Durée–Extension Projection Theory (N–Q–S): A Diagrammatic Unification with a Fixed-Point Axis

1. Background: Generalized Measurement, Channels, and Quantum Cognition

Relation to Part I. 

Part I introduced the N–Q–S three-layer architecture and the selection principle ${\eta }^{*}$ for minimal embodiments. Here we only use the following: N denotes the physical substrate/dynamics, Q the operational quantum description (states, POVMs, channels), and S the appearance/report layer; the present paper is otherwise self-contained.

Generalized measurement (POVM). 

Beyond projectors, effects $E_k$ in a POVM and instrument updates capture disturbance and order effects operationally [4,5,6,7]. Ozawa’s universally valid reformulation links error and disturbance without relying on heuristic inequalities [1].

Quantum channels and dynamics. 

Context-dependent state change is modeled by completely positive trace-preserving (CPTP) maps, with continuous-time limits governed by GKLS (Gorini–Kossakowski–Lindblad–Sudarshan) [3,8]. We use primitivity/Doeblin-type conditions to guarantee unique fixed points and mixing.

Quantum cognition. 

A growing literature explains question-order effects, interference, and contextuality using quantum probability [2,29]. Our contribution is to treat appearance as projection constraints with a fixed-point axis, and to propose falsifiable indices (ONS/PII/PAD) that bridge these traditions.

Intended audience and contribution. 

This manuscript targets readers in the mathematical foundations of quantum theory (generalized measurement/POVM, quantum channels) and quantum cognition. Contributions: (1) a measurement-first account of appearance via POVMs; (2) a fixed-point axis linking first-person (S) and third-person (N) reports; (3) testable indices (ONS/PII/PAD) with preregisterable protocols. Neuroscience is discussed only as an outlook; no domain-specific background is assumed.

Roadmap. 

Section 6 introduces the fixed-point axis and defines the context gap; Section 7 develops the minimal realization and device modeling. Appendices A–D collect the full proof of minimality, explicit mixing-rate bounds for physical channels, and the order-effect bound with attainability.

Preliminaries: standing assumptions and notation1 

We adopt the definitions of the invariant quotient $O/G$, the naming channel $w:O\to L$, and the reflection map $r_S:L\to P_S$ introduced in the companion paper; here we use only the fixed-point convergence result and notational conventions from there.

2. Notation and Standing Assumptions

POVM and instruments. 

On a finite-dimensional Hilbert space $\mathcal{H}$, a POVM $E={\left\{E_x\right\}}_x$ satisfies $E_x\ge 0$ and ${\sum }_x{E}_x=\mathbb{I}$. An instrument is a family of completely positive (CP) maps ${\left\{{\mathcal{I}}_x\right\}}_x$ with ${\sum }_x{\mathcal{I}}_x$ completely positive trace-preserving (CPTP). For a state $\rho$, outcome probabilities are $p\left(x\right)=\mathrm{Tr}\left[{\mathcal{I}}_x\left(\rho \right)\right]$.

Doeblin minorization (quantum). 

A CPTP map $\mathsf{\Phi }$ has Doeblin coefficient$\delta \in (0,1]$ if there exists a full-rank state $\sigma$ such that

$$\mathsf{\Phi }=\delta {\mathsf{\Omega }}_{\sigma }+(1-\delta )\mathsf{\Xi },{\mathsf{\Omega }}_{\sigma }\left(\rho \right)=\sigma \mathrm{Tr}\left[\rho \right],\mathsf{\Xi }\mathrm{CPTP}.$$

(1)

Then for all states $\rho ,{\rho }^{\prime }$,

$$\parallel \mathsf{\Phi }\left(\rho \right)-\mathsf{\Phi }\left({\rho }^{\prime }\right){\parallel }_1\le (1-\delta ){\parallel \rho -{\rho }^{\prime }\parallel }_1.$$

(2)

Discrete iterates and continuous-time limit. 

For a one-cycle channel $\mathsf{\Lambda }$ ("look–return"), define ${\rho }_{n+1}=\mathsf{\Lambda }\left({\rho }_n\right)$. If $\mathsf{\Lambda }$ satisfies (1), then ${\rho }_n\to {\rho }_{★}$ at geometric rate. When a step map ${\mathsf{\Lambda }}_{\mathsf{\Delta }t}$ admits ${\mathsf{\Lambda }}_{\mathsf{\Delta }t}=\mathrm{id}+\mathsf{\Delta }t\mathcal{L}+o\left(\mathsf{\Delta }t\right)$ with a GKLS generator $\mathcal{L}$, the lower bound $\gamma \ge -ln(1-\delta )/\mathsf{\Delta }t$ transfers to the semigroup $e^{t\mathcal{L}}$ under standard stability assumptions.

