence of $S$ implies that some $p \in A$ was realized; this realization event is $D$ . Determination’s necessity is the logical consequence of the concept of legitimate $L$ , not an independent causal mechanism or dynamical claim: SDT does not claim “some force drives determinations to occur”; it only claims “if we retrospectively analyze some $S \in R$ , there must be some $L$ with $D$ corresponding.”

Determination $D$ simultaneously produces two classes of objects: realized structures entering $R$ : when the selected $p \in A$ is realized, it produces $n$ realized structures $S_{1}, \dots, S_{n}$ entering $R$ ( $n \geq 1$ , determined by $L$ ‘s determination type: $n = 1$ for Types I and III; $n > 1$ for Types II and IV). The products have no intrinsic order; subscripts are merely formal markers. Trace $τ$ entering $T$ : regardless of product count $n$ , the determination produces exactly one $τ \in T$ . The trace marks the determination event itself: a determination is a single structural event, and regardless of how many $S$ it produces, the event is single, hence $τ$ is singular. $S$ and $τ$ are categorically irreducible to each other ( $S$ concerns “what is realized,” $τ$ concerns “which determination realized it”); the precise correspondence among $R$ , $T$ , $G_{D}$ is treated in §2.3.13.

§2.3.7 SDT’s characterization of “stable existence”. A common confusion concerns “stable existence” of objects across time: if $R$ only grows, what does it mean for an electron to “stably exist” rather than constantly being a new realized structure? SDT’s characterization: stable existence at the meta-level is the persistence of a particular $S \in R$ . The electron at time $t_{1}$ and the electron at $t_{2}$ correspond to the same $S$ in $R$ ; what changes between $t_{1}$ and $t_{2}$ are the determinations involving this $S$ (e.g., interactions with other particles, attribute specifications, etc.), each of which produces new $S^{'}$ in $R$ (the “evolution” of the electron through the determination chain).

This characterization preserves both stability (the electron-as- $S$ persists in $R$ ) and change (new $S^{'}$ are added through determinations). The “evolution of the electron” is thus an evolutionary chain in $G_{D}$ , not retraction-and-replacement of the electron as $R$ -member.

§2.3.8 Core legitimacy criterion of determination (non-identity of input and output). A legitimate determination $D$ must satisfy: the products $S_{1}, \dots, S_{n}$ (output) are not identical to the inputs $E$ . Formally, for any $S_{i}$ produced by $D$ , $S_{i} \neq = e$ for any $e \in E$ in the layout-content three-component sense (element identity, relational configuration, attribute specification). At least one component must differ between $S_{i}$ and any $e$ ; otherwise the “determination” is vacuous (no transition from possibility to actuality occurred).

This criterion follows from determination’s definition: “exactly one configuration in $A$ becoming a realized structure” requires the configuration to be a new structural fact, not a repetition of what was already in $R$ . If output were identical to input, no new $S$ would enter $R$ , no new $τ$ would enter $T$ , and the supposed determination would be a null event. The criterion thus formalizes determination’s productive character.

§2.3.9 The four types of determination (Types I, II, III, IV: definitions). Determinations are classified into four types by the two-dimensional cardinality $(∣ E ∣, ∣ S ∣)$ :

Type I (attribute transformation, $∣ E ∣ = 1, ∣ S ∣ = 1$ ): a single input $e$ , a single product $S^{'}$ . The product corresponds to $e$ with some attribute change (specification, despecification, or transformation).
Type II (split, $∣ E ∣ = 1, ∣ S ∣ > 1$ ): a single input $e$ , multiple products $S_{1}, \dots, S_{n}$ . The single $e$ undergoes a structural event yielding multiple independent successor structures.
Type III (merge, $∣ E ∣ > 1, ∣ S ∣ = 1$ ): multiple inputs $e_{1}, \dots, e_{m}$ , a single product $S$ . The inputs jointly form a new composite structure.
Type IV (merge-split, $∣ E ∣ > 1, ∣ S ∣ > 1$ ): multiple inputs $e_{1}, \dots, e_{m}$ , multiple products $S_{1}, \dots, S_{n}$ . The inputs jointly participate in a structural event yielding multiple independent successor structures.

The classification is exhaustive: any legitimate determination falls into exactly one of the four types (proved in the fourfold-exhaustiveness section below).

Unified preconditions. Before unfolding the internal arguments of the four types, three preconditions hold uniformly across all types. All three are concrete cashings-out of commitments already established in §2.1 ( $P$ ), §2.1.10 ( $C$ ), and §1.2.13 (consistency); they introduce no new commitments.

Precondition 1 ( $P$ does not contain $E$ ‘s current state): members of $P$ in $L = (E, P, C)$ strictly do not include $E$ ‘s current state as a configuration. When $∣ E ∣ = 1$ , each $p \in P$ describes some new specification of $e$ ; when $∣ E ∣ > 1$ , each $p \in P$ describes some new total specification of $E$ . This principle stems from $P$ ‘s ontological status as the set of unrealized possible configurations: $E$ ‘s current state belongs to $R$ , not to $P$ .

Precondition 2 (configuration- $R$ compatibility): products of any legitimate determination must be compatible with existing structures in $R$ . This compatibility is borne by $C$ ‘s dual filtering: Pathway one ( $G$ ‘s direct constraint) excludes $p$ violating meta-structural rules; Pathway two ( $G$ ‘s indirect constraint via $R$ ) excludes $p$ that would conflict with existing $S$ in $R$ . Any element of $A \subseteq P$ has passed dual filtering and, upon realization, is necessarily jointly compatible with $R$ .

Precondition 3 (joint compatibility of multiple products): for determinations with $∣ S ∣ > 1$ (Types II and IV), the $n$ products are mutually jointly compatible, enforced by $C$ . Two levels of joint compatibility: at the minimum level (an SDT meta-level commitment), product attribute contents do not directly conflict under $G$ (enforced by $C$ ); specific correlation forms (e.g., Bell correlations) belong to entity theory.

Hierarchical relation among the three preconditions: Precondition 1 governs $P$ ‘s sourcing; Precondition 2 governs the legitimacy of configurations in $A$ ( $P$ ‘s members enter $A$ only if compatible with $R$ ); Precondition 3 governs joint compatibility of products within multi-product configurations. The three form a progressive constraint structure with no overlap. Preconditions 1 and 2 apply to all four types; Precondition 3 has substantive force only for $∣ S ∣ > 1$ types (II and IV). Together, the three preconditions ensure that the §2.3.8 input ≠ output criterion holds automatically across all four types.

$∣ A ∣$ configurations: unified treatment across the four types. Each type admits two legitimate $∣ A ∣$ configurations: $∣ A ∣ > 1$ (free) and $∣ A ∣ = 1$ (constrained). The two configurations have identical determination mechanism; they differ only in $C$ ‘s filtering strength. When $∣ A ∣ > 1$ , $C$ permits multiple alternatives, and the determination freely selects from among them; when $∣ A ∣ = 1$ , $C$ contracts the possible configurations to a unique admissible configuration via $G$ ‘s direct constraint or $R$ ‘s indirect constraint, and the determination realizes the unique alternative. The two filtering mechanisms ( $G$ direct, $G$ via $R$ ) apply to type-specific objects: Type I to attribute compatibility patterns; Type III to relational configurations; Types II and IV to composite configurations (split-mode + attribute joint specification + relational configuration). $∣ A ∣$ configurations do not change the determination type; sub-cases of each type by $∣ A ∣$ remain within that type and do not constitute independent types. The internal arguments below do not repeat $∣ A ∣$ configuration discussions.

Type I internal argument (attribute transformation as the core mechanism of single-object determination). Type I’s core problem: when $E$ contains only one member $e$ , the determination produces a single product $S^{'}$ , what is the source of $P$ ‘s multi-configuration, and what kind of structural relation does $S^{'}$ bear to $e$ ?

$P$ ’s multi-configuration source comes entirely from possible new compatibility patterns of $e$ ‘s attributes. “Attribute” in SDT is formally defined as a compatibility pattern: the rule by which $e$ , under $G$ , judges joint compatibility/incompatibility with all other possible co-existing objects. An attribute is not an intrinsic property of $e$ in some isolated sense but the relational pattern $e$ exhibits within $G$ . Each $p \in P$ describes one possible new attribute compatibility pattern for $e$ ; different $p$ correspond to different new patterns. The product $S^{'}$ , as a new $R$ -member, differs from $e$ in its layout-content three-component sense (here specifically in the attribute specification component): $S^{'}$ has the new attribute pattern that $e$ did not have. Note that $e$ corresponds to $S_{e} \in R$ which by §3.1 $R$ non-retraction permanently belongs to $R$ ; the determination produces $S^{'}$ as a new $R$ -member; both $S_{e}$ and $S^{'}$ exist in $R$ as independent members. The “evolution of $e$ ” in SDT is borne by the determination chain $S_{e} \to S^{'} \to S^{''} \to \dots$ in $G_{D}$ , not by $e$ ‘s being replaced by $S^{'}$ .

Three sub-cases of Type I (specification, despecification, transformation): (i) Specification: $e$ ‘s attribute changes from unspecified to specified (e.g., quantum measurement specifying spin direction). (ii) Despecification: $e$ ‘s attribute changes from specified to unspecified (e.g., a steady state evolving into a superposition). (iii) Transformation: $e$ ‘s attribute changes from one specified value to another specified value (e.g., a chemical reaction changing the bonding state). All three sub-cases share the structural form $∣ E ∣ = 1, ∣ S ∣ = 1$ with attribute change and are jointly handled as Type I.

Type II internal argument (split as the core mechanism of single-input multi-product determination). $E = {e}$ , $∣ E ∣ = 1$ , the determination produces $n > 1$ products $S_{1}, \dots, S_{n}$ . Each $p \in P$ describes a way that ” $e$ , after a structural event, produces $n$ independent successor structures,” with $p$ internally containing the joint specification of attributes for the $n$ successor structures. $P$ ’s multi-configuration can differ along two dimensions: the specific value of $n$ , and the joint-specification mode of each successor’s attributes.

Input-product relation: Type II’s input is a single $e$ , the products are $n$ independent $R$ -members. The relation between $e$ and the $S_{i}$ is borne by the dependency relation $D$ establishes in $G_{D}$ : $D$ ‘s input $e$ corresponds to some $S_{e}$ in $R$ ( $e$ is the in- $L$ role-reference to $S_{e}$ ); $D$ ‘s products $S_{1}, \dots, S_{n}$ each enter $R$ as new members; in $G_{D}$ , $D_{e}$ connects through $D$ to the productions of each $S_{i}$ . This connection is a $G_{D}$ dependency structure, not a nesting or containment relation in $R$ (§2.2.5 indecomposability). $S_{e}$ , as an $R$ -member, by §3.1 $R$ non-retraction permanently belongs to $R$ ; $S_{e}$ in this determination has been used once as the $e$ role, and by usage exclusivity cannot serve as the $e$ role of another $L$ (the role is no longer active after use), but $S_{e}$ ‘s existence as an $R$ -member is unaffected.

Product relations: the $n$ products each exist as independent $R$ -members with no intrinsic order (subscripts are merely formal markers); each can independently serve as input to a subsequent determination (each satisfying usage exclusivity). PII distinction among the $n$ products is automatically satisfied by $p$ ‘s internal structure ( $p$ describes $n$ distinct successor structures, and after realization they must be pairwise distinct). Specific correlation forms (e.g., property correlations between decoherence products of an entangled pair) are filled in by entity theory.

Type II physical correspondent: a single object, through a structural event, separates into multiple independent successor objects; specific physical processes are judged by entity theory.

Type III internal argument (merge as the core mechanism of multi-input single-product determination). Type III’s core problem: when $E$ contains multiple members $e_{1}, e_{2}, \dots, e_{n}$ ( $n \geq 2$ ) and the determination produces a single product $S$ , what is the source of $P$ ‘s multi-configuration, and what kind of structural relation does $S$ bear to the members of $E$ ?

$P$ ’s multi-configuration source comes entirely from relational configurations among members: each $p \in P$ contains all $n$ members of $E$ (per §2.1’s “complete specification”); different $p$ differ in relational configuration, i.e., the way in which $E$ ‘s members are mutually combined. Each possible combination way constitutes a $p \in P$ . Formalization of relational configurations: the range of possibilities for relational configurations is determined by $G$ ‘s meta-structural rules; specific relation types (topological connection patterns among members, complex relational structures, directionality, binary or higher-arity relations, etc.) are given by entity theory. SDT only commits to the existence of multiple possible relational configurations; it does not regulate their specific form.

Structural features of product $S$ : Type III’s product $S$ is a new composite structure. $S$ ’s layout content describes the relational configuration established among $E$ ‘s members through this determination, entering $R$ as a new member. $S$ ’s relation to the $e_{i}$ ‘s in $R$ : $S$ and the $S_{e_{i}}$ corresponding to the $e_{i}$ ‘s each exist as independent members of $R$ ; in $G_{D}$ , the $D_{e_{i}}$ producing each $S_{e_{i}}$ connect through Type III determination $D$ to the production of $S$ . $S$ does not “contain” or “comprise” the $S_{e_{i}}$ as sub-members in $R$ ; hierarchical relations in $R$ are borne by $G_{D}$ ‘s dependency structure, not by $S$ ‘s internal “containment” (§2.2.5). $S$ , as a new $R$ -member, differs from the $S_{e_{i}}$ in layout-content three-component sense (§2.3.8 input ≠ output, borne by Precondition 1’s uniform application across the four types).

Type III does not involve attribute specification: each $e_{i}$ ‘s attribute specification within this $P$ remains constant; $P$ ‘s multi-configurations differ only in how members are mutually connected. Any attribute specification of an $e_{i}$ must precede the Type III determination with $e_{i}$ as element. Type III physical correspondent: multiple independent objects, through a structural event, jointly form a single new structure; specific physical processes are judged by entity theory.

Type IV internal argument (merge-split as the core mechanism of multi-input multi-product determination). Type IV’s core problem: when $E$ contains multiple members $e_{1}, \dots, e_{m}$ ( $m \geq 2$ ) and the determination produces multiple products $S_{1}, \dots, S_{n}$ ( $n \geq 2$ ), how is $P$ ‘s multi-configuration produced, and how does the selected $p$ simultaneously describe the input-side relational configuration and the output-side multi-product joint specification? Type IV’s $E = {e_{1}, \dots, e_{m}}$ , $∣ E ∣ > 1$ ; the determination produces $n \geq 2$ products. Each $p \in P$ describes a joint structural event: $E$ ‘s members participate jointly in the determination through some relational configuration, the determination producing $n$ independent successor structures, with $p$ internally containing joint specification of attributes for each successor. $P$ ’s different $p$ can differ along three dimensions: members’ relational configuration, product count $n$ , and joint-specification mode of product attributes.

