Physical Structural Consistency:
D-Architecture Core and Its Structural Consequences

1. Introduction

1.1 Background

D-Arch starts from simple axioms such as existence, distinction, relation, and transition, and takes only a single condition — the core invariant I_min ≡ ∃t'>t : O(x_t') ≠ ∅, namely that future selection possibility must remain — as the basis for its derivations. Every item in the framework follows logically from the maintenance condition of this invariant, and the present study addresses the Core (D0–D23) and structural consequences (SC-1~9) among them. The specific definitions of the two layers are organized in §2.

1.2 Scope and Method

This study does not aim to verify D-Arch or to correct or extend physics; it reports the correspondence observed between the two formal systems as such. That is, neither physics verifying D-Arch nor D-Arch correcting physics falls within the scope of this study.

The objects of comparison follow the condensed definitions of the D-Arch source. The 24 Core items and the 9 structural consequences — 33 items in total — serve as the units of this comparison, and for each item the D-Arch definition and necessity are compared one-to-one with structures identifiable in physics. The 10 items of the D-Arch Operating Conditions (OC) layer are not included in the scope of this paper and are treated in a separate paper.

1.3 Verdict Criteria

The verdict for each item is given in the following three categories.

• Consistent: A physical structure corresponding to the D-Arch structure exists, and no inconsistency is observed.
• Inconsistent: An inconsistency is observed between the D-Arch structure and the physical structure, or physics explicitly excludes the structure.
• Unconfirmed: No corresponding structure is confirmed in the formal system of physics. This is distinct from exclusion; physics simply does not address the structure.

For SC items, when an unconfirmed structure is included in the premises, a complete verdict is itself impossible. Such cases are also classified as Unconfirmed, and in this comparison SC-8 falls into this category.

The results of this comparison are organized in the summary table of §3, and the analysis of the distribution of unconfirmed items is treated in §5.

2. Framework

D-Arch is the set of minimal necessary structures for maintaining the core invariant I_min, and is composed of three layers: Core, structural consequences (SC), and operating conditions (OC). This study addresses the Core and structural consequences (SC) among these, while the operating conditions (OC) layer is treated in a separate paper. The Core is a set of basic structures, and the structural consequences (SC) are the logical results that follow inescapably when the Core holds. The two layers form a hierarchical dependency relation, Core → SC.

The symbols used in this document follow the definitions in §2 of Jung (2026), Structural Necessity in Selection Systems (DOI 10.5281/zenodo.19342655). Ω denotes the entire state space of the system, O(x) denotes the set of future selectable states reachable from state x under the current structure, and |O(x)| denotes the size of that set.

2.1 Core (D0–D23)

The Core consists of 24 items from D0 (Existence) to D23 (Termination), and addresses the basic structures a selection system must possess in order to exist. Each item consists of a definition of a single structural unit together with a necessity argument showing that I_min is not maintained without that structure. For example, D1 (Distinction) is the condition that a non-trivial partition must exist on Ω; without distinction, information is not defined. D8 (Boundary) is the condition that an interior/exterior decomposition structure must exist on Ω; without a boundary, the range of x in O(x) is not defined. D17 (Cost) is defined as the condition that, within D-Arch, every action carries a cost; an action with zero cost is not permitted by this definition.

2.2 Structural Consequences (SC-1~9)

Structural consequences are 9 logical results that follow inescapably if the Core holds. Each SC is derived from one Core item or a combination of several, and shows that I_min collapses if the consequence is denied. For example, SC-1 (Single-objective fixation is impossible) shows that fixing the system to a single objective causes the option space |O(x)| to shrink monotonically and I_min to collapse; SC-5 (Failure cannot be eliminated) shows that removing failure blocks restoration (D16) and the system eventually collapses; SC-9 (Complete description is impossible) states that no descriptive system can capture in full the structure to which it refers.

3. Summary Table

The physics correspondence for the 24 Core items and the 9 structural consequences (SC) — 33 items in total — is summarized in the table below. The detailed definitions and mapping basis for each item are treated in §4.

Of 33 items in total: 29 Consistent, 4 Unconfirmed (D10, D11, D14, SC-8), 0 Inconsistent.

4. Detailed Correspondence

This section organizes each of the 33 items in a six-part structure: D-Arch Definition, D-Arch Necessity, Structure in Physics, Necessity in Physics, Verdict, and References.

4.1 D0. Existence — Consistent

D-Arch Definition. Ω ≠ ∅. An axiom declaring that the space Ω of possible states is not empty. No claim is made about the properties or structure of what exists.

D-Arch Necessity. If we assume Ω = ∅, the assumption itself constitutes a state. Even when the assumption is not performed, its absence still constitutes a state. The act of denial presupposes the content being denied, so it is a performative contradiction. The negation of Ω ≠ ∅ cannot hold.

Structure in Physics. The first axiom of quantum mechanics associates a Hilbert space with each physical system, and the state of the system is described as a vector (ray) in that space (von Neumann, 1932, Ch.II; Jauch, 1968, Ch.5). In classical mechanics as well, the state space (phase space) of a system is defined by generalized coordinates and momenta (Goldstein, 2002, Ch.8).

Necessity in Physics. Without a state space, physical quantities cannot be defined, equations of motion cannot be written, and physics itself cannot be formulated.

Verdict. Without Ω ≠ ∅, description does not hold in D-Arch; in physics, without a state space, physics itself cannot be formulated.

References.

• von Neumann, Mathematical Foundations of Quantum Mechanics, 1932, Ch.II — DOI 10.1007/978-3-662-02931-8
• Jauch, Foundations of Quantum Mechanics, Addison-Wesley, 1968, Ch.5
• Goldstein, Classical Mechanics, 3rd ed., Pearson, 2002, Ch.8

4.2 D1. Distinction — Consistent

D-Arch Definition. A non-trivial partition exists on Ω. No claim is made about the cause or purpose of distinction.

D-Arch Necessity. If we assume there is no distinction, the assumption itself separates “the case where distinction exists” from “the case where it does not.” The moment D0 is accepted, the distinction between “existence” and “non-existence” is already at work. The denial of distinction is a performative contradiction, and the negation of D1 cannot hold.

Structure in Physics. In quantum mechanics, observables are defined as self-adjoint operators, and different eigenvalues correspond to different measurement outcomes. The orthogonality of eigenstates ensures that different measurement outcomes are in principle distinguishable (von Neumann, 1932, Ch.II–III; Nielsen & Chuang, 2000, Theorem 2.1). In classical mechanics as well, different points in phase space represent different physical states (Arnold, 1989, Ch.1–3).

Necessity in Physics. If states are not distinguished, measurement outcomes cannot be told apart, the initial value problem is not defined, and prediction is impossible. Physics is not possible without the distinction of states.

Verdict. Without distinction in D-Arch, information does not hold; without state distinction in physics, measurement and prediction cannot be carried out.

References.

• von Neumann, Mathematical Foundations of Quantum Mechanics, 1932, Ch.II–III — DOI 10.1007/978-3-662-02931-8
• Nielsen & Chuang, Quantum Computation and Quantum Information, Cambridge, 2000, Theorem 2.1
• Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989, Ch.1–3 — DOI 10.1007/978-1-4757-2063-1

4.3 D2. Relation — Consistent

D-Arch Definition. R ⊆ Ω × Ω, R ≠ ∅. Relations exist among states. No claim is made about the kind, direction, or strength of the relations.

D-Arch Necessity. If we assume “no relations of any kind exist among distinguished states,” then in order to judge the distinction in D1 (A ≠ B) we must reference two states simultaneously, and this reference itself constitutes a minimal relation. The act of denying relation is a performative contradiction that presupposes relation. The negation of D2 cannot hold.

Structure in Physics. Physics describes dynamics through equations of motion. Hamilton's equations in classical mechanics define a flow on phase space and mathematically connect states at different times (Landau & Lifshitz, 1976, §40). The Schrödinger equation in quantum mechanics describes the time evolution of state vectors and likewise connects states at different times (Sakurai & Napolitano, 2020, Ch.2). These equations of motion are mathematically relations between states.

Necessity in Physics. Without equations of motion, the time evolution of states cannot be described, and dynamics is not defined. This is not something Landau or Sakurai explicitly asserts; it is a logical observation that follows from the formal system of physics.

Verdict. Without relation in D-Arch, structure does not hold; without equations of motion in physics, dynamics cannot be formulated. However, while the R ⊆ Ω × Ω of D-Arch admits arbitrary relations, the equations of motion in physics carry the special structure of being deterministic (Hamiltonian) or unitary (Schrödinger). Physics realizes a special case of the relations admitted by D-Arch.

References.

• Landau & Lifshitz, Mechanics, 3rd ed., Pergamon, 1976, §40 — DOI 10.1016/C2009-0-25569-3
• Sakurai & Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge, 2020, Ch.2 — DOI 10.1017/9781108587280

4.4 D3. Transition — Consistent

D-Arch Definition. T ⊆ R, T ≠ ∅. Some relations are state changes (transitions). Time is defined as the order of transitions. No claim is made about the cause, mechanism, or directionality of transitions.

D-Arch Necessity. If we assume “there is no transition,” the assertion itself produces two states — “before the assertion” and “after the assertion” — and the passage between these two states is already a transition. The act of denying transition is a performative contradiction that performs a transition. The negation of D3 cannot hold.

Structure in Physics. In classical mechanics, Hamilton's equations define time evolution on phase space, and the state (q, p) changes with time (Landau & Lifshitz, 1976, §40). In quantum mechanics, the time evolution operator U(t) transforms state vectors with time (Sakurai & Napolitano, 2020, Ch.2). In thermodynamics, the principle of entropy increase gives directionality to irreversible processes (Callen, 1985, Ch.2).

