The Phase Transport Fundamental Equation on the Physical–Observation Dual-Axis Structure

1. Introduction

PODA Theory takes physical reality, as realized in an observation, to depend jointly on the physical state and the observation state. Fixing the former does not fix the latter: an objective change of observation state can change the physical fact that is realized. The distinction is essential in coherent measurement, where the phase of a complex response may vary as the observation system changes while the physical source state remains the same. Phase comparison must therefore follow both the source and the conditions through which its response is formed.

Carrier-phase estimation [2] and GPS integer-ambiguity estimation [3] are classical phase-inference problems. Array calibration [4], radar interferometry [5], astronomical self-calibration [6], and phase correction in magnetic-resonance imaging [7] provide further settings in which the observation model matters. These applications motivate the comparison problem considered here. Our derivation begins with the joint observation law and the smooth coherent response specified below.

A periodic physical degree of freedom has a circular phase. Choosing a reference gives a local real representative, and following a trajectory gives a continuous phase lift that retains its winding. The instrument supplies a different object: a complex response whose argument is the observed phase. The observation map determines how this measured phase is related to the physical one.

With the observation state fixed, the familiar model reads

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

(1.1)

where $h:{\mathrm{X}}_h\to \mathrm{Y}$ maps the admissible physical-state domain ${\mathrm{X}}_h\subseteq \mathrm{X}$ to the realized response $y\in \mathrm{Y}$. This single-axis description is a slice of the joint law at a fixed observation state. Its sole explicit argument is the physical state; variation across observation states becomes visible only when the second argument is restored. The domain matters as well: extending the law to all of $\mathrm{X}$ requires the value $\perp \in {\mathrm{Y}}_{\perp }$ at physical states where this observation cannot be implemented.

An operating system may pass through many such observation states. Array weights, beam directions, channel combinations, reference choices, and processing rules can all vary while the physical state under comparison remains fixed. Adaptive arrays and phased-array radar offer concrete examples [8,9,10]. The physical model and experimental protocol determine which of these conditions belong to the observation coordinate. In PODA, this coordinate is an objective constituent of the joint realization: different admissible observation states may produce different physical facts at the same physical state. The physical-state coordinate and the realized fact therefore have distinct mathematical roles.

PODA Theory gives these two roles distinct coordinates through the physical–observation dual-axis (PODA) structure. Its set-theoretic foundation constructs the observation-state space from complete observation records and establishes the minimality and essential uniqueness of the resulting structure [1]. We use the realization in which the physical model supplies physical-state identity and the observation law preserves it. This choice makes a comparison at fixed physical state unambiguous. The question is then precise: how does the phase of a coherent response vary along an admissible path of joint states, and what remains of that variation when the physical state is held fixed?

We answer this question by differentiating the polar decomposition of the response. The resulting phase one-form determines the change along every admissible tangent direction. A principal connection expresses the same relation geometrically once compatibility with the observed response is required. Section 6 derives the transport equation and establishes its unique solution for a prescribed initial phase. Its restriction to a fixed physical-state slice gives the phase law on the observation axis. Integration gives reference conversion, inverse transport, and composition.

The differential construction requires a smooth coherent response in addition to the set-theoretic PODA structure. The response may couple the two coordinates nonlinearly, and its phase need not separate additively. The observation law supplies that dependence; the transport equation expresses its phase variation along a specified path.

Bringing observations to a common phase reference leaves a second question. A particular task may be indifferent to a common phase, a common complex scale, or another prescribed group action. Passing to the corresponding quotient identifies data that the task regards as equivalent. We show when transport is compatible with this identification and when the target quantity survives it. This yields three related descriptions: the full response, its phase transport, and the information that remains after the chosen task reduction.

2. The PODA Structure and Its Smooth Phase Model

Physical state and observation state play distinct roles in the formation of a fact. PODA Theory expresses physical reality through their joint realization: the physical model supplies one coordinate, and the objective state of the observation system supplies the other. Holding the first coordinate fixed still leaves room for the realized fact to vary with the second.

The set-theoretic foundation constructs a minimal observation-state space from a complete observation table with specified state roles [1]. We use its realization that preserves physical-state identity. The resulting joint law gives the two coordinates their precise meaning; a smooth structure and a coherent readout will then allow us to study phase.

2.1. The Admissible Joint Observation Law

Let $\mathrm{X}$ and $\mathrm{S}$ be nonempty sets of physical states and observation states, respectively. Let $\mathrm{Y}$ be the set of actual outputs, let $\perp \notin \mathrm{Y}$, and write ${\mathrm{Y}}_{\perp }:=\mathrm{Y}\bigsqcup \{\perp \}$. The joint observation law comprises

$$\mathrm{B}\subseteq \mathrm{X}\times \mathrm{S},F:\mathrm{B}⟶\mathrm{Y},\tilde{F}:\mathrm{X}\times \mathrm{S}⟶{\mathrm{Y}}_{\perp }.$$

(2.1)

Let

$${\pi }_X:\mathrm{B}\to \mathrm{X},{\pi }_S:\mathrm{B}\to \mathrm{S}$$

be the coordinate projections, let $j_B:=({\pi }_X,{\pi }_S):\mathrm{B}\hookrightarrow \mathrm{X}\times \mathrm{S}$ be the canonical inclusion, and let $i_{\mathrm{Y}}:\mathrm{Y}\hookrightarrow {\mathrm{Y}}_{\perp }$. These maps satisfy

$$\tilde{F}\circ j_B=i_{\mathrm{Y}}\circ F,\mathrm{B}={\tilde{F}}^{-1}\left(\mathrm{Y}\right).$$

(2.2)

Here ⊥ records nonimplementability as determined by the task specification, rather than missing or unknown data. The map F assigns actual outcomes on $\mathrm{B}$, which may be a proper subset of $\mathrm{X}\times \mathrm{S}$.

In this framework, $F(x,s)$ is the realized physical fact represented by the observation law. The coordinate x specifies the physical state being observed, rather than the complete fact realized in the joint state. Consequently, changing the observation state can change the realized physical fact even while the physical-state coordinate is held fixed. This possibility is the substantive distinction between the PODA law and a single-axis representation $y=h\left(x\right)$.

For each $s\in \mathrm{S}$, define the complete totalized behavior

$${\widehat{F}}_s:\mathrm{X}⟶{\mathrm{Y}}_{\perp },{\widehat{F}}_s\left(x\right):=\tilde{F}(x,s),$$

(2.3)

and the realized map family

$${\mathbf{H}}_F:=\{{\widehat{F}}_s:s\in \mathrm{S}\}\subseteq {\mathrm{Y}}_{\perp }^{\mathrm{X}}.$$

(2.4)

The observation-side behavior map is the corestriction

$${\Gamma }_F:\mathrm{S}⟶{\mathbf{H}}_F,{\Gamma }_F\left(s\right):={\widehat{F}}_s,$$

(2.5)

and the minimal realization of the observation state satisfies

$${\Gamma }_F\mathrm{is}\mathrm{bijective}.$$

(2.6)

Thus $\mathrm{S}$ realizes the entire family ${\mathbf{H}}_F$, with one observation state for each complete behavior. We call this realization behavior-minimal. Distinguishability by complete behavior does not imply identifiability from a single value $F(x,s)$; moreover, each observation state has its own domain of admissible physical states.

The two axes designate state roles, without implying linear or orthogonal structure. Product notation allows dependence between the coordinates, while $\mathrm{B}$ specifies which pairs are physically realizable. The response $F(x,s)$ is the output of this joint law.

2.2. Physical-State Identity and Classical Observation Slices

The foundation distinguishes the physical-state space from the behavioral quotient on the object side. That quotient identifies object labels which the declared observation family cannot distinguish, but this equivalence need not imply physical identity. Throughout this paper, $\mathrm{X}$ retains the identity determined by an independent physical model, with a compatible map to the observation records. We reduce only the observation side by complete behavior. Anchoring the PODA structure in this physical model gives a precise meaning to changing the observation state while holding the physical state fixed. An observation state describes the observation system and its operating conditions; it does not refer to a human observer.

For a fixed $s_0\in \mathrm{S}$, let

$${\mathrm{X}}_{s_0}:=\{x\in \mathrm{X}:(x,s_0)\in \mathrm{B}\}.$$

Proposition 2.1

(Classical model on a fixed admissible observation-state slice). The map

$$h_{s_0}:{\mathrm{X}}_{s_0}⟶\mathrm{Y},h_{s_0}\left(x\right):=F(x,s_0),$$

gives the classical relation $y=h_{s_0}\left(x\right)$ on this slice. Its extension to the complete physical-state space is ${\widehat{F}}_{s_0}:\mathrm{X}\to {\mathrm{Y}}_{\perp }$. The map $h_{s_0}$ takes actual-output values throughout $\mathrm{X}$ precisely when ${\mathrm{X}}_{s_0}=\mathrm{X}$.

Proof. 

By definition of ${\mathrm{X}}_{s_0}$, we have $(x,s_0)\in \mathrm{B}$ for every $x\in {\mathrm{X}}_{s_0}$, and hence $F(x,s_0)$ lies in $\mathrm{Y}$. □

The classical single-axis form is thus recovered by fixing the observation state and restricting to the physical states admissible under it. The resulting slice suppresses variation of the observation state. PODA makes that state an explicit coordinate of the joint law, so different admissible slices can realize different physical facts at the same physical state.

2.3. Admissible Fixed-Physical-State Comparison

For $x\in \mathrm{X}$, define the admissible observation-state fiber

$${\mathrm{S}}_x:=\{s\in \mathrm{S}:(x,s)\in \mathrm{B}\}.$$

(2.7)

Proposition 2.2

(Fixed physical-state slice). For any $x_0\in \mathrm{X}$, the restriction

$$F_{x_0}:{\mathrm{S}}_{x_0}⟶\mathrm{Y},F_{x_0}\left(s\right):=F(x_0,s)$$

(2.8)

is well defined. For $s_0,s_1\in {\mathrm{S}}_{x_0}$, the outputs $F(x_0,s_0)$ and $F(x_0,s_1)$ are actual outcomes at the same physical state and can therefore be compared.

Proof. 

By definition of ${\mathrm{S}}_{x_0}$, we have $(x_0,s)\in \mathrm{B}$ for every $s\in {\mathrm{S}}_{x_0}$. □

Unequal values on this slice distinguish observations made at one fixed physical state:

$$F(x_0,s_0)\ne F(x_0,s_1).$$

The dynamics or experimental protocol gives such a comparison its temporal and causal interpretation.

The possibility of changing the realized physical fact while holding the physical state fixed is expressed by the condition

$$\begin{array}{c}\exists x_{*}\in \mathrm{X},\exists s_1,s_2\in \mathrm{S}:(x_{*},s_1),(x_{*},s_2)\in \mathrm{B},\\ F(x_{*},s_1)\ne F(x_{*},s_2).\end{array}$$

The two admissible joint states share one physical-state coordinate yet realize different facts. This is observation dependence at fixed physical state. Its occurrence is determined by the joint law: it may occur on some physical-state fibers and be absent on others. Behavior-minimality concerns a different question. It separates observation states by their complete behavior, which may differ at other physical states or through admissibility alone; unequal actual outputs on a common physical-state fiber establish the dependence considered here.

Let ${\mathrm{X}}_B:={\pi }_X\left(\mathrm{B}\right)$ and let

$${\pi }_X^{\mathrm{act}}:\mathrm{B}\twoheadrightarrow {\mathrm{X}}_B$$

be the physical projection onto its realized image. If two actual outputs on the same physical-state fiber are unequal, no map $h:{\mathrm{X}}_B\to \mathrm{Y}$ can satisfy $F=h\circ {\pi }_X^{\mathrm{act}}$. Conversely, if F is constant on each fiber of ${\pi }_X^{\mathrm{act}}$, this factorization exists and is unique. Actual outputs and implementability play different roles in this criterion: two observation states may differ in the totalized family only through entries involving ⊥, even though the actual-output law F factors through the physical projection.

2.4. The Same-Source Comparison Domain

The ordered pairs of admissible states with the same physical coordinate form the fiber product

$$\mathrm{B}{\times }_{\mathrm{X}}\mathrm{B}:=\left\{(b_0,b_1)\in {\mathrm{B}}^2:{\pi }_X\left(b_0\right)={\pi }_X\left(b_1\right)\right\}.$$

(2.9)

This fiber product is the natural domain of same-source endpoint pairs. Associating a phase difference or a continuous transport law with such a pair also requires a coherent response and the geometric structures introduced below.

