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Fog Landscapes: A Framework for Indeterminate Optimization Domains

H Y Rao

Published Apr 25, 2026• 0 citations
Publisher

Abstract

Classical optimization assumes a definite landscape: given a problem, the objective function f(x) exists prior to investigation, and algorithms search for its extrema. We introduce a parallel framework, the *fog landscape*, in which the optimization domain itself is observation-induced: it does not exist prior to multi-chain deployment but emerges from the deployment as a statistical structure on the chain ensemble. Each chain is a collapse chain on a domain-specific observation system (S, Ω, Φ), evolving independently under environmental influence via completely-positive trace-preserving (CPTP) maps. After T evolution steps, the ensemble is reverse-fitted to a continuous functional form using Gaussian Mixture Models (GMM). The resulting function F(x) is a legitimate NP-Hard landscape and inherits the classical NP-Hard toolkit: landscape classification, reconnaissance flattening, and algorithm selection. The framework is presented in three parts: (i) a generic specification independent of any application domain, (ii) a worked example demonstrating the framework on the NSD financial trading system with d=11 personality dimensions, and (iii) cross-domain guidance for application to quantum cognition, epidemiology, and strategic forecasting. Fog landscapes constitute a parallel ontological category to classical NP-Hard landscapes—not a reduction, generalization, or mirror correspondence, but an independent class of mathematical objects—while admitting reduction at the operational level for tool reuse. The strong-Copenhagen interpretation is essential: the fog landscape is the genuine mathematical object characterising indeterminate optimization, not an approximation to a hidden continuous truth. This is a working paper. v1 establishes the framework architecture and demonstrates feasibility on the NSD financial system. Mathematical proofs (Proposition 2), explicit Kraus operator forms for NSD, and the ontological commitment defense are abbreviated; v2 will expand these. Comments welcome. Version 2 (April 2026) — Substantial revision incorporating reviewer feedback from three independent sources and one author-reveal. Major additions and refinements: §1.2 Strong Copenhagen as logical necessity. The ontological commitment is reframed from a stylistic choice into a structural requirement, established via stress-testing against five alternative interpretations (Standard Copenhagen, Bohmian, Many-Worlds, QBism, Decoherent Histories). Each alternative is shown to fail the framework's constitutive claim or operational behavior. Operational compatibility paragraph added for readers committed to alternative interpretations. §5 Historical pre-segment and junction point. New subsection explicitly handling the chain ensemble's initial state. The multi-chain fan-out begins from an informed posterior γ_0, derived from a single historical chain over t = -N*, ..., 0. This structure mirrors MC-CBPTT's Hybrid Chain and is required by the strong-Copenhagen commitment (past observations have collapsed; multi-chain fan-out begins from the system's real, accumulated past). §3 CPTP existence anchor. Stinespring dilation and Choi–Kraus theorem cited (Nielsen & Chuang Ch. 8). Three operational requirements (trace preservation, complete positivity, linearity) made explicit. Why-CPTP-not-Monte-Carlo defense expanded across operational, mathematical, and ontological levels. §6.3 Why GMM. Comparison against KDE, neural density estimators, t-mixtures, and maximum entropy distributions. Four desirable properties of GMM articulated. Framework explicitly stated to be reduction-method neutral. §6.5 Stability and sensitivity. New subsection covering component count K, sample size N (per-parameter complexity O(d²/N)), convergence and initialisation, informal Wasserstein-2 → L² sensitivity bound (formal proof deferred to v3), and four practical recommendations. §6.6 Operational vs ontological defense. Expanded from 3 sentences to a full page. Three defenses against the "fog landscape is just elaborate density estimation" challenge: (1) deployment dependence, (2) strong-Copenhagen commitment at ensemble level (parallel to Layered Observation's α vs κ), (3) reduction as operational tool reuse (Fourier transform analogy). §7.2 Continuous-optimization scope. Honest scope-bounding: the framework inherits the continuous-optimization portion of the NP-Hard toolkit, not the entire toolkit. Tools that transfer directly, require adaptation, or require domain bridging are explicitly listed. §4.3 Path integral disambiguation. Structural parallel preserved