Necessity in Physics. Without state change, time evolution is not defined, and physical processes cannot be described. This is not something physics textbooks explicitly assert; it is an interpretation based on a structural similarity between the formal system of physics and the definitions of D-Arch.

Verdict. Without transition in D-Arch, time is not defined; without state change in physics, dynamics cannot be sustained. However, “time” sits in different positions on the two sides. D-Arch defines time as the order of transitions, while physics (at the textbook level) presupposes time as an external parameter and transitions occur within it. This asymmetry does not deny consistency, but it is a point where identity cannot be claimed.

References.

• Landau & Lifshitz, Mechanics, 3rd ed., Pergamon, 1976, §40 — DOI 10.1016/C2009-0-25569-3
• Sakurai & Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge, 2020, Ch.2 — DOI 10.1017/9781108587280
• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.2

4.5 D4. Indeterminacy — Consistent

D-Arch Definition. When |Ω| > 1, which state is realized as the result of a transition is not determined before the transition. No claim is made about the cause of indeterminacy or its probabilistic structure.

D-Arch Necessity. If we assume the result of a transition is determined in advance, no branching exists and only a single path remains. If the maintenance condition of the single path is broken, no alternative remains and O(x) = ∅. I_min collapses.

Structure in Physics. In quantum mechanics, measurement outcomes are predicted only probabilistically by the Born rule, and which eigenstate will be realized is not determined before measurement (Born, 1926; Sakurai & Napolitano, 2020, Ch.1). Within the axiomatic structure of standard quantum mechanics, this indeterminism cannot be removed. Some interpretations (Everett, Bohm) derive or reinterpret the Born rule, however, so the status of this indeterminism depends on interpretation.

Classical mechanics is in principle deterministic. In the chaotic regime, small differences in initial conditions diverge exponentially (Strogatz, 2015, Ch.9), but this is effective unpredictability, not genuine indeterminacy. From given initial conditions, only one trajectory is realized.

Necessity in Physics. The experimental violation of Bell's inequalities excludes local hidden-variable theories — the assumption that measurement outcomes are locally predetermined is incompatible with the predictions of quantum mechanics (Bell, 1964; Aspect, 1982). Non-local determinism (such as Bohmian mechanics) is not excluded, however, so this result does not by itself prove that “nature is non-deterministic.” Within the framework of standard quantum mechanics, the physical world is not fixed to a single path, at least at the quantum level.

Verdict. In D-Arch, single-path fixation leads to structural collapse; in physics, multiple possible outcomes are experimentally confirmed at the quantum level. They are consistent at the factual level of “the existence of multiple possible paths.”

The answer to why indeterminacy exists differs, however. D-Arch argues that “a single path is structurally insufficient because there is no alternative on failure.” Physics simply observes that “this is how nature is.” A physics structure corresponding to D4's necessity argument (structural survival) has not been confirmed.

Furthermore, classical chaos is deterministic unpredictability and is distinct from D4's “multiple states being genuinely possible.”

References.

• Born, M., Zur Quantenmechanik der Stoßvorgänge, Z. Phys. 37, 863–867, 1926 — DOI 10.1007/BF01397477
• Bell JS., On the Einstein Podolsky Rosen Paradox, Physics Physique Fizika 1(3):195–200, 1964 — DOI 10.1103/PhysicsPhysiqueFizika.1.195
• Aspect, A. et al., Experimental Realization of EPR-Bohm Gedankenexperiment, Phys. Rev. Lett. 49, 91, 1982 — DOI 10.1103/PhysRevLett.49.91
• Sakurai & Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge, 2020, Ch.1 — DOI 10.1017/9781108587280
• Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview, 2015, Ch.9 — DOI 10.1201/9780429492563

4.6 D5. Observation — Consistent

D-Arch Definition. A set Y and a mapping Obs: Ω → Y exist, with Y ⊂ Ω. Observation is a mapping from the entire state space to a smaller space, and since Y is smaller than Ω, information is necessarily lost. No claim is made about the observer, the method, or the physical realization of observation.

D-Arch Necessity. First, if we assume the mapping does not exist, it cannot be distinguished whether a transition has occurred, so O(x) is not defined and I_min collapses. Second, if we assume the mapping preserves the entire Ω without loss, multiple states are not reduced to a single outcome, no realization is determined, and O(x) is not defined. I_min collapses.

Structure in Physics. In standard quantum mechanics (von Neumann formalism), measurement is described as projection of the state vector onto an eigenstate of the observable, and the superposition information prior to projection is described as irreversibly lost (von Neumann, 1932, Ch.V–VI; Sakurai & Napolitano, 2020, Ch.1). Other interpretive frameworks (Everett, Bohm, decoherence) reconstruct measurement without the projection postulate, but the point that information accessible to the observer is reduced is maintained regardless of interpretation. In classical statistical mechanics, macroscopic variables (temperature, pressure, etc.) are defined as ensemble averages of microscopic states, and the information of individual microscopic states is not accessible from macroscopic observation (Huang, 1987, Ch.6).

Necessity in Physics. Without measurement/observation, theory is not connected to experiment. Whichever interpretive framework one adopts, the information accessible to the observer in the measurement process is less than the total state information. In statistical mechanics as well, thermodynamics cannot be described without macroscopic variables.

Verdict. Without observation in D-Arch, O(x) is not defined; without measurement in physics, theory is not connected to experiment. Both sides have factual-level consistency in that information is reduced in the observation/measurement process.

The answer to why observation is necessary differs, however. D-Arch argues that “without observation, realized and unrealized states cannot be distinguished, so the selection system is not defined.” Physics requires that “without measurement, theory is not connected to experiment.” The reasons for necessity differ.

Furthermore, D-Arch's observation is “state identification within the selection system,” while physics's measurement is “interaction between the experimental apparatus and the system.” The observer and the context of observation differ, and this asymmetry is a point where identity cannot be claimed.

References.

• von Neumann, Mathematical Foundations of Quantum Mechanics, 1932, Ch.V–VI — DOI 10.1007/978-3-662-02931-8
• Sakurai & Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge, 2020, Ch.1 — DOI 10.1017/9781108587280
• Huang, Statistical Mechanics, 2nd ed., Wiley, 1987, Ch.6

4.7 D6. Constraint — Consistent

D-Arch Definition. Γ ⊆ T, Γ ≠ ∅. Patterns that recur during transitions exist. No claim is made about the cause, purpose, or strength of constraint.

D-Arch Necessity. If we assume there are no recurring patterns, no common structure accumulates between transitions. Without accumulated regularity, a sustainable state x is not maintained, so O(x) is not defined. I_min collapses.

Structure in Physics. Physical laws repeatedly produce the same outcomes under the same conditions. This universal repeatability is the basis on which physics is possible (Feynman, The Character of Physical Law, 1965, Ch.1). Conservation laws (energy, momentum, charge, etc.) are concrete expressions of this repeatability and restrict the range of permissible state changes. Noether's theorem shows that conservation laws are derived from continuous symmetries (Noether, 1918; Goldstein, 2002, Ch.13).

Necessity in Physics. Without the repeated holding of physical laws, experiments cannot be reproduced and prediction cannot be sustained. Physics presupposes regularity in nature, and without this presupposition physics does not function as a descriptive system.

Verdict. Without recurring constraint in D-Arch, a sustainable structure cannot be maintained; without the repeated holding of laws in physics, prediction is impossible. Both sides are consistent at the factual level that “recurring transition patterns” exist.

The answer to why constraint exists differs, however. D-Arch argues that “without constraint, sustainable structure is not possible.” In physics, conservation laws are mathematical consequences of symmetries (Noether, 1918), and the universality of laws is something physics presupposes rather than explains.

References.

• Feynman, The Character of Physical Law, MIT Press, 1965, Ch.1
• Noether, E., Invariante Variationsprobleme, Nachr. Ges. Wiss. Göttingen, 235–257, 1918
• Goldstein, Classical Mechanics, 3rd ed., Pearson, 2002, Ch.13

4.8 D7. Evaluation — Consistent

D-Arch Definition. J: Ω → ℝⁿ, n ≥ 2. An evaluation structure for states exists. No claim is made about the evaluator, the content of the criteria, or the meaning of the values.

D-Arch Necessity. If we assume there is no evaluation, there is no consistency among selections, no path-maintenance structure forms, and O(x) does not persist. If we assume evaluation is single-criterion (n=1), all differences are reduced to a single axis, a single path is fixed, no alternative path remains, and O(x) = ∅. In either case, I_min collapses.

Structure in Physics. In physics, the state of a system is described not by a single number but by multiple independent physical quantities. In classical mechanics, a point in phase space consists of n generalized coordinates and n momenta (Goldstein, 2002, Ch.8), which are independent degrees of freedom not reducible to one another. In thermodynamics, the state of a system is described by multiple state variables (T, P, V, S, etc.), and a system cannot be determined by a single variable alone (Callen, 1985, Ch.1). In quantum mechanics, fully specifying the state of a system requires a complete set of commuting observables (CSCO) (Sakurai & Napolitano, 2020, Ch.1).

Necessity in Physics. If a physical system is described by only a single physical quantity, the states of the system cannot be distinguished. There exist states with the same energy but different momenta, and states with the same temperature but different pressures. It is impossible to determine the state of a system without multiple independent physical quantities.

Verdict. In D-Arch, single-axis evaluation leads to single-path fixation and the collapse of I_min; in physics, a single physical quantity cannot determine the state of a system. Both sides require multiple independent dimensions in order to describe or distinguish states.

D-Arch's “evaluation” is, however, a discriminative structure for path maintenance, while physics's “physical quantities” are quantities that describe the state of a system. In D-Arch, J is “discrimination for selection,” while in physics physical quantities describe the system independently of “selection.” The functional context differs.