Input-product relation: Type IV is a joint structural transformation within a single determination event, not a two-step ” $m$ inputs merge, then split into $n$ products” process. The selected $p$ describes $E$ ‘s members participating jointly in the determination through a specific relational configuration, the determination producing $n$ new structures as successors. The $S_{e_{i}}$ corresponding to each $e_{i}$ , by §3.1 $R$ non-retraction, permanently belongs to $R$ (each $S_{e_{i}}$ in this determination has been used once as the $e$ role, and by usage exclusivity cannot serve as the $e$ role of another $L$ , but its existence as an $R$ -member is unaffected); each $S_{j}$ enters $R$ as a new member. In $G_{D}$ , determination node $D$ is simultaneously the successor of each $D_{e_{i}}$ and the common predecessor of the productions of each $S_{j}$ . PII distinction among the multiple products is automatically guaranteed by $p$ ‘s internal structure (§2.3.13).

Irreducibility to Type III + II combination: Type IV is not the two-step combination of Type III followed by Type II. Consider two scenarios: (i) Type III + Type II two-step combination: first, Type III determination $D_{1}$ merges ${e_{1}, \dots, e_{m}}$ into a single intermediate structure $S_{mid}$ ; second, Type II determination $D_{2}$ takes $S_{mid}$ as input and produces ${S_{1}, \dots, S_{n}}$ . The intermediate $S_{mid}$ permanently remains in $R$ as an independent $R$ -member. (ii) Type IV single determination: $D$ takes ${e_{1}, \dots, e_{m}}$ directly as input and produces ${S_{1}, \dots, S_{n}}$ , with no intermediate $S_{mid}$ entering $R$ , and only a single node in $G_{D}$ . The two scenarios differ in $R$ content (scenario i has an additional $S_{mid}$ ), in $G_{D}$ topology (scenario i has two nodes in series, scenario ii has a single node), and in $T$ set (scenario i has two $τ$ , scenario ii has one). Discrimination principle: depends on whether $R$ contains the corresponding intermediate structure $S_{mid}$ as a traceable realized structure; if yes, it is a two-step combination; if no, it is a Type IV single determination. Specific physical processes are judged by entity theory.

Physical correspondent: multiple independent objects, through a structural event, jointly produce multiple independent successor objects. Specific correlations (product attribute contents influenced by input configuration; correlation forms among products) are filled in by entity theory.

Conceptual argument for determination atomicity. The strict division of labor and exhaustiveness of the four types both depend on a prior commitment: the product cardinality $∣ S ∣$ of a single determination is fixed within that determination. This subsection explicitly argues for this prior commitment from established commitments, lifting atomicity from an implicit methodological assumption to a conclusion of conceptual explication.

Argumentative dependencies: (i) §2.3.6 determination definition (exactly one $p$ from $A$ is realized); (ii) §2.1 $P$ section ( $p$ as a complete specification of $E$ ); (iii) §2.4 usage exclusivity; (iv) §1.2.13 consistency.

Step 1 ( $∣ S ∣$ fixed within a single determination): a single determination selects exactly one $p$ from $A$ (§2.3.6). $p$ , as a complete specification of $E$ (§2.1 $P$ section), has its internal structure already describing the attribute and relational-configuration specifications of all products as $p$ ‘s intrinsic content; this is not determined after $p$ is realized. “Complete specification” means all specifications lie within $p$ ; if product cardinality were determined by processes external to $p$ , the concept of “completeness” would fail. Therefore, once $p$ is realized, the products specified within $p$ enter $R$ in one operation; $∣ S ∣$ is fixed by $p$ ‘s internal specification and cannot vary during the determination process.

Step 2 (inter-type non-decomposability): with $∣ S ∣$ ‘s fixedness established by Step 1, one can further argue the impossibility of certain inter-type decompositions. Type II cannot be decomposed into multiple Type I determinations: consider decomposing " $e \to {S_{1}, \dots, S_{n}}$ " into $n$ Type I determinations. Parallel decomposition ( $n$ determinations using $e$ simultaneously as input) violates §2.4 usage exclusivity. Sequential decomposition ( $e \to S_{1}$ followed by $S_{1} \to S_{2}$ ) yields a Type I chain with a single final product, structurally distinct from Type II’s multiple parallel products. Neither decomposition can reach Type II’s structural form. Type IV cannot be decomposed into a Type III + Type II equivalence: consider decomposing " ${e_{1}, \dots, e_{m}} \to {S_{1}, \dots, S_{n}}$ " into Type III followed by Type II. Such decomposition introduces the intermediate structure $S_{mid}$ into $R$ (§2.2.5 and §3.1), adds one node to $G_{D}$ , and adds one $τ$ to $T$ . This is a structurally distinct determination form from a single Type IV determination (no intermediate structure, single $G_{D}$ node, single $T$ $τ$ ) in SDT formalization. Which form corresponds to a specific physical process is judged by entity theory (see Type IV internal-argument section); SDT meta-level confirms both as legitimate distinct structural forms.

Conclusion: determination atomicity, comprising both $∣ S ∣$ ‘s fixedness within a single determination and the non-decomposability of certain inter-type forms, is the joint internal entailment of §2.3.6 determination definition, §2.1 $P$ ‘s complete specification, §2.4 usage exclusivity, and §1.2.13 consistency. This conclusion supports the subsequent strict division-of-labor section (the four types are strictly mutually exclusive in determination mechanism) and the fourfold-exhaustiveness section (any legitimate determination belongs to exactly one of the four types).

Strict division of labor and necessity of the four types. The four types are strictly mutually exclusive in their determination mechanism: $L$ ‘s determination type is jointly determined by $∣ E ∣$ and $∣ S ∣$ two-dimensional cardinality. This strict division of labor stems from the joint working of determination atomicity (§2.3.6) and §1.2.13 consistency, and cannot be relaxed.

$P$ ’s multi-configuration source under the four types: Type I ( $∣ E ∣ = 1, ∣ S ∣ = 1$ ): $e$ ‘s attribute change modes; Type II ( $∣ E ∣ = 1, ∣ S ∣ > 1$ ): $e$ ‘s split modes plus product attribute joint specifications; Type III ( $∣ E ∣ > 1, ∣ S ∣ = 1$ ): $E$ members’ relational configurations; Type IV ( $∣ E ∣ > 1, ∣ S ∣ > 1$ ): relational configurations plus multi-product attribute joint specifications. The four types do not mix $P$ ‘s multi-configuration sources within a single determination.

Core constraint and ordering: if some $e$ in a structural event undergoes both attribute change and participation as a member of a relational configuration, this is two determinations (Type I or II first, Type III or IV second), not a single composite determination. Necessity of the ordering: Types III and IV require all $e_{i}$ in $E$ to be in $R$ with all relevant attributes locked; if some attribute is unspecified, the Type III/IV determination’s $P$ cannot give complete relational-configuration specification. The two determinations connect through §2.4 determination chain or merge.

Formal criterion of strict division: the determination type of a candidate $L$ is jointly determined by two dimensions: $∣ E ∣$ separates Types I/II from III/IV; $∣ S ∣$ separates Types I/III from II/IV. Discrimination of composite determination: if a candidate $L$ ‘s $P$ contains two configurations $p, p^{'}$ that simultaneously differ in input attribute specification and relational configuration, $L$ is not a legitimate single determination but must be temporally decomposed into multiple determinations.

Necessity of strict division: cannot be relaxed, established from two perspectives. Ground 1 ( $L$ ‘s formal definition): $∣ E ∣$ in $L$ is fixed ( $E$ is $L$ ‘s first component), $L$ is either $∣ E ∣ = 1$ or $∣ E ∣ > 1$ , with no drift between them. Ground 2 (determination atomicity): $∣ S ∣$ is determined by the structural feature of the determination event; the product count of a single determination event is fixed, with no indeterminacy between multiple $∣ S ∣$ values. The two grounds jointly entail that any candidate “mixing multi-configuration sources” does not constitute a legitimate single determination. Relation to temporal order: strict division of labor is the necessary precondition for treating determinations as single $G_{D}$ nodes (§2.4) and for the time-emergence argument (§3.3); relaxing strict division would render $G_{D}$ node structure unclear and shake the argumentative foundation.

Fourfold exhaustiveness. The previous section established the strict division of labor of the four types; this section argues that the four exhaust all legitimate determinations: no fifth type of legitimate $L$ exists. The argument enumerates by $(∣ E ∣, ∣ S ∣)$ two-dimensional cardinality. Consider any legitimate $L = (E, P, C)$ . By §2.3.5, $L$ satisfies: corresponds to at least one $S \in R$ ; $∣ A ∣ \geq 1$ ; lies within SDT description range; $P \neq = \emptyset$ .

Enumeration along $∣ E ∣$ and $∣ S ∣$ dimensions: (i) $∣ E ∣ = 0$ impossible: $E$ is the structural layer’s non-empty element set, $∣ E ∣ \geq 1$ is a definitional requirement; (ii) $∣ S ∣ = 0$ impossible: a determination by §2.3.6 produces at least one product $S$ , $∣ S ∣ = 0$ directly conflicts with the determination definition. Hence legitimate $L$ satisfies $∣ E ∣ \geq 1$ and $∣ S ∣ \geq 1$ .

Two-dimensional joint enumeration: (a) $∣ E ∣ = 1, ∣ S ∣ = 1$ $\to$ Type I (attribute transformation). $E$ ’s sole member $e$ produces a unique new $R$ -member $S^{'}$ through the determination; $P$ ‘s multi-configuration source is $e$ ‘s attribute change modes. (b) $∣ E ∣ = 1, ∣ S ∣ > 1$ $\to$ Type II (split). $e$ produces multiple new $R$ -members $S_{1}, \dots, S_{n}$ through the determination; $P$ ‘s multi-configuration source is $e$ ‘s split modes plus product attribute joint specifications. (c) $∣ E ∣ > 1, ∣ S ∣ = 1$ $\to$ Type III (merge). $e_{1}, \dots, e_{m}$ jointly produce a unique new $R$ -member $S$ as composite structure through the determination; $P$ ‘s multi-configuration source is members’ relational configurations. (d) $∣ E ∣ > 1, ∣ S ∣ > 1$ $\to$ Type IV (merge-split). $e_{1}, \dots, e_{m}$ jointly produce multiple new $R$ -members $S_{1}, \dots, S_{n}$ through the determination; $P$ ‘s multi-configuration source is relational configurations plus multi-product attribute joint specifications.

Exhaustiveness conclusion: combining $∣ E ∣ \geq 1$ and $∣ S ∣ \geq 1$ , only four legitimate cells exist in the two-dimensional cardinality space: $(∣ E ∣ = 1, ∣ S ∣ = 1)$ , $(∣ E ∣ = 1, ∣ S ∣ > 1)$ , $(∣ E ∣ > 1, ∣ S ∣ = 1)$ , $(∣ E ∣ > 1, ∣ S ∣ > 1)$ , each corresponding to one type. Any legitimate $L$ must belong to Type I, II, III, or IV; no fifth type of legitimate determination exists. Treatment of $∣ P ∣$ and $∣ A ∣$ dimensions: $∣ P ∣$ and $∣ A ∣$ are not classification dimensions but sub-configuration dimensions within each type. $∣ P ∣$ is filled by entity theory; $∣ A ∣$ is distinguished within each type into $∣ A ∣ > 1$ (free) and $∣ A ∣ = 1$ (constrained) sub-cases (see internal-argument sections and the unified $∣ A ∣$ section). Different $∣ P ∣$ and $∣ A ∣$ configurations do not change the determination type; they only affect the type’s specific form.

Decomposition of composite phenomena into determination sequences. Most physically complex phenomena involve combinations of multi-element interactions, attribute changes, and multi-product generation. SDT handles such phenomena in two cases: single Type IV determination (if the phenomenon is a single structural event directly producing multi-input multi-product, with no corresponding intermediate $S_{mid}$ in $R$ ) and determination-sequence decomposition (if the phenomenon unfolds through multi-step structural events with intermediate products as independent $R$ -members). The discrimination principle: whether $R$ contains a corresponding intermediate $S_{mid}$ as a traceable realized structure.

General principle of determination-sequence decomposition (applicable to multi-step decomposition cases): (i) identify each step’s type (Type I attribute change, Type II split, Type III merge, Type IV merge-split); (ii) establish ordering by temporal constraints (Types I and II must precede Types III and IV taking their products as elements); (iii) connect through §2.4.4 determination relations (chain, merge, split, merge-split). Specific decomposition is judged by entity theory based on the physical event’s structure.

Commitment and boundary: SDT only commits that any composite phenomenon can be decomposed into ordered combinations of Type I, II, III, IV determinations, the necessary corollary of strict division of labor and fourfold exhaustiveness; specific decomposition of each phenomenon is determined by entity theory. Decomposition of composite phenomena does not break determination atomicity: each decomposed step is an SDT-formalized atomic unit; the integral character of composite phenomena is borne by $G_{D}$ relations, not by introducing a new ontological layer above the determination level.

§2.3.10 Trace $τ$ and trace set $T$ (definition, ontological status, time-emergence preview). A trace, denoted $τ$ , is the formal marker of a determination event itself. Each determination $D$ produces exactly one $τ$ (per §2.3.6); the trace set $T$ is the set of all $τ$ corresponding to actually realized determinations. $T$ is constructed in standard ZFC and is non-empty (contains at least one $τ$ per the existence of at least one determination, which follows from the existence of at least one $S \in R$ ).

Ontological status of $τ$ : $τ$ is the formal marker of the determination event; it is not the determination itself (which is the structural event), nor the realized structure (which is in $R$ ), but the formal individuation of the event. The categorical division of labor among $R$ , $T$ , $G_{D}$ (§2.3.13) ensures that traces and realized structures occupy distinct ontological roles.

Time-emergence preview: a partial-order relation can be defined on $T$ (via §2.3.12), and this partial order corresponds to the time-emergence argument of §3.3. SDT commits at this stage to the existence of $T$ and the formalizability of the partial order; the full time-emergence argument is treated in §3.3.

§2.3.11 Two equivalent determinations produce $S$ ‘s that are structurally distinct but property-equivalent. When two determinations are equivalent (in the §2.4.5 sense: same structural pattern, different specific objects), their products $S$ and $S^{'}$ exhibit a dual feature: structurally distinct in $R$ (different element-identity components in their layout content), but property-equivalent (their compatibility patterns under $G$ are identical). The structural distinctness ensures $S \neq = S^{'}$ as members of $R$ (PII strong reading respected); the property equivalence ensures both $S$ and $S^{'}$ count equally for $C$ ‘s Pathway-two filtering of any future configuration. This dual feature is essential to the formalization of equivalence classes in §2.4.5.

