We present the genuine, defensible result of the Δ-Gate program as a tool, not a law: a single soft-threshold operator that makes the character of threshold and decoherence transitions determinable from data. Three things are claimed and no more. First, secured: the Δ-Gate measurement channel embeds exactly into a GKSL/Lindblad dephasing generator, recovering the Born statistics as the diagonal of the decohered state — that is, by decoherence, not collapse. Second, owned and new: a two-Fisher uncertainty relation relating the location-Fisher (entropy-arrow) and the parameter-Fisher (gate-sharpness) of any threshold density. Third, a canonical information-geometric framework (Fisher-information substrate, Fisher–Rao length, the 2D-Ising self-dual invariant, and the fluctuation–dissipation relation) that is correct textbook structure, honestly tagged as found, not produced. From these we assemble a diagnostic toolkit that determines a threshold's dynamical class, bifurcation type, and static-to-dynamic coupling. The limits are explicit; the universality claims of the original program are refuted separately in the companion audit (Paper II).
We present the genuine, defensible result of the Δ-Gate program as a tool, not a law: a single soft-threshold operator that makes the character of threshold and decoherence transitions determinable from data. Three things are claimed and no more. First, secured: the Δ-Gate measurement channel embeds exactly into a GKSL/Lindblad dephasing generator, recovering the Born statistics as the diagonal of the decohered state — that is, by decoherence, not collapse. Second, owned and new: a two-Fisher uncertainty relation relating the location-Fisher (entropy-arrow) and the parameter-Fisher (gate-sharpness) of any threshold density. Third, a canonical information-geometric framework (Fisher-information substrate, Fisher–Rao length, the 2D-Ising self-dual invariant, and the fluctuation–dissipation relation) that is correct textbook structure, honestly tagged as found, not produced. From these we assemble a diagnostic toolkit that determines a threshold's dynamical class, bifurcation type, and static-to-dynamic coupling. The limits are explicit; the universality claims of the original program are refuted separately in the companion audit (Paper II).
We introduce a one-parameter quantum channel — the gate channel — that continuously interpolates between unitary evolution (K=0) and projective measurement (K=1) via K(t) = 1 − exp(−Γt). Proof that the continuum limit recovers the GKSL/Lindblad master equation for dephasing exactly. Extensions to pipeline structure, general Kraus operators, and multi-environment coupling. Three falsifiable experimental protocols proposed: double-slit interferometry, NMR spin-echo decay, and polarization-state degradation. Paper II of VII in the Δ-Gate theoretical core. Part of the Δ-Gate Formalism (DOI: 10.5281/zenodo.19103765).
Introduces the coherence-time density operator ρ_K(t) as a spatially integrated, time-resolved threshold observable for quantum gate stability. Central result: a Δ-Gate remains active iff ρ_K(t) ≥ ρ_K,min. Proves Lindblad dephasing recovery in the spatially uniform limit (Proposition 2). Derives explicit collapse threshold for distance-d repetition codes (Proposition 3). Embeds all five companion papers (phase coherence, entangled tunneling, gate channel, Heisenberg-Gate, Omega operator) as projections or limits. Relativistic extension: gate lifetime dilates under Lorentz boosts with frame-invariant threshold condition. Seven falsifiable predictions proposed. Paper VI of VII — theoretical capstone of the Δ-Gate series. Part of the Δ-Gate Formalism (DOI: 10.5281/zenodo.19103765).
Abstract We identify a parameter-free instability within GKSL (Lindblad) dynamics: collapse occurs when the local von Neumann entropy IQ(x,t) exceeds the steady-state entropy D0≡S(ρss), a threshold determined entirely by the Hamiltonian and Lindblad operators. We prove this entropy overshoot is equivalent to irreversible record formation in the environment via a system--environment duality theorem. The Born rule pn=∣⟨n∣ψ⟩∣2 follows uniquely from Boltzmann weighting over quantum relative entropy to pointer states. Three auxiliary predictions are tested against >1010 measurements from public datasets and falsified, eliminating a global memory field while leaving the core criterion intact. The nulls sharpen the framework into a constraint theory: superposition is the default of a memoryless substrate; collapse occurs when record-forming subsystems force irreversible information extraction past IQ>S(ρss). On IBM Quantum hardware (Torino, 133-qubit Heron processor), the overshoot magnitude scales with the channel parameter T2/T1 across eight qubits (R=0.991, zero free parameters), independently validated in public data from ETH Z\"{u}rich. A 2.5σ signal for phase-dependent overshoot---observed on IBM Torino and replicated in the ETH data---motivates deterministic outcome selection via coherence-phase branching at the bifurcation threshold. A measurement-axis test using two qubits with different Lindblad operators yields modulation axes separated by 72.3∘, inconsistent with a global hardware artifact.