References.

• Goldstein, Classical Mechanics, 3rd ed., Pearson, 2002, Ch.8
• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.1
• Sakurai & Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge, 2020, Ch.1 — DOI 10.1017/9781108587280

4.9 D8. Boundary — Consistent

D-Arch Definition. An interior/exterior decomposition structure exists on Ω (x = (x_int, x_ext), B_t). The boundary B_t depends on the time t, and no claim is made about the form or physical realization of the boundary.

D-Arch Necessity. When observation (D5) holds, an asymmetry arises between “what is grasped” and “what is not grasped,” and this is the minimal interior/exterior decomposition. Without a boundary, x in O(x) has no range, and I_min collapses. Furthermore, since states change by transition (D3), the boundary must change with time. A fixed boundary cannot follow changing states, and I_min collapses.

Structure in Physics. Thermodynamics begins with the distinction between system and surroundings. Without this distinction, thermodynamic state variables cannot be defined and the direction of energy exchange cannot be described (Callen, 1985, Ch.1). To describe measurement in quantum mechanics, the system being observed and the measuring apparatus must be distinguished, and this distinction is a presupposition of measurement theory (von Neumann, 1932, Ch.VI). To define an ensemble in statistical mechanics, the boundary of the system must be specified (Huang, 1987, Ch.5–6).

Necessity in Physics. Without the distinction between system and surroundings, there is no object to which the first law of thermodynamics (energy conservation) can be applied. In quantum mechanics, without distinguishing the system from the apparatus, measurement outcomes cannot be defined. In physics, the boundary is a presupposition of description, and it is impossible to define a physical system without a boundary.

Verdict. Without a boundary in D-Arch, x in O(x) has no range and I_min collapses; without the system-environment distinction in physics, a physical system cannot be defined. Both sides are consistent at the factual level that the interior/exterior distinction is a presupposition of the system.

D-Arch's boundary is, however, “an access asymmetry that arises from observation,” while physics's boundary is often a modeling choice set by the analyst. Physics also has boundaries that are physically determined — event horizons, phase-transition interfaces, preferred decompositions induced by decoherence, and so on. The “difference in mode of origin” is therefore not absolute.

References.

• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.1
• von Neumann, Mathematical Foundations of Quantum Mechanics, 1932, Ch.VI — DOI 10.1007/978-3-662-02931-8
• Huang, Statistical Mechanics, 2nd ed., Wiley, 1987, Ch.5–6

4.10 D9. Selection — Consistent

D-Arch Definition. An operation that fixes some of the possible states according to a determination condition exists (Ω_{t+1} = Ω_t ∩ C_{t+1}, Ω_{t+1} ⊂ Ω_t). C does not generate states; it is a constraint that admits or excludes some of Ω. Selection does not presuppose volition, intention, or purpose.

D-Arch Necessity. If we assume no selection, the state is not fixed even after the verdict by observation (D5). If not fixed, x (the current state) in O(x) is not defined, and I_min collapses.

Structure in Physics. In quantum mechanics, measurement fixes a superposition state to a single eigenstate, and (in the standard formalism) other possibilities prior to fixation are irreversibly removed (von Neumann, 1932, Ch.V–VI). In spontaneous symmetry breaking, one of multiple symmetric states is realized and the rest are excluded — for example, the choice of magnetization direction in a ferromagnet (Weinberg, 1996, Ch.19). In thermodynamics, irreversible processes effectively reduce the number of accessible macroscopic states (Callen, 1985, Ch.2–4).

Necessity in Physics. In quantum mechanics, if measurement outcomes are not fixed, the current state of the system cannot be referenced, and subsequent time evolution cannot be described. Without spontaneous symmetry breaking, particular macroscopic ordered states (magnetism, superconductivity, etc.) do not arise. In the physical world, the fact that “one of the possible outcomes is realized” is observed in measurement, spontaneous symmetry breaking, irreversible processes, and the like.

Verdict. Without selection in D-Arch, there is no realized present; without state fixation in physics, the present of the system cannot be referenced. Both sides are consistent at the factual level that “one of the possible outcomes is fixed and the space of possibilities is reduced.”

D-Arch's selection is, however, “structural fixation for path maintenance,” while physics's state fixation is “realization by measurement or symmetry breaking.” In D-Arch, selection does not presuppose volition, and physics's state fixation is also unrelated to volition. In this respect, the two sides are aligned in being non-teleological, but the mechanism by which fixation occurs differs.

References.

• von Neumann, Mathematical Foundations of Quantum Mechanics, 1932, Ch.V–VI — DOI 10.1007/978-3-662-02931-8
• Weinberg, The Quantum Theory of Fields, Vol.II, Cambridge, 1996, Ch.19 — DOI 10.1017/CBO9781139644174
• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.2–4

4.11 D10. Attribution — Unconfirmed

D-Arch Definition. M : S → S, Self := {x | M(x) = x}. A mapping that attributes selections to a particular structure exists. No claim is made about what intrinsic properties the object of attribution may possess.

D-Arch Necessity. Without attribution, it cannot be judged whether a selection is internal to the boundary (D8) or the result of an external influence. If this judgment is not possible, the interior/exterior distinction of the boundary becomes meaningless, and the definition of the range of O(x) collapses. I_min collapses.

Structure in Physics. In fundamental physics, state changes occur, but a mapping that “attributes” them to a particular structure is not included in the formal system of physics. The point mass of Newtonian mechanics, the state vector of quantum mechanics, the system of thermodynamics — these describe states, but they do not possess a self-model (M: S → S). Physical processes do not ask “whose” process it is.

Necessity in Physics. Physics describes dynamics without attribution. Hamilton's equations and the Schrödinger equation do not require attribution to any particular structure, and the formal system of physics functions even without attribution.

Verdict. In D-Arch, attribution is a presupposition of the structure that maintains selection, but in physics no structure corresponding to attribution has been confirmed. Physics describes dynamics without attribution, and the absence of this structure does not impede the function of physics.

This is not inconsistency but unconfirmed status. Physics does not “exclude” attribution; it simply does not describe it.

4.12 D11. Integrated selection — Unconfirmed

D-Arch Definition. A structural condition exists in which multiple selection operations (D9) are integrated under a single consistent attribution (M). No claim is made about what intrinsic properties the object of integration may possess.

D-Arch Necessity. If selections attributed to the same structure are unrelated, the attribution mapping M is not consistent over time, conflicting with D10. Furthermore, if selections are not integrated, the constraints of prior selections are not reflected in subsequent selections, irreversible shrinkage is not coordinated, and a path to O(x) → ∅ opens. I_min collapses.

Structure in Physics. D11 presupposes D10 (Attribution). Since no attribution structure has been confirmed in physics under D10, no physics structure corresponding to D11 — which is built on it — can be confirmed either. Physics does have structures that integrate multiple processes into a single description — collective modes, quasiparticles, order parameters, and so on. However, these are descriptive integrations of dynamics and differ from the “integration of selections” required by D11.

Necessity in Physics. The physical necessity of D11 cannot be judged independently. D11 is a structure built on top of D10 (Attribution), and a D11-specific necessity cannot be assessed independently while no physics structure corresponding to D10 has been confirmed.

Verdict. For the same reason as D10, this is unconfirmed; this is an inherited form of unconfirmed status from D10. Physics does not exclude this structure; it simply does not describe it.

4.13 D12. Stability — Consistent

D-Arch Definition. A structure exists in which the system can maintain its path in the neighborhood of a particular region (∃ A ⊆ Ω). No claim is made about the form, scope, or mechanism of stability.

D-Arch Necessity. Without stability, the results of selection do not form a coherent path, O(x) is not maintained, and I_min collapses.

Structure in Physics. In dynamical systems theory, an attractor is a subset of state space toward which trajectories converge over time. Various types exist — point attractors, limit cycles, strange attractors, and so on (Strogatz, 2015, Ch.5–9). In thermodynamics, thermal equilibrium is the state of maximum entropy, and an isolated system converges to this state (Callen, 1985, Ch.2). In nonequilibrium dissipative structures as well, stable steady states are maintained (Nicolis & Prigogine, Self-Organization in Nonequilibrium Systems, 1977, Ch.3–4).

Necessity in Physics. Without stable states, physical structures are not maintained. Atoms, molecules, and crystal lattices are maintained in the neighborhood of attractors, while conservative systems such as planetary orbits are maintained in the neighborhood of stable invariant sets (such as KAM tori) rather than attractors. Within the range of cases mentioned above, no physical structure that persists without a stable region has been reported.

Verdict. Without stability in D-Arch, the results of selection do not form a coherent path and I_min collapses; without a stable region in physics, structure is not maintained. Both sides are consistent at the factual level that “a stable region must exist for persistence.”

D-Arch's stability is, however, “a condition for maintaining I_min,” while physics's attractor is “a mathematical property of dynamical systems.” D-Arch makes stability explicit as a condition rather than a goal, and the attractor in physics is also a result of dynamics rather than a goal. In this respect, the two sides are aligned in being non-teleological.

References.

• Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview, 2015, Ch.5–9 — DOI 10.1201/9780429492563
• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.2
• Nicolis & Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, 1977, Ch.3–4

4.14 D13. Option shrinkage — Consistent

D-Arch Definition. Selection eliminates options (selection → |O(x)| can decrease). No claim is made about the size, speed, or pattern of shrinkage.

D-Arch Necessity. If selection does not shrink options, D9 (selection = state fixation) has no effect on the path and is rendered inert, and I_min collapses.