Testable indices. 

We use three scalars computed from observed sequences: ONS (one-shot novelty), PII (projection-induced interference), and PAD (projection-aligned disturbance). Their precise definitions appear in Section 6.2; they are representation-invariant and vanish in the commutative case.

3. Related Work

Surveys and tutorials on quantum cognition provide broader context and methodology; see, e.g., [29,30,31]. We build on standard quantum measurement and channel theory (POVMs, Naimark/Stinespring dilations, Lindblad generators) while focusing on an operational bridge to order effects and context dependence in cognition.

Positioning and novelty. 

Unlike quantum-cognition models that postulate Hilbert-space interference to fit order effects, we derive tight, saturable bounds from operator-theoretic structure (Halmos blocks) and connect them to testable ONS/PII/PAD indices. Relative to Ozawa-type error–disturbance relations, our Doeblin minorization provides a complementary knob that quantifies mixing and fixed-point approach with closed-form 1-norm rates. Recent developments on quantum Doeblin/Markov–Dobrushin coefficients supply comparison baselines.

Novelty and contrasts. 

Unlike quantum-cognition models that postulate Hilbert-space interference to fit order effects (e.g., [29]), we derive tight, saturable order-effect bounds directly from operator structure (Halmos two-subspace blocks) and certify equality cases. Relative to Ozawa-type error–disturbance relations (e.g., [1]), our Doeblin minorization provides a complementary, operationally estimable knob that quantifies mixing and fixed-point approach with closed-form 1-norm rates. This yields testable indices (ONS/PII/PAD) and a monitored discrete-to-GKLS bridge, linking rigorous channel bounds to experimentally accessible summaries.

4. Bridge Diagram

Figure 1. Bridging measurement theory, channels, and quantum cognition to N–Q–S.

Figure 1. Bridging measurement theory, channels, and quantum cognition to N–Q–S.

5. Micro–Macro Duality and Ojima’s Fourfold Scheme: Positioning N–Q–S

Idea. Micro–macro duality programs reconstruct how microscopic operations are amplified into macroscopic reports via an adjoint pair (instrument → representation) and their composites. Read in this light, N–Q–S can be placed as follows:

• Q (micro): a noncommutative space of effects/operations; one `look’ is an instrument.
• S (macro/report): the update space in which outcomes are registered and compared.
• N (apparatus semantics): constraints that select admissible instruments, their coarse-grainings, and amplification mechanisms.

Composing enact (Q→S) and prepare (S→Q) yields a monad T whose iterates $T^t$ model repeated practice. The fixed-point axis is then the locus of stabilized algebras/states $\left\{{\rho }_{*}\left(\lambda \right)\right\}$ traced out as apparatus parameters $\lambda \in N$ vary. This reframes `closure’ as practice-bound and dynamic: altering N or the admissible Q moves or fractures the axis. 

Payoff. The diagrammatic reading clarifies that N–Q–S does not reify a static essence: stability is the attractor of an instrument–report loop, not a timeless object. It also explains why non-primitive regimes (multiple attractors/cycles) are natural rather than pathological, predicting switches or oscillations when amplification or coarse-graining changes.

6. Fixed-Point Axis (Self-Reflective Appearance)

Operational clarification (intrinsic/extrinsic language). 

We deliberately avoid metaphysical vocabulary and keep to an operational reading. (i) The intrinsic update flow ${\mathsf{\Theta }}_{\mathrm{self}}$ tracks how repeated practice drives S along a time-like stabilization direction—the fixed-point axis. (ii) The extrinsic visible field $E_{\mathrm{other}}$ records the apparatus-induced coordinate structure on S: choosing an instrument/POVM via N induces a decomposition (“reporting coordinates”) on S (e.g., a Bloch-sphere parametrization when a qubit-like reduction is appropriate). We do not claim that space-time is generated; rather, we assert the operational statement: apparatus parameters N endow S with a coordinate-like structure under which the iterated channel $T_{N,Q}$ converges (under primitivity) to a stabilized report state ${\rho }_{*}$; as $\lambda$ varies in N, the locus $\left\{{\rho }_{*}\left(\lambda \right)\right\}$ traces the fixed-point axis.

