Mathematical and Theoretical Foundations of the Basic Phase-Transport Equation: A Unified Differential, Covariant, and Path-Functional Theory

1. Introduction

Phase is a central precision observable in communications, satellite navigation, phased-array radar, interferometry, coherent imaging, and integrated sensing and communications. Carrier synchronization, precision ranging, target-motion inversion, coherent array combination, and phase-continuity preservation all require phase states to be read stably, compared continuously, and accumulated consistently during system evolution [1,2,3,4,5,6].

Phase originates in a periodic degree of freedom in the physical world. It is not an ordinary real-valued parameter; it is characterized by whole-cycle equivalence, reference selection, and continuous winding. A practical system does not access physical phase directly. It forms a response through an observation relation and reads phase from that response. The first foundational question is therefore how physical phase is represented after observation and how the resulting observed phase changes with the system state.

The classical observation model is

$$y=h\left(x\right),$$

(1)

where $x\in X$ is a physical state, $h:X\to Y$ is a fixed observation map, and $y\in Y$ is an observed response. When the conditions forming the response remain fixed, Equation (1) is complete and correct: it describes how variation in x is mapped to variation in y through a fixed h.

In modern coherent systems, however, the relation forming the response need not remain fixed. Array weights, beam configurations, channel combinations, reference choices, processing rules, motion of the observation apparatus, disturbances, and environmental conditions can all change the current observation map. Adaptive arrays and phased-array radar are representative examples [7,8,9]. The same physical state may generate different responses under different observation conditions, and physical-state and observation-condition variations may occur simultaneously.

If all such variations continue to be hidden inside the symbol h, the model contains no independent variable position for the current observation map. Consequently, observed-phase dynamics may combine effects from physical-state variation and observation-map variation without representing them separately. This paper therefore addresses four progressively structured questions:

(1)

How should a type-complete unified model be constructed when a fixed observation map is replaced by a family of admissible maps?

(2)

What observed-phase function is formed when physical phase passes through this observation relation, and how are circularity, reference selection, and continuity represented?

(3)

When the response is nonzero and complex and the map may be nonlinear and strongly coupled, how do the local differential and global structural properties of the complex response determine continuous phase variation, finite comparison, and the basic phase-transport equation?

(4)

Once phase has been transported to a common reference, how should task-irrelevant freedoms be removed by a prescribed group action, and when is a target parameter recoverable from the resulting quotient observation or maximal invariant?

These questions appear in different concrete forms in satellite navigation, phased-array radar, integrated sensing and communications, and other coherent systems. Models tailored to a particular system, observation configuration, or error source do not automatically provide a general structure for phase formation, continuous evolution, and cross-state comparison under a variable observation map. The present work therefore begins from the observation relation itself and identifies the common mathematical objects and validity conditions.

No linearity of the observation map is assumed, and no additive decomposition of observed phase into independent physical and observer terms is imposed. We first return to the classical model and study complete representations of variable observation-map families. We then retain only the intrinsic phase structures actually used later. Finally, the complex response generated by the derived observation relation is used to construct local phase variation, global phase comparison, the basic phase-transport equation, and task-quotient observation.

The paper thus culminates in a general theoretical task: to derive the basic phase-transport equation from the intrinsic relation among physical phase, the observation map, and the complex response under explicit domain, regularity, and structural assumptions. The equation is not defined by a coordinate formula in a selected time parameter. Its intrinsic form is a global differential functional equation on lifted paths of a principal phase bundle; ordinary differential, covariant-derivative, lifted-real-phase, and finite-transport expressions are derived as equivalent forms. Its fundamental status is established through necessity, uniqueness, gauge covariance, phase-level minimality, and completeness from local variation to finite transport.

2. From a Fixed Observation Map to a Three-Layer Observation Paradigm

This section addresses a single question: when the observation map is allowed to vary, how can physical state, current map, and observed response be represented completely in one mathematical relation? The observer and the three-layer paradigm are not introduced as prior modeling assumptions; they are named only after the relevant representation requirement has been proved. All results in this section are initially set-theoretic. Phase, complex response, fiber bundles, connections, and transport are not yet involved.

2.1. The Classical Model and Its Domain of Validity

Let X be the physical-state space and Y the response space. Given a fixed map

$$h:X⟶Y,$$

the classical observation model is

$$y=h\left(x\right),x\in X,y\in Y.$$

(2)

Here h is part of the model specification rather than a model variable. The apparatus, configuration, reference, and processing rule that form the map may all be encoded in h. Thus, when h is fixed, Equation (2) is type-complete.

When multiple observation maps are admissible, the problem changes. Let

$$\mathcal{H}\subseteq Map(X,Y)$$

(3)

denote the family of all admissible maps. A complete model must then specify both the physical state and which map is currently active. The notation $y=h\left(x\right)$ describes each fixed slice but does not provide a variable position for the current map within a single domain.

2.2. Canonical Evaluation and Joint Identifiability

The canonical representation of the family $\mathcal{H}$ is the evaluation map

$${ev}_{\mathcal{H}}:X\times \mathcal{H}⟶Y,{ev}_{\mathcal{H}}(x,h)=h\left(x\right).$$

(4)

The variables x and h have distinct types: the former is a physical state, whereas the latter identifies the current observation map.

Definition 1

(Global joint identifiability of physical state and observation map). The pair $(x,h)$ is globally jointly identifiable from the specified response data if, for all $(x,h),(x^{\prime },h^{\prime })\in X\times \mathcal{H}$,

$${ev}_{\mathcal{H}}(x,h)={ev}_{\mathcal{H}}(x^{\prime },h^{\prime })$$

implies

$$x=x^{\prime },h=h^{\prime }.$$

Proposition 1

(Injectivity criterion). The physical state and current observation map are globally jointly identifiable with respect to Y if and only if

$${ev}_{\mathcal{H}}:X\times \mathcal{H}\to Y$$

is injective.

Proof. 

This is exactly the definition of injectivity on the product domain $X\times \mathcal{H}$. □

Allowing the observation map to vary does not logically imply nonidentifiability. Identifiability depends on the evaluation map and the information contained in the response data. Regardless of injectivity, however, if the entire domain $X\times \mathcal{H}$ is under study, the current map must occupy an independent variable position. Representation necessity and joint identifiability are different questions.

2.3. Necessity of a Variable Position for the Current Map

Lemma 1

(A nontrivial map family cannot be represented completely by a single physical-state variable). If $\mathcal{H}\subseteq Map(X,Y)$ contains at least two distinct maps, then there is no single map $\tilde{h}:X\to Y$ such that

$$\tilde{h}\left(x\right)=h\left(x\right),\mathrm{for}\mathrm{every}(x,h)\in X\times \mathcal{H}.$$

(5)

Proof. 

Choose $h_0,h_1\in \mathcal{H}$ with $h_0\ne h_1$. By definition of inequality of maps, there exists $x_0\in X$ such that $h_0\left(x_0\right)\ne h_1\left(x_0\right)$. If Equation (5) held, then

$$h_0\left(x_0\right)=\tilde{h}\left(x_0\right)=h_1\left(x_0\right),$$

a contradiction. □

Remark 1

(Constrained submodels). If the system imposes a known relation $h=\sigma \left(x\right)$ with $\sigma :X\to \mathcal{H}$, then admissible states lie on the graph

$$\left\{\right(x,\sigma \left(x\right)):x\in X\}\subseteq X\times \mathcal{H}.$$

The model can then be reduced to $x\mapsto \sigma \left(x\right)\left(x\right)$. This is a constrained submodel, not a complete representation of all of $X\times \mathcal{H}$.

The second variable position follows from the requirement to represent a nontrivial map family completely, not from a prior assertion of nonidentifiability.

2.4. Joint Maps and Parametrized Map Families

Before assigning a physical interpretation to the second variable, let S be a neutral state set, $s\in S$, and consider

$$F:X\times S⟶Y.$$

(6)

For fixed s, the partial map $F(·,s):X\to Y$ is the observation map selected by that state.

Proposition 2

(Natural correspondence between joint maps and parametrized map families). Joint maps

$$F:X\times S\to Y$$

are in natural one-to-one correspondence with maps

$${\mathrm{\Gamma }}_F:S\to Map(X,Y),$$

where

$${\mathrm{\Gamma }}_F\left(s\right)=F(·,s).$$

(7)

The inverse construction is

$$F_{\mathrm{\Gamma }}(x,s)=\mathrm{\Gamma }\left(s\right)\left(x\right),$$

(8)

and

$$F={ev}_{Map(X,Y)}\circ ({id}_X\times {\mathrm{\Gamma }}_F).$$

(9)

Proof. 

Given F, define Equation (7); given $\mathrm{\Gamma }$, define Equation (8). Direct substitution gives ${\mathrm{\Gamma }}_{F_{\mathrm{\Gamma }}}=\mathrm{\Gamma }$ and $F_{{\mathrm{\Gamma }}_F}=F$. The factorization Equation (9) follows from the definition of evaluation. □

Definition 2

(State realization of an admissible map family). A joint map $F:X\times S\to Y$ is a complete state realization of $\mathcal{H}$ if

$$Im{\mathrm{\Gamma }}_F=\mathcal{H}.$$

The realization is nonredundant if ${\mathrm{\Gamma }}_F$ is injective.

Every complete realization carries the equivalence relation

$$s{\sim }_F{s}^{\prime }⟺{\mathrm{\Gamma }}_F\left(s\right)={\mathrm{\Gamma }}_F\left(s^{\prime }\right),$$

(10)

and the effective state space

$$S_{\mathrm{eff}}:=S/{\sim }_F.$$

(11)

This quotient removes only parameter redundancy that leaves the current observation map unchanged. It does not remove any possible nonidentifiability between physical state and current map.

Let

$$q_F:S\twoheadrightarrow S_{\mathrm{eff}},q_F\left(s\right)=\left[s\right]$$

(12)

denote the quotient map.

2.5. Canonical Quotient Theorem for State Realizations

Theorem 1

(State realization and canonical quotient of a variable observation-map family). Let X and Y be sets and let $\mathcal{H}\subseteq Map(X,Y)$. Then:

(1)

$F:X\times S\to Y$ completely realizes $\mathcal{H}$ if and only if there exists a surjection

$${\mathrm{\Gamma }}_F:S\twoheadrightarrow \mathcal{H}$$

such that

$$F={ev}_{\mathcal{H}}\circ ({id}_X\times {\mathrm{\Gamma }}_F).$$

(13)

(2)

A canonical nonredundant realization is

$$S_{\mathrm{can}}=\mathcal{H},{\mathrm{\Gamma }}_{\mathrm{can}}={id}_{\mathcal{H}},F_{\mathrm{can}}={ev}_{\mathcal{H}}.$$

(14)

(3)

For every complete realization, ${\mathrm{\Gamma }}_F$ induces a unique bijection

$${\overline{\mathrm{\Gamma }}}_F:S_{\mathrm{eff}}⟶\mathcal{H},{\overline{\mathrm{\Gamma }}}_F\left(\left[s\right]\right)={\mathrm{\Gamma }}_F\left(s\right),{\overline{\mathrm{\Gamma }}}_F\circ q_F={\mathrm{\Gamma }}_F,$$

(15)

and there exists a unique effective map

$$F_{\mathrm{eff}}:X\times S_{\mathrm{eff}}⟶Y$$

(16)

satisfying

$$F=F_{\mathrm{eff}}\circ ({id}_X\times q_F),F_{\mathrm{eff}}={ev}_{\mathcal{H}}\circ ({id}_X\times {\overline{\mathrm{\Gamma }}}_F).$$

(17)

(4)

$F_{\mathrm{eff}}$ is injective if and only if ${ev}_{\mathcal{H}}$ is injective.

Proof. 

Part (1) follows directly from the natural correspondence between joint maps and parametrized map families and from the definition of complete realization. Part (2) is obtained by taking the state set to be $\mathcal{H}$ and the parametrization to be the identity.

For part (3), if $\left[s\right]=\left[s^{\prime }\right]$, then Equation (10) gives ${\mathrm{\Gamma }}_F\left(s\right)={\mathrm{\Gamma }}_F\left(s^{\prime }\right)$, so ${\overline{\mathrm{\Gamma }}}_F$ is well defined. It is injective by the definition of the equivalence relation and surjective by completeness. It therefore is a bijection and satisfies ${\overline{\mathrm{\Gamma }}}_F\circ q_F={\mathrm{\Gamma }}_F$. Define

$$F_{\mathrm{eff}}(x,\left[s\right]):=F(x,s).$$

The same equivalence relation makes this definition well posed. The factorization in Equation (17) follows immediately. Uniqueness follows from the surjectivity of $q_F$.

Finally, ${id}_X\times {\overline{\mathrm{\Gamma }}}_F$ is a bijection and

$$F_{\mathrm{eff}}={ev}_{\mathcal{H}}\circ ({id}_X\times {\overline{\mathrm{\Gamma }}}_F),$$

which proves part (4). □

The theorem states that any complete representation of a variable observation-map family must contain a state position distinct from the physical-state variable and a map from that state to the admissible family. After quotienting redundancy, the effective role is determined by $\mathcal{H}$ up to canonical bijection. This is a statement about representation structure, not a set-theoretic construction of a particular physical instrument.

2.6. Observer, Observation Map, and the Three-Layer Paradigm

Only now do we assign a name to the mathematical role established above.