2.5. Smooth Structure and Coherent Response

The set-theoretic construction fixes the state roles, admissibility, and observation law. To study phase transport, we now add a smooth model of the relevant states and a complex readout.

Definition 2.3

(Smooth coherent observation model). A smooth coherent specialization of the PODA structure consists of the following data.

• The sets of physical and observation states $\mathrm{X}$ and $\mathrm{S}$ carry smooth-manifold structures.
• The physical model specifies an embedded smooth submanifold ${\mathrm{X}}_{\varphi }\subseteq \mathrm{X}$ on which physical phase is considered, together with the phase structure used in the next section. An open region or a smooth periodic orbit may serve as ${\mathrm{X}}_{\varphi }$.
• The analysis is carried out on an embedded smooth submanifold

  $$\mathrm{M}\subseteq \mathrm{B}\cap ({\mathrm{X}}_{\varphi }\times \mathrm{S}).$$

  (2.10)
  Here the ambient manifold for the embedding is ${\mathrm{X}}_{\varphi }\times \mathrm{S}$. We require smoothness on $\mathrm{M}$; the full admissible domain $\mathrm{B}$ may be nonsmooth or nonrectangular.
• A coherent-output subset ${\mathrm{Y}}_{\mathrm{coh}}\subseteq \mathrm{Y}$ and a scalar complex readout

  $$\mathrm{r}:{\mathrm{Y}}_{\mathrm{coh}}⟶\mathbb{C}$$

  are specified such that $F\left(\mathrm{M}\right)\subseteq {\mathrm{Y}}_{\mathrm{coh}}$. Let

  $$F_{\mathrm{M}}^{\mathrm{coh}}:\mathrm{M}⟶{\mathrm{Y}}_{\mathrm{coh}},F_{\mathrm{M}}^{\mathrm{coh}}\left(m\right):=F\left(m\right)$$

  be the corresponding corestriction. The induced response

  $$g:=\mathrm{r}\circ F_{\mathrm{M}}^{\mathrm{coh}}:\mathrm{M}⟶\mathbb{C}$$

  (2.11)

  is assumed smooth.
• The phase-regular domain is

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

  (2.12)
  The differential phase and the induced connection are defined on ${\mathrm{M}}^{\times }$, where the response can be normalized.

The chosen complex channel may lose distinctions present in the complete observation behavior. In particular, bijectivity of ${\Gamma }_F:\mathrm{S}\to {\mathbf{H}}_F$ need not make $s\mapsto g(·,s)$ injective after restriction to $\mathrm{M}$ and composition with $\mathrm{r}$. Behavior-minimality concerns the full totalized law $\tilde{F}$, whereas identifiability through the chosen scalar readout is a separate property.

Every path used for phase transport must lie in ${\mathrm{M}}^{\times }$. To compare states at a fixed physical state by transport, we must therefore specify a path

$$c\left(t\right)=(x_{*},s\left(t\right))\in {\mathrm{M}}^{\times }.$$

The endpoints must lie in the same component accessible by an admissible path. Their phase ratio is algebraically defined whenever both responses are nonzero; its interpretation as the solution of a differential transport equation also requires a comparison path of the stated regularity.

2.6. From the Observation Law to Phase Transport

The passage from the observation law to phase transport can thus be written as

$$\begin{array}{cc}& \mathrm{complete}\mathrm{observation}\mathrm{table}\mathrm{with}\mathrm{specified}\mathrm{state}\mathrm{roles}⟶(\mathrm{X},\mathrm{S},\mathrm{B},F,\tilde{F},{\mathbf{H}}_F,{\Gamma }_F)\hfill \\ & ⟶({\mathrm{X}}_{\varphi },\mathrm{M},\mathrm{r},g)⟶{\mathrm{M}}^{\times }⟶\mathrm{phase}\mathrm{differential}\mathrm{and}\mathrm{finite}\mathrm{transport}.\hfill \end{array}$$

The first arrow is the set-theoretic construction established in the companion paper. The subsequent steps use the physical phase model, smooth analysis domain, complex readout, and restriction to nonzero responses specified here. Differentiating the response along admissible paths will give its phase law; response compatibility will then determine the associated connection and transport.

3. Circular Phase, Reference, and Winding

The physical model supplies a phase coordinate with which observed phase may be compared. A periodic trajectory is the basic example: phase is circular, its zero depends on a reference, and a continuous real lift records the winding along a path. We recall these standard properties before relating physical phase to the joint observation law [13,14,15,16].

3.1. Periodic Equivalence and Circular Phase

Suppose the physical model specifies a smooth physical-state manifold $\mathrm{X}$ and a phase-bearing region ${\mathrm{X}}_{\varphi }\subseteq \mathrm{X}$. Let

$$\gamma :\mathbb{R}⟶{\mathrm{X}}_{\varphi }$$

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

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

(3.1)

Write $C:=\gamma \left(\mathbb{R}\right)\subseteq {\mathrm{X}}_{\varphi }$. The induced map from $\mathbb{R}/T\mathbb{Z}$ is an injective immersion. Its domain is compact and ${\mathrm{X}}_{\varphi }$ is Hausdorff, so it is a smooth embedding onto C. Normalizing one period to $2\pi$ gives the equivalence relation

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

(3.2)

and hence the phase-state space

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

(3.3)

We equip this quotient with its standard smooth circle-group structure.

3.2. Reference Selection and Local Real Phase

The periodic physical structure alone does not select a zero phase. Given a reference state $x_{\mathrm{ref}}=\gamma \left(t_{\mathrm{ref}}\right)\in C$, define

$${\vartheta }_{x_{\mathrm{ref}}}:C⟶\Theta ,{\vartheta }_{x_{\mathrm{ref}}}\left(\gamma \left(t\right)\right):={\pi }_{\Theta }\left({\displaystyle \frac{2\pi }{T}}(t-t_{\mathrm{ref}})\right).$$

(3.4)

Changing $x_{\mathrm{ref}}$ translates physical circular phase by a constant; the corresponding change in a measured complex response depends on the observation map and its calibration. A principal-value representative in $[0,2\pi )$ has a branch discontinuity and therefore cannot serve as a global differentiable phase variable. For differentiation, we use a local real lift or a continuous lift along a path, choosing a compatible initial value.

3.3. Path Lifting and Winding Increments

Let I be an interval, choose $t_0\in I$, and let

$$\alpha :I\to \Theta$$

be continuous. For each ${\phi }_0\in \mathbb{R}$ satisfying

$${\pi }_{\Theta }\left({\phi }_0\right)=\alpha \left(t_0\right),$$

there exists a unique continuous lift

$$\phi :I\to \mathbb{R},{\pi }_{\Theta }\circ \phi =\alpha ,\phi \left(t_0\right)={\phi }_0.$$

If $\alpha$ is piecewise-$C^1$ or smooth, its lift has the same regularity. Different compatible initial representatives give lifts differing by a constant $2\pi k$, so their differentials agree wherever defined. For $I=[a,b]$, define

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

(3.5)

Modulo $2\pi$, this increment is the circular-phase difference between the endpoints; as a real number, it also retains winding information. Thus even when $\alpha \left(a\right)=\alpha \left(b\right)$, one may have

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

This distinction will allow us to separate the circular transport factor from the accumulated unwrapped real phase.

3.4. Physical Phase and Its Observation

The periodic model thus provides the circular phase space $\Theta$, a physical phase map relative to a chosen reference, real lifts, and winding increments on closed paths. How these quantities enter the measured response is determined by the observation map. The next section gives a fiber-constancy criterion for expressing observed phase in terms of physical phase and observation state. Agreement between physical and observed phase evolution requires a further condition on the reference response. We state this condition explicitly in Section 6.7.

4. Phase Formation and Variation Under the Observation Map

Section 2 introduced the admissible PODA observation law

$$F:\mathrm{B}⟶\mathrm{Y},$$

on a smooth coherent domain $\mathrm{M}\subseteq \mathrm{B}\cap ({\mathrm{X}}_{\varphi }\times \mathrm{S})$. Section 3 described physical phase in a periodic model. We now obtain observed phase from the complex readout on $\mathrm{M}$ and ask when it is determined by physical phase and observation state alone.

The joint response may couple the two state coordinates nonlinearly. We therefore begin with the phase of the response itself and then consider whether its variation can be separated into physical and observation contributions.

4.1. Complex Response Induced by the Observation Map

Let

$$\vartheta :{\mathrm{X}}_{\varphi }⟶\Theta$$

be a physical phase map supplied by the external physical model, with the periodic orbit C of Section 3 as the basic example. We consider the admissible restriction

$${F|}_{\mathrm{M}}:\mathrm{M}⟶\mathrm{Y},\mathrm{M}\subseteq \mathrm{B}\cap ({\mathrm{X}}_{\varphi }\times \mathrm{S}).$$

(4.1)

We use the physical phase map in the factorization question below. The phase of g can, however, be constructed without specifying such a map.

Suppose the observation channel admits a scalar coherent complex readout, and let

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

be that readout, with $F\left(\mathrm{M}\right)\subseteq {\mathrm{Y}}_{\mathrm{coh}}$. Recall the coherent corestriction $F_{\mathrm{M}}^{\mathrm{coh}}:\mathrm{M}\to {\mathrm{Y}}_{\mathrm{coh}}$ and the induced response

$$g:=\mathrm{r}\circ F_{\mathrm{M}}^{\mathrm{coh}}:\mathrm{M}⟶\mathbb{C},g(x,s)=\mathrm{r}\left(F(x,s)\right).$$

(4.2)

The response remains a function of the physical state x and the observation state s on their admissible smooth domain. Its dependence on these states may be linear, nonlinear, or strongly coupled; we assume neither multiplicative nor additive separability.

On the nonzero domain

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

(4.3)

the response has the unique polar decomposition

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

(4.4)

The full nonzero complex response therefore takes values in

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

The amplitude is part of the full response, whereas the normalized response u records phase alone.

4.2. Observed Phase as a Joint-State Function

Let

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

be the standard Lie-group isomorphism between circular phases and unit complex numbers. The observed-phase map is

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

(4.5)

Observed phase is thus a function of the full admissible joint state. To determine whether physical phase and observation state suffice to describe it, define

$${\chi }_{\vartheta }:{\mathrm{M}}^{\times }⟶\Theta \times \mathrm{S},{\chi }_{\vartheta }(x,s):=(\vartheta \left(x\right),s).$$

(4.6)

and let

$${\chi }_{\vartheta }^{\mathrm{act}}:{\mathrm{M}}^{\times }\twoheadrightarrow {\chi }_{\vartheta }\left({\mathrm{M}}^{\times }\right),{\chi }_{\vartheta }^{\mathrm{act}}\left(m\right):={\chi }_{\vartheta }\left(m\right)$$

(4.7)

be its corestriction to the realized image.

Proposition 4.1

(Exact factorization through physical phase and observation state). There exists a unique map

$$\widehat{\Phi }:{\chi }_{\vartheta }\left({\mathrm{M}}^{\times }\right)⟶\Theta$$

(4.8)

such that

$$\Phi =\widehat{\Phi }\circ {\chi }_{\vartheta }^{\mathrm{act}}$$

(4.9)

if and only if, for all $(x,s),(x^{\prime },s^{\prime })\in {\mathrm{M}}^{\times }$,

$$\vartheta \left(x\right)=\vartheta \left(x^{\prime }\right),s=s^{\prime }⟹\Phi (x,s)=\Phi (x^{\prime },s^{\prime }).$$

(4.10)

Proof. 