Fixed-point axis falsifiability. With apparatus parameter $\lambda \in N$ varied along a preregistered path, estimate stabilized states ${\rho }_{*}\left(\lambda \right)$ (via tomography or sufficient statistics). Predictions: continuity along $\lambda$ within a regime; movement/bifurcation at regime changes; piecewise-geometric return under repeated practice. Refutation: absence of stabilization (no fixed point) under conditions predicted to be primitive; or failure of continuity where the model predicts smooth dependence.

6.1. Convergence and Fixed Points: Concise Lemmas 

6.2. Context Gap and Indices (ONS/PII/PAD) 

Context gap. For a pair of incompatible “looks” $A,B$ with (notional) projectors $(P,Q)$ and their Heisenberg evolutes $(P_t,Q_t)$ under a mixing channel ${\mathsf{\Phi }}_t$, we call

$${\mathsf{\Delta }}_{\mathrm{ctx}}\left(t\right):=\parallel [P_t,Q_t]\parallel$$

the context gap. In minimal realizations (Appendix C), ${\mathsf{\Delta }}_{\mathrm{ctx}}$ controls the order-effect magnitude and decays under primitivity (${\mathsf{\Phi }}_t\Rightarrow \mathrm{Fix}$), yielding small- t expansions that match empirical order effects.

Indices. We summarize three scalar indices used operationally throughout:

• ONS (one-shot novelty): the first-encounter gain measured at the fixed-point axis; $\mathrm{ONS}:=\mathsf{\Theta }\_\mathrm{self}/\rho$ (dimensionless; 1-homogeneous).
• PII (projection-induced interference): the signed interference term obtained from a two-look protocol; in the bridge picture it is controlled by $[P,Q]$ and vanishes when projections commute.
• PAD (projection-aligned disturbance): the component of disturbance aligned with the fixed-point axis; operationally extracted from a Lindblad-fit of the early-time channel $\mathsf{\Phi }\_t$.

These indices are invariant under representation changes discussed in Sec. 4.2.

Iterated looks are modeled by a completely positive trace-preserving (CPTP) channel $\mathsf{\Lambda }$ on $M_d\left(\mathbb{C}\right)$, or by an averaged channel $\overline{\mathsf{\Lambda }}={\sum }_i{p}_i{\mathsf{\Lambda }}_i$ when the environment randomizes looks. We also consider the one-cycle $\mathsf{\Phi }:=F\circ T\circ {\eta }^{*}$ on $\mathcal{S}$.

Lemma 1 

(Primitive ⇒ unique full-rank fixed point and exponential mixing). Let Λ be a CPTP map on density matrices over $\mathcal{H}$. Suppose Λ isprimitive: there exists $k\in \mathbb{N}$ such that ${\mathsf{\Lambda }}^k\left(X\right)$ is strictly positive for every nonzero positive operator X. Then there exists a unique full-rank fixed point ${\rho }_{★}>0$ and constants $C<\infty$, $r\in (0,1)$ such that for all states ρ,

$$\parallel {\mathsf{\Lambda }}^n\left(\rho \right)-{\rho }_{★}{\parallel }_1\le C{r}^n.$$

(3)

Proof. 

(1) Noncommutative Perron–Frobenius.). Primitivity implies irreducibility and aperiodicity. By the quantum Perron–Frobenius theory for CP maps, the spectral radius 1 of $\mathsf{\Lambda }$ is a simple eigenvalue with a strictly positive eigenvector ${\rho }_{★}$ (normalized to $\mathrm{Tr}{\rho }_{★}=1$). No other peripheral eigenvalues exist.

(2) Spectral gap and exponential convergence.) Decompose the space into the fixed-point axis and its trace-zero complement. On the complement, the spectral radius is $<1$. Thus there is $r\in (0,1)$ and $C<\infty$ such that (3) holds (take C from a Jordan bound or use the induced $1\to 1$ norm).