Definition 3

(Observer and observation map). An observer realizing the admissible family $\mathcal{H}$ is a pair

$$O=(S_O,{\mathrm{\Gamma }}_O),{\mathrm{\Gamma }}_O:S_O\twoheadrightarrow \mathcal{H},$$

(18)

where $S_O$ is the observer-state space and $\eta \in S_O$ an observer state. The associated observation map is

$$F_O:X\times S_O⟶Y,F_O(x,\eta )={\mathrm{\Gamma }}_O\left(\eta \right)\left(x\right).$$

(19)

A single observer object O is fixed throughout the paper. Symbols such as ${\eta }_0,{\eta }_1,{\eta }_r,{\eta }_c$ denote different states of that same observer, not different observers. When the observer state changes, the slice

$$h_{\eta }:={\mathrm{\Gamma }}_O\left(\eta \right):X\to Y$$

may change, while O and $F_O$ remain the same objects.

Definition 4

(Three-layer observation paradigm). The relation determined by the physical-state space X, the observer $O=(S_O,{\mathrm{\Gamma }}_O)$, the response space Y, and the observation map $F_O:X\times S_O\to Y$,

$$(x,\eta )\stackrel{F_O}{\to }y=F_O(x,\eta ),$$

(20)

is called the three-layer observation paradigm.

The term “three-layer” refers to three classes of objects with distinct theoretical roles: physical state, observer, and observed response. It is not a serial map $X\to O\to Y$. The physical state x and observer state $\eta$ jointly enter $F_O$. The product $X\times S_O$ supplies independent variable positions only; it does not assume physical, statistical, or dynamical independence.

The derivation chain is

$$\begin{array}{cc}\hfill y=h\left(x\right)& ⟹\mathcal{H}\subseteq Map(X,Y)⟹{ev}_{\mathcal{H}}:X\times \mathcal{H}\to Y,\hfill \\ & ⟹\mathrm{state}\mathrm{realization}\mathrm{and}\mathrm{canonical}\mathrm{quotient}⟹O=(S_O,{\mathrm{\Gamma }}_O)⟹y=F_O(x,\eta ).\hfill \end{array}$$

Thus the three-layer paradigm is a consequence of complete representation of a variable observation-map family, not a structure imposed in advance.

2.7. Fixed Observer-State Slice: Recovery of the Classical Model

Proposition 3

(Classical model on a fixed observer-state slice). For any fixed ${\eta }_0\in S_O$, define

$${\iota }_{{\eta }_0}:X\to X\times S_O,{\iota }_{{\eta }_0}\left(x\right)=(x,{\eta }_0),$$

and

$$h_{{\eta }_0}:=F_O\circ {\iota }_{{\eta }_0}:X\to Y.$$

(21)

Then

$$h_{{\eta }_0}\left(x\right)=F_O(x,{\eta }_0)={\mathrm{\Gamma }}_O\left({\eta }_0\right)\left(x\right),$$

(22)

and the classical form $y=h_{{\eta }_0}\left(x\right)$ is recovered.

Proof. 

The claim follows directly from the definition of $F_O$. □

The classical model and the three-layer paradigm are not competing descriptions. The former is the physical-state–response relation on a fixed observer-state slice; the latter is the complete relation when the current observation map must also vary explicitly.

2.8. Fixed Physical-State Slice: Explicit Observer-State Dependence

The three-layer paradigm also provides a slice absent from the classical notation: the physical state is fixed while the state of the same observer varies.

Proposition 4

(Fixed physical-state slice). For fixed $x_0\in X$, define

$$j_{x_0}:S_O\to X\times S_O,j_{x_0}\left(\eta \right)=(x_0,\eta ),$$

and

$${\Lambda }_{x_0}:=F_O\circ j_{x_0}:S_O\to Y.$$

(23)

Then

$${\Lambda }_{x_0}\left(\eta \right)=F_O(x_0,\eta )={\mathrm{\Gamma }}_O\left(\eta \right)\left(x_0\right).$$

(24)

Consequently, even when the physical state $x_0$ and observer object O remain fixed, different observer states may yield

$$F_O(x_0,{\eta }_0)\ne F_O(x_0,{\eta }_1).$$

(25)

Proof. 

Equation (24) follows by definition. If ${\mathrm{\Gamma }}_O\left({\eta }_0\right)\left(x_0\right)\ne {\mathrm{\Gamma }}_O\left({\eta }_1\right)\left(x_0\right)$, then Equation (25) follows. □

This proposition states only that observer state is an independent variable capable of changing the observed response. It does not yet introduce phase, nor does it interpret observer state as an observable. Its role is to provide the typed variable structure needed to compare responses formed from the same physical state under different states of the same observer.

2.9. Conclusion and Boundary of the Section

Starting from a fixed observation map, this section has derived the admissible family $\mathcal{H}$, the evaluation map ${ev}_{\mathcal{H}}$, a neutral state realization $(S,F)$, the effective quotient $S_{\mathrm{eff}}$, the observer $O=(S_O,{\mathrm{\Gamma }}_O)$, and the observation map

$$F_O:X\times S_O\to Y.$$

The three-layer observation paradigm emerges from this chain. A fixed observer-state slice recovers the classical model, whereas a fixed physical-state slice makes observer-state effects explicit.

All results in this section are set-theoretic. No topology or smooth structure has yet been imposed on X, $S_O$, or Y, and no assumption has been made that the response carries phase. The next section returns to the physical world and records only the minimal phase structures needed later; the subsequent sections then study how phase is represented after the observation map.

3. Minimal Mathematical Structure of Physical Phase

This section retains only the phase structures actually used in the later construction of observed phase, the phase connection, and the basic phase-transport equation. Circular phase and path lifting are standard results; their general theory is not redeveloped here [12,13,14,15].

3.1. Periodic Equivalence and Circular Phase

Let the physical-state space be a smooth manifold X, and let

$$\gamma :\mathbb{R}⟶X$$

be an oriented periodic trajectory with fundamental period $T>0$ such that

$$\gamma (t+T)=\gamma \left(t\right),\gamma \left(t\right)=\gamma \left(t^{\prime }\right)⟺t-t^{\prime }\in T\mathbb{Z},\dot{\gamma }\left(t\right)\ne 0.$$

(26)

Write $C:=\gamma \left(\mathbb{R}\right)\subseteq X$. After normalizing one physical period to $2\pi$, phase coordinates obey

$$\theta \sim {\theta }^{\prime }⟺\theta -{\theta }^{\prime }\in 2\pi \mathbb{Z},$$

(27)

so the phase-state space is

$$\mathrm{\Theta }:=\mathbb{R}/\left(2\pi \mathbb{Z}\right),\pi :\mathbb{R}\to \mathrm{\Theta },\pi \left(\theta \right)=\left[\theta \right].$$

(28)

Only this circular state space and its standard smooth group structure are used below.

3.2. Reference Selection and Local Real Phase

The periodic physical structure does not determine a zero phase. Once a reference state $p_0=\gamma \left(t_0\right)\in C$ is selected, define

$${\vartheta }_{p_0}:C⟶\mathrm{\Theta },{\vartheta }_{p_0}\left(\gamma \left(t\right)\right):=\left[{\displaystyle \frac{2\pi }{T}}(t-t_0)\right].$$

(29)

Changing the reference state translates this circular phase by a constant and does not change the phase differential or transport law constructed later. A principal-value representative may be selected in $[0,2\pi )$, but it is discontinuous at its branch cut and therefore cannot serve as a global differentiable phase variable. Every real phase used below is either a local lift or a continuous lift along a path.

3.3. Path Lifting and Winding Increments

For any continuous circular path

$$\alpha :I\to \mathrm{\Theta },$$

and any prescribed initial real phase, there exists a unique continuous lift

$$\phi :I\to \mathbb{R},\pi \circ \phi =\alpha .$$

Lifts associated with different initial representatives differ by a constant $2\pi k$ and therefore have the same differential. If $I=[a,b]$, define

$${\mathrm{\Delta }}_{\alpha }\phi :=\phi \left(b\right)-\phi \left(a\right).$$

(30)

Modulo $2\pi$, this is the endpoint circular-phase difference, while as a real number it retains winding information. In particular, even if $\alpha \left(a\right)=\alpha \left(b\right)$, one may have

$${\mathrm{\Delta }}_{\alpha }\phi =2\pi n,n\in \mathbb{Z}.$$

This distinction is used later to separate a circular transport factor from an unwrapped real phase accumulation.

3.4. Boundary of Later Use

Only four interfaces are retained from the standard theory: the circular phase space $\mathrm{\Theta }$, a physical phase map after reference selection, local or pathwise continuous real lifts, and winding increments on closed paths. Whether an observed response carries phase, how that phase depends on physical and observer states, and how its differential and finite transport are determined will follow from the complex response induced by the observation map.

4. Phase Formation and Variation under the Observation Map

Section 2 established the observation map

$$F_O:X\times S_O⟶Y,$$

whereas Section 3 supplied the circular structure of physical phase. The present section connects these two lines: it identifies the observed phase formed after the observation map and derives its variation under changes in physical state and observer state.

The observation map may be nonlinear and coupled. Observed phase is neither assumed equal to physical phase nor assumed to decompose into independent physical and observer terms. The phase law will be obtained directly from the complex response generated by the observation map. Fiber bundles, connections, and transport are not yet introduced.

4.1. Complex Response Induced by the Observation Map

Let $X_{\varphi }\subseteq X$ be a domain on which physical phase is defined and let

$$\vartheta :X_{\varphi }⟶\mathrm{\Theta }$$

denote a physical phase map. The periodic orbit C of Section 3 is the basic prototype. Restrict the observation map to the phase-bearing domain,

$$F_O:X_{\varphi }\times S_O⟶Y.$$

(31)

Suppose the observation channel admits a scalar coherent complex readout. Let

$$Y_{\mathrm{coh}}\subseteq Y,\mathfrak{r}:Y_{\mathrm{coh}}⟶\mathbb{C}$$

be that readout. Choose an admissible joint-state domain

$$\mathcal{M}\subseteq X_{\varphi }\times S_O,F_O\left(\mathcal{M}\right)\subseteq Y_{\mathrm{coh}},$$

and define

$$g:=\mathfrak{r}\circ \left(F_O{|}_{\mathcal{M}}\right):\mathcal{M}⟶\mathbb{C},g(x,\eta )=\mathfrak{r}\left(F_O(x,\eta )\right).$$

(32)

This is the central response function. It retains the dependence of the observed result on both physical state x and observer state $\eta$. The function g may be linear, nonlinear, time-varying, or strongly coupled; explicit time can be incorporated into the joint state. No multiplicative or additive separability is assumed.

On the nonzero domain

$${\mathcal{M}}^{\times }:=\{(x,\eta )\in \mathcal{M}:g(x,\eta )\ne 0\},$$

(33)

the response has the unique polar decomposition

$$g(x,\eta )=a(x,\eta )u(x,\eta ),a:=\left|g\right|>0,u:={\displaystyle \frac{g}{\left|g\right|}}\in \mathrm{U}\left(1\right).$$

(34)

Thus a full nonzero complex response lies in

$${\mathbb{C}}^{\times }\cong {\mathbb{R}}_{>0}\times \mathrm{U}\left(1\right).$$

Amplitude and phase must be distinguished; only the normalized response u is a pure phase state.

4.2. Map from Physical Phase to Observed Phase

Let

$$\iota :\mathrm{\Theta }⟶\mathrm{U}\left(1\right),\iota \left(\left[\theta \right]\right)=e^{\mathrm{i}\theta }$$

be the unit-complex realization of circular phase. Define the observed-phase map by

$$\mathrm{\Phi }:{\mathcal{M}}^{\times }⟶\mathrm{\Theta },\mathrm{\Phi }:={\iota }^{-1}\circ u.$$

(35)

Suppose that on an open set $U\subseteq X_{\varphi }$ the physical state admits local coordinates $(q,\theta )$, where $\theta \in I\subseteq \mathbb{R}$ is a local real lift of $\vartheta$ and q denotes the remaining local degrees of freedom. Then any local real lift of the observed phase can be written on the corresponding joint neighborhood as

$${\phi }_{\mathrm{\Phi }}(x,\eta )=\psi (q,\theta ,\eta ).$$

(36)

Globally, the circular relation remains the map $\mathrm{\Phi }:{\mathcal{M}}^{\times }\to \mathrm{\Theta }$. Equation (36) is used only where the stated coordinates exist; no global product decomposition of $X_{\varphi }$ into a phase variable and remaining variables is assumed.

The functional form of observed phase is determined by g. In general it cannot be assumed to be the identity, a linear map, or a phase translation. In particular, one cannot postulate

$$\mathrm{\Phi }(x,\eta )=\vartheta \left(x\right)+\beta \left(\eta \right).$$

The observer state does not create a second independent phase; it changes the function $g(x,\eta )$ and thereby changes the observed-phase map $\mathrm{\Phi }(x,\eta )$.