Structure in Physics. According to the second law of thermodynamics, the entropy of an isolated system increases or remains constant. While entropy increase actually expands the number of accessible microstates, the macroscopic paths by which the system can return are reduced — the system converges to equilibrium and does not spontaneously return to a prior nonequilibrium macrostate (Callen, 1985, Ch.2–4). In quantum mechanics, measurement fixes a superposition state to a single eigenstate, and (in the standard formalism) other possibilities are irreversibly removed (von Neumann, 1932, Ch.V).

Necessity in Physics. Without irreversibility, the direction of thermodynamic time is not defined, and the directionality of macroscopic processes disappears. Strictly reversible macroscopic processes are close to an idealization, and actual macroscopic processes appear effectively irreversible.

Verdict. In D-Arch, selection irreversibly shrinks options; in physics, irreversible processes shrink the macroscopically realizable paths. Both sides are consistent at the factual level that “fixation reduces possibilities.”

D-Arch's option shrinkage is, however, a structural cost built into the definition of selection, while physics's irreversibility is a statistical result of overwhelmingly high probability. In D-Arch, shrinkage is a logical necessity, while in physics it is a statistical necessity (in principle reversible by Poincaré recurrence, fluctuation theorems, and the like).

References.

• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.2–4
• von Neumann, Mathematical Foundations of Quantum Mechanics, 1932, Ch.V — DOI 10.1007/978-3-662-02931-8

4.15 D14. Meta-evaluation — Unconfirmed

D-Arch Definition. J_meta: a function that evaluates the evaluation function J. A higher-level evaluation that evaluates evaluation itself. No claim is made about the form, depth, or operation of meta-evaluation.

D-Arch Necessity. With only a single evaluation, revision of evaluation criteria is impossible. If the evaluation structure is fixed while constraint patterns change (D6), it cannot reflect the current state, and the criteria are effectively reduced to a single one (D7 degeneration), so I_min collapses.

Structure in Physics. In fundamental physics, physical laws do not evaluate or revise themselves (Weinberg, 1995, Ch.1–2). Newton's laws of motion, Maxwell's equations, the Schrödinger equation — these describe the dynamics of systems but do not include a structure that recursively examines the descriptive criteria themselves. Renormalization changes effective parameters with scale (Wilson, 1983), but this is an adjustment of parameter values within a fixed Lagrangian structure, not a re-examination of the evaluation criteria (the framework) themselves.

Necessity in Physics. Physical laws are not selection systems, so the problem of evaluation criteria reducing to a single one (D7 degeneration) does not arise. A structure corresponding to meta-evaluation is therefore not required by the formal system of physics, and its absence does not impede the function of physics.

Verdict. In D-Arch, meta-evaluation is a necessary structure for the revision of criteria, but in physics no structure corresponding to it has been confirmed. Physical laws do not evaluate themselves, and the absence of this structure does not impede the function of physics. Like D10 (Attribution) and D11 (Integrated selection), this is unconfirmed. Unlike D11, however, the unconfirmed status of D14 does not inherit from another item's unconfirmed status; it is a direct unconfirmed status arising from the absence of a self-evaluation structure as a separate structure. Physics does not exclude meta-evaluation; it simply does not describe it.

References.

• Weinberg, The Quantum Theory of Fields, Vol.I, Cambridge, 1995, Ch.1–2 — DOI 10.1017/CBO9781139644167
• Wilson, K.G., The renormalization group and critical phenomena, Rev. Mod. Phys. 55, 583, 1983 — DOI 10.1103/RevModPhys.55.583

4.16 D15. Threshold — Consistent

D-Arch Definition. ∃Θ s.t. |O(x)| < Θ → collapse risk. When options fall below the threshold value Θ, system collapse risk arises. No claim is made about the size, form, or mode of detection of the threshold.

D-Arch Necessity. Without a threshold, the shrinkage states of O(x) cannot be distinguished, and it cannot be judged whether collapse is imminent. In a structure where this judgment is not possible, collapse cannot be avoided, so I_min collapses.

Structure in Physics. In physics, a critical point is a boundary at which the qualitative behavior of a system changes. In phase transitions, when the temperature crosses the critical value the phase changes (Stanley, Introduction to Phase Transitions and Critical Phenomena, 1971, Ch.1–2), and in dynamical systems, when a parameter crosses a bifurcation point the structure of attractors changes qualitatively (Strogatz, 2015, Ch.3–4). At thermodynamic stability limits (spinodals), metastable states can no longer be sustained (Callen, 1985, Ch.8).

Necessity in Physics. In systems with phase transitions, bifurcations, or stability limits, qualitative transitions occur under particular conditions, and the boundary of these transitions is the critical point. Systems whose qualitative behavior never changes as parameters vary are uncommon.

Verdict. In D-Arch, when options fall below the threshold, collapse risk arises; in physics, when a parameter crosses a critical point, the qualitative behavior of the system changes. Both sides are consistent at the factual level that “crossing a boundary changes the character of the system.”

D-Arch's threshold is, however, “a lower bound on the number of options,” while physics's critical point is “a bifurcation in parameter space.” In D-Arch, falling below the threshold means “collapse,” while in physics, crossing a critical point need not be collapse but can be “transition to another state.” The meaning of threshold differs.

References.

• Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford, 1971, Ch.1–2
• Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview, 2015, Ch.3–4 — DOI 10.1201/9780429492563
• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.8

4.17 D16. Restoration — Consistent

D-Arch Definition. Rest: collapse risk → attempt to recover options. When collapse risk arises, an action that attempts to recover options exists. Restoration does not guarantee its outcome, and no claim is made about its performer, method, or whether it succeeds.

D-Arch Necessity. Without restoration, option shrinkage proceeds in only one direction. By D13, selection can eliminate options, and by D15, a threshold exists; without a structure to reverse O(x) approaching the threshold, the path O(x) = ∅ cannot be blocked. I_min collapses.

Structure in Physics. In physics, systems in stable equilibrium return to equilibrium after disturbance. According to Le Chatelier's principle, when an external change is applied to a system in equilibrium, the system responds in a direction that offsets the change (Callen, 1985, Ch.8). The fluctuation-dissipation theorem shows that fluctuations near equilibrium are damped by dissipative processes (Kubo, 1966). In nonequilibrium dissipative structures as well, disturbances near a steady state are damped and the system returns to the steady state (Nicolis & Prigogine, 1977, Ch.3).

Necessity in Physics. An equilibrium state without a return mechanism after disturbance is not stable, and a stable structure necessarily has restoring forces against disturbance. This is intrinsic to the definition of stability.

Verdict. Without restoration in D-Arch, option depletion is irreversible; without restoring forces in physics, a stable state is not maintained. Both sides are consistent at the factual level that “recovery after disturbance must be possible.”

D-Arch's restoration is, however, “the reopening of the option space,” while physics's restoring force is mainly “the return to an attractor or to equilibrium.” Since physics also includes cases of transition to a new stable state after a bifurcation, however, “return to a prior state” is not the only form of restoration in physics. The emphasis of restoration differs, but it is not an absolute dichotomy.

References.

• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.8
• Kubo, R., The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255, 1966 — DOI 10.1088/0034-4885/29/1/306
• Nicolis & Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, 1977, Ch.3

4.18 D17. Cost — Consistent

D-Arch Definition. ∀a, Cost(a) > 0. A structural resource consumed when an action is performed; always positive. No claim is made about the kind, unit, or measurement of cost.

D-Arch Necessity. If an action with zero cost exists, no structural restriction applies to it. Without such restriction, selection can occur at arbitrary speed and frequency, and the shrinkage process is uncontrolled. In uncontrolled shrinkage, the threshold (D15) and restoration (D16) cannot be structurally maintained, and the path O(x) = ∅ cannot be blocked. I_min collapses.

Structure in Physics. In physics, dissipative processes, control and driving, measurement, and irreversible processes involve resource constraints or energy budgets. The first law of thermodynamics declares that the energy of a closed system is conserved, and exchange between system and surroundings is described in the form of an energy budget (Callen, 1985, Ch.1–2). In dissipative processes, part of the useful energy is irreversibly converted to heat (Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed., 1980, §13–15).

Necessity in Physics. In processes involving resources — dissipation, driving, measurement, and the like — no change exempt from the energy budget is known. Energy conservation is among the most universally treated laws in physics.

Verdict. In D-Arch, an action without cost is not structurally permitted; in physics as well, in processes involving resources, no change without an energy cost is known. Among D0–D23, this is one of the more direct correspondences with physics.

D-Arch's “cost” is, however, a general structure not limited to energy, while physics's “energy cost” is a particular realization through energy conservation.

References.

• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.1–2
• Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed., Pergamon, 1980, §13–15 — DOI 10.1016/C2009-0-24487-4

4.19 D18. Delay — Consistent

D-Arch Definition. t(effect) > t(cause). Between two causally connected events there is an interval in their order. No claim is made about the size, cause, or physical realization of the interval.

D-Arch Necessity. Without delay, the order of cause and effect is not defined, and the causal relation between D9 (Selection) and D16 (Restoration) is not formed. State changes do not accumulate, so O(x) cannot be maintained, and I_min collapses.

Structure in Physics. In special relativity, the propagation speed of causal signals cannot exceed the speed of light c. This is among the most fundamental constraints in physics (Einstein, 1905; Weinberg, 1995, Ch.2). A finite time interval necessarily exists between cause and effect, and instantaneous causal influence is excluded by the light-cone structure. In classical mechanics as well, finite propagation speeds (acoustic, elastic, electromagnetic waves, etc.) impose causal delay.

Necessity in Physics. If the speed-of-light limit is violated, the causal order can be reversed depending on the observer (time-order reversal), and the causal structure of physics collapses. Finite propagation speed is the basis of the causal consistency of physics.