(3) 1-norm contraction under Doeblin.) If, in addition, $\mathsf{\Lambda }$ admits (1), then (2) yields the explicit bound $\parallel {\mathsf{\Lambda }}^n\left(\rho \right)-{\mathsf{\Lambda }}^n\left({\rho }^{\prime }\right){\parallel }_1\le {(1-\delta )}^n{\parallel \rho -{\rho }^{\prime }\parallel }_1$. Taking ${\rho }^{\prime }={\rho }_{★}$ gives $C=1$ and $r=1-\delta$.    □

Proposition 1 

(Doeblin minorization ⇒ primitivity and geometric mixing). Let $\overline{\mathsf{\Lambda }}={\sum }_i{p}_i{\mathsf{\Lambda }}_i$ be a convex combination of CPTP maps. Assume there exist $\epsilon \in (0,1)$ and a full-rank state σ such that

$$\overline{\mathsf{\Lambda }}\left(X\right)\ge \epsilon \mathrm{Tr}\left[X\right]\sigma forallX\ge 0,X\ne 0.$$

Then $\overline{\mathsf{\Lambda }}$ is primitive and for the unique fixed point ${\rho }_{★}$,

$$\parallel {\overline{\mathsf{\Lambda }}}^n\left(\rho \right)-{\rho }_{★}{\parallel }_1\le {(1-\epsilon )}^n{\parallel \rho -{\rho }_{★}\parallel }_1.$$

Proof. 

(1) Primitivity.). The minorization implies strict positivity after one step on any nonzero positive operator; hence $\overline{\mathsf{\Lambda }}$ is primitive.

(2) Contraction.) For states $\rho ,{\rho }^{\prime }$, write $\overline{\mathsf{\Lambda }}=\epsilon {\mathsf{\Omega }}_{\sigma }+(1-\epsilon )\mathsf{\Xi }$ with CPTP $\mathsf{\Xi }$. Then $\parallel \overline{\mathsf{\Lambda }}\left(\rho \right)-\overline{\mathsf{\Lambda }}\left({\rho }^{\prime }\right){\parallel }_1\le (1-\epsilon ){\parallel \rho -{\rho }^{\prime }\parallel }_1$, and iterate.    □

Remark 1 

(Continuous time & hypocoercivity). For a Lindblad generator $\mathcal{L}$, primitivity of $e^{t\mathcal{L}}$ yields $\parallel e^{t\mathcal{L}}\left(\rho \right)-{\rho }_{*}{\parallel }_1\le C{e}^{-\gamma t}$; recenthypocoercivityresults furnish mixing-time bounds withouta priorispectral-gap estimates.

Figure 2. Fixed-point dynamics of iterated look (Section 5).

Figure 2. Fixed-point dynamics of iterated look (Section 5).

7. Framework and Minimal Realization

Physical picture and a toy example. 

Equation (1) says that a “look” does not create or destroy the total budget of internal temporal tension (durée) plus externalized report (extension); it only redistributes it along the fixed-point axis. Consider a 1D phase-ramp state $\varphi \left(x\right)=kx$ with density $\rho$. Before the look, ${\mathsf{\Theta }}_{\mathrm{self}}=\rho \left|k\right|$ and $E_{\mathrm{other}}=\rho |cos\phi |$ (e.g., $\phi ={\displaystyle \frac{\pi }{2}}$ gives $E_{\mathrm{other}}=0$). A weak look can be modeled as a small tilting channel ${\mathsf{\Phi }}_{\epsilon }$ that rotates $\phi \mapsto \phi -\epsilon$. To first order in $\epsilon$, $E_{\mathrm{other}}$ increases by $\rho |sin\phi |\epsilon$ while ${\mathsf{\Theta }}_{\mathrm{self}}$ decreases linearly in $\epsilon$ so that the sum obeys Eq. (1). This small-time linearization matches the empirical “first look” gain and the decay under mixing established by our convergence lemmas.

Minimality rationale. 

Among positive scalars built from $\psi =R{e}^{i\phi }$, we justify ${\mathsf{\Theta }}_{\mathrm{self}}=\rho \parallel \nabla \phi \parallel$ and $E_{\mathrm{other}}=\rho |cos\phi |$ as minimal densities consistent with: (i) gauge invariance in phase differences; (ii) additivity on disjoint regions; and (iii) dimensional consistency for transport fluxes. In closed regions we obtain

$${\partial }_t({\mathsf{\Theta }}_{\mathrm{self}}-E_{\mathrm{other}})+\nabla ·(F_{\mathrm{self}}-J_{\mathrm{other}})=0,$$

(4)

while at look boundaries, $\delta$-type jumps reallocate durée to extension.

