Bilinear observation problems arise in many physical and information-theoretic settings, where observables and states enter multiplicatively. Rank-based diagnostics are commonly used in such problems to assess the effective dimensionality accessible to observation, often under the implicit assumption that rank deficiency can be resolved through numerical refinement. Here we examine this assumption by analyzing the rank and nullity of a bilinear observation operator under systematic tolerance variation. Rather than focusing on a specific reconstruction algorithm, we study the operator directly and identify extended rank plateaus that persist across broad tolerance ranges. These plateaus indicate stable dimensional deficits that are not removed by refinement procedures applied within a fixed problem definition. To investigate the origin of this behavior, we resolve the nullspace into algebraic sectors defined by the block structure of the variables. The nullspace exhibits a pronounced but nonexclusive concentration in specific sectors, revealing an organized internal structure rather than uniform dimensional loss. Comparing refinement with explicit modification of the problem formulation further shows that rank recovery in the reported setting requires a change in the structure of the observation problem itself. Here, "problem modification" refers to changes that alter the bilinear observation structure (e.g., admissible operator/state families or coupling constraints), in contrast to refinements that preserve the original formulation such as tolerance adjustment and numerical reparameterizations. Together, these results delineate limits of rank recovery in bilinear observation problems and clarify the distinction between numerical refinement and problem modification in accessing effective dimensional structure.
Bilinear observation problems arise in many physical and information-theoretic settings, where observables and states enter multiplicatively. Rank-based diagnostics are commonly used in such problems to assess the effective dimensionality accessible to observation, often under the implicit assumption that rank deficiency can be resolved through numerical refinement. Here we examine this assumption by analyzing the rank and nullity of a bilinear observation operator under systematic tolerance variation. Rather than focusing on a specific reconstruction algorithm, we study the operator directly and identify extended rank plateaus that persist across broad tolerance ranges. These plateaus indicate stable dimensional deficits that are not removed by refinement procedures applied within a fixed problem definition. To investigate the origin of this behavior, we resolve the nullspace into algebraic sectors defined by the block structure of the variables. The nullspace exhibits a pronounced but nonexclusive concentration in specific sectors, revealing an organized internal structure rather than uniform dimensional loss. Comparing refinement with explicit modification of the problem formulation further shows that rank recovery in the reported setting requires a change in the structure of the observation problem itself. Here, "problem modification" refers to changes that alter the bilinear observation structure (e.g., admissible operator/state families or coupling constraints), in contrast to refinements that preserve the original formulation such as tolerance adjustment and numerical reparameterizations. Together, these results delineate limits of rank recovery in bilinear observation problems and clarify the distinction between numerical refinement and problem modification in accessing effective dimensional structure.
Arc II — Observation Paper 4 Title Correlation Degradation, Rank Collapse, and Operational Horizons in Observational Physics Zenodo DOI 10.5281/zenodo.18188547 Description This paper develops a general operational theory of correlation loss in physical observation, showing how finite accessibility leads to systematic degradation of correlation rank across measurement channels. Rather than treating decoherence, noise, or irreversibility as dynamical effects, the analysis reframes them as consequences of constrained inference under finite observational capacity. Using rank as a protocol-independent measure of accessible correlation, the paper demonstrates that successive observations generically induce rank collapse: a reduction in the dimensionality of distinguishable joint states accessible to observers. When rank collapse becomes irreversible, an operational horizon forms, beyond which no experimental refinement can recover lost correlations. The framework applies uniformly to quantum measurement, classical signal processing, and cosmological observation. It clarifies why certain observational limits appear absolute despite differing experimental designs, and why increased resolution or repeated measurement cannot bypass fundamental accessibility constraints. By identifying horizons with rank collapse rather than spacetime boundaries, the paper unifies measurement limits, decoherence, and observational irreversibility within a single operational structure. Keywords correlation rank; decoherence; measurement limits; operational horizons; information loss; observational physics Notes Arc II — Observation (core synthesis paper) Introduces rank collapse as a universal operational phenomenon Horizons defined by irrecoverable correlation loss Applies to quantum, classical, and cosmological observations Bridges measurement theory and horizon physics
Translator This work addresses a foundational but rarely examined question in physics: under what conditions does an observable exist at all? Treating finite resolution as a structural feature rather than a technical limitation, the paper shows that observable existence is not guaranteed and must be assessed relative to the refinement structure induced by realistic probes. Within the framework of Finite Resolution Physics (FRP), a refinement-stability requirement is introduced as a necessary condition for observable existence. This requirement is independent of statistical typicality, dynamical modeling, or spacetime and field ontologies. Applying it to a realistic interferometric probe specified strictly at the acquisition level, the paper proves that commonly assumed scalar quantities such as interferometric phase and visibility fail to remain stable under admissible refinement paths and therefore do not exist as probe-invariant observables within FRP. In addition to the structural result, the paper discusses reported interferometric experiments in which extracted phase- or visibility-like quantities exhibit dependence on detector resolution, spatial averaging, or reconstruction procedures. Without modeling these experiments or predicting their detailed behavior, the FRP analysis clarifies why such non-stabilization is not paradoxical: once an observable fails to exist structurally, finite resolution imposes no requirement that numerical procedures applied to experimental data should converge. The analysis is complemented by a conceptual connection to Operational Reflexive Systems, which provide an algebraic account of operational realizability. Overall, the paper establishes a principled distinction between calculable quantities and observables, and shows how finite resolution alone can rule out observables rather than merely limit their precision.