Verdict. In D-Arch, instantaneous causal influence is impossible; in physics, instantaneous causal influence is excluded by the speed-of-light limit. Together with D17 (Cost), this is among the more direct correspondences with physics.

D-Arch's “delay” is, however, the general form “a temporal interval between cause and effect,” while physics's speed-of-light limit is a particular realization that imposes a lower bound on this delay.

References.

• Einstein, A., Zur Elektrodynamik bewegter Körper, Ann. Phys. 17, 891–921, 1905 — DOI 10.1002/andp.19053221004
• Weinberg, The Quantum Theory of Fields, Vol.I, Cambridge, 1995, Ch.2 — DOI 10.1017/CBO9781139644167

4.20 D19. Closure boundary — Consistent

D-Arch Definition. Ω_local ⊂ Ω, ∂Ω_local ≠ ∅. There are structural limits on the state space that the system can access. No claim is made about the form, size, or permeability of the boundary.

D-Arch Necessity. If the entire Ω is accessible, there is no reason for the loss in observation (D5) to hold, and the irreversible shrinkage of selection (D9) is immediately reflected globally, rendering the boundary (D8) meaningless. The range of O(x) is not defined, so I_min collapses.

Structure in Physics. In physics, locality is a fundamental principle. The light-cone structure of special relativity restricts the spacetime region that an event can causally access (Einstein, 1905; Weinberg, 1995, Ch.2). In quantum field theory, locality is expressed as the condition that operators at spacelike-separated points commute (Weinberg, 1995, Ch.3). Quantum entanglement violates Bell inequalities, but this does not allow superluminal signaling, and field-theoretic locality (the commutativity condition for spacelike-separated operators) is preserved. In statistical mechanics as well, an observer or a subsystem cannot globally access the entire state space, and actual descriptions are restricted to limited degrees of freedom and observables.

Necessity in Physics. If locality is removed, superluminal causation becomes possible and the relativistic causal structure collapses. Locality is the basis of the causal consistency and informational structure of physics.

Verdict. In D-Arch, if the entire Ω is accessible, observation and boundary become meaningless; in physics, without locality, the causal structure collapses. Both sides are consistent at the factual level that “the accessible region is restricted.”

D-Arch's closure boundary is, however, the general form “an access restriction on state space,” while physics's locality is an access restriction by “the light-cone structure of spacetime.” The formal structures of the two are consistent, but since D-Arch's form is more abstract than physics's locality, identity cannot be claimed.

References.

• Einstein, A., Zur Elektrodynamik bewegter Körper, Ann. Phys. 17, 891–921, 1905 — DOI 10.1002/andp.19053221004
• Weinberg, The Quantum Theory of Fields, Vol.I, Cambridge, 1995, Ch.2–3 — DOI 10.1017/CBO9781139644167

4.21 D20. Overheating — Consistent

D-Arch Definition. Single objective + acceleration → Θ approach. A structural tendency to maximize a single criterion and thereby eliminate other paths. No claim is made about the cause, speed, or target of overheating.

D-Arch Necessity. By D7, evaluation has multiple criteria. If only a single criterion is followed, selection is biased, and by D13, paths are irreversibly eliminated. The loss of multiple paths leaves insufficient alternatives for restoration (D16) to operate near the threshold (D15), so I_min collapses.

Structure in Physics. The state of a physical system is described by multiple independent physical quantities (confirmed in D7). When one of these multiple axes dominates the dynamics, other degrees of freedom are eliminated and the system becomes unstable. An undamped oscillator driven at its resonant frequency is dominated by a single mode and its amplitude diverges (Strogatz, 2015, Ch.5).

Necessity in Physics. In physical systems where a single axis dominates the dynamics, divergence or collapse appears in the absence of buffering by other degrees of freedom. The undamped resonance example above is one instance of this pattern.

Verdict. In D-Arch, single-axis dominance plus acceleration approaches Θ; in physics, the dominance of a single variable or constraint induces divergence or collapse. They are consistent at the factual level that “the dominance of one among multiple axes induces instability.”

References.

• Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview, 2015, Ch.5 — DOI 10.1201/9780429492563

4.22 D21. Buffering — Consistent

D-Arch Definition. Dispersion / postponement / diversification → maintenance of O(x). No claim is made about the form, strength, or operation of buffering.

D-Arch Necessity. Without buffering, the bias of D20 is not controlled and accumulates. By D13, elimination is irreversible, so multiple paths are gradually removed. The alternatives for restoration (D16) near the threshold (D15) disappear, and I_min collapses.

Structure in Physics. In physics, thermal buffering absorbs temperature variations and suppresses abrupt state changes in a system. Systems with large heat capacity show more gradual temperature changes in response to energy input (Callen, 1985, Ch.3). In dynamical systems, damping reduces the amplitude of oscillations and prevents divergence (Strogatz, 2015, Ch.5). In dissipative structures, energy dispersion relaxes local excess and maintains the steady state (Nicolis & Prigogine, 1977, Ch.4).

Necessity in Physics. Physical systems without buffering or damping diverge or are destroyed under external driving. An undamped oscillator driven at its resonant frequency increases in amplitude without bound. Physical structures that persist include some form of buffering.

Verdict. Without buffering in D-Arch, option shrinkage accelerates; without damping or buffering in physics, the system diverges. Both sides are consistent at the factual level that “a structure suppressing abrupt change is necessary.”

D-Arch's buffering is, however, a structure for “the maintenance of selection diversity,” while physics's damping or buffering is a result that follows from the thermodynamic properties of a system without purpose. In D-Arch, buffering is a functional structure; in physics, buffering is a response property of the system.

References.

• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.3
• Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview, 2015, Ch.5 — DOI 10.1201/9780429492563
• Nicolis & Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, 1977, Ch.4

4.23 D22. Non-intervention — Consistent

D-Arch Definition. Buffering must be implemented through the structure itself, not through external intervention. No claim is made about the form or mode of implementation.

D-Arch Necessity. Buffering that depends on external intervention is not a property internal to the structure, and when intervention stops, buffering also stops. The accumulation of bias in D21 resumes, paths are eliminated, and I_min collapses.

Structure in Physics. The stability of physical systems is maintained by the intrinsic dynamics of the system itself, without external control. Return to attractors, Le Chatelier responses, the damping of fluctuations — these all operate without external intervention, by the intrinsic structure of the system. The self-organization of dissipative structures maintains internal structure autonomously when an external energy flow is provided (Nicolis & Prigogine, 1977, Ch.4–5).

Necessity in Physics. Physical laws do not presuppose an external controller. Dynamics is determined by the intrinsic structure of the system and operates without intervention at every moment. This is a principle of dynamical autonomy in physics.

Verdict. In D-Arch, buffering must operate structurally without external intervention; in physics, dynamics operates intrinsically without an external controller. Both sides are consistent at the factual level of “maintenance by autonomous structure.”

References.

• Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview, 2015, Ch.6–7 — DOI 10.1201/9780429492563
• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.8
• Nicolis & Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, 1977, Ch.4–5

4.24 D23. Termination — Consistent

D-Arch Definition. Termination as completion rather than collapse includes transition to a higher-level system. No claim is made about the properties, structure, or form of the higher-level system.

D-Arch Necessity. If every termination is collapse, I_min cannot function as a general maintenance criterion. Termination as completion (rather than collapse) must be structurally possible.

Structure in Physics. In physics, the termination of a system is not necessarily collapse. A phase transition is the termination of one phase and the beginning of another. Reaching thermodynamic equilibrium is the “termination” of a nonequilibrium structure, but energy and matter are conserved (Callen, 1985, Ch.2).

Necessity in Physics. As long as energy conservation holds, at least in the cases listed above, termination appears as rearrangement or transition into another structure.

Verdict. In D-Arch, if every termination is collapse, the maintenance criterion does not function; in physics, complete annihilation is excluded by energy conservation. Both sides are consistent at the factual level that “termination includes transition.”

D-Arch's termination, however, makes “transition to a higher-level system” explicit, while physics's transitions are not necessarily to a “higher” level. In radioactive decay, the products are not “higher” than the original nucleus. The directionality of “higher transition” is not confirmed in physics.

References.

• Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, Ch.2

The following SC-1~9 are structural consequences derived from the Core (D0–D23), so their item labels differ from those of the D items. Each SC item is organized in a six-part structure: D-Arch Consequence, Derivation Premises, Correspondence in Physics, Consistency, Verdict, and References.

4.25 SC-1. Single-objective fixation is impossible — Consistent

D-Arch Consequence. When fixed to a single objective, I_min is eroded. A system cannot permanently pursue a single criterion only.

Derivation Premises. D7 (evaluation has multiple criteria) + D13 (irreversible elimination by selection) + D16 (restoration requires paths outside the objective) + D20 (overheating → Θ approach) → fixation to a single criterion → irreversible removal of paths outside the objective → restoration is limited → I_min is eroded by D20.

Correspondence in Physics. If D7 (multiple independent pressures or energy sources) + D13 (irreversible order of nuclear-burning stages) + D16 (restoration by degeneracy pressure) + D20 (runaway by single-axis dominance) exist in a physical system, then a violation of D7 (single-degree-of-freedom dominance) should proceed via D13 (irreversible elimination of alternative energy sources), see D16 blocked (the degeneracy-pressure limit exceeded), and culminate in the D20 runaway pattern.

The form most structurally similar to D20 overheating is the gravitational collapse of massive stars. In stars, single-degree-of-freedom dominance is not active fixation but a resulting convergence by the sequential exhaustion of energy sources; this is the structural correspondence to a D7 violation, not a semantic equivalence.