§2.3.12 Concrete formalization of the partial order on traces. Given $τ_{i}, τ_{j} \in T$ , define $τ_{i} ≺ τ_{j}$ iff $D_{j}$ ‘s $L_{j}$ has some $e \in E_{j}$ equal to some product $S$ of $D_{i}$ (per the §2.4 dependency definition). Equivalently: $τ_{i} ≺ τ_{j}$ iff there is a directed path from node $D_{i}$ to node $D_{j}$ in $G_{D}$ . This partial order is the concretization of the partial order induced by $G_{D}$ on $T$ .

The partial order is irreflexive ( $τ \neq ≺ τ$ , since no determination’s product is its own input), antisymmetric ( $τ_{i} ≺ τ_{j}$ implies not $τ_{j} ≺ τ_{i}$ , by $G_{D}$ ‘s acyclicity), and transitive ( $τ_{i} ≺ τ_{j}$ and $τ_{j} ≺ τ_{k}$ imply $τ_{i} ≺ τ_{k}$ , by composition of directed paths). The partial order on $T$ is therefore well-defined and provides the formal basis for the time-emergence argument of §3.3.

§2.3.13 Categorical division of labor among $T$ , $G_{D}$ , and $R$ . $R$ , $T$ , $G_{D}$ are three independent formal levels each handling a distinct categorical fact; this categorical division of labor is the internal structure of SDT’s formalization, with direct supporting role for §1.2 PII strong reading’s strict applicability range, §1.3 Branch-four Black response, and §2.4 equivalent structural layer treatment.

Three-level division of labor: $R$ (realized-structure set) handles “what is realized”; its members $S$ are realized structures. $T$ (trace set) handles “which determination realized it”; its members $τ$ are markers of determination events. $G_{D}$ (determination dependency graph; formalized in §2.4) handles “structural dependencies among determinations”; its structure is a directed acyclic graph.

Correspondence relations among the three: $T$ and $G_{D}$ nodes are in strict bijection (each determination $D$ adds one $τ$ to $T$ and one node to $G_{D}$ ). $R$ ’s correspondence with $T$ / $G_{D}$ is not bijective: each determination adds $n$ new members to $R$ ( $n \geq 1$ , determined by $L$ ‘s determination type), so each $τ$ corresponds to a non-empty realized-structure set ${S_{1}, \dots, S_{n}} \subseteq R$ ; each $S \in R$ uniquely belongs to the product set corresponding to some $τ$ . When $n = 1$ (Types I and III), the correspondence reduces to a bijection between $R$ and $T$ ; when $n > 1$ (Types II and IV), one $τ$ corresponds to multiple $S$ , but each $S$ is still uniquely traceable to that $τ$ . Preservation of traceability: any $R$ -member is uniquely traceable to the determination $D$ producing it and its trace $τ$ ; the “determination event partial order” upon which the time-emergence argument (§3.3) depends is borne by the $T$ / $G_{D}$ level, unaffected by $R$ -side product-count variations.

Division of labor of identity criteria: the three independently bear identity criteria. $R$ uses §1.2.11 PII strong reading (layout-content three components: element identity, relational configuration, attribute specification) to judge whether two $S$ are identical; $T$ bears trace identity by determination event identity itself; $G_{D}$ bears node identity by node topological position in DAG. PII strong reading applies only to $R$ ; $T$ and $G_{D}$ are not constrained by PII.

Application of PII to multi-product determinations: the multiple $S_{1}, \dots, S_{n}$ produced by Type II and Type IV determinations, as $R$ -members, are pairwise distinct under PII strong reading; this is automatically guaranteed by the internal structure of $p \in A$ as a complete specification of $E$ ( $p$ describing the specification of $n$ objects necessarily yields pairwise-distinct specifications under PII), without requiring additional formal commitment.

Dual feature of equivalent determination products: by §2.4 equivalent structural layer and usage exclusivity jointly applied, the two products $S$ and $S^{'}$ of equivalent determinations exhibit a dual feature: at the $R$ level structurally distinct (layout-content $E$ -identity component differs, by PII strong reading judged as two distinct $R$ -members); at the property level equivalent ( $C$ ‘s filtering rules are equivalent, products’ compatibility patterns under $G$ are identical, belong to the same equivalence class). “Structurally distinct + property-equivalent” guarantees PII strictness at the $R$ level while bearing the formal significance of equivalence-class classification. At $T$ level each equivalent determination corresponds to a new $τ$ ; at $G_{D}$ level it corresponds to a new node, independently bearing determination event individuality.

Application value of categorical division of labor: the standard quantum mechanics treatment of “indistinguishable particles” (assigning particle individuality to relational structure rather than the particle itself), Black’s (1952) mirror-universe challenge (with core presupposition “individuality must be borne by structure itself”) are both handled from this division-of-labor perspective: the former is homologous with the division-of-labor structure; the latter’s core presupposition is directly rejected by this division (individuality is distributed across $R$ , $T$ , $G_{D}$ as three independent levels). Categorical division of labor is not an ad hoc response to specific challenges but the internal organizational principle of SDT’s formal framework.

§2.4 Structural Relations among Determinations

§2.4.1 Section overview. Within a single $L$ , the formal apparatus of $E$ , $P$ , $C$ , $A$ , $D$ , $τ$ is sufficient. But the realized totality contains many determinations, and structural relations among them require additional apparatus. This subsection introduces the dependency relation between determinations, usage exclusivity of $e$ , the four basic forms of inter-determination structural organization (chain, merge, split, merge-split), the equivalent structural layer concept, and the determination dependency graph $G_{D}$ as the integrated formal object. After this subsection, SDT’s full formal framework is in place; Chapter 3 unfolds the core arguments on this basis.

§2.4.2 Determination dependency relation (definition, ontological anchor, unidirectionality). Two determinations $D_{i}$ and $D_{j}$ ( $i \neq = j$ ) stand in the dependency relation $D_{i} \to D_{j}$ iff there exists some $e \in E_{j}$ (where $E_{j}$ is $L_{j}$ ‘s element set) such that $e$ is some product of $D_{i}$ (i.e., $e$ corresponds to some $S \in R$ that was produced by $D_{i}$ ). Read informally: $L_{j}$ ‘s element set contains some element produced by $D_{i}$ , hence $D_{j}$ ‘s structural context depends on $D_{i}$ having occurred.

Ontological anchor: the dependency relation falls on §2.1.2 element- $E$ ’s commitment that each $e$ corresponds to some realized $S$ . Since $e \in E_{j}$ requires $e$ to correspond to some realized $S$ , and $S$ was produced by some determination $D_{i}$ , $L_{j}$ ‘s legitimacy (per §2.3.5) presupposes $D_{i}$ ‘s occurrence. Dependency thus has direct ontological grounding.

Unidirectionality: dependency is unidirectional. If $D_{i} \to D_{j}$ , then $D_{j} \neq \to D_{i}$ . This unidirectionality follows from the asymmetric nature of ” $e$ being a product of $D$ ”: if $e \in E_{j}$ is a product of $D_{i}$ , then $D_{j}$ ‘s context formed after $D_{i}$ ‘s realization (i.e., $D_{i}$ ‘s product was the input to $D_{j}$ ); $D_{j}$ ‘s product cannot have been $D_{i}$ ‘s input, since $D_{i}$ already realized prior to $D_{j}$ . Unidirectionality is the formal precondition of $G_{D}$ ‘s acyclicity (§2.4.7).

Multi-product determinations and dependency: when $D_{i}$ is a multi-product determination (Type II or IV), $D_{i}$ has products $S_{1}, \dots, S_{n}$ . Dependency $D_{i} \to D_{j}$ holds iff some $e \in E_{j}$ equals some product of $D_{i}$ (any one of $S_{1}, \dots, S_{n}$ suffices). Different $D_{j}$ may depend on different products of $D_{i}$ .

§2.4.3 Usage exclusivity of $e$ (argument and consequences). Usage exclusivity: each realized $S \in R$ can serve as the role $e$ in at most one $L$ . If $S$ has served as $e$ in $L_{j}$ (i.e., participated in $D_{j}$ as input), $S$ cannot serve as $e$ in any other $L_{k}$ ( $k \neq = j$ ). Usage exclusivity is not an arbitrary stipulation but follows from §1.2.13 consistency together with the role-vs-membership distinction: " $e$ " denotes the role of an element within a specific structural context, and $S$ playing that role in $D_{j}$ is part of $D_{j}$ ‘s structural event; $S$ playing the same role in another $D_{k}$ would entail two distinct structural events with $S$ in the same role, but $S$ ‘s role-instance is tied to a specific event (each role-occurrence is event-specific).

Argumentative basis of usage exclusivity (precisification): the argument above moves quickly from ” $S$ playing the same role in two events” to “role-instance specificity is violated.” “Two distinct structural events with $S$ in the same role” does not by itself immediately equate to “two mutually exclusive states”; two $L$ ‘s could in principle handle different aspects of $e$ . This precisification makes the conceptual basis of usage exclusivity explicit. The argument depends on the joint application of four established commitments: (i) §2.1 $P$ ‘s complete-specification requirement ( $p$ is a complete specification of $E$ , covering all attributes of $e$ and all relational-configuration specifications); (ii) §2.3.8 input ≠ output (a legitimate determination’s product must structurally differ from the input; each determination changes $e$ ‘s state); (iii) §1.2.13 consistency; (iv) §2.3.13 categorical division of labor among $R$ , $T$ , $G_{D}$ (each $D$ corresponds to a unique $τ$ and a unique $G_{D}$ node; each $D$ produces a unique combination of new $R$ -members).

Precise argument: suppose $e$ is simultaneously a member of $E$ in $L_{1}$ and $L_{2}$ ( $L_{1} \neq = L_{2}$ ). By (i), $p_{1} \in A_{1}$ and $p_{2} \in A_{2}$ both cover all of $e$ ‘s attributes; there is no ” $L_{1}$ handles only $e$ ‘s attribute a, $L_{2}$ handles only $e$ ‘s attribute b” partial-specification form. By (ii), $D_{1}$ and $D_{2}$ each transform $e$ into products $S_{1}, S_{2}$ . Consider two cases. Case 1 ( $S_{1} = S_{2}$ ): two independent determinations $D_{1}, D_{2}$ produce the same $R$ -member. But $D_{1}, D_{2}$ are independent determination events; by (iv), each corresponds to an independent $τ$ and an independent $G_{D}$ node. Two independent nodes corresponding to the production of the same $R$ -member would invalidate the bijection $D \leftrightarrow τ \leftrightarrow node \leftrightarrow R$ -member. Case 2 ( $S_{1} \neq = S_{2}$ ): $e$ is simultaneously transformed by two determinations into two different states. But $e$ as a specific object in $R$ , its “transformed state” is borne by a specific $S$ in $R$ (by PII strong reading, only one $S$ can correspond), and cannot simultaneously correspond to two mutually exclusive $S_{1}, S_{2}$ , violating (iii). Neither case is possible.

Conclusion: usage exclusivity is the joint internal entailment of complete specification + input ≠ output + consistency + categorical division of labor, derived directly from established commitments and not an additional formal constraint. This precisification makes the argumentative basis of §2.4.3 transparent and avoids reliance on intuitive jumps.

Important clarification: usage exclusivity concerns role-instances, not $R$ -membership. After serving as $e$ in $D_{j}$ , $S$ remains an $R$ -member (by §3.1 non-retraction); $S$ ‘s role as $e$ in $D_{j}$ is exhausted, but $S$ can still be referenced in other formal contexts (e.g., in the analysis of $G_{D}$ structure). The role $e$ in any new $L$ would refer to a different $R$ -member, not to $S$ .

Consequences: usage exclusivity entails that $G_{D}$ ‘s structure is well-defined at the role level: each $S$ ‘s “use” as $e$ corresponds to a unique edge in $G_{D}$ (from the determination producing $S$ to the determination consuming $S$ as $e$ ). Without usage exclusivity, $S$ could be edge-source to multiple downstream determinations, making the dependency structure ambiguous.

§2.4.4 Determination chain, merge, split, merge-split (definition, structural organization, four forms of determination relations). Based on dependency unidirectionality and usage exclusivity, the structural organization of determinations exhibits four basic forms. These four correspond to §2.3.9’s four determination types: chain corresponds to single-source single-product continuous determination relations; merge corresponds to multi-source convergence to single product; split corresponds to single-source dispersal to multiple products; merge-split corresponds to multi-source convergence with simultaneous multi-product dispersal. This subsection unfolds the four forms by $(source count, product count)$ two-dimensional structure.

Determination chain (continuous relation of Type I nodes): a determination chain is a sequence of Type I determinations $D_{i_{1}} \to D_{i_{2}} \to \dots \to D_{i_{n}}$ linked by dependency, where each $D_{i_{k}}$ is a Type I determination ( $∣ E_{i_{k}} ∣ = 1$ and $∣ S_{i_{k}} ∣ = 1$ ), and each $D_{i_{k}}$ ( $k > 1$ ) depends only on the immediately preceding $D_{i_{k - 1}}$ , i.e., the unique $e \in E_{i_{k}}$ comes from $D_{i_{k - 1}}$ ‘s unique product. The two constraints “depends only” and “Type I node” jointly define the determination chain: Type II/III/IV determinations do not belong within a determination chain. Treatment of Type II determinations: Type II determinations serve as split-origin points and are branching points for determination chains in $G_{D}$ ; after a Type II determination, its products $S_{1}, \dots, S_{n}$ each initiate independent subsequent chains (if their subsequent determinations are Type I) or participate in merge/merge-split (Types III/IV). Treatment of Type III determinations: Type III determinations serve as merge points where multiple chains converge; the merged product $S$ initiates a new chain. Treatment of Type IV determinations: Type IV determinations serve simultaneously as merge and split points. Determination chains are therefore SDT’s single-source single-product continuous determination form; multi-object interactions are expressed through chain combinations connected by Type II/III/IV nodes. Structural consequence of pure determination chains: each determination on a pure chain is Type I; therefore the chain corresponds to a sequence of attribute changes for the same object: each determination causes some attribute change (Type I includes attribute specification, despecification, transformation; see §2.3.9 Type I internal argument), with the product $S$ serving as the unique $e$ role of the next determination, continuing to bear attribute changes.

Merge (multi-source single-product convergence relation): merge is the multi-source convergence form of determination chain. $L_{j}$ ’s $E_{j} = {e_{1}, e_{2}, \dots, e_{m}}$ ( $m \geq 2$ ); each $e$ comes from a distinct prior determination. $D_{j}$ depends simultaneously on $D_{k_{1}}, D_{k_{2}}, \dots, D_{k_{m}}$ ; multiple chains converge at $D_{j}$ , which produces a single product $S$ as a new composite structure. Merge corresponds to Type III determinations. Merge is compatible with usage exclusivity: each $e$ comes from a unique prior determination; different $e$ in $E_{j}$ share no prior $S$ . Physical correspondent: multiple independently evolving objects converge into a common structural layer through merge, forming a new relational configuration as composite $R$ -member within that layer’s determination.