4.3. Local Phase-Variation Model

Assume from this point that $X_{\varphi }$ and $S_O$ are smooth manifolds, that $\mathcal{M}$ is a smooth submanifold of $X_{\varphi }\times S_O$, and that g is smooth. Let

$$c\left(t\right)=\left(x\left(t\right),\eta \left(t\right)\right)\in {\mathcal{M}}^{\times }$$

(37)

be a smooth joint-state path and write $g_c\left(t\right):=g\left(c\left(t\right)\right)$. Since $g_c\left(t\right)\ne 0$, choose a continuous real phase $\phi \left(t\right)$ along the path such that

$$g_c\left(t\right)=a_c\left(t\right)e^{\mathrm{i}\phi \left(t\right)},a_c\left(t\right)>0.$$

(38)

Proposition 5

(Local phase variation after the observation map). Along every path satisfying Equation (37),

$$\dot{\phi }\left(t\right)=Im\left({\displaystyle \frac{{\dot{g}}_c\left(t\right)}{g_c\left(t\right)}}\right).$$

(39)

If, on the neighborhood under consideration, $\mathcal{M}$ is open in $X_{\varphi }\times S_O$, or if its tangent space admits the corresponding physical–observer splitting, then

$$\dot{\phi }=Im\left[{\displaystyle \frac{D_{x}g(x,\eta )\left[\dot{x}\right]+D_{\eta }g(x,\eta )\left[\dot{\eta }\right]}{g(x,\eta )}}\right].$$

(40)

Proof. 

From Equation (38),

$${\displaystyle \frac{{\dot{g}}_c}{g_c}}={\displaystyle \frac{{\dot{a}}_c}{a_c}}+\mathrm{i}\dot{\phi }.$$

Taking the imaginary part gives Equation (39). The chain rule gives

$${\dot{g}}_c=D_{x}g(x,\eta )\left[\dot{x}\right]+D_{\eta }g(x,\eta )\left[\dot{\eta }\right],$$

which yields Equation (40). □

Equation (40) is a general local model for phase variation after observation. It does not require linearity of g or functional separability of physical-state and observer-state effects. The two differential terms add only because the joint tangent vector decomposes linearly as

$$(\dot{x},\dot{\eta })=(\dot{x},0)+(0,\dot{\eta }).$$

Whenever this tangent splitting is available, define at the same point $(x,\eta )$

$$\begin{array}{cc}\hfill A_X(x,\eta )\left[v_x\right]& :=Im\left({\displaystyle \frac{D_{x}g(x,\eta )\left[v_x\right]}{g(x,\eta )}}\right),\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill A_O(x,\eta )\left[v_{\eta }\right]& :=Im\left({\displaystyle \frac{D_{\eta }g(x,\eta )\left[v_{\eta }\right]}{g(x,\eta )}}\right).\hfill \end{array}$$

(42)

Then

$$\dot{\phi }=A_X(x,\eta )\left[\dot{x}\right]+A_O(x,\eta )\left[\dot{\eta }\right].$$

(43)

Both $A_X$ and $A_O$ generally depend on the full joint state. Equation (43) is a decomposition of differential directions, not a global additive decomposition of the phase function.

4.4. Fixed Slices and Same-Source Double-State Phase

For fixed observer state ${\eta }_0$, Equation (40) reduces to

$$\dot{\phi }=Im\left({\displaystyle \frac{D_{x}g(x,{\eta }_0)\left[\dot{x}\right]}{g(x,{\eta }_0)}}\right),$$

(44)

which describes the observed-phase variation produced by physical-state change under a fixed observer state.

For fixed physical state $x_{*}$,

$$\dot{\phi }=Im\left({\displaystyle \frac{D_{\eta }g(x_{*},\eta )\left[\dot{\eta }\right]}{g(x_{*},\eta )}}\right),$$

(45)

which describes phase variation produced by changes in the state of the same observer while the physical state remains fixed.

For fixed $x_{*}$ and two states ${\eta }_r,{\eta }_c$ of the same observer, define

$${\mathrm{\Delta }}_{\mathrm{ss}}\mathrm{\Phi }(x_{*};{\eta }_c,{\eta }_r):=\mathrm{\Phi }(x_{*},{\eta }_c)-\mathrm{\Phi }(x_{*},{\eta }_r)\in \mathrm{\Theta }.$$

(46)

This is the same-source double-state phase. Its unit-complex representation is

$$R_{\mathrm{ss}}(x_{*};{\eta }_c,{\eta }_r):=u(x_{*},{\eta }_c)u{(x_{*},{\eta }_r)}^{-1}={\displaystyle \frac{g(x_{*},{\eta }_c)g{(x_{*},{\eta }_r)}^{*}}{|g(x_{*},{\eta }_c)\left|\right|g(x_{*},{\eta }_r)|}}\in \mathrm{U}\left(1\right).$$

(47)

This phase relation is not an error or contamination term. It relates two legitimate observed-phase states formed from the same physical state by two states of the same observer. Because g may be nonlinear and coupled, the relation generally still depends on $x_{*}$. At this stage it is only a finite endpoint relation; it is not yet called transport. A transport interpretation requires a consistent pathwise accumulation law and a proof of existence and uniqueness for the associated comparison rule.

4.5. Conclusion and Next Step

The construction in this section is

$$F_O⟶g=\mathfrak{r}\circ F_O⟶g=\left|g\right|u⟶\mathrm{\Phi }={\iota }^{-1}\circ u.$$

The differential properties of the complex response yield

$$\dot{\phi }=Im\left({\displaystyle \frac{\dot{g}}{g}}\right)=Im\left[{\displaystyle \frac{D_{x}g\left[\dot{x}\right]+D_{\eta }g\left[\dot{\eta }\right]}{g}}\right].$$

Thus the local differential structure of the response function determines observed-phase variation, whereas zero sets and winding determine whether the local law can be accumulated continuously. The next section distinguishes the full response fiber ${\mathbb{C}}^{\times }\cong {\mathbb{R}}_{>0}\times \mathrm{U}\left(1\right)$ from the pure phase factor $\mathrm{U}\left(1\right)$ and constructs the global geometry associated with the local phase law.

5. Complex-Response Bundle, Phase Reduction, and Phase Connection

Section 4 produced the scalar complex response

$$g:\mathcal{M}⟶\mathbb{C}$$

and the local phase law on its nonzero domain,

$$\dot{\phi }=Im\left({\displaystyle \frac{\dot{g}}{g}}\right).$$

We now identify the global geometry corresponding to that law. The order of construction is dictated by the response itself: first retain the full complex response, including amplitude and phase; next reduce it by polar decomposition to a pure phase structure; finally determine the phase connection from the already derived local phase law. Finite transport is deferred to the next section.

5.1. Full Complex-Response Bundle and Nonzero Principal Response Bundle

Over the complete admissible joint-state domain $\mathcal{M}$, define the complex line response bundle

$${\pi }_L:L:=\mathcal{M}\times \mathbb{C}⟶\mathcal{M},{\pi }_L(m,z)=m,$$

(48)

and the response section

$${\sigma }_g:\mathcal{M}⟶L,{\sigma }_g\left(m\right):=(m,g\left(m\right)).$$

(49)

The section may vanish; the full response and its zeros are therefore retained.

Let

$$B:={\mathcal{M}}^{\times }=\{m\in \mathcal{M}:g\left(m\right)\ne 0\}.$$

(50)

Here m abbreviates the joint state $(x,\eta )$. Restricting L to B and removing the zero section yields

$${\pi }_{\times }:P_{\times }:=B\times {\mathbb{C}}^{\times }⟶B,{\mathbb{C}}^{\times }:=\mathbb{C}\setminus \left\{0\right\}.$$

(51)

Under the right multiplication

$$(m,z)·\lambda =(m,z\lambda ),\lambda \in {\mathbb{C}}^{\times },$$

$P_{\times }$ is a trivial principal ${\mathbb{C}}^{\times }$ bundle, and ${\sigma }_g{|}_B$ is a global section. The full nonzero response fiber is therefore not $\mathrm{U}\left(1\right)$ but

$${\mathbb{C}}^{\times }\cong {\mathbb{R}}_{>0}\times \mathrm{U}\left(1\right).$$

The factor $\mathrm{U}\left(1\right)$ represents only pure phase.

5.2. Amplitude–Phase Decomposition and the Principal Phase Bundle

Define

$$\begin{array}{cc}\hfill {\pi }_{\mathrm{amp}}:P_{\mathrm{amp}}& :=B\times {\mathbb{R}}_{>0}⟶B,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\pi }_{\mathrm{ph}}:P_{\mathrm{ph}}& :=B\times \mathrm{U}\left(1\right)⟶B.\hfill \end{array}$$

(53)

The right action on $P_{\mathrm{ph}}$ is

$$(m,u)·v=(m,uv),u,v\in \mathrm{U}\left(1\right).$$

Fiberwise polar decomposition gives the diffeomorphism

$$\mathcal{Q}:P_{\times }⟶P_{\mathrm{amp}}{\times }_B{P}_{\mathrm{ph}},\mathcal{Q}(m,z)=\left(m,\left|z\right|,{\displaystyle \frac{z}{\left|z\right|}}\right).$$

(54)

Hence

$$P_{\times }\cong P_{\mathrm{amp}}{\times }_B{P}_{\mathrm{ph}}.$$

(55)

The response section decomposes into

$$\begin{array}{cccc}\hfill {\sigma }_a\left(m\right)& :=(m,a(m\left)\right),\hfill & \hfill a\left(m\right)& :=\left|g\right(m\left)\right|,\hfill \end{array}$$

(56)

$$\begin{array}{cccc}\hfill {\sigma }_u\left(m\right)& :=(m,u(m\left)\right),\hfill & \hfill u\left(m\right)& :={\displaystyle \frac{g\left(m\right)}{\left|g\right(m\left)\right|}}.\hfill \end{array}$$

(57)

Thus $P_{\times }$ carries the full nonzero response, whereas $P_{\mathrm{ph}}$ carries only normalized phase. The phase connection will be defined on $P_{\mathrm{ph}}$, not on a mistakenly phase-only representation of the full response bundle.

5.3. Differential of the Full Response and the Phase One-Form

On the multiplicative Lie group ${\mathbb{C}}^{\times }$, define the complex Maurer–Cartan form

$${\omega }_{{\mathbb{C}}^{\times }}:=z^{-1}\mathrm{d}z.$$

(58)

On $\mathrm{U}\left(1\right)$ define the standard real phase form

$${\omega }_{\mathrm{U}\left(1\right)}:=Im\left(u^{-1}\mathrm{d}u\right)=-\mathrm{i}{u}^{-1}\mathrm{d}u.$$

(59)

If $z=au$ with $a>0$ and $u\in \mathrm{U}\left(1\right)$, then

$${\omega }_{{\mathbb{C}}^{\times }}=\mathrm{d}loga+\mathrm{i}{\omega }_{\mathrm{U}\left(1\right)}.$$

(60)

This decomposition separates amplitude and phase variation at the differential level.

Pulling the forms back along the response section gives

$${\displaystyle \frac{\mathrm{d}g}{g}}=\mathrm{d}loga+\mathrm{i}A,$$

(61)

where

$$A:=u^{*}{\omega }_{\mathrm{U}\left(1\right)}=Im\left({\displaystyle \frac{\mathrm{d}g}{g}}\right)\in {\mathrm{\Omega }}^1\left(B\right).$$

(62)

The one-form A is the observed-phase one-form induced by the complex response. It is defined on the joint nonzero state domain and is not itself a principal connection. For $v\in T_{m}B$,

$$A_m\left(v\right)=Im\left({\displaystyle \frac{\mathrm{d}{g}_m\left(v\right)}{g\left(m\right)}}\right).$$

(63)

Along a joint-state path $c\left(t\right)$,

$$\dot{\phi }\left(t\right)=A_{c\left(t\right)}\left(\dot{c}\left(t\right)\right),$$

(64)

which recovers the local phase law of the previous section. Hence A contains the first-order phase response in every admissible joint-state direction.

5.4. The Phase Connection Determined by the Local Phase Law

The one-form A assigns a phase rate to each infinitesimal displacement on the base. To compare phase-fiber elements over different base points, one must specify which total-space tangent directions represent continuous preservation of the same phase reference. This is the role of a connection [10,11].

Let ${pr}_{\mathrm{U}\left(1\right)}:P_{\mathrm{ph}}\to \mathrm{U}\left(1\right)$ be the second projection and identify $\mathfrak{u}\left(1\right)$ with $\mathbb{R}$ by $\mathrm{i}a\leftrightarrow a$. A real one-form $\mathrm{\Omega }$ on the principal $\mathrm{U}\left(1\right)$ bundle is a principal phase connection if it reproduces each fundamental vertical generator, $\mathrm{\Omega }\left(a^{\#}\right)=a$, and is invariant under the right $\mathrm{U}\left(1\right)$ action.

No arbitrary connection is selected. Instead, its horizontal condition is required to reproduce the already established local phase law. For any base path $c\left(t\right)$ and lift

$$\tilde{c}\left(t\right)=\left(c\left(t\right),\zeta \left(t\right)\right)\in P_{\mathrm{ph}},$$

horizontality must be equivalent to

$${\omega }_{\mathrm{U}\left(1\right)}\left(\dot{\zeta }\left(t\right)\right)=A_{c\left(t\right)}\left(\dot{c}\left(t\right)\right).$$

(65)

This is a compatibility requirement with the response-induced law, not the introduction of an independent phase dynamics.

Theorem 2

(Existence and uniqueness of the compatible phase connection for a given response). On $P_{\mathrm{ph}}=B\times \mathrm{U}\left(1\right)$ there exists a unique principal $\mathrm{U}\left(1\right)$ connection whose horizontal condition is equivalent to Equation (65). It is

$${\mathrm{\Omega }}_g={pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}-{\pi }_{\mathrm{ph}}^{*}A.$$

(66)

Proof. 

