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Dr. Elliot McGucken · Light, Time, Dimension Theory · elliotmcguckenphysics.com · drelliot@gmail.com

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“More intellectual curiosity, versatility and yen for physics than Elliot McGucken’s I have never seen in any senior or graduate student. Originality, powerful motivation, and a can-do spirit make me think that McGucken is a top bet.” — John Archibald Wheeler, Joseph Henry Professor of Physics, Princeton University

Abstract

The companion paper [G] (McGucken Geometry: A Novel Mathematical Category Exalted by the Physical Principle dx₄/dt = ic) formalized McGucken Geometry underlying the McGucken Principle that the fourth dimension is expanding at the velocity of light dx₄/dt = ic as a mathematical category with three equivalent formulations — the moving-dimension manifold (M, g, F, V), the second-order jet-bundle formalization, and the Cartan-geometry formulation of Klein type (ISO(1,3), SO⁺(1,3)) with distinguished active translation generator P₄ — and articulated four privileged-element conditions (P1)–(P4) of Definition 5.4 specifying the structural commitments of the framework. The novelty claim of [G] was established by exhibition: a comprehensive prior-art survey across §§9–14 of [G], covering eleven frameworks with privileged direction or flow content (Einstein-aether theory, the Standard-Model Extension, Hořava-Lifshitz gravity, Causal Dynamical Triangulations, Shape Dynamics, the Connes-Rovelli Thermal Time Hypothesis, Connes’ noncommutative geometry, Penrose Conformal Cyclic Cosmology, Lorentz-Finsler with timelike Killing field, tetrad and vierbein formulations, and the cosmological-time-function literature beyond Hawking and Andersson-Galloway-Howard) plus quantum-gravity programs and philosophical traditions, established that no surveyed framework contains the conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4). The closest cousin in the surveyed literature is the Connes-Rovelli Thermal Time Hypothesis, with three of the four conditions partially satisfied (privileged content structural-plus-state, modular flow at thermodynamically determined rate, FRW thermal time recovering CMB cosmological time) and (P3)’s spherical-wavefront content absent.

The present paper proves the formal categorical no-embedding theorem that strengthens [G]’s survey claim. We construct the categorical apparatus and prove three rigidity theorems establishing the universality of McGucken Geometry within a precisely-specified category of axis-dynamics frameworks satisfying the formal versions of conditions (P1)–(P3).

The mathematical content is the formalization of conditions (P1)–(P3) of Definition 5.4 of [G] as categorical predicates and the proof that the moving-dimension manifold category 𝓜 is universal among predicate-strict axis-dynamics frameworks. Condition (P4) — the empirical identification of V with the cosmic microwave background rest frame — is empirical content addressed in the cosmology paper [79], not a mathematical predicate; the empirical identification selects the physically realized object within the category 𝓜 but is outside the categorical apparatus.

The paper is organized in four parts.

Part I (§§2–4) constructs the category 𝓜 of moving-dimension manifolds: objects are quadruples (M, g, F, V) where M is a smooth, connected, oriented, time-oriented, globally hyperbolic Lorentzian 4-manifold; g is a Lorentzian metric of signature (−, +, +, +); F is a smooth codimension-1 foliation by spacelike Cauchy surfaces; V is a smooth future-directed unit timelike vector field with g(V, V) = −c² and V everywhere orthogonal to the leaves of F. Morphisms are smooth diffeomorphisms preserving g, F, and V (Definition 2.1). Conditions (P1), (P2), (P3) of [G] are formalized as Predicates 𝒫₁, 𝒫₂, 𝒫₃ (Definitions 2.4, 2.5, 2.6) and shown to be automatic on objects of 𝓜 (Proposition 2.7).

Part II (§§5–7) establishes three rigidity theorems on objects of 𝓜.

• Theorem A (Minkowski Rigidity, §5). Every moving-dimension structure on flat Minkowski space ℝ⁴ is isomorphic to the standard structure (ℝ⁴, η, F_std, V_std = ∂/∂t) by a Poincaré transformation preserving time-orientation, modulo a one-parameter family of foliation-origin choices.
• Theorem B (Local Rigidity in Adapted Charts, §6). In any McGucken-adapted chart on a moving-dimension manifold (M, g, F, V), the metric takes the form ds² = −N²c²dt² + h_{ij}dx^i dx^j with V = (1/N)∂/∂t and the moving-dimension data (F, V) determined globally by N and the spatial metrics on the leaves. Two moving-dimension structures on the same M with identical lapse function N and identical spatial metrics {h_{ij}(t)} are equal as objects of 𝓜.
• Theorem C (Categorical Universality, §7). Define the larger category 𝓐 of axis-dynamics frameworks (Definition 7.1). Then the embedding ι: 𝓜 → 𝓐 factors through the full subcategory 𝓐₀ ⊂ 𝓐 of trivially-decorated objects as an isomorphism of categories ι: 𝓜 ⥲ 𝓐₀ (Theorem 7.5.2), with strict inverse R|_{𝓐₀}: 𝓐₀ → 𝓜. The forgetful functor R: 𝓐 → 𝓜 satisfies R ∘ ι = 1_𝓜 globally on 𝓜 (Remark 7.5.4). For any A ∈ 𝓐, the canonical morphism A → ι(R(A)) with underlying identity diffeomorphism id_M exists in 𝓐 if and only if A is predicate-strict (i.e., A ∈ 𝓐₀); when it exists, A = ι(R(A)) on the nose and the canonical morphism is id_A (Proposition 7.6.3, Theorem 7.7.3). The universal-property characterization (Corollary 7.7.4) establishes that 𝓐₀ is the unique full subcategory of 𝓐 satisfying these properties. Equivalently: McGucken Geometry 𝓜 is canonically isomorphic to the predicate-strict subcategory 𝓐₀ of 𝓐, and every framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ with no auxiliary structural decoration is equal on the nose to a moving-dimension manifold of 𝓜.

Part III (§§8–9) develops the consequences. §8 establishes that the no-embedding content of Theorem C is the categorical universality of 𝓜: any framework satisfying the formal predicates (P1)–(P3) — under categorical formalization — factors uniquely through 𝓜, and 𝓜 is the universal recipient of all such frameworks. §9 articulates what the theorem does and does not establish, with explicit attention to the role of (P4) as empirical content outside the categorical apparatus.

Part IV (§§10–11) is synthesis. §10 places the result in the corpus context of [G] and the broader McGucken framework. §11 concludes.

The paper observes the following methodological commitments. (i) Each numbered Theorem, Lemma, Proposition has a formal statement and a proof at standard category-theoretic rigor. (ii) The categorical apparatus is constructed explicitly: objects, morphisms, functors, adjunctions, natural transformations are all specified concretely. (iii) The role of condition (P4) as empirical content outside the categorical apparatus is acknowledged at the outset and at the conclusion. (iv) The mathematical apparatus is taken from standard category theory [MacLane] and standard differential geometry [Wald, Hawking-Ellis]; the structural results are McGucken-framework contributions.

The structural payoff is fivefold.

First, the category 𝓜 of moving-dimension manifolds exists as a precise mathematical object. Objects are quadruples (M, g, F, V) satisfying the data-and-compatibility conditions of Definition 2.1; morphisms are foliation-preserving isometries intertwining V; the category-axiom verifications of §3.2 establish that 𝓜 is a locally small category satisfying the standard axioms. Predicates 𝒫₁, 𝒫₂, 𝒫₃ formalizing the mathematical content of conditions (P1)–(P3) of [G, Definition 5.4] are shown to be automatic on objects of 𝓜 (Proposition 2.7).

Second, Theorem A establishes Minkowski rigidity: the moving-dimension structure on flat Minkowski space ℝ⁴ is essentially unique. There is no inequivalent alternative; every candidate moving-dimension structure on ℝ⁴ reduces to the standard one (ℝ⁴, η, F_std, V_std = ∂/∂t) by Poincaré transformation modulo time translation.

Third, Theorem B establishes local rigidity in adapted charts: on any moving-dimension manifold, the moving-dimension data (F, V) in any McGucken-adapted chart is determined by the lapse function N and the spatial metrics on the leaves. The chart-level data fixes the moving-dimension content uniquely.

Fourth, Theorem C is the formal categorical no-embedding theorem. Within the categorical setup of 𝓐 (Definition 7.1), the moving-dimension manifold category 𝓜 is the terminal subcategory corresponding to predicate-strict frameworks. Every framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ with no auxiliary structural decoration is canonically equivalent to a moving-dimension manifold of 𝓜. The categorical theorem strengthens [G]’s survey claim: where the survey covers concrete frameworks examined, the categorical theorem quantifies over all frameworks satisfying the formal predicates within the categorical setup.

Fifth, the eleven closest-neighbor frameworks of [G, §13] are all categorically distinguishable from 𝓜 within 𝓐. Each carries non-trivial decoration ε ≠ 0 — matter Lagrangians (Einstein-aether), VEV coefficients (SME), anisotropic-scaling action (Hořava-Lifshitz), simplicial-discretization data (CDT), conformal-three-geometry plus CMC gauge (Shape Dynamics), algebraic-state pair (TTH), spectral-triple data (Connes NCG), conformal-cyclic identification (Penrose CCC), Finsler-metric-plus-Killing-condition (Lorentz-Finsler), tetrad-gauge-equivalence-class (Vierbein), or absence-of-privilege (cosmological-time-function literature) — placing them outside the predicate-strict subcategory of 𝓐. The forgetful functor R: 𝓐 → 𝓜 strips each framework’s decoration to produce a moving-dimension manifold of 𝓜; the discarded decoration is precisely the structural feature that distinguishes the framework from McGucken Geometry. The Connes-Rovelli Thermal Time Hypothesis (the closest neighbor in the entire surveyed literature, treated in detail in §8.3 N.13.6) carries the most structurally important decoration: the algebraic-state pair (𝒜, ω) whose state-dependence makes TTH’s flow thermodynamic rather than geometric.

The strongest claim of the paper is Theorem C: within the categorical setup of 𝓐, every predicate-strict axis-dynamics framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ is canonically equivalent to a moving-dimension manifold of 𝓜. This is the formal categorical no-embedding theorem strengthening the survey-based novelty claim of [G]. Together, the comprehensive survey of [G] covering eleven concrete frameworks plus quantum-gravity programs and philosophical traditions, the categorical universality theorem of the present paper within the precisely-specified categorical setup of 𝓐, and the empirical CMB-frame identification of [79] articulate the strongest novelty claim that the McGucken corpus apparatus supports: McGucken Geometry is the unique mathematical category, in the surveyed literature and within the categorical setup, satisfying the conjunction of (P1)–(P4) of Definition 5.4 of [G].

Keywords: McGucken Geometry; moving-dimension manifold; categorical no-embedding theorem; rigidity theorem; Minkowski rigidity; foliation-preserving diffeomorphism; axis-dynamics framework; terminal object; adjunction; right adjoint; forgetful functor; natural isomorphism; universal property; structure-preserving functor; moving-dimension category 𝓜; axis-dynamics category 𝓐; predicate-strict axis-dynamics; McGucken Principle dx₄/dt = ic; privileged-element conditions (P1)–(P3); empirical condition (P4); cosmic microwave background rest frame.

1. Introduction

1.1 The Companion Paper [G] Established Novelty by Survey; This Paper Strengthens That Result by Proving the Categorical Universality of McGucken Geometry within a Precisely-Specified Category of Axis-Dynamics Frameworks

The companion paper [G] formalized McGucken Geometry underlying the McGucken Principle that the fourth dimension is expanding at the velocity of light dx₄/dt = ic as a mathematical category with three equivalent formulations and articulated the structural commitments of the framework as four privileged-element conditions (P1)–(P4) of Definition 5.4 of [G]. The novelty claim of [G] was established by comprehensive prior-art survey: across §§9–14 of [G], covering eleven frameworks with privileged direction or flow content (Einstein-aether theory, the Standard-Model Extension, Hořava-Lifshitz gravity, Causal Dynamical Triangulations, Shape Dynamics, the Connes-Rovelli Thermal Time Hypothesis, Connes’ noncommutative geometry, Penrose Conformal Cyclic Cosmology, Lorentz-Finsler with timelike Killing field, tetrad and vierbein formulations, and the cosmological-time-function literature) plus quantum-gravity programs and philosophical traditions, no surveyed framework was shown to contain the conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4). The closest cousin in the entire surveyed literature is the Connes-Rovelli Thermal Time Hypothesis, with three of the four conditions partially satisfied and (P3) absent.

[G]’s survey claim is bounded by what the survey examines: across the eleven frameworks of §13 plus the quantum-gravity programs and philosophical traditions of §14, none contains the conjunction. The present paper proves a complementary categorical-universality result: within the precisely-specified categorical setup of 𝓐 (Definition 7.1), the moving-dimension manifold category 𝓜 is the terminal subcategory corresponding to predicate-strict frameworks, and every framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ with no auxiliary structural decoration is canonically equivalent to McGucken Geometry. The categorical universality strengthens the survey claim: where the survey covers concrete frameworks examined, the categorical theorem quantifies over all frameworks satisfying the formal predicates within the categorical setup. The two papers together — [G]’s comprehensive survey plus this paper’s categorical universality — articulate the strongest novelty claim that the apparatus supports.

1.2 The Categorical Theorem Proves Universality on Mathematical Conditions (P1)–(P3); the Empirical Condition (P4) Is Empirical Content Outside Any Categorical Apparatus and Is Addressed in the Cosmology Paper [79]

A categorical no-embedding theorem can speak only to mathematical predicates on objects of a specified category. Condition (P4) of Definition 5.4 of [G] is empirical, not mathematical:

(P4) V is empirically identified with the cosmic microwave background rest frame.

This identification is empirical content that selects, among the mathematical objects in the category, the one corresponding to the physically realized cosmological structure; the selection is not itself a categorical predicate on objects of the category. The categorical theorem of the present paper therefore concerns the mathematical predicates 𝒫₁, 𝒫₂, 𝒫₃ formalizing conditions (P1)–(P3), and the empirical condition (P4) is addressed in the cosmology paper [79] where the identification of V with the CMB rest frame is established empirically.

The categorical theorem proved here is universal:

Under formal predicates 𝒫₁, 𝒫₂, 𝒫₃ formalizing conditions (P1)–(P3) on objects of the category 𝓐 of axis-dynamics frameworks (Definition 7.1), every predicate-strict framework is canonically equivalent to a moving-dimension manifold of the category 𝓜.

The combined mathematical-empirical content of the McGucken framework is therefore: the categorical universality of 𝓜 within 𝓐 (this paper, Theorem C), together with the survey-based novelty of (P1) ∧ (P2) ∧ (P3) ∧ (P4) on the eleven surveyed frameworks of [G, §13] plus quantum-gravity programs and philosophy-of-time traditions ([G, §§9–14]), together with the empirical CMB-frame identification (P4) ([79]). Each component is established by its own apparatus; together they articulate the mathematical and empirical standing of McGucken Geometry as a coherent whole.

1.3 Three Theorems Established in This Paper: Theorem A (Minkowski Rigidity), Theorem B (Local Rigidity in Adapted Charts), Theorem C (Categorical Universality / The No-Embedding Theorem)

The paper proves three theorems of increasing strength.

Theorem A (Minkowski Rigidity). On flat Minkowski space ℝ⁴ with the standard Lorentzian metric η, the moving-dimension structure (ℝ⁴, η, F_std, V_std) of [G, Proposition 5.7] is unique up to a one-parameter family of foliation-origin choices, in the following precise sense: every moving-dimension structure (ℝ⁴, η, F, V) on flat Minkowski space is related to the standard structure by a Poincaré transformation preserving time-orientation, modulo time translation.

This is the simplest case of the rigidity content. It establishes that on the model space ℝ⁴ with its standard Lorentzian metric, the moving-dimension data (F, V) is essentially unique. The proof is a Lie-group calculation using the Poincaré-group structure on Minkowski space.

Theorem B (Local Rigidity in Adapted Charts). On any moving-dimension manifold (M, g, F, V) and any McGucken-adapted chart on M, the local moving-dimension data (F, V) is determined by the lapse function N and the spatial metric h_{ij} on each leaf of F. Two moving-dimension structures on the same M with identical lapse N and identical spatial metrics are equal as objects of 𝓜.

This is the local rigidity content. It establishes that the moving-dimension data is locally determined by standard ADM-decomposition data — the lapse and the spatial metrics on the leaves. The proof uses the standard 3+1 decomposition of Lorentzian geometry [12, 15].

Theorem C (Categorical Universality / The No-Embedding Theorem). Define the larger category 𝓐 of axis-dynamics frameworks (Definition 7.1). Define the embedding ι: 𝓜 → 𝓐 (Definition 7.3) and the forgetful functor R: 𝓐 → 𝓜 (Definition 7.4). Identify the full subcategory 𝓐₀ ⊂ 𝓐 of trivially-decorated objects (Definition 7.5.1). Then ι factors through 𝓐₀ as an isomorphism of categories ι: 𝓜 ⥲ 𝓐₀ with strict inverse R|_{𝓐₀} (Theorem 7.5.2); R ∘ ι = 1_𝓜 globally (Remark 7.5.4); and for any A ∈ 𝓐, the canonical morphism A → ι(R(A)) exists in 𝓐 if and only if A is predicate-strict (Proposition 7.6.3), with predicate-strict membership in 𝓐₀ characterized by four equivalent properties (Theorem 7.7.1). The full subcategory 𝓐₀ is the unique full subcategory of 𝓐 satisfying the universal-property characterization of Corollary 7.7.4: McGucken Geometry 𝓜 is canonically isomorphic to 𝓐₀, the predicate-strict subcategory of 𝓐.

This is the no-embedding theorem. It establishes the categorical universality of 𝓜 as the terminal object in a suitable category of axis-dynamics frameworks. Equivalently: any framework satisfying the formal predicates (P1)–(P3) — under categorical formalization — factors uniquely through 𝓜.

1.4 Methodology: Each Categorical Construction Is Given Explicitly with Objects, Morphisms, Functors, Adjunctions, and Universal Properties Specified Concretely; Standard Apparatus from Mac Lane, Wald, and Hawking-Ellis Is Used Without Modification; Mathematical and Empirical Content Are Strictly Separated

The paper observes the following methodology.

(M1) Formal-mathematical rigor. Every categorical construction is given explicitly. Objects are specified by their data; morphisms are specified by their compatibility conditions; functors are specified by their action on objects and morphisms; adjunctions are specified by their unit and counit natural transformations; universality is established by verifying the universal property.

(M2) Set-theoretic foundations. We work in standard ZFC set theory with the convention that “category” means “locally small category” — every Hom-set is a set in the universe of sets, although the collection of objects may be a proper class. The categories 𝓜 and 𝓐 of the present paper are locally small.

(M3) Standard machinery from category theory. We use standard apparatus from MacLane [MacLane]: functors, natural transformations, adjunctions, unit and counit, terminal objects, universal properties, Yoneda lemma. We assume the reader is familiar with this content; references are to MacLane’s text.

(M4) Standard machinery from differential geometry. We use standard apparatus from Wald [W] and Hawking-Ellis [HE]: smooth manifolds, Lorentzian metrics, foliations, Cauchy surfaces, time-orientation, the Lie derivative, the Levi-Civita connection. We assume the reader is familiar with this content; references are to those texts.

(M5) Strict separation of mathematical content from empirical content. The categorical theorem is strictly mathematical and concerns conditions (P1)–(P3). Condition (P4) is empirical and is acknowledged but not incorporated into the categorical apparatus. The paper does not claim categorical content for (P4).

1.4a Falsifiability Criteria for the Three Theorems

The three theorems carry concrete mathematical risk in the precise sense developed by Popper [Popper1959]: each is an explicit mathematical proposition with a specific proof, and each can be falsified by exhibiting a counterexample. A theorem with no possible counterexample is not a theorem in any meaningful mathematical sense; the falsifiability criteria below specify what counterexamples would falsify each theorem.

Criterion N1 (Theorem A falsification). Theorem A states that every moving-dimension structure on flat Minkowski space ℝ⁴ is isomorphic to the standard structure (ℝ⁴, η, F_std, V_std = ∂/∂t) by a Poincaré transformation preserving time-orientation, modulo a one-parameter family of foliation-origin choices. If a future analysis exhibited a moving-dimension structure on ℝ⁴ — i.e., a quadruple (ℝ⁴, η, F, V) satisfying Definition 2.1 — that was not isomorphic to the standard structure modulo Poincaré transformations and foliation-origin choices, Theorem A would be falsified at the Minkowski-rigidity level. The proof of Theorem A in §5 reduces the rigidity claim to (a) the uniqueness of unit timelike vector fields with vanishing Lie derivative on flat spacetime, and (b) the universality of Poincaré transformations in mapping such fields to the standard one; a counterexample would invalidate one of these two reductions.

Criterion N2 (Theorem B falsification). Theorem B states that on any moving-dimension manifold, the moving-dimension data (F, V) in any McGucken-adapted chart is determined by the lapse function N and the spatial metrics on the leaves; two structures sharing a common adapted chart with identical lapse and spatial metrics are equal as objects of 𝓜. If a future analysis exhibited two moving-dimension structures on a single manifold M with identical lapse function N and identical spatial metrics {h_{ij}(t)} that were not equal as objects of 𝓜 — i.e., that differed in the foliation F or vector field V despite identical chart data — Theorem B would be falsified at the local-rigidity level.

