Complex-Time (CT) theory posits that physical time is not exhausted by a single real parameter t, but includes an internal rotation phase φ such that the time coordinate is τ = t + iφ. In this ontology, φ is not a mere mathematical artifact of complex amplitudes: it is a physical degree of freedom that mediates a rotation between two logically distinct time aspects—future-pending (fp) time that retains unrealized alternatives, and past-fixed (pf) time that corresponds to realized events and stable records. Dynamic quantum circuits with mid-circuit measurement (MCM) are an unusually direct laboratory for CT because they contain repeated, addressable fp→pf conversion events *inside a single circuit*.Here we analyze IBM superconducting quantum-processor data and show that the target-qubit equatorial coherence z ≡ ⟨σx⟩ + i⟨σy⟩ undergoes a deterministic complex rotation whose angle is set primarily by the number M of spectator mid-circuit measurements. In Dataset A (4096 shots; 5 repeats), the complex ratio w(M) ≡ z_MEAS(M)/z_DELAY(M) displays an event-additive phase shift Δθ_eff(M) = arg w(M) that grows from ≈ 0 at M = 0–1 to Δθ_eff(M=8) ≈ 2.279 rad, with an average increment of δ ≈ 0.323 rad per added measurement. In Dataset B (1024 shots), inserting a virtual-Z compensation Rz(comp×M) on the target yields linear response d(Δθ_eff)/d(comp) ≈ M and demonstrates coherent controllability of the rotation.To probe whether the effect is reducible to schedule-level artifacts alone, Dataset C (CT3 Layer3) uses identical classical control-flow (if-structure) while changing only the spectator’s internal phase coherence: mode A forces |+⟩ before each mid-measurement, whereas mode B allows a |+⟩/|−⟩ mixture to accumulate. A small but systematic phase difference Δθ_AB(M) = θ_A(M) − θ_B(M) emerges (≈ 0.06 rad at M = 4–8) alongside a large change in spectator ⟨σx⟩, consistent with an operational coupling between the driven phase rotation and an internal i-phase degree of freedom.We interpret these results as operational evidence that an i-axis phase coordinate is physically actionable: it can be incremented by pf-creating events and steered by coherent frame updates. This provides a concrete, hardware-grounded route for testing CT claims about the physical reality of the imaginary-time phase. Keywords: complex time; τ = t + iφ; i-phase; past-fixed / future-pending; dynamic circuits; mid-circuit measurement; complex-plane diagnostics; virtual-Z compensation.
Complex-Time (CT) theory posits that physical time is not exhausted by a single real parameter t, but includes an internal rotation phase φ such that the time coordinate is τ = t + iφ. In this ontology, φ is not a mere mathematical artifact of complex amplitudes: it is a physical degree of freedom that mediates a rotation between two logically distinct time aspects—future-pending (fp) time that retains unrealized alternatives, and past-fixed (pf) time that corresponds to realized events and stable records. Dynamic quantum circuits with mid-circuit measurement (MCM) are an unusually direct laboratory for CT because they contain repeated, addressable fp→pf conversion events inside a single circuit.Here we analyze IBM superconducting quantum-processor data and show that the target-qubit equatorial coherence z ≡ ⟨σx⟩ + i⟨σy⟩ undergoes a deterministic complex rotation whose angle is set primarily by the number M of spectator mid-circuit measurements. In Dataset A (4096 shots; 5 repeats), the complex ratio w(M) ≡ z_MEAS(M)/z_DELAY(M) displays an event-additive phase shift Δθ_eff(M) = arg w(M) that grows from ≈ 0 at M = 0–1 to Δθ_eff(M=8) ≈ 2.279 rad, with an average increment of δ ≈ 0.323 rad per added measurement. In Dataset B (1024 shots), inserting a virtual-Z compensation Rz(comp×M) on the target yields linear response d(Δθ_eff)/d(comp) ≈ M and demonstrates coherent controllability of the rotation.To probe whether the effect is reducible to schedule-level artifacts alone, Dataset C (CT3 Layer3) uses identical classical control-flow (if-structure) while changing only the spectator’s internal phase coherence: mode A forces |+⟩ before each mid-measurement, whereas mode B allows a |+⟩/|−⟩ mixture to accumulate. A small but systematic phase difference Δθ_AB(M) = θ_A(M) − θ_B(M) emerges (≈ 0.06 rad at M = 4–8) alongside a large change in spectator ⟨σx⟩, consistent with an operational coupling between the driven phase rotation and an internal i-phase degree of freedom.We interpret these results as operational evidence that an i-axis phase coordinate is physically actionable: it can be incremented by pf-creating events and steered by coherent frame updates. This provides a concrete, hardware-grounded route for testing CT claims about the physical reality of the imaginary-time phase. Keywords: complex time; τ = t + iφ; i-phase; past-fixed / future-pending; dynamic circuits; mid-circuit measurement; complex-plane diagnostics; virtual-Z compensation.