For $\tilde{c}\left(t\right)=(c\left(t\right),\zeta \left(t\right))$,

$${\mathrm{\Omega }}_g\left(\dot{\tilde{c}}\right)={\omega }_{\mathrm{U}\left(1\right)}\left(\dot{\zeta }\right)-A\left(\dot{c}\right).$$

Thus ${\mathrm{\Omega }}_g\left(\dot{\tilde{c}}\right)=0$ if and only if Equation (65) holds. The first term reproduces the fundamental vertical generator, while the second vanishes on vertical vectors. Both terms are right invariant, so ${\mathrm{\Omega }}_g$ is a principal connection.

Any other principal connection differs from ${pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}$ by the pullback of a base one-form,

$$\tilde{\mathrm{\Omega }}={pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}+{\pi }_{\mathrm{ph}}^{*}\beta .$$

If its horizontal condition must also give ${\omega }_{\mathrm{U}\left(1\right)}\left(\dot{\zeta }\right)=A\left(\dot{c}\right)$, then $\beta =-A$, and hence $\tilde{\mathrm{\Omega }}={\mathrm{\Omega }}_g$. □

The observed-phase section ${\sigma }_u\left(m\right)=(m,u\left(m\right))$ satisfies

$${\sigma }_u^{*}{\mathrm{\Omega }}_g=u^{*}{\omega }_{\mathrm{U}\left(1\right)}-A=0.$$

(67)

Thus its horizontality is a consequence of the constructed connection, not an independent condition imposed before the construction.

5.5. Flatness, Zeros, and Validity Boundary

Because $\mathrm{U}\left(1\right)$ is Abelian and $\mathrm{d}{\omega }_{\mathrm{U}\left(1\right)}=0$, and because $A=u^{*}{\omega }_{\mathrm{U}\left(1\right)}$,

$$\mathrm{d}A=0.$$

(68)

The curvature is therefore

$${\mathcal{F}}_g:=\mathrm{d}{\mathrm{\Omega }}_g=-{\pi }_{\mathrm{ph}}^{*}\left(\mathrm{d}A\right)=0.$$

(69)

This flatness follows because the phase form is induced by a globally defined nonzero complex response; it is not a universal property of arbitrary principal $\mathrm{U}\left(1\right)$ connections. Since $u:B\to \mathrm{U}\left(1\right)$ is a global section, the connection is pure gauge. Circular transport around a closed loop is the identity, although a continuous real lift may accumulate $2\pi n$. The circular factor and unwrapped real accumulation are distinct.

Three boundaries must be kept separate:

(1)

If $g\left(m\right)=0$, the complex line response bundle and response section still exist, but $u=g/\left|g\right|$ does not. No phase section, phase one-form, or phase connection is canonically induced at that point.

(w)

If $g\left(m\right)\ne 0$ but $A_m=0$, phase remains readable; its first-order variation simply vanishes in all admissible directions at that point.

(3)

The phase connection governs comparison in the pure phase fiber and does not replace amplitude evolution, which is described independently by

$$Re\left({\displaystyle \frac{\mathrm{d}g}{g}}\right)=\mathrm{d}log\left|g\right|.$$

5.6. Conclusion and Next Step

The construction is

$$F_O⟶g⟶P_{\times }\cong P_{\mathrm{amp}}{\times }_B{P}_{\mathrm{ph}}⟶A⟶{\mathrm{\Omega }}_g.$$

The full response fiber is ${\mathbb{C}}^{\times }$; the phase fiber is $\mathrm{U}\left(1\right)$; $A=Im(\mathrm{d}g/g)$ is a base one-form; and ${\mathrm{\Omega }}_g$ is the unique principal phase connection reproducing the response-induced local phase law. The next section uses the horizontal condition ${\mathrm{\Omega }}_g\left(\dot{\tilde{c}}\right)=0$ to define the basic phase-transport equation, prove its fundamental status, and derive same-source double-state inverse transport.

6. Basic Phase-Transport Equation: Unified Differential, Covariant, and Path-Functional Forms

Section 5 constructed, on the nonzero joint-state domain

$$B={\mathcal{M}}^{\times }=\{m\in \mathcal{M}:g\left(m\right)\ne 0\},$$

the observed-phase one-form

$$A_g:=Im\left({\displaystyle \frac{\mathrm{d}g}{g}}\right)=u^{*}{\omega }_{\mathrm{U}\left(1\right)},u:={\displaystyle \frac{g}{\left|g\right|}},$$

(70)

and the unique compatible principal $\mathrm{U}\left(1\right)$ connection

$${\mathrm{\Omega }}_g={pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}-{\pi }_{\mathrm{ph}}^{*}{A}_g.$$

(71)

No additional phase-dynamics assumption is introduced here. Instead, the horizontal condition of this unique compatible connection is promoted to a global fundamental equation for phase transport. Ordinary time-differential, covariant-derivative, real-phase, and finite path-functional forms will be proved equivalent to, or uniquely determined by, that equation.

The word “fundamental” is used in a precise sense. The equation is the minimal first-order phase law uniquely induced by the nonzero complex response formed through the observation map. It is equivalent to horizontal lifting in the principal phase bundle; prescribed initial data determine finite phase transport completely; and its form is independent of local phase coordinates, arbitrary zero-phase references, and positive amplitude rescaling. It is not a dynamical equation for the physical state or observer state. It is the geometric–kinematic equation that determines how the phase fiber is carried along a given joint-state path.

6.1. Lifted Path Space and Global Differential Functional

Let $I=[t_0,t_1]\subset \mathbb{R}$ be compact and define the piecewise-$C^1$ base-path space

$${\mathcal{C}}^1(I,B):=\{c:I\to B:c\mathrm{is}\mathrm{piecewise}{C}^1\}.$$

(72)

The corresponding lifted path space on ${\pi }_{\mathrm{ph}}:P_{\mathrm{ph}}\to B$ is

$${\mathcal{C}}_{\pi }^1(I,P_{\mathrm{ph}}):=\left\{\tilde{c}:I\to P_{\mathrm{ph}}:\tilde{c}\mathrm{is}\mathrm{piecewise}{C}^1,{\pi }_{\mathrm{ph}}\circ \tilde{c}\in {\mathcal{C}}^1(I,B)\right\}.$$

(73)

For $\tilde{c}\in {\mathcal{C}}_{\pi }^1(I,P_{\mathrm{ph}})$, write

$$c:={\pi }_{\mathrm{ph}}\circ \tilde{c}.$$

Let

$${\mathrm{\Omega }}_{\mathrm{pc}}^1\left(I\right):=\{f\left(t\right)\mathrm{d}t:f\mathrm{is}\mathrm{piecewise}\mathrm{continuous}\mathrm{on}I\}.$$

(74)

If a path is only piecewise $C^1$, the pullback of a smooth one-form is generally only piecewise continuous at the break points.

Definition 5

(Phase-transport differential functional). The compatible connection ${\mathrm{\Omega }}_g$ defines

$${\mathfrak{E}}_g:{\mathcal{C}}_{\pi }^1(I,P_{\mathrm{ph}})⟶{\mathrm{\Omega }}_{\mathrm{pc}}^1\left(I\right),{\mathfrak{E}}_g\left[\tilde{c}\right]:={\tilde{c}}^{*}{\mathrm{\Omega }}_g.$$

(75)

The operator ${\mathfrak{E}}_g$ is called the phase-transport differential functional induced by g.

Here “functional” means that the input is an entire lifted path and the output is a one-form on the parameter interval. It is not an action functional in the calculus of variations and carries no optimization principle. Since I is one-dimensional, there is a unique piecewise continuous function $e_{g,\tilde{c}}:I\to \mathbb{R}$ such that

$${\mathfrak{E}}_g\left[\tilde{c}\right]=e_{g,\tilde{c}}\left(t\right)\mathrm{d}t.$$

Thus a one-form equation on path space is equivalent to a pointwise first-order equation on each $C^1$ subinterval, with continuous concatenation across break points.

6.2. The Basic Phase-Transport Equation and Equivalent Forms

Definition 6

(Basic phase-transport equation). Let $g:B\to {\mathbb{C}}^{\times }$ be a smooth nonzero complex response and let ${\mathrm{\Omega }}_g$ be its unique compatible phase connection. The global differential functional equation

$${\mathfrak{E}}_g\left[\tilde{c}\right]={\tilde{c}}^{*}{\mathrm{\Omega }}_g=0$$

(PT)

for $\tilde{c}\in {\mathcal{C}}_{\pi }^1(I,P_{\mathrm{ph}})$ is called thebasic phase-transport equation.

Equation (PT) is independent of a local trivialization of the phase bundle and of any chosen real phase branch. It is the intrinsic global form. The standard differential representations are now shown to be equivalent to it.

Theorem 3

(Equivalent representations of the basic phase-transport equation). Let $c\in {\mathcal{C}}^1(I,B)$ and write, in the trivial principal phase bundle,

$$\tilde{c}\left(t\right)=\left(c\left(t\right),\zeta \left(t\right)\right),\zeta \left(t\right)\in \mathrm{U}\left(1\right).$$

The following statements are equivalent. Equations containing time derivatives hold on each $C^1$ subinterval, equivalently almost everywhere except at finitely many break points.

(1)

Global differential-functional form:

$${\tilde{c}}^{*}{\mathrm{\Omega }}_g=0.$$

(76)

(2)

Local group-valued differential form:

$${\omega }_{\mathrm{U}\left(1\right)}\left(\dot{\zeta }\left(t\right)\right)=A_{g,c\left(t\right)}\left(\dot{c}\left(t\right)\right)=Im\left({\displaystyle \frac{\frac{\mathrm{d}}{\mathrm{d}t}g\left(c\left(t\right)\right)}{g\left(c\right(t\left)\right)}}\right).$$

(77)

(3)

Covariant-derivative form:

$$D_t^{(g,c)}\zeta :=\dot{\zeta }-\mathrm{i}{A}_g\left(\dot{c}\right)\zeta =0.$$

(78)

(4)

Continuous real-phase form:if $\psi :I\to \mathbb{R}$ is a continuous lift with $\zeta =e^{\mathrm{i}\psi }$, then

$$\mathrm{d}\psi =c^{*}{A}_g,\dot{\psi }\left(t\right)=Im\left({\displaystyle \frac{\frac{\mathrm{d}}{\mathrm{d}t}g\left(c\left(t\right)\right)}{g\left(c\right(t\left)\right)}}\right).$$

(79)

(5)

Physical–observer directional form:if, near the path, $\mathcal{M}$ is open in $X_{\varphi }\times S_O$, or if g extends to a product neighborhood and $\dot{c}=(\dot{x},\dot{\eta })$ admits the corresponding splitting, then

$$\dot{\psi }=Im\left[{\displaystyle \frac{D_{x}g(x,\eta )\left[\dot{x}\right]+D_{\eta }g(x,\eta )\left[\dot{\eta }\right]}{g(x,\eta )}}\right].$$

(80)

Proof. 

From Equation (71),

$${\tilde{c}}^{*}{\mathrm{\Omega }}_g={\zeta }^{*}{\omega }_{\mathrm{U}\left(1\right)}-c^{*}{A}_g.$$

Thus Equation (76) is equivalent to ${\zeta }^{*}{\omega }_{\mathrm{U}\left(1\right)}=c^{*}{A}_g$, which evaluated on ${\partial }_t$ gives Equation (77). Since

$${\omega }_{\mathrm{U}\left(1\right)}=-\mathrm{i}{\zeta }^{-1}\mathrm{d}\zeta ,$$

that equation is equivalent to $-\mathrm{i}{\zeta }^{-1}\dot{\zeta }=A_g\left(\dot{c}\right)$, hence to Equation (78). If $\zeta =e^{\mathrm{i}\psi }$, then $-\mathrm{i}{\zeta }^{-1}\dot{\zeta }=\dot{\psi }$, giving Equation (79). The chain rule gives

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}g\left(c\left(t\right)\right)=D_{x}g(x,\eta )\left[\dot{x}\right]+D_{\eta }g(x,\eta )\left[\dot{\eta }\right],$$

which yields Equation (80). □

The addition in Equation (80) arises from linearity of the tangent vector and differential. It does not imply that observed phase globally decomposes into independent physical and observer phase functions. The intrinsic objects remain the joint-state path c, the response $g\circ c$, and the lifted path $\tilde{c}$.

6.3. Fundamentality Theorem

Theorem 4

(Fundamental status of the basic phase-transport equation). Given a smooth nonzero response $g:B\to {\mathbb{C}}^{\times }$ and the standard radian normalization, Equation (PT) has the following properties.

(1)

Necessary induction.The equation is uniquely induced through

$$g⟼u={\displaystyle \frac{g}{\left|g\right|}}⟼A_g=u^{*}{\omega }_{\mathrm{U}\left(1\right)}⟼{\mathrm{\Omega }}_g$$

and contains no independent empirical phase term.

(2)

Geometric equivalence and uniqueness.A lifted path satisfies the equation if and only if it is horizontal for the unique compatible connection ${\mathrm{\Omega }}_g$. Any principal $\mathrm{U}\left(1\right)$ connection reproducing the same local phase law for every base path equals ${\mathrm{\Omega }}_g$.

(3)

Uniqueness of the first-order law.If $K\in {\mathrm{\Omega }}^1\left(B\right)$ satisfies, for every $C^1$ path c,

$${\omega }_{\mathrm{U}\left(1\right)}\left({\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}u\left(c\left(t\right)\right)\right)=K_{c\left(t\right)}\left(\dot{c}\left(t\right)\right),$$

then $K=A_g$. No different intrinsic first-order law reproduces the same observed-phase variation in every joint-state direction.

