This dataset presents the full empirical realisation of the Forensic Inversion Protocol (RE-INVERT), validating the Collapse Criterion governing inference failure in multi-witness systems. The figure integrates six panels (A–F), each corresponding to a structural component of the underlying geometric inference framework.Panel A demonstrates accurate recovery of interaction amplitudes under adjoint reconstruction, confirming numerical stability even near collapse. Panel B quantifies the competition between geometric stability, via the minimum eigenvalue of the Fisher Information Metric λ_min(t), and cumulative environmental distortion D(t). Panel C tracks the evolution of the dimensionless collapse parameter Γ(t), showing smooth convergence toward the critical threshold. Panel D empirically confirms loss of identifiability as Γ → 1, with reconstructed states exhibiting extreme dispersion. Panel E analyses the role of regularisation ε, demonstrating that bias–variance trade-offs cannot compensate for geometric degeneration. Panel F provides regime classification across rigid, stable, transitional, and collapsed phases.The results establish that inference failure arises from geometric degeneration of the Fisher metric, rather than numerical instability or noise amplification. The collapse parameter Γ(t) emerges as a universal control variable governing the transition from identifiable to non-identifiable regimes.This dataset supports and extends the theoretical results presented in:Trace Forensics — Paper 31: Fisher-Geometric Inference under Cascade RefinementTrace Forensics — Paper 2: Quantum Forensics via Environmental Bow WavesTrace Forensics — Paper 5: Multi-Witness Inference Foundations
This dataset presents the full empirical realisation of the Forensic Inversion Protocol (RE-INVERT), validating the Collapse Criterion governing inference failure in multi-witness systems. The figure integrates six panels (A–F), each corresponding to a structural component of the underlying geometric inference framework.Panel A demonstrates accurate recovery of interaction amplitudes under adjoint reconstruction, confirming numerical stability even near collapse. Panel B quantifies the competition between geometric stability, via the minimum eigenvalue of the Fisher Information Metric λ_min(t), and cumulative environmental distortion D(t). Panel C tracks the evolution of the dimensionless collapse parameter Γ(t), showing smooth convergence toward the critical threshold. Panel D empirically confirms loss of identifiability as Γ → 1, with reconstructed states exhibiting extreme dispersion. Panel E analyses the role of regularisation ε, demonstrating that bias–variance trade-offs cannot compensate for geometric degeneration. Panel F provides regime classification across rigid, stable, transitional, and collapsed phases.The results establish that inference failure arises from geometric degeneration of the Fisher metric, rather than numerical instability or noise amplification. The collapse parameter Γ(t) emerges as a universal control variable governing the transition from identifiable to non-identifiable regimes.This dataset supports and extends the theoretical results presented in:Trace Forensics — Paper 31: Fisher-Geometric Inference under Cascade RefinementTrace Forensics — Paper 2: Quantum Forensics via Environmental Bow WavesTrace Forensics — Paper 5: Multi-Witness Inference Foundations
Scale-Dependent Quantum Inference via Environmental Transformation Description This record contains the updated and finalized version of the paper titled Scale-Dependent Quantum Inference via Environmental Transformations. This version represents a unified and fully polished synthesis of the conceptual frameworks introduced in the author’s earlier works: Thought Experiment on Photon Pulse Reconstruction in Extreme Environments (Paper 1) Quantum Forensics: Inferring Photon States via Environmental Bow Waves (Paper 2) The updated paper formalizes the relationship between a photon’s initial state and an observable environmental signal through a transformation operator �, where the environment is treated as an active encoding medium rather than a passive source of noise. A central contribution of this version is the explicit integration of scale-dependent behavior via a complexity parameter �, which governs the degree to which environmental transformations preserve or obscure information about the underlying photon state. As environmental complexity increases, the transformation becomes increasingly non-invertible, leading to reduced mutual information and Fisher information between the photon state and the observable signal. The paper develops the interpretation of environmental responses as structured perturbations, described as “bow waves,” which represent distributed signatures of photon–medium interactions. These signatures form the basis of a quantum forensic approach, in which the photon state is inferred indirectly through its environmental imprint. A toy model is included to illustrate these principles in a simplified one-dimensional system consisting of a photon interacting with a chain of stochastic scatterers. The model demonstrates how recursive interactions and stochastic feedback lead to progressive information loss as system size increases. This updated version improves the clarity, internal consistency, and formal presentation of the framework. It refines terminology, strengthens the mathematical and conceptual structure, and aligns the exposition into a coherent, submission-ready document suitable for archival and dissemination. Version Notes This version supersedes the previous draft of the same work hosted under this record. The following updates have been incorporated: Full integration of conceptual material from related foundational works (Paper 1 and Paper 2) into a unified framework Refinement and standardization of terminology across the document, including consistent use of: “environmental transformations” “quantum forensics” “bow waves” “scale parameter �” Improved clarity and precision