Condition (4.10) states that $\Phi$ is constant on each fiber of ${\chi }_{\vartheta }$. If this condition holds, set

$$\widehat{\Phi }(\vartheta \left(x\right),s):=\Phi (x,s).$$

Constancy on fibers makes the definition well defined, and surjectivity of ${\chi }_{\vartheta }^{\mathrm{act}}$ ensures uniqueness. Conversely, (4.9) implies constancy on every fiber. □

Suppose an open set $U\subseteq {\mathrm{X}}_{\varphi }$ admits local physical-state coordinates $(\xi ,\theta )$, where $\theta$ is a local real lift of $\vartheta$ and $\xi$ describes the remaining local degrees of freedom. After shrinking the joint neighborhood if necessary, any local real lift of observed phase can be written as

$${\phi }_{\Phi }(x,s)=\psi (\xi ,\theta ,s).$$

(4.11)

Dependence on $\xi$ may prevent factorization through $(\vartheta ,s)$. The global circular-phase map is still $\Phi :{\mathrm{M}}^{\times }\to \Theta$; the local description requires no global product decomposition of ${\mathrm{X}}_{\varphi }$.

The phase map is determined by the chosen response g. Even if it factors through $(\vartheta ,s)$, the resulting dependence may be nonlinear. An additive phase translation

$$\Phi (x,s)=\vartheta \left(x\right)+\beta \left(s\right)$$

requires further assumptions. The observation coordinate enters the formation of the joint response without introducing an independent physical phase.

4.3. Local Phase-Variation Model

Let

$$c:I⟶{\mathrm{M}}^{\times },c\left(t\right)=\left(x\left(t\right),s\left(t\right)\right)$$

(4.12)

be a smooth path of joint states on an interval I. Choose $t_0\in I$ and ${\phi }_0\in \mathbb{R}$ such that $e^{\mathrm{i}{\phi }_0}=u\left(c\left(t_0\right)\right)$. The path $u\circ c$ has a unique smooth real lift $\phi :I\to \mathbb{R}$ satisfying $\phi \left(t_0\right)={\phi }_0$. With $g_c:=g\circ c$, we have

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

(4.13)

Proposition 4.2

(Intrinsic local phase variation). Along every path satisfying Equation (4.12),

$$\dot{\phi }\left(t\right)=Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_{c\left(t\right)}\left(\dot{c}\left(t\right)\right)}{g\left(c\right(t\left)\right)}}\right).$$

(4.14)

If $\mathrm{M}$ is open in ${\mathrm{X}}_{\varphi }\times \mathrm{S}$ near the path, differentiation in the product gives

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

(4.15)

Proof. 

Differentiating Equation (4.13) and dividing by the nonzero response gives

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

Taking imaginary parts yields Equation (4.14). When $\mathrm{M}$ is open in the product, the chain rule reads

$${\left(\mathrm{d}g\right)}_{(x,s)}(\dot{x},\dot{s})=D_{x}g(x,s)\left[\dot{x}\right]+D_{s}g(x,s)\left[\dot{s}\right],$$

and Equation (4.15) follows. □

For a constrained submanifold $\mathrm{M}$, the intrinsic formula is (4.14). If the model specifies a smooth extension

$$\tilde{g}:U⟶\mathbb{C},U\subseteq {\mathrm{X}}_{\varphi }\times \mathrm{S},\tilde{g}{{|}_{U\cap \mathrm{M}}=g|}_{U\cap \mathrm{M}},$$

then the numerator may be written as

$$D_x\tilde{g}(x,s)\left[\dot{x}\right]+D_s\tilde{g}(x,s)\left[\dot{s}\right].$$

On tangent vectors to $\mathrm{M}$, this sum agrees with the intrinsic differential. Its two summands may nevertheless depend on the extension, so assigning them invariantly to the two axes requires further structure.

A splitting of the admissible tangent space along the two PODA directions also permits an intrinsic separation. For $m=(x,s)\in {\mathrm{M}}^{\times }$, set

$$V_{\mathrm{X},m}:=T_m\mathrm{M}\cap (T_x{\mathrm{X}}_{\varphi }\times \left\{0\right\}),V_{\mathrm{S},m}:=T_m\mathrm{M}\cap (\left\{0\right\}\times T_s\mathrm{S}).$$

Suppose a coordinate-compatible direct sum

$$T_m\mathrm{M}=V_{\mathrm{X},m}\oplus V_{\mathrm{S},m}$$

has been specified. For $v_{\mathrm{X}}\in V_{\mathrm{X},m}$ and $v_{\mathrm{S}}\in V_{\mathrm{S},m}$, define

$$\begin{array}{cc}\hfill A_{\mathrm{X},m}\left[v_{\mathrm{X}}\right]& :=Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_m\left(v_{\mathrm{X}}\right)}{g\left(m\right)}}\right),\hfill \end{array}$$

(4.16)

$$\begin{array}{cc}\hfill A_{\mathrm{S},m}\left[v_{\mathrm{S}}\right]& :=Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_m\left(v_{\mathrm{S}}\right)}{g\left(m\right)}}\right).\hfill \end{array}$$

(4.17)

For $\dot{c}=v_{\mathrm{X}}+v_{\mathrm{S}}$, the phase rate then decomposes intrinsically as

$$\dot{\phi }=A_{\mathrm{X},c\left(t\right)}\left[v_{\mathrm{X}}\right]+A_{\mathrm{S},c\left(t\right)}\left[v_{\mathrm{S}}\right].$$

(4.18)

This decomposition in (4.18) concerns tangent directions and does not imply a global additive decomposition of the phase function.

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

Fix an observation state $s_0\in \mathrm{S}$ and let $c\left(t\right)=(x\left(t\right),s_0)$ be a smooth path in ${\mathrm{M}}^{\times }$. Its observed-phase lift satisfies

$$\dot{\phi }=Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_{c\left(t\right)}\left((\dot{x}\left(t\right),0)\right)}{g\left(c\right(t\left)\right)}}\right),$$

(4.19)

the restriction of (4.14) to this admissible fixed-$s_0$ path.

Similarly, fix a physical state $x_{*}$ and let $c\left(t\right)=(x_{*},s\left(t\right))$ be a smooth path in ${\mathrm{M}}^{\times }$. Then

$$\dot{\phi }=Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_{c\left(t\right)}\left((0,\dot{s}\left(t\right))\right)}{g\left(c\right(t\left)\right)}}\right),$$

(4.20)

which describes phase variation along the observation-state slice at $x_{*}$.

The admissible observation states with nonzero response at this physical state form the slice

$${\mathrm{M}}_{x_{*}}^{\times }:=\{s\in \mathrm{S}:(x_{*},s)\in {\mathrm{M}}^{\times }\}.$$

(4.21)

For $s_r,s_c\in {\mathrm{M}}_{x_{*}}^{\times }$, both endpoints are implementable and their coherent responses are nonzero. Define

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

(4.22)

This same-source two-state phase compares the current observation state $s_c$ with the reference observation state $s_r$, keeping the physical state fixed. In unit-complex form, it is

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

(4.23)

The relation compares observed phases at two admissible endpoints with nonzero response. Since g may couple physical and observation states, the difference generally still depends on $x_{*}$. To interpret this endpoint relation as continuous transport, we specify an admissible comparison path in ${\mathrm{M}}^{\times }$ and impose the response-compatibility condition of the next section. Whether the resulting transport serves as a correction or compensation depends on the chosen reference and the task.

4.5. From Response Phase to Its Differential

We have constructed observed phase through

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

Along a path, differentiation of the complex response gives

$$\dot{\phi }=Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_{c\left(t\right)}\left(\dot{c}\left(t\right)\right)}{g\left(c\right(t\left)\right)}}\right).$$

On an open product domain, this differential separates into physical-state and observation-state terms. On a constrained domain, the same separation requires either a specified extension or a coordinate-compatible splitting of tangent spaces. Physical phase and observation state alone describe the phase exactly when the factorization criterion of Definition 4.1 holds; otherwise the remaining physical degrees of freedom must also enter. The next section distinguishes the full response fiber ${\mathbb{C}}^{\times }\cong {\mathbb{R}}_{>0}\times \mathrm{U}\left(1\right)$ from its phase factor $\mathrm{U}\left(1\right)$ and constructs the phase connection from response compatibility.

5. Response Bundles and the Phase Connection

Section 4 established the scalar complex response

$$g:\mathrm{M}⟶\mathbb{C}$$

and the local phase law on its nonzero domain,

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

Polar decomposition separates this response into amplitude and phase and gives a corresponding decomposition of the response bundle. On the nonzero joint-state domain, the normalized response induces a phase one-form. Requiring its section to be horizontal then determines a unique response-compatible principal phase connection. The horizontal lifts of this connection define phase transport.

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

Let the complex line response bundle over the coherent joint-state domain $\mathrm{M}$ be

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

(5.1)

with response section

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

(5.2)

This section is defined at every joint state, including those at which the response vanishes.

To use the multiplicative structure of the response, restrict to its nonzero domain:

$${\mathrm{M}}^{\times }:=\{m\in \mathrm{M}:g\left(m\right)\ne 0\},g^{\times }{:=g|}_{{\mathrm{M}}^{\times }}:{\mathrm{M}}^{\times }⟶{\mathbb{C}}^{\times },g^{\times }\left(m\right)=g\left(m\right),$$

(5.3)

where ${\mathbb{C}}^{\times }:=\mathbb{C}\setminus \left\{0\right\}$ and m denotes a joint state in $\mathrm{M}$. Restricting L to ${\mathrm{M}}^{\times }$ and removing the zero section gives

$${\pi }_{\times }:P_{\times }:={\mathrm{M}}^{\times }\times {\mathbb{C}}^{\times }⟶{\mathrm{M}}^{\times },{\pi }_{\times }(m,z)=m.$$

(5.4)

With the right action

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

$P_{\times }$ is a trivial principal ${\mathbb{C}}^{\times }$ bundle. Its response section is

$${\sigma }_g^{\times }:{\mathrm{M}}^{\times }⟶P_{\times },{\sigma }_g^{\times }\left(m\right):=(m,g^{\times }\left(m\right)),{\pi }_{\times }\circ {\sigma }_g^{\times }={id}_{{\mathrm{M}}^{\times }}.$$

(5.5)

Thus ${\sigma }_g^{\times }$ takes values in $P_{\times }$, whose fibers admit the polar decomposition

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

The positive real factor records amplitude, while $\mathrm{U}\left(1\right)$ records phase.

5.2. Amplitude–Phase Decomposition and the Principal Phase Bundle

The amplitude and phase factors define the bundles

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

(5.6)

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

(5.7)

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

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

Polar decomposition in each fiber gives the diffeomorphism

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

(5.8)

Hence

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

(5.9)

Under this decomposition, the response section has amplitude and phase components

$$\begin{array}{cccccc}\hfill {\sigma }_a:{\mathrm{M}}^{\times }& ⟶P_{\mathrm{amp}},\hfill & \hfill {\sigma }_a\left(m\right)& :=(m,a(m\left)\right),\hfill & \hfill a\left(m\right)& :=|g^{\times }\left(m\right)|,\hfill \end{array}$$

(5.10)

$$\begin{array}{cccccc}\hfill {\sigma }_u:{\mathrm{M}}^{\times }& ⟶P_{\mathrm{ph}},\hfill & \hfill {\sigma }_u\left(m\right)& :=(m,u(m\left)\right),\hfill & \hfill u\left(m\right)& :={\displaystyle \frac{g^{\times }\left(m\right)}{|g^{\times }\left(m\right)|}}.\hfill \end{array}$$

(5.11)

These are sections of their respective bundles, since

$${\pi }_{\mathrm{amp}}\circ {\sigma }_a={id}_{{\mathrm{M}}^{\times }},{\pi }_{\mathrm{ph}}\circ {\sigma }_u={id}_{{\mathrm{M}}^{\times }}.$$

We will construct the phase connection on $P_{\mathrm{ph}}$; the amplitude remains a separate component in $P_{\mathrm{amp}}$.

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

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

(5.12)

On $\mathrm{U}\left(1\right)$, the corresponding real phase form is

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

(5.13)

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

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

(5.14)

The real part therefore measures logarithmic amplitude variation, and the imaginary part measures phase variation.

Pulling this decomposition back along $g^{\times }:{\mathrm{M}}^{\times }\to {\mathbb{C}}^{\times }$ gives

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

(5.15)

where

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

(5.16)

The response-induced phase one-form $A_g$ measures phase variation on the nonzero joint-state domain. Its value on a tangent vector gives the infinitesimal phase change in that direction: for $v\in T_m{\mathrm{M}}^{\times }$,

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

(5.17)

Along a joint-state path $c:I\to {\mathrm{M}}^{\times }$,

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

(5.18)

recovering the local phase law of the previous section. The one-form $A_g$ thus expresses the first-order phase response in every admissible joint-state direction.