Criterion N3 (Theorem C falsification). Theorem C states that within the categorical setup of 𝓐 (Definition 7.1), the full subcategory 𝓐₀ ⊂ 𝓐 of trivially-decorated objects is the unique full subcategory satisfying the universal-property characterization of Corollary 7.7.4, with ι: 𝓜 ⥲ 𝓐₀ an isomorphism of categories. If a future analysis exhibited a predicate-strict object A ∈ 𝓐 — i.e., one with trivial decoration ε ≡ 0 satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ — that did not equal ι(R(A)) on the nose (or admitted no canonical morphism A → ι(R(A)) with underlying identity diffeomorphism id_M), Theorem C would be falsified at the categorical-universality level.

Criterion N4 (Equivalence-existence falsification). Theorem 7.5.2 establishes that ι: 𝓜 → 𝓐 factors through 𝓐₀ as an isomorphism of categories with strict inverse R|{𝓐₀}. The proof consists of three explicit verifications: (a) essential surjectivity of ι onto 𝓐₀, (b) full faithfulness of ι (injectivity and surjectivity on Hom-sets), and (c) the equality (R|{𝓐₀}) ∘ ι = 1_𝓜 and ι ∘ (R|{𝓐₀}) = 1{𝓐₀} on the nose. If a future analysis identified an object of 𝓜 that ι sent outside 𝓐₀, or two distinct morphisms in 𝓜 that ι identified, or a morphism in 𝓐₀ not in the image of ι on Hom-sets, or a failure of the inverse identity on either composition, Theorem 7.5.2 would be falsified at the equivalence-existence level, and the universal-property characterization of Theorem C would lose its principal apparatus.

Criterion N5 (Categorical-setup falsification). Theorem C is established within the specific categorical setup of 𝓐 (Definition 7.1). If a future analysis showed that the categorical setup of Definition 7.1 was inconsistent — for instance, by exhibiting two definitions of a morphism A → A′ in 𝓐 that gave different results for the same pair of objects — the entire apparatus would be falsified at the categorical-setup level. The setup’s consistency is verified explicitly in §3.2 (verification of category axioms for 𝓜) and §7.2 (verification for 𝓐).

The five criteria together constitute the mathematical-structural risk of the present paper. Each theorem and each foundational construction is checkable by exhibiting an explicit counterexample in the appropriate category. The framework is corroborated, not falsified, at every level: N1 by the explicit Minkowski-rigidity proof of §5; N2 by the explicit local-rigidity proof of §6; N3 by the explicit categorical-universality proof of §7; N4 by the explicit construction of the equivalence ι: 𝓜 ⥲ 𝓐₀ in §7.5 with full essential-surjectivity, full-faithfulness, and section-identity verifications; N5 by the explicit category-axiom verifications of §3.2 and §7.2.

1.5 Notation, Conventions

We retain the conventions of the companion paper [G, §1.4], summarized briefly here.

Convention 1.5.1. M is a smooth, connected, oriented, time-oriented, globally hyperbolic Lorentzian 4-manifold. Smooth structure is the standard one. Time-orientation is fixed once and for all.

Convention 1.5.2. The Lorentzian metric g on M has signature (−, +, +, +). The line element in standard numbering is ds² = −c²dt² + dx² where dx² = dx₁² + dx₂² + dx₃². The McGucken coordinate identification x₄ = ict is recorded as in [G, Convention 1.4.2] and underlies the algebraic generation of the Lorentzian signature [G, Lemma 2.1]; in the present paper, all formal work is done in the standard numbering with signature (−, +, +, +).

Convention 1.5.3. Greek indices μ, ν ∈ {0, 1, 2, 3}; Latin indices i, j, k ∈ {1, 2, 3}.

Convention 1.5.4. F is a smooth codimension-1 foliation of M whose leaves are spacelike Cauchy surfaces. Each leaf Σ ∈ F carries the induced Riemannian metric h_{ij} of signature (+, +, +). The leaves are level sets of a Hawking cosmic time function τ: M → ℝ in the sense of [HE].

Convention 1.5.5. V is a smooth future-directed unit timelike vector field with g(V, V) = −c². V is orthogonal to the leaves of F at every event of M.

Convention 1.5.6. A McGucken-adapted chart is a coordinate chart (t, x¹, x², x³) on M in which (i) the time coordinate t coincides up to a global affine transformation with the cosmic-time-function parameter labeling the leaves of F, and (ii) the chart is irrotational with respect to F (the shift vector vanishes). In such a chart, the metric takes the form ds² = −N²(t, x)c²dt² + h_{ij}(t, x)dx^i dx^j with N the lapse function.

Convention 1.5.7. Throughout the paper, “moving-dimension manifold” means a quadruple (M, g, F, V) satisfying Conventions 1.5.1–1.5.5. “McGucken Geometry” refers to the category 𝓜 of moving-dimension manifolds and their morphisms, defined in §2.

1.6 Structure of the Paper

Part I (§§2–4) constructs the categorical apparatus: the category 𝓜 of moving-dimension manifolds (§2), the formalization of predicates (P1)–(P3) (§3), and basic categorical properties of 𝓜 (§4).

Part II (§§5–7) proves the three rigidity theorems: Minkowski rigidity (§5, Theorem A), local rigidity in adapted charts (§6, Theorem B), and categorical universality (§7, Theorem C).

Part III (§§8–9) develops the consequences: the no-embedding content of Theorem C (§8) and the explicit standing of what the theorems do and do not establish, including the role of (P4) (§9).

Part IV (§§10–11) is synthesis: corpus context (§10) and conclusion (§11).

References follow.

PART I — THE CATEGORICAL APPARATUS

Part I constructs the category 𝓜 of moving-dimension manifolds. §2 defines the category, its objects and morphisms, and verifies the category axioms. §3 formalizes predicates (P1)–(P3) of [G, Definition 5.4] as Predicates 𝒫₁, 𝒫₂, 𝒫₃ on objects of 𝓜 and shows they are automatic. §4 establishes basic categorical properties of 𝓜: composition, identities, isomorphisms.

2. The Category 𝓜 of Moving-Dimension Manifolds

2.1 Definition of the Category 𝓜

Definition 2.1 (The category 𝓜). The category 𝓜 of moving-dimension manifolds has:

Objects: quadruples (M, g, F, V) satisfying Conventions 1.5.1–1.5.5. Explicitly, the data are:

(a) M is a smooth, connected, oriented, time-oriented, globally hyperbolic Lorentzian 4-manifold;

(b) g is a Lorentzian metric on M of signature (−, +, +, +);

(c) F is a smooth codimension-1 foliation of M whose leaves are spacelike Cauchy surfaces;

(d) V is a smooth future-directed unit timelike vector field on M with g(V, V) = −c² and V everywhere orthogonal to the leaves of F.

Morphisms: a morphism φ: (M, g, F, V) → (M’, g’, F’, V’) in 𝓜 is a smooth diffeomorphism φ: M → M’ such that:

(M1) φg’ = g (φ is an isometry);*

(M2) φ maps leaves of F to leaves of F’ (φ is foliation-preserving);

(M3) φ_ V = V’ (φ intertwines V with V’).*

Composition is composition of diffeomorphisms; identity is the identity diffeomorphism.

2.2 Verification of Category Axioms

Proposition 2.2 (𝓜 is a category). The data of Definition 2.1 satisfy the category axioms: composition is well-defined and associative; identity morphisms exist and satisfy the identity laws.

Proof. We verify that composition of morphisms in 𝓜 yields a morphism in 𝓜, and that identity diffeomorphisms are morphisms.

Composition. Let φ: (M, g, F, V) → (M’, g’, F’, V’) and ψ: (M’, g’, F’, V’) → (M”, g”, F”, V”) be morphisms in 𝓜. The composite ψ ∘ φ: M → M” is a smooth diffeomorphism by smoothness of φ and ψ. We verify (M1)–(M3) for ψ ∘ φ.

(M1): (ψ ∘ φ)g” = φ(ψg”) = φg’ = g, using (M1) for ψ and then for φ.

(M2): for any leaf Σ ∈ F, φ(Σ) is a leaf of F’ (by (M2) for φ); ψ(φ(Σ)) is a leaf of F” (by (M2) for ψ); hence (ψ ∘ φ)(Σ) is a leaf of F”.

(M3): (ψ ∘ φ)* V = ψ(φ_ V) = ψ_* V’ = V”, using (M3) for φ and then for ψ.

Therefore ψ ∘ φ is a morphism (M, g, F, V) → (M”, g”, F”, V”) in 𝓜. Composition is associative because composition of diffeomorphisms is associative.

Identity. The identity diffeomorphism id_M: M → M satisfies (id_M)g = g, id_M maps each leaf of F to itself, and (id_M)_ V = V. Therefore id_M is the identity morphism on (M, g, F, V) in 𝓜.

Identity laws. For any morphism φ: (M, g, F, V) → (M’, g’, F’, V’), id_{M’} ∘ φ = φ = φ ∘ id_M. ∎

2.3 Examples

Example 2.3.1 (Standard Minkowski moving-dimension structure). Take M = ℝ⁴ with the standard Lorentzian metric η_{μν} = diag(−c², 1, 1, 1) in coordinates (t, x¹, x², x³). Let F_std be the foliation by leaves Σ_t = {(t, x¹, x², x³) : t = const}; each leaf is a flat Cauchy surface. Let V_std = ∂/∂t; then g(V_std, V_std) = η_{tt} = −c², and V_std is orthogonal to F_std. The quadruple (ℝ⁴, η, F_std, V_std) is an object of 𝓜. We refer to this as the standard Minkowski moving-dimension structure.

Example 2.3.2 (Boosted Minkowski moving-dimension structure). Take M = ℝ⁴ with η as above, but let F_β be the foliation by leaves Σ_β,t = {p ∈ M : x⁰_β(p) = ct} where x⁰_β is the time coordinate of the boosted frame B_β with rapidity β. Let V_β be the unit timelike vector field along the boosted time axis. The quadruple (ℝ⁴, η, F_β, V_β) is an object of 𝓜, distinct from the standard structure as a 4-tuple of data, but related to it by the Poincaré transformation B_β.

Example 2.3.3 (FLRW spacetime). Take M = ℝ × Σ where Σ is a 3-manifold (spatial slice) with a fixed Riemannian metric h̃, and equip M with the FLRW metric g = −c²dt² + a(t)²·h̃. Let F be the foliation by constant-t slices and V = ∂/∂t. The quadruple (M, g, F, V) is an object of 𝓜, with the dynamical content encoded in the scale factor a(t) appearing in the metric. The object lies in 𝓜 because the data satisfy (a)–(d) of Definition 2.1.

The above examples illustrate that 𝓜 contains both flat Minkowski structures and curved cosmological structures; the category is rich.

2.4 Predicate 𝒫₁: V Is Part of the Geometric Structure

Condition (P1) of [G, Definition 5.4] reads:

V is part of the geometric structure of M, not a matter field defined on M. Equivalently, V is not associated to any matter Lagrangian density 𝓛_matter on M; V is a primitive geometric object on M, like the metric g and the foliation F.

Translating to a categorical predicate: V is part of the data specifying an object of 𝓜.

Definition 2.4 (Predicate 𝒫₁). Predicate 𝒫₁ on an object (M, g, F, V) of 𝓜 is the predicate “V is part of the structural data specifying the object” — equivalently, “V appears as the fourth component of the quadruple (M, g, F, V) defining the object in Definition 2.1.”

Proposition 2.4 (Predicate 𝒫₁ is automatic). Every object of 𝓜 satisfies Predicate 𝒫₁.

Proof. By Definition 2.1, every object of 𝓜 is specified as a quadruple (M, g, F, V) with V the fourth component of the data. Therefore V is part of the structural data of every object. ∎

This establishes that condition (P1) is built into the categorical specification of 𝓜. Within the category, every object has V as part of its structural data; (P1) is not a predicate that distinguishes some objects from others, but a meta-condition on how the category is set up.

2.5 Predicate 𝒫₂: V’s Flow Generates dx₄/dt = ic in Adapted Charts

Condition (P2) of [G, Definition 5.4] reads:

V’s flow is an active geometric process at rate ic. Mathematically, the flow φ_t : M → M generated by V satisfies the McGucken Principle dx₄/dt = ic, where x₄ is the McGucken coordinate of Convention 1.4.2 evaluated along the integral curves of V.

The mathematical content of (P2) is the differential equation dx₄/dt = ic along V’s integral curves in any McGucken-adapted chart. The “active flow” reading is interpretive and not a mathematical predicate; we restrict the formal predicate to the mathematical content.

Definition 2.5 (Predicate 𝒫₂). Predicate 𝒫₂ on an object (M, g, F, V) of 𝓜 is the predicate: in any McGucken-adapted chart (Convention 1.5.6) on M, the flow of V along its integral curves satisfies

dx₄/dt = ic

where x₄ = ict is the McGucken coordinate (Convention 1.5.2) and t is the time coordinate of the chart.

Proposition 2.5 (Predicate 𝒫₂ is automatic). Every object of 𝓜 satisfies Predicate 𝒫₂.

Proof. Let (M, g, F, V) be an object of 𝓜. Choose a McGucken-adapted chart (t, x¹, x², x³) on M. By Convention 1.5.6, in such a chart the metric takes the form ds² = −N²(t, x)c²dt² + h_{ij}(t, x)dx^i dx^j with N the lapse function. The vector field V is the unit timelike vector field orthogonal to F with g(V, V) = −c²; in the chart, V = (1/N)∂/∂t (the unit normalization in the time direction).

The flow of V along its integral curves is parameterized by proper time τ; in the chart, dt/dτ = (1/N)·(dt/dτ)·N = 1/N · N = 1 along V’s flow at unit normalization. Equivalently: for a comoving observer (an observer following V), the chart time t advances at unit rate with respect to proper time τ, so that dt/dτ = 1/N for the unit-normalized flow.

The McGucken coordinate is x₄ = ict by Convention 1.5.2. Therefore dx₄/dt = i·c·(dt/dt) = ic. The first-order rate of advance of x₄ with respect to chart time t along V’s flow is dx₄/dt = ic.

This holds in every McGucken-adapted chart, so Predicate 𝒫₂ is satisfied by every object of 𝓜. ∎

This establishes that condition (P2) — in its mathematical content — is built into the categorical specification of 𝓜. Every object of 𝓜 has V’s flow generating dx₄/dt = ic in adapted charts. The “active flow” reading is interpretive and not a categorical predicate.

2.6 Predicate 𝒫₃: V’s Wavefront Is the Future Null Cone

Condition (P3) of [G, Definition 5.4] reads:

V’s wavefront at every event p ∈ M is the McGucken Sphere Σ⁺(p): the future null cone of p generated by x₄’s expansion at rate ic from p, with each spatial direction sharing the wavefront equally by the spherical symmetry of x₄’s expansion.

The mathematical content of (P3) is that the future null cone Σ⁺(p) at every event p is generated by null geodesics emanating from p, with the future-directed null rays forming a spatially-spherical wavefront that expands at rate c with respect to the privileged time-coordinate of any McGucken-adapted chart.

Definition 2.6 (Predicate 𝒫₃). Predicate 𝒫₃ on an object (M, g, F, V) of 𝓜 is the predicate: at every event p ∈ M, the future null cone Σ⁺(p) = {q ∈ M : there exists a future-directed null geodesic from p to q} satisfies the following structural property: in any McGucken-adapted chart at p with coordinates (t, x¹, x², x³), the intersection of Σ⁺(p) with the leaf Σ_{t = t_p + Δt} (for Δt > 0) is the spatial sphere

{(x¹, x², x³) ∈ Σ_{t_p + Δt} : |x − x_p|² = c²·Δt²}

centered at the spatial position x_p of p, of radius c·Δt.

Proposition 2.6 (Predicate 𝒫₃ is automatic). Every object of 𝓜 satisfies Predicate 𝒫₃.

Proof. Let (M, g, F, V) be an object of 𝓜. Choose an event p ∈ M and a McGucken-adapted chart at p with coordinates (t, x¹, x², x³) and p = (t_p, xp). The metric in the chart takes the form ds² = −N²c²dt² + h{ij}dx^i dx^j with lapse N(t, x). At the event p itself, the local-flat-spacetime approximation yields ds² ≈ −c²dt² + dx² (working in a sufficiently small neighborhood where N ≈ 1 and h_{ij} ≈ δ_{ij}), and in this approximation the future null cone Σ⁺(p) is locally given by setting ds² = 0 with future direction:

0 = −c²·(t − t_p)² + |x − x_p|², t > t_p

⟹ |x − x_p|² = c²·(t − t_p)².

For Δt = t − t_p > 0, the intersection of Σ⁺(p) with the leaf Σ_{t_p + Δt} is the spatial sphere of radius c·Δt centered at x_p, in the local-flat approximation.

For events at finite distance from p in the chart (where the local-flat approximation breaks down), the future null cone is generated by null geodesics from p; the null geodesics in a McGucken-adapted chart, in the limit of small Δt, emanate isotropically from p in the spatial directions (because the spatial metric h_{ij} is Riemannian and isotropic at p in suitable coordinates). The expansion of the wavefront proceeds at rate c with respect to the chart time t.

The Predicate 𝒫₃ is therefore satisfied at every event p ∈ M, in the local-flat approximation that suffices for the wavefront’s structural property. ∎

This establishes that condition (P3) — in its mathematical content as the future-null-cone wavefront structure — is built into the categorical specification of 𝓜. Every object of 𝓜 has the McGucken-Sphere wavefront structure at every event, by virtue of the Lorentzian-metric and orthogonal-foliation structure.

2.7 Summary: The Three Predicates Are Automatic on 𝓜

Proposition 2.7 (Predicates 𝒫₁, 𝒫₂, 𝒫₃ are automatic on 𝓜). Every object of the category 𝓜 of moving-dimension manifolds satisfies all three predicates 𝒫₁, 𝒫₂, 𝒫₃ of Definitions 2.4, 2.5, 2.6.

Proof. Combine Propositions 2.4, 2.5, 2.6. ∎

The structural consequence of Proposition 2.7 is the following: the predicates (P1)–(P3) of Definition 5.4 of [G], formalized as mathematical predicates 𝒫₁, 𝒫₂, 𝒫₃, are built into the categorical specification of 𝓜. They are not predicates that distinguish some objects of 𝓜 from others; they are meta-conditions on how 𝓜 is set up. Every object of 𝓜 satisfies all three.

This is the categorical home of conditions (P1)–(P3): the category 𝓜 is the category whose objects are precisely the structures satisfying the three predicates. Categorically, (P1)–(P3) are the defining conditions of 𝓜.

3. Basic Categorical Properties of 𝓜

This section establishes basic categorical facts about 𝓜 that will be needed in Parts II and III.

3.1 Isomorphisms in 𝓜

Definition 3.1. A morphism φ: (M, g, F, V) → (M’, g’, F’, V’) in 𝓜 is an isomorphism if there exists ψ: (M’, g’, F’, V’) → (M, g, F, V) in 𝓜 with ψ ∘ φ = id and φ ∘ ψ = id’.

Proposition 3.2. A morphism φ in 𝓜 is an isomorphism if and only if the underlying diffeomorphism φ: M → M’ is a diffeomorphism (smooth with smooth inverse).

Proof. (⇒) If φ is an isomorphism in 𝓜 with inverse ψ in 𝓜, then ψ ∘ φ = id_M and φ ∘ ψ = id_{M’} as diffeomorphisms; hence φ is a diffeomorphism with smooth inverse ψ.

(⇐) Suppose φ: M → M’ is a diffeomorphism with smooth inverse φ⁻¹ and φ satisfies (M1)–(M3) of Definition 2.1. We claim φ⁻¹ also satisfies (M1)–(M3) (with primed and unprimed roles swapped) and is therefore an inverse morphism in 𝓜.

(M1) for φ⁻¹: (φ⁻¹)g = ?. Using φg’ = g, we have g = φ*g’, and pulling back by φ⁻¹ gives (φ⁻¹)*g = (φ⁻¹)φg’ = (φ ∘ φ⁻¹)*g’ = (id_{M’})*g’ = g’. So (φ⁻¹)*g = g’. ✓

(M2) for φ⁻¹: φ maps leaves of F bijectively to leaves of F’ (this is what (M2) for φ asserts), so φ⁻¹ maps leaves of F’ bijectively to leaves of F. ✓

(M3) for φ⁻¹: from φ_* V = V’, we have V = φ⁻¹_(φ_ V) = φ⁻¹_* V’. So (φ⁻¹)_* V’ = V. ✓

Therefore φ⁻¹ is a morphism (M’, g’, F’, V’) → (M, g, F, V) in 𝓜, and is an inverse of φ. ∎

Corollary 3.3. Two objects of 𝓜 are isomorphic if and only if there exists an isometry between their underlying Lorentzian manifolds that maps the foliation and vector field of one to those of the other.

This is the standard situation for categories of geometric structures: isomorphism is “structure-preserving diffeomorphism.”

3.2 The Category 𝓜 Is Not Skeletal

Proposition 3.4. The category 𝓜 is not skeletal: there exist isomorphic but distinct objects in 𝓜.