Abstract Complex-Time (CT) theory posits that physical time is not exhausted by a single real parameter t, but includes an internal rotation phase φ such that the time coordinate is τ = t + iφ. In this ontology, φ is not a mere mathematical artifact of complex amplitudes: it is a physical degree of freedom that mediates a rotation between two logically distinct time aspects—future-pending (fp) time that retains unrealized alternatives, and past-fixed (pf) time that corresponds to realized events and stable records. Dynamic quantum circuits with mid-circuit measurement (MCM) are an unusually direct laboratory for CT because they contain repeated, addressable fp→pf conversion events *inside a single circuit*.Here we analyze IBM superconducting quantum-processor data and show that the target-qubit equatorial coherence z ≡ ⟨σx⟩ + i⟨σy⟩ undergoes a deterministic complex rotation whose angle is set primarily by the number M of spectator mid-circuit measurements. In Dataset A (4096 shots; 5 repeats), the complex ratio w(M) ≡ z_MEAS(M)/z_DELAY(M) displays an event-additive phase shift Δθ_eff(M) = arg w(M) that grows from ≈ 0 at M = 0–1 to Δθ_eff(M=8) ≈ 2.279 rad, with an average increment of δ ≈ 0.323 rad per added measurement. In Dataset B (1024 shots), inserting a virtual-Z compensation Rz(comp×M) on the target yields linear response d(Δθ_eff)/d(comp) ≈ M and demonstrates coherent controllability of the rotation.To probe whether the effect is reducible to schedule-level artifacts alone, Dataset C (CT3 Layer3) uses identical classical control-flow (if-structure) while changing only the spectator’s internal phase coherence: mode A forces |+⟩ before each mid-measurement, whereas mode B allows a |+⟩/|−⟩ mixture to accumulate. A small but systematic phase difference Δθ_AB(M) = θ_A(M) − θ_B(M) emerges (≈ 0.06 rad at M = 4–8) alongside a large change in spectator ⟨σx⟩, consistent with an operational coupling between the driven phase rotation and an internal i-phase degree of freedom.We interpret these results as operational evidence that an i-axis phase coordinate is physically actionable: it can be incremented by pf-creating events and steered by coherent frame updates. This provides a concrete, hardware-grounded route for testing CT claims about the physical reality of the imaginary-time phase. Keywords: complex time; τ = t + iφ; i-phase; past-fixed / future-pending; dynamic circuits; mid-circuit measurement; complex-plane diagnostics; virtual-Z compensation.