Registration Theory proposes that physical reality begins not with objects situated in a pre-given spacetime, but with interactions that create and preserve distinguishable relations—registrations—from which objects, information, dimensionality, time, and effective geometry may emerge. This paper examines one finite-dimensional mathematical component of that broader program. Given a positive-semidefinite registration operator R and a fixed linear resolution map A_S associated with a system S, it studies the resolved operator D_S = A_S R A_S†. The paper establishes that D_S is well-defined, self-adjoint, and positive semidefinite; derives its factorization, rank identity, and rank bounds; and proves continuous dependence on perturbations of R and A_S. Under the additional assumption that A_S is contractive, the trace of D_S cannot exceed the trace of R, with equality when A_S is isometric on the support of R. The principal result concerns threshold effective rank: the number of eigenvalues of D_S above a specified threshold. This count is locally stable when the threshold is separated from the spectrum by a positive gap. Explicit counterexamples demonstrate why the stated assumptions and strict perturbation bounds matter. The results do not derive spacetime, physical dimensionality, Lorentzian signature, or an empirical law. The threshold effective rank is a spectral statistic rather than a physical spatial dimension. The paper instead establishes a limited mathematical prerequisite for future work connecting system-relative resolution to effective dimensionality and geometry. This version revises and supersedes v1, previously titled “Existence and Stability of Finite Pre-Geometric Resolution Structures in Registration Theory.” The core construction and theorem package are unchanged, while the trace theorem has been refined to match the verified result and the exposition, framing, limitations, and physical motivation have been substantially revised.
When does operational distinguishability contain sufficient information to identify physical structure? We address this question through a resource-relative framework of Operational Informational Distinguishability (OID), designed to investigate the mathematical consistency of identifiability across quantum states, information geometry, and physical processes. An operational observer is characterized by (\mathcal{D}=(\mathcal{A},\mathcal{M},\mathcal{R},\delta)), whose labelled outcome distributions define an operational signature and induce equivalence classes of physically indistinguishable realizations. We establish an exact reconstruction criterion: a proposed geometric assignment factors uniquely through the operational signature if and only if it is constant on these equivalence classes. This provides a necessary-and-sufficient condition for operational reconstruction, while separating identifiability from the additional requirements of local non-degeneracy and stability. For faithful finite-dimensional quantum-state manifolds, we establish the measurement–information-geometric hierarchy[g^{(\mathcal{M})}\preceq g^{\mathrm{SLD}}\preceq g^{\mathrm{BKM}}relating measurement-restricted Fisher information, the symmetric logarithmic derivative metric, and the Bogoliubov–Kubo–Mori metric. We clarify the directional attainability of the SLD bound and its distinction from simultaneous multiparameter attainability. As an explicit algebraic result, we characterize the operator span generated by diagonal phase modulation of cyclic-shift eigenstate projectors in the one-excitation sector: it is precisely the constant-diagonal Hermitian subspace, of dimension (N^{2}-N+1), leaving (N-1) independent population directions unresolved. Adjoining site-basis projectors restores informational completeness. This idealized construction is connected to experimentally realized entangled measurements of (W) states. We further extend operational distinguishability to quantum processes through tester-restricted signatures and divergences, explicitly separating signalling from intervention-defined causal influence. Finally, we introduce accumulated information-geometric path length as a candidate operational clock parameter, without identifying it with physical time. By relating these constructions to holographic information-distance bounds, bulk entanglement, conditional mutual information and recovery, and recent approaches to emergent geometry and informational time, we formulate a central testable question: can a common resource-relative distinguishability structure remain identifiable, stable, and compositionally consistent across states, processes, geometry, temporal ordering, and causality? The framework makes both constructive reconstruction and mathematically explicit obstructions possible, providing a falsifiable methodological program rather than presupposing the emergence or unification of physical structure.
This document is part of the Informational Mechanics (IM) framework and is published in the IM-D (Core Physics) series within the IM v0.3 corpus. The paper develops a structural account of quantum entanglement under regime-limited admissibility constraints. Rather than introducing new dynamics, hidden variables, or collapse mechanisms, it analyzes entanglement as a consequence of structural non-factorizability arising from bilinear composition and tensor minimality within admissible operator regimes. Building on IM-B004 (tensor minimality), IM-C004 (regime-conditional structural principles), and IM-E004 (operator-defined outcome weights), the analysis shows that Bell-inequality violations constrain classical factorizability assumptions without implying superluminal signaling or ontological inseparability. The account preserves the formal structure of quantum mechanics and treats locality and no-signaling conditions as operator-level constraints. No new axioms, empirical fits, or modifications to established physical theories are introduced. The contribution is classificatory and structural: it clarifies how entanglement and Bell correlations emerge within regime-bounded admissibility while explicitly identifying the structural assumptions under which the account would require revision.
We introduce a contrast-based recoverability diagnostic for relativistic and mediated-entanglement experiments. The diagnostic compares a homogeneous reference spectrum with spectra obtained after controlled parameter variation, species change, hidden-mode access, finite-record feedback, or null intervention. In the minimal two-qubit setting, proper-time and transport effects are locally recoverable dressings. For n-constituent configuration sectors, one-round k-block recovery gives a scale-resolved spectrum. A diagonal phase is exactly recoverable at block size k precisely when it factorizes over a partition whose blocks have size at most k. Homogeneous occupation shells provide the symmetry null; heterogeneous sectors test the factorization of the observed phase. The four-constituent, four-level case is the smallest compact example with nontrivial block hierarchy, shell leakage, and constituent ambiguity. A staged fermion--fermion (FF)/boson--fermion (BF) comparison separates ordinary parameter nonidentity from species and statistics constraints. Extensions to unused levels, redundant modules, and larger connected clusters remove specific diagnostic blind spots. A finite classical record is bounded by a record-overlap contraction theorem, while a coherent hidden mode is not. The framework organizes which observed phases, visibility losses, and mediated-entanglement witnesses are locally recoverable, block-recoverable, hidden-mode recoverable, or record-limited.