(4)

Initial-value completeness.For every base path c and every initial fiber state $(c\left(t_0\right),{\zeta }_0)$, the equation has a unique solution and determines the phase state at every point of the path and the finite endpoint transport.

(5)

Path-functional and composition completeness.The equation uniquely induces a real phase-increment functional, a circular transport functional, and a fiber transport operator. These satisfy identity on constant paths, inversion under path reversal, and composition under path concatenation.

(6)

Reparametrization invariance.For every $C^1$ diffeomorphism $\lambda :J\to I$, $\tilde{c}$ satisfies the equation if and only if $\tilde{c}\circ \lambda$ does. The equation is independent of the particular time parameter.

(7)

Local phase-gauge covariance.For any smooth $q:B\to \mathrm{U}\left(1\right)$, let

$$g^{\prime }:=qg,u_{g^{\prime }}=qu,A_{g^{\prime }}=A_g+q^{*}{\omega }_{\mathrm{U}\left(1\right)},{\zeta }^{\prime }=q\left(c\right)\zeta .$$

The transformed equation has exactly the same form. Constant q corresponds to a zero-phase reference change; state-dependent q is a local phase-gauge transformation.

(8)

Phase-level minimality.For every positive smooth $\rho :B\to {\mathbb{R}}_{>0}$, $g^{\prime }=\rho g$ induces the same equation. The equation retains the first-order information required for phase transport while removing arbitrary positive amplitude rescaling; it does not require linearity, separability, or a particular engineering realization.

Proof. 

Part (1) follows from Equations (70) and (71) and the existence–uniqueness result for the compatible connection.

For part (2),

$${\mathrm{\Omega }}_g\left(\dot{\tilde{c}}\right)={\omega }_{\mathrm{U}\left(1\right)}\left(\dot{\zeta }\right)-A_g\left(\dot{c}\right).$$

Any other principal connection in the present trivialization has the form

$$\tilde{\mathrm{\Omega }}={pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}+{\pi }_{\mathrm{ph}}^{*}\beta .$$

If its horizontal condition reproduces ${\omega }_{\mathrm{U}\left(1\right)}\left(\dot{\zeta }\right)=A_g\left(\dot{c}\right)$ for every path, then $\beta =-A_g$, so $\tilde{\mathrm{\Omega }}={\mathrm{\Omega }}_g$.

For part (3), choose any $m\in B$ and $v\in T_{m}B$, and a local $C^1$ path with $c\left(0\right)=m$ and $\dot{c}\left(0\right)=v$. Since $A_g=u^{*}{\omega }_{\mathrm{U}\left(1\right)}$,

$$K_m\left(v\right)={\omega }_{\mathrm{U}\left(1\right)}\left(\mathrm{d}{u}_m\left(v\right)\right)={\left(A_g\right)}_m\left(v\right).$$

Because m and v are arbitrary, $K=A_g$.

For part (4), define

$$\zeta \left(t\right):={\zeta }_{0}u{\left(c\left(t_0\right)\right)}^{-1}u\left(c\left(t\right)\right).$$

(81)

The identity $A_g=u^{*}{\omega }_{\mathrm{U}\left(1\right)}$ verifies that this is a solution; uniqueness of horizontal lifts makes it the only solution.

Part (5) follows by evaluating Equation (81) at endpoints, reversed paths, and concatenated paths. For part (6), functoriality of pullback gives

$${(\tilde{c}\circ \lambda )}^{*}{\mathrm{\Omega }}_g={\lambda }^{*}\left({\tilde{c}}^{*}{\mathrm{\Omega }}_g\right).$$

Since $\lambda$ is a diffeomorphism, either side vanishes if and only if the other does. For part (7), commutativity of $\mathrm{U}\left(1\right)$ gives

$$u_{g^{\prime }}^{*}{\omega }_{\mathrm{U}\left(1\right)}=u^{*}{\omega }_{\mathrm{U}\left(1\right)}+q^{*}{\omega }_{\mathrm{U}\left(1\right)},$$

and

$${\omega }_{\mathrm{U}\left(1\right)}\left({\dot{\zeta }}^{\prime }\right)=q^{*}{\omega }_{\mathrm{U}\left(1\right)}\left(\dot{c}\right)+{\omega }_{\mathrm{U}\left(1\right)}\left(\dot{\zeta }\right),$$

so the form of the equation is unchanged.

For part (8), if $g^{\prime }=\rho g$ and $\rho >0$, then

$${\displaystyle \frac{g^{\prime }}{|g^{\prime }|}}={\displaystyle \frac{g}{\left|g\right|}}=u,Im\left({\displaystyle \frac{\mathrm{d}{g}^{\prime }}{g^{\prime }}}\right)=Im\left({\displaystyle \frac{\mathrm{d}\rho }{\rho }}+{\displaystyle \frac{\mathrm{d}g}{g}}\right)=A_g.$$

Therefore $A_g$, ${\mathrm{\Omega }}_g$, and the equation are unchanged. □

Remark 2

(Theoretical level of the equation). The basic phase-transport equation is not an empirical endpoint formula inferred from finite phase ratios. Its logical order is

$$F_O⟶g⟶A_g⟶{\mathrm{\Omega }}_g⟶{\mathfrak{E}}_g\left[\tilde{c}\right]=0⟶\mathrm{finite}\mathrm{transport}\mathrm{solutions}.$$

Finite phase differences, endpoint factors, and same-source inverse transport are solutions or slice consequences, not the basis of the definition.

Remark 3

(Exact boundary of universality). The terms “fundamental” and “unified” apply to the explicitly specified model class: the coherent channel is represented by a globally smooth scalar nonzero response $g:B\to {\mathbb{C}}^{\times }$, and the path remains in B. Within this class, Equation (PT) is the unique compatible first-order phase-transport equation. Multicomponent responses may be treated channelwise or by a joint bundle extension. Nontrivial principal bundles, only locally defined responses, and zero crossings require additional patching or singular structures and are not covered unconditionally by the present trivial-bundle theorem.

6.4. Finite Path Functionals and Transport Operator

The basic equation uniquely induces three path objects.

Definition 7

(Phase-increment functional, transport functional, and transport operator). For $c\in {\mathcal{C}}^1(I,B)$, define

$${\mathrm{\Delta }}_g\left[c\right]:={\int }_c{A}_g={\int }_{c}Im\left({\displaystyle \frac{\mathrm{d}g}{g}}\right)\in \mathbb{R},$$

(82)

$${\mathcal{T}}_g\left[c\right]:=exp\left(\mathrm{i}{\mathrm{\Delta }}_g\left[c\right]\right)=exp\left(\mathrm{i}{\int }_c{A}_g\right)\in \mathrm{U}\left(1\right),$$

(83)

and, using fiber coordinates in $P_{\mathrm{ph}}=B\times \mathrm{U}\left(1\right)$,

$${\mathcal{P}}_c^g:{\left(P_{\mathrm{ph}}\right)}_{c\left(t_0\right)}⟶{\left(P_{\mathrm{ph}}\right)}_{c\left(t_1\right)},{\mathcal{P}}_c^g\left(c\left(t_0\right),{\zeta }_0\right):=\left(c\left(t_1\right),{\zeta }_0{\mathcal{T}}_g\left[c\right]\right).$$

(84)

From Equation (81),

$${\mathcal{T}}_g\left[c\right]=u{\left(c\left(t_0\right)\right)}^{-1}u\left(c\left(t_1\right)\right)={\displaystyle \frac{g{\left(c\left(t_0\right)\right)}^{*}g\left(c\left(t_1\right)\right)}{|g\left(c\left(t_0\right)\right)\left|\right|g\left(c\left(t_1\right)\right)|}}.$$

(85)

Thus Equations (82) and (84) are finite-solution representations of the basic equation. If $c_1$ runs from $m_0$ to $m_1$, $c_2$ from $m_1$ to $m_2$, and $c_2*c_1$ means first $c_1$ and then $c_2$, then

$$\begin{array}{cc}\hfill {\mathcal{P}}_{c_2*c_1}^g& ={\mathcal{P}}_{c_2}^g\circ {\mathcal{P}}_{c_1}^g,\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill {\mathcal{T}}_g[c_2*c_1]& ={\mathcal{T}}_g\left[c_2\right]{\mathcal{T}}_g\left[c_1\right],\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill {\mathcal{T}}_g\left[\overline{c}\right]& ={\mathcal{T}}_g{\left[c\right]}^{-1},\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill {\mathcal{T}}_g\left[c_{\mathrm{const}}\right]& =1.\hfill \end{array}$$

(89)

Because $A_g$ is induced by the global normalized response u, the circular transport functional depends only on endpoints. The real increment retains winding. For two paths $c_1,c_2$ with the same endpoints,

$${\mathrm{\Delta }}_g\left[c_1\right]-{\mathrm{\Delta }}_g\left[c_2\right]=2\pi n,n\in \mathbb{Z},$$

(90)

while ${\mathcal{T}}_g\left[c_1\right]={\mathcal{T}}_g\left[c_2\right]$. Unwrapped phase accumulation and circular fiber transport must therefore remain distinct.

6.5. Same-Source Double-State Inverse Transport on a Fixed Physical-State Slice

Fix $x_{*}$ and define

$$B_{x_{*}}:=\{\eta \in S_O:(x_{*},\eta )\in B\}.$$

(91)

Let ${\eta }_r,{\eta }_c\in B_{x_{*}}$ lie in the same path component, and choose any piecewise-$C^1$ path

$${\gamma }_{r\to c}:[0,1]\to B_{x_{*}},{\gamma }_{r\to c}\left(0\right)={\eta }_r,{\gamma }_{r\to c}\left(1\right)={\eta }_c.$$

The same-source double-state factor is

$$R_{\mathrm{ss}}(x_{*};{\eta }_c,{\eta }_r):=u(x_{*},{\eta }_c)u{(x_{*},{\eta }_r)}^{-1}.$$

(92)

It is the finite solution of the basic equation along the fixed-physical-state path $s\mapsto (x_{*},{\gamma }_{r\to c}\left(s\right))$ from ${\eta }_r$ to ${\eta }_c$. Because the connection is induced by a global normalized response, the circular transport factor depends only on the endpoints.

Theorem 5

(Same-source double-state inverse phase transport). The unique inverse phase-transport factor from the current observer state ${\eta }_c$ to the reference state ${\eta }_r$ is

$$T_{{\eta }_r\leftarrow {\eta }_c}\left(x_{*}\right):=R_{\mathrm{ss}}{(x_{*};{\eta }_c,{\eta }_r)}^{-1}=u(x_{*},{\eta }_r)u{(x_{*},{\eta }_c)}^{-1}.$$

(93)

It satisfies

$$T_{{\eta }_r\leftarrow {\eta }_c}\left(x_{*}\right)u(x_{*},{\eta }_c)=u(x_{*},{\eta }_r),$$

(94)

and in complex-response coordinates

$$T_{{\eta }_r\leftarrow {\eta }_c}\left(x_{*}\right)={\displaystyle \frac{g(x_{*},{\eta }_r)g{(x_{*},{\eta }_c)}^{*}}{|g(x_{*},{\eta }_r)\left|\right|g(x_{*},{\eta }_c)|}}.$$

(95)

Proof. 

By inversion of the finite transport functional, the factor from current to reference is the inverse of the reference-to-current factor, giving Equation (93). Substitution gives Equation (94). If $T^{\prime }$ also satisfies $T^{\prime }u(x_{*},{\eta }_c)=u(x_{*},{\eta }_r)$, right multiplication by $u{(x_{*},{\eta }_c)}^{-1}$ gives $T^{\prime }=T_{{\eta }_r\leftarrow {\eta }_c}$. Equation (95) follows from $u=g/\left|g\right|$. □

If ${\eta }_r$ and ${\eta }_c$ lie in different path components of $B_{x_{*}}$, Equation (93) remains a valid algebraic conversion between two nonzero endpoint phases, but it cannot be interpreted as a solution of the basic equation along a continuous path within the current domain.

Inverse transport does not remove an artificially appended phase error. It converts the legitimate observed phase formed at the current observer state into the representation formed at the reference observer state for the same physical state. The factor generally depends on $x_{*}$ and cannot be simplified a priori to a constant depending only on ${\eta }_r$ and ${\eta }_c$.

Phase conversion and full complex-response conversion are different. Phase inverse transport gives

$$T_{{\eta }_r\leftarrow {\eta }_c}\left(x_{*}\right)g(x_{*},{\eta }_c)=\left|g(x_{*},{\eta }_c)\right|u(x_{*},{\eta }_r),$$

(96)

which retains the current amplitude. Recovering the full reference response additionally requires

$$g(x_{*},{\eta }_r)={\displaystyle \frac{\left|g\right(x_{*},{\eta }_r\left)\right|}{\left|g\right(x_{*},{\eta }_c\left)\right|}}{T}_{{\eta }_r\leftarrow {\eta }_c}\left(x_{*}\right)g(x_{*},{\eta }_c).$$

(97)

Thus inverse phase transport belongs to the pure phase level and does not replace amplitude conversion at the full-response level.