Dynamical collapse models secure the occurrence of definite outcomes, and decoherence theory explains the dynamical stability of preferred bases. Two structural questions are frequently run together in discussions of what these accounts leave open: which set of channels serves as the collapse basis, and what distinguishes the realised outcome among the admissible ones. We separate the two and give each a variational formulation that is provably consistent with standard quantum statistics. For the second question we define the outcome cost of a target outcome law as the minimal relative entropy, on the path space of the collapse dynamics, of any trajectory measure realising that law. We prove that this cost equals the Kullback–Leibler divergence from the Born distribution, so that the Born law is the unique zero-cost outcome law and the cost of a single outcome k is its surprisal −log pk. For continuous quantum-non-demolition collapse we show that the least-cost dynamics conditioned on outcome k is the Doob h-transform of the collapse dynamics, and that −log pk equals the expected integrated squared steering drift. For the first question we prove that, under pure-dephasing system–environment coupling, the coarsest channel set with zero decoherence cost is the eigen-decomposition of the coupling observable, and we state a compatibility condition that collapse operators must meet to share this basis. Finally, we show that sharp-threshold mathematics applies naturally to the registration of an outcome in redundant environmental records, not to the selection of the outcome itself. The framework introduces no new dynamics, does not derive the Born rule, and makes no empirical predictions beyond those of the underlying collapse model.
Abstract Fisher information is a central diagnostic in quantum readout, sensing, and open-system dynamics, but a positive Fisher derivative is not by itself a non-Markovianity witness. This paper does not introduce a new universal monotonicity measure. Instead, it establishes an encoding-aware diagnostic rule for deciding when Fisher growth can be interpreted as information backflow and when it should be read as ordinary parameter imprinting or measurement readability. The rule separates three operational layers: input-encoded optimized Fisher geometry, channel-parameter Fisher information, and fixed-readout Fisher information. For input-encoded parameters propagated by parameter-independent CP-divisible dynamics, optimized quantum Fisher information has the expected Lyapunov contraction interpretation; a revival is therefore a genuine backflow candidate. For channel-encoded parameters, Fisher information can grow even under a Markovian semigroup because the dynamics progressively writes the process parameter into the output state. A calibrated binary readout makes the measurement layer explicit. Dephasing, amplitude-damping, and two-qubit X-state examples then show the quantum-information payoff: Fisher readability, coherence, concurrence, and complementarity coincide in special calibrated sectors but separate in general. In particular, amplitude-damped X-state slices contain regions with vanishing concurrence but nonzero phase Fisher readability. The resulting criterion is an interpretation rule, not a replacement for established non-Markovianity witnesses: Fisher growth becomes causally meaningful only after the encoding, dynamical map, and readout layer have been fixed.
We work in the universal information hydrodynamics (UIH) framework developed in two companion papers, where reversible quantum dynamics and irreversible response are organised in a local metriplectic geometry generated by a convex free energy, a Fisher metric, and an antisymmetric no work channel. In this setting the irreversible drift is always a Fisher gradient flow in a weighted H−1 geometry, equipped with a cost entropy inequality and a curvature coercivity bound that are invariant under the addition of any compatible reversible dynamics. In the present paper we show that this structure has a canonical realisation for Fokker Planck equations, finite Markov chains and GKLS semigroups, and that it can be reconstructed operationally from process and state tomography on real quantum hardware. First, starting from overdamped Langevin dynamics and its Fokker Planck generator, we identify a canonical Fisher Dirichlet operator G=Qdiag(π) that plays the role of the symmetric mobility on a periodic lattice and saturates the cost entropy inequality mode by mode. We then demonstrate that in a wide class of thermal and driven GKLS models the symmetric part of the real generator in the stationary BKM metric has a density block that coincides exactly with this classical Fisher operator, including nonreversible chains and random qubit and qutrit ensembles, and that the global decay of entropy and Fisher energy is governed by the Markov spectral gap while the Fisher gap controls only local curvature. Finally, we implement a suite of IBM Quantum experiments on a superconducting qubit that realises the full UIH picture on hardware. Process tomography of idle circuits yields an effective real generator K in the Pauli basis whose metric adjoint split in the BKM metric at the device stationary state produces a clean metriplectic decomposition K=G+J, with G metric symmetric and J metric skew. Independent idle depths are consistent with a single time homogeneous generator, the dissipative spectrum of G sets an information theoretic speed limit for BKM Fisher decay, and a curvature test shows that the same BKM metric is the local Hessian of quantum relative entropy at the stationary state. All claims are backed by a reproducible Python archive that includes constructive Fokker Planck to Markov to GKLS lifts, hypocoercive decay diagnostics, and hardware K tomography. Taken together, the results in this paper show that this programme is not only mathematically coherent, but experimentally visible, showing strong signatures of universality.