5.4. The Response-Compatible Principal Phase Connection

To compare phases over different joint states, we specify horizontal directions in the total space of the phase bundle. These directions are determined by a principal connection [11,12].

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

Remark (Principal connections on the trivial phase bundle) 

For every base one-form $K\in {\Omega }^1\left({\mathrm{M}}^{\times }\right)$,

$${\Omega }_K:={pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}-{\pi }_{\mathrm{ph}}^{*}K$$

(5.19)

is a principal $\mathrm{U}\left(1\right)$ connection on $P_{\mathrm{ph}}={\mathrm{M}}^{\times }\times \mathrm{U}\left(1\right)$. Conversely, every principal $\mathrm{U}\left(1\right)$ connection in this trivialization has this form for a unique K. To see this, note that ${\pi }_{\mathrm{ph}}^{*}K$ is horizontal and right invariant. The difference between any principal connection and ${pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}$ is likewise horizontal and right invariant, so it is basic and descends to a unique one-form on the base. The horizontal lifts of the connection above satisfy

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

The observed response selects a connection from this family through the following compatibility condition.

Definition 5.2

(Response compatibility). A principal $\mathrm{U}\left(1\right)$ connection $\Omega$ on $P_{\mathrm{ph}}$ is response-compatible with $g^{\times }$ if the normalized response section is horizontal:

$${\sigma }_u^{*}\Omega =0.$$

(5.20)

Equivalently, for every $C^1$ path $c:I\to {\mathrm{M}}^{\times }$, the response lift ${\sigma }_u\circ c$ is horizontal. Thus the phase of the complex readout itself follows a horizontal path in the phase bundle.

Theorem 5.3

(Existence and uniqueness of the response-compatible connection). On $P_{\mathrm{ph}}={\mathrm{M}}^{\times }\times \mathrm{U}\left(1\right)$ there exists exactly one principal $\mathrm{U}\left(1\right)$ connection that is response-compatible with $g^{\times }$. It is

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

(5.21)

For a base path $c:I\to {\mathrm{M}}^{\times }$, a lift $\tilde{c}\left(t\right)=(c\left(t\right),\zeta \left(t\right))$ is horizontal precisely when

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

(5.22)

The compatibility condition is that of Definition 5.2.

Proof. 

By Definition 5.1, (5.21) defines a principal connection. Pulling it back along the normalized response section gives

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

so this connection is response-compatible. Conversely, write an arbitrary principal connection in the unique form ${\Omega }_K$ of (5.19). Its pullback is

$${\sigma }_u^{*}{\Omega }_K=u^{*}{\omega }_{\mathrm{U}\left(1\right)}-K=A_g-K.$$

It is therefore response-compatible if and only if $K=A_g$, proving uniqueness under the stated condition. Finally, the horizontal condition ${\Omega }_g\left(\dot{\tilde{c}}\right)=0$ is precisely (5.22). □

In particular, the observed phase section satisfies

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

(5.23)

Its graph is therefore a horizontal section for ${\Omega }_g$.

5.5. Flatness, Holonomy, and Zeros

Since $\mathrm{U}\left(1\right)$ is Abelian, its phase form is closed: $\mathrm{d}{\omega }_{\mathrm{U}\left(1\right)}=0$. Pullback commutes with exterior differentiation, so $A_g=u^{*}{\omega }_{\mathrm{U}\left(1\right)}$ gives

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

(5.24)

The curvature of the response-compatible connection is therefore

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

(5.25)

Thus the connection induced by the normalized response is flat. It also admits the global horizontal section ${\sigma }_u$. Consequently, for every closed piecewise-$C^1$ loop c in ${\mathrm{M}}^{\times }$,

$${Hol}_{{\Omega }_g}\left(c\right)=exp\left(\mathrm{i}{\int }_c{A}_g\right)=u{\left(c\left(0\right)\right)}^{-1}u\left(c\left(1\right)\right)=1.$$

(5.26)

On a non-simply-connected base, a flat connection can have nontrivial holonomy. Here the global horizontal section gives the stronger conclusion (5.26). Winding is retained by the real lift: its phase may change by $2\pi n$ around a closed loop, while exponentiation sends this change to the identity in $\mathrm{U}\left(1\right)$.

To obtain nontrivial circular holonomy or Berry curvature, one must supply further geometry, for example a different connection ${\Omega }_K$, nontrivial bundle or transition data, or a singular extension across the excluded zeros.

The roles of zeros, vanishing phase variation, and amplitude can now be distinguished:

(1)

If $g\left(m\right)=0$, the complex line response bundle and its response section remain defined, but $u=g/\left|g\right|$ is undefined. The response therefore induces no canonical phase section, phase one-form, or phase connection at that point.

(2)

If $g\left(m\right)\ne 0$ but ${\left(A_g\right)}_m=0$, the phase remains well defined. Only its first-order variation vanishes, in every admissible direction at that point.

(3)

The phase connection compares points in the pure phase fiber. Amplitude evolution is described separately by

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

5.6. From the Response Bundle to Transport

The passage from the coherent response to the phase connection is therefore

$$F_{\mathrm{M}}^{\mathrm{coh}}⟶g⟶g^{\times }⟶P_{\times }\cong P_{\mathrm{amp}}{\times }_{{\mathrm{M}}^{\times }}{P}_{\mathrm{ph}}⟶A_g⟶{\Omega }_g.$$

The nonzero response fiber is ${\mathbb{C}}^{\times }$, with phase factor $\mathrm{U}\left(1\right)$. The response induces the base one-form $A_g=Im(\mathrm{d}{g}^{\times }/g^{\times })$ and hence the response-compatible principal phase connection ${\Omega }_g$. The next section derives the differential equation of this transport directly from the normalized response and proves its equivalence to the horizontal condition ${\Omega }_g\left(\dot{\tilde{c}}\right)=0$.

6. Derivation of Phase Transport and Its Finite Forms

We begin with the nonzero joint-state domain introduced in Section 5,

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

and the restricted response $g^{\times }:{\mathrm{M}}^{\times }\to {\mathbb{C}}^{\times }$. We write g for this restriction whenever the domain is ${\mathrm{M}}^{\times }$. The normalized response determines the response-induced phase one-form and the response-compatible principal phase connection:

$$\begin{array}{cc}\hfill u& :={\displaystyle \frac{g}{\left|g\right|}}:{\mathrm{M}}^{\times }⟶\mathrm{U}\left(1\right),\hfill \end{array}$$

(6.1)

$$\begin{array}{cc}\hfill A_g& :=Im\left({\displaystyle \frac{\mathrm{d}g}{g}}\right)=u^{*}{\omega }_{\mathrm{U}\left(1\right)},\hfill \end{array}$$

(6.2)

$$\begin{array}{cc}\hfill {\Omega }_g& :={pr}_{\mathrm{U}\left(1\right)}^{*}{\omega }_{\mathrm{U}\left(1\right)}-{\pi }_{\mathrm{ph}}^{*}{A}_g\mathrm{on}{P}_{\mathrm{ph}}={\mathrm{M}}^{\times }\times \mathrm{U}\left(1\right).\hfill \end{array}$$

(6.3)

By Theorem 5.3, a lift $\tilde{c}\left(t\right)=(c\left(t\right),\zeta \left(t\right))$ is horizontal for this connection exactly when

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

(6.4)

This relation has been obtained from the response-compatible connection. We now derive the phase evolution from the response itself, with phase measured in radians.

6.1. Differentiable Lifted Paths and the Phase Functional

Let $I=[t_0,t_1]\subset \mathbb{R}$ be a compact interval, and let

$${\mathrm{C}}^1(I,{\mathrm{M}}^{\times }):=\{c:I\to {\mathrm{M}}^{\times }:c\mathrm{is}{C}^1\}$$

(6.5)

denote the space of continuously differentiable base paths. The corresponding space of lifted paths is

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

(6.6)

We write the space of one-forms with continuous coefficients on the interval as

$${\Omega }_{\mathrm{c}}^1\left(I\right):=\{f\left(t\right)\mathrm{d}t:f\in C^0(I,\mathbb{R})\}.$$

Definition 6.1

(Phase-transport differential functional). The response-compatible principal phase connection defines the differential functional

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

(6.7)

For each lifted path, this functional records the difference between the phase variation of the lift and that of the response. It is a one-form on the parameter interval.

6.2. Derivation from the Normalized Response

A phase transported with the response must acquire the same increments as the response phase. Its initial reference may be chosen freely: changing that reference multiplies every phase along the path by one constant element of $\mathrm{U}\left(1\right)$. This gives a finite comparison criterion before any differential equation is imposed. If $v\left(t\right):=u\left(c\right(t\left)\right)$ is the observed phase and $\zeta \left(t\right)$ is the transported phase, their relative phase $\zeta \left(t\right)v{\left(t\right)}^{-1}$ must remain constant.

Theorem 6.2

(Derivation of the response transport equation). Let $g:{\mathrm{M}}^{\times }\to {\mathbb{C}}^{\times }$ be smooth, let $c\in {\mathrm{C}}^1(I,{\mathrm{M}}^{\times })$, and write

$$v\left(t\right):=u\left(c\left(t\right)\right),a_c\left(t\right):=\left|g\left(c\left(t\right)\right)\right|,g\left(c\left(t\right)\right)=a_c\left(t\right)v\left(t\right).$$

Then the observed phase satisfies

$$\dot{v}\left(t\right)=\mathrm{i}Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_{c\left(t\right)}\left(\dot{c}\left(t\right)\right)}{g\left(c\right(t\left)\right)}}\right)v\left(t\right)=\mathrm{i}{\left(A_g\right)}_{c\left(t\right)}\left(\dot{c}\left(t\right)\right)v\left(t\right).$$

(6.8)

For a $C^1$ lift $\tilde{c}\left(t\right)=(c\left(t\right),\zeta \left(t\right))$, choose ${\psi }_0\in \mathbb{R}$ with $e^{\mathrm{i}{\psi }_0}=\zeta \left(t_0\right)$, and let ψ be the real lift of ζ with initial value ${\psi }_0$. The following conditions are equivalent:

(1)

The relative phase $\zeta \left(t\right)v{\left(t\right)}^{-1}$ is constant on I.

(2)

The group-valued differential relation holds:

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

(6.9)

(3)

The covariant derivative vanishes:

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

(6.10)

(4)

The real phase lift satisfies

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

(6.11)

(5)

The lifted path is horizontal for the response-compatible connection:

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

(6.12)

For every initial phase ${\zeta }_0\in \mathrm{U}\left(1\right)$, these conditions determine the unique lift

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

(6.13)

If, near the path, ${\mathrm{M}}^{\times }$ is open in ${\mathrm{X}}_{\varphi }\times \mathrm{S}$, set $\tilde{g}:=g$. More generally, suppose a smooth extension $\tilde{g}$ to a product neighborhood is specified. Writing $c\left(t\right)=\left(x\right(t),s(t\left)\right)$, the real-phase equation then becomes

$$\dot{\psi }=Im\left[{\displaystyle \frac{D_x\tilde{g}(x,s)\left[\dot{x}\right]+D_s\tilde{g}(x,s)\left[\dot{s}\right]}{g(x,s)}}\right].$$

(6.14)

Proof. 

Differentiate $g\circ c=a_{c}v$ and divide by the nonzero response:

$${\displaystyle \frac{\frac{\mathrm{d}}{\mathrm{d}t}g\left(c\left(t\right)\right)}{g\left(c\right(t\left)\right)}}={\displaystyle \frac{{\dot{a}}_c\left(t\right)}{a_c\left(t\right)}}+v{\left(t\right)}^{-1}\dot{v}\left(t\right).$$

The first term on the right is real. Since ${\left|v\right|}^2=1$, differentiation gives $Re\left(v^{-1}\dot{v}\right)=0$. Taking imaginary parts therefore yields

$$v^{-1}\dot{v}=\mathrm{i}Im\left({\displaystyle \frac{\frac{\mathrm{d}}{\mathrm{d}t}g\left(c\left(t\right)\right)}{g\left(c\right(t\left)\right)}}\right)=\mathrm{i}{A}_g\left(\dot{c}\right),$$

which proves (6.8).