Split (single-source multi-product dispersal relation): split is the single-source multi-product form of determination chain. $L_{j}$ ’s $E_{j} = {e}$ , $∣ E_{j} ∣ = 1$ ; $D_{j}$ produces $n > 1$ products $S_{1}, \dots, S_{n}$ as independent $R$ -members each entering $R$ . In $G_{D}$ , $D_{j}$ has one in-edge (connecting to $e$ ‘s producing determination $D_{prev}$ ) and multiple out-edges connecting to the subsequent determinations of the post-split products. Split corresponds to Type II determinations. Split is compatible with usage exclusivity: $e$ as single input participates only in $D_{j}$ ; each product $S_{i}$ enters $R$ as new $R$ -member independently and can independently participate in subsequent determinations. Physical correspondent: a single object, after a structural event, produces multiple independent successor structures, each unfolding its own evolution.

Merge-split (multi-source multi-product complex relation): merge-split is the composite form of merge and split. $L_{j}$ ’s $E_{j} = {e_{1}, \dots, e_{m}}$ ( $m \geq 2$ ); $D_{j}$ produces $n > 1$ products $S_{1}, \dots, S_{n}$ as independent $R$ -members. In $G_{D}$ , $D_{j}$ has $m$ in-edges and $n$ out-edges. Merge-split corresponds to Type IV determinations. Merge-split is compatible with usage exclusivity: input side has no shared $e$ ; output side has no shared $S$ ; exclusivity holds at both ends. Discrimination from Type III + II combination: merge-split (Type IV) and the two-step combination (Type III + Type II) differ in $G_{D}$ topology: the former is a single node, the latter two nodes in series; specific discrimination details are in §2.3.9 Type IV internal argument.

Exhaustiveness of the four forms: the four determination-relation forms (chain, merge, split, merge-split) are strictly exhaustive by $(source count, product count)$ two-dimensional cardinality, in strict correspondence with §2.3.9’s four determination types. Any multi-determination structure can be characterized as combinations of determination chains connected via merge, split, merge-split: this is SDT’s basic structural form in the multi-determination case, and the four forms jointly exhaust the legitimate ways of multi-determination organization.

§2.4.5 Equivalent structural layer (definition, derivation path, basis of classification). The previous subsections handled determination-dependency relations, usage exclusivity, determination chain, and merge: all at the level of specific $L$ and specific $D$ . This subsection introduces a concept at a different level: structural-pattern equivalence among $L$ , namely equivalent structural layers. $L_{1}$ and $L_{2}$ are equivalent iff they have the same structural pattern but different specific objects.

Formally, $L_{1} = (E_{1}, P_{1}, C_{1})$ and $L_{2} = (E_{2}, P_{2}, C_{2})$ are equivalent iff: (a) $∣ E_{1} ∣ = ∣ E_{2} ∣$ and the member types of $E_{1}$ and $E_{2}$ are in one-to-one correspondence (i.e., there exists a structural-isomorphism mapping each $e \in E_{1}$ to some $e \in E_{2}$ , preserving member structural types); (a’) same determination type: $L_{1}$ and $L_{2}$ are of the same determination type per §2.3.9 four-type classification (Type I, II, III, or IV); equivalently, $∣ S_{1} ∣ = ∣ S_{2} ∣$ (since $∣ E ∣$ is already established by (a), $∣ S ∣$ equivalence places $L_{1}$ and $L_{2}$ in the same cell of the $(∣ E ∣, ∣ S ∣)$ two-dimensional classification); (b) the configurations of $P_{1}$ and $P_{2}$ correspond structurally under the mapping (each $p$ of $P_{1}$ corresponds to some $p$ of $P_{2}$ , both giving the same relational-configuration and attribute-specification pattern after abstracting from specific member identities, particularly with corresponding $p$ describing the same product cardinality $∣ S ∣$ ); (c) the filtering rules of $C_{1}$ and $C_{2}$ are equivalent under the mapping (i.e., $G$ ‘s direct requirements and $G$ ‘s indirect requirements via $R$ produce equivalent filtering results on the two $L$ , even though $R$ ‘s specific influence on each may differ).

By the above three (or four, including (a’)) criteria, equivalence is reflexive, symmetric, transitive (reflexive: any $L$ is equivalent to itself under identity mapping; symmetric: isomorphism is invertible; transitive: isomorphisms compose); hence the equivalence relation among $L$ partitions all legitimate $L$ into equivalence classes, each consisting of $L$ with the same structural pattern but different specific objects.

Ontological status of equivalence classes: equivalence classes are mathematical products of an equivalence relation on the set of $L$ ; they are not independent ontological objects introduced by SDT. An equivalence class in SDT is the formal object “all $L$ sharing this structural pattern”; it inherits its existence from the existence of its members. SDT does not commit to equivalence classes existing independently of their members (e.g., as Platonic forms or as universals).

Basis of classification: the equivalence relation provides SDT with a formal mechanism for identifying “types of structural events”, e.g., “the type of all electron-spin measurements” is an equivalence class containing all $L$ structurally equivalent to one another in the relevant sense. This classification mechanism is essential to entity-theoretic applications of SDT (where physical theories specify which $L$ are equivalent under physical symmetries).

§2.4.6 Dual feature of equivalent determination products (structurally distinct + property-equivalent). When two determinations are equivalent, their products $S, S^{'}$ exhibit a dual feature established in §2.3.11: structurally distinct in $R$ (different element-identity in layout content), property-equivalent (identical compatibility patterns under $G$ ). This dual feature is foundational for $C$ -neutrality (§2.2.11) and for the application of equivalence classes to entity theories. The full discussion is in §2.3.11; this subsection only marks the dual feature as established.

§2.4.7 Determination dependency graph (DAG characterization, ontological status, formal foundation for time emergence). The inter-determination structural relations established in the previous subsections (dependency relation, usage exclusivity, chain, merge, split, merge-split, equivalent structural layer) can be integrated into a unified formal object: the determination dependency graph, denoted $G_{D}$ . $G_{D}$ ’s node set is all determinations ${D_{1}, D_{2}, \dots}$ ( $D$ as the formal marker of a determination event has specific identity and serves as a node in the graph); the directed edge set is given by the dependency relation: there is a directed edge from $D_{i}$ to $D_{j}$ iff $D_{i} \to D_{j}$ (i.e., some $e \in E_{j}$ equals some product of $D_{i}$ ).

Edge counts under multi-product determinations: each node $D$ ‘s out-edge count in $G_{D}$ depends on how many of $D$ ‘s products are subsequently used. Single-product determinations (Types I, III): each node has at most a few out-edges, each corresponding to the unique product going to a different subsequent $L$ (if the unique product serves as input to multiple subsequent $L$ , this would violate usage exclusivity, so in fact single-product determinations have at most one out-edge per node). Multi-product determinations (Types II, IV): each node’s products can serve as inputs to different subsequent $L$ , hence can have multiple out-edges, each corresponding to one product going to one subsequent $L$ . In-edge count depends on $L_{D}$ ‘s $∣ E_{D} ∣$ : nodes with $∣ E_{D} ∣ = 1$ (Types I, II) have one in-edge; nodes with $∣ E_{D} ∣ > 1$ (Types III, IV) have multiple in-edges, each corresponding to the producing determination of some $e \in E_{D}$ .

$G_{D}$ has two key structural properties. Directionality comes from the unidirectionality of dependency relation (§2.4.2): if $D_{i} \to D_{j}$ , then there is no $D_{j} \to D_{i}$ ; each edge’s direction is determined by the inherent asymmetry of the dependency relation. Acyclicity comes from the joint working of usage exclusivity (§2.4.3) and dependency unidirectionality: suppose $G_{D}$ contains a cycle $D_{i_{1}} \to D_{i_{2}} \to \dots \to D_{i_{n}} \to D_{i_{1}}$ ; then $D_{i_{1}}$ depends on $D_{i_{n}}$ (via the last edge), but $D_{i_{n}}$ also indirectly depends on $D_{i_{1}}$ via the chain (via the first $n - 1$ edges), constituting a circular dependency violating dependency unidirectionality. Preservation of acyclicity under multi-product: multi-product determinations do not break acyclicity, since each edge is still given unidirectionally by dependency; multi-product only allows nodes to have more out-edges, not reverse-tracing from some product to its producing determination. The two properties together make $G_{D}$ a directed acyclic graph (DAG).

Ontological status of $G_{D}$ : it is not a new ontological object introduced by SDT but the integrated expression of the inter-determination structural relations established in previous subsections. The graph’s nodes ( $D$ ) and edges (dependency relation) are already-established formal objects; the graph structure is merely the unified characterization of relations among them.

§2.4.8 DAG structural-order properties (summary). $G_{D}$ as DAG has the structural-order property that any two distinct nodes either stand in a partial-order relation (directed path between them) or are partial-order-incomparable (no directed path either way). This property is the formal foundation of time emergence in §3.3: time as the partial-order structure of $G_{D}$ ‘s growth. The full time-emergence argument is treated in §3.3; this subsection only marks the structural-order property as the formal precondition.

Chapter 3. Core Unfolding and Arguments

Methodological note. Chapter 2 has established all conceptual foundations of SDT’s formal framework: elements $E$ , possible configurations $P$ , compatibility conditions $C$ , global constraint $G$ , realized-structure set $R$ , admissible configurations $A$ , structural layer $L$ , determination $D$ , trace $τ$ , trace set $T$ , determination dependency graph $G_{D}$ (with DAG structural-order property). This chapter unfolds SDT’s core arguments on these conceptual foundations: starting from the axiom and the concepts established in Chapter 2, through conceptual explication it reveals the necessary intensions already contained within SDT’s axiom and concept definitions. The character of this chapter’s work is explication, not “derivation” of new conclusions from independent premises. Explication as a philosophical mode of work has a deep tradition: Euclidean geometry unfolds the entire geometric system from axioms, Spinoza’s Ethics unfolds the whole of ethics from axioms, and Carnap (1950, Logical Foundations of Probability) uses “explication” specifically for “the precisification of concepts and the revelation of their necessary intensions.” SDT as a formalized ontological theory adopts explication as its core mode of work, in alignment with this tradition. Position of SDT’s path: it should be made clear that SDT, as one specific path of unfolding from the axiom, is not the unique legitimate explication of that axiom. In principle, the same axiom can be unfolded into different meta-structural theories (e.g., alternative paths centered on “states” or “relational nets”). SDT chooses “determination” as its core dynamical concept, $L = (E, P, C)$ as its basic analytic unit, and $G_{D}$ as its integrative structure: this is SDT’s theoretical choice, whose value rests on the path’s own coherence and explanatory power, not on a status of “unique legitimacy.” Similarly, Euclidean and non-Euclidean geometries are both legitimate paths unfolded from a common geometric basis; their values do not exclude each other.

Classification of the three core arguments and one structural corollary. This chapter contains three core arguments and one structural corollary, classified by argumentative character:

Argument 1: Conceptual explication of $R$ ‘s non-retraction, unfolding $R$ ‘s non-retraction directly from the joint internal content of $R$ ‘s definition and the determination concept; this is pure conceptual explication.
Argument 2: Conceptual explication of $G$ ‘s constancy, unfolding $G$ ‘s non-rewritability from the universally quantified internal structure of $G$ ‘s definition; the main body is conceptual explication, supplemented by the joint application of Argument 1 as the closure of evasion paths.
Argument 3: Constructive explication of time emergence, starting from the established concepts of $G_{D}$ and $R$ ‘s monotonic growth, constructs all features of time (order, division, direction, passage, simultaneity); this is constructive explication.
§3.4: The derived status of spatial distance (structural corollary), starting from the time structure established by Argument 3, defines the zero-carrier concept and gives the structural definition of spatial distance and the spacetime topological trichotomy; the topological conclusions are pure SDT corollaries, while the quantitative form and the speed upper bound are SDT + $G$ joint conclusions.

The dependency relations among the three core arguments: Argument 1 unfolds independently from the axiom and concepts; Argument 2 depends on Argument 1’s conclusion as the closure of evasion paths; Argument 3 depends on Argument 1 to provide $R$ ‘s monotonic growth and on §2.4 to provide $G_{D}$ ‘s DAG structural-order property. §3.4 depends on Argument 3 for time order, and its quantitative form additionally requires $G$ input. Unified declaration of argumentative strength: the three core arguments are all conceptually necessary; §3.4’s topological conclusions are likewise conceptually necessary, while its quantitative conclusions are joint outputs of SDT + $G$ . Following §1.2’s methodological principle on the use of existing terms, “conceptually necessary” in SDT carries the technical meaning “directly analytically derivable from SDT’s internal axiom and concept definitions, independent of any empirical content.” SDT borrows the Carnapian tradition’s “explication” concept as its technical method of unfolding; it does not commit to any specific philosophical position on “conceptual necessity” (Kantian transcendental synthesis, Kripkean a posteriori necessity, Carnapian strong analyticity, Chalmersian two-dimensional semantics, etc.). The conclusions of the three arguments and the structural corollary necessarily hold within SDT’s framework; readers attempting to map SDT’s “conceptual necessity” onto some specific philosophical tradition will find that SDT is methodologically close to the Carnapian explication tradition while remaining ontologically independent.

The academic positioning of Chapter 3’s work. SDT, as a meta-structural theory, contributes at the meta-level framework, conceptual structure, and formalization tools, not at the content level of classical philosophical disputes (modal ontology, metaphysics of time, ontology of rules, etc.). The three core explications and one structural corollary in Chapter 3 provide: a complete revelation of SDT’s conceptual intensions starting from a single axiom, making SDT’s internal coherence as a self-contained theory open to scrutiny. The chapter concludes with an integrative statement on exhaustiveness: realized states are wholly composed of structure, a statement that fulfills the conjunctive claim of §1.2 consists of; this is a natural consequence of the three explications and the structural corollary together with the overall axiomatic architecture, and is not unfolded as an independent argument.

§3.1 Argument 1: Conceptual Explication of $R$ ‘s Non-Retraction

§3.1.1 Statement of the proposition and argument form. Argument 1 asserts: the realized-structure set $R$ does not retract; once $S$ enters $R$ , $S$ remains in $R$ permanently. The argument form is conceptual explication: from the joint internal content of $R$ ‘s definition (§2.2.5) and the determination concept (§2.3.6), $R$ ‘s non-retraction is derived as a conceptual consequence, not as an additional empirical or metaphysical commitment.

§3.1.2 Core unfolding (joint internal content of $R$ ‘s definition and the determination concept). $R$ is defined as the set of all realized structures (§2.2.5). A determination $D$ is the formal marker of “exactly one configuration in $A$ becoming a realized structure” (§2.3.6); each determination produces $S$ entering $R$ . The question of $R$ ‘s non-retraction reduces to the question: can some $S$ that has entered $R$ leave $R$ ?