6.6. Pointwise Conversion of Dynamic Observations to a Reference State

Suppose the system follows

$$t⟼\left(x\left(t\right),{\eta }_c\left(t\right)\right)$$

and fix a reference state ${\eta }_r$ of the same observer. Whenever

$$g(x\left(t\right),{\eta }_c\left(t\right))\ne 0,g(x\left(t\right),{\eta }_r)\ne 0,$$

define

$$T_r\left(t\right):=u(x\left(t\right),{\eta }_r)u{(x\left(t\right),{\eta }_c\left(t\right))}^{-1}.$$

(98)

Then

$$u_{\mathrm{tr}}\left(t\right):=T_r\left(t\right)u(x\left(t\right),{\eta }_c\left(t\right))=u(x\left(t\right),{\eta }_r).$$

(99)

Equation (99) is first a pointwise algebraic identity. If, for each t, ${\eta }_r$ and ${\eta }_c\left(t\right)$ lie in the same path component of $B_{x\left(t\right)}$ and a continuously varying family of connecting paths can be selected, it is also the continuous application of the basic equation over a family of fixed-physical-state slices. The converted path equals exactly the observed-phase path on the reference-state slice,

$$x\left(t\right)⟼u(x\left(t\right),{\eta }_r).$$

If ${\eta }_r$ is fixed and the corresponding function is smooth, a continuous real phase satisfies

$${\dot{\phi }}_r\left(t\right)=Im\left({\displaystyle \frac{D_{x}g(x\left(t\right),{\eta }_r)\left[\dot{x}\left(t\right)\right]}{g(x\left(t\right),{\eta }_r)}}\right).$$

(100)

This is the observed-phase evolution under the reference observer state. Equality with intrinsic physical-phase evolution requires a phase-faithfulness or calibration condition on the reference slice and does not follow from transport alone.

6.7. Position of the ${\mathsf{G}}_0$–${\mathsf{G}}_1$–${\mathsf{G}}_2$ hierarchy

Two completed representation levels can now be named.

Definition 8

(${\mathsf{G}}_0$ full complex-response level). The structure

$${\mathsf{G}}_0:=\left(\mathcal{M},F_O{|}_{\mathcal{M}},\mathfrak{r},g,L,{\sigma }_g\right)$$

(101)

formed by the admissible joint-state domain, observation map, coherent readout, full complex response, response bundle, and response section is called the ${\mathsf{G}}_0$ full complex-response level. It retains amplitude, phase, the zero-response set, and the identities of the physical and observer variables.

Definition 9

(${\mathsf{G}}_1$ phase-differential and transport level). The structure

$${\mathsf{G}}_1:=\left(B,u,A_g,P_{\mathrm{ph}},{\mathrm{\Omega }}_g,{\mathfrak{E}}_g,{\left\{D_t^{(g,c)}\right\}}_c,{\mathrm{\Delta }}_g,{\mathcal{T}}_g,{\left\{{\mathcal{P}}_c^g\right\}}_c\right)$$

(102)

is called the ${\mathsf{G}}_1$ phase-differential and transport level. It retains first-order phase variation, covariant evolution, and cross-state comparison, but not full amplitude or phase structure at zero-response points.

The ${\mathsf{G}}_0$–${\mathsf{G}}_1$–${\mathsf{G}}_2$ hierarchy is not the same classification as the three-layer observation paradigm. The three-layer paradigm separates physical world, observer, and observed world. The ${\mathsf{G}}_j$ hierarchy separates representation and reduction levels of the same observation problem.

Section 7 constructs ${\mathsf{G}}_2$ after phase transport is complete. It requires a task-prescribed group action, quotient observation, maximal invariants, and recoverability. A single scalar complex channel cannot by itself determine which freedoms a task should remove; ${\mathsf{G}}_2$ is determined jointly by the retained task quantity, the nuisance group action, and the joint data structure.

6.8. Conclusion and Validity Boundary

The chain completed in this section is

$$\begin{array}{cc}\hfill F_O& ⟶g⟶A_g⟶{\mathrm{\Omega }}_g⟶{\mathfrak{E}}_g\left[\tilde{c}\right]=0,\hfill \\ \hfill & ⟶\left(D_t^{(g,c)}\zeta =0,\mathrm{d}\psi =c^{*}{A}_g\right)⟶\left({\mathrm{\Delta }}_g,{\mathcal{T}}_g,{\mathcal{P}}_c^g\right),\hfill \\ \hfill & ⟶\mathrm{same}-\mathrm{source}\mathrm{double}-\mathrm{state}\mathrm{inverse}\mathrm{transport}.\hfill \end{array}$$

(103)

Equation (PT) is thereby established as the basic phase-transport equation. Its status follows not from nomenclature but because it is the unique global differential-functional form of the first-order phase law induced by the observation map; it is equivalent to horizontal lifting by the unique compatible connection; its group-valued, covariant, and real-phase forms are equivalent; prescribed initial data determine finite path transport completely; it is gauge covariant and invariant under positive amplitude rescaling.

The equation is valid while paths remain in the nonzero response domain B, with the required smoothness of the response and path, and while the transported object is the normalized phase fiber. At a crossing of $g=0$, response-induced phase, the phase one-form, and the phase connection cease to be defined; model-specific local expansions, reinitialization, or other extensions are then required. The equation transports phase along a given base path; it does not predict the path or replace amplitude evolution. A same-source endpoint factor is a continuous transport solution only if a connecting path exists within the fixed-physical-state nonzero slice. Full response recovery still requires amplitude information. Task reduction, maximal invariants, and recoverability are developed next.

7. ${\mathsf{G}}_2$: Task-Quotient Observation, Maximal Invariants, and Recoverability

Section 6 unified phase comparison across joint-state points through horizontal lifting by the unique compatible connection and obtained finite path transport and same-source double-state inverse transport. Once this reference unification is complete, a different question remains: for a prescribed task, which variations in the observed data must be retained, and which represent task-irrelevant coordinates, common scales, or common references that should be removed from the final representation?

Neither the complex response g nor the basic phase-transport equation can answer this question alone. The same transport-unified data may serve different tasks, and different tasks may regard different freedoms as irrelevant. Consequently, ${\mathsf{G}}_2$ is neither imposed a priori nor generated automatically from ${\mathsf{G}}_0$ or ${\mathsf{G}}_1$. It is determined only after phase transport, jointly by the task-specific data object, the transformation group to be removed, and the recoverability of the target. We first construct the quotient set-theoretically and add smooth-quotient hypotheses only when differential geometry is required.

7.1. Task Data after Transport Unification

Let ${\mathcal{D}}_{\mathrm{task}}$ collect the physical states, observer states, times, channels, or data blocks admissible for a particular task. The full response of ${\mathsf{G}}_0$ and the phase transport of ${\mathsf{G}}_1$ can be assembled into a task-specific transport-unified data map

$$Z_{\mathrm{tr}}:{\mathcal{D}}_{\mathrm{task}}⟶\mathcal{Z},d⟼Z_{\mathrm{tr}}\left(d\right),$$

(104)

where $\mathcal{Z}$ is the task data space. It may consist of a multitime sequence, a multichannel vector, a multisource complex-response matrix, a collection of relative phases, or another joint data structure. No assumption is made that $\mathcal{Z}$ is a single complex scalar; nontrivial task quotients generally require several related observation components.

The term “transport-unified” means that every phase component requiring comparison across observer states has been converted to the common reference prescribed by the task. It does not mean that amplitude, channel gains, or other freedoms have already been removed. ${\mathsf{G}}_1$ establishes consistent phase-reference comparison, whereas ${\mathsf{G}}_2$ performs task-dependent quotient reduction. The two operations are distinct.

7.2. Task-Group Action, Equivalence, and Quotient Observation

Definition 10

(Task-irrelevant transformation group). Let $\mathcal{K}$ be a group with a left action on $\mathcal{Z}$,

$$\rho :\mathcal{K}\times \mathcal{Z}⟶\mathcal{Z},\rho (k,z)=k·z.$$

(105)

If the task declares all data on the same orbit equivalent, so that $k·z$ and z are indistinguishable with respect to the target quantity, then $\mathcal{K}$ is called the task-irrelevant transformation group.

Task irrelevance is part of the task definition; it is not guaranteed by the notation of a group action. Whether a common complex scale, common phase, channel reweighting, or coordinate transformation should be removed must be specified by the target under study.

Definition 11

(Task equivalence and quotient observation space). For $z_1,z_2\in \mathcal{Z}$, define

$$z_1{\sim }_{\mathcal{K}}{z}_2⟺\exists k\in \mathcal{K}\mathrm{such}\mathrm{that}{z}_2=k·z_1.$$

(106)

The class

$${\left[z\right]}_{\mathcal{K}}:=\mathcal{K}·z=\{k·z:k\in \mathcal{K}\}$$

(107)

is the task orbit of z. The set of all orbits is the quotient observation space

$${\mathcal{Q}}_{\mathcal{K}}:=\mathcal{Z}/\mathcal{K},q_{\mathcal{K}}:\mathcal{Z}\to {\mathcal{Q}}_{\mathcal{K}},q_{\mathcal{K}}\left(z\right)={\left[z\right]}_{\mathcal{K}}.$$

(108)

The group axioms make ${\sim }_{\mathcal{K}}$ an equivalence relation. The quotient is initially a set. If $\mathcal{Z}$ is topological and the action is continuous, it carries the quotient topology. If $\mathcal{K}$ is a Lie group, $\mathcal{Z}$ a smooth manifold, and the action smooth, free, and proper, then $\mathcal{Z}/\mathcal{K}$ has a unique smooth-manifold structure for which $q_{\mathcal{K}}$ is a smooth submersion [12]. If the action has nontrivial stabilizers or is not proper, the quotient may be singular or stratified and should not be forced into the category of ordinary manifolds.

7.3. Factorization of Invariants and Maximal Invariants

Definition 12

(Task invariant). A map

$$I:\mathcal{Z}⟶\mathcal{W}$$

(109)

is a task invariant with respect to $\mathcal{K}$ if

$$I(k·z)=I\left(z\right),k\in \mathcal{K},z\in \mathcal{Z}.$$

(110)

Theorem 6

(Quotient factorization of task invariants). A map $I:\mathcal{Z}\to \mathcal{W}$ is $\mathcal{K}$-invariant if and only if there exists a unique map

$$\overline{I}:{\mathcal{Q}}_{\mathcal{K}}⟶\mathcal{W}$$

(111)

such that

$$I=\overline{I}\circ q_{\mathcal{K}}.$$

(112)

Proof. 

If I is invariant, define $\overline{I}\left({\left[z\right]}_{\mathcal{K}}\right):=I\left(z\right)$. If ${\left[z_1\right]}_{\mathcal{K}}={\left[z_2\right]}_{\mathcal{K}}$, then $z_2=k·z_1$ for some k, and invariance gives $I\left(z_2\right)=I\left(z_1\right)$. Thus the definition is independent of the representative and Equation (112) holds. Surjectivity of $q_{\mathcal{K}}$ gives uniqueness.

Conversely, if $I=\overline{I}\circ q_{\mathcal{K}}$, then

$$I(k·z)=\overline{I}\left({[k·z]}_{\mathcal{K}}\right)=\overline{I}\left({\left[z\right]}_{\mathcal{K}}\right)=I\left(z\right),$$

so I is invariant. □

The theorem shows that an invariant is not an object parallel to the quotient; every invariant is necessarily a representation of quotient information. The quotient observation $q_{\mathcal{K}}\left(z\right)$ retains all orbit-level information, whereas a particular invariant provides coordinates or features on that information.

Definition 13

(Maximal invariant). A task invariant $I:\mathcal{Z}\to \mathcal{W}$ is maximal if

$$I\left(z_1\right)=I\left(z_2\right)⟺z_1{\sim }_{\mathcal{K}}{z}_2.$$

(113)

Theorem 7

(Equivalence between maximal invariants and quotient observations). Let $I:\mathcal{Z}\to \mathcal{W}$ be invariant and let $\overline{I}$ be the induced map from 6. Then I is maximal if and only if

$$\overline{I}:{\mathcal{Q}}_{\mathcal{K}}⟶I\left(\mathcal{Z}\right)$$

(114)

is a bijection. Thus the range of a maximal invariant is canonically equivalent to the quotient observation space.

Proof. 

The map $\overline{I}$ is always surjective onto $I\left(\mathcal{Z}\right)$. If I is maximal and $\overline{I}\left(\left[z_1\right]\right)=\overline{I}\left(\left[z_2\right]\right)$, then $I\left(z_1\right)=I\left(z_2\right)$, hence $\left[z_1\right]=\left[z_2\right]$; therefore $\overline{I}$ is injective.

Conversely, if $\overline{I}$ is bijective, then $I\left(z_1\right)=I\left(z_2\right)$ is equivalent to $\overline{I}\left(\left[z_1\right]\right)=\overline{I}\left(\left[z_2\right]\right)$, and injectivity is equivalent to $\left[z_1\right]=\left[z_2\right]$, that is, $z_1{\sim }_{\mathcal{K}}{z}_2$. □

A maximal invariant is not a maximal list of features. It distinguishes exactly the task orbits: all prescribed nuisance freedoms are removed, and no distinct orbits are additionally merged.