7.1. Look-as-Projection and Devices

PVM (hard look):$p=\left(\mathsf{\Pi }\rho \right)$ with üders update $\rho \mapsto \mathsf{\Pi }\rho \mathsf{\Pi }/\left(\mathsf{\Pi }\rho \right)$. POVM (soft look): effects $\left\{E_k\right\}$ with instrument updates $\rho \mapsto {\sum }_{k,\alpha }{M}_{k,\alpha }\rho M_{k,\alpha }^{†}$. POVMs admit Naimark/Stinespring dilations to a PVM in a larger space; compression is 1-Lipschitz.

7.2. Observer-Neutral Note

No generalized external observer is postulated. Operationally, the role is played by boundary measurements (PVM/POVM) and—at the ensemble level—by an averaged CPTP channel $\overline{\mathsf{\Lambda }}$. Objectivity is expressed by invariance of fixed points of $\overline{\mathsf{\Lambda }}$ and, when applicable, hypocoercive mixing to them; noncommutativity excludes a single globally consistent context, whereas representation independence guarantees that empirical statements do not depend on a chosen representation.

8. Methods: Behavioral Protocol (Pre-Registered)

Design. 

Within-subject repeated-measures with two yes/no items $(A,B)$. Sequences $(A\to B)$ and $(B\to A)$ are interleaved across repetitions $t=1,\dots ,T$ with random scheduling.

Outcome. 

For each t, we estimate the order-effect ${\mathsf{\Delta }}_t$ (difference between cross-conditional response probabilities or logits, depending on the chosen scale). The theory predicts (i) $|{\mathsf{\Delta }}_t|\le \parallel [P,Q]\parallel$ and (ii) convergence of ${\mathsf{\Delta }}_t$ to a fixed point (typically 0) under non-selective üders-type updates.

Analysis. 

We fit exponential-decay curves to ${\mathsf{\Delta }}_t$ and test bound violations. Individual differences enter via $(\theta ,p)$-like parameters. Deviations from the predicted convergence or bound constitute direct falsification.

Preregistration and Data. 

We will preregister hypotheses, design, exclusion criteria, and analysis plans; materials and code will be released alongside the camera-ready version (see “Data & code.”).

Figure 3. Suggested parameter regimes for double-slit/MZI chains: visibility $V\left(t\right)\sim e^{-\mathsf{\Gamma }t}$; Doeblin mixing adds a lower bound $\kappa \approx p/\mathsf{\Delta }t$.

Figure 3. Suggested parameter regimes for double-slit/MZI chains: visibility $V\left(t\right)\sim e^{-\mathsf{\Gamma }t}$; Doeblin mixing adds a lower bound $\kappa \approx p/\mathsf{\Delta }t$.

Proposition 2 

(EDR-anchored control of bipartite order effects). Let $(E^A,F^B)$ be local effects (PVMs or POVMs via Naimark dilation) measured on subsystems A and B by instruments ${\mathcal{M}}_A,{\mathcal{M}}_B$. Denote by ${\mathsf{\Delta }}_{AB}\left(\rho \right)$ the sequential order effect between $(A\to B)$ and $(B\to A)$ for a bipartite state ρ. Then

$${\mathsf{\Delta }}_{AB}\left(\rho \right)={\mathsf{\Delta }}_A^{\mathrm{loc}}\left(\rho \right)+{\mathsf{\Delta }}_B^{\mathrm{loc}}\left(\rho \right)+{\mathsf{\Delta }}^{\mathrm{nonloc}}\left(\rho \right),$$

where the local parts obey the same Ozawa-type error–disturbance upper bounds as in the single-system case (via Naimark dilation for POVMs), while the genuinely nonlocal remainder is correlation-suppressed:

$$|{\mathsf{\Delta }}^{\mathrm{nonloc}}\left(\rho \right)|\le \frac{1}{2}∥[E^A\otimes I,I\otimes F^B]∥{∥\rho -{\rho }_A\otimes {\rho }_B∥}_1.$$

Proof sketch. 

Apply the single-system EDR-to-order-effect bridge (Part I, Sec. 3) to the local margins, using Naimark dilation for POVMs. For the remainder, expand the commutator on $A\otimes B$ and bound the correlation part by $\parallel \rho -{\rho }_A\otimes {\rho }_B{\parallel }_1$.    □

Appendix A.1. A Semigroup Stability Tool (Trotter–Kato)

Theorem A2. 

