We develop a reproducible diagnostic framework for modular phase transitions in RG-proxy flows of quantum states. The diagnostics are derived from the modular generator Kρ = −log ρ and include (i) spectral quantile coordinates kq = −log λq(ρ), (ii) a normalized commutator probe L(ρ;O) = ∥[Kρ,O]∥F /(∥Kρ∥F ∥O∥F ), and (iii) a running exponent ν extracted from sliding log–log fits of L(t). We introduce pre-registered stability rules using bootstrap over seeds and model-selection criteria (AICc/BIC/LOOCV), converting qualitative phase narratives into PASS/FAIL statements. Using a reference experiment (dim=64; Nvis = 5, Nhid = 1; Gibbs-chaotic family with mixing parameter p), we demonstrate robustness across observables (near vs far), normalization choices, and mild noise channels, and we identify explicit domain limitations (failure on extreme random-pure families). Finally, we explain how these diagnostics interface with phase/locality criteria through spectral-flow markers and provide a practical route to numerical phase mapping under modular RG-proxy dynamics.
We develop a reproducible diagnostic framework for modular phase transitions in RG-proxy flows of quantum states. The diagnostics are derived from the modular generator Kρ = −log ρ and include (i) spectral quantile coordinates kq = −log λq(ρ), (ii) a normalized commutator probe L(ρ;O) = ∥[Kρ,O]∥F /(∥Kρ∥F ∥O∥F ), and (iii) a running exponent ν extracted from sliding log–log fits of L(t). We introduce pre-registered stability rules using bootstrap over seeds and model-selection criteria (AICc/BIC/LOOCV), converting qualitative phase narratives into PASS/FAIL statements. Using a reference experiment (dim=64; Nvis = 5, Nhid = 1; Gibbs-chaotic family with mixing parameter p), we demonstrate robustness across observables (near vs far), normalization choices, and mild noise channels, and we identify explicit domain limitations (failure on extreme random-pure families). Finally, we explain how these diagnostics interface with phase/locality criteria through spectral-flow markers and provide a practical route to numerical phase mapping under modular RG-proxy dynamics.
We introduce a new computational diagnostic family, MSRO (Modular–Spectral RG Observables), designed to detect emergent geometry, critical/string-like windows, and nongeometric regimes in modular RG-proxy flows. MSRO integrates spectral profile observables k(q) = −log λq(ρ), modular commutator diagnostics (normalized commutators, running exponents ν, and closure quality), and reproducible calibration rules from phase mapping. We define a spectral-only MSRO tier that predicts geometry stability and phase labels using only modular-spectral features, and a full MSRO tier that provides an “instrument panel” including locality / partition markers and backreaction response coefficients. Using a reference experimental design (dim=64 mixed/Gibbs-like family; (p, κ) grid; bootstrap seeds), we specify out-of-sample validation, robustness checks, and explicit failure domains. MSRO is presented as a measurable instrument rather than an ontological claim: every emergence statement is paired with reproducible diagnostics, uncertainty quantification, and domain-restricted validity.
We propose a conservative definition of “string-like” criticality within a modular foundations program: not as a microscopic ontological primitive, but as a dynamical regime in which no stable phase partition persists under RG-proxy modular flow. We define partition instability markers based on (i) the switching rate of P(t), (ii) the flatness of the partition landscape Φ(P; ρ) measured by gaps between near-optimal partitions, and (iii) the persistence of nonlocality through elevated TP . We connect these markers to modular spectral diagnostics from Paper D (spectral quantiles kq, commutator probes L, running exponents ν) and propose a reproducible PASS/FAIL detection protocol with data-driven thresholds calibrated from baseline percentiles. A demonstrative case study (Variant A) uses the dim=64 Gibbs-chaotic mixing path and a critical zoom around px ≈ 0.04 to illustrate how partition flatness and switching emerge alongside spectral-tail anomalies, providing a practical route to identifying critical regimes in modular RG-proxy flows.