but explicitly delimited: density matrix ensemble aggregation does not involve phase coherence or interference effects. §8.2 NSD Kraus operators. Decomposition K_k(E, ψ) = K_k^(E) · P_ψ made explicit. Eleven Digital Exhaust correction operators and six personality filters described in detail. HMM forward update written as explicit Kraus form K_1^(E) = diag(√B) · √A. Full matrix entries deferred to v3 Appendix A. Definition 2 (Chain ensemble) and Definition 3 (Fog landscape as deployment-indexed empirical measure on personality space). The fog landscape is now a measure, not a function—the function F(x) emerges only after GMM reduction. Proposition 2. Proof added (using Bishop Ch. 2). Property (4) "admits optimization tools" demoted to Remark; the proposition covers only mathematical properties of F(x). Figure 2 expanded to four panels: (A) historical chain converging to junction, (B) junction point γ_0, (C) multi-chain fan-out, (D) ensemble at t = T as fog landscape measure μ_T,C. Bibliography expanded: Nielsen & Chuang (CPTP), McLachlan & Peel (GMM theory), MOEH paper, MC-CBPTT paper, LANCER 2023 (landscape surrogate comparison). v3 substantially expands the framework's mathematical foundations and programme integration. New: formal Wasserstein-2 sensitivity theorem with full proof (Theorem 1, Appendix B); explicit Kraus operator construction for the NSD test case with Lüders-rule formalisation (Appendix A); fourth Defense identifying the GMM reduction operator as a collapse operator in the unified programme language (§6.7); consolidated notation table (Appendix C). Sharpened: §3.1 density matrix grounding now anchored in four sister papers (Layered Observation, Five-Step Methodology, MOEH, MCCO); §1.2 strong-Copenhagen position now empirically anchored via the consensus interference parameter measurement κ̂=0.3606 (p<0.001); §10 limitations expanded from 3 to 7 substantive items; §11 closing now positions the paper within the FOG-1/2/3/4 series. Two reference bugs fixed: bibkey collision between MC-CBPTT and MCCO; MOEH paper title corrected. New external citations: Pitzer 2012 (FLA) and Mersmann 2011 (ELA). Mathematical correctness fix in §8.2/Appendix A: per-personality update reframed as Lüders rule with explicit renormalisation (the v2 multiplicative form K=K^(E)·P_ψ does not preserve trace; v3 makes the renormalisation explicit, consistent with the diagonal Lüders rule of MOEH Theorem 1).

DOI
10.5281/zenodo.19766683

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Classical optimization assumes a definite landscape: given a problem, the objective function f(x) exists prior to investigation, and algorithms search for its extrema. We introduce a parallel framework, the fog landscape, in which the optimization domain itself is observation-induced: it does not exist prior to multi-chain deployment but emerges from the deployment as a statistical structure on the chain ensemble. Each chain is a collapse chain on a domain-specific observation system (S, Ω, Φ), evolving independently under environmental influence via completely-positive trace-preserving (CPTP) maps. After T evolution steps, the ensemble is reverse-fitted to a continuous functional form using Gaussian Mixture Models (GMM). The resulting function F(x) is a legitimate NP-Hard landscape and inherits the classical NP-Hard toolkit: landscape classification, reconnaissance flattening, and algorithm selection. The framework is presented in three parts: (i) a generic specification independent of any application domain, (ii) a worked example demonstrating the framework on the NSD financial trading system with d=11 personality dimensions, and (iii) cross-domain guidance for application to quantum cognition, epidemiology, and strategic forecasting. Fog landscapes constitute a parallel ontological category to classical NP-Hard landscapes—not a reduction, generalization, or mirror correspondence, but an independent class of mathematical objects—while admitting reduction at the operational level for tool reuse. The strong-Copenhagen interpretation is essential: the fog landscape is the genuine mathematical object characterising indeterminate optimization, not an approximation to a hidden continuous truth. This is a working paper. v1 establishes the framework architecture and demonstrates feasibility on the NSD financial system. Mathematical proofs (Proposition 2), explicit Kraus operator forms for NSD, and the ontological commitment defense are abbreviated; v2 will expand these. Comments welcome. Version 2 (April 2026) — Substantial revision incorporating reviewer feedback from three independent sources and one author-reveal. Major additions and refinements: §1.2 Strong Copenhagen as logical necessity. The ontological commitment