Consistency. D7 violation (multiple evaluation axes → single convergence) + via D13 (irreversible elimination) — In their evolution, massive stars sequentially employ thermal pressure, radiation pressure, and in later stages electron degeneracy pressure as the mechanisms opposing gravity, supported by nuclear-reaction energy sources (H, He, C, Ne, O, Si). Nuclear burning proceeds irreversibly in the order H→He→C→Ne→O→Si→Fe, and once the fuel of each stage is exhausted it cannot be reversed. Iron (Fe) is the maximum of binding energy per nucleon, so further exothermic nuclear fusion is impossible (Burbidge et al. 1957; Woosley, Heger & Weaver 2002).

D16 blocked (restoration failure) — After nuclear fuel is exhausted, electron degeneracy pressure operates as the last restoring mechanism against gravity. However, when the mass of the iron core exceeds the Chandrasekhar limit (~1.4 solar masses), electron degeneracy pressure cannot support gravity (Chandrasekhar 1931).

D20 entry (runaway) — When restoration fails, gravity becomes the single dominating axis and the core collapses in free fall — this is a core-collapse supernova or black hole formation (Bethe 1990; Janka 2012).

Verdict. The physical structures corresponding to each premise of SC-1 (D7, D13, D16, D20) have been independently confirmed, and the sequential process corresponding to the chain SC-1 predicts (multiple evaluation axes → dominance of a single criterion → irreversible elimination of alternative paths → exceeding the restoration limit → entry into runaway) is observed in the evolution and collapse of massive stars. This is, however, the structural interpretation of D-Arch; physics describes this process through stellar evolution and core-collapse dynamics.

References.

• Burbidge EM, Burbidge GR, Fowler WA, Hoyle F., Synthesis of the Elements in Stars, Rev. Mod. Phys. 29(4):547–650, 1957 — DOI 10.1103/RevModPhys.29.547
• Woosley SE, Heger A, Weaver TA., The evolution and explosion of massive stars, Rev. Mod. Phys. 74(4):1015–1071, 2002 — DOI 10.1103/RevModPhys.74.1015
• Chandrasekhar S., The Maximum Mass of Ideal White Dwarfs, Astrophys. J. 74:81, 1931 — DOI 10.1086/143324
• Bethe HA., Supernova mechanisms, Rev. Mod. Phys. 62(4):801–866, 1990 — DOI 10.1103/RevModPhys.62.801
• Janka H-T., Explosion Mechanisms of Core-Collapse Supernovae, Annu. Rev. Nucl. Part. Sci. 62:407–451, 2012 — DOI 10.1146/annurev-nucl-102711-094901

4.26 SC-2. Omniscient optimization is impossible — Consistent

D-Arch Consequence. Optimization based on global information is structurally impossible.

Derivation Premises. D19 (Ω_local ⊂ Ω, no global access) + D5 (O: Ω→Y, lossy observation) → global information not accessible + observation loss → omniscient optimization is impossible.

Correspondence in Physics. If D19 (light-cone locality — information propagation speed ≤ c) + D5 (projection in quantum measurement — lossy observation) exist in a physical system, then it should be structurally impossible to simultaneously obtain complete information about the global state.

Consistency. D19 (no global access) — By the light-cone structure of special relativity, information cannot be exchanged between spacelike-separated events. An observer can access only the interior of their own past light cone, and access to the simultaneous state of the entire universe is excluded in principle.

D5 (lossy observation) — Even within the accessible local region, quantum measurement projects and fixes a state, and in this process the information of the conjugate observable is irreversibly destroyed. The Heisenberg uncertainty principle (Δx·Δp ≥ ℏ/2) shows that this loss is a structural constraint, not a technical limitation (Heisenberg 1927).

Combined — The global region is not accessible (D19), and even within the accessible range, complete information cannot be obtained (D5). Omniscient optimization is doubly blocked. The “completeness” requirement raised by the EPR paper (Einstein, Podolsky & Rosen 1935) was shown to be unsatisfiable if locality is maintained, by Bell's theorem and the Aspect experiment.

Verdict. The physics structures corresponding to each premise of SC-2 — relativistic locality (D19) and the lossiness of quantum measurement (D5) — have been independently confirmed, and phenomena corresponding to the consequence SC-2 predicts (the structural impossibility of optimization based on global information) are observed. This is, however, the structural interpretation of D-Arch; physics describes this constraint through relativistic causality and quantum measurement theory.

References.

• Heisenberg W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43(3–4):172–198, 1927 — DOI 10.1007/BF01397280
• Einstein A, Podolsky B, Rosen N., Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?, Phys. Rev. 47(10):777–780, 1935 — DOI 10.1103/PhysRev.47.777
• Bell JS., On the Einstein Podolsky Rosen Paradox, Physics Physique Fizika 1(3):195–200, 1964 — DOI 10.1103/PhysicsPhysiqueFizika.1.195
• Aspect A, Dalibard J, Roger G., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49(25):1804–1807, 1982 — DOI 10.1103/PhysRevLett.49.1804

4.27 SC-3. Selection speed has a structural upper bound — Consistent

D-Arch Consequence. A structural speed limit exists for selection. If the elimination speed exceeds the restoration efficiency, the system enters a collapse path.

Derivation Premises. D17 (C(a)>0, cost) + D18 (t(effect)>t(cause), delay) + D19 (Ω_local ⊂ Ω, locality) → cost + delay + locality together impose a structural upper bound on selection speed.

Correspondence in Physics. If D17 (energy budget in finite-time driving, dissipation, and control processes involving resources) + D18 (upper bound on causal propagation speed) + D19 (light-cone locality) exist in a physical system, then as process speed increases, irreversible losses increase and efficiency declines, and there should be a structural upper bound on the process speed itself.

Consistency. D17 + D18 (cost + delay → speed-efficiency tradeoff) — Finite-time thermodynamics has established a fundamental tradeoff between process speed and efficiency. Reversible processes (Carnot efficiency) are achieved only in the infinitely slow quasi-static limit, and at finite speed additional entropy is necessarily produced. The efficiency at maximum power is always lower than the Carnot efficiency (Curzon & Ahlborn 1975; Andresen, Salamon & Berry 1984). This tradeoff has been proven to be a universal constraint that does not depend on a specific heat-engine design (Shiraishi, Saito & Tasaki 2016).

D18 (causal speed limit) — By special relativity, no causal influence can propagate faster than the speed of light c (Einstein 1905). This is a fundamental speed limit applying to information, energy, and matter alike.

D17 + D18 (quantum speed limits) — In quantum mechanics as well, a fundamental lower bound exists on the time required for state transitions. The Mandelstam-Tamm bound imposes a minimum time for state transitions via energy uncertainty (Mandelstam & Tamm 1945), and the Margolus-Levitin bound imposes an upper bound on computation speed via the average energy (Margolus & Levitin 1998).

D19 (locality → delayed threshold detection) — As confirmed in SC-2, locality means global information is not accessible, so it cannot be immediately detected that the threshold (D15) is being approached. Detection delay postpones the timing of restoration, and by D18 (delay), late restoration is more costly and less effective.

Verdict. The physical structures corresponding to each premise of SC-3 (D17, D18, D19) have been independently confirmed, and phenomena corresponding to the consequence derived from SC-3 (a structural upper bound on process speed) are confirmed in physics: the thermodynamic speed-efficiency tradeoff (D17+D18), the relativistic speed-of-light limit (D18), quantum speed limits (D17+D18), and delayed threshold detection (D19). This is, however, the structural interpretation of D-Arch; physics describes these constraints respectively through finite-time thermodynamics, special relativity, and quantum evolution limits.

References.

• Curzon FL, Ahlborn B., Efficiency of a Carnot engine at maximum power output, Am. J. Phys. 43(1):22–24, 1975 — DOI 10.1119/1.10023
• Andresen B, Salamon P, Berry RS., Thermodynamics in finite time, Phys. Today 37(9):62–70, 1984 — DOI 10.1063/1.2916405
• Shiraishi N, Saito K, Tasaki H., Universal Trade-Off Relation between Power and Efficiency for Heat Engines, Phys. Rev. Lett. 117:190601, 2016 — DOI 10.1103/PhysRevLett.117.190601
• Einstein A., Zur Elektrodynamik bewegter Körper, Ann. Phys. 17, 891–921, 1905 — DOI 10.1002/andp.19053221004
• Mandelstam L, Tamm I., The Uncertainty Relation Between Energy and Time in Non-relativistic Quantum Mechanics, J. Phys. (USSR) 9:249–254, 1945
• Margolus N, Levitin LB., The maximum speed of dynamical evolution, Physica D 120(1–2):188–195, 1998 — DOI 10.1016/S0167-2789(98)00054-2

4.28 SC-4. Diversity retention is mandatory — Consistent

D-Arch Consequence. A system is driven toward maintaining diversity. Diversity is not a value but a stability condition.

Derivation Premises. D4 (multiple paths) + D13 (irreversible elimination) + D16 (restoration) + D17 (cost) + D18 (delay) + D19 (locality) → vulnerability of a single path + irreversible shrinkage + restricted restoration + locality → the retention of diversity becomes structurally necessary.

Correspondence in Physics. If D4 (quantum indeterminism — multiple paths) + D13 (entropy increase — irreversible elimination) + D16 (return to equilibrium — restoration) + D17 (energy cost) + D18 (speed-of-light limit — delay) + D19 (locality) exist in a physical system, then single-state fixation should be structurally unstable, and diversity (multiple microstates) should be a stability condition.