The negative answer follows from the modal sense of “realized.” Recall (§1.2.8) that “realized” in SDT is read as the modal-state marker “having crossed the boundary from possibility into actuality.” Once $S$ has crossed this boundary, $S$ has been actualized: the modal fact “this state of affairs has been realized” holds. For $S$ to “leave $R$ ” would require this modal fact to cease holding, i.e., for “this state of affairs has been realized” to become false.

But “this state of affairs has been realized” is a fact about a past (or, in the modal sense, completed) actualization event. Once such an event has occurred, no subsequent event can undo the fact that it occurred. The most a subsequent event can do is to further specify the realized structure (e.g., produce a new $S^{'}$ with different attributes) or to cease the physical instantiation of the realized structure (e.g., a particle decaying into other particles); neither of these undoes the fact that the original realization occurred.

This is the conceptual core: $R$ is a record of realization events. Records cannot retract because they capture facts about past events, and past events cannot become non-events. $R$ ’s non-retraction is therefore a conceptual consequence of $R$ ‘s definition as the realized-structure set together with the modal reading of “realized.”

§3.1.3 Nature of the argument. The argument is conceptual, not empirical or metaphysical. It does not claim that retraction is empirically impossible (no empirical observation could establish such a strong claim); it claims that retraction is conceptually incoherent given $R$ ‘s definition. An ” $R$ that retracts” is not a different kind of $R$ : it is a contradiction in terms (a record that is not a record). SDT’s commitment to non-retraction is thus a commitment to the coherence of the concept of $R$ , not to an additional substantive thesis.

§3.1.4 Auxiliary clarification on ” $R$ only grows, never shrinks”. A possible misreading: ” $R$ only grows” suggests that $R$ has a temporal index and that growth happens “in time.” On SDT’s account, this is not quite right. $R$ does not grow “in time” in any pre-given sense; rather, time itself emerges from the partial-order structure of $G_{D}$ ‘s growth (Argument 3, §3.3). The “growth of $R$ ” is the unfolding of new determinations adding new $S$ , and the partial-order of these additions (in $G_{D}$ ) is what time consists of.

This auxiliary clarification matters because it preempts a circular-reasoning objection: if $R$ ‘s growth presupposes time, and time emerges from $R$ ‘s growth, the account would be circular. The non-circular reading is: the partial-order structure of determinations is given by $G_{D}$ ‘s DAG structure (§2.4); this partial-order structure is what we mean by “time” in SDT. $R$ ’s monotonic growth is the partial-order-respecting accumulation of $S$ , not a temporally-indexed sequence. The two formulations describe the same phenomenon at different levels of abstraction.

§3.1.5 Technical reading of “realized”. A potential confusion concerns the technical reading of “realized.” In everyday usage, “realized” suggests a temporally past event (the realization happened before now). On SDT’s technical reading (§1.2.8), “realized” is the modal-state marker of “having crossed the boundary from possibility to actuality,” with no specific temporal index. A state of affairs is realized iff the modal fact “this state of affairs has crossed into actuality” holds; the temporal index of the realization (when it crossed) is a separate question, addressed by $G_{D}$ ‘s partial-order structure.

This technical reading supports non-retraction without invoking temporality: even without a pre-given temporal framework, “having crossed into actuality” is a modal fact that, once holding, continues to hold (it cannot become un-crossed). Non-retraction is thus a feature of the modal logic of “realized,” not of any temporal structure.

§3.1.6 Structural consequence ( $R$ ‘s monotonic growth). From non-retraction, $R$ ‘s monotonic growth follows: any new realization adds new $S$ to $R$ without removing any existing member. Formally, at any stage of $G_{D}$ ‘s unfolding, $R (stage n + 1) \supseteq R (stage n)$ where the inclusion is strict iff at least one determination occurred between stages $n$ and $n + 1$ .

Monotonic growth has two structural consequences relevant to subsequent arguments. (a) It supports the unidirectionality of $G_{D}$ ‘s growth (Argument 3, §3.3): $G_{D}$ as a DAG can only acquire new nodes and edges, not lose existing ones. (b) It supports the directionality of $A$ ‘s evolution (§2.2.9): $A$ may contract as $R$ grows, but $A$ ‘s contractions are not retractions of any prior $A$ : each $A$ is the filter applied at a specific stage, and contractions reflect the strengthening of $C$ ‘s Pathway-two filtering as $R$ accumulates.

§3.1.7 Distinction between the physical level and the modal level (with specific-case analysis). A common objection to non-retraction concerns physical decay: when an unstable particle decays, the original particle “ceases to exist”: doesn’t this contradict non-retraction? The response distinguishes the physical level from the modal level. At the physical level, the unstable particle’s specific physical instantiation ceases (it is no longer present in the lab); at the modal level, the realization fact “the particle was realized” remains permanent: the historical fact of its realization does not vanish with the particle’s decay.

$R$ in SDT operates at the modal level: it records the modal fact of realization, not the physical persistence of the realized object. The decay of an unstable particle adds new $S$ to $R$ (the decay products) without removing the original $S$ (the original particle’s realization fact). $G_{D}$ records this as the original particle’s $D$ producing a successor $D^{'}$ for the decay event, with both $D$ and $D^{'}$ as nodes in $G_{D}$ .

Specific cases: (a) Particle decay: the unstable particle’s $S$ remains in $R$ ; the decay determination produces decay-product $S$ ‘s; $G_{D}$ records the dependency $D_{particle} \to D_{decay}$ . (b) Annihilation: a particle-antiparticle pair “annihilates” into photons. At the modal level, the pair’s $S$ ‘s remain in $R$ ; the annihilation determination produces the photon $S$ ‘s. (c) Quantum measurement: a superposed state is “collapsed” by measurement to a specific eigenstate. At the modal level, the original superposed-state $S$ remains in $R$ ; the measurement determination produces the post-measurement $S$ . In all cases, non-retraction holds at the modal level despite the apparent “loss” of the prior physical state.

§3.1.8 Compatibility, possible objections, and boundary declarations. Compatibility with §2.4 (DAG structural order): $R$ ‘s non-retraction is compatible with $G_{D}$ ‘s DAG structure: indeed, it is required for the structure to be well-defined (a retracting $R$ would correspond to a DAG with disappearing nodes, which is not a well-defined DAG). Compatibility with §3.2 (Argument 2): non-retraction is independent of $G$ ‘s constancy and does not depend on Argument 2.

Possible objections: (a) “What about the heat death of the universe: doesn’t reality eventually ‘end’?” Response: heat death is a specific physical hypothesis about the long-term behavior of physical systems; even under heat death, the modal facts of past realizations remain. SDT’s non-retraction concerns modal facts, not physical persistence. (b) “What if the universe were time-symmetric: retraction in one time-direction could be growth in the reverse direction?” Response: SDT does not commit to a specific physical theory of time-symmetry. Even in a time-symmetric physical theory, the modal fact “this state of affairs has been realized” is a fact about the modal status of the state, not about its physical temporal location; this fact does not flip under time-reversal.

Boundary declarations: Argument 1 is modal-conceptual, not physical-empirical. SDT’s commitment to non-retraction is a commitment about the modal status of realized states, not about the physical persistence of any particular realized object. The argument does not contradict any specific physical theory; it operates at a different level (the meta-structural / modal level).

§3.2 Argument 2: Conceptual Explication of $G$ ‘s Constancy

§3.2.1 Statement of the proposition and argument form. Argument 2 asserts: the global constraint $G$ is constant; $G$ does not change across determinations. The argument form is conceptual explication, paralleling Argument 1: from the joint internal content of $G$ ‘s definition (§2.2.2) and SDT’s other commitments, $G$ ‘s constancy is derived as a conceptual consequence.

The argument has two paths. Path one (conceptual explication of $G$ ‘s definition itself) shows that $G$ ‘s definition entails its constancy. Path two (joint application of Argument 1’s conclusion) shows that any change in $G$ would conflict with $R$ ‘s non-retraction. Both paths support the same conclusion; together they make the argument robust.

§3.2.2 Path one (conceptual explication of $G$ ‘s definition itself). $G$ is defined (§2.2.2) as the meta-structural rule governing global compatibility. The uniqueness of $G$ (§2.2.3) entails that there is exactly one $G$ at any meta-temporal stage. The question of $G$ ‘s constancy reduces to: can the unique $G$ at one stage differ from the unique $G$ at another stage?

Suppose $G$ at stage 1 ( $G^{(1)}$ ) differs from $G$ at stage 2 ( $G^{(2)}$ ). Then there exists some configuration $p$ that $G^{(1)}$ judges compatible (or incompatible) with $R$ , while $G^{(2)}$ judges differently. But $G$ governs $R$ ‘s consistency (per §1.2.13 and §2.2.2). If $R$ ‘s consistency-judgments change between stages, then $R$ at stage 1 is consistent under one rule and $R$ at stage 2 is consistent under a different rule. $R$ ’s consistency is not a stage-relative notion: a set is either consistent or it is not.

A possible response: “Maybe $R$ at stage 2 is judged consistent under $G^{(2)}$ but not under $G^{(1)}$ .” But this would mean $R$ at stage 2 contains structures incompatible under $G^{(1)}$ : structures that, by $G^{(1)}$ ‘s rules, should not coexist. By non-retraction, the stage-1 members of $R$ are still in $R$ at stage 2; if some stage-1 member is incompatible with stage-2 additions under $G^{(1)}$ , then under $G^{(1)}$ , $R$ at stage 2 is inconsistent. But $G^{(1)}$ is no longer the active rule at stage 2, so the inconsistency is not detected. This means $R$ at stage 2 is “consistent” only by the standard of a rule different from the rule under which its earlier members were realized. The notion of consistency thereby becomes incoherent across stages.

The path-one argument concludes: $G$ must be constant, otherwise consistency-judgments become stage-relative and the notion of $R$ ‘s consistency becomes incoherent.

§3.2.3 Path two (joint application of Argument 1’s conclusion). Path one’s argument has a key precondition: “the realized structures from before the change still exist.” This is exactly Argument 1’s conclusion ( $R$ ‘s non-retraction). Path two makes this dependency explicit: if $R$ retracted, $G$ ‘s past consistency-judgments could be “erased” along with the retracted structures, and the path-one argument would not go through. But Argument 1 establishes non-retraction; therefore the path-one argument is valid.

Path two also yields a stronger conclusion: $G$ ‘s constancy is not merely a stipulation about the rule, but a structural feature of the SDT framework’s coherence. The framework requires (via Argument 1) that $R$ be a permanent record of realizations; any change in $G$ would either render this record inconsistent (under the changed $G$ ) or render past consistency judgments meaningless. Both alternatives undermine the framework’s intelligibility.

§3.2.4 Type-token distinction between $G$ and physical constants (core conceptual clarification). Argument 2’s conclusion ( $G$ is strictly constant) faces a frequent objection: physics observes that “constants” (e.g., the fine-structure constant, the cosmological constant) may vary slightly across cosmological scales, and theories of varying constants (Magueijo 2003) are subjects of legitimate research. Doesn’t $G$ ‘s strict constancy contradict this physics?

The response distinguishes the type level ( $G$ ) from the token level (physical constants). $G$ is the meta-structural rule (the type of compatibility judgments); physical constants are specific tokens of compatibility within $G$ ‘s rule space. A theory with varying physical constants holds that the specific token-values of compatibility (e.g., the strength of the fine-structure interaction) may vary, while the type-rule (the form of the law including the variable parameter) remains constant.

For example, even in a theory where the fine-structure constant $α$ varies in space-time, the form of the law (e.g., “fine-structure interaction obeys QED with parameter $α (x, t)$ ”) remains constant. $G$ at the type level corresponds to the form of the law; the variation of $α$ is at the token level (specific values of the parameter), not the type level. SDT’s $G$ constancy is at the type level; varying-constant physical theories are at the token level. The two are compatible.

§3.2.5 Compatibility, other objections, and boundary declarations. Compatibility with §2.4 (DAG structural order): $G$ ‘s constancy supports $G_{D}$ ‘s well-definedness: a varying $G$ would yield different DAG structures at different stages, undermining the structural-order property.

Other objections: (a) “What about quantum-gravity scales where the laws of physics may be different?” Response: at the level of physical theory, this is an open question. At the level of SDT’s $G$ , the type-level rule remains constant; what may differ at quantum-gravity scales are specific token-values within the rule space. (b) “What if the universe undergoes a phase transition that changes the laws?” Response: phase transitions in physics typically change specific parameters (token values), not the form of the laws (type rules). $G$ as type rule remains constant through such transitions.

Boundary declarations: Argument 2 commits to $G$ ‘s constancy at the type level. SDT does not specify the token-level values; these are filled by entity theories. SDT also does not take a stance on whether the type-level rule is finitely or infinitely specifiable, uniformly applicable, or fully accessible to inquiry. These are entity-theoretic questions.

§3.3 Argument 3: Constructive Explication of Time Emergence

§3.3.1 Statement of the proposition and argument form. Argument 3 asserts: time emerges as the partial-order structure of $G_{D}$ ‘s unfolding. The argument form is constructive: SDT does not derive time from prior temporal concepts (which would be circular) but constructs the structural features of time (order, direction, past/present/future, simultaneity, passage) from the structural features of $G_{D}$ (DAG structure, modal asymmetry, partial-order incomparability, monotonic growth).

The argument’s construction unfolds in steps: (i) identifying the partial-order structure of $G_{D}$ with temporal order; (ii) deriving past/present/future from $G_{D}$ ‘s DAG structure plus monotonic growth; (iii) deriving modal asymmetry and time direction from $R$ ‘s non-retraction plus $∣ A ∣ > 1$ openness; (iv) deriving time passage from $G_{D}$ ‘s continual extension; (v) deriving simultaneity (or its limits) from partial-order incomparability.

The conclusion is not “time is identical to $G_{D}$ ” in some reductive sense; it is “the structural features traditionally attributed to time are bestowed by $G_{D}$ ‘s structural features.” Time as an independent entity is not what SDT commits to; what SDT commits to is the full bestowal of time-features by $G_{D}$ .

§3.3.2 Bridge from constructive description to ontological identity. A potential objection: “constructing time-features from $G_{D}$ is one thing; identifying time with $G_{D}$ is another. Why does the construction support identification?” The response invokes Occam’s razor: if all the structural features of time are already bestowed by $G_{D}$ , then positing time as an independent entity is redundant: it adds no explanatory or descriptive power that $G_{D}$ does not already provide.

This is SDT’s via negativa on time: SDT does not claim “what time is” in some positive metaphysical sense; SDT claims that the widespread concepts of absolute time as background, eternalism, metric time as fundamental, time as a continuous parameter, and measurement as generating time are all redundant given $G_{D}$ ‘s structural bestowal. Time as the partial-order structure of $G_{D}$ ‘s unfolding is the minimal commitment compatible with all observable temporal phenomena.