We develop a mathematically rigorous and experimentally testable framework for the classification of physical regimes based on the modular structure of quantum states. The central object of the theory is the modular generator K = −log ρ, which provides a unified language for spectral, geometric, and information-theoretic diagnostics. We introduce a universal instrument panel (MSRO: Modular–Spectral RG Observables) that integrates spectral quantiles, commutator-based response functions, partition-based locality diagnostics, and information-geometric structures. A key requirement of the framework is strong portability: the same diagnostic protocol applies across multiple domains—including open quantum systems, information geometry, entanglement-based models, and RG-like flows—without domain-specific retuning. The main result of the work is the derivation of a universal and experimentally testable scaling law for the modular response signal: ν(λ) ∼ 1 log λ, which characterizes critical regimes in a wide class of quantum systems. We prove that this scaling emerges from spectral asymptotics of the modular operator and establish a strict equivalence between spectral, geometric, and information-theoretic descriptions of criticality: ν ↔ k(q) ↔ I(λ) ↔ K ↔ R ↔ Δ. The framework is supported by functional analytic results (including Fr´echet differentiability of log ρ), operator inequalities (Golden–Thompson, monotonicity of relative entropy), and stability estimates. We further provide a concrete experimental protocol for quantum simulators, including statistical error bounds and Fisher information constraints, making the prediction directly testable. Beyond regime classification, the results suggest a new perspective on criticality as a geometric phenomenon associated with vanishing modular curvature. The approach opens pathways toward large-N limits, quantum field extensions, and the discovery of new classes of non-standard critical behavior.
Building on the canonical emergence of the density operator ρ from normalized probabilistic structure (Paper A), we develop the next canonical layer: modular generators and CPTP-compatible evolution. For faithful states, the modular generator Kρ := −log ρ is well-defined, unitary covariant, and additive under tensor products. Introducing a phase reference state σ yields the relative modular generator Kρ|σ = −log ρ + log σ, naturally tied to the relative entropy D(ρ∥σ). We define the minimal unitary “relative modular flow” ρ˙ = −i[Kρ|σ, ρ], and then derive a physically consistent completion by augmenting the flow with GKSL dissipators, including an entropy-relaxing channel with fixed point σ and an optional classicalization channel implementing decoherence in a pointer algebra. The resulting equation constitutes a conservative, testable backbone for subsequent work on informational phases and spectral diagnostics.
Building on Paper A (canonical emergence of the density operator) and Paper B (relative modular dynamics and CPTP completion), we introduce phase structure and the emergence of locality in a conservative, information-theoretic manner. A phase is specified by an access structure encoded by an optimal partition P , the corresponding accessible algebra AF , and a pointer/center structure ZF . The reference state σF within a macro-constraint class is defined canonically via a maximum-entropy (MaxEnt) construction under macroscopic constraints extracted from AF . We emphasize the equivalence between MaxEnt references and relative-entropy projections, which provides a canonical and minimal notion of phase stability. We introduce a canonical information-geometric metric (BKM/Fishertype) on the state manifold and connect these constructions to modular spectral diagnostics used in numerical protocols, such as spectral quantiles kq = −log λq and commutator-based probes.
We propose a strong notion of portability for a unified “instrument panel” intended to operate across four domains: open quantum systems (OQS), information geometry (IG), entanglement-based emergent geometry/gravity (EG), and renormalization group (RG) flow. The panel is defined solely from a density operator ρ and its modular generator K = −log ρ (and, when needed, a visible reduction ρvis). It combines spectral profile coordinates, observable-facing modular commutator diagnostics, operational locality/criticality markers based on optimal correlation factorization, correlation-graph geometry with stability certification, and optional backreaction links. Strong portability is formulated as a transport contract: the same panel, the same gates, and the same validation protocol are applied across domains with only minimal domain adapters that declare the flow coordinate and the visible subsystem. We provide an explicit portability scorecard, robustness checks, and a strict failure-domain taxonomy to prevent overclaim.