For stationary, stable linear systems, quantum form (symplectic form, complex structure, Hilbert inner product) is not a set of external postulates but geometric architecture already carried by the classical action and its flow; read through a local projection, for which decoherence is the explicit candidate, only one postulate remains: that states are sections. The Born measure is not derived here. This is a foundations paper. It adds no new dynamics, and most of its mathematics is known; its contribution is to arrange known results inside one ontology and to check what that ontology can and cannot carry. The ontology has a static space of consistent configurations, a single actualized history, a many-to-one projection Π and a selection operator A, and was proposed to account for quantum structure, the arrow of time and the Born rule. Taking each claim to the smallest system on which it can be computed, we find that the natural explicit candidate for Π, a projection onto locally accessible degrees of freedom, is standard decoherence theory; we do not show that Π must take this form. For linear systems with stable dynamics the quantum structure the ontology presupposes is reconstructed from the classical action: the symplectic form from the boundary term, the complex structure from the flow, and the Hilbert inner product from their combination. Stability of the linearized dynamics, rather than positivity of the action's Hessian, is the condition for quantum form to exist. Given the section postulate, the topological invariants of the ontology become the integrality conditions of geometric quantization; under the preservation assumption of Proposition 1, a dynamics that couples subsystems admits a single value of ħ; and if topology fixes dimensionless constants at all, it can do so only at a boundary, the measured values being images under renormalization-group flow (an illustrated pattern, not a derivation). The hidden sector splits into a gauge part and a temporally accessible part, which exposes a dilemma for structural accounts of irreversibility. A global selection cannot signal under three stated conditions, the central one being that its criterion reads only unconditioned reduced states in the causal past; we conjecture that selection is triggered by redundant environmental records. A hypothesis that the history begins at a regular horizon is tested against the Einstein equations in homogeneous models. Each result is labelled as known, synthesis or conjecture. The code-backed checks are listed in Tables 2–4 and reproduced by the attached scripts (run_all.py, with assertions on the quoted values; tested on Python 3.11/Linux and Python 3.14/Windows). Of the original framework's three predictions, one coincides with standard quantum mechanics in its safe form, one is withdrawn, and one lacks a mechanism and is bounded by binary-pulsar timing. How to check this paper: every result carries a status tag (known / synthesis / conjecture / withdrawn / scoped numerical evidence) collected in the ledger (Table 1). Code-backed claims are listed with their IDs in Tables 2–4. Unzip scripts_v15.zip and run python run_all.py (a few minutes; --rederive also redoes the symbolic Bianchi IX derivation); it asserts the numbers quoted in the text and exits non-zero on any mismatch. Corrections are welcome.
For stationary, stable linear systems, quantum form (symplectic form, complex structure, Hilbert inner product) is not a set of external postulates but geometric architecture already carried by the classical action and its flow; read through a local projection, for which decoherence is the explicit candidate, only one postulate remains: that states are sections. The Born measure is not derived here. This is a foundations paper. It adds no new dynamics, and most of its mathematics is known; its contribution is to arrange known results inside one ontology and to check what that ontology can and cannot carry. The ontology has a static space of consistent configurations, a single actualized history, a many-to-one projection Π and a selection operator A, and was proposed to account for quantum structure, the arrow of time and the Born rule. Taking each claim to the smallest system on which it can be computed, we find that the natural explicit candidate for Π, a projection onto locally accessible degrees of freedom, is standard decoherence theory; we do not show that Π must take this form. For linear systems with stable dynamics the quantum structure the ontology presupposes is reconstructed from the classical action: the symplectic form from the boundary term, the complex structure from the flow, and the Hilbert inner product from their combination. Stability of the linearized dynamics, rather than positivity of the action's Hessian, is the condition for quantum form to exist. Given the section postulate, the topological invariants of the ontology become the integrality conditions of geometric quantization; under the preservation assumption of Proposition 1, a dynamics that couples subsystems admits a single value of ħ; and if topology fixes dimensionless constants at all, it can do so only at a boundary, the measured values being images under renormalization-group flow (an illustrated pattern, not a derivation). The hidden sector splits into a gauge part and a temporally accessible part, which exposes a dilemma for structural accounts of irreversibility. A global selection cannot signal under three stated conditions, the central one being that its criterion reads only unconditioned reduced states in the causal past; we conjecture that selection is triggered by redundant environmental records. A hypothesis that the history begins at a regular horizon is tested against the Einstein equations in homogeneous models. Each result is labelled as known, synthesis or conjecture. The code-backed checks are listed in Tables 2–4 and reproduced by the attached scripts (run_all.py, with assertions on the quoted values; tested on Python 3.11/Linux and Python 3.14/Windows). Of the original framework's three predictions, one coincides with standard quantum mechanics in its safe form, one is withdrawn, and one lacks a mechanism and is bounded by binary-pulsar timing. How to check this paper: every result carries a status tag (known / synthesis / conjecture / withdrawn / scoped numerical evidence) collected in the ledger (Table 1). Code-backed claims are listed with their IDs in Tables 2–4. Unzip scripts_v15.zip and run python run_all.py (a few minutes; --rederive also redoes the symbolic Bianchi IX derivation); it asserts the numbers quoted in the text and exits non-zero on any mismatch. Corrections are welcome.