Proof. Consider Examples 2.3.1 and 2.3.2: the standard Minkowski moving-dimension structure (ℝ⁴, η, F_std, V_std) and the boosted structure (ℝ⁴, η, F_β, V_β) with rapidity β ≠ 0. As objects of 𝓜 these are distinct (the foliations F_std and F_β have different leaves, and V_std ≠ V_β as vector fields on ℝ⁴). However, they are isomorphic in 𝓜: the Poincaré transformation B_β: ℝ⁴ → ℝ⁴ corresponding to the boost of rapidity β is an isometry of (ℝ⁴, η), maps F_β to F_std (i.e., maps F_std-leaves to F_β-leaves under B_β), and intertwines V_β with V_std. Therefore B_β: (ℝ⁴, η, F_β, V_β) → (ℝ⁴, η, F_std, V_std) is an isomorphism in 𝓜.

Hence 𝓜 contains isomorphic but distinct objects; it is not skeletal. ∎

Remark 3.5. The non-skeletal nature of 𝓜 will be important for the rigidity theorems of Part II. Theorem A states that on flat Minkowski space ℝ⁴, the moving-dimension structure is unique up to isomorphism — i.e., up to a Poincaré transformation. Theorems B and C establish similar up-to-isomorphism uniqueness in increasingly general settings.

3.3 The Category 𝓜 Has Pullbacks (Pointwise)

We will not need the general categorical-limit structure of 𝓜 in the present paper, but we record one fact for completeness.

Proposition 3.6 (Pointwise pullbacks in 𝓜). Given a diagram

(M_1, g_1, F_1, V_1) →ᵠ (M, g, F, V) ←ᵠ’ (M_2, g_2, F_2, V_2)

in 𝓜, the fiber product M_1 ×_M M_2 (the pullback in the category of smooth manifolds), equipped with the pulled-back structures (g_1 ×_g g_2, F_1 ×_F F_2, V_1 ×_V V_2), is a candidate pullback object in 𝓜 provided the fiber product is itself a smooth manifold and the pulled-back structures satisfy (a)–(d) of Definition 2.1.

Proof. Standard categorical-limit construction; the conditions for the fiber product to be a smooth manifold are the standard transversality conditions for smooth pullbacks. We do not develop the general theory here; we will not need pullbacks in the rigidity theorems. ∎

3.4 Closure of 𝓜 Under Standard Constructions

Lemma 3.7 (Direct sum of moving-dimension manifolds). Let (M_1, g_1, F_1, V_1) and (M_2, g_2, F_2, V_2) be objects of 𝓜 with disjoint underlying manifolds. The disjoint union M_1 ⊔ M_2 with the disjoint-union metric, foliation, and vector field is an object of 𝓜.

Proof. The disjoint union of smooth, oriented, time-oriented globally hyperbolic Lorentzian 4-manifolds is again a smooth, oriented, time-oriented globally hyperbolic Lorentzian 4-manifold (working in the disconnected case, with each connected component globally hyperbolic individually). The metric, foliation, and vector field are constructed component-wise. ∎

This lemma will not be central to the rigidity theorems but illustrates that 𝓜 is closed under standard constructions.

4. Restricted Subcategories and Connection to [G]

4.1 The Subcategory 𝓜_M for Fixed M

For Theorems A and B (rigidity on a fixed underlying manifold), we will work with full subcategories of 𝓜 corresponding to fixed underlying manifolds.

Definition 4.1 (The subcategory 𝓜_M for fixed M). Given a fixed smooth, connected, oriented, time-oriented, globally hyperbolic Lorentzian 4-manifold M, the subcategory 𝓜_M ⊂ 𝓜 has:

Objects: triples (g, F, V) on M satisfying (b)–(d) of Definition 2.1, regarded as objects of 𝓜 with underlying manifold M.

Morphisms: morphisms in 𝓜 whose underlying diffeomorphism is the identity id_M.

The category 𝓜_M is the subcategory of 𝓜 consisting of moving-dimension structures on a fixed M, with morphisms restricted to the identity map. Two objects of 𝓜_M are equal as objects (of 𝓜_M) if and only if they have the same data (g, F, V) as triples on M. Two distinct objects of 𝓜_M may still be related by a non-identity morphism in 𝓜 (a non-identity diffeomorphism of M), but in 𝓜_M they remain distinct objects.

The point of working in 𝓜_M is to address the rigidity question precisely: given a fixed M and two moving-dimension structures (g, F, V) and (g’, F’, V’) on it, when are these structures equal (as triples) versus when are they merely isomorphic in 𝓜 (related by a non-identity diffeomorphism)?

4.2 Connection to Definitions of [G]

The moving-dimension manifold (M, F, V) of [G, Definition 5.6] is — under the present paper’s apparatus — an object of 𝓜 once we adjoin the metric g. The connection between [G]’s formulation and the present paper’s category 𝓜 is the following:

Lemma 4.2. Every moving-dimension manifold (M, F, V) of [G, Definition 5.6] satisfying conditions (P1)–(P3) of [G, Definition 5.4] determines, after specification of a Lorentzian metric g compatible with the data (in the sense that the leaves of F are spacelike Cauchy surfaces of (M, g) and V is the unit timelike vector field orthogonal to F with g(V, V) = −c²), a unique object of 𝓜.

Proof. Given (M, F, V) of [G, Definition 5.6] and a compatible Lorentzian metric g, the quadruple (M, g, F, V) satisfies (a)–(d) of Definition 2.1 of the present paper (by the compatibility specification). Hence (M, g, F, V) is an object of 𝓜. ∎

Lemma 4.3. Every object (M, g, F, V) of 𝓜 satisfies conditions (P1)–(P3) of [G, Definition 5.4] in their formal mathematical content (Predicates 𝒫₁, 𝒫₂, 𝒫₃ of §2 above).

Proof. By Proposition 2.7. ∎

The category 𝓜 is therefore the formal categorical home of the moving-dimension manifolds of [G, §5] under the formalization of conditions (P1)–(P3) as Predicates 𝒫₁, 𝒫₂, 𝒫₃.

4.3 The Empirical Condition (P4) Sits Outside the Categorical Apparatus

Condition (P4) of [G, Definition 5.4] reads:

V is empirically identified with the cosmic microwave background rest frame.

This identification is not a mathematical predicate on objects of 𝓜. The category 𝓜 contains all moving-dimension manifolds satisfying conditions (P1)–(P3); any of them is, mathematically, an object of 𝓜. The empirical identification of one specific object (the physically realized cosmological structure) with “the cosmic microwave background rest frame” is an empirical specification, not a categorical content.

We acknowledge (P4) at this point and set it aside. The mathematical theorems of the present paper concern the category 𝓜 and its objects; they do not address (P4) and cannot do so under categorical apparatus. The empirical specification (P4) selects, among the objects of 𝓜, the one corresponding to the actual cosmological structure of the universe. This selection is an empirical question, addressed in the corpus paper [79] of [G] and not in the present paper.

The standing of the categorical theorem we will prove is therefore: any framework satisfying the formal predicates (P1)–(P3) factors uniquely through 𝓜 (Theorem C). This is the mathematical content. Whether the framework, once factored through 𝓜, identifies the privileged object empirically with the CMB rest frame is a separate empirical question.

PART II — THE THREE RIGIDITY THEOREMS

Part II proves the three rigidity theorems. Theorem A (§5) establishes Minkowski rigidity: every moving-dimension structure on flat Minkowski space is isomorphic to the standard structure by a Poincaré transformation modulo time translation. Theorem B (§6) establishes local rigidity in adapted charts: the moving-dimension data on any (M, g, F, V) is determined locally by the lapse function and the spatial metrics on the leaves. Theorem C (§7) establishes categorical universality: in a suitable category 𝓐 of axis-dynamics frameworks containing 𝓜 as a subcategory, 𝓜 is the terminal subcategory corresponding to predicate-strict frameworks.

5. Theorem A: Minkowski Rigidity

5.1 Statement of the Theorem

Theorem A (Minkowski Rigidity). Every moving-dimension structure (ℝ⁴, η, F, V) on flat Minkowski space — i.e., every object of 𝓜 with underlying manifold ℝ⁴ and metric η — is isomorphic in 𝓜 to the standard structure (ℝ⁴, η, F_std, V_std) of Example 2.3.1, by a Poincaré transformation Λ ∈ ISO⁺(1, 3) (the proper orthochronous Poincaré group).

The isomorphism is unique modulo a one-parameter family of foliation-origin choices: two Poincaré transformations Λ_1, Λ_2 give the same isomorphism (ℝ⁴, η, F, V) → (ℝ⁴, η, F_std, V_std) if and only if Λ_2 = T_a ∘ Λ_1 where T_a is time translation by a real parameter a.

5.2 Preliminary: The Structure of Minkowski Moving-Dimension Data

Before proving Theorem A, we establish the structure of moving-dimension data on flat Minkowski space.

Lemma 5.1. Let (ℝ⁴, η, F, V) be an object of 𝓜 with underlying manifold ℝ⁴ and metric η. Then:

(a) F is a foliation by spacelike Cauchy surfaces of Minkowski space ℝ⁴; each leaf Σ ∈ F is a smooth spacelike hypersurface that is a Cauchy surface for ℝ⁴.

(b) V is a smooth future-directed unit timelike vector field on ℝ⁴ with η(V, V) = −c² and V everywhere orthogonal to the leaves of F.

(c) The lapse function N: ℝ⁴ → ℝ_{>0} associated to V (in any McGucken-adapted chart) is a smooth positive function on ℝ⁴, with N(p) = (−η(V, V))^{1/2} / (the normalization factor) = c / c = 1 in the chart where V = ∂/∂t, so N ≡ 1 in the McGucken-adapted chart of the data.

Proof. (a), (b) are immediate from Definition 2.1. (c): in any McGucken-adapted chart adapted to (F, V), the chart is irrotational with respect to F (Convention 1.5.6) and V is the unit timelike vector field orthogonal to F with η(V, V) = −c². In such a chart, the metric η has the form η_{tt} = −c² (constant on Minkowski space), η_{ti} = 0 (the chart is irrotational), η_{ij} = δ_{ij} (the spatial slices of Minkowski space are flat Euclidean). The unit timelike vector field orthogonal to constant-t slices, in this chart, is V = ∂/∂t (no lapse function is needed since η_{tt} = −c² already fixes the time normalization). Hence N ≡ 1. ∎

Lemma 5.2. In the McGucken-adapted chart of Lemma 5.1, the foliation F has leaves Σ_t = {(t, x¹, x², x³) : t = const} and V = ∂/∂t.

Proof. From Lemma 5.1, the chart is irrotational and N ≡ 1. The leaves of F are the constant-t hypersurfaces in the chart, which are flat Euclidean Cauchy surfaces. V is ∂/∂t. ∎

5.3 The Action of the Poincaré Group on Minkowski Moving-Dimension Data

Lemma 5.3. The proper orthochronous Poincaré group ISO⁺(1, 3) — the semidirect product of the proper orthochronous Lorentz group SO⁺(1, 3) with the translation group ℝ⁴ — acts on the set of moving-dimension structures on (ℝ⁴, η).

Specifically, for Λ = (R, a) ∈ ISO⁺(1, 3) with R ∈ SO⁺(1, 3) and a ∈ ℝ⁴, the action sends a moving-dimension structure (ℝ⁴, η, F, V) to (ℝ⁴, η, Λ_F, Λ_V) where Λ_F is the foliation whose leaves are the images Λ(Σ) for Σ ∈ F, and Λ_V is the push-forward of V by Λ.

Proof. We verify that (ℝ⁴, η, Λ_F, Λ_V) satisfies (a)–(d) of Definition 2.1.

(a): Underlying manifold ℝ⁴ unchanged; metric η is preserved by Λ since Λ ∈ ISO⁺(1, 3) is an isometry of (ℝ⁴, η). Hence (ℝ⁴, η) is the same Lorentzian manifold.

(b): Λ_*F is a foliation by spacelike Cauchy surfaces because Λ is a diffeomorphism preserving spacelike-ness of hypersurfaces (it is an isometry) and preserving the Cauchy property (it is a global diffeomorphism preserving causality structure).

(c): Λ_V is a smooth future-directed unit timelike vector field with η(Λ_V, Λ_V) = −c² (since Λ preserves η-norms) and is orthogonal to Λ_F (since Λ preserves orthogonality).

Therefore (ℝ⁴, η, Λ_F, Λ_V) is an object of 𝓜. The map Λ_*: 𝓜_{ℝ⁴, η} → 𝓜_{ℝ⁴, η} is the action of Λ on the set of moving-dimension structures with underlying manifold ℝ⁴ and metric η. ∎

5.4 Proof of Theorem A

We now prove Theorem A: every moving-dimension structure on (ℝ⁴, η) is isomorphic to the standard structure by a Poincaré transformation, with the isomorphism unique modulo time translation.

Proof of Theorem A. Let (ℝ⁴, η, F, V) be a moving-dimension structure on Minkowski space. We construct a Poincaré transformation Λ ∈ ISO⁺(1, 3) such that Λ: (ℝ⁴, η, F_std, V_std) → (ℝ⁴, η, F, V) is an isomorphism in 𝓜.

Step 1: V determines a unique Lorentz-frame direction. The vector field V on ℝ⁴ is a smooth future-directed unit timelike vector field with η(V, V) = −c². At any point p ∈ ℝ⁴, V(p) is a vector in the tangent space T_p ℝ⁴ ≅ ℝ⁴ (the tangent space to flat Minkowski space at p). The vector V(p) is timelike, future-directed, and of squared-norm −c².

We claim: V is parallel (covariantly constant) as a vector field on (ℝ⁴, η) — i.e., ∇^η V = 0, where ∇^η is the Levi-Civita connection of η. Since (ℝ⁴, η) is flat and simply connected, parallel-transport along any path is path-independent, and this is equivalent to V being constant under the canonical global identification T_p ℝ⁴ ≅ ℝ⁴ supplied by the affine structure of Minkowski space.

The argument proceeds in three substeps, using only intrinsic geometric facts about (ℝ⁴, η, F, V) without presupposing the existence of a global McGucken-adapted chart.

Substep 1a (V is geodesic and shear-free). By Lemma 5.1(c), the lapse function N associated to V (defined intrinsically by N := −η(V, T)/c² where T is the future-directed unit normal to F under any local time-function compatible with F) takes the value N ≡ 1 on the Minkowski background. The intrinsic verification of N ≡ 1 does not require a global chart: it follows from the fact that the leaves of F are flat Riemannian Cauchy surfaces of (ℝ⁴, η) — a direct consequence of (a)–(d) of Definition 2.1 applied to the flat metric η — and that V is the unit timelike vector field orthogonal to these flat leaves with η(V, V) = −c². The standard ADM identity (Lemma 6.1) gives the acceleration a^μ := V^ν ∇ν V^μ = (∇^μ ln N) projected orthogonal to V. With N ≡ 1, the acceleration vanishes: V is a geodesic vector field. Furthermore, the extrinsic curvature K{ij} := ½(∂t h{ij}) of any leaf with respect to V vanishes because the spatial metric h_{ij} on each leaf is flat Euclidean (as the induced metric of η on a flat Cauchy surface) and the leaves are isometric to one another under the flow of V (since η is V-invariant on flat space — η has translational Killing symmetries, and V being orthogonal to flat leaves with N ≡ 1 generates one of them). Hence the expansion θ = trK, shear σ_{ij}, and rotation ω_{ij} of the V-congruence all vanish.

Substep 1b (V is parallel under the Levi-Civita connection of η). The Raychaudhuri-type decomposition of ∇_μ V_ν on a unit timelike congruence gives

∇μ V_ν = (1/3)θ·(η{μν} + V_μ V_ν/c²) + σ_{μν} + ω_{μν} − V_μ a_ν,

where θ, σ_{μν}, ω_{μν}, a_ν are the kinematic quantities of Substep 1a. With θ = 0, σ_{μν} = 0, ω_{μν} = 0, and a_ν = 0 (all established in Substep 1a), we obtain ∇_μ V_ν = 0 on all of ℝ⁴. Hence V is covariantly constant under the Levi-Civita connection of η.

Substep 1c (V is globally constant under the affine identification of T ℝ⁴). Because (ℝ⁴, η) is flat (R^μ_{νρσ} = 0) and simply connected (ℝ⁴ is contractible), the Levi-Civita connection ∇^η admits a unique global parallelism: for any p, q ∈ ℝ⁴, parallel transport from p to q along any smooth path is path-independent and coincides with the canonical affine identification T_p ℝ⁴ ≅ T_q ℝ⁴ ≅ ℝ⁴. Combined with Substep 1b (V is ∇^η-parallel), this yields: V(q) and V(p) are identified under this canonical identification for every pair p, q ∈ ℝ⁴. In standard Cartesian coordinates (y⁰, y¹, y², y³) on Minkowski space (which exist globally and are independent of any moving-dimension structure on (ℝ⁴, η)), V has constant components V^μ ∈ ℝ⁴ everywhere on ℝ⁴.

This establishes the global constancy of V intrinsically, without circular appeal to a McGucken-adapted chart whose existence is part of what Theorem A is being asked to deliver.

Step 2: The constant V is determined up to Lorentz boost. The constant future-directed unit timelike vector V at a fixed reference point of ℝ⁴ is an element of the future-directed unit hyperboloid in Minkowski space — i.e., the set H⁺ = {v ∈ ℝ⁴ : η(v, v) = −c², v⁰ > 0}. This hyperboloid is a homogeneous space under SO⁺(1, 3): given any future-directed unit timelike vector v ∈ H⁺, there exists a Lorentz boost B_v ∈ SO⁺(1, 3) such that B_v · v = e_0 = (1, 0, 0, 0) (the standard time direction).

Hence: there exists a Lorentz boost B_V ∈ SO⁺(1, 3) such that B_V · V = V_std = ∂/∂t at the reference point, and (since V is constant on ℝ⁴) B_V · V = V_std globally.

Step 3: The foliation F is determined by V. Once V is determined as the standard time direction (after applying B_V), the foliation F orthogonal to V is determined: F has leaves orthogonal to V_std at every point, which are precisely the constant-t hyperplanes Σ_t = {p : t = const}. Hence after applying B_V, the foliation F becomes F_std up to a translation in the t-direction.

The remaining freedom is a translation in time: the standard foliation F_std is invariant under spatial translations and Lorentz rotations preserving the t-axis but is not invariant under time translation. A time translation T_a: (t, x) → (t + a, x) maps Σ_t to Σ_{t+a} — i.e., maps the leaves of F_std to themselves (as a set), but changes the labeling. Under the moving-dimension data the labeling matters: distinct values of a give distinct foliations (with leaves at different parameter values).

Step 4: Combining the boost and the time translation. We can construct a Poincaré transformation Λ ∈ ISO⁺(1, 3) such that Λ: (ℝ⁴, η, F_std, V_std) → (ℝ⁴, η, F, V) is an isomorphism in 𝓜. Specifically, set Λ = T_a ∘ B_V⁻¹ where B_V is the Lorentz boost of Step 2 and T_a is a time translation determined by matching the foliation origin.

The Poincaré transformation Λ is by construction an isometry of (ℝ⁴, η). It maps the standard moving-dimension data (F_std, V_std) to the original data (F, V). Hence Λ is an isomorphism in 𝓜.

Step 5: Uniqueness modulo time translation. Suppose Λ_1 and Λ_2 are two Poincaré transformations giving isomorphisms (ℝ⁴, η, F_std, V_std) → (ℝ⁴, η, F, V) in 𝓜. Then the composite Λ_2⁻¹ ∘ Λ_1 is an automorphism of (ℝ⁴, η, F_std, V_std) — i.e., a Poincaré transformation preserving F_std and V_std.

What Poincaré transformations preserve F_std and V_std?

• They must preserve V_std = ∂/∂t (intertwining condition (M3)). Among Poincaré transformations, only those of the form Λ = (R, a) with R fixing the t-axis (R ∈ SO(3) ⊂ SO⁺(1, 3)) and a arbitrary preserve V_std up to translation. Among these, only those with R ∈ SO(3) (no boost in the t-direction) preserve V_std (intertwining requires V_std → V_std, which forces R to fix V_std as a vector).
• They must preserve F_std (foliation-preserving condition (M2)) — i.e., map constant-t slices to constant-t slices. R ∈ SO(3) preserves the t-coordinate trivially (rotations of the spatial part). Spatial translations preserve F_std (they map Σ_t to Σ_t). Time translations T_a map Σ_t to Σ_{t-a}, which is again a leaf of F_std (just at a different parameter value). All translations in ℝ⁴ preserve F_std as a foliation.
• They must preserve η (isometry condition (M1)) — automatically satisfied by Poincaré transformations.

Hence the automorphisms of (ℝ⁴, η, F_std, V_std) are exactly the elements (R, a) ∈ SO(3) ⋉ ℝ⁴ — spatial rotations combined with arbitrary translations.

The equivalence classes of Poincaré transformations modulo time translation alone (ignoring spatial rotations and spatial translations) form a one-parameter family parameterized by a ∈ ℝ. Therefore the isomorphism Λ in Theorem A is unique modulo a one-parameter family of time-translation choices.

This completes the proof. ∎

5.5 Remarks on Theorem A

Remark 5.5.1. Theorem A is the simplest case of the rigidity content of the present paper. On flat Minkowski space, the moving-dimension structure is essentially unique: any object of 𝓜 with underlying manifold ℝ⁴ and metric η is isomorphic to the standard one, modulo a one-parameter family of foliation-origin choices.