is reframed from a stylistic choice into a structural requirement, established via stress-testing against five alternative interpretations (Standard Copenhagen, Bohmian, Many-Worlds, QBism, Decoherent Histories). Each alternative is shown to fail the framework's constitutive claim or operational behavior. Operational compatibility paragraph added for readers committed to alternative interpretations. §5 Historical pre-segment and junction point. New subsection explicitly handling the chain ensemble's initial state. The multi-chain fan-out begins from an informed posterior γ_0, derived from a single historical chain over t = -N*, ..., 0. This structure mirrors MC-CBPTT's Hybrid Chain and is required by the strong-Copenhagen commitment (past observations have collapsed; multi-chain fan-out begins from the system's real, accumulated past). §3 CPTP existence anchor. Stinespring dilation and Choi–Kraus theorem cited (Nielsen & Chuang Ch. 8). Three operational requirements (trace preservation, complete positivity, linearity) made explicit. Why-CPTP-not-Monte-Carlo defense expanded across operational, mathematical, and ontological levels. §6.3 Why GMM. Comparison against KDE, neural density estimators, t-mixtures, and maximum entropy distributions. Four desirable properties of GMM articulated. Framework explicitly stated to be reduction-method neutral. §6.5 Stability and sensitivity. New subsection covering component count K, sample size N (per-parameter complexity O(d²/N)), convergence and initialisation, informal Wasserstein-2 → L² sensitivity bound (formal proof deferred to v3), and four practical recommendations. §6.6 Operational vs ontological defense. Expanded from 3 sentences to a full page. Three defenses against the "fog landscape is just elaborate density estimation" challenge: (1) deployment dependence, (2) strong-Copenhagen commitment at ensemble level (parallel to Layered Observation's α vs κ), (3) reduction as operational tool reuse (Fourier transform analogy). §7.2 Continuous-optimization scope. Honest scope-bounding: the framework inherits the continuous-optimization portion of the NP-Hard toolkit, not the entire toolkit. Tools that transfer directly, require adaptation, or require domain bridging are explicitly listed. §4.3 Path integral disambiguation. Structural parallel preserved but explicitly delimited: density matrix ensemble aggregation does not involve phase coherence or interference effects. §8.2 NSD Kraus operators. Decomposition K_k(E, ψ) = K_k^(E) · P_ψ made explicit. Eleven Digital Exhaust correction operators and six personality filters described in detail. HMM forward update written as explicit Kraus form K_1^(E) = diag(√B) · √A. Full matrix entries deferred to v3 Appendix A. Definition 2 (Chain ensemble) and Definition 3 (Fog landscape as deployment-indexed empirical measure on personality space). The fog landscape is now a measure, not a function—the function F(x) emerges only after GMM reduction. Proposition 2. Proof added (using Bishop Ch. 2). Property (4) "admits optimization tools" demoted to Remark; the proposition covers only mathematical properties of F(x). Figure 2 expanded to four panels: (A) historical chain converging to junction, (B) junction point γ_0, (C) multi-chain fan-out, (D) ensemble at t = T as fog landscape measure μ_T,C. Bibliography expanded: Nielsen & Chuang (CPTP), McLachlan & Peel (GMM theory), MOEH paper, MC-CBPTT paper, LANCER 2023 (landscape surrogate comparison). v3 substantially expands the framework's mathematical foundations and programme integration. New: formal Wasserstein-2 sensitivity theorem with full proof (Theorem 1, Appendix B); explicit Kraus operator construction for the NSD test case with Lüders-rule formalisation (Appendix A); fourth Defense identifying the GMM reduction operator as a collapse operator in the unified programme language (§6.7); consolidated notation table (Appendix C). Sharpened: §3.1 density matrix grounding now anchored in four sister papers (Layered Observation, Five-Step Methodology, MOEH, MCCO); §1.2 strong-Copenhagen position now empirically anchored via the consensus interference parameter measurement κ̂=0.3606 (p<0.001); §10 limitations expanded from 3 to 7 substantive items; §11 closing now positions the paper within the FOG-1/2/3/4 series. Two reference bugs fixed: bibkey collision between MC-CBPTT and MCCO; MOEH paper title corrected. New external citations: Pitzer 2012 (FLA) and Mersmann 2011 (ELA). Mathematical correctness fix in §8.2/Appendix A: per-personality update reframed as Lüders rule with explicit renormalisation (the v2 multiplicative form K=K^(E)·P_ψ does not preserve trace; v3 makes the renormalisation explicit, consistent with the diagonal Lüders rule of MOEH Theorem 1).
