This work introduces the geodesometry of coherence, a minimal discrete dynamical extension of the Hamming Decay Theorem in quantum state spaces. The Hamming Decay law establishes that off-diagonal elements of a density matrix decay exponentially with Hamming distance under local noise channels. However, this metric does not account for the physical structure through which coherence is lost. In this work, the Hamming hypercube is endowed with anisotropic physical weights on its edges, representing local noise, coupling asymmetry, or entropic cost. A minimal 3-qubit model with non-uniform weights (3, 2, 1) demonstrates that pairs of states with equal Hamming distance can exhibit different accumulated transition costs, leading to a falsifiable prediction: equal Hamming distance does not necessarily imply equal decoherence rate under anisotropic conditions. Explicit GO/KILL conditions are defined, and a minimal experimental protocol is proposed. The formulation is deliberately restricted to the quantum discrete sector and does not claim extension to gravity, cosmology, or gauge theories. It is presented as a falsifiable intermediate step between: Hamming geometry (metric structure), geodesometry (discrete dynamics), and a potential continuous effective action Seff[C]S_{\mathrm{eff}}[C]Seff[C] within the Radial Coherential Dynamics (RCD) program. This work is intended as a minimal, testable framework rather than a complete theory.
This work introduces the geodesometry of coherence, a minimal discrete dynamical extension of the Hamming Decay Theorem in quantum state spaces. The Hamming Decay law establishes that off-diagonal elements of a density matrix decay exponentially with Hamming distance under local noise channels. However, this metric does not account for the physical structure through which coherence is lost. In this work, the Hamming hypercube is endowed with anisotropic physical weights on its edges, representing local noise, coupling asymmetry, or entropic cost. A minimal 3-qubit model with non-uniform weights (3, 2, 1) demonstrates that pairs of states with equal Hamming distance can exhibit different accumulated transition costs, leading to a falsifiable prediction: equal Hamming distance does not necessarily imply equal decoherence rate under anisotropic conditions. Explicit GO/KILL conditions are defined, and a minimal experimental protocol is proposed. The formulation is deliberately restricted to the quantum discrete sector and does not claim extension to gravity, cosmology, or gauge theories. It is presented as a falsifiable intermediate step between: Hamming geometry (metric structure), geodesometry (discrete dynamics), and a potential continuous effective action Seff[C]S_{\mathrm{eff}}[C]Seff[C] within the Radial Coherential Dynamics (RCD) program. This work is intended as a minimal, testable framework rather than a complete theory.
Two recent results suggest Hamming distance organizes quantum coherence dynamics, but no unified framework connects them. We establish the Hamming Persistence Principle: quantum correlations at Hamming distance persist with amplitude scaling as e−κd, where κ depends on context but the geometric structure is universal. The principle unies two independent discoveries: (1) In open systems under local noise, density matrix elements decay as ρij(t) = ρij(0)e−γH(i,j)ta theorem following from locality of interactions. (2) In isolated systems, Strasberg et al. observed numerically that o-diagonal elements of the decoherence functional correlate negatively with Hamming distance, but noted they had no elaborate theoretical explanation. We provide that explanation: both phenomena arise because physical interactions are local, making Ham-ming distancethe minimum number of single-bit operations connecting two congurationsthe dynamically privileged metric. Numerical verication conrms the framework: (i) Coherence magnitude decreases with Hamming distance(r =−0.26, p < 10−70), reproducing Strasberg's Figure 12. (ii) The decay coecient scales as α ≈0.5 ln D(r= 0.965), supporting the locality derivation. (iii) Maverick histories that deviate from Born statistics cluster at smaller Hamming distance (3/4 tests p < 0.05) and are 2×more coherent with each otherexplaining why they recohere together. The synthesis resolves why entanglement fails to predict decoherence rates: entanglement ignores Hamming geometry. It also suggests a geometric origin for branch selection: histories spread across Hamming space (large D∗H) are stable, while clustered histories (small D∗H) recohere away. The classical world may emerge not from environmental monitoring but from the intrinsic geometry of conguration space.
Book IV — Persistence is part of the Radial Coherential Dynamics (RCD) research program and focuses on the physical conditions under which coherence and order can persist in open systems subject to noise and dissipation. Building on cross-platform experimental results obtained in January 2026—where GHZ entangled states exhibited approximately 50% improved coherence retention across incompatible quantum hardware—this book introduces the concept of structural persistence: the idea that certain correlation geometries resist decoherence not by isolation or active correction, but by minimizing their entropic attack surface. The work reframes the central question from “How do we create coherence?” to “What kinds of coherence can survive?” and examines three interconnected elements:(i) empirical persistence thresholds (e.g., the observed ~99.5% fidelity threshold),(ii) thermodynamic openness via entropy sinks (including the cosmological horizon), and(iii) the role of state geometry in optimizing dissipation. No new experiments are presented. Instead, the book consolidates existing data, evaluates compatibility with established physics (Landauer’s principle, Gibbons–Hawking thermodynamics, Jacobson’s derivation of gravity), and explicitly delineates what is known, hypothesized, and not yet derived. Book IV does not claim a new theory of gravity, consciousness, or cosmology. It establishes a disciplined conceptual framework and a falsifiable research program, identifying open gaps and proposing concrete directions for future experimental and theoretical tests. This volume emphasizes methodological modesty: compatibility is not proof, but without compatibility, no path forward exists.