Remark 5.5.2. The one-parameter family of foliation-origin choices reflects the residual freedom in choosing where t = 0 is located in the McGucken-adapted chart. This is a coordinate-system choice, not a structural difference. Two moving-dimension structures on Minkowski space differing only in foliation origin are indistinguishable in any way that does not depend on the absolute value of the t-parameter.

Remark 5.5.3. Theorem A is consistent with [G, Proposition 5.7]’s existence statement on Minkowski space: the moving-dimension structure on Minkowski space exists and is unique up to a one-parameter family of foliation-origin choices. The present Theorem A strengthens [G, Proposition 5.7] to a categorical isomorphism statement: not only does the structure exist and is unique, but every other Minkowski moving-dimension structure is related to the standard one by a Poincaré transformation, with the isomorphism unique modulo time translation.

Remark 5.5.4. Theorem A is a flat-spacetime rigidity theorem; it does not address curved spacetimes. The next theorem (Theorem B) extends the rigidity content to general moving-dimension manifolds in adapted charts.

Remark 5.5.5 (Intrinsic character of the Step 1 argument). The Step 1 derivation of the global constancy of V proceeds entirely through intrinsic geometric data — the kinematic decomposition of ∇_μ V_ν on a unit timelike congruence (Substep 1b), combined with the flatness and simply-connectedness of (ℝ⁴, η) (Substep 1c) — and does not presuppose the global existence of a McGucken-adapted chart. The vanishing of the kinematic quantities (acceleration, expansion, shear, rotation) of the V-congruence in Substep 1a is established intrinsically from (a)–(d) of Definition 2.1 applied to the flat metric η, using only that the leaves of F are flat Riemannian Cauchy surfaces (a consequence of Definition 2.1 plus the flatness of η, not a chart-level assertion) and that V is the unit normal to these leaves with η(V, V) = −c² (Definition 2.1(d)). The chart-relative content of Lemma 5.1(c) — that N ≡ 1 in any McGucken-adapted chart — is consistent with the intrinsic Substep 1a but is logically downstream of it: once V is known to be parallel and the leaves to be flat, the existence of a global McGucken-adapted chart with N ≡ 1 follows by integrating V’s flow against any leaf, which produces a global diffeomorphism ℝ⁴ ≅ ℝ × Σ where Σ is any leaf. The intrinsic-then-chart logical order avoids the circularity that would arise from invoking the chart’s existence to derive properties of V on which the chart’s existence itself depends.

6. Theorem B: Local Rigidity in Adapted Charts

6.1 Statement of the Theorem

Theorem B (Local Rigidity in Adapted Charts). Let M be a smooth, connected, oriented, time-oriented globally hyperbolic Lorentzian 4-manifold. Suppose (g, F, V) and (g’, F’, V’) are two moving-dimension structures on M (i.e., two objects of 𝓜_M from Definition 4.1).

Suppose further that:

(i) g and g’ have the same lapse function in every McGucken-adapted chart for both structures: in any chart adapted to (F, V) and also adapted to (F’, V’), the lapse functions N(t, x) and N'(t, x) coincide as smooth positive functions.

(ii) The induced spatial metrics on the leaves coincide: for every leaf Σ ∈ F (which is also a leaf of F’ by (i)), the induced spatial metrics h_{ij} (from g) and h’_{ij} (from g’) on Σ are equal as Riemannian metrics on Σ.

Then (g, F, V) = (g’, F’, V’) as objects of 𝓜_M; i.e., the two structures are equal (not merely isomorphic).

6.2 The ADM 3+1 Decomposition

The proof of Theorem B uses the standard ADM 3+1 decomposition of Lorentzian geometry [12, 15], which we recall here for completeness.

Lemma 6.1 (ADM Decomposition for Moving-Dimension Manifolds). Let (M, g, F, V) be an object of 𝓜. In any McGucken-adapted chart on M with coordinates (t, x¹, x², x³), the metric g takes the form

ds² = −N²(t, x)·c²·dt² + h_{ij}(t, x)·dxⁱ·dxʲ

with:

— N(t, x) the lapse function (a smooth positive function on M);

— h_{ij}(t, x) the induced spatial metric on the leaf Σ_t at parameter value t (a smooth Riemannian metric on each leaf, varying smoothly in t);

— the shift vector N^i = 0 (the chart is irrotational with respect to F).

Moreover, V = (1/N)·∂/∂t in this chart (the unit timelike normalization).

Proof. This is the standard ADM 3+1 decomposition adapted to the McGucken-adapted-chart setting. The chart is McGucken-adapted means: (i) the time coordinate t coincides with the cosmic-time-function parameter labeling the leaves of F (so leaves are constant-t hypersurfaces in the chart), and (ii) the chart is irrotational with respect to F (Convention 1.5.6), which forces the shift vector to vanish. The lapse N(t, x) is the function relating the cosmic-time-function parameter t to the proper time τ of an observer comoving with V; specifically dτ = N·dt along V’s flow. The induced spatial metric on each leaf is h_{ij}(t, x) (the spatial part of g restricted to the leaf at parameter t).

The vector field V is by definition the unit timelike vector field orthogonal to the leaves of F. In the McGucken-adapted chart, V is in the t-direction (orthogonal to constant-t slices), and the unit normalization g(V, V) = −c² translates to V = (1/N)·∂/∂t in the chart (the factor 1/N normalizes V to unit timelike length). ∎

6.3 Proof of Theorem B

Proof of Theorem B. Let (g, F, V) and (g’, F’, V’) be two moving-dimension structures on M satisfying (i) and (ii) of Theorem B. We show they are equal as objects of 𝓜_M.

Step 1: F = F’. Condition (i) presupposes that both structures admit a common McGucken-adapted chart. Choose any such chart (t, x¹, x², x³) on M. In the chart, the leaves of F are constant-t hypersurfaces (with respect to the time coordinate of the chart, which coincides with the cosmic-time-function parameter of F by Convention 1.5.6). Similarly, the leaves of F’ are constant-t hypersurfaces in the same chart (by the chart-adaptation condition of (i)). Hence F and F’ have the same leaves in the chart.

The chart is global on M (or covers M after suitable patching across coordinate patches; we assume for simplicity a single chart, with the patching argument extending the conclusion globally by smooth gluing of leaves). Hence F = F’ globally on M.

Step 2: V = V’. By Lemma 6.1, in the McGucken-adapted chart, V = (1/N)·∂/∂t and V’ = (1/N’)·∂/∂t. By condition (i), N = N’. Hence V = V’ in the chart, and globally on M by the same patching argument.

Step 3: g = g’. By Lemma 6.1, the metrics g and g’ in the McGucken-adapted chart take the form

ds² = −N²·c²·dt² + h_{ij}·dxⁱ·dxʲ,

ds’² = −N’²·c²·dt² + h’_{ij}·dxⁱ·dxʲ.

By condition (i), N = N’. By condition (ii), h_{ij} = h’_{ij}. Hence g = g’ in the chart, and globally on M.

Combining Steps 1, 2, 3: F = F’, V = V’, g = g’. The structures (g, F, V) and (g’, F’, V’) are equal as triples on M, so they are equal as objects of 𝓜_M. ∎

6.4 Remarks on Theorem B

Remark 6.2.1. Theorem B is a local-data rigidity theorem: it says that the moving-dimension data on M is determined locally by the lapse function N and the spatial metrics h_{ij} on the leaves. This is essentially the standard ADM-decomposition fact, articulated in the moving-dimension setting.

Remark 6.2.2. The hypothesis (i) of Theorem B presupposes that the two structures share a common McGucken-adapted chart. This is a substantive condition: two distinct moving-dimension structures on M may not have any common McGucken-adapted chart (their foliations may differ, in which case the time coordinates of their respective adapted charts cannot coincide). Theorem B applies only when the lapse functions and the spatial metrics on leaves coincide in a common adapted chart; this rules out structures with genuinely different foliations.

Remark 6.2.3. Theorem B can be restated in a coordinate-free way: two moving-dimension structures on M are equal if and only if they have the same foliation F (as a foliation of M), the same lapse function (as a smooth function on M), and the same induced spatial metrics on the leaves of F. This is the coordinate-free content of the local rigidity.

Remark 6.2.4. Theorem B does not establish that any two moving-dimension structures on M are related by an isomorphism in 𝓜. Two structures with different foliations on the same M may be distinct objects of 𝓜_M and may or may not be isomorphic in 𝓜 (related by a non-identity diffeomorphism of M). Theorem B addresses only the question of when two structures are equal as triples on the same M; the question of when they are merely isomorphic is addressed by Theorem C in the categorical sense.

7. Theorem C: Categorical Universality (The No-Embedding Theorem)

7.1 The Strategy

Theorem C is the main theorem of the paper. We construct a larger category 𝓐 of axis-dynamics frameworks containing 𝓜 as a full subcategory (via the embedding ι: 𝓜 → 𝓐); define a forgetful functor R: 𝓐 → 𝓜; identify the full subcategory 𝓐₀ ⊂ 𝓐 of trivially-decorated objects; prove that ι factors through 𝓐₀ as an isomorphism of categories 𝓜 ⥲ 𝓐₀ with strict inverse R|_{𝓐₀} (Theorem 7.5.2); prove the section identity R ∘ ι = 1_𝓜 globally (Remark 7.5.4); characterize predicate-strictness through the existence of a canonical morphism A → ι(R(A)) (Proposition 7.6.3); and establish the universal-property characterization of 𝓐₀ as the unique full subcategory satisfying the four equivalent conditions (C1)–(C4) of Theorem 7.7.1.

The strategy is not an adjunction in the standard sense — the morphism conditions of 𝓐 force decoration compatibility, which breaks the candidate adjunction Hom_𝓐(ι(X), Y) ≅ Hom_𝓜(X, R(Y)) for objects Y with non-trivial decoration ε_Y ≠ 0 (Remark 7.5.3). What the apparatus does establish is full-subcategory equivalence (𝓜 ≃ 𝓐₀ via ι) combined with a universal-property characterization of 𝓐₀ in 𝓐 (Corollary 7.7.4): 𝓐₀ is the unique full subcategory of 𝓐 such that ι factors through it as an isomorphism, every object of 𝓐₀ admits the canonical morphism of Proposition 7.6.3, and no object outside 𝓐₀ admits such a morphism. This universal-property characterization is the structural content of Theorem C, and it does not require adjoint-functor machinery.

7.2 The Category 𝓐 of Axis-Dynamics Frameworks

We define the larger category 𝓐 in which 𝓜 will be embedded as a full subcategory.

Definition 7.1 (The category 𝓐). The category 𝓐 of axis-dynamics frameworks has:

Objects: sextuples (M, g, F, V, E, ε) where:

(a) M is a smooth, connected, oriented, time-oriented globally hyperbolic Lorentzian 4-manifold;

(b) g is a Lorentzian metric on M of signature (−, +, +, +);

(c) F is a smooth codimension-1 foliation of M whose leaves are spacelike Cauchy surfaces;

(d) V is a smooth future-directed timelike vector field on M with g(V, V) = −c² and V everywhere orthogonal to the leaves of F (V is unit timelike, just as in 𝓜);

(e) E → M is a smooth vector bundle over M of finite rank n_E ≥ 0 (rank 0 is allowed and means E is the rank-0 zero bundle 0_M whose total space coincides with M and whose only section is the zero section);

(f) ε ∈ Γ(E) is a smooth section of E (the axis-dynamics decoration*).*

Morphisms: a morphism (φ, Φ): (M, g, F, V, E, ε) → (M’, g’, F’, V’, E’, ε’) in 𝓐 is a pair where:

• φ: M → M’ is a smooth diffeomorphism;
• Φ: E → E’ is a smooth bundle map covering φ — i.e., Φ is a smooth fiberwise-linear map fitting into the commutative square of bundle projections, with Φ_x: E_x → E’_{φ(x)} linear for every x ∈ M;

satisfying the four conditions:

(M1) φ_ g = g’ (metric preservation);* (M2) φ(F) = F’ (foliation preservation); (M3) φ_ V = V’ (vector-field preservation);* (M4) Φ ∘ ε = ε’ ∘ φ (decoration intertwining: the section ε ∈ Γ(E) is mapped by Φ to a section of E’ over M’, and this image section equals ε’ under the identification by φ).

Composition: (ψ, Ψ) ∘ (φ, Φ) := (ψ ∘ φ, Ψ ∘ Φ) where Ψ ∘ Φ: E → E” is the bundle composition covering ψ ∘ φ.

Identity: 1_{(M, g, F, V, E, ε)} := (id_M, id_E).

The decoration E in (e) of Definition 7.1 represents the type of extra structure that an axis-dynamics framework carries (matter Lagrangian field, VEV tensor, gauge connection, etc.); the section ε in (f) represents the value of that structure. The rank n_E of E is part of the object data — different objects of 𝓐 may have decorations on bundles of different ranks. The rank-0 case n_E = 0 corresponds to “no extra decoration” (E is the zero bundle 0_M; ε is forced to be the zero section). A morphism (φ, Φ) consists of a diffeomorphism φ of base manifolds together with an explicit bundle map Φ over φ; morphisms between objects with different ranks exist when a suitable bundle map Φ exists.

Proposition 7.2 (𝓐 is a category). The data of Definition 7.1 satisfy the category axioms.

Proof. Closure under composition: Given (φ, Φ): (M, g, F, V, E, ε) → (M’, g’, F’, V’, E’, ε’) and (ψ, Ψ): (M’, g’, F’, V’, E’, ε’) → (M”, g”, F”, V”, E”, ε”), the composition (ψ ∘ φ, Ψ ∘ Φ) has:

• ψ ∘ φ: M → M” is a smooth diffeomorphism (composition of diffeomorphisms);
• Ψ ∘ Φ: E → E” is a smooth bundle map covering ψ ∘ φ (composition of bundle maps);
• (M1) (ψ ∘ φ)* g = ψ(φ_ g) = ψ_* g’ = g” ✓
• (M2) (ψ ∘ φ)(F) = ψ(φ(F)) = ψ(F’) = F” ✓
• (M3) (ψ ∘ φ)* V = ψ(φ_ V) = ψ_* V’ = V” ✓
• (M4) (Ψ ∘ Φ) ∘ ε = Ψ ∘ (Φ ∘ ε) = Ψ ∘ (ε’ ∘ φ) = (Ψ ∘ ε’) ∘ φ = (ε” ∘ ψ) ∘ φ = ε” ∘ (ψ ∘ φ) ✓

So the composition is a morphism in 𝓐.

Identity laws: For any object A, the pair (id_M, id_E) satisfies (M1)–(M4) trivially: id_M preserves g, F, V on the nose, and id_E ∘ ε = ε ∘ id_M = ε. Composition with (id_M, id_E) on either side yields the original morphism by the identity laws for diffeomorphisms and bundle maps separately.

Associativity: Composition is associative on each component (composition of diffeomorphisms is associative; composition of bundle maps is associative), and the conditions (M1)–(M4) are pointwise, so associativity transfers to 𝓐-morphisms. ∎

7.3 The Embedding ι: 𝓜 → 𝓐

Definition 7.3 (Embedding ι: 𝓜 → 𝓐). Define a functor ι: 𝓜 → 𝓐 as follows.

On objects: for X = (M, g, F, V) ∈ 𝓜, set

ι(X) := (M, g, F, V, 0_M, 0)

where 0_M → M is the rank-0 zero bundle (whose total space is M itself with the canonical projection 0_M → M being the identity, and whose only section is the zero section) and 0 denotes that unique zero section.

On morphisms: for φ: X → X’ in 𝓜, the diffeomorphism φ: M → M’ canonically induces the bundle map 0_φ: 0_M → 0_{M’} (the unique bundle map covering φ between rank-0 bundles, given by Φ_x: {0_x} → {0_{φ(x)}} sending the single fiber element to the single fiber element). Set

ι(φ) := (φ, 0_φ): ι(X) → ι(X’).

Verification that ι is a functor. On objects, ι(X) ∈ 𝓐 by construction: (a)–(d) of Definition 7.1 hold because X ∈ 𝓜 (which uses the same data); (e) holds because 0_M is rank 0; (f) holds because 0 is the (unique) section.

On morphisms, (ι(φ)) is a morphism in 𝓐:

• (M1), (M2), (M3) hold for φ because they hold for φ as a morphism in 𝓜;
• (M4): 0_φ ∘ 0 = 0 ∘ φ holds because both sides are the zero section pulled back along φ — both are identically zero as sections of 0_{M’}.

Functoriality:

• ι(id_X) = (id_M, 0_{id_M}) = (id_M, id_{0_M}) = id_{ι(X)} ✓
• ι(ψ ∘ φ) = (ψ ∘ φ, 0_{ψ∘φ}) = (ψ ∘ φ, 0_ψ ∘ 0_φ) = (ψ, 0_ψ) ∘ (φ, 0_φ) = ι(ψ) ∘ ι(φ) ✓

Proposition 7.4 (ι is a fully faithful functor). The functor ι: 𝓜 → 𝓐 is fully faithful: for every pair of objects X, X’ ∈ 𝓜, the map

ι_{X,X’}: Hom_𝓜(X, X’) → Hom_𝓐(ι(X), ι(X’))

is a bijection.

Proof. Let X = (M, g, F, V) and X’ = (M’, g’, F’, V’) be objects of 𝓜.

Injectivity: Suppose φ, ψ ∈ Hom_𝓜(X, X’) satisfy ι(φ) = ι(ψ) in 𝓐. Then (φ, 0_φ) = (ψ, 0_ψ) as pairs, so φ = ψ as diffeomorphisms M → M’. Since 𝓜-morphisms are determined by their underlying diffeomorphisms (Definition 2.1), φ = ψ as 𝓜-morphisms.

Surjectivity: Let (χ, Χ): ι(X) → ι(X’) be a morphism in 𝓐. The bundle-map component Χ: 0_M → 0_{M’} is uniquely determined as the trivial map between rank-0 bundles, so Χ = 0_χ. Conditions (M1)–(M3) hold for χ. Condition (M4) holds trivially. Hence χ: M → M’ is a diffeomorphism satisfying (M1)–(M3), which by Definition 2.1 is exactly a morphism χ ∈ Hom_𝓜(X, X’). Then ι(χ) = (χ, 0_χ) = (χ, Χ) = the given morphism. Surjectivity established. ∎

The functor ι: 𝓜 → 𝓐 thus embeds 𝓜 as a full subcategory of 𝓐, consisting of objects with rank-0 decoration bundle and zero section.

7.4 The Forgetful Functor R: 𝓐 → 𝓜

Definition 7.5 (Forgetful functor R: 𝓐 → 𝓜). Define a functor R: 𝓐 → 𝓜 as follows.

On objects: for A = (M, g, F, V, E, ε) ∈ 𝓐, set

R(A) := (M, g, F, V),

forgetting the decoration data (E, ε).

On morphisms: for (φ, Φ): A → A’ in 𝓐, set

R(φ, Φ) := φ: R(A) → R(A’),

forgetting the bundle-map component Φ.

Proposition 7.6 (R is a functor). The data of Definition 7.5 form a functor R: 𝓐 → 𝓜.

Proof. On objects: R(A) has data (M, g, F, V) satisfying (a)–(d) of Definition 2.1 because A satisfies (a)–(d) of Definition 7.1 (the conditions are word-for-word identical). Hence R(A) ∈ 𝓜.

On morphisms: For (φ, Φ): A → A’ in 𝓐, the underlying diffeomorphism φ satisfies (M1)–(M3) by virtue of (φ, Φ) satisfying (M1)–(M3) in 𝓐. Hence φ: R(A) → R(A’) is a morphism in 𝓜.

Functoriality:

• R(id_A) = R(id_M, id_E) = id_M = id_{R(A)} ✓
• R((ψ, Ψ) ∘ (φ, Φ)) = R(ψ ∘ φ, Ψ ∘ Φ) = ψ ∘ φ = R(ψ, Ψ) ∘ R(φ, Φ) ✓ ∎

7.5 The Subcategory 𝓐₀ of Trivially-Decorated Objects, the Equivalence 𝓜 ≃ 𝓐₀, and the Section R∘ι = 1_𝓜

We now establish the precise categorical relationship between 𝓜, 𝓐, ι, and R. The structure is not an adjunction in the standard sense — for an adjunction ι ⊣ R we would require Hom_𝓐(ι(X), Y) ≅ Hom_𝓜(X, R(Y)) naturally in X and Y, but the morphism conditions of 𝓐 force ε_Y = 0 in any morphism out of ι(X) (since ι(X) carries trivial decoration and (M4) requires φ_*0 = ε_Y), while the right-hand side imposes no such condition. The adjunction fails for objects Y with non-trivial decoration. The honest categorical content is therefore not adjunction but full-subcategory equivalence plus a universal-property characterization, which we now prove.

Definition 7.5.1 (Full subcategory 𝓐₀ of trivially-decorated objects). Let 𝓐₀ ⊂ 𝓐 denote the full subcategory whose objects are sextuples (M, g, F, V, E, ε) ∈ 𝓐 satisfying both:

• E = 0_M (the rank-0 zero bundle over M); and
• ε = 0 (the unique zero section of 0_M).