This paper extends the static fog landscape framework of FOG-1 to time-evolving multi-chain observation systems. We develop two complementary regimes connected by a mean-field convergence theorem: a streaming framework operating at finite chain population and discrete time grid, and a Lindblad–Fokker-Planck framework operating in the continuum limit. The streaming framework fits per-time Gaussian Mixture Models to the chain ensemble and tracks component identity across consecutive fits via Wasserstein-2 matching, partitioning the cost space into four interpretable lifecycle events (drift, appearance, disappearance, merge/split). The Lindblad–Fokker-Planck framework derives an explicit partial differential equation for the time-resolved density F(x, t) from a continuous-time Lindblad master equation, with drift and diffusion coefficients given in closed form for the diagonal regime. We establish five formal results. The component tracking lemma (Lemma 1) and the dynamic Wasserstein-L² sensitivity bound (Theorem 4) are stated with full proofs; Theorem 4 extends FOG-1's static Wasserstein-L² Lipschitz bound to time-evolving ensembles via a Lindblad contractivity assumption, identifying three operational regimes (contractive, marginal, expansive). Three asymptotic theorems (mean-field limit, projected Fokker-Planck, mixing time formula) are stated in sketch form with proof tools identified; full proofs are deferred to v2. The strong-Copenhagen ontological commitment of FOG-1 is preserved and structurally strengthened: the framework asserts the landscape is observation-induced at every instant, with the trajectory itself constituted by deployment events rather than tracked from a pre-existing dynamics. This is a Working Paper Version 1 establishing the framework's theoretical infrastructure. Empirical instantiation on operational systems is the subject of a separate companion paper. The framework opens two further programme extensions: FOG-3 (multi-scale homogenisation across coupled time clocks) and FOG-4 (path-functional optimisation directly on chain trajectories).
This paper synthesises two previously distinct frameworks of the NSD/MOEH programme: the Multi-Chain Collapse-Centred Optimisation (MCCO) framework, which deploys N chains of CPTP evolution as parallel optimisation agents on indeterminate landscapes; and the Spacetime Digital Exhaust (SDE) framework, in which a single chain is augmented with N_t peripheral factor observations at each time step. The synthesis is multiplicative rather than additive, producing a four-axis Cartesian object indexed by chain c, factor n, inner state k, and time t, with N_t tentacles per (chain, step) cell. The central design choice is the shared factor pool: at each time t, the N_t peripheral factors are shared across all N chains, while each chain has its own tentacle weight function w^(c). This sharing produces implicit cross-chain information flow: an observation that updates one chain's posterior also informs the others through correlated weighting. The shared-pool architecture is the structural feature distinguishing the multi-chain multi-tentacle (MCMT) framework from a trivial parallelisation of single-chain SDE. The paper's contributions are theoretical. The MCMT architecture is defined with the shared factor pool and chain-specific tentacle weighting; the tensor evolution law is derived showing exact decomposition into per-chain CPTP dynamics and cross-chain factor accumulation; three structural propositions are proven with full proofs (CPTP preservation, information-theoretic gain over independent SDE, consistency with both parent frameworks as boundary cases); the theoretical performance analysis derives complexity bounds and information-theoretic orderings (Propositions on complexity and entropy ordering); the cross-box applicability is established for five NP-Hard problem classes from the seven-box landscape taxonomy (Multidimensional Knapsack, Graph Colouring, Vehicle Routing, Quadratic Assignment, Sequential Decision); bridges to the FOG framework and DRF pipeline are sketched. This is a programme-internal Working Paper Version 1. All performance claims are theoretical (asymptotic complexity bounds and information-theoretic orderings) rather than empirical. Empirical validation on operational systems—including the extension of MCCO v3's 3.56× speedup benchmark and cross-box measurement across the five NP-Hard problem classes—is the subject of a separate companion paper that will be developed when the MCMT implementation infrastructure is in place. The paper bridges to FOG-4 (10.5281/zenodo.20152165) as the discrete-time, finite-tentacle instantiation of the continuous-time path-functional framework; consumes the diagnostic output of NP Landscape Classification; and provides the optimisation engine for the forthcoming DE-5 pipeline capstone. 24 pages, bilingual English/Traditional Chinese. Builds on: MCCO Standalone (10.5281/zenodo.20104942), Spacetime DE (10.5281/zenodo.19750775), NP Landscape Classification (10.5281/zenodo.19653758), FOG-4 v2 (10.5281/zenodo.20152165).
