This work introduces an operational geometric framework for analyzing Eisenbud–Wigner–Smith (EWS) time delay in non-smooth quantum regimes. Without modifying standard quantum mechanics, we define scale-dependent geometric proxies compatible with non-smooth metric-measure structures and connect them to spectral shift mechanisms via Krein theory. Two scaling regimes are distinguished and expressed through a falsifiable law τ(E;ℓ) ∝ (1 + X(ℓ))^γ. The framework explicitly preserves unitarity, no-signaling, and relativistic causality. Concrete falsifiability criteria (Types I–III) are formulated. Order-of-magnitude comparison with attosecond chronoscopy suggests compatibility with observed delay enhancements in reduced-dimensional systems. The work introduces no new fundamental fields; it reframes time-delay observables in terms of coarse-grained geometric descriptors.
This work introduces an operational geometric framework for analyzing Eisenbud–Wigner–Smith (EWS) time delay in non-smooth quantum regimes. Without modifying standard quantum mechanics, we define scale-dependent geometric proxies compatible with non-smooth metric-measure structures and connect them to spectral shift mechanisms via Krein theory. Two scaling regimes are distinguished and expressed through a falsifiable law τ(E;ℓ) ∝ (1 + X(ℓ))^γ. The framework explicitly preserves unitarity, no-signaling, and relativistic causality. Concrete falsifiability criteria (Types I–III) are formulated. Order-of-magnitude comparison with attosecond chronoscopy suggests compatibility with observed delay enhancements in reduced-dimensional systems. The work introduces no new fundamental fields; it reframes time-delay observables in terms of coarse-grained geometric descriptors.
Scope (minimal thesis): The record contains (i) the main preprint (PDF), (ii) LaTeX source (fully reproducible), and (iii) an extended “explainer” PDF (intuition-first, minimal prerequisites, with figures and tables). Key idea: effective degradation of geometric regularity—defined without smooth curvature—can act as a temporal bottleneck, increasing measurable quantum time delays. Operational proxies: (1) volume-growth/ball-measure deficit proxy , and (2) spectral/heat-kernel descriptors including spectral dimension . Mechanism anchor: geometry-induced spectral modifications can shift scattering phase through spectral shift theory (Kreĭn / Birman–Yafaev viewpoint), hence modifying EWS delays. Test program: controlled families where a geometric scale is varied at fixed probe conditions; fit versus and apply falsifiability Types I–III. Non-claims: no new gravity, no QM violation, no “free energy”, no universal constant assumed a priori. Reproducibility: theoretical/conceptual; no proprietary data required.
We develop a metric-measure geometric reinterpretation of quantum entanglement in which nonlocal correlations are encoded in the intrinsic geometry of composite state space rather than in dynamical propagation. Using the Bures metric and CPTP coarse-graining semigroups, we define a scale-dependent effective dimension of the entangled-state manifold. We propose that geometric observables along the renormalization flow may exhibit long-range dependence (LRD) characterized by a Hurst exponent H ∈ (1/2,1). A conjectured infrared attractor H* ≈ 0.65 is formulated as a falsifiable hypothesis. The framework preserves no-signaling and does not modify quantum mechanics. A direct link to the variance (RMS) of Eisenbud–Wigner–Smith time delay is proposed, providing operational accessibility of geometric LRD effects.
A generalized formulation of non-relativistic quantum mechanics is developed within multidimensional geometric (NG) frameworks characterized by a power-law dispersion relation E∝∣p∣j, where j=N−1. Starting from the generalized Minkowski distance in Lj-normed spaces, the conventional quadratic kinetic structure of three-dimensional geometry is extended to higher-order spatial derivatives, yielding a consistent j-th order Schrödinger equation. The formalism is applied to free particles and to particles confined within a one-dimensional infinite potential well for 2G, 3G, 4G, and 5G geometries. While plane-wave solutions and translational invariance are preserved, the spectral structure is modified, with bound-state energies scaling as (2n+1)j, leading to cubic and quartic growth in higher geometries. The corresponding eigenfunctions exhibit mixed exponential, trigonometric, and hyperbolic forms determined by the roots of negative unity. A generalized probability framework based on j-fold conjugation is introduced, ensuring a real-valued probability density and consistent expectation values. Despite these generalizations, the Heisenberg uncertainty principle is preserved. The formulation presents quantum mechanics as a geometry-dependent theory in which dispersion relations, spectral properties, and probabilistic structure emerge from the underlying spatial metric.