Version: v1.0 DESCRIPTION: We introduce Coherence Field Dynamics (CFD), a theoretical framework that extends holographic quantum gravity by promoting quantum coherence — the off-diagonal density matrix elements — to an independent geometric degree of freedom. While standard holographic frameworks (AdS/CFT, Ryu-Takayanagi) encode bulk geometry through entanglement entropy alone, CFD posits that coherence parameterizes effective bulk depth, complementary to entanglement's encoding of connectivity. The coherence field φ(p,γ) satisfies a wave equation in information-geometric parameter space, generating a Lorentzian metric that degenerates at a critical decoherence threshold γ_c. The framework predicts the existence of such a threshold from geometric stability analysis; its specific numerical value is system-dependent. Connections are established between information geometry, MERA tensor networks, and holographic entropy. The variational principle coupling coherence to an effective gravitational action is presented as an analogue gravity model (c and G are effective parameters), not a claim about physical 4D spacetime. Key results: Theoretical framework: coherence field equation, information-geometric metric, variational principle, and five testable predictionsCompanion experimental work demonstrates a fidelity threshold at γ ≈ 0.535 on IonQ Forte-1 (F = 0.988 baseline, collapse to F ≈ 0 under dephasing) and consistency with the dephasing response on Pasqal FRESNEL_CAN1 QPU (noise ratio 1.13×). These results are consistent with CFD's predicted critical behavior, though they do not uniquely distinguish CFD from standard noise models. The specific threshold value (0.535) is determined by the experimental circuit geometry, not derived from CFD aloneFour predictions remain open and experimentally accessible on current NISQ hardware: coherence-depth complementarity, MERA-decoherence correspondence, coherence-modulated entropy scaling, and hardware demonstration of non-trivial reversibilityThis paper presents the theoretical framework. The detailed experimental protocols and hardware data are reported in the companion publication "Cross-Platform Noise Thresholds in Quantum Teleportation: Trapped-Ion and Neutral-Atom Architectures" (see Related Identifiers).
Quantum decoherence is traditionally modeled through empirical decay laws in which observable coherence decreases with time. While these models successfully describe the dynamics of open quantum systems, they do not by themselves explain the physical destination of the apparently lost coherence. Motivated by the Structured Quantum Noise (SQN) framework, which models noise as a structured quantum subsystem rather than a scalar attenuation parameter, this paper proposes an information-theoretic interpretation of decoherence as coherence redistribution into environmental degrees of freedom. We introduce the concept of a Noise Black Hole, a hierarchical environmental structure through which coherence and entanglement diffuse into progressively less accessible sectors while remaining globally conserved. A minimal coherence-retention model shows that the familiar exponential decoherence law emerges naturally from repeated information redistribution rather than being imposed phenomenologically. The framework further predicts that deviations from ideal exponential decay arise from differences in environmental structure, memory, coupling, and backflow pathways, suggesting that measured decoherence curves may provide indirect probes of otherwise inaccessible environmental Hilbert-space dynamics. Rather than modifying conventional open-system theory, the proposed framework provides a possible information-theoretic explanation for the observed form of decoherence while generating experimentally testable predictions.
Quantum decoherence—the loss of coherence in a quantum system due to environmental coupling—has been described since the 1980s as a continuous exponential decay of the off-diagonal elements of the density matrix. This mathematical description captures the statistical average of many events, not the underlying mechanism of decoherence itself. Based on the Structural Axiom System (Lin Xiaohei, 2026), this paper proposes: **decoherence is not a continuous, smooth decay, but a discrete process composed of indivisible micro-coupling events.** Each micro-coupling event corresponds to a structural unit—a binary asymmetric relation, the smallest configurational unit—taking shape between system and environment. Each step of decoherence removes one integer share of quantum coherence; macroscopic "gradual decoherence" is the statistical smoothing of many discrete steps. This paper provides: 1. A quantitative formula for the step height: κ = g × d, where g is the micro-coupling strength and d is the structural unit directionality, both independently calibratable. 2. A decomposable verification prediction: changing photon frequency → g changes, but d remains unchanged → the normalized step shape remains invariant. 3. A complete experimental design: single-photon path superposition state + precision environmental coupling + quantum state tomography. If this experiment succeeds, the structural unit will become the first directly observed "configurational quantum"—the empirical confirmation of the smallest unit of existence. This would be the ultimate resolution of the quantum-classical transition problem, comparable in significance to Brownian motion confirming the existence of atoms.
Random circuit sampling (RCS) on noisy quantum hardware is widely modeled as exhibiting exponential decay of fidelity metrics (e.g., XEB, HOG) with circuit depth, converging toward Haar-random statistics beyond a characteristic noise-limited scale. This work introduces a minimal phenomenological recurrence for a coarse-grained local coherence observable defined on the qubit connectivity graph. The recurrence reproduces standard exponential decay in unstructured (fully scrambled) circuits but predicts delayed thermalization and a nonzero plateau when repeated local motifs are engineered into circuit compilation. The model introduces a single topology-dependent memory parameter α(µ), where µ denotes motif density. A transcritical bifurcation occurs when α(µ) exceeds the effective per-layer decoherence rate γ, producing a calculable plateau height and build-up depth scale. A matched-pair experimental protocol—holding qubit count, two-qubit gate count per layer, and reported hardware error rates fixed while varying only motif density—provides a direct falsification criterion. Absence of statistically significant topology dependence would rule out the model in its minimal form. The recurrence is an effective coarse-grained description at the metric level and does not modify underlying quantum channel structure or violate complete positivity.