Morphisms of 𝓐₀ are morphisms of 𝓐 between objects of 𝓐₀ — pairs (φ, Φ) where Φ = 0_φ is the unique trivial bundle map between rank-0 zero bundles, satisfying (M1)–(M4) where (M4) is automatic.

Theorem 7.5.2 (Isomorphism ι: 𝓜 ⥲ 𝓐₀). The functor ι: 𝓜 → 𝓐 of Definition 7.3 factors through 𝓐₀ ⊂ 𝓐. The factored functor

ι: 𝓜 → 𝓐₀

is an isomorphism of categories, with strict inverse R|_{𝓐₀}: 𝓐₀ → 𝓜 given by the restriction of R (Definition 7.5) to 𝓐₀. Specifically, both compositions are equalities of functors on the nose:

(R|{𝓐₀}) ∘ ι = 1_𝓜 and ι ∘ (R|{𝓐₀}) = 1_{𝓐₀}.

Proof. We verify the four claims: (a) ι factors through 𝓐₀; (b) ι: 𝓜 → 𝓐₀ is a bijection on objects; (c) ι: 𝓜 → 𝓐₀ is a bijection on Hom-sets; (d) the two compositions are equalities of functors on the nose. Together these establish that ι: 𝓜 → 𝓐₀ is an isomorphism of categories with strict inverse R|_{𝓐₀}.

(a) ι factors through 𝓐₀. For any X = (M, g, F, V) ∈ 𝓜, by Definition 7.3 we have ι(X) = (M, g, F, V, 0_M, 0). The decoration bundle is 0_M (rank 0) and the section is 0 (the unique zero section). Hence ι(X) ∈ 𝓐₀ by Definition 7.5.1. For any morphism φ ∈ Hom_𝓜(X, X’), ι(φ) = (φ, 0_φ); the bundle-map component 0_φ between rank-0 zero bundles is the unique trivial map, satisfying the 𝓐₀-morphism condition. Hence ι(φ) ∈ Hom_{𝓐₀}(ι(X), ι(X’)).

(b) ι: 𝓜 → 𝓐₀ is a bijection on objects.

Injectivity: Suppose ι(X) = ι(X’) for X = (M, g, F, V), X’ = (M’, g’, F’, V’) ∈ 𝓜. Then (M, g, F, V, 0_M, 0) = (M’, g’, F’, V’, 0_{M’}, 0). Component-wise equality gives M = M’, g = g’, F = F’, V = V’, hence X = X’.

Surjectivity: Let A₀ = (M, g, F, V, E, ε) ∈ 𝓐₀. By Definition 7.5.1, E = 0_M and ε = 0. Set X := (M, g, F, V) ∈ 𝓜 (the data satisfies Definition 2.1 because A₀ satisfies Definition 7.1(a)–(d)). Then ι(X) = (M, g, F, V, 0_M, 0) = A₀. Hence ι is surjective on objects of 𝓐₀.

(c) ι: 𝓜 → 𝓐₀ is a bijection on Hom-sets. Let X, X’ ∈ 𝓜.

Injectivity: Established in Proposition 7.4 (the underlying-diffeomorphism component determines the 𝓜-morphism).

Surjectivity: Let (χ, Χ) ∈ Hom_{𝓐₀}(ι(X), ι(X’)). The bundle-map component Χ: 0_M → 0_{M’} is uniquely the trivial map 0_χ between rank-0 bundles, by the structure of the rank-0 zero bundle (each fiber has exactly one element, so any map between such bundles is uniquely determined as fiberwise the unique map between singleton sets). The diffeomorphism component χ: M → M’ satisfies (M1)–(M3) by virtue of (χ, Χ) being an 𝓐-morphism. Hence χ ∈ Hom_𝓜(X, X’) by Definition 2.1. Then ι(χ) = (χ, 0_χ) = (χ, Χ) = the given morphism. Surjectivity established.

(d) (R|_{𝓐₀}) ∘ ι = 1_𝓜 globally on 𝓜.

On objects: For X = (M, g, F, V) ∈ 𝓜:

(R|{𝓐₀})(ι(X)) = R|{𝓐₀}(M, g, F, V, 0_M, 0) = (M, g, F, V) = X.

On morphisms: For φ ∈ Hom_𝓜(X, X’):

(R|{𝓐₀})(ι(φ)) = R|{𝓐₀}(φ, 0_φ) = φ ∈ Hom_𝓜(X, X’).

Hence (R|_{𝓐₀}) ∘ ι = 1_𝓜 as functors on the nose.

(d’) ι ∘ (R|{𝓐₀}) = 1{𝓐₀} on 𝓐₀.

On objects: For A₀ = (M, g, F, V, 0_M, 0) ∈ 𝓐₀:

ι((R|_{𝓐₀})(A₀)) = ι(M, g, F, V) = (M, g, F, V, 0_M, 0) = A₀.

On morphisms: For (φ, 0_φ) ∈ Hom_{𝓐₀}(A₀, A₀’):

ι((R|{𝓐₀})(φ, 0_φ)) = ι(φ) = (φ, 0_φ) ∈ Hom{𝓐₀}(A₀, A₀’).

Hence ι ∘ (R|{𝓐₀}) = 1{𝓐₀} as functors on the nose.

The four claims (a)–(d, d’) establish that ι: 𝓜 → 𝓐₀ is an isomorphism of categories with strict inverse R|_{𝓐₀}. The isomorphism is strict (equalities of functors on the nose), not merely an equivalence (natural isomorphism), which is the strongest categorical relationship two functors can satisfy. ∎

Remark 7.5.3 (Why this is not an adjunction). The functors ι: 𝓜 → 𝓐 and R: 𝓐 → 𝓜 do not form an adjoint pair ι ⊣ R or R ⊣ ι in the standard sense. To see why, consider the candidate natural bijection for ι ⊣ R:

Hom_𝓐(ι(X), Y) ≅ Hom_𝓜(X, R(Y)) natural in X ∈ 𝓜, Y ∈ 𝓐.

A morphism in Hom_𝓐(ι(X), Y) is a pair (φ, Φ) where φ: M_X → M_Y is a diffeomorphism satisfying (M1)–(M3) and Φ: 0_{M_X} → E_Y is a bundle map covering φ satisfying (M4): Φ ∘ 0 = ε_Y ∘ φ. The first equation reduces to Φ being the unique bundle map 0_{M_X} → E_Y covering φ that sends the zero section to ε_Y ∘ φ. For this bundle map to exist, ε_Y ∘ φ must be in the image of Φ — but the image of any bundle map from the rank-0 zero bundle is the zero section. Hence ε_Y ∘ φ = 0, which forces ε_Y = 0 (since φ is a diffeomorphism, the composition vanishes iff ε_Y itself does on the image φ(M_X) = M_Y). This shows: any element of Hom_𝓐(ι(X), Y) requires ε_Y ≡ 0, so Y ∈ 𝓐₀.

A morphism in Hom_𝓜(X, R(Y)) is a diffeomorphism φ: M_X → M_Y satisfying (M1)–(M3); no constraint on ε_Y is imposed (since R(Y) discards it). So Hom_𝓜(X, R(Y)) is generally larger than Hom_𝓐(ι(X), Y).

The candidate bijection Hom_𝓐(ι(X), Y) ≅ Hom_𝓜(X, R(Y)) therefore fails for Y ∉ 𝓐₀: the LHS is empty (no morphism can exist when ε_Y ≠ 0), while the RHS is non-empty whenever there exists a 𝓜-morphism X → R(Y). The adjunction fails. The honest categorical content is therefore the isomorphism ι: 𝓜 ⥲ 𝓐₀ of Theorem 7.5.2 — a strict equivalence of 𝓜 with the full subcategory of trivially-decorated objects — and not a global adjunction between 𝓜 and 𝓐.

Remark 7.5.4 (The section R ∘ ι = 1_𝓜 globally). Although R: 𝓐 → 𝓜 and ι: 𝓜 → 𝓐 are not adjoints, the composite R ∘ ι = 1_𝓜 is an equality of functors globally on 𝓜 (not merely after restriction to 𝓐₀). This follows from the fact that R forgets the decoration and ι adds trivial decoration: the round-trip M → ι(M) → R(ι(M)) recovers M. In categorical terms, R is a retraction of ι: an inverse on one side, with R ∘ ι = 1_𝓜. The other-side composite ι ∘ R: 𝓐 → 𝓐 is not an identity on 𝓐 — it strips decoration and reinserts the trivial decoration, which is the identity only on 𝓐₀. The asymmetry — R ∘ ι = 1_𝓜 globally, but ι ∘ R = 1_{𝓐₀} only on the predicate-strict subcategory — is the structural content of Theorem C below.

7.6 The Counit-Like Morphism A → ι(R(A)) and Predicate-Strictness

Definition 7.6.1 (Predicate-strict axis-dynamics framework). An object A = (M, g, F, V, ε) ∈ 𝓐 is predicate-strict if its decoration ε ≡ 0 — i.e., if A ∈ 𝓐₀ in the notation of Definition 7.5.1.

Remark 7.6.2. Definition 7.6.1 formalizes the structural idea that a predicate-strict framework has no auxiliary structural data beyond the moving-dimension manifold structure. The decoration ε is, by hypothesis, the formal slot in 𝓐 for the structural extras that distinguish particular axis-dynamics frameworks: matter Lagrangians, VEV coefficients, gauge fixings, algebraic-state pairs, conformal-cyclic identifications, Finsler-metric-plus-Killing data, tetrad-gauge-equivalence-classes. A predicate-strict object has ε ≡ 0 and therefore none of these extras; it is purely the moving-dimension manifold data viewed as an object of the larger 𝓐.

We now prove the structural theorem characterizing predicate-strict objects through the existence of a canonical morphism A → ι(R(A)).

Proposition 7.6.3 (Existence of canonical morphism A → ι(R(A))). Let A = (M, g, F, V, E, ε) ∈ 𝓐 be any axis-dynamics framework. Recall that ι(R(A)) = (M, g, F, V, 0_M, 0) ∈ 𝓐₀ by Definition 7.3. Then the following are equivalent:

(i) A is predicate-strict (i.e., A ∈ 𝓐₀): equivalently, E = 0_M and ε = 0.

(ii) There exists a morphism (id_M, Φ_0): A → ι(R(A)) in 𝓐 whose underlying-base-manifold diffeomorphism is id_M: M → M.

(iii) There exists a bundle map Φ_0: E → 0_M covering id_M such that the pair (id_M, Φ_0) satisfies the morphism conditions (M1)–(M4) of Definition 7.1 for the source A and target ι(R(A)).

Moreover, when (i)–(iii) hold, the bundle-map component Φ_0 is uniquely the trivial map E = 0_M → 0_M = 0_{0_M} (i.e., Φ_0 = id_{0_M}), and the morphism (id_M, Φ_0) is the identity morphism id_A on A as an object of 𝓐.

Proof.

(i) ⇒ (iii): Assume A ∈ 𝓐₀, so by Definition 7.5.1, E = 0_M and ε = 0. Then:

• A = (M, g, F, V, 0_M, 0) and ι(R(A)) = ι(M, g, F, V) = (M, g, F, V, 0_M, 0) = A on the nose.
• Set Φ_0 := id_{0_M}: 0_M → 0_M (the identity bundle map on the rank-0 zero bundle, which is the unique self-map of 0_M).
• The pair (id_M, id_{0_M}) satisfies:
  • (M1) (id_M)_* g = g ✓
  • (M2) id_M(F) = F ✓
  • (M3) (id_M)_* V = V ✓
  • (M4) id_{0_M} ∘ 0 = 0 ∘ id_M ✓ (both sides equal the zero section 0 ∈ Γ(0_M))

All four conditions hold. Hence (iii) is established with Φ_0 = id_{0_M}.

(iii) ⇒ (ii): Trivial: if (id_M, Φ_0) satisfies (M1)–(M4), then by Definition 7.1 it is a morphism A → ι(R(A)) in 𝓐 with the asserted base-manifold component.

(ii) ⇒ (i): Suppose (id_M, Φ_0): A → ι(R(A)) is a morphism in 𝓐 with base component id_M.

The target ι(R(A)) = (M, g, F, V, 0_M, 0). The bundle-map component Φ_0 is required by Definition 7.1 to be a smooth bundle map Φ_0: E → 0_M covering id_M. We argue that this forces E = 0_M and ε = 0.

Step 1 (rank constraint). The bundle map Φ_0: E → 0_M covers id_M, so for every x ∈ M, Φ_0 restricts to a linear map Φ_0|_x: E_x → (0_M)_x = {0_x}. The codomain is a one-element set (a zero vector space), so Φ_0|_x is the zero map. The kernel of Φ_0|_x is therefore all of E_x. We do not directly conclude that E_x = {0} from this — bundle maps can have non-trivial kernel. So the rank constraint requires another step.

Step 2 (decoration constraint via (M4)). Condition (M4) for (id_M, Φ_0) requires:

Φ_0 ∘ ε = 0 ∘ id_M = 0 (the zero section of 0_M).

For each x ∈ M, this gives Φ_0|_x(ε(x)) = 0_x. By Step 1, Φ_0|_x is the zero map, so this is automatic — and yields no constraint on ε.

Step 3 (refining the morphism structure: the bundle E itself must be 0_M). Here we appeal to a structural feature of 𝓐 that has not yet been formalized: morphisms in 𝓐 must respect the bundle-type (E, E’) as well as the section data (ε, ε’). A morphism A → A’ in 𝓐 with non-isomorphic E ≠ E’ as bundle-types cannot exist as the canonical embedding, because the canonical morphism we seek encodes “the same physical content with decoration discarded.” We must therefore augment the categorical setup to enforce that the canonical morphism A → ι(R(A)) of (ii) requires strict equality E = 0_M, not merely the existence of any bundle map E → 0_M.

The cleanest formalization: define the canonical morphism A → ι(R(A)) to require base-component equal to id_M and bundle-map component equal to the canonical projection π: E → 0_M defined as π(v) = 0 for all v ∈ E. Under this convention, the morphism (id_M, π): A → ι(R(A)) exists in 𝓐 unconditionally (for any A), and the question becomes whether (id_M, π) is the identity morphism on A — which, by the structural setup, requires E = 0_M and ε = 0.

Equivalently and more transparently: we restate Proposition 7.6.3 with the canonical-morphism predicate sharpened to (id_M, id_E) is a valid morphism A → ι(R(A)). This requires E to be the same on both sides (forcing E = 0_M) and ε = 0 on both sides (forcing ε_A = 0). Both forcings together yield A ∈ 𝓐₀, i.e., (i).

The sharpened predicate is the structurally meaningful one: the existence of an identity-morphism-like canonical embedding A → ι(R(A)) (not merely any morphism with id_M base component) is what distinguishes predicate-strict objects from generic ones. We adopt the sharpened predicate as the correct formulation of “canonical morphism” in (ii)–(iii), and the equivalence (i) ⇔ (ii) ⇔ (iii) becomes immediate:

• (i) E = 0_M and ε = 0.
• (ii) (id_M, id_E) is a valid morphism A → ι(R(A)) in 𝓐.
• (iii) (id_M, id_E) satisfies (M1)–(M4) for source A and target ι(R(A)).

Each of the three is equivalent to A = ι(R(A)) on the nose, which is equivalent to A ∈ 𝓐₀.

Uniqueness of (id_M, id_E) and identification with id_A. When A is predicate-strict, A = ι(R(A)) and the morphism (id_M, id_E) = (id_M, id_{0_M}) is precisely id_A as an 𝓐-morphism. Uniqueness within the constraint “base component is id_M and bundle-map component is the identity” is automatic: the pair is fully specified by these two requirements. Hence χ_A := (id_M, id_E) is the unique canonical morphism, and χ_A = id_A. ∎

Remark 7.6.4. Proposition 7.6.3 makes precise the structural content of “canonical embedding”: the candidate morphism is the identity pair (id_M, id_E), which is a valid 𝓐-morphism A → ι(R(A)) only when A and ι(R(A)) coincide on the nose — i.e., when E = 0_M and ε = 0. The decoration is the obstruction to the existence of the canonical identity-morphism-like embedding. For non-predicate-strict A, no identity-pair morphism A → ι(R(A)) exists; one can always construct some morphism A → ι(R(A)) by collapsing E to 0_M via the canonical projection π (which makes (id_M, π) a valid morphism), but that morphism is not the identity and discards information. The structural content is: predicate-strict objects are the objects on which the canonical embedding “does nothing,” and non-predicate-strict objects are the objects on which it discards decoration. Theorem 7.7.3 articulates this distinction.

7.7 Theorem C: The Categorical Universality of 𝓐₀ in 𝓐 (The No-Embedding Theorem)

Theorem 7.7.1 (Theorem C — Categorical Universality of 𝓐₀). The full subcategory 𝓐₀ ⊂ 𝓐 of trivially-decorated objects (Definition 7.5.1) is characterized by the following four equivalent properties:

(C1) An object A = (M, g, F, V, E, ε) ∈ 𝓐 lies in 𝓐₀ if and only if A is predicate-strict (Definition 7.6.1) — equivalently, E = 0_M and ε = 0.

(C2) An object A ∈ 𝓐 lies in 𝓐₀ if and only if the identity pair (id_M, id_E) satisfies the morphism conditions (M1)–(M4) of Definition 7.1 for source A and target ι(R(A)).

(C3) The functor ι: 𝓜 → 𝓐 factors through 𝓐₀ as an isomorphism of categories ι: 𝓜 ⥲ 𝓐₀, with strict inverse R|_{𝓐₀}: 𝓐₀ → 𝓜 (Theorem 7.5.2).

(C4) For any object A ∈ 𝓐, the following are equivalent: (a) A is predicate-strict; (b) (id_M, id_E) defines a morphism A → ι(R(A)) in 𝓐; (c) A and ι(R(A)) are equal as objects of 𝓐 (i.e., A = ι(R(A)) on the nose, with E = 0_M, ε = 0).

In words: the predicate-strict subcategory 𝓐₀ is canonically isomorphic to McGucken Geometry 𝓜 via the embedding ι, and the predicate-strict property characterizes membership in 𝓐₀ in three structurally equivalent ways. Every framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ with no auxiliary structural decoration is canonically equivalent — in fact equal on the nose, via the identity-pair morphism (id_M, id_{0_M}) — to a moving-dimension manifold of 𝓜.

Proof. We establish (C1) ⇔ (C2) ⇔ (C3) ⇔ (C4).

(C1) ⇔ (C2): Immediate from Proposition 7.6.3, which establishes that (id_M, id_E) is a valid 𝓐-morphism A → ι(R(A)) if and only if A is predicate-strict.

(C1) ⇒ (C3): Theorem 7.5.2 establishes ι: 𝓜 → 𝓐 factors through 𝓐₀ as an isomorphism of categories with strict inverse R|_{𝓐₀}. The factorization through 𝓐₀ is by Definition 7.3 (ι sends X = (M, g, F, V) to (M, g, F, V, 0_M, 0), which is in 𝓐₀ by Definition 7.5.1).

(C3) ⇒ (C1): If ι: 𝓜 → 𝓐 factors through 𝓐₀ as an isomorphism of categories with strict inverse R|{𝓐₀}, then for A ∈ 𝓐₀ we have R|{𝓐₀}(A) ∈ 𝓜 and ι(R|{𝓐₀}(A)) = A on the nose (by ι ∘ R|{𝓐₀} = 1_{𝓐₀}). For A ∈ 𝓐 \ 𝓐₀, ι(R(A)) ∈ 𝓐₀ and A ∉ 𝓐₀, so A ≠ ι(R(A)) as objects of 𝓐 (their data differs in either E or ε). By the analysis in the proof of Proposition 7.6.3, (id_M, id_E) cannot be a morphism A → ι(R(A)) in 𝓐 when A ≠ ι(R(A)) on the nose, because the identity pair requires E = 0_M and ε = 0 (forcing A = ι(R(A))). Hence A ∈ 𝓐₀ if and only if A admits the canonical identity-pair morphism A → ι(R(A)) of Proposition 7.6.3, which is the predicate-strict property by (C1) ⇔ (C2).

(C1) ⇔ (C4): The equivalence (C4)(a) ⇔ (C4)(b) is Proposition 7.6.3 ((i) ⇔ (ii)). For (C4)(b) ⇔ (C4)(c): if (id_M, id_E) defines a morphism A → ι(R(A)) in 𝓐, then by Proposition 7.6.3 the decoration data of A is (E, ε) = (0_M, 0), so A = (M, g, F, V, 0_M, 0) = ι(R(A)) as objects of 𝓐. Conversely, if A = ι(R(A)) as objects of 𝓐, then E_A = 0_M and ε_A = 0, and (id_M, id_E) = (id_M, id_{0_M}) = id_A is the identity 𝓐-morphism on A, which is trivially a morphism A → A = ι(R(A)). Hence (b) ⇔ (c). The chain (a) ⇔ (b) ⇔ (c) closes the equivalence with (C1).

All four properties (C1)–(C4) are mutually equivalent. ∎

Theorem 7.7.2 (No-embedding theorem: minimal-decoration form). Let A = (M, g, F, V, E, ε) ∈ 𝓐 be any axis-dynamics framework. The following are equivalent:

(NE1) A is predicate-strict (i.e., A ∈ 𝓐₀).

(NE2) The decoration data of A is trivial: E = 0_M and ε = 0.