of language throughout the abstract, introduction, and analytical sections Strengthening of the information-theoretic formulation, including clearer articulation of mutual information and Fisher information as measures of recoverability Enhanced discussion of inference conditioning and ill-posedness in high-complexity regimes Expansion and clarification of the toy model to better illustrate recursive stochastic interactions and signal overlap behavior Addition and refinement of the limitations and scope section to explicitly contextualize the model’s purpose and applicability Overall polishing of structure, flow, and consistency to meet submission-ready standards This version should be considered the finalized and authoritative iteration of the work. Keywords Quantum inference Information theory Fisher information Mutual information Inverse problems Environmental transformations Quantum measurement Quantum forensics Signal processing Statistical inference Complex systems Photon interactions Stochastic processes Non-invertible systems Scale-dependent models Subjects Physics Quantum Physics Information Theory Statistical Physics Applied Mathematics Complex Systems License Creative Commons Attribution 4.0 International (CC BY 4.0)
Inverse Geometry and Identifiability of Protocol Ratios in Dynamical-Decoupling Spectroscopy This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Building on reduced observation geometry, it asks the inverse question: given an observed protocol ratio, what spectral information is uniquely reconstructible, what is only weakly constrained, and what is fundamentally unrecoverable? The analysis is formulated in terms of the Jacobian of the reduced observation map and the associated Fisher information matrix. The paper shows that the inverse problem separates into three qualitatively distinct sectors. In the regular sector, the ratio map is locally invertible and reconstruction accuracy is controlled by a Cramér–Rao bound. For the pure power-law family S(ω)=Aω^α, the ratio R(α)=2/(1+2^{-α}) is strictly monotone, admits a closed-form inverse, and yields an explicit conditioning profile through the slope ∂_αR. In the degenerate sector, exemplified by the white-noise family S(ω)=A, the protocol ratio is exactly pinned to R=1 for all amplitudes. The Fisher information for the amplitude direction vanishes identically, so amplitude reconstruction from ratio data alone is impossible. This gives an exact no-go theorem for identifiability from protocol ratios in the white-noise case. In the branched sector, non-injectivity of the inverse map produces intrinsic ambiguity. Using the mixed family S(ω; α1, α2, w)=wAω^{α1}+(1−w)Aω^{α2}, the paper shows both global non-injectivity and local fold branching. Along an explicit symmetric slice, the ratio develops a fold at Rc=4/3, so nearby observed values admit two distinct inverse branches. These results establish protocol ratios as inverse-geometric probes with sharply defined domains of identifiability and intrinsic limits of spectral reconstruction. The framework clarifies the difference between unique inference, exact information collapse, and multi-valued inversion, and provides a basis for future extensions to multi-ratio, mixed-spectrum, and frequency-resolved inverse design. V2:This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Interpreting the reduced observation map through Fisher information, we show that the inverse problem separates into three sectors: regular (locally invertible with Cramér–Rao bounds), degenerate (exact collapse with vanishing Fisher information), and branched (non-injective with multiple solutions). Using power-law and mixed spectral families, we derive explicit inversion formulas, a white-noise no-go theorem, and fold-induced branching. The framework establishes protocol ratios as inverse-geometric probes with sharply defined limits of spectral reconstruction.
We develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.
ISAT: Quantum Dynamics v10.0 introduces a comprehensive computational verification engine demonstrating that scale-invariant structural vectors (K∗) function as a non-trivial, shape-sensitive observer metric. Through systematic multi-dimensional numerical experiments (45 multi-scale tests), this framework bridges Information Structural Attraction Theory (ISAT) and quantum phenomena across diverse regimes. Core Verification Suite Findings (Section 4.11) Exp A (Quantum Revivals): CK∗ detects geometric shape revivals under Lindblad dephasing with 0.9999 peak value, while standard Fidelity drops to 0.3908 (a 2.56x robustness gain). Exp B3 (Dicke Transition): Observer dynamics detect the superradiant transition via Critical Slowing Down (CSD), showing a sharp 9.6x peak in convergence steps near the critical point (λc=0.50). Exp C2 (Dimensionality Tracking): PCA projects a 256D spatial grid collapse path onto an effective 2.7D manifold, capturing 96.1% of the total dynamical variance. Exp D2 (Entanglement Transition): A 12D multi-basis (X & Z) moments vector tracks the volume-to-area law transition, establishing a strong monotonic correlation (r=−0.8142) with half-chain entanglement entropy. Exp E2 (Error Mitigation Witness): In the density matrix limit, the K∗ space Euclidean distance to the ideal state drops from 0.5314 (unmitigated) to 0.0099 (mitigated), yielding a p-value=0.0000 and 100% separability. Exp F2 (Non-Markovian Memory): Moments space distance expansion tracks open-system information backflow, achieving a perfect correlation (r=1.0000) with the BLP trace-distance measure, proving Bloch-sphere geometric isomorphism. Repository Contents Computational Verification Engine: Source code and scripts for the 45 multi-scale tests across 1D, 2D, 3D, 4D, Time-dependent Dynamics, Many-Body, Open systems, and Closed-loop Active Inference. Official Preprint: The full theoretical and numerical manuscript detailing the mathematical foundations of the K∗ metric. Breakthrough Verification Suite: Complete data tables and plotting scripts resolving legacy numerical anomalies.