Consistency. D4 + D13 (multiple paths + irreversible shrinkage → diversity is stable) — By the Boltzmann entropy S = k ln W, thermal equilibrium corresponds to the macrostate in which the number of accessible microstates (W) is maximal (Boltzmann 1877). The second law of thermodynamics states that an isolated system evolves irreversibly toward states with larger W. A single microstate (W = 1) has S = 0 and an extremely low statistical probability of realization. Jaynes (1957) formalized this through the maximum entropy principle: the equilibrium distribution is the one that maximizes microstate diversity under the given constraints.

D16 (existence of restoration mechanisms) — In physical systems, restoration mechanisms operate against deviations from equilibrium. Le Chatelier's principle states that when an external change is applied to a system in equilibrium, it responds in a direction that offsets the change. This restoration is, however, structurally limited by D17 and D18.

D17 + D18 (cost + delay of restoration → restoration of extreme states is impossible) — Returning from a high-diversity state (high entropy) to a highly ordered state (low entropy) requires a structural cost. As one example of an information-erasure cost, Landauer's principle showed that erasing one bit of information dissipates a minimum of kT ln 2 of energy (Landauer 1961; experimentally confirmed by Bérut et al. 2012). By the third law of thermodynamics (the unattainability principle), an extreme low-entropy state cannot be reached in a finite number of operations (Nernst 1906; Masanes & Oppenheim 2017) — meaning that restoration would require an infinite amount of time (D18).

D19 (locality → no fixation of single-path optimality) — As confirmed in SC-2, locality means global information cannot be accessed, so the determination “this single path is best” is structurally impossible.

Verdict. The physical structures corresponding to each premise of SC-4 (D4, D13, D16, D17, D18, D19) have been independently confirmed, and phenomena corresponding to the consequence derived from SC-4 (diversity is a stability condition, and single-state fixation is structurally unstable) are observed. This is, however, the structural interpretation of D-Arch; physics describes these phenomena through statistical mechanics and the laws of thermodynamics.

References.

• Boltzmann L., Ueber die Beziehung eines allgemeinen mechanischen Satzes zum zweiten Hauptsatze der mechanischen Waermetheorie, Sitzungsber. Kais. Akad. Wiss. Wien, Math.-Naturwiss. Cl. 76:373–435, 1877
• Jaynes ET., Information Theory and Statistical Mechanics, Phys. Rev. 106(4):620–630, 1957 — DOI 10.1103/PhysRev.106.620
• Landauer R., Irreversibility and Heat Generation in the Computing Process, IBM J. Res. Dev. 5(3):183–191, 1961 — DOI 10.1147/rd.53.0183
• Bérut A. et al., Experimental verification of Landauer's principle linking information and thermodynamics, Nature 483:187–189, 2012 — DOI 10.1038/nature10872
• Nernst W., Ueber die Berechnung chemischer Gleichgewichte aus thermischen Messungen, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1–40, 1906
• Masanes L, Oppenheim J., A general derivation and quantification of the third law of thermodynamics, Nat. Commun. 8:14538, 2017 — DOI 10.1038/ncomms14538

4.29 SC-5. Failure cannot be eliminated — Consistent

D-Arch Consequence. Failure cannot be structurally eliminated. Failure is a necessary condition rather than a mere allowance.

Derivation Premises. D16 (restoration includes the possibility of search and failure) + D15 (|O(x)| < Θ → collapse) + D19 (omniscient judgment is impossible) → eliminating failure → restoration becomes impossible → collapse cannot be blocked when Θ is approached. Failure is a necessary condition.

Correspondence in Physics. If D16 (return to equilibrium — restoration through fluctuation) + D15 (phase-transition critical points) + D19 (locality) exist in a physical system, then processes should not be sustainable without loss or fluctuation, and removing fluctuation should be equivalent to removing restoration.

Consistency. D16 (restoration includes fluctuation → the possibility of failure is intrinsic) — The fluctuation-dissipation theorem proves that a system's return to equilibrium (dissipation) and spontaneous fluctuation are structurally coupled (Callen & Welton 1951; Kubo 1966). For dissipation (restoration) to operate, fluctuation must exist, and fluctuation is a deviation from the current state — that is, a search with uncertain outcome. Removing fluctuation also removes dissipation, which corresponds to a restriction on the accessible phase-space paths — that is, a contraction of the action space.

D15 (critical points → suppressing fluctuations makes collapse unavoidable) — Near phase-transition critical points, fluctuations diverge at all scales. Renormalization group theory shows that critical phenomena essentially require multi-scale fluctuations (Wilson 1975). When fluctuations are suppressed, the system loses the structural mechanism for crossing the critical point.

Second law of thermodynamics (irreversibility → no loss-free process) — By the second law of thermodynamics, every actual process produces entropy. A reversible process with zero loss is an infinitely slow quasi-static idealization and cannot actually be realized. The Jarzynski equality (1997) and the Crooks fluctuation theorem (1999) quantify the work and entropy-production distributions of nonequilibrium processes and the average irreversibility constraint.

D19 (locality → no optimal-path selection) — As confirmed in SC-2, by locality the omniscient judgment of “selecting only the optimal path without failure” is structurally impossible.

Verdict. The physical structures corresponding to each premise of SC-5 (D16, D15, D19) have been independently confirmed, and phenomena corresponding to the consequence derived from SC-5 (processes cannot be sustained without fluctuation or loss) are observed. This is, however, the structural interpretation of D-Arch; physics describes these phenomena through the fluctuation-dissipation theorem and the second law of thermodynamics. D-Arch's “failure” (an uncertain outcome of selection) and physics's “fluctuation / irreversible loss” are a structural correspondence, not a semantic equivalence.

References.

• Callen HB, Welton TA., Irreversibility and Generalized Noise, Phys. Rev. 83(1):34–40, 1951 — DOI 10.1103/PhysRev.83.34
• Kubo R., The fluctuation-dissipation theorem, Rep. Prog. Phys. 29(1):255–284, 1966 — DOI 10.1088/0034-4885/29/1/306
• Wilson KG., The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47(4):773–840, 1975 — DOI 10.1103/RevModPhys.47.773
• Jarzynski C., Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett. 78(14):2690–2693, 1997 — DOI 10.1103/PhysRevLett.78.2690
• Crooks GE., Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60(3):2721–2726, 1999 — DOI 10.1103/PhysRevE.60.2721

4.30 SC-6. Judgment must be distributed — Consistent

D-Arch Consequence. Judgment cannot be concentrated in a single center. The structure of judgment is necessarily distributed.

Derivation Premises. D19 (Ω_local ⊂ Ω, locality) → Case A: if judgment is centralized while complying with D19, the whole cannot be judged using only local information. Case B: if D19 is bypassed, this conflicts with SC-4 (diversity) and SC-5 (failure cannot be eliminated). Judgment is distributed.

Correspondence in Physics. If D19 (light-cone locality) exists in a physical system, dynamics should be determined locally, and a central controller that simultaneously coordinates the global state cannot exist.

Consistency. D19 (locality → local determination of dynamics) — In quantum field theory, microcausality requires that field operators at spacelike-separated points commute, and the cluster decomposition principle ensures that the physics of sufficiently distant systems is independent (Weinberg 1995, Ch.4; Haag & Kastler 1964). The dynamics of a physical system is determined by local conditions alone and does not require a global coordinate frame or a central controller.

D19 (locality → emergence of macroscopic structure) — Macroscopic properties (temperature, pressure, phase transitions) emerge from local microscopic interactions without central coordination. Anderson (1972) showed that symmetry breaking and emergence produce qualitative differences across scales that are not reducible, and Kadanoff's (1966) block-spin method shows the mechanism by which local interactions generate macroscopic behavior.

Case B (D19 cannot be bypassed) — By the no-signaling theorem, even quantum correlations (entanglement) cannot be used for superluminal information transfer. This result follows generally from the linearity of quantum mechanics, and Ghirardi, Rimini & Weber (1980) gave an early proof of the impossibility of superluminal transmission through the measurement process. Simultaneous access to global information is structurally blocked even within quantum mechanics.

Verdict. The core premise of SC-6 (D19) corresponds to physical structures that have been independently confirmed, and phenomena corresponding to the consequences derived from SC-6 (distributed judgment, absence of a central controller) — local dynamics, emergence, the no-signaling theorem — are observed. This is, however, the structural interpretation of D-Arch; physics describes these phenomena through quantum field-theoretic locality and statistical-mechanical emergence.

References.

• Weinberg S., The Quantum Theory of Fields, Vol. I: Foundations, Cambridge, 1995, Ch.4 — DOI 10.1017/CBO9781139644167
• Haag R, Kastler D., An Algebraic Approach to Quantum Field Theory, J. Math. Phys. 5(7):848–861, 1964 — DOI 10.1063/1.1704187
• Anderson PW., More Is Different, Science 177(4047):393–396, 1972 — DOI 10.1126/science.177.4047.393
• Kadanoff LP., Scaling laws for Ising models near T_c, Physics Physique Fizika 2(6):263–272, 1966 — DOI 10.1103/PhysicsPhysiqueFizika.2.263
• Ghirardi GC, Rimini A, Weber T., A general argument against superluminal transmission through the quantum mechanical measurement process, Lett. Nuovo Cimento 27(10):293–298, 1980 — DOI 10.1007/BF02817189

4.31 SC-7. Identity cannot remain fixed — Consistent

D-Arch Consequence. A system cannot maintain a fixed identity. Identity is not something to be preserved but a variable structure.

Derivation Premises. D3 (transition exists) + D6 (constraint patterns change) + D16 (restoration includes the search for new paths) + D23 (termination as transition) → constraint change → fixed identity blocks necessary actions → A(x) shrinks → O(x) shrinks + restoration paths blocked + completion paths blocked → collapse.