§3.3.3 Structure of $G_{D}$ : temporal order and past/present/future. The temporal order of SDT corresponds to $G_{D}$ ‘s partial-order structure: $τ_{i} ≺ τ_{j}$ (i.e., $D_{i}$ is in the temporal past of $D_{j}$ ) iff there is a directed path from $D_{i}$ to $D_{j}$ in $G_{D}$ . By §2.3.12, this partial-order is irreflexive, antisymmetric, and transitive: it is a well-defined partial-order on $T$ .

Past, present, future: at any stage of $G_{D}$ ‘s unfolding, past is the set of $G_{D}$ -nodes already drawn (determinations already realized); present is the set of $G_{D}$ -nodes with no out-edges (determinations realized but not yet succeeded by further determinations); future is the region into which $G_{D}$ will extend (potential determinations not yet realized). The past is fully recorded in $G_{D}$ ‘s already-drawn structure; the present is the boundary of the drawn structure; the future is the modal openness allowed by the unfilled $A$ ‘s of present-stage determinations.

This structural reading of past/present/future avoids the standard pitfalls of A-theory and B-theory. A-theory’s concern with the objective division of past/present/future is met: $G_{D}$ ‘s drawn structure objectively partitions into past (with out-edges already drawn), present (no out-edges yet), and future (not yet drawn). B-theory’s concern with reducibility of temporal properties to relations is met: all temporal properties are reducible to $G_{D}$ ‘s structural relations. Both concerns are satisfied without the trade-offs typical of A/B-theory positions.

§3.3.4 Modal asymmetry and the passage of time. The previous subsection established the structural definitions of past/present/future. This subsection argues for the modal asymmetry between them and derives time passage from $G_{D}$ ‘s continual growth. The argument for modal asymmetry: past and future differ fundamentally in modal property; this difference follows entirely from SDT’s already-established commitments.

The past is modally determinate: every determination in the past has been completed; its product $S$ has determinately entered $R$ (§1.2 Reality’s determinacy), and is irrevocable (Argument 1, $R$ non-retraction). ” $S_{0}$ was realized” is a fact already occurred; this fact cannot be altered by any subsequent determination. The drawn portion of $G_{D}$ is therefore fixed: drawn nodes cannot be deleted, drawn edges cannot be removed, the partial-order structure of the past cannot be modified.

The future is modally open: future determinations have not yet occurred. If some future determination has $∣ A ∣ > 1$ (multiple admissible configurations), which $p \in A$ is ultimately realized has not yet been determined; multiple configurations coexist as genuine open possibilities. The region into which $G_{D}$ will extend is therefore indeterminate: although dependency relations give structural constraints (which determinations must precede which), the specific product $S$ of each determination depends on which $p \in A$ is selected, and this selection is unpredictable before the determination occurs.

Structural source of the asymmetry: this modal asymmetry is not an additional commitment of SDT but a joint consequence of $R$ non-retraction (Argument 1) and the configuration openness of $∣ A ∣ > 1$ (§2.1 $P$ section and §2.3 determination section). $R$ non-retraction locks the past; $∣ A ∣ > 1$ opens the future; the two together produce the asymmetry from “locked” to “open.” This asymmetry is the formalized source of time direction in SDT: $G_{D}$ ‘s growth direction (from drawn to about-to-be-drawn) corresponds to the modal direction from “determinate” to “indeterminate.”

Two-layer distinction of time direction: SDT’s time direction has two layers, which must be strictly distinguished to avoid misjudging argumentative strength. Layer 1 (structural direction, unconditional): $G_{D}$ as a DAG has directionality (edge direction is irreversible, a direct consequence of §2.4 acyclicity); $R$ ‘s monotonic growth (Argument 1) gives $G_{D}$ ‘s unfolding a determinate “forward” direction (from drawn to about-to-be-drawn). This layer depends only on Argument 1 and §2.4’s DAG property and holds unconditionally: even if all legitimate $L$ are $∣ A ∣ = 1$ , the structural direction still exists, because $R$ still grows monotonically and $G_{D}$ remains a directed acyclic graph. Layer 2 (modal asymmetry, conditional): the modal asymmetry from “determinate past” to “open future” depends on the existence of legitimate $L$ with $∣ A ∣ > 1$ ; if all $L$ are $∣ A ∣ = 1$ , modal asymmetry vanishes and Layer 2 degenerates to Layer 1 (pure structural direction). Relation between the two layers: Layer 1 is SDT’s meta-level unconditional commitment; Layer 2 arises on the basis of Layer 1 jointly with the entity-theoretic fact $∣ A ∣ > 1$ . SDT’s time direction holds unconditionally as Layer 1; conditionally as Layer 2 (only under the condition that $∣ A ∣ > 1$ exists). This two-layer distinction corresponds to the formal grading in the Mathematical Certification Matrix: Layer 1 corresponds to Theorem 3.3-a (Past Locked) and the §2.4 DAG theorems (Grade A, unconditional); Layer 2 corresponds to Statement 3.3-b (Future Open, Grade A−, conditional). Significance of the two-layer distinction: current physical descriptions (quantum mechanics, thermodynamics, breakdown of classical determinism) widely indicate that $∣ A ∣ > 1$ exists, making Layer 2 physically operative; but the argumentative strength of SDT as a meta-structural theory does not depend on this physical fact: even under some extreme deterministic entity theory where Layer 2 degenerates, Layer 1 still preserves the formal foundation of SDT’s time direction.

Precisification of the conditionality of modal asymmetry: clause (c) of Argument 3, modal asymmetry, strictly depends on the condition “there exists at least one future determination with $∣ A ∣ > 1$ .” If all future determinations have $∣ A ∣ = 1$ (i.e., the future is wholly filtered by $C$ to a unique admissible configuration), modal asymmetry vanishes and time direction degenerates to “pure structural order”: the partial-order still exists (dependency relations among determinations are unchanged), but the directionality from “determinate” to “indeterminate” vanishes, leaving only the formal structure of the partial-order itself. SDT meta-level does not guarantee this condition holds; it is a conclusion of entity theory rather than a meta-level commitment of SDT. Physically, $∣ A ∣ > 1$ corresponds to the breakdown of classical determinism, quantum open possibilities (superposition measurement), the multiple possible directions of thermodynamic sub-processes, and so on, and is widely present in current physical descriptions; but SDT as a meta-structural theory does not take this physical fact as its own commitment. This degenerate situation is an acceptable limiting form within SDT’s framework and does not constitute a refutation of Argument 3: Argument 3’s claim is that, under the condition that $∣ A ∣ > 1$ determinations exist, modal asymmetry and time direction strictly hold; whether this condition is universally satisfied at the entity-theoretic level, SDT does not commit.

This structural account of passage avoids McTaggart’s paradox (McTaggart 1908): the apparent contradictions in attributing both A-properties and B-properties to events. On SDT’s account, A-properties (past/present/future) are stage-relative features of $G_{D}$ at any stage of its unfolding; B-properties (earlier/later) are partial-order features of $G_{D}$ as a whole. The two sets of properties are at different levels (stage-relative vs. graph-level) and do not conflict.

§3.3.5 Passage of time (further specification). The passage of time is the increase in $G_{D}$ ‘s drawn structure. At one stage, $G_{D}$ has drawn structure $G_{D}^{(1)}$ ; at a later stage, $G_{D}^{(2)} \supset G_{D}^{(1)}$ . The passage is the relation $G_{D}^{(1)} \subset G_{D}^{(2)}$ . This relation is well-defined on the graph level and corresponds to the experiential notion of time passing.

No external clock: SDT’s account of passage does not invoke an external clock or temporal background. The passage is internal to $G_{D}$ ‘s growth. Different “rates” of passage (e.g., relativistic time dilation) correspond to different rates of $G_{D}$ extension along different determination chains; SDT meta-level does not specify rates: these are entity-theoretic.

§3.3.6 Simultaneity, ontological positioning, and the reconstruction of traditional time philosophy. Simultaneity in SDT corresponds to partial-order incomparability in $G_{D}$ : two determinations $D_{i}, D_{j}$ are “simultaneous” iff neither $D_{i} ≺ D_{j}$ nor $D_{j} ≺ D_{i}$ . This reduces simultaneity to a partial-order feature, in line with relativistic physics (Einstein 1905; cf. Hawking and Ellis 1973), which already recognizes simultaneity as frame-relative, not absolute.

SDT does not commit to a privileged frame’s simultaneity. Simultaneity in SDT is not absolute; it is the partial-order incomparability relation. In specific physical theories (e.g., relativity), specific frames may pick out specific simultaneity-classes, but SDT meta-level only commits to the partial-order incomparability structure.

Reconstruction of traditional time philosophy: SDT’s account reconstructs the central commitments of A-theory and B-theory in $G_{D}$ terms (§3.3.4), avoiding the trade-offs typical of A/B positions. The reconstruction also reframes the simultaneity debate: the question is not “is simultaneity absolute?” but “is the partial-order incomparability relation specifiable independent of frame?” The latter is a question with a clear answer in relativistic physics (no, simultaneity is frame-relative).

§3.3.7 Relation to traditional time philosophy. SDT’s structural emergentism of time relates to traditional time-philosophy positions as follows. (a) A-theory (Mellor 1998; Zimmerman 2005): A-theory’s central commitment to the objective past/present/future division is met by $G_{D}$ ‘s drawn-structure partition. SDT does not require A-theory’s stronger commitment to the present’s metaphysical privilege. (b) B-theory: B-theory’s central commitment to the reducibility of temporal properties to relations is met by all temporal features being structural features of $G_{D}$ . SDT does not require B-theory’s stronger commitment to the eternalist totality. (c) Growing block: the growing-block view’s commitment to a growing past plus a fixed present is structurally close to SDT’s account, but SDT does not commit to the metaphysics of “growth” as a primitive notion. (d) Presentism: presentism’s commitment to the metaphysical priority of the present is not endorsed by SDT: SDT treats past, present, and future as different aspects of $G_{D}$ , not as different ontological tiers.

SDT’s position is therefore not a standard A-, B-, growing-block, or presentist position; it is structural emergentism, a position that bestows all temporal features through structural features of $G_{D}$ and does not depend on any specific traditional commitment.

§3.3.8 Relation to causal set theory. Causal set theory (Sorkin 2005; Rideout and Sorkin 2000) shares with SDT the structural-discrete approach to time emergence: time as the partial-order structure of a discrete event set (the causal set). SDT and causal set theory have important structural similarities and differences.

Structural similarities: (a) discrete event set with partial order; (b) emergent time from partial-order structure; (c) DAG-like structure (with acyclicity).

Structural differences: (i) Foundational object: causal set theory takes discrete event sets as basic ontology (events first, relations characterize causal structure); SDT takes determination-dependency relations as basic (relations first, $G_{D}$ is the integrated formal expression). (ii) Node nature: causal-set events are atomic without internal structure; SDT determination nodes carry full $L = (E, P, C)$ internal structure with type distinction (Type I attribute transformation, Type II split, Type III merge, Type IV merge-split, §2.3.9). (iii) Order source: causal set’s order comes from causal relations (presupposing some timelike notion); SDT’s order comes from determination-dependency (no spacetime or timelike presupposition). (iv) Discreteness source: causal set theory’s discreteness is an assumption (a basic conjecture for quantum gravity); SDT’s discreteness is an argument conclusion ( $G_{D}$ as determination event set is naturally discrete since each determination is a countable structural event).

The four structural differences make SDT not a variant or special case of causal set theory but an independent framework with different ontological positioning. The two share the structural similarity “discrete structure induces temporal order” while bearing different ontological commitments and adopting different formal strategies.

§3.3.9 Compatibility and boundary. Dependency on §2.4 (DAG structural order): Argument 3 depends on §2.4 as the structural foundation for time emergence. The DAG dependency relations make different determinations occupy different topological positions in $G_{D}$ ; this structural distinction gives partial-order substantive content. The necessary occurrence of determinations makes $G_{D}$ necessarily continuously extended. If §2.4’s DAG structure does not hold (all determinations have no structural distinction), $G_{D}$ degenerates into an unordered event set, and temporal order cannot emerge. Dependency on Argument 1 ( $R$ non-retraction): Argument 3 depends on Argument 1 as the basis for modal asymmetry. $R$ ’s non-retraction locks the past (completed determinations cannot be revoked); $R$ ‘s monotonic growth gives $G_{D}$ a direction of growth (from drawn to about-to-be-drawn). If Argument 1 fails ( $R$ retractable), the past becomes modifiable, modal asymmetry vanishes, and time direction loses its formal source. Relation to Argument 2 ( $G$ constancy): Argument 2 serves as a stable background for Argument 3. $G$ constancy ensures $G_{D}$ ‘s growth occurs against a stable rule background; $C$ ‘s filtering rules do not change with determinations (though $C$ ‘s specific filtering results vary with $R$ ), enabling time emergence to be undisturbed by rule changes. Argument 3 does not directly depend on Argument 2’s conclusion as an argumentative premise, but if $G$ were modifiable, $G_{D}$ ‘s growth rule could differ in different regions, undermining the unity of temporal order. Fulfillment of §2.3 trace-section preview: §2.3 trace section previewed “a partial-order relation can be defined among traces; this partial-order corresponds to the formalization of temporal order in SDT.” Argument 3 fully cashes out this preview: the trace partial-order ( $τ_{i} ≺ τ_{j} ⟺ D_{i} ≺ D_{j}$ ) is identical to the partial-order induced by $G_{D}$ and is the temporal order in SDT.

Boundary declaration: Argument 3 commits to the structural formalization of time: temporal order, past/present/future, modal asymmetry, time passage, simultaneity all acquire rigorous definition as structural attributes of $G_{D}$ . Argument 3 does not commit to: temporal-interval metric (how the “distance” between two $G_{D}$ -nodes is quantified, determined by entity theory); specific correspondence between physical time and SDT time (how relativistic time dilation, quantum-mechanical time evolution map to $G_{D}$ structure, determined by entity theory); whether time has a beginning or end (whether $G_{D}$ has an initial or final node, determined by entity-theoretic analysis of specific cosmology). Spatial distance’s derived status, as a structural corollary of Argument 3, is treated separately in §3.4. SDT as meta-structural theory provides the structural skeleton of spacetime; specific metric and physical correspondences are established by entity theories on this skeleton. Self-propelling cycles describe $G_{D}$ ‘s continual-growth mechanism, not the initial-state account; $R$ ‘s non-emptiness follows directly from the axiom (the axiom asserts “reality consists of…”, entailing at least the existence of realized structures); whether $G_{D}$ has a root node and the specific properties of initial determinations are determined by entity theory.

§3.3.10 Possible objections and final boundary declaration. Argument 3 may face five categories of objection that require explicit response.