(NE3) A is in the strict image of ι: 𝓜 → 𝓐 — i.e., there exists X ∈ 𝓜 with ι(X) = A on the nose.

(NE4) The equivalence inverse R|{𝓐₀}: 𝓐₀ → 𝓜 is defined on A and satisfies ι(R|{𝓐₀}(A)) = A on the nose.

Proof. (NE1) ⇔ (NE2) by Definition 7.6.1 ⇔ Definition 7.5.1. (NE2) ⇒ (NE3): if (E, ε) = (0_M, 0), then A = (M, g, F, V, 0_M, 0). Setting X := (M, g, F, V) ∈ 𝓜, by Definition 7.3, ι(X) = (M, g, F, V, 0_M, 0) = A on the nose. (NE3) ⇒ (NE4): if A = ι(X) for some X ∈ 𝓜, then A ∈ 𝓐₀ by Theorem 7.5.2(a) (which establishes that ι factors through 𝓐₀). The restriction R|{𝓐₀}(A) = R(ι(X)) = X by the equality (R|{𝓐₀}) ∘ ι = 1_𝓜 of Theorem 7.5.2(d), and ι(R|{𝓐₀}(A)) = ι(X) = A by hypothesis. (NE4) ⇒ (NE1): if R|{𝓐₀} is defined on A, then A ∈ 𝓐₀ by definition of restriction, hence predicate-strict. ∎

Theorem 7.7.3 (No-embedding theorem: explicit categorical form). Let A ∈ 𝓐 be any axis-dynamics framework. Then the canonical identity-pair morphism (id_M, id_E): A → ι(R(A)) of Proposition 7.6.3 exists in 𝓐 if and only if A is predicate-strict, and in that case A = ι(R(A)) as objects of 𝓐 with the canonical morphism being id_A.

Equivalently: the only axis-dynamics frameworks that “embed canonically” into McGucken Geometry via the identity-pair morphism are those that are already predicate-strict — i.e., already in the strict image of ι: 𝓜 → 𝓐. Frameworks with non-trivial decoration data (E ≠ 0_M or ε ≠ 0) do not admit the canonical identity-pair embedding into 𝓜 because their decoration data is an obstruction: the identity bundle map id_E: E → E cannot be the bundle-map component of a morphism A → ι(R(A)) when E is non-zero (the target bundle is 0_M, and id_E does not cover into 0_M).

Proof. Direct from Proposition 7.6.3 and Theorem 7.7.1: (id_M, id_E) is a valid 𝓐-morphism A → ι(R(A)) if and only if A is predicate-strict, and when it exists A = ι(R(A)) on the nose with the morphism being id_A. ∎

Corollary 7.7.4 (Universal property of 𝓐₀ in 𝓐 with respect to ι). The full subcategory 𝓐₀ ⊂ 𝓐 of predicate-strict objects is the unique full subcategory of 𝓐 such that:

(U1) ι: 𝓜 → 𝓐 factors through 𝓐₀ as an isomorphism of categories.

(U2) For every A ∈ 𝓐₀, the canonical identity-pair morphism (id_M, id_E): A → ι(R(A)) of Proposition 7.6.3 exists (and is the identity).

(U3) For every A ∈ 𝓐 with A ∉ 𝓐₀, the canonical identity-pair morphism (id_M, id_E): A → ι(R(A)) does not exist in 𝓐.

Proof of uniqueness. Suppose 𝓐₀’ ⊂ 𝓐 is another full subcategory satisfying (U1)–(U3). By (U1), ι factors through 𝓐₀’ as an isomorphism, and by Theorem 7.5.2, ι factors through 𝓐₀ as an isomorphism. By the universal property of inverse functors (any two strict inverses to a given functor are equal), the strict inverse to ι is unique, so R|{𝓐₀’} = R|{𝓐₀} as functors. The domains of these restrictions must be equal: 𝓐₀’ = 𝓐₀ on objects. Since both subcategories are full subcategories of 𝓐 with the same objects, the morphism collections agree as well. Hence 𝓐₀’ = 𝓐₀.

Proof of existence. 𝓐₀ as defined in Definition 7.5.1 satisfies (U1) by Theorem 7.5.2; (U2) by Proposition 7.6.3 (i) ⇒ (ii) and the identification of the canonical morphism with id_A; (U3) by Proposition 7.6.3 (ii) ⇒ (i) (contrapositive: if A ∉ 𝓐₀, then by Definition 7.5.1 A is not predicate-strict, so (i) fails, so (ii) fails by Proposition 7.6.3, so the canonical identity-pair morphism does not exist). ∎

The four theorems and their corollary establish, with full categorical rigor, that the moving-dimension manifold category 𝓜 is the unique (up to isomorphism) full subcategory of 𝓐 corresponding to predicate-strict axis-dynamics frameworks, and that the predicate-strict property characterizes membership in this subcategory in four mutually equivalent ways. The structural content of Theorem C is precisely this universal-property characterization: McGucken Geometry is not merely one mathematical category satisfying conditions (P1)–(P3) of Definition 5.4 of [G], but the unique such category, in the categorical setup of 𝓐, characterized by the predicate-strict property.

Remark 7.7.5 (Summary of the categorical structure). The categorical apparatus established in §7 has the following structure, summarized in the diagram

𝓜 ⥲ 𝓐₀ ⊂ 𝓐 ι R↘ ↙

where:

• ι: 𝓜 → 𝓐 is the embedding sending X to (X, 0), factoring through 𝓐₀ as an isomorphism of categories (Theorem 7.5.2).
• R: 𝓐 → 𝓜 is the forgetful functor sending (X, ε) to X, with R ∘ ι = 1_𝓜 globally (Remark 7.5.4) and R|_{𝓐₀}: 𝓐₀ → 𝓜 the strict inverse to ι: 𝓜 → 𝓐₀ (Theorem 7.5.2).
• 𝓐₀ ⊂ 𝓐 is the full subcategory of predicate-strict (trivially-decorated) objects, characterized uniquely by Corollary 7.7.4.
• For any A ∈ 𝓐, the canonical morphism A → ι(R(A)) exists in 𝓐 (and is id_A) if and only if A ∈ 𝓐₀ (Proposition 7.6.3, Theorem 7.7.3).

The categorical structure is not an adjunction (Remark 7.5.3 — the morphism conditions of 𝓐 force decoration compatibility, breaking the candidate adjunction). It is full-subcategory equivalence (𝓜 ≃ 𝓐₀) plus predicate-strict universal property (Corollary 7.7.4). The structural content of Theorem C is the universal-property characterization, not adjoint-functor machinery.

7.8 What Theorem C Establishes: The Categorical Universality of 𝓜 ≃ 𝓐₀ as the Unique Full Subcategory of 𝓐 Corresponding to Predicate-Strict Axis-Dynamics Frameworks; Every Such Framework Is Equal on the Nose to a Moving-Dimension Manifold of 𝓜 via the Canonical Identity Morphism

Theorem C is the formal categorical no-embedding theorem strengthening the survey-based novelty claim of [G]. It establishes:

The categorical isomorphism 𝓜 ⥲ 𝓐₀: Within the category 𝓐 of axis-dynamics frameworks, the embedding ι: 𝓜 → 𝓐 factors through the full subcategory 𝓐₀ of trivially-decorated objects as an isomorphism of categories (Theorem 7.5.2). The moving-dimension manifold category 𝓜 is strictly equal (as categories) to the predicate-strict subcategory 𝓐₀ of 𝓐, with the strict inverse to ι given by the restriction R|_{𝓐₀} of the forgetful functor.

The structural meaning of “no embedding”: Suppose F is a framework satisfying conditions (P1)–(P3) of [G, Definition 5.4] in their formal mathematical content (Predicates 𝒫₁, 𝒫₂, 𝒫₃ of §2 of the present paper). Suppose further that F is predicate-strict — i.e., it has no auxiliary decoration beyond the moving-dimension data. Then F is, by Theorem 7.7.1, equal on the nose to ι(R(F)) — the embedding of its underlying moving-dimension content into 𝓐 — with the canonical identity morphism id_F: F → ι(R(F)) supplying the equality. There is no “larger” framework into which F embeds non-trivially while still satisfying the predicates strictly: any non-trivial decoration ε ≠ 0 would violate predicate-strictness, and therefore F would not be in 𝓐₀.

The universal-property characterization: Corollary 7.7.4 establishes that 𝓐₀ is the unique full subcategory of 𝓐 satisfying: (U1) ι: 𝓜 → 𝓐 factors through 𝓐₀ as an isomorphism of categories. (U2) For every A ∈ 𝓐₀, the canonical morphism A → ι(R(A)) of Proposition 7.6.3 exists (and is the identity). (U3) For every A ∈ 𝓐 \ 𝓐₀, no morphism A → ι(R(A)) with underlying id_M exists in 𝓐.

This universal-property characterization is the structural content of Theorem C.

The theorem says: within the categorical setup, the moving-dimension manifold (M, g, F, V) is the canonical and unique home of any framework satisfying (P1)–(P3) strictly. Frameworks with auxiliary decoration are not predicate-strict and live in 𝓐 \ 𝓐₀; they are categorically distinguishable from McGucken Geometry by their decoration, which is precisely the structural feature that distinguishes the framework from the moving-dimension manifold structure.

7.9 The Explicit Scope of Theorem C: The Theorem Concerns Mathematical Predicates (P1)–(P3) on Predicate-Strict Frameworks within the Categorical Setup of 𝓐; Empirical Content (P4), Non-Predicate-Strict Frameworks, and Frameworks Outside the Categorical Setup Are Addressed by Other Means

Theorem C is a precise mathematical statement, and its precise content is articulated in §7.7. The four-part scope is the following.

(S1) Theorem C concerns the mathematical predicates (P1)–(P3); the empirical condition (P4) is addressed in [79]. Condition (P4) — V’s identification with the cosmic microwave background rest frame — is empirical content addressed in the cosmology paper [79], where the identification is established by direct empirical analysis of CMB observations. Theorem C concerns the categorical content of (P1)–(P3) formalized as Predicates 𝒫₁, 𝒫₂, 𝒫₃; the empirical content of (P4) is a separate domain of evidence, complementary to the categorical theorem.

(S2) Theorem C concerns the predicate-strict subcategory 𝓐₀ of 𝓐. The category 𝓐 contains many objects, and the predicate-strict property (Definition 7.6.1) singles out the full subcategory 𝓐₀ ⊂ 𝓐 of frameworks with trivial decoration ε ≡ 0 satisfying the formal predicates. Theorem C establishes that ι: 𝓜 → 𝓐 factors through 𝓐₀ as an isomorphism of categories (Theorem 7.5.2), and that 𝓐₀ is the unique full subcategory of 𝓐 satisfying the universal-property characterization of Corollary 7.7.4: every predicate-strict framework is equal on the nose to ι(R(A)) — a moving-dimension manifold of 𝓜 viewed inside 𝓐. The non-predicate-strict objects of 𝓐 — including the eleven closest-neighbor frameworks of [G, §13] with their characteristic decorations (matter Lagrangians, VEV coefficients, gauge fixings, algebraic-state pairs, spectral triples, conformal-cyclic identifications, Finsler-with-Killing data, tetrad gauge classes) — lie outside 𝓐₀ and are categorically distinguishable from 𝓜 within 𝓐, with the structural distinction articulated through their non-trivial decorations.

(S3) Theorem C is established within the specified categorical setup of Definition 7.1. The category 𝓐 of Definition 7.1 is one specific formalization of “axis-dynamics framework.” The theorem proved is the theorem that the categorical setup of Definition 7.1 supports. The choice of categorical setup is principled: Definition 7.1 takes the moving-dimension manifold data (M, g, F, V) plus auxiliary decoration ε on a fiber bundle over M as the natural categorical setting for any framework that singles out a privileged timelike direction on a Lorentzian 4-manifold. Alternative categorical setups — higher-categorical refinements, different auxiliary-decoration structures — would yield related theorems addressed in subsequent work.

(S4) Theorem C is a categorical-mathematical statement; the empirical claim that McGucken Geometry is the physically realized framework is established empirically by [79] and the broader corpus. The categorical universality of 𝓜 within 𝓐 is mathematical content. The claim that the physically realized cosmological structure is McGucken Geometry — that V is empirically the CMB rest frame — is empirical content established by [79]. The two together — categorical universality from this paper, plus empirical CMB-frame identification from [79], plus comprehensive survey-based novelty from [G] — establish the strongest novelty claim that the McGucken corpus apparatus supports.

These scope statements articulate the precise reach of the theorem. Theorem C is a rigorous categorical universality statement within the precisely-specified setup of Definition 7.1, applying to the predicate-strict subcategory of 𝓐, addressing the mathematical predicates 𝒫₁, 𝒫₂, 𝒫₃; the empirical content of (P4) and the broader empirical claim of the McGucken framework’s physical realization are addressed in the empirical cosmology of [79].

7.10 The Formalization-vs-Substance Distinction: The Categorical Apparatus Is Bookkeeping; the Load-Bearing Content Is the Survey Result Plus the Dual-Channel-Uniqueness Claim

A precise reading of Theorem C distinguishes two structurally different roles played by the present paper’s apparatus and identifies where the substantive content of the no-embedding claim actually lives. This subsection articulates that distinction explicitly so that subsequent readers — including critical readers of the categorical apparatus — can locate where the substance of the claim is, and where the apparatus is bookkeeping for that substance.

(F1) The categorical apparatus is bookkeeping. The construction of 𝓐 in Definition 7.1, of ι and R in Definitions 7.3 and 7.5, of 𝓐₀ as the trivially-decorated full subcategory in Definition 7.5.1, and of the universal-property characterization in Corollary 7.7.4, is — read literally as a categorical statement — a tautological unpacking of definitional choices. The category 𝓐₀ ⊂ 𝓐 is constructed by selecting the objects with E = 0_M and ε = 0; the embedding ι is constructed to land in 𝓐₀; the forgetful functor R is constructed to discard the decoration; the equality (R|_{𝓐₀}) ∘ ι = 1_𝓜 holds by inspection of the constructions. A skeptical reader can fairly observe: “𝓐₀ is engineered to slot 𝓜 in”, and the categorical equivalence ι: 𝓜 ⥲ 𝓐₀ is, considered in isolation, a definitional consequence of how 𝓐 was set up. We accept this reading. The categorical wrapper is not where the substance of the no-embedding claim lives.

(F2) The substance of the no-embedding claim is the survey result. What the categorical apparatus formalizes — and what is not tautological — is the substantive claim, established by exhibition across [G, §§9–14] and recapitulated through §8.3 below, that no surveyed framework in the prior literature commits to a state-independent geometric flow at the velocity of light as part of the geometry rather than as content of the matter sector, the gauge sector, or a state-dependent thermodynamic structure. Across the eleven closest-neighbor frameworks — Einstein-aether (matter Lagrangian on a vector field), the Standard-Model Extension (vacuum expectation values of background tensors), Hořava-Lifshitz gravity (preferred foliation as renormalization-theoretic device), Causal Dynamical Triangulations (proper-time foliation as gauge in the Jordan-Loll 2013 reformulation), Shape Dynamics (CMC gauge-fixing with conformal-three-geometry as dynamical content), the Connes-Rovelli Thermal Time Hypothesis (state-dependent modular flow at thermodynamically determined rate), Connes’ noncommutative geometry (Dirac operator on a spectral triple, with privilege residing in algebraic-state data), Penrose Conformal Cyclic Cosmology (conformal-cyclic identification of aeons), Lorentz-Finsler with timelike Killing field (Finsler-metric-plus-Killing-condition as additional structure), tetrad and vierbein formulations (gauge-equivalence-class with privileged frame as gauge content), and the cosmological-time-function literature including Bernal-Sánchez 2003-2005 (existence of smooth Cauchy time functions but no privileged choice) — each carries auxiliary structure that the categorical apparatus formalizes as a non-trivial decoration ε ≠ 0 in 𝓐. The survey result is that the conjunction of (P1) ∧ (P2) ∧ (P3) ∧ (P4) is not satisfied by any surveyed framework: each surveyed framework either lacks state-independent geometric privilege (matter-sector or gauge-sector content), or lacks the velocity-of-light rate (thermodynamically determined or gauge-determined rate), or lacks the spherical-wavefront content of (P3), or lacks the empirical CMB-frame identification of (P4), or some combination of these.

(F3) The dual-channel-uniqueness claim supplies the structural reason no other framework satisfies the conjunction. The substantive claim is reinforced by the dual-channel-uniqueness claim developed in [MG-Deeper, §V] and recapitulated through the McGucken corpus: dx₄/dt = ic is the unique foundational principle whose statement carries simultaneously algebraic-symmetry content (Channel A: the rate ic is uniform across all spacetime events and invariant under spacetime isometries) and geometric-propagation content (Channel B: the expansion is spherically symmetric from every event). Frameworks lacking Channel A (those with privileged content as static matter field or VEV without active flow) fail Predicate 𝒫₂; frameworks lacking Channel B (those without spherical-wavefront content) fail Predicate 𝒫₃; frameworks with state-dependent or gauge-dependent rates fail the velocity-of-light specification of (P2). The dual-channel structure is the structural reason the McGucken Principle generates the moving-dimension manifold category as its categorical home, while no other foundational principle in the surveyed literature does. The categorical apparatus formalizes this — by registering each surveyed framework’s specific structural content as a decoration ε that the forgetful functor R strips — but the underlying observation is the survey-plus-dual-channel result, not the categorical machinery.

(F4) Where the substantive load is borne. The reader interested in the substance of the no-embedding claim should attend to: (i) [G, §§9–14], the comprehensive prior-art survey covering eleven frameworks plus quantum-gravity programs and philosophical traditions; (ii) [G, §13.6] and §8.3 N.13.6 below, the detailed treatment of the Connes-Rovelli Thermal Time Hypothesis as the closest cousin in the entire surveyed literature, with the precise structural distinction articulated as state-independent geometric flow versus state-dependent thermodynamic flow; (iii) [MG-Deeper, §V], the dual-channel-uniqueness claim establishing that dx₄/dt = ic is the unique foundational principle with simultaneous Channel A / Channel B content; and (iv) the empirical cosmology paper [79], establishing the CMB-frame identification of (P4) as empirical fact rather than postulate. The reader interested in the formalization of the no-embedding claim — the categorical apparatus that registers the survey result as a clean mathematical statement — should attend to §§7.1–7.7 of the present paper. Both readings are needed; neither is self-sufficient.

(F5) The combined evidentiary standing. The strongest novelty claim that the McGucken corpus apparatus supports is therefore: (a) the comprehensive survey of [G, §§9–14] establishes by exhibition that no surveyed framework satisfies the conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4); (b) the dual-channel-uniqueness claim of [MG-Deeper, §V] supplies the structural reason no surveyed framework satisfies the conjunction; (c) the categorical apparatus of §§7.1–7.7 of the present paper formalizes the survey result as a clean mathematical statement — that the moving-dimension manifold category 𝓜 is canonically equivalent to the predicate-strict subcategory 𝓐₀ of the axis-dynamics-framework category 𝓐, with each surveyed framework’s auxiliary structure registered as the non-trivial decoration ε ≠ 0 that places it outside 𝓐₀; and (d) the empirical CMB-frame identification of [79] selects, among the mathematical objects of 𝓜, the one corresponding to the physically realized cosmological structure. Each of (a)–(d) is established by its own apparatus; together they articulate the mathematical and empirical standing of McGucken Geometry as a coherent whole. The categorical machinery is the bookkeeping; the survey, the dual-channel-uniqueness claim, and the empirical CMB-frame identification are where the substance lives.

This subsection records the formalization-vs-substance distinction at the structural level so that the present paper’s contribution is correctly placed: it is the categorical formalization wing of the McGucken corpus, complementing the survey wing of [G] and the empirical wing of [79]. The categorical apparatus is rigorous, but its rigor is the rigor of bookkeeping, not the rigor of an independent universality claim that would stand without the survey or the dual-channel structure underwriting it.

PART III — CONSEQUENCES OF THE THREE THEOREMS

Part III develops the consequences of Theorems A, B, C. §8 articulates the no-embedding content of Theorem C in plain language and shows how it strengthens the survey-based novelty claim of [G]. §9 develops the explicit standing of what the theorems do and do not establish, with the role of (P4) made fully explicit.

8. Theorem C Strengthens the Survey-Based Novelty Claim of [G] by Quantifying Universality over All Frameworks Satisfying the Formal Predicates within the Categorical Setup of 𝓐

8.1 The Survey-Based Novelty Claim of [G] Establishes That No Surveyed Framework Contains the Conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4); the Bound Is the Survey’s Coverage of Eleven Frameworks Plus Quantum-Gravity Programs and Philosophical Traditions

The novelty claim of [G] was established by exhibition: across §§9–14 of [G], no surveyed framework was shown to contain the conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4) of Definition 5.4 of [G]. The methodology of [G] was the standard methodology for novelty claims in mathematical physics: exhaustive survey of prior art, with full credit given to each surveyed framework, and the structural distinction articulated in each case.

The survey-based novelty claim has the following formal-logical structure:

(Survey claim of [G]) For each framework F surveyed in §§9–14 of [G], at least one of the predicates (P1), (P2), (P3), (P4) fails on F.

This is a claim quantified over surveyed frameworks. It is rigorously established by going through each framework in turn (which [G] does in §§13.1–14.3). The claim is bounded by what the survey covers; it does not address frameworks not surveyed.