7.4. Recoverability Criterion for Target Parameters

Let the target be

$$\tau :{\mathcal{D}}_{\mathrm{task}}⟶\mathcal{T},$$

(115)

where $\mathcal{T}$ is the target space. The transport-unified observation followed by quotient projection is

$$\overline{Z}:=q_{\mathcal{K}}\circ Z_{\mathrm{tr}}:{\mathcal{D}}_{\mathrm{task}}⟶{\mathcal{Q}}_{\mathcal{K}}.$$

(116)

Definition 14

(Recoverability from the ${\mathsf{G}}_2$ quotient observation). The target τ is recoverable from the ${\mathsf{G}}_2$ quotient observation if there exists

$$R_{\tau }:\overline{Z}\left({\mathcal{D}}_{\mathrm{task}}\right)⟶\mathcal{T}$$

(117)

such that

$$\tau =R_{\tau }\circ \overline{Z}.$$

(118)

Theorem 8

(Necessary and sufficient condition for quotient-observation recoverability). The target τ is recoverable from Equation (116) if and only if, for all $d_1,d_2\in {\mathcal{D}}_{\mathrm{task}}$,

$$Z_{\mathrm{tr}}\left(d_1\right){\sim }_{\mathcal{K}}{Z}_{\mathrm{tr}}\left(d_2\right)⟹\tau \left(d_1\right)=\tau \left(d_2\right).$$

(119)

Equivalently, τ must be constant on every fiber of $\overline{Z}$.

Proof. 

If Equation (118) holds and the two observations lie on the same task orbit, then $\overline{Z}\left(d_1\right)=\overline{Z}\left(d_2\right)$, and therefore

$$\tau \left(d_1\right)=R_{\tau }\left(\overline{Z}\left(d_1\right)\right)=R_{\tau }\left(\overline{Z}\left(d_2\right)\right)=\tau \left(d_2\right).$$

Conversely, assume Equation (119). For $q\in \overline{Z}\left({\mathcal{D}}_{\mathrm{task}}\right)$, choose any d with $\overline{Z}\left(d\right)=q$ and define $R_{\tau }\left(q\right):=\tau \left(d\right)$. Constancy on fibers makes this independent of the choice of d, and Equation (118) follows. □

If I is maximal, the theorem is equivalently stated by the existence of a unique recovery map ${\tilde{R}}_{\tau }$ on $I\left(Z_{\mathrm{tr}}\left({\mathcal{D}}_{\mathrm{task}}\right)\right)$ such that

$$\tau ={\tilde{R}}_{\tau }\circ I\circ Z_{\mathrm{tr}}.$$

(120)

A maximal invariant therefore retains all information that remains recoverable after quotienting. Whether a particular target is recoverable still depends on its constancy over quotient-observation fibers.

7.5. Compatibility between Phase Transport and Task Quotienting

By default, ${\mathsf{G}}_2$ acts on data whose phase references have already been unified by ${\mathsf{G}}_1$. Only under an additional equivariance condition can “quotient first, then transport” be made equivalent to “transport first, then quotient.”

Let ${\mathcal{Z}}_0$ and ${\mathcal{Z}}_1$ be task data spaces carrying actions of $\mathcal{K}$, and let

$$P:{\mathcal{Z}}_0⟶{\mathcal{Z}}_1$$

(121)

be a data transformation assembled from a family of ${\mathsf{G}}_1$ phase-transport operators.

Definition 15

(Transport–task-group equivariance). The transformation P is equivariant if

$$P(k·z)=k·P\left(z\right),k\in \mathcal{K}.$$

(122)

Theorem 9

(Induced transport on the quotient). If P satisfies Equation (122), there exists a unique map

$$\overline{P}:{\mathcal{Z}}_0/\mathcal{K}⟶{\mathcal{Z}}_1/\mathcal{K}$$

(123)

such that

$$q_{\mathcal{K},1}\circ P=\overline{P}\circ q_{\mathcal{K},0},$$

(124)

and

$$\overline{P}\left({\left[z\right]}_{\mathcal{K}}\right)={\left[P\left(z\right)\right]}_{\mathcal{K}}.$$

(125)

If P is an equivariant bijection, then $\overline{P}$ is a bijection. If a family of transports satisfies path composition, the induced quotient transports satisfy the same composition law.

Proof. 

Define Equation (125). If $z_2=k·z_1$, equivariance gives $P\left(z_2\right)=k·P\left(z_1\right)$, so the definition is independent of representatives. The commuting relation is immediate, and uniqueness follows from surjectivity of $q_{\mathcal{K},0}$.

If P is an equivariant bijection, then $P^{-1}$ is equivariant; the same construction yields $\overline{P^{-1}}$, which is inverse to $\overline{P}$. Composition follows from $\left[P_2\left(P_1\left(z\right)\right)\right]={\overline{P}}_2\left({\overline{P}}_1\left(\left[z\right]\right)\right)$. □

This theorem is the precise interface between ${\mathsf{G}}_1$ and ${\mathsf{G}}_2$. Without equivariance, the orbit obtained after transport may depend on the selected representative, so no well-defined quotient transport exists. In that case the reference must first be unified through the basic phase-transport equation, and the task group and quotient must then be defined on the unified data space.

7.6. Nontriviality Boundary and a Complex-Projective Prototype

A single complex scalar is generally insufficient for a nontrivial ${\mathsf{G}}_2$ quotient.

Proposition 6

(Collapse of a single complex scalar under full complex scaling). Let $\mathcal{Z}={\mathbb{C}}^{\times }$ and let $\mathcal{K}={\mathbb{C}}^{\times }$ act by multiplication. The action is transitive, and hence

$${\mathbb{C}}^{\times }/{\mathbb{C}}^{\times }=\{*\}.$$

(126)

Every task invariant is constant and no nontrivial quotient information remains.

Proof. 

For arbitrary $z_1,z_2\in {\mathbb{C}}^{\times }$, choose $\lambda =z_2{z}_1^{-1}$. Then $z_2=\lambda z_1$, so all points lie on one orbit. □

A nontrivial ${\mathsf{G}}_2$ must use a richer joint data object or remove a smaller group. A canonical prototype is common-complex-scale reduction of a multicomponent observation. Let

$${\mathcal{Z}}_n:={\mathbb{C}}^n\setminus \left\{0\right\},n\ge 2,$$

(127)

and let ${\mathbb{C}}^{\times }$ act diagonally,

$$\lambda ·(z_1,\dots ,z_n):=(\lambda z_1,\dots ,\lambda z_n).$$

(128)

Then

$${\mathcal{Z}}_n/{\mathbb{C}}^{\times }={\mathbb{CP}}^{n-1}.$$

(129)

The quotient removes common amplitude and common phase while retaining the relative complex direction.

On the chart

$$U_1:=\{z\in {\mathcal{Z}}_n:z_1\ne 0\},$$

define

$$I_1\left(z\right):=\left({\displaystyle \frac{z_2}{z_1}},\dots ,{\displaystyle \frac{z_n}{z_1}}\right).$$

(130)

The map $I_1$ is invariant under common complex scaling and is maximal on $U_1$. Indeed, if $I_1\left(z\right)=I_1\left(w\right)$, choosing $\lambda =w_1/z_1$ yields $w=\lambda z$. Standard projective transition functions make the charts compatible and together represent ${\mathbb{CP}}^{n-1}$.

Equation (129) is a prototype, not the universal definition of ${\mathsf{G}}_2$. If only common phase is task irrelevant, one should take $\mathcal{K}=\mathrm{U}\left(1\right)$ and retain amplitude. If channel permutations, common gains, or other structures are also irrelevant, the corresponding action must be used. Different task groups produce different quotients; there is no task-independent unique ${\mathsf{G}}_2$.

7.7. Formal definition of ${\mathsf{G}}_2$ and the three-level hierarchy

Definition 16

(${\mathsf{G}}_2$ task-quotient observation level). Given the admissible task domain ${\mathcal{D}}_{\mathrm{task}}$, the transport-unified observation map $Z_{\mathrm{tr}}:{\mathcal{D}}_{\mathrm{task}}\to \mathcal{Z}$, the task-irrelevant group $\mathcal{K}$, and its action ρ, define

$${\mathsf{G}}_2:=\left({\mathcal{D}}_{\mathrm{task}},Z_{\mathrm{tr}},\mathcal{Z},\mathcal{K},\rho ,q_{\mathcal{K}},{\mathcal{Q}}_{\mathcal{K}}\right).$$

(131)

If an explicit maximal invariant I is available, it may be included as an equivalent coordinate representation of the quotient. A recovery map $R_{\tau }$ depends on a particular target τ and is not part of the target-independent minimal definition of ${\mathsf{G}}_2$.

The three representation and reduction levels are therefore

$$\begin{array}{cc}\hfill {\mathsf{G}}_0& :\mathrm{full}\mathrm{complex}\mathrm{response}\text{:}\mathrm{amplitude},\mathrm{phase},\mathrm{and}\mathrm{zeros};\hfill \\ \hfill {\mathsf{G}}_1& :\mathrm{phase}\mathrm{differential}\mathrm{and}\mathrm{cross}-\mathrm{state}\mathrm{transport};\hfill \\ \hfill {\mathsf{G}}_2& :\mathrm{task}\mathrm{quotient},\mathrm{maximal}\mathrm{invariants},\mathrm{and}\mathrm{recoverability}.\hfill \end{array}$$

(132)

Their logical order is

$${\mathsf{G}}_0⟶{\mathsf{G}}_1⟶Z_{\mathrm{tr}}⟶{\mathsf{G}}_2.$$

(133)

${\mathsf{G}}_0$ contains full response information that later levels cannot recreate; ${\mathsf{G}}_1$ establishes consistent phase transfer across joint states; ${\mathsf{G}}_2$ removes the task-prescribed freedoms. These are not the physical-world–observer–observed-world layers; they are representation, transport, and task-reduction levels of one observation problem.

Proposition 7

(Task dependence of ${\mathsf{G}}_2$). For a fixed transport-unified data space $\mathcal{Z}$, different task-irrelevant groups ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ generally induce different equivalence relations and quotient spaces. Thus ${\mathsf{G}}_0$ and ${\mathsf{G}}_1$ do not uniquely determine ${\mathsf{G}}_2$ unless the task-group action is specified.

Proof. 

On $\mathcal{Z}={\left({\mathbb{C}}^{\times }\right)}^n$, if ${\mathcal{K}}_1=\mathrm{U}\left(1\right)$ removes only common phase, component amplitudes remain in the quotient. If ${\mathcal{K}}_2={\mathbb{C}}^{\times }$ removes common complex scale, common amplitude is removed as well. The orbit decompositions and quotients differ. □

7.8. Relative Completeness and Exact Information Loss

Completeness must be stated relative to an explicit data domain, representation map, and class of questions. ${\mathsf{G}}_0$ is not an identifiability theorem reconstructing all physical states from one complex value; ${\mathsf{G}}_1$ does not retain amplitude and is undefined at zero response; and ${\mathsf{G}}_2$ deliberately identifies data on the same task orbit. We first give a uniform definition.

Definition 17

(Representation completeness relative to a problem class). Let $\mathcal{D}$ be a data domain, $R:\mathcal{D}\to \mathcal{R}$ a representation map, and $\mathfrak{F}$ a class of target maps. The representation R is complete for $\mathfrak{F}$ if, for every $f\in \mathfrak{F}$, there exists a unique map

$$\overline{f}:R\left(\mathcal{D}\right)⟶{\mathcal{W}}_f$$

such that

$$f=\overline{f}\circ R.$$

(134)

This means that R retains all information needed to answer every problem in $\mathfrak{F}$. It does not imply losslessness for other problems or invertibility of R on its data domain.

Fix a nonzero joint-state domain B and write

$$\mathcal{G}\left(B\right):=C^{\infty }(B,{\mathbb{C}}^{\times }),{\mathcal{R}}_{+}\left(B\right):=C^{\infty }(B,{\mathbb{R}}_{>0}),$$

(135)

where ${\mathcal{R}}_{+}\left(B\right)$ acts on $\mathcal{G}\left(B\right)$ by pointwise multiplication. Define

$$R_1:\mathcal{G}\left(B\right)⟶C^{\infty }(B,\mathrm{U}\left(1\right)),R_1\left(g\right):=u_g:={\displaystyle \frac{g}{\left|g\right|}}.$$

(136)

Proposition 8

(Exact quotient structure from ${\mathsf{G}}_0$ to the full ${\mathsf{G}}_1$). The map $R_1$ is surjective, and for $g_1,g_2\in \mathcal{G}\left(B\right)$,

$$R_1\left(g_1\right)=R_1\left(g_2\right)⟺\exists !\rho \in {\mathcal{R}}_{+}\left(B\right)\mathrm{such}\mathrm{that}{g}_2=\rho g_1.$$

(137)

Consequently, $R_1$ induces the canonical bijection

$$\mathcal{G}\left(B\right)/{\mathcal{R}}_{+}\left(B\right)\cong C^{\infty }(B,\mathrm{U}\left(1\right)).$$

(138)

Thus, after fixing the nonzero domain and joint-state identity, the full ${\mathsf{G}}_1$ removes exactly positive amplitude functions and retains all circular phase data.

Proof. 