Objection 1: Does $G_{D}$ ‘s “continual growth” itself presuppose time? The word “continual” in everyday usage suggests temporal duration. Response: $G_{D}$ ‘s growth does not presuppose time; “growth” here is a structural statement: $R$ ‘s member set goes from one state to a state containing more members, and $G_{D}$ ‘s node set goes from one state to a state containing more nodes. This sequence of states does not need to “unfold in time”; it is time itself. “Continual” is merely a description of this structural evolution (determinations continually being completed, new nodes continually joining), and it borrows no external time framework. If a reader insists that “any growth-description necessarily presupposes time,” this projects everyday-language temporalized readings onto SDT’s structural language; SDT’s formalization does not bear this projection.

Objection 2: $G_{D}$ ‘s partial-order admits multiple distinct linearizations (topological orderings); which one corresponds to the “true temporal order”? Response: SDT does not claim that time is a linear order. Time order in SDT is a partial-order: some pairs of determinations stand in precedence relations (comparable), others do not (incomparable, i.e., “simultaneous”). This is SDT’s conclusion, not a defect. The intuition that “time is a linear order” comes from a single observer’s experience (the events a single observer experiences do form a linear order), but the event-set across multiple observers need not form a linear order (in relativity, spacelike-separated events have no global linear order). SDT derives a partial-order rather than a linear order from its own structure, converging with this structural feature of relativity. If entity theory requires a linear order (as in some non-relativistic physical descriptions), this can be obtained by selecting a specific path (determination chain) in $G_{D}$ , yielding a linear order on that chain; this is entity theory’s specialization of SDT’s partial-order, not an internal problem of SDT.

Objection 3: Does “the present” as $R$ ‘s current content conflict with $R$ non-retraction? $R$ non-retraction means $R$ ‘s content only grows and does not shrink, so is “ $R$ ‘s current content” always “the totality of all realized structures”? If so, the “present” would always be all of $R$ , and the past/present/future distinction would vanish. Response: this objection conflates $R$ ‘s membership with $G_{D}$ ‘s boundary. All members of $R$ are indeed permanently retained ( $R$ non-retraction), but the boundary of $G_{D}$ extends along the partial-order direction; each time a new node is added on $G_{D}$ ‘s partial-order (corresponding to a new $S$ entering $R$ and a new node joining $G_{D}$ ), $G_{D}$ ‘s boundary expands, and the dependency conditions of some determinations previously outside $G_{D}$ become satisfied. The “present” is not $R$ itself ( $R$ contains all realized structures, including those of completed determinations and those just added) but $G_{D}$ ‘s current boundary: formally, the determination region corresponding to the set of nodes with no out-edges in $G_{D}$ (consistent with this argument’s earlier definition of “present”). The “past” comprises nodes in $G_{D}$ with out-edges (those completed determinations depended upon by other nodes), and the “future” is the region beyond $G_{D}$ ‘s boundary (determinations not yet drawn and structures not yet realized). $R$ non-retraction together with $G_{D}$ ‘s monotonic growth ensures that the boundary extends only along the partial-order direction and does not retract, giving the past/present/future division a clear structural basis.

Objection 4: Is SDT’s theory of time too abstract to interface with physical time? Response: SDT as a meta-structural theory is designed to provide a structural skeleton, not physical detail. SDT’s formalization of time (partial-order, past/present/future, modal asymmetry, passage, simultaneity) provides the structural conditions any time theory must satisfy, not a complete description of specific physical time. How relativistic time, quantum-mechanical time, the thermodynamic arrow of time, and the like map onto SDT’s $G_{D}$ structure is the work of entity theory. SDT’s contribution lies in this: prior to any entity theory, it derives the structural skeleton of time from the axiom, providing a unified formal foundation on which subsequent physical interfacing can rely.

Objection 5: Does $G_{D}$ ‘s unfolding presuppose some ontological position of determinism or randomism? One line of questioning holds that $G_{D}$ ‘s continual unfolding must fall into one of two ontological positions: either each new node’s addition is predetermined by more fundamental facts (determinism, another way of stating the block universe), or each new node’s addition is purely random selection without any constraint (absolute randomness); the two positions respectively introduce additional commitments (globally pre-written facts vs. arbitrary possible jumps), and SDT must take a stance. Response: SDT reduces neither to determinism nor to absolute randomness. Rejection of determinism: at determination $D$ , when $∣ A ∣ > 1$ , $C$ does not predetermine which $p \in A$ is realized; per §2.3.6’s argument on determination legitimacy, no more fundamental “already-selected” fact closes admissible configurations in $A$ to a unique one. Strong-determinism positions like the block universe require future-corresponding $G_{D}$ nodes to already exist as established facts; SDT rejects this requirement: future-corresponding nodes have not yet been generated, do not belong to $R$ , and do not belong to $G_{D}$ . Rejection of absolute randomness: $A$ is not the set of arbitrary possibilities but the result of $P$ filtered by $C$ ‘s two pathways; per Pathway Two, new configurations must be compatible with all realized structures in $R$ , making $∣ A ∣$ monotonically non-increasing as $G_{D}$ grows (§2.2 corresponding section); per §2.4 usage exclusivity, the elements of new determinations must be selected from $R$ , so each determination is grounded in existing structure. These constraints jointly exclude an “arbitrarily reconfigurable at each moment” picture of absolute openness. SDT’s ontological position can therefore be stated as: the future remains open under historical constraint. $∣ A ∣ > 1$ reflects the current degree of openness; $∣ A ∣$ ‘s monotonic non-increase reflects the accumulation of historical constraint; openness and constraint are two faces of the same structural mechanism in SDT ( $C$ ‘s two pathways plus $R$ ‘s monotonic growth plus usage exclusivity), not two independent principles. This position may be called constrained randomness (or restricted freedom): it neither denies the genuine ontological openness when $∣ A ∣ > 1$ , nor claims that this openness can be detached from historical constraint. This is consistent with the experiential stability of “the world continues to develop, not undergoing groundless resets at each moment.” SDT does not migrate to any additional commitments beyond Arguments 1, 2, 3 and the §2 formalism on this question; constrained randomness is a synthetic consequence of existing structural mechanisms, not a new ontological assumption.

Objection 6: Does identifying time with $G_{D}$ leave out the felt experience of time-passage? Some readers, particularly from the Husserlian or Bergsonian phenomenological traditions, may object that SDT’s structural account misses the qualitative felt character of time experience. Response: SDT’s account does not pretend to capture the phenomenology of time experience; it captures the structural features of time. Phenomenology operates at a different level (the experiential level) and may require additional theory; SDT does not claim universal coverage of all aspects of time. The structural account and the phenomenological account address different questions: SDT asks “what is the formal structure that any time-bearing reality must satisfy?”; phenomenology asks “what is it like, from a first-person perspective, to experience time?” These are not in competition. SDT’s structural foundation is compatible with phenomenological supplementation by entity-theoretic accounts of conscious experience; SDT does not commit to or against any specific phenomenological theory.

Final boundary declaration: Argument 3 is structural-conceptual, not phenomenological-empirical. SDT commits to the structural formalization of time; phenomenology of time experience and specific physical theories of time are at other levels. The argument is also conditional: it depends on the existence of $∣ A ∣ > 1$ determinations for modal asymmetry; without such determinations, the argument degenerates to structural order without modal content (see §3.3.4 conditionality precisification).

§3.3.11 Final boundary declaration (concluding remark). The three core arguments (Arguments 1, 2, 3) together establish the structural-conceptual foundation of SDT: $R$ is a permanent record (Argument 1); $G$ is a constant rule (Argument 2); time is the partial-order structure of $G_{D}$ ‘s unfolding (Argument 3). These three arguments are mutually supportive and jointly underwrite the conceptual coherence of SDT. The structural corollary of §3.4 (derived spatial distance) builds on this foundation.

§3.4 The Derived Status of Spatial Distance (Structural Corollary)

§3.4.1 Statement of the proposition, argument form, and epistemic status. §3.4 asserts: spatial distance is a derived structural quantity, derivable from the time structure (§3.3) plus zero-carrier propagation. The argument form is structural derivation: spatial distance is constructed from the partial-order structure of $G_{D}$ plus the formal definition of zero-carrier (an entity transmitting structural relation across $G_{D}$ ). Spatial distance is not a primitive structural feature of SDT; it is derived from time structure plus a specific structural mechanism.

Epistemic status: §3.4 is a structural corollary, not a fourth core argument. It depends on Argument 3 (time emergence) and on a formal definition (zero-carrier). The argument’s commitment is weaker than the three core arguments: it depends on Argument 3’s conclusion plus a new formal mechanism. The conclusion is a structural derivation, not a self-standing argument about reality.

§3.4.2 Zero-carrier and the structural definition of spatial distance. Let $D_{em}$ and $D_{abs}$ be two determinations in $G_{D}$ with $D_{em} ≺ D_{abs}$ . The propagation from $D_{em}$ to $D_{abs}$ is zero-carrier propagation (null carrier propagation) iff some product $S_{D_{em}}$ of $D_{em}$ directly serves as the element $e$ of $L_{D_{abs}}$ participating in $D_{abs}$ , with no intermediate determination node taking $S_{D_{em}}$ as input, i.e., the carrier’s own chain length is 1: $D_{em} \to D_{abs}$ is a direct edge in $G_{D}$ with no intermediate $L$ inserted with the product as element. The zero-carrier has no local determinations during transmission; it exists in $R$ in the form of $S_{D_{em}}$ awaiting entry into the next $L$ , with proper time not increasing during this period. This definition uses only existing $G_{D}$ and $R$ concepts; no new primitive object is introduced.

Zero-carrier under multi-product determinations: when $D_{em}$ is a multi-product determination (Type II or IV), $D_{em}$ has multiple products $S_{1}, \dots, S_{n}$ . Zero-carrier propagation refers to a specific product $S_{k}$ as the carrier: $S_{k}$ directly serves as $L_{D_{abs}}$ ‘s element $e$ ; other products $S_{j}$ ( $j \neq = k$ ) are unrelated to this zero-carrier propagation and have their own subsequent propagations. “Which product serves as zero-carrier” is specified by entity theory based on the specific physical process; SDT meta-level only commits to the formal identifiability of the zero-carrier definition, not to which specific product corresponds.

Definition of spatial distance: let $D_{abs}$ be the merge node of zero-carrier propagation and the target’s local chain, i.e., $S_{D_{em}}$ together with some local-chain end product serves as $L_{D_{abs}}$ ‘s elements. Define the target’s local chain length $λ_{B} (D_{em}, D_{abs})$ as the number of nodes on B’s local determination chain $C_{B}$ satisfying both: (a) the node is a predecessor of $D_{abs}$ in $G_{D}$ ( $v ⪯ D_{abs}$ ); (b) the node is not a predecessor of $D_{em}$ in $G_{D}$ ( $v \neq ⪯ D_{em}$ ). Formally, $λ_{B} (D_{em}, D_{abs}) := ∣ {v \in C_{B} : v ⪯ D_{abs} and v \neq ⪯ D_{em}} ∣$ . In typical cases (B-chain nodes all $⪯$ -incomparable with $D_{em}$ ), this reduces to $D_{abs}$ ‘s total predecessor count on B-chain; when some B-chain nodes lie in $D_{em}$ ‘s causal past, the formula correctly subtracts nodes already in $D_{em}$ ‘s causal past. $λ_{B}$ is a pure DAG topological quantity, computable for any position accepting $G_{D}$ ‘s partial-order structure. Structural distance is defined as $d_{S} (A, B) := λ_{B} (D_{em}, D_{abs})$ , with units of determination-steps: spatial distance between two points equals the number of determination steps experienced by the target’s local chain during zero-carrier propagation.

§3.4.3 Minimum value of the carrier’s proper determination count and asymmetry (pure SDT conclusion). The carrier’s chain length is 1 (zero-carrier definition); this is the minimum value for any inter-point structural relation. Asymmetry: $λ_{B} (D_{em}, D_{abs})$ counts B-chain nodes; the analogous quantity for A would be $λ_{A}$ , which counts A-chain nodes between $D_{em}$ ‘s producing chain and $D_{em}$ . The two are asymmetric because they count different chains. This asymmetry corresponds to the directionality of zero-carrier propagation (from emitter to absorber).

§3.4.4 Spacetime topological trichotomy (pure SDT conclusion). The structural relations between two determinations $D_{i}, D_{j}$ in $G_{D}$ partition into three categories: (a) timelike: $D_{i} ≺ D_{j}$ or $D_{j} ≺ D_{i}$ (directed path between them); (b) lightlike: zero-carrier propagation (direct edge in $G_{D}$ with no intermediate carrier-input); (c) spacelike: $G_{D}$ -incomparable (no directed path either way). This trichotomy is purely structural (no metric needed) and corresponds to the relativistic spacetime classification (cf. Hawking and Ellis 1973). Boundary: SDT meta-level commits only to the trichotomy structure; the specific metric (proper time on timelike paths, proper distance on spacelike) is filled by entity theory plus $G$ .

§3.4.5 Quantitative form (requires $G$ ). Among the topological conclusions above, the zero-carrier definition, structural distance $d_{S} = λ_{B}$ , and spacetime trichotomy are all pure SDT conclusions; carrier proper-time minimum value of 0 is a pure SDT conclusion; $c$ as a speed upper bound is an SDT + $G$ joint conclusion (depending on cross-chain step comparability). The quantitative form of spatial distance requires $G$ to provide three inputs: first, step-rates across different determination chains are comparable (making $λ_{B}^{sub} > λ_{B}^{null}$ hold and thus making $c$ a speed upper bound); second, all zero-carriers (regardless of emitter motion state) have the same propagation ratio ( $c$ constant); SDT defines what a zero-carrier is, but cannot derive from the axiom which physical entities satisfy this definition, the physical identification of zero-carriers (e.g., photons) is provided by $G$ ; third, the conversion relation between determination steps and physical-time units. After $G$ provides these three inputs, the quantitative form of spatial distance is $d (A, B) = c \cdot λ_{B} (D_{em}, D_{abs})$ . $c$ here bears three conceptually independent roles: structural consequence of carrier proper-time minimization (pure SDT), speed upper bound (SDT + $G$ ), time-space unit conversion coefficient ( $G$ input). The three roles should not be conflated. This structure is exactly parallel to SDT’s mode of treating quantum mechanics: pure SDT provides the structural skeleton (existence of Bell-inequality violation, irreversibility of measurement); $G$ provides the quantitative content (Tsirelson upper bound $22$ , specific collapse probabilities).

$G$ ’s role grading in §3.4. §3.4’s conclusions do not depend on $G$ in a single uniform way; the dependence falls into two categories that must be strictly distinguished to clarify the argumentative strength of each conclusion. $G$ ’s meta-level properties (guaranteed by SDT’s meta-level arguments): (i) $G$ exists (§2.2.2); (ii) $G$ is unique (§2.2.3); (iii) $G$ is constant (Argument 2). These three are properties any legitimate $G$ must satisfy, guaranteed by SDT’s own arguments and not depending on any specific entity theory. $G$ ’s special-content commitments (filled by entity theory): (iv) cross-chain homogeneity, i.e., step-rates across different determination chains are comparable; (v) physical identification of zero-carriers, i.e., which physical entities satisfy the structural definition “self-chain length 1”; (vi) specific values of physical constants. These three are content provided by some specific entity theory’s instantiation of $G$ ; different entity theories may provide different fillings.

