This preprint introduces a dynamical qutrit coherence-cycle witness based on calibrated pairwise visibility measurements in a three-path interferometer. The central idea is that pairwise visibility data contain a geometry of normalized qutrit coherences that is not captured by scalar coherence measures alone. For nonzero path populations, the population-calibrated two-path visibility equals the normalized off-diagonal coherence of the reduced path qutrit. The squared normalized coherences define the cycle expression Scyc = kappa_12^2 + kappa_23^2 - kappa_13^2. Classical jointly diagonalizable overlap geometries obey the boundary Scyc <= 1. A statistically certified value Scyc > 1 therefore indicates a nonclassical cyclic arrangement of pair coherences, rather than merely a large amount of coherence. The paper proves the physical qutrit range Scyc <= 5/4, gives explicit positive semidefinite qutrit examples with equal scalar coherence but different cycle geometry, constructs a completely positive random-unitary dephasing model with time-dependent geometry windows, and formulates a finite-data certification protocol including calibration and covariance propagation. The static overlap-cycle inequality is imported from prior overlap-cycle work; the new contribution is its calibrated realization as a path-qutrit coherence-geometry witness. The proposed witness is intended as a measurement-efficient diagnostic of normalized coherence geometry in multipath interferometry and open qutrit dynamics. It is not a full state-tomography method and does not by itself constitute a complete contextuality experiment; stronger interpretations require the additional operational assumptions stated in the manuscript.
This preprint introduces a dynamical qutrit coherence-cycle witness based on calibrated pairwise visibility measurements in a three-path interferometer. The central idea is that pairwise visibility data contain a geometry of normalized qutrit coherences that is not captured by scalar coherence measures alone. For nonzero path populations, the population-calibrated two-path visibility equals the normalized off-diagonal coherence of the reduced path qutrit. The squared normalized coherences define the cycle expression Scyc = kappa_12^2 + kappa_23^2 - kappa_13^2. Classical jointly diagonalizable overlap geometries obey the boundary Scyc <= 1. A statistically certified value Scyc > 1 therefore indicates a nonclassical cyclic arrangement of pair coherences, rather than merely a large amount of coherence. The paper proves the physical qutrit range Scyc <= 5/4, gives explicit positive semidefinite qutrit examples with equal scalar coherence but different cycle geometry, constructs a completely positive random-unitary dephasing model with time-dependent geometry windows, and formulates a finite-data certification protocol including calibration and covariance propagation. The static overlap-cycle inequality is imported from prior overlap-cycle work; the new contribution is its calibrated realization as a path-qutrit coherence-geometry witness. The proposed witness is intended as a measurement-efficient diagnostic of normalized coherence geometry in multipath interferometry and open qutrit dynamics. It is not a full state-tomography method and does not by itself constitute a complete contextuality experiment; stronger interpretations require the additional operational assumptions stated in the manuscript.
A reduced qutrit does not only lose coherence under environmental coupling; it loses coherence with a geometry. This preprint shows that the pairwise coherence structure of a qutrit contains a directly accessible fingerprint of environmental distinguishability. By normalizing the three pair coherences with respect to the corresponding populations, one obtains a reduced-state map that identifies which pairs of qutrit alternatives remain mutually indistinguishable, which pairs are resolved by the environment, and which sectors may be protected or rephased. The method is deliberately economical. It does not attempt full bath reconstruction, process-tensor tomography, or spectral noise reconstruction. Instead, it uses only the reduced qutrit populations and pairwise coherences, or equivalently calibrated pairwise visibilities in an interferometric implementation. The resulting three-component distinguishability map gives the environmental structure as seen by the qutrit. Under additional model assumptions it can be interpreted as an environmental record-overlap map, a scalar coupling-distance diagnostic, a vector-noise metric, or a pair-selective memory fingerprint. The reconstruction is relational rather than microscopic. Different reservoirs may be indistinguishable to the qutrit if they generate the same reduced coherence geometry. The paper therefore identifies the equivalence class of environmental structures visible to the three-level probe. Explicit examples illustrate scalar Gaussian dephasing, protected pairs, vector or correlated-noise geometry, pair-selective recoherence, and safe analysis windows when populations or coherences approach a noise floor.