Let ${\left\{{\mathsf{\Lambda }}_{\mathsf{\Delta }t}\right\}}_{\mathsf{\Delta }t>0}$ be step maps on the trace class ${\mathcal{T}}_1\left(\mathcal{H}\right)$ such that ${\mathsf{\Lambda }}_{\mathsf{\Delta }t}=\mathrm{id}+\mathsf{\Delta }t\mathcal{L}+o\left(\mathsf{\Delta }t\right)$ with a GKLS generator $\mathcal{L}$. Assume resolvent convergence for some $\lambda >0$, $R(\lambda ,{\mathsf{\Lambda }}_{\mathsf{\Delta }t}-\mathrm{id})\to R(\lambda ,\mathcal{L})$ in the strong operator topology, and uniform contractivity on ${\mathcal{T}}_1$. Then ${\mathsf{\Lambda }}_{\mathsf{\Delta }t}^{\lfloor t/\mathsf{\Delta }t\rfloor }\stackrel{s}{\to }{e}^{t\mathcal{L}}$ for each $t\ge 0$. Moreover, if $\parallel {\mathsf{\Lambda }}_{\mathsf{\Delta }t}^n\left(\rho \right)-{\rho }_{★}{\parallel }_1\le e^{-{\gamma }_{\mathsf{\Delta }t}n\mathsf{\Delta }t}{\parallel \rho -{\rho }_{★}\parallel }_1$ with ${\gamma }_{\mathsf{\Delta }t}\to \gamma \ge 0$, then $\parallel e^{t\mathcal{L}}\left(\rho \right)-{\rho }_{★}{\parallel }_1\le e^{-\gamma t}{\parallel \rho -{\rho }_{★}\parallel }_1$.

Proof sketch. 

Apply the Trotter–Kato theorem for contraction semigroups on Banach spaces (e.g., [32,33]) to ${\mathcal{T}}_1\left(\mathcal{H}\right)$. Uniform dissipativity yields strong convergence of ${\mathsf{\Lambda }}_{\mathsf{\Delta }t}^{\lfloor t/\mathsf{\Delta }t\rfloor }$ to $e^{t\mathcal{L}}$. The exponential rate transfer follows by an $\epsilon$–$\delta$ argument using semigroup convergence and uniform envelopes.    □

Appendix B.1. Admissible Class

Let $\psi =R{e}^{i\phi }$ with $R\ge 0$, $\phi \in \mathbb{R}$ on a domain $\mathsf{\Omega }\subset {\mathbb{R}}^n$ with $R,\phi$ locally Lipschitz and $\rho =R^2\in L_{\mathrm{loc}}^1$. A pair $(\mathsf{\Theta },E)$ of nonnegative densities is admissible if:

(A1)

Locality: $\mathsf{\Theta }\left(x\right),E\left(x\right)$ depend on $\psi \left(x\right)$ and its first derivatives at x only (a.e.).

(A2)

Gauge invariance in phase differences: $\phi \mapsto \phi +\mathrm{const}$ leaves $\mathsf{\Theta },E$ unchanged.

(A3)

Additivity: For disjoint measurable sets $A,B$, ${\int }_{A\cup B}\mathsf{\Theta }={\int }_A\mathsf{\Theta }+{\int }_B\mathsf{\Theta }$ and likewise for E.

(A4)

Positive 1-homogeneity: $\mathsf{\Theta }(x;\lambda \nabla \phi )=\lambda \mathsf{\Theta }(x;\nabla \phi )$ for $\lambda \ge 0$; $E(x;\lambda u)=\lambda E(x;u)$ for the externalization scalar u.

(A5)

Rotational invariance (isotropy): $\mathsf{\Theta }$ depends on $\nabla \phi$ only through ${\parallel \nabla \phi \parallel }_2$; E depends on $\psi$ through $|cos\phi |$ (i.e., $|\Re (\psi \left)\right|/\parallel \psi \parallel$).

Remark A1. 

If isotropy is dropped, minimality holds with respect to the smallest admissible symmetric gauge replacing ${\parallel ·\parallel }_2$.

Appendix B.2. Statement and Uniqueness up to Trivial Modifications

Define the functionals

$${\mathcal{J}}_{\mathsf{\Theta }}[\phi ;\rho ]={\int }_{\mathsf{\Omega }}{\rho \left(x\right)f(\parallel \nabla \phi \left(x\right)\parallel }_2)dx,{\mathcal{J}}_E[\psi ;\rho ]={\int }_{\mathsf{\Omega }}\rho \left(x\right)g\left(\right|\Re \left(\psi \left(x\right)\right)\left|\right)dx,$$

(A1)

where $f,g:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ are even, convex, 1-homogeneous, and $f\left(0\right)=g\left(0\right)=0$.