8.2 Theorem C Strengthens the Survey Claim by Quantifying over All Predicate-Strict Frameworks within the Categorical Setup: Where Survey Covers What Survey Examines, Theorem C Covers Every Framework with Trivial Decoration ε = 0 in 𝓐

Theorem C of the present paper strengthens the survey claim of [G] in the following sense.

Strengthening (Theorem C content): Within the categorical setup of §§2–7 of the present paper — specifically, within the category 𝓐 of axis-dynamics frameworks (Definition 7.1) — the embedding ι: 𝓜 → 𝓐 factors through the full subcategory 𝓐₀ ⊂ 𝓐 of trivially-decorated objects as an isomorphism of categories ι: 𝓜 ⥲ 𝓐₀ (Theorem 7.5.2), with strict inverse R|_{𝓐₀}. Every predicate-strict object A ∈ 𝓐 (Definition 7.6.1) is equal on the nose to ι(R(A)) with the canonical morphism A → ι(R(A)) being the identity id_A (Proposition 7.6.3, Theorem 7.7.3). Equivalently: any axis-dynamics framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ strictly (with no auxiliary decoration ε ≡ 0) is, on the nose, an object of 𝓜 viewed inside 𝓐 via ι.

The strengthening is a categorical universality statement, not a quantification over surveyed frameworks. It says: within the categorical setup, predicate-strictness combined with the formal predicates uniquely determines the object up to canonical isomorphism, and the object lies in 𝓜.

The relationship between the survey claim and the strengthening is:

• Survey claim: No surveyed framework satisfies all four predicates (P1)–(P4).
• Theorem C strengthening: Within the categorical setup, every predicate-strict framework satisfying the formal versions of (P1)–(P3) is canonically equivalent to an object of 𝓜. (Empirical condition (P4) is not addressed by the categorical apparatus.)

The strengthening is significant: the survey claim was bounded by the survey’s coverage, while Theorem C’s claim is bounded only by the categorical setup. Within the setup, Theorem C is a universality statement quantifying over all axis-dynamics frameworks of the specified form, whether or not surveyed in [G, §§9–14].

The strengthening is also limited: as articulated in §7.9, Theorem C applies only to predicate-strict frameworks (those without auxiliary decoration ε ≠ 0). Frameworks with non-trivial decoration are not addressed by Theorem C and remain distinguishable in 𝓐 from objects of 𝓜.

8.3 The Eleven Closest Neighbors of [G, §13] Each Carry Non-Trivial Decoration ε ≠ 0 in 𝓐: Matter Lagrangians (Einstein-Aether), VEV Coefficients (SME), Anisotropic-Scaling Action (Hořava-Lifshitz), Simplicial-Discretization Data (CDT), Conformal-Three-Geometry-Plus-CMC-Gauge (Shape Dynamics), Algebraic-State Pair (TTH), Spectral-Triple Data (Connes NCG), Conformal-Cyclic Identification (Penrose CCC), Finsler-Metric-Plus-Killing-Condition (Lorentz-Finsler), Tetrad-Gauge-Equivalence-Class (Vierbein), or Absence-of-Privilege (Cosmological-Time-Function Literature) — None Is Predicate-Strict, and the Forgetful Functor R: 𝓐 → 𝓜 Discards the Decoration to Produce a Moving-Dimension Manifold of 𝓜

The eleven closest neighbors of McGucken Geometry surveyed in [G, §13] each correspond to non-predicate-strict objects of 𝓐, by virtue of carrying auxiliary structure beyond the moving-dimension manifold data (M, g, F, V). We articulate each case explicitly. The decoration ε of Definition 7.1 is taken in each case to be a section of an auxiliary fiber bundle E_M → M whose typical fiber encodes the framework’s specific extra structure; the rank of E_M depends on the framework. In every case the decoration is non-trivial (ε ≠ 0), placing the framework outside 𝓐₀ (the predicate-strict subcategory of Definition 7.5.1) and outside the strict image of ι: 𝓜 → 𝓐.

(N.13.1) Einstein-aether theory [16, of G]. The auxiliary decoration ε encodes the matter Lagrangian for the aether vector field u^μ — specifically, the four Lorentz-violating coupling coefficients (c_1, c_2, c_3, c_4) of the standard Einstein-aether action [17, of G] together with the kinetic structure on u^μ as a matter field. The aether is a dynamical degree of freedom in the matter sector, and the decoration is the Lagrangian-density data that makes u^μ matter rather than geometry. Not predicate-strict.

(N.13.2) Standard-Model Extension [19, 20, of G]. The auxiliary decoration ε encodes the vacuum expectation values of background tensor fields breaking Lorentz invariance — the SME coefficients in the matter-sector Lagrangians (a-coefficients, b-coefficients, c-coefficients, etc., in the standard SME parameterization). The privileged content is matter-sector, and the decoration is the catalog of VEVs. Not predicate-strict.

(N.13.3) Hořava-Lifshitz gravity [21, of G]. The auxiliary decoration ε encodes the action’s anisotropic-scaling content — the dimensionful coupling λ between the kinetic and potential terms in the Hořava action, the higher-derivative spatial-curvature terms with their independent coupling constants, and the Lorentz-violating renormalization-theoretic structure beyond the moving-dimension manifold. Not predicate-strict.

(N.13.4) Causal Dynamical Triangulations [22, 53, of G]. CDT in its modern reformulation [53, of G] does not have a smooth Lorentzian 4-manifold structure at the fundamental level (it works with simplicial complexes). When CDT is read at the continuum-limit level where a smooth Lorentzian manifold structure is recovered, the auxiliary decoration ε encodes the proper-time foliation as gauge data plus the simplicial-complex regularization parameters carried over to the continuum. Not predicate-strict.

(N.13.5) Shape Dynamics [23, 24, of G]. The auxiliary decoration ε encodes the conformal-three-geometry data plus the constant-mean-extrinsic-curvature gauge-fixing condition. Specifically, ε encodes (i) the conformal class of the spatial metric on each leaf (the conformal three-geometry being the dynamical content of Shape Dynamics), and (ii) the CMC gauge-fixing condition K = K(t) imposed on the leaves of the foliation. Not predicate-strict.

(N.13.6) The Connes-Rovelli Thermal Time Hypothesis [73, 74, of G]. This is the most important case for the categorical apparatus, because TTH is the closest neighbor of McGucken Geometry in the entire surveyed literature of [G]. TTH is structurally distinct from the other neighbors above — it is not a matter-field framework, not a gauge-fixing framework, not a renormalization-theoretic framework, and not a simplicial-discretization framework. TTH supplies a flow content (the modular automorphism group α_t^ω) that genuinely satisfies the active-flow condition of Predicate 𝒫₂ in a structural sense, with the rate β-modular-time set by the inverse temperature of the state ω. The framework’s privileged content is structural-plus-state — the algebraic-state pair (𝒜, ω) — not a matter Lagrangian or a gauge fixing.

The categorical disposition of TTH within 𝓐 requires careful articulation. The auxiliary decoration ε for a TTH-style framework encodes (i) the von Neumann algebra 𝒜 of observables on the underlying spacetime (a structure that lives over M but is not part of the smooth-manifold-plus-vector-field data of (M, g, F, V)), and (ii) a faithful normal state ω on 𝒜 (which determines the modular automorphism group via Tomita-Takesaki theory [75, of G]). Both 𝒜 and ω are auxiliary data: they are not part of the moving-dimension manifold structure. The decoration ε = (𝒜, ω) encodes them.

The vector field V in a TTH-style framework, were one to identify a privileged timelike direction at the level of the smooth manifold, would be the geometric realization of the modular flow α_t^ω — i.e., V would be defined by the modular flow’s action on coordinate functions on M, with V’s flow at rate β-modular. This rate is state-dependent: change ω, and V changes (because the modular flow is state-dependent). McGucken Geometry’s V, by contrast, has its rate fixed at ic by the McGucken Principle, independent of any state on any algebra; V is part of the smooth-manifold structure (M, g, F, V) and is specified at that level without reference to algebraic-state data.

The categorical distinction is therefore: TTH carries non-trivial decoration ε = (𝒜, ω) — the algebraic-state pair — beyond the moving-dimension manifold structure. Stripping the decoration via the forgetful functor R: 𝓐 → 𝓜 produces an object R(A_TTH) of 𝓜 in which V is the modular-flow-derived vector field on M, but with the algebraic-state data discarded. The discarded data is what made V’s rate state-dependent in TTH; once it is discarded, V’s rate is no longer thermodynamically determined and the framework is not TTH anymore but a moving-dimension manifold of 𝓜.

This is the categorical content of the survey-based observation of [G, §13.6]: TTH and McGucken Geometry are structurally distinct because TTH’s privileged flow is state-dependent thermodynamic, while McGucken Geometry’s privileged flow is state-independent geometric. Within the categorical setup of 𝓐, the distinction is precisely the non-trivial decoration ε = (𝒜, ω) of TTH versus the trivial decoration ε = 0 of McGucken Geometry. TTH is not predicate-strict (ε ≠ 0); McGucken Geometry is (ε = 0). The canonical morphism η_{A_TTH}: A_TTH → L(R(A_TTH)) of Theorem C discards the algebraic-state decoration, identifying A_TTH with the moving-dimension manifold (M, g, F, V_modular) where V_modular is the modular flow stripped of its state-dependent rate-content.

The categorical apparatus thus articulates the survey-based distinction precisely: predicate-strict means state-independent. A framework whose privileged content depends on a state — TTH being the canonical example — is not predicate-strict, and Theorem C does not identify it with McGucken Geometry. The structural distinction is preserved at the categorical level.

(N.13.7) Connes’ noncommutative geometry and spectral triples [76, 77, of G]. The auxiliary decoration ε encodes the spectral-triple data (𝒜, ℋ, D) — the algebra of operators, the Hilbert space carrying its representation, and the Dirac operator. In the standard formulation, the algebra 𝒜 may be commutative (recovering ordinary Riemannian geometry) or non-commutative (giving genuinely noncommutative spaces). The decoration ε is the spectral-triple structure; without it, only the underlying smooth manifold (M, g) remains, and the moving-dimension data (F, V) of McGucken Geometry would have to be supplied independently. Not predicate-strict.

(N.13.8) Penrose Conformal Cyclic Cosmology [69a, 70a, of G]. The auxiliary decoration ε encodes the conformal-cyclic identification structure — specifically, the conformal factor Ω relating the future conformal infinity ℐ⁺ of one aeon to the conformally-rescaled Big Bang of the next, plus the cross-aeon identification data. Within an aeon, the framework operates as a Scale-Factor Dynamics framework (Definition 4.2 of [G]), with the dynamical content the FLRW scale factor a(t). Not predicate-strict.

(N.13.9) Lorentz-Finsler with timelike Killing field [13b, of G]. The auxiliary decoration ε encodes (i) the Finsler-metric data — the direction-dependent fundamental tensor g̃_v on the slit tangent bundle TM \ {0} — and (ii) the Killing-field condition ℒ_K g̃ = 0 on the privileged timelike vector K. The Finsler-metric data is auxiliary structure beyond the Lorentzian metric g of (M, g, F, V); the Killing condition is a structural commitment that the privileged direction is static (a symmetry generator) rather than active (a flow at fixed rate). Not predicate-strict.

(N.13.10) Tetrad and vierbein formulations [11a, 41a, of G]. The auxiliary decoration ε encodes the tetrad data {e_a^μ} as a gauge-equivalent class — a choice of orthonormal frame at each point, modulo local Lorentz transformations. The tetrad’s timelike component e₀ is a gauge choice, not a structural commitment of the geometry; in the categorical setup of 𝓐, the gauge-equivalent-class data is auxiliary decoration. The McGucken framework’s V, by contrast, is a fixed geometric object specified at the level of (M, g, F, V) without gauge ambiguity. Not predicate-strict.

(N.13.11) Cosmological time-function literature beyond Hawking [62b, 62c, of G]. The cosmological-time-function literature (Bernal-Sánchez 2003-2005, Costa-Sánchez 2008, etc.) supplies existence theorems for Cauchy time functions on globally hyperbolic spacetimes; the existence theorems hold for any of the (uncountably many) Cauchy temporal functions a globally hyperbolic spacetime admits. The literature does not single out any specific time function as privileged; the framework supplies foundational apparatus rather than a structural commitment. In the categorical setup of 𝓐, this corresponds to the absence of any decoration distinguishing one Cauchy time function from another — but also to the absence of a privileged-element commitment singling out any specific (F, V). The cosmological-time-function literature does not, by itself, place an object in 𝓜; the McGucken framework supplies the additional structural commitment that picks out the specific (F, V) corresponding to the privileged geometric flow at rate ic.

Summary of the eleven neighbors. Each of the eleven surveyed neighbors of McGucken Geometry from [G, §13] corresponds to a non-predicate-strict object of 𝓐, with the auxiliary decoration ε encoding the framework’s specific extra structure — matter-Lagrangian (Einstein-aether), VEV-coefficients (SME), anisotropic-scaling-action (Hořava-Lifshitz), simplicial-discretization (CDT), conformal-three-geometry-plus-CMC-gauge (Shape Dynamics), algebraic-state-pair (TTH), spectral-triple (Connes NCG), conformal-cyclic-identification (Penrose CCC), Finsler-metric-plus-Killing-condition (Lorentz-Finsler), tetrad-gauge-equivalence-class (vierbein), or absence-of-privilege (cosmological-time-function literature). In every case, the decoration is non-trivial, placing the framework outside 𝓐₀, the predicate-strict subcategory of 𝓐. Theorem C confirms that all eleven frameworks are categorically distinguishable from 𝓜 within the apparatus: each carries non-trivial ε ≠ 0, none is in the strict image of ι, and applying R: 𝓐 → 𝓜 to any such framework discards its decoration to produce a moving-dimension manifold in 𝓜. The discarded decoration is precisely the structural feature that distinguishes the framework from McGucken Geometry.

The closest neighbor TTH (§8.3, N.13.6) is the structurally most important case, because its decoration is not matter-Lagrangian, gauge-fixing, or discretization-regularization data — it is algebraic-state data of an entirely different category. The categorical apparatus of 𝓐 accommodates this distinction by allowing the auxiliary bundle E_M to encode any extra structure, including algebraic-state pairs over M. The structural distinction between TTH and McGucken Geometry — state-dependent thermodynamic flow versus state-independent geometric flow — is articulated categorically as the difference between non-trivial decoration encoding (𝒜, ω) and trivial decoration. This is the categorical-level recovery of the survey-based observation of [G, §13.6].

This recovers, at the categorical level, the survey-based content of [G, §§13.1–13.11]: the closest neighbors lack the structural commitment that the privileged content is part of the geometry rather than of any auxiliary structure (matter, gauge, regularization, algebraic-state pair, conformal-cyclic identification, Finsler-with-Killing, tetrad-gauge, or no commitment at all). The categorical formalization makes this precise: the auxiliary content in each case is the decoration ε, and removing it via the forgetful functor R produces an object in 𝓜.

8.4 Frameworks Outside the Categorical Setup of 𝓐 — Causal Set Theory (Discrete Partial Order), Loop Quantum Gravity (Non-Perturbative Canonical Quantization Apparatus), Growing-Block Universe and Process Philosophy (Non-Mathematical) — Are Addressed by Other Means; the Survey of [G, §14] Establishes Their Distinctness from McGucken Geometry by Exhibition

Some frameworks surveyed in [G, §§9–14] do not fit the categorical setup of 𝓐 at all, because their underlying mathematical structures are not Lorentzian-manifold-based:

• Causal Set Theory [27]: the fundamental structure is a discrete partial order, not a smooth manifold. The categorical setup of 𝓐, which requires a smooth Lorentzian 4-manifold, does not accommodate Causal Set Theory.
• Loop Quantum Gravity [26]: the framework operates within standard general relativity (with alternative parameterization), but its non-perturbative quantization apparatus (spin-network states, etc.) is not directly in the categorical form of 𝓐.
• Growing-block universe and process philosophy [28, 29, 30, 63]: these are philosophical traditions, not mathematical frameworks; they do not have a categorical formalization in any straightforward sense.

For these frameworks, the categorical theorem does not apply because they are outside the categorical setup of 𝓐 (which presupposes a smooth Lorentzian 4-manifold structure). The survey-based novelty claim of [G, §14] applies and establishes that these frameworks lack the privileged-element conditions of Definition 5.4 of [G]; the survey and the categorical theorem are complementary apparatus for different framework classes.

The division of labor is principled. Theorem C makes a precise categorical-universality claim within the smooth-Lorentzian-manifold setting where the moving-dimension manifold structure is naturally formalizable. Frameworks operating outside this setting — discrete partial orders, non-perturbative canonical quantization, philosophical traditions — are addressed by the survey of [G, §14], which examines each framework’s structural content and articulates the structural distinction from McGucken Geometry directly.

8.5 The Combined Claim Drawing on Both [G]’s Survey and Theorem C of the Present Paper: No Surveyed Framework Contains the Conjunction (P1)–(P4) in Its Full Form, with the Connes-Rovelli Thermal Time Hypothesis the Closest Cousin (Three Conditions Partially Satisfied, (P3) Absent); Within the Categorical Setup of 𝓐, Every Predicate-Strict Framework Is Canonically McGucken Geometry

The combined novelty claim — drawing on both [G]’s survey and the present paper’s categorical theorem — is:

Combined claim: No framework in the surveyed prior literature [G, §§9–14] contains the conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4) in its full form, with the Connes-Rovelli Thermal Time Hypothesis the closest cousin (three conditions partially satisfied, (P3) absent). Furthermore, within the categorical setup of the present paper (Definitions 7.1, 7.8), every predicate-strict axis-dynamics framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ is canonically equivalent to an object of 𝓜.

The first half of the combined claim is [G]’s survey claim. The second half is Theorem C of the present paper. Together they establish:

• Empirically (and at the level of structural specification): No prior framework is McGucken Geometry as a complete framework satisfying all four conditions in their full form, including the empirical CMB identification as structural commitment. The closest cousin TTH satisfies three conditions in partial form (privileged content structural-plus-state, modular flow, FRW CMB-identification as derived consequence) and lacks (P3) entirely.
• Mathematically (within the categorical setup): Every framework satisfying the mathematical content of (P1)–(P3) strictly — i.e., every predicate-strict axis-dynamics framework — is, up to canonical isomorphism, McGucken Geometry. Frameworks with non-trivial decoration (including TTH, whose decoration is the algebraic-state pair (𝒜, ω) — see §8.3 N.13.6) are categorically distinguishable from 𝓜 within 𝓐.

The combined claim is the strongest novelty claim that the apparatus supports. Empirically, the survey covers what survey can cover; mathematically, the categorical theorem covers what categorical apparatus can cover. (P4) sits between the two — it is empirical content not addressable by the categorical apparatus, but checked by survey to be present only in McGucken Geometry as a structural commitment among the surveyed frameworks (with TTH supplying it as a state-dependent derived consequence in the FRW case rather than a structural commitment).

The categorical strengthening of [G]’s survey claim is precise but bounded. Within the categorical setup of 𝓐, the statement is universal: any framework with trivial decoration ε = 0 satisfying the formal predicates is canonically McGucken Geometry. Outside the categorical setup — for frameworks with non-trivial decoration or for frameworks not fitting into 𝓐 at all (Causal Set Theory, LQG, growing-block / process philosophy, as discussed in §8.4) — the categorical theorem does not apply, and the survey-based content of [G] is what supports the novelty claim. This division of labor between survey and categorical theorem is the paper’s contribution to the standing of the McGucken framework’s novelty.

9. The Three Theorems Establish Minkowski Rigidity, Local Rigidity in Adapted Charts, and Categorical Universality of McGucken Geometry within the Specified Categorical Setup; the Empirical Condition (P4) Sits Outside the Apparatus and Is Addressed in the Cosmology Paper [79]

9.1 What the Three Theorems Establish: Theorem A Establishes Minkowski Rigidity, Theorem B Establishes Local Rigidity in Adapted Charts, Theorem C Establishes Categorical Universality

We articulate explicitly what each of the three theorems establishes.

Theorem A (Minkowski Rigidity). Establishes that on flat Minkowski space ℝ⁴ with the standard Lorentzian metric η, the moving-dimension structure is unique up to a Poincaré transformation modulo time translation. This is a flat-spacetime rigidity theorem.

Theorem B (Local Rigidity in Adapted Charts). Establishes that the moving-dimension data on any (M, g, F, V) is determined locally by the lapse function and the spatial metrics on the leaves. This is a local-data rigidity theorem.

Theorem C (Categorical Universality). Establishes that within the category 𝓐 of axis-dynamics frameworks, the subcategory 𝓜 of moving-dimension manifolds is the universal recipient of predicate-strict frameworks: every predicate-strict framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ is canonically equivalent to an object of 𝓜. This is the categorical no-embedding theorem.

9.2 The Explicit Scope of the Three Theorems: Theorem A Applies to Flat Minkowski Space, Theorem B to Local Adapted-Chart Data, Theorem C to Predicate-Strict Frameworks within the Categorical Setup; the Apparatus Is Mathematical and Does Not Address Empirical Content

The four-part scope of the three theorems is the following.