We place several known indicators of Kochen--Specker contextuality -- the KCBS correlator χ, the contextual fraction $\CF$, the Shannon-entropic n-cycle inequality of Chaves and Fritz, and the operational commutator witness D of Paper~I -- into a single projector-geometric framework organized around the overlap matrix $\Tcal_{ij} = d^{-1}\tr[(\hat P_i \hat Q_j)^2]$, where P^i and Q^j are the joint-eigenspace projectors of the two compatible observable pairs within a measurement context. The state-independent scalar content of $\Tcal$ is carried by two independent contractions: the mutual information energy $E = \sum_{ij}\Tcal_{ij}$ of Paper~I (equivalently, its logarithmic form S2=−log2E), and the Maassen--Uffink extremal overlap $c_\MU = \max_{i,j}|\langle a_i,b_i | c_j,b_j\rangle|$. We prove that S2 is non-increasing under coarse-graining, that $S_2(\Gcal) > 0$ is a necessary configuration-level condition for observable contextuality, and that the additive composition $S_2(\Gcal) = \sum_αS_2(\Gcal_α)$ is exact for the KCBS pentagon. We further show that in the spin-1 realization of the KCBS pentagon, a shared ms=0 eigenstate in each context forces $c_\MU = 1$, rendering every Maassen--Uffink-type bound trivial -- a structural mechanism that makes explicit why outcome-entropic uncertainty relations based on $c_\MU$ are silent on KCBS contextuality, while S2≈2.7266~bits throughout. Applied to KCBS and CHSH, the framework identifies regimes in which every state-dependent witness considered here is silent yet $S_2(\Gcal) > 0$ by an amount set by the projector geometry alone.
We identify the qutrit as the minimal Hilbert-space dimension in which coherence amountand coherence sharing become operationally distinguishable. The starting point is theexact purity decomposition into population predictability, Hilbert–Schmidt coherence, andmixedness. Although this closure is a state-space identity, in dimension three it selects aEuclidean coherence coordinate that can be compared nontrivially with normalized ℓ1coherence. Their mismatch defines a positive deficit Δℓ1 that vanishes if and only if thethree pairwise coherence moduli are equal. The deficit therefore diagnosescoherence-sharing anisotropy rather than total coherence magnitude alone. We show thatΔℓ1 separates two physically distinct mechanisms of coherence reduction: populationanisotropy in symmetric channels and branch asymmetry in balanced qutrits. In a purifiedqutrit–meter model the same Euclidean language acquires a dynamical interpretationbecause reduced mixedness equals normalized linear-entropy entanglement across thesystem–meter cut. We clarify the resource-theoretic status of the diagnostic by showingthat it is not a coherence monotone, but a witness of unequal sharing among pairwisechannels. Finally, we give a reduced-state reconstruction protocol based on populationsand three embedded SU(2) Ramsey blocks, supported by finite-sample simulations. Thesame quantity has a direct three-path interferometric implementation in terms of pairwisefringe visibilities and calibrated path intensities, where it measures anisotropicmutual-coherence sharing rather than a universal violation of classical coherence theory.
We investigate the entanglement structure of bipartite two-qutrit pure states from both geometric and operational perspectives.Using the eigenvalues of the reduced density matrix, we analyze how symmetric polynomials characterize pairwise and genuinely three-level correlations. We show that the determinant of the coefficient matrix defines a natural, rank-sensitive geometric invariant that vanishes for all rank-2 states and is nonzero only for rank-3 entangled states. An explicit analytic constraint relating this determinant-based invariant to the I-concurrence is derived, thereby defining the physically accessible region of two-qutrit states in invariant space. Furthermore, we establish an operational correspondence with three-path optical interferometry and analyze conditional visibility and predictability in a qutrit quantum erasure protocol, including the effects of unequal path transmittances. Numerical demonstrations confirm the analytic results and the associated complementarity relations. These findings provide a unified geometric and operational framework for understanding two-qutrit entanglement.