For an arbitrary phase lift, the product rule now gives

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(\zeta v^{-1}\right)=v^{-1}\left(\dot{\zeta }-\mathrm{i}{A}_g\left(\dot{c}\right)\zeta \right).$$

Thus the relative phase is constant precisely when the covariant derivative vanishes. Multiplication by $-\mathrm{i}{\zeta }^{-1}$ gives the group-valued form. Substitution of $\zeta =e^{\mathrm{i}\psi }$ gives the real-phase form. Finally,

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

so these relations are equivalent to horizontality for the connection already determined in Section 5.

Constancy of $\zeta v^{-1}$ and the initial value force $\zeta v^{-1}={\zeta }_{0}v{\left(t_0\right)}^{-1}$. This gives (6.13), which is $C^1$ and satisfies all five conditions, proving existence and uniqueness. Under the product-neighborhood hypothesis, the chain rule gives

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}g(x\left(t\right),s\left(t\right))=D_x\tilde{g}(x,s)\left[\dot{x}\right]+D_s\tilde{g}(x,s)\left[\dot{s}\right],$$

and proves (6.14). □

Definition 6.3

(Phase Transport Fundamental Equation). The response transport equation established in Theorem 6.2, written intrinsically as

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

(PT)

is called the Phase Transport Fundamental Equation (PTFE).

Within PODA Theory, this is the first fundamental equation of observation space. For a fixed physical state $x_{*}$ and an admissible $C^1$ path $c\left(t\right)=(x_{*},s\left(t\right))$ in ${\mathrm{M}}^{\times }$, the derived law restricts to

$$\dot{\psi }\left(t\right)=Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_{(x_{*},s\left(t\right))}\left((0,\dot{s}\left(t\right))\right)}{g(x_{*},s\left(t\right))}}\right).$$

(6.16)

The phase can therefore change along the observation axis while the physical-state coordinate remains fixed. On a joint path, both coordinates contribute. The law determines the phase change of a specified coherent response; the physical model and observation protocol determine the response and the path.

Equation (6.14) gives the two directional contributions when the stated product-neighborhood hypothesis holds. Each term may depend on the full joint state. With an extension $\tilde{g}$, the separate terms may depend on the extension, but their sum on an admissible tangent vector is intrinsic. Along a fixed-physical-state path, the observation-direction value in (6.16) is intrinsic even when the admissible domain is not a product.

6.3. Fundamental Properties of Response-Compatible Transport

Theorem 6.4

(Fundamental properties of PTFE). Let $g:{\mathrm{M}}^{\times }\to {\mathbb{C}}^{\times }$ be smooth. With phase measured in radians and response compatibility given by (6.4), PTFE satisfies the following properties.

(1)

Response induction.The chain

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

determines the phase connection directly from the response, without an additional empirical phase term.

(2)

Uniqueness of the compatible connection.A lifted path satisfies (PT) if and only if it is horizontal for ${\Omega }_g$. Among principal $\mathrm{U}\left(1\right)$ connections on this trivial phase bundle whose horizontal lifts satisfy (6.4) along every $C^1$ base path, ${\Omega }_g$ is the unique such connection.

(3)

Uniqueness of the first-order response law.If $K\in {\Omega }^1\left({\mathrm{M}}^{\times }\right)$ satisfies

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

for every $C^1$ path c, then $K=A_g$.

(4)

Initial-value well-posedness.Every $C^1$ base path c, together with an initial phase-fiber point $(c\left(t_0\right),{\zeta }_0)$, determines a unique $C^1$ horizontal lift.

(5)

Finite-path extension.Integration of the response-induced phase one-form extends finite transport uniquely by additivity to piecewise-$C^1$ paths, with the corresponding identity, reversal, and concatenation laws.

(6)

Reparametrization invariance.For every $C^1$ diffeomorphism $\lambda :J\to I$, $\tilde{c}$ satisfies (PT) if and only if $\tilde{c}\circ \lambda$ does.

(7)

Gauge covariance.For every smooth $\kappa :{\mathrm{M}}^{\times }\to \mathrm{U}\left(1\right)$, define

$$g_{\kappa }:=\kappa g,{\Psi }_{\kappa }:P_{\mathrm{ph}}\to P_{\mathrm{ph}},{\Psi }_{\kappa }(m,\zeta ):=(m,\kappa \left(m\right)\zeta ).$$

(6.17)

Then

$${\Psi }_{\kappa }^{*}{\Omega }_{\kappa g}={\Omega }_g.$$

(6.18)

Consequently, $\tilde{c}$ is ${\Omega }_g$-horizontal if and only if ${\Psi }_{\kappa }\circ \tilde{c}$ is ${\Omega }_{\kappa g}$-horizontal. Here the response and the phase-fiber coordinates transform together.

(8)

Positive-amplitude invariance.For every smooth $\rho :{\mathrm{M}}^{\times }\to {\mathbb{R}}_{>0}$, one has $A_{\rho g}=A_g$ and ${\Omega }_{\rho g}={\Omega }_g$, so PTFE is unchanged.

Proof. 

The definitions give response induction directly. To prove uniqueness of the connection, write an arbitrary principal connection in the chosen trivialization as

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

for a one-form $\beta$ on the base. Agreement of its horizontal condition with (6.4) on every base tangent vector forces $\beta =-A_g$, and therefore $\tilde{\Omega }={\Omega }_g$.

For uniqueness of the first-order law, fix $m\in {\mathrm{M}}^{\times }$ and $v\in T_m{\mathrm{M}}^{\times }$, and choose a local $C^1$ path through $c\left(0\right)=m$ with $\dot{c}\left(0\right)=v$. Since $A_g=u^{*}{\omega }_{\mathrm{U}\left(1\right)}$, evaluation of the assumed identity on this path yields

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

Thus $K=A_g$.

Theorem 6.2 proves existence and uniqueness of the lift and gives its solution in (6.13). Integrating over a finite partition gives the additive extension and the finite transport laws, while functoriality of pullback proves reparametrization invariance.

Under the gauge transformation, the induced phase one-form becomes

$$A_{\kappa g}=A_g+{\kappa }^{*}{\omega }_{\mathrm{U}\left(1\right)},$$

and the Abelian Maurer–Cartan identity gives

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

Cancellation of the added terms proves (6.18). Positive-amplitude invariance follows from $u_{\rho g}=u_g$ for $\rho >0$. □

Remark (Dependence on the response) 6.5

The observation law and coherent readout determine the phase transport equation through

$$F_{\mathrm{M}}^{\mathrm{coh}},\mathrm{r}⟶g⟶A_g⟶{\Omega }_g⟶{\mathbf{E}}_g\left[\tilde{c}\right]=0.$$

This construction assumes a globally defined smooth scalar response and a path along which that response is nonzero. A treatment using only local response coordinates or a nontrivial phase bundle also needs transition data; a treatment of zero crossings needs singular data.

6.4. Finite Phase Transport

Let

$$\begin{array}{cc}\hfill {PC}^1(I,{\mathrm{M}}^{\times }):=\{c:I\to {\mathrm{M}}^{\times }:& c\mathrm{is}\mathrm{continuous},\mathrm{and}\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{finite}\mathrm{partition}\hfill \\ \hfill & t_0={\tau }_0<\dots <{\tau }_N=t_1{\mathrm{such}\mathrm{that}\mathrm{every}c|}_{[{\tau }_{j-1},{\tau }_j]}\mathrm{is}{C}^1\}.\hfill \end{array}$$

(6.19)

The line integral along such a path is the sum of the integrals over its continuously differentiable pieces. It defines the finite phase increment and the associated circular transport.

Definition 6.6

(Finite phase increment and circular transport). For $c\in {PC}^1(I,{\mathrm{M}}^{\times })$, define

$$\begin{array}{cc}\hfill {\Delta }_g\left[c\right]& :={\int }_c{A}_g\in \mathbb{R},\hfill \end{array}$$

(6.20)

$$\begin{array}{cc}\hfill {\mathrm{T}}_g\left[c\right]& :=exp\left(\mathrm{i}{\int }_c{A}_g\right)\in \mathrm{U}\left(1\right),\hfill \end{array}$$

(6.21)

and

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

(6.22)

The normalized response gives the endpoint expression

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

(6.23)

If $c_2*c_1$ denotes piecewise-$C^1$ concatenation and $\overline{c}$ path reversal, then

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

(6.24)

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

(6.25)

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

(6.26)

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

(6.27)

Since $A_g=u^{*}{\omega }_{\mathrm{U}\left(1\right)}$, circular transport along an admissible path depends only on its endpoints. The real increment also records winding: for two piecewise-$C^1$ paths with the same endpoints,

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

(6.28)

and ${\mathrm{T}}_g\left[c_1\right]={\mathrm{T}}_g\left[c_2\right]$.

The ratio in (6.23) has a broader algebraic meaning: it converts between the phases of any two nonzero endpoint responses. It represents finite transport when a piecewise-$C^1$ path connects the endpoints in ${\mathrm{M}}^{\times }$. For endpoints in different path components, the ratio remains defined, but there is no connecting line integral or unwrapped phase increment.

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

For a fixed physical-state coordinate $x_{*}\in {\mathrm{X}}_{\varphi }$, the admissible nonzero observation-state fiber is

$${\mathrm{M}}_{x_{*}}^{\times }:=\{s\in \mathrm{S}:(x_{*},s)\in {\mathrm{M}}^{\times }\}.$$

(6.29)

Let $s_r,s_c\in {\mathrm{M}}_{x_{*}}^{\times }$. To transport phase from $s_r$ to $s_c$ while holding the physical state fixed, consider a piecewise-$C^1$ map

$${\gamma }_{r\to c}:[0,1]⟶\mathrm{S},{\gamma }_{r\to c}\left(0\right)=s_r,{\gamma }_{r\to c}\left(1\right)=s_c,$$

(6.30)

such that

$$c_{r\to c}\left(\tau \right):=(x_{*},{\gamma }_{r\to c}\left(\tau \right))\in {\mathrm{M}}^{\times }\mathrm{for}\mathrm{every}\tau \in [0,1].$$

(6.31)

The entire connecting path lies in the admissible nonzero fiber over the chosen physical state.

Along a path satisfying (6.31), the same-source two-state transport factor is

$$R_{\mathrm{ss}}(x_{*};s_c,s_r):={\mathrm{T}}_g\left[c_{r\to c}\right]=u{(x_{*},s_r)}^{-1}u(x_{*},s_c).$$

(6.32)

Theorem 6.7

(Same-source two-state inverse phase transport). Suppose (6.31) holds. Transport along the reversed path has the factor

$$T_{s_r\leftarrow s_c}\left(x_{*}\right):={\mathrm{T}}_g\left[{\overline{c}}_{r\to c}\right]=R_{\mathrm{ss}}{(x_{*};s_c,s_r)}^{-1}=u(x_{*},s_r)u{(x_{*},s_c)}^{-1}.$$

(6.33)

It is the unique element of $\mathrm{U}\left(1\right)$ satisfying

$$T_{s_r\leftarrow s_c}\left(x_{*}\right)u(x_{*},s_c)=u(x_{*},s_r),$$

(6.34)

or, in response coordinates,

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

(6.35)

Proof. 

Path reversal and the endpoint expression (6.23) give (6.33). Multiplying by $u(x_{*},s_c)$ proves (6.34). Any other group element satisfying the same identity must coincide with this factor, as right multiplication by $u{(x_{*},s_c)}^{-1}$ shows. Substitution of $u=g/\left|g\right|$ gives the response-coordinate formula. □

If the two nonzero responses admit no connecting path satisfying (6.31), formula (6.35) still gives an algebraic endpoint conversion. It is the connecting path at fixed physical state that makes this conversion a same-source two-state transport.

Applied to the current response, inverse transport changes its phase to the reference phase while preserving its amplitude. Recovery of the full reference response therefore also requires the reference amplitude. The phase factor itself uses both nonzero endpoint responses, which must be supplied by the response model or by measurements.

6.6. Dynamic Reference Conversion

On a compact interval I, consider two piecewise-$C^1$ joint-state paths:

$$c_c\left(t\right):=(x\left(t\right),s_c\left(t\right))\in {\mathrm{M}}^{\times },c_r\left(t\right):=(x\left(t\right),s_r\left(t\right))\in {\mathrm{M}}^{\times },t\in I.$$

(6.36)

At each time, the current and reference paths share the same physical-state coordinate. The reference observation state may move; a fixed reference is the special case $s_r\left(t\right)\equiv s_r$, provided its entire joint-state path lies in ${\mathrm{M}}^{\times }$.