Dependency classification of §3.4’s conclusions: (a) the structural definition of zero-carrier, spacetime topological trichotomy, and carrier proper-time minimum value of 0 depend only on SDT itself (§2.4 DAG + §3.3 time order), not on any property of $G$ . These are pure SDT conclusions. (b) $c$ as a speed upper bound depends on $G$ ‘s special commitment (iv) (cross-chain homogeneity). If $G$ does not guarantee cross-chain homogeneity, the concept of “upper bound” loses meaning. (c) $c$ being constant depends on $G$ ‘s special commitment (v) (physical identification of zero-carriers); all physical entities satisfying the structural definition have identical propagation ratio. (d) The quantitative form of spatial distance $d = c \cdot λ_{B}$ depends jointly on $G$ ‘s special commitments (iv), (v), (vi).

Argumentative-strength grading: type (a) conclusions enjoy the conceptual-necessity strength of SDT itself; type (b)–(d) conclusions have strength equal to “SDT + specific entity theory” joint strength, varying with entity theory. The current physics instantiation of $G$ (special and general relativity + Standard Model of particle physics) satisfies (iv)–(vi), giving §3.4 complete quantitative predictive power under the current physical description. But if a future entity theory’s instantiation of $G$ differs from the current one (e.g., some quantum gravity candidate theories may modify $c$ ‘s constancy), type (b)–(d) conclusions will adjust accordingly, while type (a) conclusions remain unchanged. This grading makes the argumentative strength of §3.4 transparent: readers can clearly identify which conclusions are SDT meta-level commitments (not relaxable) and which are SDT + current-physics joint outputs (varying with entity theory).

§3.4.6 Possible objections. (a) “What about distances between two points that have not yet exchanged a zero-carrier? Are they ‘undefined’?” Response: in SDT, only structural relations established by zero-carrier propagation receive a structural distance. Two points without such a propagation have $⪯$ -incomparable status (spacelike); spatial distance for them is filled in by entity theory’s metric structure (which extends beyond the determination-count formalism). SDT’s $d_{S}$ is the structural skeleton; the full metric is entity-theoretic. (b) “Doesn’t this account require entity-theoretic content for completeness?” Response: yes, and SDT acknowledges this. SDT is a meta-structural theory, not a complete physical theory.

§3.4.7 Compatibility and boundary. Compatibility with §3.3 (Argument 3): §3.4 depends on §3.3’s time emergence; the partial-order structure underlying $λ_{B}$ is the time structure of §3.3. Compatibility with relativistic spacetime: §3.4’s spacetime trichotomy aligns with relativistic spacetime classification (timelike, lightlike, spacelike). Boundary: §3.4 commits only to the structural skeleton (trichotomy plus determination-count form); specific metric content is filled by entity theory plus $G$ .

§3.4.8 Exhaustiveness (integrative statement). The three core arguments of §3 plus the structural corollary of §3.4 establish four key conclusions of SDT: $R$ ‘s non-retraction (§3.1), $G$ ‘s constancy (§3.2), time’s structural emergence (§3.3), spatial distance’s derived status (§3.4). The four conclusions cover the structural ontology, modal direction, rule background, and spacetime skeleton of SDT’s framework. These conclusions are traditionally handled by separate metaphysical theories (time philosophy, space philosophy, modal metaphysics, philosophy of natural law), each independently. SDT derives them from the same axiom, providing a unified formal basis for traditionally independent metaphysical domains.

Chapter 4. Summary and Outlook

§4.1 Theoretical Summary

Starting from a single ontological axiom (reality consists of distinctly differentiated structure), SDT completes a self-contained theoretical construction in three stages. Stage one (Chapter 1): the four constituents of the axiom (reality, structure, consists of, distinctly differentiated) are placed within established philosophical traditions and explicated; the undeniability of each constituent is established through self-refutation arguments; the scope of the theory (reality coherently describable) is explicitly delineated. Stage two (Chapter 2): the axiom’s ontological commitments are unfolded into eleven formal concepts: elements $E$ , possible configurations $P$ , compatibility conditions $C$ , the global constraint $G$ , the realized-structure set $R$ , admissible configurations $A$ , the structural layer $L$ , determination $D$ , trace $τ$ , the trace set $T$ , and the determination dependency graph $G_{D}$ . Each concept retraces an ontological anchor in the axiom and introduces no commitments beyond the axiom. Stage three (Chapter 3): three core arguments are derived from this conceptual foundation: $R$ ‘s non-retraction and monotonic growth (Argument 1), $G$ ‘s constancy (Argument 2), and the emergence of time (Argument 3); a structural corollary derives the structural definition of spatial distance from Argument 3 (§3.4), introducing the zero-carrier concept, the upper-bound character of $c$ , and the spacetime topological trichotomy.

The dependency relations among the three core arguments are explicit: Argument 1 is independent; Argument 2 depends on Argument 1; Argument 3 depends on both Argument 1 and §2.4. §3.4 depends on Argument 3, and its quantitative form requires input from $G$ . The strength stratification of the arguments is therefore clear. The exhaustiveness statement confirms that the three core arguments together with the structural corollary cover SDT’s central structural theses at the current level of formalization. The entire argumentative chain (from axiom to concepts to arguments) borrows no specific physical theory, depends on no empirical data, and presupposes no concept of time or space. Time, as the emergent conclusion drawn from the unfolding of $G_{D}$ ‘s partial-order structure, and space, as a derived quantity of the time structure together with zero-carrier propagation, are the terminal results of this chain.

§4.2 Theoretical Contributions

SDT’s theoretical contributions can be characterized at four levels.

§4.2.1 First, structural emergentism of time. SDT’s treatment of time transcends the traditional A-theory / B-theory dichotomy in time philosophy. A-theory’s central concerns (the objective division of past, present, and future, and the direction of time) are fulfilled in SDT: the past is the structure of determinations already drawn in $G_{D}$ ; the present is the set of nodes in $G_{D}$ with no out-edges; the future is the region into which $G_{D}$ will extend; modal asymmetry ( $R$ ‘s non-retraction together with the openness of $∣ A ∣ > 1$ ) supplies the direction of time. B-theory’s central concerns (the reducibility of temporal properties to relations) are also fulfilled in SDT: temporal order is identical to $G_{D}$ ‘s dependency partial order, and all temporal properties are reducible to $G_{D}$ ‘s structural relations. SDT can simultaneously fulfill both sets of concerns because its starting point lies outside the A/B-theory problem framework: every dimension of time (order, division, direction, passage, simultaneity) emerges in unified form from the same structural basis: the continual growth of $G_{D}$ . This structural emergentism is SDT’s distinctive position in the philosophy of time.

§4.2.2 Second, logical irreversibility. $R$ ’s non-retraction (Argument 1) is grounded in the modal irreversibility of “realized”; in the context of irreversibility philosophy, it can be characterized as a new type of irreversibility, namely logical irreversibility. This irreversibility differs from the three types known in physics: (i) statistical irreversibility (Boltzmann’s second law of thermodynamics (Boltzmann 1877): reversible but extremely improbable); (ii) thermodynamic irreversibility (Prigogine’s far-from-equilibrium dissipative structures (Prigogine 1980): requiring sustained energy input); (iii) initial-condition irreversibility (the Albert-Price low-entropy hypothesis on cosmological initial conditions (Albert 2000; Price 1996): dependent on specific boundary conditions). SDT’s logical irreversibility is unconditional: it does not depend on statistical probability, far-from-equilibrium conditions, or cosmological initial conditions, but is derived directly from the modal structure of the concept of “realized” itself. This fourth kind of irreversibility is SDT’s independent contribution to the philosophy of irreversibility.

§4.2.3 Third, the derived status of space. The structural corollary of §3.4 establishes the ontological positioning of spatial distance in SDT: space is not an independent foundational quantity but a joint derived quantity of the time structure ( $G_{D}$ ‘s partial order) and the structure of zero-carrier propagation. This positioning directly supports SDT’s via negativa strategy: the conception of absolute space as an independent container is redundant within SDT’s framework, symmetric in strategy with Argument 3’s treatment of absolute time. §3.4 simultaneously provides the spacetime topological trichotomy (lightlike / timelike / spacelike) as a pure SDT corollary, the upper-bound character of $c$ as an SDT + $G$ joint conclusion, and the metric structure of space as established by $G$ on the topological skeleton. This treatment pattern (SDT giving the structural skeleton, $G$ giving the quantitative content) runs through SDT’s entire treatment of time and space, instantiating the division of labor between SDT as a meta-structural theory and entity theories.

§4.2.4 Fourth, the unifying power of a single axiom. From the single axiom reality consists of distinctly differentiated structure, SDT derives DAG structural-order, the irreversibility of reality, the stability of the rule background, the emergence of time, and the derived status of space. The five conclusions cover the structural order of determinations (§2.4), ontological direction (Argument 1), the rule background (Argument 2), and the spacetime skeleton (Argument 3 and §3.4). In traditional metaphysics, these conclusions are typically handled by separate theories: philosophy of time, philosophy of space, modal metaphysics, and metaphysics of natural law: each operating independently. SDT derives them from the same axiom, providing a unified formal foundation for several traditionally independent metaphysical domains. This unifying power is not the goal of theoretical design (SDT begins from the axiom without presupposing what is to be derived) but a structural consequence implicit in the axiom; the richness of the axiom is reflected in the breadth of its argumentative consequences.

§4.3 Boundaries and Future Directions

The scope of SDT has already been delineated in §1.4: reality coherently describable. This section, on that basis, marks two classes of content that SDT does not address.

§4.3.1 The division of labor between SDT and entity theories. All conclusions of SDT’s arguments are qualitative structural propositions: DAG structural-order, $R$ ‘s non-retraction, $G$ ‘s constancy, the emergence of time from the unfolding of $G_{D}$ ‘s partial-order structure, and the derivation of spatial distance from time structure and zero-carrier propagation. SDT commits to no quantitative content: the specific rate of determination occurrence, the specific rate of $R$ ‘s growth, the specific rule form of $G$ , the temporal metric between determinations in $G_{D}$ , the dimensionality and geometric structure of space, the specific numerical value of $c$ . These quantitative contents are established by entity theories on SDT’s structural skeleton. The relation between SDT and entity theories is not competitive: SDT provides the structural conditions that any theory describing a world with definite, irreversible spacetime order must satisfy; entity theories give specific quantitative descriptions within these conditions.

§4.3.2 Mathematical formalization. SDT’s current formalization operates at the set-theoretic level (sets, functions, and graph structures within the ZFC background). Deeper mathematical formalization, for example, characterizing SDT’s axiom in terms of the model class within model theory, or expressing SDT’s concepts in category-theoretic language: is the principal path for raising the rigor of the theory. The primary aim of mathematical formalization is to lift SDT’s arguments from “structural necessity” (necessary within SDT’s own formalism) to “mathematical necessity” (rigorously verifiable within mathematical structure), so that SDT’s conclusions acquire mathematical expression independent of SDT’s terminology.

§4.4 Application Prospects

SDT’s formal framework exhibits application potential in the following directions.

§4.4.1 Quantum foundations. From the same axiom, SDT offers a preliminary structural reading of the central interpretive puzzles of quantum mechanics: the measurement problem, nonlocality, contextuality, irreversibility, and irreducible randomness can be uniformly handled as different aspects of a single structural logic, without introducing new assumptions or free parameters beyond quantum mechanics. The detailed development of this application direction is the primary objective of SDT’s subsequent work.

§4.4.2 Irreversibility. SDT’s logical irreversibility (Argument 1) is distinct from the three types of irreversibility known in physics (statistical / thermodynamic / initial-condition) and constitutes a fourth, independent kind. This distinction makes an independent contribution to the philosophy of irreversibility and to research on the direction of time; its detailed argumentation is an important direction for SDT’s subsequent work.

§4.4.3 Further directions. SDT’s conceptual framework may engage in dialogue with the following research areas: causal set theory (structural similarity in discrete partial-order structures and spacetime emergence), process philosophy (conceptual resonance between determinations as structurally ontologically prior events and Whitehead’s “actual entities” tradition (Whitehead 1929)), and informational foundations (structural analogy between $R$ ‘s monotonic growth and the non-erasability of information). These dialogue directions are at a preliminary identification stage; their development depends on the deepening of SDT’s mathematical formalization and the advancement of cross-domain collaboration.

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SDT Corpus

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Structural Determination Theory v5.9

Chapter 1. The Axiom

§1.1 Statement of the Axiom

§1.2 Explication of the Axiom

§1.3 Self-Refutation Arguments

§1.4 The Scope of the Axiom

Chapter 2. Core Concepts

§2.1 Elements and Possible Configurations

§2.2 The Constraint System

§2.3 Structural Layers and Dynamical Concepts

§2.4 Structural Relations among Determinations

Chapter 3. Core Unfolding and Arguments

§3.1 Argument 1: Conceptual Explication of $R$ ‘s Non-Retraction

§3.2 Argument 2: Conceptual Explication of $G$ ‘s Constancy

§3.3 Argument 3: Constructive Explication of Time Emergence

§3.4 The Derived Status of Spatial Distance (Structural Corollary)

Chapter 4. Summary and Outlook

§4.1 Theoretical Summary

§4.2 Theoretical Contributions

§4.3 Boundaries and Future Directions

§4.4 Application Prospects

References

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Table of Contents

Backlinks

SDT Corpus

Explorer

Structural Determination Theory v5.9

Chapter 1. The Axiom

§1.1 Statement of the Axiom

§1.2 Explication of the Axiom

§1.3 Self-Refutation Arguments

§1.4 The Scope of the Axiom

Chapter 2. Core Concepts

§2.1 Elements and Possible Configurations

§2.2 The Constraint System

§2.3 Structural Layers and Dynamical Concepts

§2.4 Structural Relations among Determinations

Chapter 3. Core Unfolding and Arguments

§3.1 Argument 1: Conceptual Explication of R‘s Non-Retraction

§3.2 Argument 2: Conceptual Explication of G‘s Constancy

§3.3 Argument 3: Constructive Explication of Time Emergence

§3.4 The Derived Status of Spatial Distance (Structural Corollary)

Chapter 4. Summary and Outlook

§4.1 Theoretical Summary

§4.2 Theoretical Contributions

§4.3 Boundaries and Future Directions

§4.4 Application Prospects

References

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Table of Contents

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§3.1 Argument 1: Conceptual Explication of $R$ ‘s Non-Retraction

§3.2 Argument 2: Conceptual Explication of $G$ ‘s Constancy