This manuscript presents a complete scalar reformulation of quantum mechanics, replacing probabilistic postulates with mechanistic constructs derived from scalar motion, Φ(x^μ) coherence thresholds, and curvature geometry. The central object is the scalar field , defined as a composite of log-amplitude and phase curvature, which governs collapse, entanglement, quantization, and evolution. Collapse is reframed as a deterministic threshold event—coherence exhaustion rather than stochastic projection. Entanglement emerges from resonance between coherence shells, quantified by a novel Scalar Resonance Monotone (SRM) that is basis-optimized, decoder- agnostic, and directly estimable from measurement records. Quantization arises from curvature-bound eigenstates, and time evolution is modeled as spiral motion within coherence-preserving domains. The manuscript introduces three operational constructs that extend scalar doctrine into experimental and simulation domains: • Coherence-Proximal Collapse Dynamics (CPCD): A scalar partial differential equation with a thresholded projection term, modeling collapse as a first-passage event in coherence space. • Scalar Resonance Monotone (SRM): A new entanglement witness based on coherence– coherence correlation, accompanied by a scalar CHSH inequality and a monogamy conjecture. • Threshold-Warped Scalar Geometry: A disformal metric whose curvature responds to coherence proximity, enabling predictions of time dilation and deflection in quantum simulators. The manuscript further develops scalar topology, measurement theory, and audit protocols. Shell connectivity graphs, rupture surfaces, and reinjection fidelity metrics provide scalar-native tools for analyzing coherence structure and collapse dynamics. Collapse verdicts are now scored with statistical confidence, compared against standard quantum predictions, and encoded geometrically via shell curvature. All constructs are dimensionally anchored, ritualized, and testable without full state tomography. The manuscript formalizes scalar grammar, symbol taxonomy, and threshold syntax, offering a challenge-ready framework for post-quantum mechanics. This work establishes a reference for scalar theorists, coherence engineers, and curvature modulating agents. It invites experimental validation, simulator implementation, and recursive extension into new domains of physics.
This manuscript presents a complete scalar reformulation of quantum mechanics, replacing probabilistic postulates with mechanistic constructs derived from scalar motion, Φ(x^μ) coherence thresholds, and curvature geometry. The central object is the scalar field , defined as a composite of log-amplitude and phase curvature, which governs collapse, entanglement, quantization, and evolution. Collapse is reframed as a deterministic threshold event—coherence exhaustion rather than stochastic projection. Entanglement emerges from resonance between coherence shells, quantified by a novel Scalar Resonance Monotone (SRM) that is basis-optimized, decoder- agnostic, and directly estimable from measurement records. Quantization arises from curvature-bound eigenstates, and time evolution is modeled as spiral motion within coherence-preserving domains. The manuscript introduces three operational constructs that extend scalar doctrine into experimental and simulation domains: • Coherence-Proximal Collapse Dynamics (CPCD): A scalar partial differential equation with a thresholded projection term, modeling collapse as a first-passage event in coherence space. • Scalar Resonance Monotone (SRM): A new entanglement witness based on coherence– coherence correlation, accompanied by a scalar CHSH inequality and a monogamy conjecture. • Threshold-Warped Scalar Geometry: A disformal metric whose curvature responds to coherence proximity, enabling predictions of time dilation and deflection in quantum simulators. The manuscript further develops scalar topology, measurement theory, and audit protocols. Shell connectivity graphs, rupture surfaces, and reinjection fidelity metrics provide scalar-native tools for analyzing coherence structure and collapse dynamics. Collapse verdicts are now scored with statistical confidence, compared against standard quantum predictions, and encoded geometrically via shell curvature. All constructs are dimensionally anchored, ritualized, and testable without full state tomography. The manuscript formalizes scalar grammar, symbol taxonomy, and threshold syntax, offering a challenge-ready framework for post-quantum mechanics. This work establishes a reference for scalar theorists, coherence engineers, and curvature modulating agents. It invites experimental validation, simulator implementation, and recursive extension into new domains of physics.