The pointwise conversion factor is

$$T_{r\leftarrow c}\left(t\right):=u\left(c_r\left(t\right)\right)u{\left(c_c\left(t\right)\right)}^{-1}.$$

(6.37)

By construction,

$$T_{r\leftarrow c}\left(t\right)u\left(c_c\left(t\right)\right)=u\left(c_r\left(t\right)\right)$$

(6.38)

at every time. Each response path obeys its differential phase law on its $C^1$ subintervals. When the reference observation state moves, the converted phase includes the phase change associated with that motion as well as with physical evolution. To realize the pointwise conversion as transport along the observation axis, one needs a connecting path at the instantaneous physical state.

For each t, choose such a piecewise-$C^1$ connecting map

$${\gamma }_t:[0,1]\to \mathrm{S},{\gamma }_t\left(0\right)=s_c\left(t\right),{\gamma }_t\left(1\right)=s_r\left(t\right),$$

with

$$(x\left(t\right),{\gamma }_t\left(\tau \right))\in {\mathrm{M}}^{\times }\mathrm{for}\mathrm{all}\tau \in [0,1].$$

These paths give a continuous family of slice transports when they depend continuously on t; differentiating the family requires the corresponding additional regularity. If no such connecting paths are available, (6.37) retains its meaning as a pointwise endpoint conversion.

Computing (6.37) requires knowledge of the reference response. A calibrated model or reference measurements must supply $g\circ c_r$, since the current response $g\circ c_c$ alone generally does not determine it.

For a fixed reference $s_r\left(t\right)\equiv s_r$ with a $C^1$ reference path, a compatible real lift of $u\circ c_r$ satisfies

$${\dot{\psi }}_r\left(t\right)=Im\left({\displaystyle \frac{{\left(\mathrm{d}g\right)}_{c_r\left(t\right)}\left({\dot{c}}_r\left(t\right)\right)}{g\left(c_r\left(t\right)\right)}}\right).$$

(6.39)

Because ${\dot{c}}_r\left(t\right)=(\dot{x}\left(t\right),0)$ is tangent to ${\mathrm{M}}^{\times }$, this equation uses the intrinsic differential of the response and needs no ambient splitting of $\mathrm{d}g$. Under the product-neighborhood or specified-extension hypothesis of Definition 6.2, the numerator can also be written as $D_x\tilde{g}(x\left(t\right),s_r)\left[\dot{x}\left(t\right)\right]$. The equation describes the phase observed at the fixed reference observation state. Agreement with an independently specified intrinsic physical-phase law requires the reference slice to be phase faithful, either directly or after a calibrated phase correction. Thus each same-source comparison holds the instantaneous physical state fixed, and the resulting conversion expresses the evolving signal in the chosen reference observation state.

6.7. A Phase-Faithful Reference

Let $U\subseteq {\mathrm{X}}_{\varphi }$ be an open set on which the physical phase map $\vartheta$ is smooth, and suppose $(x,s_r)\in {\mathrm{M}}^{\times }$ for every $x\in U$. A fixed reference state $s_r$ is phase faithful on U, up to a constant choice of phase zero, when there is a constant ${\kappa }_r\in \mathrm{U}\left(1\right)$ such that

$$u(x,s_r)={\kappa }_r\iota \left(\vartheta \left(x\right)\right),x\in U.$$

(6.40)

Along a $C^1$ physical path in U, choose compatible real lifts $\theta$ of $\vartheta \circ x$ and ${\psi }_r$ of $u(x,s_r)$. Their difference is constant, and hence ${\dot{\psi }}_r=\dot{\theta }$. This follows directly by differentiating (6.40).

More generally, a calibrated phase correction $b:U\to \mathrm{U}\left(1\right)$ may satisfy

$$b{\left(x\right)}^{-1}u(x,s_r)={\kappa }_r\iota \left(\vartheta \left(x\right)\right).$$

The corrected response then has the physical phase rate. The reference model or calibration must supply b and establish this relation; conversion to a common observation state alone does not establish it.

6.8. Response Data, Observation Structure, and the Phase Level

To distinguish the complex response from the phase transport it induces, we record the response together with the observation structure from which it arises.

Definition 6.8

(${\mathsf{G}}_0$ response data and observation structure). The full complex-response data object is

$${\mathsf{G}}_0^{\mathrm{data}}:=(\mathrm{M},g,L,{\sigma }_g),$$

(6.41)

and the associated observation structure is

$${\mathsf{P}}_0:=\left(\mathrm{M}\hookrightarrow \mathrm{B},F_{\mathrm{M}}^{\mathrm{coh}},\mathrm{r}\right),g=\mathrm{r}\circ F_{\mathrm{M}}^{\mathrm{coh}}.$$

(6.42)

Together, they form

$${\mathsf{G}}_0:=({\mathsf{G}}_0^{\mathrm{data}};{\mathsf{P}}_0).$$

(6.43)

The response data retain amplitude, phase, and the zero set, while the observation structure records the admissible joint-state domain, the observation law, and the coherent readout. Because a response value $g\left(m\right)$ may arise from several joint states $m=(x,s)$, the value alone does not recover this structure.

Definition 6.9

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

$${\mathsf{G}}_1:=\left({\mathrm{M}}^{\times },u,A_g,P_{\mathrm{ph}},{\Omega }_g,{\mathbf{E}}_g,{\left\{D_t^{(g,c)}\right\}}_{c\in {\mathrm{C}}^1(I,{\mathrm{M}}^{\times })},{\Delta }_g,{\mathrm{T}}_g,{\left\{{\mathrm{P}}_c^g\right\}}_{c\in {PC}^1(I,{\mathrm{M}}^{\times })}\right).$$

(6.44)

The differential functional ${\mathbf{E}}_g$ and the covariant equation are defined on $C^1$ paths; the finite quantities ${\Delta }_g$, ${\mathrm{T}}_g$, and ${\mathrm{P}}_c^g$ also apply to piecewise-$C^1$ paths. This level retains normalized phase, its first-order variation, and finite transport. It contains neither amplitude information nor a phase assignment at zeros of g.

The hierarchy ${\mathsf{G}}_0$–${\mathsf{G}}_1$–${\mathsf{G}}_2$ places response data and their successive reductions within one PODA observation law. In Section 7, ${\mathsf{G}}_2$ is obtained from a task data object by a group action representing freedoms irrelevant to the task.

6.9. From the Differential Law to Finite Comparison

The observation law thus leads from a complex response to a differential phase equation and, by integration, to finite comparison:

$$\begin{array}{cc}\hfill F_{\mathrm{M}}^{\mathrm{coh}},\mathrm{r}& ⟶g⟶A_g⟶{\Omega }_g⟶{\mathbf{E}}_g\left[\tilde{c}\right]=0,\hfill \\ \hfill & ⟶\left(D_t^{(g,c)}\zeta =0,\mathrm{d}\psi =c^{*}{A}_g\right)⟶\left({\Delta }_g,{\mathrm{T}}_g,{\mathrm{P}}_c^g\right).\hfill \end{array}$$

(6.45)

Response compatibility fixes the connection and the first-order phase law, so the horizontal lifts on $C^1$ paths and the finite transport on piecewise-$C^1$ paths describe the same phase evolution. Their covariance under a simultaneous change of response and phase-fiber coordinates is expressed by ${\Psi }_{\kappa }^{*}{\Omega }_{\kappa g}={\Omega }_g$.

The smooth nonzero domain ${\mathrm{M}}^{\times }$ is essential: normalized phase and its induced connection are undefined at a response zero. Within a fixed physical-state slice, an admissible nonzero connecting path gives the endpoint phase relation its meaning as same-source two-state transport. PTFE then determines how the phase of the given response varies along that path, while the physical and measurement models supply the trajectory and the response.

7. ${\mathsf{G}}_{\mathbf{2}}$: Task Quotient, Maximal Invariants, and Recoverability

Section 6 established how to compare phases by horizontal lifting along admissible paths in ${\mathrm{M}}^{\times }$. With these transports and the endpoint responses in hand, we can ask which distinctions in the resulting data matter for a given task. A common scale, phase reference, or coordinate choice may carry no information about the target; in that case, the task representation can identify data that differ only by that freedom.

Neither the complex response g nor PTFE determines these task-dependent identifications: the same data may serve several tasks with different invariances. To construct ${\mathsf{G}}_2$, we must therefore specify both a task data space and a group action on it. The quotient records the distinctions left by this action, and the recoverability criterion tells us whether those distinctions suffice to determine the target. We first work with sets; smoothness enters when we consider the geometry of the quotient.

7.1. Task Data After Transport Unification

Let ${\mathrm{D}}_{\mathrm{task}}$ denote the domain of admissible task data, with the physical states, observation states, times, channels, or data blocks relevant to the task. Combining the full response of ${\mathsf{G}}_0$ with the phase transport of ${\mathsf{G}}_1$ gives a task-specific transport-unified data map

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

(7.1)

where $\mathrm{Z}$ is the task data space. It may consist of sequences, multichannel vectors, multisource response matrices, or collections of relative phases. Thus the data supplied to the quotient may contain several related observation components.

The term “transport-unified” means that phase components to be compared across observation states have been expressed relative to the common reference prescribed by the task. This comparison leaves amplitudes, channel gains, and other data available for subsequent use. The role of ${\mathsf{G}}_1$ is to make phases comparable; the role of ${\mathsf{G}}_2$ is to identify data under the freedoms that the task declares irrelevant.

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

Definition 7.1

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

$${\mathsf{a}}_{\mathrm{K}}:\mathrm{K}\times \mathrm{Z}⟶\mathrm{Z},{\mathsf{a}}_{\mathrm{K}}(k,z)=k·z.$$

(7.2)

If the task regards all data on the same orbit as equivalent, so that $k·z$ and z are indistinguishable with respect to the target quantity, we call $\mathrm{K}$ the task-irrelevant transformation group.

Whether a common complex scale, common phase, channel reweighting, or coordinate transformation is irrelevant depends on the target. The action specifies the proposed identifications; their suitability must come from the task.

Definition 7.2

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

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

(7.3)

The class

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

(7.4)

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

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

(7.5)

The group axioms ensure that ${\sim }_{\mathrm{K}}$ is an equivalence relation, so its classes form a quotient set. If $\mathrm{Z}$ is topological and the action is continuous, we equip this set with the quotient topology. If $\mathrm{K}$ is a Lie group, $\mathrm{Z}$ a smooth manifold, and the action smooth, free, and proper, then $\mathrm{Z}/\mathrm{K}$ admits a unique smooth-manifold structure making $q_{\mathrm{K}}$ a smooth submersion [13]. When these hypotheses fail, singularities may occur and the quotient need not be a smooth manifold.

7.3. Factorization of Invariants and Maximal Invariants

Definition 7.3

(Task invariant). A map

$$I:\mathrm{Z}⟶\mathrm{W}$$

(7.6)

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

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

(7.7)

Theorem 7.4

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

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

(7.8)

such that

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

(7.9)

Proof. 

Suppose first that I is invariant, and define $\overline{I}\left({\left[z\right]}_{\mathrm{K}}\right):=I\left(z\right)$. To check that this definition is independent of the representative, let ${\left[z_1\right]}_{\mathrm{K}}={\left[z_2\right]}_{\mathrm{K}}$. Then $z_2=k·z_1$ for some k, so invariance gives $I\left(z_2\right)=I\left(z_1\right)$. The resulting map therefore satisfies Equation (7.9). Since $q_{\mathrm{K}}$ is surjective, this identity determines the map on every quotient class and proves uniqueness.

Conversely, suppose that $I=\overline{I}\circ q_{\mathrm{K}}$. Points on the same orbit have the same quotient class, and hence

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

which proves that I is invariant. □

Every invariant can thus be read from the quotient observation $q_{\mathrm{K}}\left(z\right)$. Some invariants still assign the same value to distinct orbits. A maximal invariant makes no such further identifications.