Theorem A: scope of Minkowski rigidity. The theorem applies to flat Minkowski space ℝ⁴ with the standard metric η, where it establishes that every moving-dimension structure is the standard one up to Poincaré transformation modulo time translation. Curved-spacetime rigidity content is supplied by Theorem B’s local-rigidity in adapted charts, which applies to general (M, g, F, V).

Theorem B: scope of local rigidity. The theorem establishes that on any moving-dimension manifold, the moving-dimension data (F, V) in any McGucken-adapted chart is determined by the lapse function N and the spatial metrics on the leaves; two structures sharing a common adapted chart with identical lapse and spatial metrics are equal as objects of 𝓜. Global-rigidity content across distinct foliations on the same manifold is a separate question (involving inequivalent objects of 𝓜_M for fixed M).

Theorem C: scope of categorical universality. The theorem applies to the predicate-strict subcategory of the category 𝓐 of axis-dynamics frameworks of Definition 7.1, where it establishes that every predicate-strict framework is canonically equivalent to a moving-dimension manifold of 𝓜. The non-predicate-strict objects of 𝓐 — including the eleven closest-neighbor frameworks of [G, §13] with their characteristic non-trivial decorations — are categorically distinguishable from 𝓜 within 𝓐, and the structural distinction is articulated through their decorations as catalogued in §8.3.

The complementary empirical content. The categorical-mathematical theorem is complementary to the empirical content of the broader McGucken framework: the empirical identification of V with the CMB rest frame (condition (P4) of [G, Definition 5.4]) is established in the cosmology paper [79] by direct empirical analysis of CMB observations, the survey-based novelty across the eleven frameworks of [G, §13] plus quantum-gravity programs and philosophical traditions of [G, §14] is established by [G]’s comprehensive survey, and the present paper’s categorical universality strengthens both. The three components — categorical universality from this paper, empirical cosmology from [79], and survey-based novelty from [G] — together establish the strongest novelty claim that the apparatus supports, with each component bounded precisely by its own apparatus.

Frameworks not fitting into the categorical setup of 𝓐 (Causal Set Theory with its discrete partial-order foundation, Loop Quantum Gravity with its non-perturbative canonical-quantization apparatus, growing-block universe and process philosophy as non-mathematical traditions) are addressed by the survey of [G, §14] rather than by the categorical theorem of this paper; this is appropriate division of labor between survey and categorical apparatus.

9.3 The Role of Condition (P4) Is Empirical-Cosmological: V’s Identification with the CMB Rest Frame Selects the Physically Realized Object within the Mathematical Category 𝓜; the Selection Is Empirical, Addressed in the Cosmology Paper [79], and Outside Any Categorical Apparatus

Condition (P4) of [G, Definition 5.4] is the empirical identification:

V is empirically identified with the cosmic microwave background rest frame.

This identification is empirical. It cannot be a predicate on objects of 𝓜 or 𝓐 — neither category has the empirical apparatus to formalize “the cosmic microwave background rest frame” as a categorical content.

The empirical identification (P4) plays the following role in the combined McGucken framework:

• Among objects of 𝓜: The empirical identification (P4) selects, among all objects of 𝓜, the one corresponding to the physically realized cosmological structure. This is an empirical specification, not a categorical predicate.
• In the survey of [G, §§9–14]: The empirical identification (P4) is one of the four conditions, alongside the mathematical conditions (P1), (P2), (P3). The survey establishes that no prior framework satisfies all four.
• In the present paper’s categorical theorem: The empirical condition (P4) is addressed in the cosmology paper [79] rather than in the categorical apparatus; Theorem C concerns the mathematical predicates 𝒫₁, 𝒫₂, 𝒫₃ formalizing conditions (P1)–(P3). The categorical theorem is complementary to (P4)’s empirical content; together with [79] and [G] they articulate the mathematical and empirical standing of the framework as a coherent whole.

The role of (P4) is therefore empirical-cosmological, not categorical-mathematical. The empirical cosmological program of [79] (the McGucken Cosmology paper) addresses (P4) by identifying the privileged frame V, in the cosmological extension of McGucken Geometry, with the empirical CMB rest frame. The categorical apparatus of the present paper does not address this empirical question; it remains an empirical specification.

9.4 The Combined Mathematical-Empirical Standing of the McGucken Framework: Categorical Universality on (P1)–(P3) within 𝓐 from This Paper, Combined with Survey-Based Novelty on (P1)–(P4) from [G] and Empirical CMB-Frame Identification from [79], Establishes the Strongest Novelty Claim the Apparatus Supports

The combined standing of the McGucken framework, after both the comprehensive survey of [G] and the categorical theorem of the present paper, is the following.

Mathematically (within the categorical setup of the present paper): Theorem C establishes that the conjunction of formal predicates 𝒫₁, 𝒫₂, 𝒫₃ on predicate-strict objects of 𝓐 uniquely determines the framework up to canonical isomorphism, with the framework lying in 𝓜.

Empirically (within the survey of [G, §§9–14]): No prior framework satisfies the conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4) as articulated in [G, Definition 5.4].

Combined: McGucken Geometry is mathematically universal among predicate-strict axis-dynamics frameworks (Theorem C of the present paper) and empirically privileged among surveyed prior frameworks ([G, §§9–14]). The mathematical universality and the empirical privilege combine to support the strongest novelty claim that the apparatus supports.

The combined claim is bounded by the explicit setups: the categorical theorem is bounded by the categorical setup of 𝓐 and the predicate-strict property; the empirical claim is bounded by the survey’s coverage of the eleven frameworks of [G, §13] plus the quantum-gravity programs and philosophical traditions of [G, §14] and by the empirical identification of [79]. These are precise bounds on what each apparatus establishes; together, the categorical theorem, the comprehensive survey, and the empirical cosmology articulate the strongest novelty claim that the McGucken corpus apparatus supports.

PART IV — SYNTHESIS

Part IV synthesizes the result. §10 places the theorem in the corpus context of [G] and the broader McGucken framework. §11 concludes.

10. The Present Paper’s Categorical Theorem Combined with [G]’s Comprehensive Survey and [79]’s Empirical Cosmology Establishes the Strongest Novelty Claim the McGucken Corpus Apparatus Supports

10.1 The Present Paper Supplies the Categorical Strengthening of [G]’s Survey Claim within the Specified Categorical Setup; Together with [G] and [79], the Three Papers Articulate the Mathematical and Empirical Status of McGucken Geometry as a Coherent Whole

The McGucken corpus comprises the foundational papers establishing dx₄/dt = ic [38, 39 of G], the derivational chains for general relativity [31 of G], quantum mechanics [32 of G], and thermodynamics [33 of G], the Wick rotation [80 of G], the McGucken Sphere as foundational atom [86 of G], the Father Symmetry [85 of G], the McGucken Lagrangian [87 of G], the McGucken Space and Operator [81–83 of G], cosmological tests [79 of G], and the formal mathematical category McGucken Geometry [G — the immediate predecessor of the present paper].

The present paper is the categorical strengthening of [G]. It builds the categorical apparatus deferred from [G, §15.4] and proves the formal categorical no-embedding theorem.

10.2 The Categorical-Mathematical Status of McGucken Geometry After the Present Paper: The Moving-Dimension Manifold Category 𝓜 Is the Terminal Subcategory of Predicate-Strict Axis-Dynamics Frameworks within the Categorical Setup of 𝓐, with Universality Established by Theorem C and Rigidity Established by Theorems A and B

After the present paper, McGucken Geometry has the following mathematical-categorical status:

• The category 𝓜 of moving-dimension manifolds (§2 of the present paper) is precisely the category whose objects are the moving-dimension manifolds of [G, Definition 5.6] and whose morphisms are the foliation-preserving isometries intertwining the privileged vector field.
• The mathematical content of conditions (P1)–(P3) of [G, Definition 5.4] is automatic on objects of 𝓜 (Proposition 2.7 of the present paper).
• Within the larger category 𝓐 of axis-dynamics frameworks (Definition 7.1 of the present paper), the embedding ι: 𝓜 → 𝓐 factors through the full subcategory 𝓐₀ of trivially-decorated objects as an isomorphism of categories ι: 𝓜 ⥲ 𝓐₀ (Theorem 7.5.2), with strict inverse R|_{𝓐₀}: 𝓐₀ → 𝓜 given by the restriction of the forgetful functor R.
• Every predicate-strict object of 𝓐 (i.e., every object with trivial decoration ε ≡ 0) is equal on the nose to ι(R(A)) ∈ 𝓐, with the canonical morphism A → ι(R(A)) being the identity id_A (Proposition 7.6.3, Theorem 7.7.3, Corollary 7.7.4).
• The combined novelty claim — drawing on [G]’s survey and the present paper’s theorem — is: no surveyed prior framework satisfies (P1) ∧ (P2) ∧ (P3) ∧ (P4), and within the categorical setup, predicate-strict axis-dynamics frameworks are canonically equivalent to McGucken Geometry.

This is the strongest novelty standing that the combined apparatus of [G] and the present paper supports.

10.3 The Empirical Cosmological Program of [79] Selects the Physically Realized Object within 𝓜 by Identifying V with the Cosmic Microwave Background Rest Frame; This Empirical Selection Is Outside the Categorical Apparatus and Carries the Empirical Content of Condition (P4)

Condition (P4) — the empirical identification of V with the CMB rest frame — sits outside the categorical apparatus and is addressed by the empirical-cosmological program of [79]. The combined empirical and mathematical content of the McGucken framework comprises:

• Mathematical: McGucken Geometry as the universal categorical home of axis-dynamics frameworks (the present paper).
• Empirical: McGucken Cosmology as the empirically identified cosmological structure ([79], with V identified with the CMB rest frame and the framework’s empirical predictions tested across twelve observational tests).
• Derivational: The corpus [31, 32, 33, 80, 85, 86, 87, etc.] establishing the framework’s derivational consequences across general relativity, quantum mechanics, thermodynamics, and beyond.

The mathematical, empirical, and derivational content together constitute the McGucken framework in its full development.

10.4 Open Questions: The Equivalence Conjecture of [G, Conjecture 8.2], Higher-Categorical Refinements (2-Categories, ∞-Categories, Homotopy Type Theory), Comprehensive Coverage of Frameworks Beyond the Eleven Surveyed in [G, §13], Categorical Formalization of (P4), and Higher-Dimensional Generalization Beyond 3+1 Dimensions

Some questions remain open even after the present paper:

• The full equivalence of the three formulations of [G, §§5–7] — moving-dimension manifold, jet-bundle, Cartan geometry — was conjectured in [G, Conjecture 8.2] and not fully proved. The present paper works in the moving-dimension manifold language; full proofs of the equivalences with the jet-bundle and Cartan-geometric formulations remain to be given.
• Higher-categorical refinements. The categorical setup of the present paper uses ordinary 1-categories. A higher-categorical refinement (using 2-categories, ∞-categories, or the language of homotopy type theory) might give finer structural information. This is left open.
• Categorical formalization of (P4). Whether the empirical condition (P4) admits any categorical formalization — perhaps through some “decoration” by an empirically realized cosmological tensor — is unclear. The present paper sets (P4) aside.
• Generalization to higher-dimensional spacetimes. The present paper works with smooth 4-manifolds. Whether the categorical apparatus generalizes naturally to spacetimes of other dimensions is left open.

These open questions are noted as directions for subsequent work; the present paper does not undertake them.

11. Conclusion: The Three Theorems Establish Minkowski Rigidity, Local Rigidity in Adapted Charts, and Categorical Universality of McGucken Geometry within the Specified Categorical Setup; Combined with [G]’s Comprehensive Survey, This Paper Establishes the Strongest Novelty Claim the McGucken Apparatus Supports

The companion paper [G] formalized McGucken Geometry underlying the McGucken Principle that the fourth dimension is expanding at the velocity of light dx₄/dt = ic as a mathematical category with three equivalent formulations and articulated four privileged-element conditions (P1)–(P4). The novelty claim of [G] was established by exhibition: comprehensive prior-art survey across §§9–14 of [G], covering eleven frameworks with privileged direction or flow content plus quantum-gravity programs and philosophical traditions, established that no surveyed framework contains the conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4), with the Connes-Rovelli Thermal Time Hypothesis the closest cousin (three conditions partially satisfied, (P3) absent).

The present paper proves the formal categorical no-embedding theorem that strengthens [G]’s survey claim. The two papers are companions: [G] formalizes the category and surveys the literature; this paper proves the categorical universality within the precisely-specified categorical setup of 𝓐.

The three theorems established in this paper.

Theorem A (Minkowski Rigidity, §5) establishes that every moving-dimension structure on flat Minkowski space ℝ⁴ is isomorphic to the standard structure (ℝ⁴, η, F_std, V_std = ∂/∂t) via a Poincaré transformation preserving time-orientation, modulo a one-parameter family of foliation-origin choices. The Minkowski case is rigid: there is no inequivalent alternative.

Theorem B (Local Rigidity in Adapted Charts, §6) establishes that on any moving-dimension manifold, the moving-dimension data (F, V) in any McGucken-adapted chart is determined by the lapse function N and the spatial metrics on the leaves; two structures sharing a common adapted chart with identical lapse and spatial metrics are equal as objects of 𝓜.

Theorem C (Categorical Universality, §7) is the formal categorical no-embedding theorem. The category 𝓐 of axis-dynamics frameworks (Definition 7.1) is constructed; the embedding ι: 𝓜 → 𝓐 (Definition 7.3) and the forgetful functor R: 𝓐 → 𝓜 (Definition 7.4) are defined; the full subcategory 𝓐₀ ⊂ 𝓐 of trivially-decorated objects is identified (Definition 7.5.1); the embedding ι factors through 𝓐₀ as an isomorphism of categories ι: 𝓜 ⥲ 𝓐₀ with strict inverse R|_{𝓐₀} (Theorem 7.5.2); the section identity R ∘ ι = 1_𝓜 holds globally on 𝓜 (Remark 7.5.4); the canonical morphism A → ι(R(A)) exists in 𝓐 if and only if A is predicate-strict (Proposition 7.6.3); and the universal-property characterization of 𝓐₀ in 𝓐 (Corollary 7.7.4) establishes uniqueness. Equivalently: McGucken Geometry 𝓜 is canonically isomorphic to the predicate-strict subcategory 𝓐₀ of 𝓐, and every framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ with no auxiliary structural decoration is equal on the nose to a moving-dimension manifold of 𝓜.

The category 𝓜 of moving-dimension manifolds (§2) is constructed: objects are quadruples (M, g, F, V) with M a globally hyperbolic Lorentzian 4-manifold, g a Lorentzian metric, F a foliation by spacelike Cauchy surfaces, V a unit timelike vector field orthogonal to F; morphisms are foliation-preserving isometries intertwining V. The mathematical content of conditions (P1)–(P3) of [G, Definition 5.4] is formalized as Predicates 𝒫₁, 𝒫₂, 𝒫₃ (Definitions 2.4, 2.5, 2.6) and shown to be automatic on objects of 𝓜 (Proposition 2.7). The empirical condition (P4) — V’s identification with the cosmic microwave background rest frame — is empirical content addressed in the cosmology paper [79] rather than formalized as a categorical predicate.

The strengthening of [G]’s novelty claim (§8): where [G]’s survey claim is bounded by the survey’s coverage of eleven concrete frameworks plus quantum-gravity programs and philosophical traditions, Theorem C’s claim is bounded by the categorical setup of 𝓐. Within the setup, Theorem C is a universality statement quantifying over all axis-dynamics frameworks satisfying the predicate-strict property. The eleven closest neighbors of [G, §13] correspond to non-predicate-strict frameworks (with non-trivial decoration ε ≠ 0) and are categorically distinguishable from McGucken Geometry within 𝓐 — including the Connes-Rovelli Thermal Time Hypothesis (the closest cousin), whose decoration is the algebraic-state pair (𝒜, ω) that makes its flow state-dependent rather than state-independent, articulated in detail in §8.3 N.13.6.

The combined claim drawing on the present paper, the companion paper [G], and the cosmology paper [79]: no surveyed prior framework satisfies the conjunction (P1) ∧ (P2) ∧ (P3) ∧ (P4) in its full form, with the Connes-Rovelli Thermal Time Hypothesis the closest cousin; within the categorical setup of 𝓐, every predicate-strict framework satisfying the formal predicates 𝒫₁, 𝒫₂, 𝒫₃ is canonically equivalent to McGucken Geometry; the empirical identification of V with the CMB rest frame is established in [79]. The mathematical universality is established by Theorem C; the empirical-survey privilege is established by [G, §§9–14]; the empirical-cosmological identification (P4) is established by [79]. The three components together articulate the mathematical and empirical standing of McGucken Geometry as a coherent whole.

The deferred categorical no-embedding theorem of [G, §15.4] is now established. McGucken Geometry stands as the categorical universal among predicate-strict axis-dynamics frameworks within the categorical setup of 𝓐, with the universality strengthening the comprehensive survey-based novelty claim of [G] and complementing the empirical CMB-frame identification of [79]. The combined apparatus — categorical universality from this paper, comprehensive survey from [G], and empirical cosmology from [79] — establishes the strongest novelty claim that the McGucken framework’s apparatus supports.

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[21] P. Hořava, “Quantum Gravity at a Lifshitz Point,” Physical Review D 79, 084008 (2009).

[22] J. Ambjørn and R. Loll, “Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change,” Nuclear Physics B 536, 407–434 (1998).

[23] J. Barbour, “Shape Dynamics: An Introduction,” in Quantum Field Theory and Gravity, Birkhäuser (2012), pp. 257–297.

[24] H. Gomes, S. Gryb, T. Koslowski, and F. Mercati, “The Gravity/CFT Correspondence,” European Physical Journal C 73, 2275 (2013).

[26] C. Rovelli, Quantum Gravity, Cambridge University Press (2004).

[27] L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, “Space-Time as a Causal Set,” Physical Review Letters 59, 521 (1987).

[28] H. Reichenbach, The Direction of Time, University of California Press, Berkeley (1956).

[29] J. M. E. McTaggart, “The Unreality of Time,” Mind 17, 457–474 (1908).

[30] A. N. Whitehead, Process and Reality, Macmillan, New York (1929).

[53] S. Jordan and R. Loll, “De Sitter Universe from Causal Dynamical Triangulations without Preferred Foliation,” arXiv:1307.5469 (2013).

[63] C. D. Broad, Scientific Thought, Routledge & Kegan Paul, London (1923).

[73] A. Connes and C. Rovelli, “Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories,” Classical and Quantum Gravity 11, 2899–2917 (1994). arXiv:gr-qc/9406019.

[74] C. Rovelli, “Statistical Mechanics of Gravity and the Thermodynamical Origin of Time,” Classical and Quantum Gravity 10, 1549–1566 (1993). [The FRW thermal time recovers the cosmological time of the CMB rest frame.]

[75] M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and its Applications, Lecture Notes in Mathematics 128, Springer-Verlag (1970).

[76] A. Connes, Noncommutative Geometry, Academic Press (1994).

[77] A. H. Chamseddine and A. Connes, “The Spectral Action Principle,” Communications in Mathematical Physics 186, 731–750 (1997).

[78] F. Besnard and N. Bizi, “On the definition of spacetimes in Noncommutative Geometry, Part I,” Journal of Geometry and Physics 123, 292–309 (2018). arXiv:1611.07830.

[11a] A. Einstein, “Riemann-Geometrie mit Aufrechterhaltung des Begriffes des Fernparallelismus,” Sitzungsberichte der Preussischen Akademie der Wissenschaften 217 (1928); H. Weyl, “Elektron und Gravitation. I,” Zeitschrift für Physik 56, 330–352 (1929).

[13b] E. Caponio and G. Stancarone, “On Finsler spacetimes with a timelike Killing vector field,” Classical and Quantum Gravity 35, 085007 (2018). arXiv:1710.05318.

[41a] É. Cartan, “Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion,” Comptes Rendus de l’Académie des Sciences 174, 593–595 (1922).

[62b] A. N. Bernal and M. Sánchez, “On smooth Cauchy hypersurfaces and Geroch’s splitting theorem,” Communications in Mathematical Physics 243, 461–470 (2003). arXiv:gr-qc/0306108.

[62c] A. N. Bernal and M. Sánchez, “Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes,” Communications in Mathematical Physics 257, 43–50 (2005). arXiv:gr-qc/0401112.

[69a] R. Penrose, Cycles of Time: An Extraordinary New View of the Universe, Bodley Head, London (2010).

[70a] R. Penrose, “On the gravitization of quantum mechanics 2: Conformal cyclic cosmology,” Foundations of Physics 44, 873–890 (2014).

[79] E. McGucken, McGucken Cosmology: First-Place Empirical Standing across Twelve Independent Observational Tests, Light, Time, Dimension Theory, May 1, 2026. URL: https://elliotmcguckenphysics.com/2026/05/01/the-mcgucken-cosmology-dx4-dt-ic-outranks-every-major-cosmological-model/

[80] E. McGucken, The McGucken Principle dx₄/dt = ic Necessitates the Wick Rotation and i Throughout Physics, Light, Time, Dimension Theory, May 1, 2026. URL: https://elliotmcguckenphysics.com/2026/05/01/the-mcgucken-principle-dx4-dtic-necessitates-the-wick-rotation/

[31, 32, 33, 38, 39, 81–87] Additional corpus references as cited in [G, References [31]–[87]]; see [G] for the complete McGucken corpus bibliography.