Correspondence in Physics. If D3 (time evolution) + D6 (changing constraint patterns — symmetry breaking) + D16 (search for new paths — phase-transition nucleation) + D23 (transition under energy conservation) exist in a physical system, then a fixed “identity” (a fixed symmetry structure, a fixed particle species) should be unable to adapt to changes of constraint and should be unstable because it lacks structural paths of transition.

Consistency. D3 + D6 (transition + constraint change → change of symmetry structure) — The “identity” of physical systems is determined by symmetry structure — particle species are classified by representations of gauge groups, and the symmetry structure determines which interactions are permitted or forbidden. Spontaneous symmetry breaking fundamentally alters this structure — through a phase transition, a system passes into a qualitatively different symmetry class than before (Nambu 1960; Goldstone 1961). In the electroweak unified theory, symmetry breaking endows particles with mass, and the very “identity” of a particle is a function of the symmetry structure (Weinberg 1967; Higgs 1964).

D6 (constraint change → particle conversion by the weak interaction) — In the weak interaction, particle species are not conserved. In beta decay (neutron → proton + electron + antineutrino), quark-flavor conversion, and the like, particles are converted into other particle species (Weinberg 1967). This shows that the fixation of “identity” is structurally excluded by changes of constraint (interaction).

D16 (restoration requires the search for paths outside the current identity) — The transition from a metastable state to a stable state (phase-transition nucleation) requires the system to explore configurations outside its current symmetry class. Nucleation theory shows that this process occurs through fluctuations that cross an energy barrier (Langer 1967). When fixed to the current “identity,” this transition path is blocked.

D23 (termination = transition → passage to a higher-level structure) — By energy conservation, the termination of a physical system is always accompanied by transition to another structure (confirmed in §4.24). In particle conversion, phase transitions, and stellar evolution, the termination of one “identity” appears as the beginning of another. When identity is fixed, this transition path is closed and termination becomes possible only in the form of collapse.

Verdict. The physical structures corresponding to each premise of SC-7 (D3, D6, D16, D23) have been independently confirmed, and phenomena corresponding to the consequence derived from SC-7 (the variability of identity) — spontaneous symmetry breaking, particle conversion, phase-transition nucleation, and termination as structural transition — are observed. This correspondence is restricted to the structural property of variability of symmetry structure. This is, however, the structural interpretation of D-Arch; physics describes these phenomena through symmetry breaking and particle physics. D-Arch's “identity” (the fixity of permission/prohibition rules) and physics's “symmetry structure” are a structural correspondence, not a semantic equivalence.

References.

• Nambu Y., Quasi-Particles and Gauge Invariance in the Theory of Superconductivity, Phys. Rev. 117(3):648–663, 1960 — DOI 10.1103/PhysRev.117.648
• Goldstone J., Field theories with «Superconductor» solutions, Nuovo Cimento 19(1):154–164, 1961 — DOI 10.1007/BF02812722
• Higgs PW., Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13(16):508–509, 1964 — DOI 10.1103/PhysRevLett.13.508
• Weinberg S., A Model of Leptons, Phys. Rev. Lett. 19(21):1264–1266, 1967 — DOI 10.1103/PhysRevLett.19.1264
• Langer JS., Theory of the condensation point, Ann. Phys. 41(1):108–157, 1967 — DOI 10.1016/0003-4916(67)90200-X

4.32 SC-8. Coupled failure theorem — Unconfirmed

D-Arch Consequence. D10 (Attribution), D21 (Buffering), D22 (Non-intervention), and D23 (Termination) are each independently necessary, but if they are not coupled, the system converges to one of the collapse types.

Derivation Premises. D10 (Attribution) + D21 (Buffering) + D22 (Non-intervention) + D23 (Termination) → coupling failure → four-type collapse.

Correspondence in Physics. D21 (thermal buffering and damping) + D22 (intrinsic dynamics) + D23 (transition under energy conservation) have been confirmed consistent in §4.22, §4.23, and §4.24. However, D10 (Attribution) is unconfirmed in §4.11.

Consistency. Among the four premises of SC-8, D21, D22, and D23 have each been confirmed consistent with physics, but D10 has no corresponding structure confirmed in the formal system of physics. The consequence of SC-8, derived from the combination of the four premises, cannot be judged independently as long as D10 is unconfirmed. Partial consistency among the components cannot substitute for a verdict on the combined consequence as a whole.

Verdict. Of the four premises, three (D21, D22, D23) are confirmed consistent and one (D10) is unconfirmed. Since the chain is not complete, the overall verdict is suspended; this is an inherited form of unconfirmed status from D10. It belongs to the same inherited unconfirmed category as D11. Physics does not exclude SC-8; one of its premises is simply not described.

4.33 SC-9. Complete description is impossible — Consistent

D-Arch Consequence. No descriptive system can fully encompass the space Ω of possible states.

Derivation Premises. D0 (Ω ≠ ∅) + D1 (Distinction) + D9 (selection → partial fixation) + D13 (selection → irreversible shrinkage of options) + D19 (Ω_local ⊂ Ω, no global access) → ℒ operates by distinction and selection and at each moment can access only the contracted Ω_local. It cannot encompass the entirety of Ω.

Correspondence in Physics. If D0 (existence of state space) + D1 (distinction of eigenstates) + D9 (state fixation by measurement) + D13 (irreversible shrinkage by entropy increase) + D19 (light-cone locality) exist in a physical system, then no descriptive system should be able to completely describe the physical state space.

Consistency. D9 + D1 (selection + distinction → description is always partial) — The Heisenberg uncertainty principle (Δx·Δp ≥ ℏ/2) structurally excludes the simultaneous exact description of conjugate observables (Heisenberg 1927). Fixing one observable (D9) destroys the information of the conjugate observable, and this is a constraint derived from the mathematical structure of quantum mechanics, not a technical limitation.

D9 + D19 (selection + locality → no hidden variables) — The Kochen-Specker theorem proves that assigning context-independent definite values to all observables simultaneously is incompatible with quantum mechanics (Kochen & Specker 1967). That is, “complete prior description” (the simultaneous fixation of all observables) is structurally excluded. Combined with Bell's theorem (1964), it is also impossible to fully reproduce quantum mechanics with local hidden variables.

D13 (selection → irreversible shrinkage of the option space) — Whenever measurement or observation is performed, the option space is irreversibly contracted (confirmed in §4.14). As long as a descriptive system depends on measurement, every description at any given moment is about an already contracted subspace, and it cannot return to the full option space prior to contraction. This corresponds to the projection postulate of quantum measurement (von Neumann) and to thermodynamic irreversibility.

D0 + D1 + D19 (incompleteness of formal systems) — Gödel's incompleteness theorem proves that in any consistent formal system that includes arithmetic, there exist true propositions that cannot be proved within the system (Gödel 1931). As long as a physical theory is described by a formal system that encodes arithmetic, it inherits this constraint. Chaitin (1982) reaffirmed this from the perspective of algorithmic information theory: the truth or falsity of sufficiently complex propositions cannot be decided by any axiom system.

Self-reference — SC-9 also applies to D-Arch itself. This has the same self-referential structure as Gödel's theorem when applied to mathematics itself. D-Arch cannot fully describe itself, and is not a closed meta-theory.

Verdict. The physical structures corresponding to each premise of SC-9 (D0, D1, D9, D13, D19) have been independently confirmed, and phenomena corresponding to the consequence derived from SC-9 (the impossibility of complete description) — the uncertainty principle, Kochen-Specker contextuality, the irreversibility of measurement, and the incompleteness of formal systems — are observed. This is, however, the structural interpretation of D-Arch; physics describes these constraints through the mathematical structure of quantum mechanics and through formal logic.

References.

• Heisenberg W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43(3–4):172–198, 1927 — DOI 10.1007/BF01397280
• Kochen S, Specker EP., The Problem of Hidden Variables in Quantum Mechanics, J. Math. Mech. 17(1):59–87, 1967 — DOI 10.1512/iumj.1968.17.17004
• Bell JS., On the Einstein Podolsky Rosen Paradox, Physics Physique Fizika 1(3):195–200, 1964 — DOI 10.1103/PhysicsPhysiqueFizika.1.195
• Gödel K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38(1):173–198, 1931 — DOI 10.1007/BF01700692
• Chaitin GJ., Gödel's theorem and information, Int. J. Theor. Phys. 21(12):941–954, 1982 — DOI 10.1007/BF02084159

5. Discussion

5.1 Convergence on Self-Referential Structure

The four unconfirmed items — D10 (Attribution), D11 (Integrated selection), D14 (Meta-evaluation), and SC-8 (Coupled failure theorem) — converge on a single theme: self-referential structure. The reasons for their unconfirmed status are not identical, however. D11 and SC-8 inherit the unconfirmed status of D10, while D14 is unconfirmed through a separate structure: the absence of self-evaluation.

5.2 Meta-Disclaimer

No inconsistency was observed in this comparison, but it is difficult to read this result as strong agreement between D-Arch and physics. The derivation of the D-Arch framework, this mapping work, and the writing of this document all depend on AI, so the possibility of omission or bias remains; this comparison is also an observation within the scope of the condensed definitions of the D-Arch source. SC-9 (Incompleteness of description) applies to this document itself and makes explicit that this comparison cannot be a complete verification of D-Arch.

6. Conclusion

Comparing the 33 items in total — D-Architecture's 24 Core items and 9 structural consequences (SC) — with physics, 29 cases of consistency and 4 cases of unconfirmed status were observed, while no inconsistency was observed. The 4 unconfirmed cases converge on a single theme: self-referential structure. This comparison is not a work in which D-Arch proposes new physics or verifies or corrects existing physics. Two facts together mark what this observation can report: that a structural framework derived purely by logic could be compared with the formal system of physics without contradiction, and that the gaps remaining in this comparison concentrate on a single theme — self-reference.