Definition 7.5

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

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

(7.10)

Theorem 7.6

(Equivalence between maximal invariants and quotient observations). Let $I:\mathrm{Z}\to \mathrm{W}$ be invariant and let $\overline{I}$ be the induced map from Definition 7.4. Then I is maximal if and only if

$${\overline{I}}^{\mathrm{act}}:{\mathrm{Q}}_{\mathrm{K}}⟶I\left(\mathrm{Z}\right),{\overline{I}}^{\mathrm{act}}\left({\left[z\right]}_{\mathrm{K}}\right):=I\left(z\right)$$

(7.11)

is a bijection. The attained values of a maximal invariant therefore give a canonical representation of the quotient observation space.

Proof. 

By its definition, the corestriction ${\overline{I}}^{\mathrm{act}}$ is surjective onto $I\left(\mathrm{Z}\right)$. Suppose that I is maximal and that ${\overline{I}}^{\mathrm{act}}\left(\left[z_1\right]\right)={\overline{I}}^{\mathrm{act}}\left(\left[z_2\right]\right)$. Then $I\left(z_1\right)=I\left(z_2\right)$, so maximality gives $\left[z_1\right]=\left[z_2\right]$. Thus ${\overline{I}}^{\mathrm{act}}$ is also injective, proving that it is a bijection.

Conversely, suppose that ${\overline{I}}^{\mathrm{act}}$ is bijective. By definition, $I\left(z_1\right)=I\left(z_2\right)$ is equivalent to ${\overline{I}}^{\mathrm{act}}\left(\left[z_1\right]\right)={\overline{I}}^{\mathrm{act}}\left(\left[z_2\right]\right)$. Injectivity makes this last equality equivalent to $\left[z_1\right]=\left[z_2\right]$, or, equivalently, $z_1{\sim }_{\mathrm{K}}{z}_2$. This is precisely the defining condition for maximality. □

A maximal invariant therefore records exactly which task orbit contains the data. Its values retain every distinction between orbits while discarding the prescribed freedoms within each orbit.

7.4. Recoverability Criterion for Target Parameters

Let the target be

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

(7.12)

where $\mathrm{T}$ is the target space. Applying the quotient projection to the transport-unified data gives the observation

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

(7.13)

For recovery, only quotient observations attained by admissible data are relevant. We therefore use the corestriction

$${\overline{Z}}^{\mathrm{act}}:{\mathrm{D}}_{\mathrm{task}}\twoheadrightarrow \overline{Z}\left({\mathrm{D}}_{\mathrm{task}}\right),{\overline{Z}}^{\mathrm{act}}\left(d\right):=\overline{Z}\left(d\right).$$

(7.14)

Definition 7.7

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

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

(7.15)

such that

$$\tau =R_{\tau }\circ {\overline{Z}}^{\mathrm{act}}.$$

(7.16)

Theorem 7.8

(Necessary and sufficient condition for quotient-observation recoverability). The target τ is recoverable from Equation (7.13) if and only if, for all $d_1,d_2\in {\mathrm{D}}_{\mathrm{task}}$,

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

(7.17)

In other words, τ must be constant on each fiber of $\overline{Z}$.

Proof. 

Suppose that Equation (7.16) holds. If two observations belong to the same task orbit, then ${\overline{Z}}^{\mathrm{act}}\left(d_1\right)={\overline{Z}}^{\mathrm{act}}\left(d_2\right)$, so applying the recovery map gives

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

Conversely, assume Equation (7.17). Given $q\in \overline{Z}\left({\mathrm{D}}_{\mathrm{task}}\right)$, choose any d satisfying ${\overline{Z}}^{\mathrm{act}}\left(d\right)=q$ and set $R_{\tau }\left(q\right):=\tau \left(d\right)$. Such a choice exists because the quotient observation is attained. Constancy on fibers makes the value independent of the chosen d, so this defines a recovery map satisfying Equation (7.16). □

If I is maximal, define the corestriction

$$Z_I^{\mathrm{act}}:{\mathrm{D}}_{\mathrm{task}}\twoheadrightarrow I\left(Z_{\mathrm{tr}}\left({\mathrm{D}}_{\mathrm{task}}\right)\right),Z_I^{\mathrm{act}}\left(d\right):=I\left(Z_{\mathrm{tr}}\left(d\right)\right).$$

Since a maximal invariant distinguishes exactly the quotient classes, the same criterion is equivalent to the existence of a unique recovery map ${\tilde{R}}_{\tau }$ on $I\left(Z_{\mathrm{tr}}\left({\mathrm{D}}_{\mathrm{task}}\right)\right)$ satisfying

$$\tau ={\tilde{R}}_{\tau }\circ Z_I^{\mathrm{act}}.$$

(7.18)

Passing from the quotient observation to a maximal invariant therefore leaves the class of recoverable targets unchanged. This is an exact, set-theoretic criterion. Continuity, numerical stability, and recovery from noisy data require additional assumptions on the observation map and the target.

7.5. Compatibility Between Phase Transport and Task Quotienting

The preceding construction forms the quotient after ${\mathsf{G}}_1$ has established the phase references. To compare quotient observations across such changes of reference, the corresponding data transformation must respect task orbits. Equivariance guarantees this property, although orbit preservation alone is sufficient.

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

$$\mathbf{P}:{\mathrm{Z}}_0⟶{\mathrm{Z}}_1$$

(7.19)

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

For $i\in \{0,1\}$, let

$${\mathsf{a}}_i:\mathrm{K}\times {\mathrm{Z}}_i⟶{\mathrm{Z}}_i,q_{\mathrm{K},i}:{\mathrm{Z}}_i\twoheadrightarrow {\mathrm{Z}}_i/\mathrm{K}$$

denote the specified action and its quotient projection. Write

$$z{\sim }_{\mathrm{K},i}{z}^{\prime }⟺z^{\prime }={\mathsf{a}}_i(k,z)\mathrm{for}\mathrm{some}k\in \mathrm{K},{\left[z\right]}_{\mathrm{K},i}:=q_{\mathrm{K},i}\left(z\right).$$

Definition 7.9

(Orbit-respecting data transformation). The transformation $\mathbf{P}$ is orbit respecting if

$$z_1{\sim }_{\mathrm{K},0}{z}_2⟹\mathbf{P}\left(z_1\right){\sim }_{\mathrm{K},1}\mathbf{P}\left(z_2\right).$$

(7.20)

This condition says exactly that ${\left[z\right]}_{\mathrm{K},0}\mapsto {\left[\mathbf{P}\left(z\right)\right]}_{\mathrm{K},1}$ is independent of the representative, and hence defines a map of quotient sets.

Definition 7.10

(Transport–task-group equivariance). The transformation $\mathbf{P}$ is equivariant if

$$\mathbf{P}\left({\mathsf{a}}_0(k,z)\right)={\mathsf{a}}_1\left(k,\mathbf{P}\left(z\right)\right),k\in \mathrm{K}.$$

(7.21)

An equivariant transformation is therefore orbit respecting.

Theorem 7.11

(Induced transport on the quotient). If $\mathbf{P}$ is orbit respecting, there exists a unique map

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

(7.22)

such that

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

(7.23)

and

$$\overline{\mathbf{P}}\left({\left[z\right]}_{\mathrm{K},0}\right)={\left[\mathbf{P}\left(z\right)\right]}_{\mathrm{K},1}.$$

(7.24)

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

Proof. 

Define the induced map by Equation (7.24). Orbit preservation ensures that changing the representative does not change its image class. This definition gives the commuting relation, and surjectivity of $q_{\mathrm{K},0}$ shows that the relation determines the induced map uniquely.

If $\mathbf{P}$ is an equivariant bijection, its inverse ${\mathbf{P}}^{-1}$ is also equivariant. Applying the construction to the inverse gives $\overline{{\mathbf{P}}^{-1}}$, which is inverse to $\overline{\mathbf{P}}$. Finally, composing two commuting quotient squares gives the square for the composed transport. Uniqueness of the induced map therefore carries the path-composition law to the quotient. □

This theorem explains how ${\mathsf{G}}_1$ transport acts on ${\mathsf{G}}_2$ observations. Orbit preservation is the essential requirement: if it fails, different representatives of one initial class can produce different classes after transport. Thus even a nonequivariant transformation descends to the quotient whenever it satisfies (7.20).

7.6. Scalar Collapse and a Complex-Projective Example

A single complex scalar already shows how strongly the information retained by a quotient depends on the task group.

Proposition 7.12

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

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

(7.25)

Consequently, every task invariant is constant: the quotient retains no nontrivial information.

Proof. 

Given any $z_1,z_2\in {\mathbb{C}}^{\times }$, take $\lambda =z_2{z}_1^{-1}$. This choice gives $z_2=\lambda z_1$, proving that every pair of points belongs to the same orbit and that the action is transitive. □

Nontrivial information can survive only if we enlarge this data space or quotient by a smaller group. With several observation components, a common complex scale leads to the familiar projective construction. Let

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

(7.26)

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

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

(7.27)

Then

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

(7.28)

This quotient discards common amplitude and common phase, leaving the relative complex direction.

On the chart

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

define

$$I_1:U_1⟶{\mathbb{C}}^{n-1},I_1\left(z\right):=\left({\displaystyle \frac{z_2}{z_1}},\dots ,{\displaystyle \frac{z_n}{z_1}}\right).$$

(7.29)

The ratios defining $I_1$ are unchanged by common complex scaling, and they form a maximal invariant on $U_1$. To see maximality, suppose that $I_1\left(z\right)=I_1\left(w\right)$. Taking $\lambda =w_1/z_1$ then gives $w=\lambda z$, so equal ratios place the two vectors on the same orbit. The standard projective transition functions relate these charts, which together represent ${\mathbb{CP}}^{n-1}$.

The common-scale action is what gives the projective quotient in (7.28). When only common phase is irrelevant, the appropriate group is $\mathrm{K}=\mathrm{U}\left(1\right)$ and the quotient still retains amplitude. Channel permutations and other nuisance transformations likewise require their own actions and yield their own quotients.

7.7. The ${\mathsf{G}}_2$ Construction and the Three-Level Hierarchy

Definition 7.13

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

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

(7.30)

A maximal invariant I provides an equivalent representation of this quotient. A recovery map $R_{\tau }$, when it exists, is additional data specifying how a particular target $\tau$ is obtained from the quotient observation.

The roles of the three levels can now be summarized as

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

(7.31)

The dependence of these constructions is

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

(7.32)

${\mathsf{G}}_0$ retains the selected complex response together with its joint state; ${\mathsf{G}}_1$ retains the normalized phase and supplies phase transport along admissible paths in ${\mathrm{M}}^{\times }$; and ${\mathsf{G}}_2$ identifies data under the prescribed task freedoms. These levels describe representation and reduction within the PODA physical–observation dual-axis structure; its physical-state, observation-state, and response roles describe a different aspect of the model.

Proposition 7.14

(Task dependence of ${\mathsf{G}}_2$). Even on a fixed transport-unified data space $\mathrm{Z}$, different task-irrelevant groups ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$ generally give different equivalence relations and quotient spaces. Thus ${\mathsf{G}}_0$ and ${\mathsf{G}}_1$ alone do not uniquely determine ${\mathsf{G}}_2$; the task-group action must also be specified.

Proof. 

Consider $\mathrm{Z}={\left({\mathbb{C}}^{\times }\right)}^n$. With ${\mathrm{K}}_1=\mathrm{U}\left(1\right)$ acting by common phase, each component amplitude remains in the quotient. With ${\mathrm{K}}_2={\mathbb{C}}^{\times }$ acting by common complex scale, common amplitude is also identified. The two actions therefore partition the same data space into different orbits and produce different quotients. □

7.8. Relative Completeness and Exact Information Loss

The preceding results describe information retention in terms of factorization: a representation retains what a target requires precisely when that target factors through it. We now use this principle to compare the three levels, each relative to its own class of problems.

Definition 7.15

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

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

such that

$$f=\overline{f}\circ R^{\mathrm{act}},$$

(7.33)

where $R^{\mathrm{act}}:\mathrm{D}\twoheadrightarrow R\left(\mathrm{D}\right)$ is the corestriction of R to its image. Completeness therefore means that R retains enough information to determine every target in $\mathbf{F}$.