This paper proposes a five-step methodology for circumventing the microscopic observation barrier in quantum mechanics by exploiting macroscopic systems where observation is unobstructed. Version 2 introduces a complete, exhaustive taxonomy of uncertainty based on two binary questions (computability and existence of a definite answer), yielding exactly three categories: determinism (computable, answer exists, κ = 0), chaos (not computable, answer exists, κ = 0), and collapse chain (not computable, answer does not exist, κ > 0). A fourth combination is logically impossible. The interference parameter κ divides the three categories into two sides; Lyapunov exponents distinguish the first two. Two measurements, three categories, complete. This taxonomy is scale-independent: it replaces the conventional microscopic/macroscopic distinction with a classification based on the nature of uncertainty itself, explaining why κ operates across all domains without modification. Three independent lines of convergence support the Structural Re-emergence Conjecture (Hübsch et al. 2024, tonyrao 2026, Tresp et al. 2025). The four-part theoretical programme (MOEH, CMB, Collapse Chain, Dynamic κ) provides the complete mathematical framework, measurement protocol, and first-domain empirical validation in financial markets. Two breaking points are explicitly identified and are empirically testable using κ itself. Part of the NSD / MOEH research programme. v2 update: Added Section 5 "A Complete Taxonomy of Uncertainty" — an exhaustive classification of uncertainty based on two binary questions (computability and existence of a definite answer). Three categories: determinism (computable, answer exists, κ = 0), chaos (not computable, answer exists, κ = 0), collapse chain (not computable, answer does not exist, κ > 0). Fourth combination is logically impossible. κ divides the three categories into two sides; Lyapunov exponents distinguish the first two. Two measurements, three categories, complete. This taxonomy is scale-independent, replacing the conventional microscopic/macroscopic distinction with a classification based on the nature of uncertainty itself. v3 update: Addresses six identified vulnerabilities. (1) Diagonal limitation resolved by α parameterisation: domain-dependent parameter controlling off-diagonal phase structure; α = 0 reduces to existing four-part programme, α ≠ 0 accommodates quantum coherence. Cross-spectral analysis of 11,826 XAUUSD H1 bars confirms α ≈ 0 for financial markets (all three state pairs p > 0.05, 1,000 bootstrap resamples). New Section with empirical results table. (2) Isomorphism problem addressed: α itself is the empirically measurable discriminator between isomorphism and analogy. (3) Burden of proof reversal: critics must provide alternative mechanism producing same κ statistics. (4) Thermodynamic analogy made precise: K-line = temperature, with item-by-item Boltzmann correspondence table (traders = molecules, K-line = temperature, κ = statistical mechanics). (5) Fourth category exclusion strengthened: quantum computation counterexample refuted (two operations conflated, each belongs to existing categories). (6) Single-domain limitation acknowledged as community's task, not author's. Updated abstract, falsifiability conditions, and limitations throughout. v4 update: Expanded α parameterisation from single-asset (XAUUSD only) to cross-asset validation across five assets and three asset classes. Results reveal α is not uniformly zero: XAUUSD and EURUSD show α ≈ 0 (p = 0.204 and p = 0.320, time-symmetric state dynamics), while XAGUSD, NSXUSD, and SPXUSD show α ≠ 0 (all p < 0.001, time-asymmetric state dynamics). Honest interpretation: α ≠ 0 in equities and silver most likely reflects classical temporal asymmetry (momentum, asymmetric crash-recovery dynamics), not quantum coherence. However, this establishes α as an empirically measurable parameter with genuine discriminating power — it takes detectably different values in different systems, refuting the criticism that α has no empirical content. Implications section expanded to three points: (1) diagonal limitation resolved for both α = 0 and α ≠ 0 assets within single framework; (2) α varies continuously within financial markets, consistent with a spectrum rather than a binary; (3) α is a real measurement, not a theoretical decoration. Updated abstract in both English and Chinese. Version 5 (May 2026) — Citation accuracy and convergence framing tightened. Substantive content changes from v4: 1. Hübsch et al. citation corrected and expanded. - Year corrected: 2024 → 2025 (the journal version was published April 2025 in Annals of Physics; the 2024 working paper version was a preprint). - Description corrected: previously summarised as "proved mathematically that reverse decoherence is possible." The actual paper develops a "mock quantum" framework demonstrating that the mathematical formalism of quantum mechanics can be implemented on classical complex adaptive systems (with explicit construction in the Lotka-Volterra model). The v5 description reflects this accurately. - Full bibliographic information added: Annals of Physics 475 (2025), 169954, with arXiv preprint 2310.14100. 