This work presents a strengthened formulation of Delayed Definability Field Theory (DDFT), a unification architecture in which spacetime geometry, quantum probabilistic structure, and gauge dynamics emerge from a single pre-geometric principle: the stabilization and delayed realization of definable relational difference. The framework is designed to address four major weaknesses that commonly limit foundational unification attempts. First, the definability field is no longer treated as a symbolic placeholder, but is defined on a measurable relational substrate as a bounded symmetric field on pair space. Second, the continuum problem is no longer hidden inside an unspecified coarse-graining map; instead, the theory introduces discrete closure geometry on simplicial approximants, a Regge-type closure-gravity action, and a controlled continuum route under bounded-defect and Γ-convergence assumptions. Third, the quantum sector is strengthened by replacing a purely schematic probability assignment with a projective-history distance formalism whose Fubini–Study limit yields exact Born-weight recovery. Fourth, the gauge sector is reformulated more carefully through closure-consistency constraints, anomaly freedom, chirality, and minimal redundancy, narrowing the admissible low-energy symmetry toward the Standard Model pattern. In DDFT, time is not primitive but emerges as the order of admissible realized presents. Geometry is not assumed but reconstructed from stabilized relational covariance. Gravity appears as the low-defect continuum compression of closure inconsistency, while quantum behavior appears when delayed realization remains non-unique across admissible histories. Gauge connections arise as the minimal structures required to preserve definability content under local representational redundancy. The present paper does not claim a final mathematically closed derivation of every physical sector. Instead, it provides a substantially more formal, explicit, and testable framework than earlier versions, with clearly identified assumptions, bottlenecks, and recovery limits. It also formulates concrete phenomenological windows, including energy-dependent propagation delays, curvature-induced interference asymmetries, black-hole information leakage, and delayed corrections to exact Schrödinger compression. The central thesis of the work is that spacetime, quantum probability, and gauge-consistent physical law can all be understood as different regimes of a single delayed definability principle. In this sense, DDFT is proposed not as a decorative synthesis, but as a more rigorous mother framework beneath relativity, quantum theory, and emergent field structure.
This work introduces a resolution-dependent geometric framework—Quantum Fractal Geometry (QFG)—in which the effective physical description of systems depends explicitly on observational scale. Within this framework, entangled quantum systems are modelled as unified geometric structures at deep resolution, characterised by a scale-dependent metric g_{μν}(ℓ) and a geometric phase field φ(x,ℓ) valued on S¹. Measurement is interpreted as a resolution-dependent projection of this unified structure onto detector-scale observables. A symmetry-constrained coarse-graining map is introduced as the minimal fixed point of resolution flow, identifying antipodal phase configurations (S¹ → S¹/ℤ₂). Under isotropy, continuity, and convex-linearity, the quantum correlation function E(a,b) = −cos(θ_a − θ_b) is recovered as the unique rotationally invariant statistic compatible with the projection structure, saturating the Tsirelson bound. Within QFG, Bell inequality violation arises not from nonlocal signalling or measurement dependence, but from the failure of scale-independent separability. Locality (in the operational no-signalling sense) and measurement independence are preserved. The framework further outlines a programme for deriving the Born rule from resolution-induced measure structure and identifies falsifiable predictions, including resolution-dependent deviations from ideal cosine correlations. This work should be understood as establishing the minimal geometric structure required for the QFG interpretation of Bell correlations, with full dynamical derivations deferred to future work.