For any $u\in C^{\infty }(B,\mathrm{U}\left(1\right))$, choosing $g=u$ gives $R_1\left(g\right)=u$, so $R_1$ is surjective. If $g_2=\rho g_1$ with $\rho >0$, normalization gives the same unit phase. Conversely, if $u_{g_1}=u_{g_2}$, define $\rho :=|g_2|/|g_1|$. Then $\rho$ is positive and smooth, $g_2=\rho g_1$, and uniqueness follows because $g_1$ is nowhere zero. Therefore the fibers of $R_1$ are precisely the ${\mathcal{R}}_{+}\left(B\right)$ orbits. □

Theorem 10

(Relative completeness of ${\mathsf{G}}_0$–${\mathsf{G}}_1$–${\mathsf{G}}_2$). Fix a selected scalar coherent readout and response $g:\mathcal{M}\to \mathbb{C}$, the full ${\mathsf{G}}_1$ on the nonzero domain B, and a ${\mathsf{G}}_2$ determined by $Z_{\mathrm{tr}}:{\mathcal{D}}_{\mathrm{task}}\to \mathcal{Z}$, a task group $\mathcal{K}$, and its action. Then:

(1)

Typed response-graph completeness of ${\mathsf{G}}_0$.Let

$${\mathrm{\Gamma }}_g:=\{(m,g\left(m\right)):m\in \mathcal{M}\}\subseteq L,R_0:={\sigma }_g:\mathcal{M}\to {\mathrm{\Gamma }}_g.$$

(139)

Then $R_0$ is a bijection. Every problem defined on the typed response graph ${\mathrm{\Gamma }}_g$ therefore factors uniquely through $R_0$. This is graph completeness of the joint-state–full-response relation, not an identifiability statement that reconstructs m from the scalar value $g\left(m\right)$ alone.

(2)

Phase-problem completeness of ${\mathsf{G}}_1$.The full ${\mathsf{G}}_1$ retains $u_g$, and $A_g$, ${\mathrm{\Omega }}_g$, the basic phase-transport equation, the covariant derivative, and all finite transports are uniquely determined by $u_g$. Hence every model-level phase problem invariant under positive amplitude rescaling,

$$\mathcal{F}:\mathcal{G}\left(B\right)\to \mathcal{W},\mathcal{F}\left(\rho g\right)=\mathcal{F}\left(g\right),$$

factors uniquely through $R_1$. Conversely, if only the differential–transport substructure $A_g,{\mathrm{\Omega }}_g,{\mathcal{T}}_g$ is retained and $u_g$ is omitted, one constant phase anchor per path component is required to recover $u_g$.

(3)

Task-invariant completeness of ${\mathsf{G}}_2$.The quotient projection $q_{\mathcal{K}}:\mathcal{Z}\to \mathcal{Z}/\mathcal{K}$ is complete for all $\mathcal{K}$-invariant problems: every task invariant factors uniquely through it; maximal invariants are bijectively equivalent to the quotient; and a target τ is recoverable exactly when it is constant on each fiber of $q_{\mathcal{K}}\circ Z_{\mathrm{tr}}$.

(4)

Relative completeness of the full hierarchy.${\mathsf{G}}_0$, the full ${\mathsf{G}}_1$, and ${\mathsf{G}}_2$ are complete, respectively, for typed full-response-graph problems, phase problems invariant under positive amplitude scaling, and quotient invariants and recoverable targets under the prescribed task group. This statement is restricted to these problem classes. It does not assert that all phase tasks are amplitude independent or that a task quotient can recover orbit-internal freedoms that it deliberately removes.

Proof. 

For part (1), $R_0^{-1}={\pi }_L{|}_{{\mathrm{\Gamma }}_g}$, so $R_0$ is a bijection and the factorization statement follows. For part (2), 8 shows that the fibers of $R_1$ are exactly the positive-amplitude orbits, and every problem constant on those orbits factors uniquely through $R_1$. Moreover,

$$A_g=u_g^{*}{\omega }_{\mathrm{U}\left(1\right)},{\mathrm{\Omega }}_g={pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}-{\pi }_{\mathrm{ph}}^{*}{A}_g,$$

so all remaining ${\mathsf{G}}_1$ objects are determined by $u_g$. If all finite transports are given, then for any $m\in B$ and $v\in T_{m}B$, a short path $c_v$ recovers

$${\left(A_g\right)}_m\left(v\right)=-\mathrm{i}{\left.{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\right|}_{t=0}{\mathcal{T}}_g\left[c_v{|}_{[0,t]}\right].$$

(140)

A phase anchor $u\left(m_{\alpha }\right)$ on a path component, together with the transport functional, gives

$$u\left(m\right)=u\left(m_{\alpha }\right){\mathcal{T}}_g\left[c\right],$$

(141)

where c connects $m_{\alpha }$ to m. Closed-loop circular transport is the identity, so the result is path independent. Part (3) follows from Theorems 6–8. Part (4) combines the first three statements within their explicitly stated problem classes. □

Proposition 9

(Constant-phase anchor loss in the differential–transport substructure). Let $g_1,g_2:B\to {\mathbb{C}}^{\times }$ be smooth. The following are equivalent:

(i)

$A_{g_1}=A_{g_2}$;

(ii)

${\mathcal{T}}_{g_1}\left[c\right]={\mathcal{T}}_{g_2}\left[c\right]$ for every piecewise-$C^1$ path c;

(iii)

On each path component $B_{\alpha }$ there exist a constant $q_{\alpha }\in \mathrm{U}\left(1\right)$ and a unique positive smooth function ${\rho }_{\alpha }:B_{\alpha }\to {\mathbb{R}}_{>0}$ such that

$$g_2={\rho }_{\alpha }{q}_{\alpha }{g}_1\mathrm{on}{B}_{\alpha }.$$

(142)

If the responses induce the same full ${\mathsf{G}}_1$, which includes $u_g$, then every $q_{\alpha }=1$. Hence the full ${\mathsf{G}}_1$ loses only positive amplitude, whereas the substructure containing only $A_g$ and transport also loses one constant phase anchor on each path component.

Proof. 

The equivalence of (i) and (ii) follows from ${\mathcal{T}}_g\left[c\right]=exp\left(\mathrm{i}{\int }_c{A}_g\right)$ and Equation (140). Assume (i) and define $h:=u_{g_2}{u}_{g_1}^{-1}:B\to \mathrm{U}\left(1\right)$. Since $\mathrm{U}\left(1\right)$ is Abelian,

$$h^{*}{\omega }_{\mathrm{U}\left(1\right)}=A_{g_2}-A_{g_1}=0.$$

Because ${\omega }_{\mathrm{U}\left(1\right)}$ is a linear isomorphism from $T_u\mathrm{U}\left(1\right)$ to $\mathbb{R}$ at every point, $\mathrm{d}h=0$. Therefore h is constant, equal to $q_{\alpha }$, on each path component. Taking ${\rho }_{\alpha }=|g_2|/|g_1|$ gives (iii). The converse follows because a constant phase factor and a positive amplitude factor do not change $A_g$. If the full ${\mathsf{G}}_1$ representations agree, then $u_{g_1}=u_{g_2}$, so $q_{\alpha }=1$. □

Proposition 10

(Exact orbit loss from transport-unified data to ${\mathsf{G}}_2$). For $z_1,z_2\in \mathcal{Z}$,

$$q_{\mathcal{K}}\left(z_1\right)=q_{\mathcal{K}}\left(z_2\right)⟺z_1{\sim }_{\mathcal{K}}{z}_2.$$

(143)

Therefore quotient reduction from the data space of $Z_{\mathrm{tr}}$ to ${\mathsf{G}}_2$ removes exactly the task-group orbit directions and does not merge distinct orbits. This should not be abbreviated as a direct quotient of ${\mathsf{G}}_1$ itself, because the input to ${\mathsf{G}}_2$ is the task data $Z_{\mathrm{tr}}$ assembled jointly from ${\mathsf{G}}_0$ and ${\mathsf{G}}_1$.

Proof. 

The equivalence is the definition of the quotient projection and orbit relation. If a maximal invariant is used as coordinates for ${\mathsf{G}}_2$, Theorem 7 likewise guarantees complete separation of distinct orbits. □

Remark 4

(Boundary of completeness). The completeness in Theorem 10 is relative. If the readout $\mathfrak{r}:Y_{\mathrm{coh}}\to \mathbb{C}$ is not injective, ${\mathsf{G}}_0$ cannot recover information in the original response space that never entered the selected complex channel; graph completeness also does not identify m from $g\left(m\right)$ alone. ${\mathsf{G}}_1$ does not describe amplitude evolution and is undefined at $g=0$; its differential–transport substructure requires phase anchors. ${\mathsf{G}}_2$ cannot recover orbit-internal freedoms that the task group has deliberately identified. These boundaries arise from representation type, the nonzero domain, and quotient definition, not from an undeclared information loss.

7.9. Conclusion and Validity Boundary

This section has established task reduction after the basic phase-transport equation. The task-group action defines orbit equivalence, and the quotient projection supplies the complete orbit-level observation. Every invariant factors uniquely through the quotient; maximal invariants are canonically equivalent to it; recoverability is equivalent to constancy on quotient-observation fibers; and equivariant transport induces a unique quotient transformation commuting with quotient projection. The relative-completeness theorem and the three exact-loss propositions identify the information boundaries of full phase reduction, differential–transport representation, and task quotienting.

The validity of ${\mathsf{G}}_2$ requires an explicitly specified joint data object, task-irrelevant action, and target space. The quotient is generally first a set or topological quotient and becomes a smooth manifold only under additional hypotheses such as a smooth, free, and proper action. Explicit maximal invariants depend on the action. Insufficient joint data or a transitive action produces a degenerate quotient. A task quotient cannot recover information deliberately identified on an orbit and cannot repair the failure of ${\mathsf{G}}_1$ at zero-response crossings.

The theoretical chain from variable observation maps through the basic phase-transport equation to quotient observation and recoverability is now complete. Applications to satellite navigation, phased-array radar, integrated sensing and communications, and other coherent systems may specify $Z_{\mathrm{tr}}$, $\mathcal{K}$, a maximal invariant, and a recovery map without altering the general structure.

8. Conclusions

Starting from $y=h\left(x\right)$, this paper did not assume a three-layer paradigm, a complex phase structure, a fiber bundle, a connection, or a transport equation. The requirement to represent variable observation maps completely led successively to an observer-state variable, the three-layer observation paradigm, the complex response, its nonzero phase domain, amplitude–phase reduction of the response bundle, the observed-phase one-form, and the unique compatible phase connection. On this foundation,

$${\mathfrak{E}}_g\left[\tilde{c}\right]={\tilde{c}}^{*}{\mathrm{\Omega }}_g=0$$

was established as the basic phase-transport equation.

The equation is a first-order global differential functional equation on lifted paths of the principal phase bundle. It is independent of a local real phase branch and of a particular time coordinate. The local group-valued equation, covariant-derivative equation, continuous real-phase equation, and physical–observer directional expansion are equivalent forms. For prescribed initial data, it uniquely determines

$${\mathrm{\Delta }}_g\left[c\right]={\int }_c{A}_g,$$

$${\mathcal{T}}_g\left[c\right]=exp\left(\mathrm{i}{\int }_c{A}_g\right),$$

and the fiber transport operator ${\mathcal{P}}_c^g$. Finite endpoint factors are therefore solutions of the basic equation rather than separately imposed formulas.

Its fundamental status is supported by Theorem 4: the equation is uniquely generated by the nonzero complex response induced by the observation map; it is equivalent to horizontal lifting under the unique compatible connection; no different intrinsic first-order law reproduces the same observed-phase variation in every joint-state direction; prescribed initial phase determines finite transport and its identity, inverse, and composition structures; the equation is covariant under local phase-gauge transformations, with constant gauges representing zero-phase reference changes; and it is invariant under positive amplitude rescaling. Same-source double-state phase and inverse transport on fixed-physical-state slices follow directly from finite solutions.

The theoretical boundary is equally explicit. The equation is a geometric–kinematic phase-transport law, not a dynamical equation for physical or observer states and not an amplitude-recovery law. Its canonical domain is the nonzero complex-response domain.

On top of phase transport, a ${\mathsf{G}}_2$ task-quotient observation level was constructed. A task-irrelevant group action defines orbit equivalence and the quotient observation space. All task invariants arise from the same quotient structure; a maximal invariant distinguishes exactly the orbits; and target recoverability is equivalent to constancy on quotient-observation fibers. Under equivariance, ${\mathsf{G}}_1$ transport induces a unique transformation on the ${\mathsf{G}}_2$ quotient. Quotienting a single complex scalar by full complex scale collapses to one point, while common-complex-scale quotienting of multicomponent data yields a complex-projective observation. Nontrivial ${\mathsf{G}}_2$ structures therefore depend essentially on task and joint data.

A large and highly differentiated body of phase research has been developed, covering phase generation, measurement, recovery, unwrapping, synchronization, noise modeling, and geometric accumulation. The foundational problem addressed here is therefore not an absence of phase research in general, but the need for a unified mathematical formulation of phase formation, identifiability, and consistent cross-state transport when the observation map varies with the observer state. Starting from the complex response induced by the observation map, the basic phase-transport equation unifies local phase differentials, compatible connections, horizontal lifting, and finite path transport within a single theoretical structure.

The resulting hierarchy consists of ${\mathsf{G}}_0$ full complex response, ${\mathsf{G}}_1$ phase differential and transport, and ${\mathsf{G}}_2$ task quotient and recoverability. Relative completeness was proved for explicit problem classes. ${\mathsf{G}}_0$ is complete for the typed response graph but does not identify a joint state from the complex value alone. The full ${\mathsf{G}}_1$ is precisely the quotient by positive amplitude functions; its differential–transport substructure additionally lacks one constant phase anchor per path component. ${\mathsf{G}}_2$ is complete for orbit-level invariants and quotient-recoverable targets, and its quotient removes exactly task-group orbit freedoms. This hierarchy remains categorically distinct from the physical-world–observer–observation-world paradigm and provides a unified mathematical and theoretical foundation for zero-response crossings, multichannel relative structures, and maximal-invariant constructions in satellite navigation, radar, and integrated sensing and communications.