Living Field Theory (LFT) proposes that physical structure is described fundamentally by a single real field together with constraints on its admissible histories. This paper investigates the emergence of quantum-mechanical structure within the coherent sector of that framework. A phase-active real coherent carrier equipped with a compatible positive quadratic form is shown to admit an effective complex structure and associated symplectic geometry without requiring a primitive complex field. Predictive equivalence under nonzero coherent scaling yields projective state geometry without requiring a dynamically selected universal fixed-action shell. Stable detector outcomes are represented by disjoint families of admissible Living-Field histories. Their weighted history measure is therefore additive under physical refinement. When the corresponding detector refinement occurs coherently before stable record formation, projective detector weights satisfy x=x1+x2x=x_1+x_2. Requiring a common continuous response law P=f(x)P=f(x) to remain consistent under such refinement gives f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y). Together with f(0)=0f(0)=0 and f(1)=1f(1)=1, this uniquely yields f(x)=xf(x)=x, giving the Born weighting rule Pi=∣⟨ei∣Ψ⟩∣2P_i=|\langle e_i|\Psi\rangle|^2 for normalized states and orthonormal detector channels. The work also distinguishes pre-record coherent refinement from post-record subdivision and examines several unsuccessful stronger emergence routes, including universal fixed-action-shell selection, norm preservation as sufficient for unitary dynamics, and derivation of Hamiltonian time evolution from a positive fluctuation Hessian alone. The result is a conditional derivation of Born weighting within the LFT coherent-sector bridge. It does not claim an unconditional derivation of quantum mechanics from the minimal Living-Field ontology. The remaining problem is localized to the primitive provenance of the required coherent carrier, compatible metric, detector-response structure, and accessible coherent refinements.
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
We develop a derivation of dynamics of Quantum Mechanics and measurement within a fundamentally timeless and scale-invariant framework. Building on a prior reconstruction of quantum kinematics from renormalization consistency, we extend the analysis to dynamical evolution and the measurement process using three minimal principles: a global constraint (timelessness), a variational formulation, and resolution independence. We show that relational time emerges through conditional decomposition of the global state, leading to an effective Schrödinger evolution under suitable conditions on clock subsystems. Renormalization group (RG) transformations, treated as maps on the state space induced by coarse-graining, generate an effective non-unitary dynamics for reduced states. Under standard assumptions of weak coupling and scale separation, this dynamics takes Lindblad form, providing a structural origin for decoherence. A central result of this work is the formulation of a stability selection principle: configurations contributing to observable physics correspond to those that are simultaneously stationary under the action and stable under RG flow. This principle unifies dynamical selection (via the variational structure) and structural selection (via renormalization), and provides a mechanism for the emergence of classical behavior. Measurement is described as an effective sector dynamics arising from decoherence and coarse-grained evolution. The Lindblad equation admits a stochastic representation in which the system evolves between dynamically independent sectors, leading to an effective description of definite outcomes without introducing a fundamental collapse postulate. The Born rule follows from the unique quadratic measure consistent with refinement structure and is realized dynamically within this framework. Together, these results provide a structural account under minimal assumptions of quantum dynamics, decoherence, and measurement as emergent features of a timeless, renormalization-consistent theory governed by stability across scales.