Theorem A3 

(Minimality with isotropy and 1-homogeneity). Within (A1)–(A5), the unique (up to positive constants, measure-zero changes, and divergence-free flux terms) minimizers are

$${\mathsf{\Theta }}_{min}={\kappa }_{\mathsf{\Theta }}{\rho \parallel \nabla \phi \parallel }_2,E_{min}={\kappa }_E\rho |cos\phi |,$$

(A2)

with constants ${\kappa }_{\mathsf{\Theta }},{\kappa }_E>0$.

Proof. 

Step 1 (structure of f). By isotropy, $\mathsf{\Theta }$ is governed by a rotation-invariant Minkowski functional; hence $f(\parallel ·{\parallel }_2)$. By 1-homogeneity and convexity, $f\left(t\right)=c_{\mathsf{\Theta }}t$. Among isotropic gauges, the Euclidean norm is pointwise minimal (up to scaling), giving ${\mathsf{\Theta }}_{min}={\kappa }_{\mathsf{\Theta }}\rho {\parallel \nabla \phi \parallel }_2$.

Step 2 (structure of g). By isotropy, E depends on $\psi$ only through $|cos\phi |$ (i.e., $|\Re (\psi \left)\right|/\parallel \psi \parallel$). A convex even 1-homogeneous scalar is $g\left(x\right)=c_E\left|x\right|$, hence $E=c_E\rho |cos\phi |$.

Step 3 (uniqueness). Adding divergence-free flux terms or modifying on measure-zero sets leaves volume integrals unchanged; constants fix units.    □

Remark A2 

(Worked example: phase kinks). For $\phi \left(x\right)=\alpha \mathrm{sgn}\left(x\right)$ (mollified) in 1D, ${\int \rho \parallel \nabla \phi \parallel }_2\approx 2\alpha \rho \left(0\right)$; any strictly convex f with $f\left(t\right)\ge ct$ yields a larger cost unless f is linear.

Figure A1. Among isotropic gauges, the Euclidean norm is the minimal choice up to scaling; other symmetric norms are larger pointwise.

Figure A1. Among isotropic gauges, the Euclidean norm is the minimal choice up to scaling; other symmetric norms are larger pointwise.

Appendix D.1. Order-Effect Convergence from CSV

Figure A4. Order-effect time series ${\mathsf{\Delta }}_t$ with an exponential bound fit $e^{-t}$ (fallback uses a default ).

Figure A4. Order-effect time series ${\mathsf{\Delta }}_t$ with an exponential bound fit $e^{-t}$ (fallback uses a default ).

Provenance. 

The CSV data/order_effect_convergence.csv is generated by scripts/generate_convergence.py (deterministic; default $\gamma =0.10$). Columns: t,delta. The bound overlay uses $e^{-\gamma t}$ with $\gamma$ set in the preamble.

Appendix D.2. Heatmap from CSV

Figure A5. Convergence proxy $R(\theta ,p)$ on a $\theta \times p$ grid. Colors encode the magnitude of R (darker = slower mixing). Diagnostic visualization only; not used in proofs.

Figure A5. Convergence proxy $R(\theta ,p)$ on a $\theta \times p$ grid. Colors encode the magnitude of R (darker = slower mixing). Diagnostic visualization only; not used in proofs.

Provenance. 

The grid CSV data/heatmap_proxy.csv is provided with this submission (columns: theta,p,R). The heatmap is diagnostic and not used in proofs.

Figure A6. Equality cases of the tight order-effect bound. (a) Halmos two-subspace principal angle $\theta$ for $P=|u\rangle \langle u|$ and $Q=|v\rangle \langle v|$. (b) Bloch-sphere rendering where the measurement axes differ by $2\theta$. These geometries realize the saturation conditions used in the proof of the main bound.

Figure A6. Equality cases of the tight order-effect bound. (a) Halmos two-subspace principal angle $\theta$ for $P=|u\rangle \langle u|$ and $Q=|v\rangle \langle v|$. (b) Bloch-sphere rendering where the measurement axes differ by $2\theta$. These geometries realize the saturation conditions used in the proof of the main bound.