Abstract We introduce a complex-time framework in which the imaginary unit i is interpreted not as a formal algebraic device but as the generator of a physically meaningful temporal transformation. In this formulation, time is represented as a complex quantity T = pf + i·fp with |T| = 1, where pf (past-fixed) denotes the real component representing actualized, recorded history, and fp (future-pending) denotes the imaginary component representing undetermined possibilities. The constraint |T| = 1 expresses conservation of total temporal magnitude: as possibilities are determined (fp decreases), actualized history accumulates (pf increases). Temporal evolution is described by continuous rotation in the complex-time plane, governed by dT/dθ = iT, while a non-Hermitian extension dT/dθ = iT − γT (γ > 0) describes dissipative processes and the emergence of the arrow of time. Within this framework, quantum measurement is identified with a discrete, irreversible real-axis projection—the conversion of fp into pf—that constitutes a single step of time’s advance from the imaginary to the real axis. This identification yields three concrete predictions for quantum circuits: (i) measurement-induced effects depend on measurement count M rather than free-evolution time T (T-independence); (ii) irreversibility accumulates linearly with M in time-reversal protocols; and (iii) the elementary duration δ of one real-axis projection is implementation-dependent. The framework provides a unified structural basis for the continuity of motion, quantum uncertainty, and macroscopic irreversibility, suggesting that these phenomena arise from a single generative process: the continuous, irreversible conversion of future-pending structure into past-fixed record. Keywords: complex time; past-fixed / future-pending decomposition; quantum measurement; irreversibility; arrow of time; uncertainty principle
We present complex-time (CT) theory, a framework in which the fundamental structures of quantum mechanics are derived from three physical postulates concerning the internal structure of time. The postulates are: (P1) physical time possesses two orthogonal components — past-fixed (pf), representing actualized, irreversible records, and future-pending (fp), representing physically real but undetermined possibilities; (P2) pf and fp are orthogonal (pf fp); and (P3) the total temporal magnitude is conserved (pf + fp = 1). From these three postulates alone, we show that: (i) the state space is the unit circle and the dynamical group is , so that the imaginary unit arises as the unique generator of temporal evolution — not as a mathematical convenience but as a consequence of temporal geometry; (ii) the Schrödinger equation follows from the identification of energy with the rate of phase rotation in the pf–fp plane, with the Planck constant playing the role of a natural constant intrinsic to the pf/fp structure; (iii) the de Broglie–Einstein relation holds universally — for massive and massless particles alike — because the wave structure belongs to time itself, not to any particular particle; (iv) the uncertainty principle is a geometric consequence of the impossibility of fully eliminating fp while preserving ; (v) the complex structure of Hilbert space, the inner product, and the Born rule all follow from the geometry of fp pf projection; and (vi) energy quantization emerges from a “real-axis sampling” rule without imposing discreteness on the underlying dynamics. We further show that the CT framework provides unified descriptions of double-slit interference, quantum tunneling, unstable-state decay widths, and the structural origin of the arrow of time. Preliminary experimental predictions for quantum-circuit observables are stated, and applications to cosmological dark-energy phenomenology, galaxy rotation curves, and Lorentzian path-integral contour selection are outlined with references to dedicated companion papers. The central thesis is that quantum mechanics is not a theory of particles with mysterious wave-like properties, but a consequence of the internal structure of time — time whose wave-like phase (fp) is projected, irreversibly and discretely, onto the real axis (pf) in every act of measurement. Keywords: complex time; temporal structure; imaginary unit; quantum mechanics derivation; past-fixed / future-pending decomposition; Born rule; energy quantization; measurement; irreversibility; arrow of time
ABSTRACT We report a systematic, measurement-induced irreversibility in superconducting quantum circuits that is not accounted for by any tested conventional noise model. Using IBM Quantum processors (ibm_marrakesh and ibm_kingston), we implement a time-reversal protocol in which mid-circuit measurements are inserted between a unitary U and its inverse U†. The fidelity deficit ΔF = F(dummy) − F(meas) grows linearly with measurement count M, reaching ΔF = 0.235 (22.2σ) and ΔF = 0.164 (24.0σ) at M = 8 on the two processors. Three independent controls rule out conventional noise sources: (1) T-independence of the spectator-qubit phase shift (dA ≈ +0.10 at T = 40 and 80 μs) eliminates crosstalk and leakage; (2) a non-Markovian symmetry test (spread = 1.5σ) eliminates decoherence asymmetry; and (3) cross-backend reproducibility of the linear ΔF(M) scaling eliminates hardware-specific measurement noise. The three observations share a common structural feature: the irreversibility is tied to discrete measurement events rather than free-evolution time, accumulates proportionally to their count, and reproduces across independent hardware. We report these observations as experimental evidence consistent with the interpretation that the imaginary unit i in the Schrödinger equation reflects a physically real internal structure of time, and that quantum measurement drives an irreversible projection of that structure onto the real axis. This interpretation is developed from first principles in Sec. II and connected to an existing theoretical framework [13] in Sec. VI. Keywordsquantum computing, mid-circuit measurement, complex time, irreversibility, IBM Quantum, time-reversal protocol