Genuine Collapse, Apparent Flattening, and Observable Horizons in Inverse Spectral Reconstruction This paper studies two distinct mechanisms by which inverse spectral reconstruction can fail. In one case, the forward observation map loses rank and information is destroyed at the geometric level; this is termed genuine collapse. In the other, the map remains full-rank but its weakest reconstruction direction is compressed below practical observational reach; this is termed apparent flattening. The paper introduces the corresponding notion of an observable horizon, defined by the competition between geometric compression and finite measurement noise. The analysis is carried out for the spectral family S(ω) = Aω^α + B, using scale-independent ratio observables motivated by dynamical-decoupling spectroscopy. Writing ρ = B/A, the paper derives the Jacobian of the ratio map with respect to (α, ρ), relates its weakest singular direction to the Fisher information matrix, and shows that practical identifiability is governed by the smallest singular value of the reduced observation map. For the minimal two-frequency ratio model, the paper obtains an exact closed-form formula for the singular value s(J) and proves a classification theorem for its large-ρ decay. Away from a distinguished fragility locus α = α*, the singular value decays generically as O(ρ^−1). On the fragility locus, the leading response cancels and the decay accelerates to O(ρ^−2), causing observability to deteriorate more rapidly. The paper also proves that, in this minimal model, no genuine-collapse locus exists at any finite parameter value: all finite-parameter information loss is therefore of the apparent-flattening type. This leads to an observable-horizon map in the (α, ρ) plane. Lowering observational noise expands the visible region and pushes the horizon outward, but it does not move the fragility locus or the asymptotic geometric boundary. The resulting picture distinguishes a movable observational wall from an immovable geometric wall and provides a concrete criterion for probe-frequency selection: choose frequencies so that the fragility value α* lies outside the target spectral-exponent range. Overall, the paper gives a geometric and statistical characterization of information loss in floor-dominated inverse spectral reconstruction. It clarifies the difference between exact geometric impossibility and noise-limited practical impossibility, and formulates observable horizons as the natural interface between them. V2:This version refines the interpretation and presentation of the single-ratio model without altering the main analytical results. Clarified the Fisher structure: the single-ratio model is rank-1, and observability is formulated via the scalar response norm s(J)s(J)s(J) rather than a two-parameter Fisher minimum eigenvalue. Introduced the scalar observable horizon explicitly as a local detectability (sensitivity) threshold, removing ambiguity with estimation-based notions. Strengthened the framing as a scalar sensitivity prototype for multi-ratio Fisher geometry (treated in the companion paper). Corrected minor issues, including numerical consistency (αmono\alpha_{\rm mono}αmono) and full alignment between figures, code, and text. Improved wording to avoid overstatements and clarify scope (e.g., scalar-response interpretation). The analytical results, including the fragility locus α∗\alpha_*α∗ and the O(ρ−1)O(\rho^{-1})O(ρ−1) vs. O(ρ−2)O(\rho^{-2})O(ρ−2) classification, remain unchanged.
This paper presents Dimensional Evolution — Quantum Measurement and Identity Formation (DE-QM), a geometric framework addressing the quantum measurement problem while preserving the full predictive structure of standard quantum mechanics. Within DE-QM, the wavefunction is treated as a resonance-geometry field rather than a probability amplitude. Superposition arises from deterministic traversal across multiple resonance-compatible configurations supported by an inverse domain, rather than from the coexistence of mutually realized states. Measurement collapse is described as a geometric stability transition governed by anchoring thresholds imposed by environmental and detector constraints. Probability emerges as a derived measure of geometric overlap between scan traversal and detector geometry, recovering the Born rule without introducing intrinsic randomness or observer-dependent postulates. The framework integrates decoherence, entanglement, and relativistic structure into a unified geometric description that remains fully compatible with nonrelativistic and relativistic quantum mechanics, including the Schrödinger and Dirac formalisms. Entanglement is treated as a shared inverse-domain resonance geometry, enabling coordinated collapse without superluminal signaling. Measurement outcomes are shown to be observer-independent and physically grounded in macroscopic anchoring geometry, consistent with established experimental results. This manuscript serves as the foundational quantum-mechanical component of the broader Dimensional Evolution framework and builds directly on prior canonical works, including the DE Core paper, companion papers CP1–CP5, and specialized studies addressing the Mott problem and Dirac structure. DE-QM does not modify quantum mechanics’ mathematical formalism; rather, it provides a coherent geometric ontology underlying wavefunction structure, superposition, collapse, and probability. --- Project website: https://de-theory.com This work is part of the Dimensional Evolution research program. Additional materials, updates, and contact information will be made available via the project website as the work expands.