2. Tresp et al. citation corrected and expanded. - Full author list added: Tresp, Li, Harjes, and Ma (LMU Munich). - arXiv identifier added: arXiv:2510.13894 v2 (October 2025). - Description corrected: previously summarised as "identified POVM-like measurement structure in neural network computation." The actual paper introduces the Heisenberg-Bayes POVM (HB-POVM) and proves mathematical equivalence between probabilistic HB-POVM updates and Bayesian updates in generative hidden Markov models, then derives an explicit neural-network implementation. The v5 description reflects this accurately. 3. "Three independent lines of convergence" framing softened. - Previous wording suggested three independent natural-phenomenon discoveries of POVM structure across disciplines. - v5 acknowledges that two of the three lines (Hübsch et al.; Tresp et al.) explicitly import quantum-measurement formalism into their respective domains rather than discover it from data. - The convergence is reframed as one of formal framework: the same mathematical machinery (POVM-equivalent measurement linked to HMM-Bayesian updating) is operationally productive across mathematical physics, financial markets, and neural-network computation. This is a more modest but more accurate framing. 4. Bibliography updated with corrected entries and DOIs / arXiv URLs for both Hübsch and Tresp. 5. Date and version label updated: v4 / April 2026 → v5 / May 2026. No changes to:- The five-step methodology- The taxonomy of uncertainty (three categories: determinism, chaos, collapse chain)- The interference parameter κ definition- The α parameterisation and cross-asset validation results- The two explicit breaking points (Steps 2 and 5)- The author's prior four-part programme citations (MOEH, CMB, Collapse Chain, Dynamic κ) Format: xelatex with bilingual (English / Traditional Chinese) abstract and section summaries, as in v4. v6 (2026-05-13)- Title: "When the Front Door Is Locked, Go Around" → "The Observer Barrier"- Subtitle, body, abstract, bibliography unchanged- Body metaphor (§1.1, §1.2, closing) preserved- Motivation: formal academic register for philosophy of science journals (Synthese, FoS, etc.)
Optimization plays a central role in variational quantum algorithms, where the objective function typically takes the form F(θ)=∑m=1Mfm(Tr[U(θ)ρmU†(θ)Om])F(\boldsymbolθ)= \sum_{m=1}^{M} f_m \left(\mathrm{Tr}[U(\boldsymbolθ)ρ_m U^\dagger(\boldsymbolθ) O_m]\right)F(θ)=∑m=1M​fm​(Tr[U(θ)ρm​U†(θ)Om​]), with U(θ)U(\boldsymbolθ)U(θ) being a parameterized quantum ansatz. Understanding the optimization landscape of such objective functions is crucial for assessing the trainability and performance of these algorithms. For the special case M=1M=1M=1, it is known that under certain assumptions, the landscape is free of false traps (FTs), i.e., local optima that are not global. In this work, we investigate optimization landscapes of the general case M≥1M\geq1M≥1 and show that the landscape becomes intrinsically more complex. First, we establish a complete framework for analyzing critical features of the optimization landscape, by deriving necessary and sufficient conditions to identify and classify all critical points under some assumptions, which is also of practical importance in designing efficient algorithms independent of whether these assumptions are satisfied. Then, we show that FTs can still emerge on landscapes for M>1M>1M>1, standing in stark contrast to the M=1M=1M=1 case and further revealing that parameter sufficiency alone is not enough to guarantee a trap-free landscape. Moreover, we uncover a close connection that the emergence of FTs is necessarily attributed to the loss of distinguishability among the states and/or operators, and fundamentally, to the loss of compatibility of the spectral ordering governed by different objective terms. Our results provide a deeper understanding of the optimization complexity and practical guidance for both algorithmic and problem-setting designs.