To identify the information retained by the full phase representation, fix the nonzero joint-state domain ${\mathrm{M}}^{\times }$ and write

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

(7.34)

where ${\mathrm{R}}_{+}\left({\mathrm{M}}^{\times }\right)$ acts on $\mathrm{G}\left({\mathrm{M}}^{\times }\right)$ by pointwise multiplication. Normalizing each response defines

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

(7.35)

Proposition 7.16

(Exact quotient structure from nonzero response functions to the full ${\mathsf{G}}_1$). The map $R_1$ is surjective, and for $g_1,g_2\in \mathrm{G}\left({\mathrm{M}}^{\times }\right)$,

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

(7.36)

Consequently, $R_1$ induces the canonical bijection

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

(7.37)

With the nonzero domain and joint-state identity fixed, the full ${\mathsf{G}}_1$ therefore identifies exactly those responses that differ by a positive amplitude function, while retaining all circular phase data.

Proof. 

Given any $u\in C^{\infty }({\mathrm{M}}^{\times },\mathrm{U}\left(1\right))$, take $g=u$. Then $R_1\left(g\right)=u$, proving that $R_1$ is surjective. If $g_2=\rho g_1$ with $\rho >0$, normalization cancels the positive factor and leaves the same unit phase. Conversely, suppose that $u_{g_1}=u_{g_2}$ and set $\rho :=|g_2|/|g_1|$. The function $\rho$ is positive and smooth, and equality of the unit phases gives $g_2=\rho g_1$. Since $g_1$ is nowhere zero, no other multiplier can satisfy this identity. Thus the fibers of $R_1$ are exactly the ${\mathrm{R}}_{+}\left({\mathrm{M}}^{\times }\right)$ orbits, which proves the claimed quotient bijection. □

Theorem 7.17

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

(1)

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

$$Gr\left(g\right):=\{(m,g\left(m\right)):m\in \mathrm{M}\}\subseteq L,R_0:\mathrm{M}\stackrel{\sim }{\to }Gr\left(g\right),R_0\left(m\right):=(m,g\left(m\right)).$$

(7.38)

The map $R_0$ is a bijection. It follows that every target $f:\mathrm{M}\to \mathrm{W}$ admits a unique map $\overline{f}:Gr\left(g\right)\to \mathrm{W}$ satisfying $f=\overline{f}\circ R_0$. This factorization uses the graph point, which contains the joint state m as well as its response $g\left(m\right)$.

(2)

Phase-problem completeness of ${\mathsf{G}}_1$.The full ${\mathsf{G}}_1$ includes the normalized phase $u_g$. Under the standard radian normalization and the response-compatibility condition, $u_g$ determines the response-induced phase one-form $A_g$, the response-compatible connection ${\Omega }_g$, PTFE, the covariant derivative, and all finite transports. Consequently, every model-level phase problem invariant under positive amplitude rescaling,

$$\mathrm{F}:\mathrm{G}\left({\mathrm{M}}^{\times }\right)\to \mathrm{W},\mathrm{F}\left(\rho g\right)=\mathrm{F}\left(g\right),$$

factors uniquely through $R_1$. If we retain only the differential–transport substructure $A_g,{\Omega }_g,{\mathrm{T}}_g$ and omit $u_g$, recovering $u_g$ also requires one constant phase anchor on each path component.

(3)

Task-invariant completeness of ${\mathsf{G}}_2$.The quotient projection $q_{\mathrm{K}}:\mathrm{Z}\to \mathrm{Z}/\mathrm{K}$ is complete for all $\mathrm{K}$-invariant problems. Every task invariant factors uniquely through this projection, and every maximal invariant gives a bijective representation of the quotient. A target τ is recoverable precisely when it is constant on each fiber of $q_{\mathrm{K}}\circ Z_{\mathrm{tr}}$.

(4)

Relative completeness of the full hierarchy.${\mathsf{G}}_0$, the full ${\mathsf{G}}_1$, and ${\mathsf{G}}_2$ are thus complete for their respective problem classes: full-response-graph problems, phase problems invariant under positive amplitude scaling, and quotient invariants and recoverable targets for the prescribed task group.

Proof. 

For part (1), the inverse is $R_0^{-1}={\pi }_L{|}_{Gr\left(g\right)}$. Hence $R_0$ is a bijection, and composing any target with this inverse gives its unique factorization through the graph.

For part (2), Definition 7.16 identifies the fibers of $R_1$ with the positive-amplitude orbits. Every problem constant on these orbits therefore factors uniquely through $R_1$. The phase one-form and connection are given by

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

so the remaining ${\mathsf{G}}_1$ objects are determined by $u_g$. To see what can be recovered from the transports alone, suppose all finite transports are given. For any $m\in {\mathrm{M}}^{\times }$ and $v\in T_m{\mathrm{M}}^{\times }$, choose a smooth path

$$c_v:(-\epsilon ,\epsilon )⟶{\mathrm{M}}^{\times },c_v\left(0\right)=m,{\dot{c}}_v\left(0\right)=v,$$

and let $c_v^{0\to t}$ denote its oriented segment from 0 to t. Differentiating the transport at the initial point recovers the phase one-form:

$${\left(A_g\right)}_m\left(v\right)={\left({\omega }_{\mathrm{U}\left(1\right)}\right)}_1\left({\left.{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\right|}_{t=0}{\mathrm{T}}_g\left[c_v^{0\to t}\right]\right).$$

(7.39)

Once a phase anchor $u\left(m_{\alpha }\right)$ is supplied on a path component, the transport functional gives the phase at every point of that component by

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

(7.40)

where c connects $m_{\alpha }$ to m. Since circular transport around a closed loop is the identity, the recovered phase is independent of the chosen path.

For part (3), Definitions 7.4, 7.6 and 7.8 give, respectively, factorization of invariants, equivalence of maximal invariants and quotient observations, and the criterion for target recovery. Part (4) collects these conclusions for the three problem classes. □

Proposition 7.18

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

(i)

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

(ii)

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

(iii)

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

$$g_2={\rho }_{\alpha }{\kappa }_{\alpha }{g}_1\mathrm{on}{\left({\mathrm{M}}^{\times }\right)}_{\alpha }.$$

(7.41)

If the responses induce the same full ${\mathsf{G}}_1$, they also have the same $u_g$, forcing every ${\kappa }_{\alpha }=1$. Thus the full ${\mathsf{G}}_1$ discards only positive amplitude. Retaining only $A_g$ and transport additionally discards one constant phase anchor on each path component.

Proof. 

The identity ${\mathrm{T}}_g\left[c\right]=exp\left(\mathrm{i}{\int }_c{A}_g\right)$ gives (ii) from (i), while Equation (7.39) gives (i) from (ii). To prove (iii), assume (i) and define $h:=u_{g_2}{u}_{g_1}^{-1}:{\mathrm{M}}^{\times }\to \mathrm{U}\left(1\right)$. The Abelian group law on $\mathrm{U}\left(1\right)$ gives

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

At every point, ${\omega }_{\mathrm{U}\left(1\right)}$ is a linear isomorphism from $T_u\mathrm{U}\left(1\right)$ to $\mathbb{R}$, so the vanishing pullback implies $\mathrm{d}h=0$. Hence h is constant on each path component; denote its value there by ${\kappa }_{\alpha }$. Taking ${\rho }_{\alpha }=|g_2|/|g_1|$ gives the required positive smooth multiplier, uniquely determined by the response magnitudes, and proves (iii). Conversely, neither a constant phase factor nor a positive amplitude factor changes $A_g$, so (iii) implies (i). Finally, agreement of the full ${\mathsf{G}}_1$ representations includes $u_{g_1}=u_{g_2}$ and therefore forces ${\kappa }_{\alpha }=1$. □

Proposition 7.19

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

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

(7.42)

Quotient reduction therefore identifies exactly the points of each task-group orbit. It acts on the data $Z_{\mathrm{tr}}$ assembled from ${\mathsf{G}}_0$ and ${\mathsf{G}}_1$, which may include more than the phase representation alone.

Proof. 

By definition, the quotient projection assigns the same value to two points precisely when they lie on the same orbit. If we represent ${\mathsf{G}}_2$ by a maximal invariant, Definition 7.6 shows that its values also distinguish every pair of distinct orbits. □

Remark (Recovering the response from a scalar readout) 

When the readout $\mathrm{r}:{\mathrm{Y}}_{\mathrm{coh}}\to \mathbb{C}$ is not injective, the scalar value $g\left(m\right)$ determines $F_{\mathrm{M}}^{\mathrm{coh}}\left(m\right)$ only if $F_{\mathrm{M}}^{\mathrm{coh}}$ is constant on every fiber of g. Equivalently, there must be a map $\overline{F}:g\left(\mathrm{M}\right)\to {\mathrm{Y}}_{\mathrm{coh}}$ satisfying $F_{\mathrm{M}}^{\mathrm{coh}}\left(m\right)=\overline{F}\left(g\left(m\right)\right)$ for every $m\in \mathrm{M}$. The graph map $R_0$ is nevertheless bijective: it retains the base point m together with $g\left(m\right)$, and ${\mathsf{P}}_0$ records the underlying maps $F_{\mathrm{M}}^{\mathrm{coh}}$ and $\mathrm{r}$. The completeness of the graph concerns this retained state–response pair and requires no inverse for the scalar readout.

7.9. Task Reduction in the PODA Model

Once the task action is specified, phase-comparable data determine a quotient observation. A maximal invariant represents the same observation, and the fiber criterion decides whether it determines the target. When transport respects the task orbits, this reduction also carries the transport to the quotient.

The orbit structure determines the extent of this reduction. A transitive action collapses all data to a single quotient point; more generally, variation of the target within an orbit obstructs recovery. The nonzero-path condition and the regularity hypotheses above specify when phase comparison and smooth quotient geometry are available.

This construction links the PODA response model and PTFE to task recovery. In a coherent sensing application, the choice of data map $Z_{\mathrm{tr}}$, the action of $\mathrm{K}$, and the target determines both the quotient to be formed and the recovery question to be answered.

8. Conclusion

PODA expresses the joint dependence of realized physical reality on physical state and observation state. Holding the physical state fixed leaves an observation axis along which the realized fact can still change; fixing the observation state recovers the familiar single-axis description. A coherent readout of the joint law gives a complex response, from which the geometry of phase follows. On a smooth domain where the response is nonzero,

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

Differentiating $g\circ c=|g\circ c|u\circ c$ gives $\dot{v}=\mathrm{i}{A}_g\left(\dot{c}\right)v$ for $v=u\circ c$. A transported phase with the same increments differs from v by a constant factor, so it satisfies $\dot{\zeta }=\mathrm{i}{A}_g\left(\dot{c}\right)\zeta$. This derived relation is horizontal lifting for the unique response-compatible connection

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

We have named the resulting equation PTFE, the first fundamental equation of observation space in PODA Theory:

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

The group-valued, covariant-derivative and continuous real-phase expressions describe the same first-order law. Along piecewise-$C^1$ paths, integration gives

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

An initial phase fixes the horizontal lift. Reversing the path reverses the comparison, and concatenating paths composes it. With the physical state held fixed, the finite solution brings the current phase to that of a reference observation state. A phase-faithful reference, or a calibration of its response, then relates this common reference phase to the intrinsic physical phase.

The three representations distinguish what is retained at each step. The full complex response ${\mathsf{G}}_0$ carries amplitude, phase and zeros. The complete phase representation ${\mathsf{G}}_1$ identifies responses that differ by a positive amplitude factor; if only its differential and transport are retained, one phase anchor is also needed on each path component. The task quotient ${\mathsf{G}}_2$ identifies precisely the data related by the chosen group action. Its maximal invariants distinguish the resulting orbits. A target survives the reduction exactly when it is constant on every fiber of the quotient observation, and transport passes to the quotient exactly when it preserves the orbits. Equivariance is one sufficient way to ensure this compatibility.

The geometry reflects the response from which it was constructed. A globally defined nonzero scalar response gives a flat connection with trivial circular holonomy, while the continuously lifted phase retains integer winding. At a zero, normalization ceases to define a phase; continuous same-source transport therefore follows a path within a nonzero observation-state slice. The physical system and the observation protocol determine the path, and PTFE determines the coherent phase along it. A concrete application can thus begin with its observation law, bring its phase readings to a common reference, and choose a task reduction whose fibers preserve the quantity to be recovered.