Abstract This paper extends the framework developed in Part I: Complex-Time Geometry and Phase Symmetry by examining the ontological origin and physical meaning of the imaginary rotation eiθeiθ that sustains temporal continuity.While Part I defined time as an intrinsic complex-phase rotation linking quantum mechanics and relativity, the present work investigates what the imaginary component ii itself represents in the emergence of temporal direction and measurement. It is proposed that ii introduces an orthogonality between “being” (real component) and “becoming” (imaginary component).Temporal flow is interpreted as the continuous projection of undetermined future phases into determined past fixation.The increase of entropy and the arrow of time naturally arise as the cumulative effect of these projections—the gradual loss of imaginary-phase freedom. Within this complex-time geometry, quantum measurement is redefined as past fixation, the act that converts potential states within the imaginary domain into realized events on the real axis.A superposed quantum state therefore does not lack existence; rather, it exists within the imaginary-phase continuum as an undetermined future.Observation performs the transition Im(ψ)→Re(ψ)Im(ψ)→Re(ψ), turning potentiality into actuality. Quantum entanglement is reinterpreted as a temporally coherent relation already realified at its creation.Because the correlation itself is fixed in the real domain, no information transfer or superluminal signaling is required—the apparent simultaneity is the result of a shared past fixation.Once realified, the relation loses its imaginary component (dτ=0dτ=0), existing as a null-temporal entity outside ordinary spacetime.Consequently, the problem of nonlocality dissolves as a misunderstanding of temporal structure rather than spatial distance. At the macroscopic level, matter experiences time as the aggregate rhythm of countless microscopic realificationsamong interacting particles.Each crossing of eiθeiθ through the real axis imprints a discrete mark of actuality.The apparent continuity of classical time emerges as the statistical accumulation of these quantized projections—a self-recording process through which the universe continuously writes its own history. Ultimately, this study concludes that time is not an external parameter but the internal oscillation of existence itself—the ongoing conversion between potential and actual that underlies all physical and informational processes. Keywords:complex-time geometry; temporal projection; quantum measurement; entropy; arrow of time; quantum entanglement; nonlocality; ontology of time; self-recording process of reality
Abstract We propose a two-component structure of physical time based on a distinction between Past-fixed (pf) and future-pending (fp) contributions. pf time encodes events that have already been realized and recorded, while fp time encodes those possibilities that have not yet become facts but may do so. This structure is represented by a complex-time variable and a normalized complex amplitude with , where pf and fp denote, respectively, the pf and fpcomponents. In this framework the imaginary unit is interpreted as a generator of rotations between pf and fp, rather than as a purely formal algebraic device. We show that unitary quantum dynamics can be understood as the conservation of the combined pf – fp weight, and that the Schrödinger equation describes rotations of the complex-time amplitude with angular velocity . Measurement processes are described as projections from fp to pf, and we argue that the Born rule arises from natural symmetry and additivity requirements on the frequencies of such future-to-past conversions. For two-level systems, equal fp alternatives lead to equal pf frequencies; for rational amplitude ratios, fp can be decomposed into equal “future packets” whose uniform projection into pf yields probabilities proportional to the squared amplitudes. In higher dimensions, Gleason’s theorem implies that these conditions uniquely single out the usual Born rule. We further show that discrete energy spectra can be viewed as a consequence of sampling the complex-time rotation only at those instants where lies on the real axis, , at which fp vanishes and the corresponding mode is fully pf. This leads to a quantization condition that reproduces Planck-type discrete energies for simple systems. Finally, we interpret tunnelling and evanescent domains as manifestations of fp structure: from the pf perspective an infinite barrier still enforces , but from the fp perspective a particle that has not yet appeared in that pf spacetime may exhibit fp amplitudes that transiently extend into the barrier region. Taken together, these results suggest that the imaginary unit can be interpreted as a “future factor” in quantum theory, and that a complex-time geometry provides a coherent route to connect quantum dynamics, probability, and discrete spectra with the temporal status of physical reality. Running Title: What is Time? Complex-Time Geometry Key words: complex time; past-fixed and future-pending; time in quantum mechanics; Born rule; unitary dynamics; quantum tunnelling; quantum foundations