This paper argues that quantum mechanics is not a separate postulate system to be reconciled with spacetime physics. It is the boundary algebra of a non-invertible projection: the algebraic structure that a globally coherent bulk field is forced to produce when it becomes observable. The bulk field is a Majorana spinor, Ξ = CΞ̄, on M^5 = R^4 × S^1(τ), governed by a single conservation law. The projection Π onto the observable boundary is not an external measurement device added afterward: it is the form the field must take in order to admit an observable image at all. What survives Π defines the observable sector; what enters ker Π is lost structurally, not thermodynamically. From this single mechanism, four boundary structures emerge: the Hilbert space of quantum mechanics, physical time, matter stability, and the Born rule. These are four boundary readings of one conservation law, K_τ = ξη = α^−1, resolved by the same tangential dynamics on the invariant surface Σ. Observable states are equivalence classes [Ξ] = Ξ + ker Π. The Born rule is the canonical quadratic probability measure induced by the projection geometry. Measurement irreversibility is the non-injectivity of Π, with non-vanishing protected by the transcendence of π. Physical time is not assumed in advance: it emerges as the output of chiral projection through a conformal map from the compact cyclic fibre to the irreversible real line. The Wheeler–DeWitt equation is not a paradox but the correct timeless description of the bulk. Spacetime emerges after quantum mechanics, not before it. The fine-structure constant is not a free parameter. It is the unique observable residue forced by the self-coherent projection structure. The double-slit interference pattern, the Born rule, and measurement irreversibility are derived as structural corollaries of this paper in the companion work doi:10.5281/zenodo.19654630
Quantum mechanics describes the potential future; general relativity describes the actualized past; the present is the process of actualization that turns the one into the other. That single move is Time–Actualization (TA), the foundational ontology set out here. TA assumes the quantum formalism and the relativistic causal structure, and derives neither: it adds no new physics, makes no new prediction, and asks to be judged on coherence and unifying power rather than on a measurement. It proposes a common ontological reading of measurement, EPR and the structural arrow of time. D5 locates the einselecting environment in the event's causal past; the couplings within that past remain imported, and the system/environment split is relocated rather than removed. In detail. The ontology is a minimal triplet (ρ, ν, ⟨P,≺⟩): a positive normalized state ρ on a local net of algebras (the conditional future potential), a primitive tempo ν (the rate of becoming), and a growing causal set ⟨P,≺⟩ (the past). What is not derived, and is named as such throughout, is the Born rule, Einstein's equations, and the constants. Eight core hypotheses yield seven consequences. A structural arrow of time, prior to thermodynamics (D1). A partial dissolution of the measurement problem (D2), with definiteness posited and the trigger open. Conditional dissolutions of EPR and the signalling worry (D3–D4), requiring a propagation law. A preferred basis anchored in the couplings carried by the causal past (D5). A conditional thermodynamic reading of the structural arrow (D6), requiring a posited bridge to physical reset, and further thermal assumptions for its value. And a conditional coherence lemma (D7): the actualization dynamics is generally covariant and no-signalling once the substrate carries the standard locality of physics. D7 claims less than the Rideout–Sorkin Bell causality condition, which is a screening-off analogue of Bell's local causality and is not inherited. We then develop the proposed physics of the tempo ν: a correspondence with a local quantum speed scale and a saturation choice, both posited. Quantum speed limits alone do not bound an ontic event count. The missing bridge must specify event resolution, additive counting and an applicable composable speed bound. The substrate cadence that reconstructs proper time is kept distinct from a system's tempo; their operational connection remains to be supplied. TA aims at empirical equivalence to quantum mechanics on a causal-set substrate. We label the admissibility requirement P5; its verification, including instrumented multi-time statistics, remains open. No distinctive prediction is claimed. The causal-set "swerve" is inherited from discreteness, generic and not specific to actualization. We close with the open dynamics, its system/substrate interface, the conditional constitution of systems, and the inherited limits of the programme.
We develop a conditionally closed reconstruction of quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime--vacuum substrate, and persistent causal source--response coupling. Wave functions, Hilbert spaces, complex amplitudes, density operators, the Born rule, commutation relations, Fock spaces, path integrals, completely positive maps, and Markovian dynamics are introduced only as derived mathematical or operational structures, not as additional physical axioms. The construction distinguishes the actual finite carrier and its causal history from the quantum state used to represent preparations, transformations, and measurement probabilities. It derives unitary dynamics from self-adjoint Hamiltonians, formulates spectral observables and probability currents, and states the explicit equilibrium, additivity, noncontextuality, and calibration conditions required for the Born representation. Measurement is described through instruments, pointer dynamics, stable records, weak values, continuous monitoring, diffusive and counting trajectories, and operational error--disturbance relations. Quantum trajectories remain record-conditioned predictive representations rather than automatically ontic paths. Composite systems, identical particles, spin, entanglement entropy, purification, steering, Bell correlations, dense coding, and no-broadcasting are developed on declared tensor-product and symmetry domains. Open-system dynamics are formulated through causal Volterra equations, memory kernels, initial-slip terms, auxiliary-mode realizations, and unitary dilations. Existence, positivity, complete positivity, trace preservation, no-signalling, and energy closure are treated as logically distinct properties. Semiclassical pseudodifferential calculus, controlled path-integral convergence, Ehrenfest-time bounds, semiclassical measures, decoherence, and macroscopic-record stability define the quantum--classical interface. Topological phases, defect-bound states, relativistic wave equations, infrared dressing, and effective quantum-field-theory domains are incorporated with explicit applicability and error conditions. Numerical spectral certification, statistical model comparison, twenty-five benchmark models, theorem-status ledgers, and a complete experimental statistical protocol provide falsification and replication criteria. The resulting framework preserves standard quantum mechanics on its validated domain while isolating finite-core and historical corrections as conditional, quantitatively bounded, and experimentally testable extensions. **Keywords** Quantum foundations; topological quantum mechanics; historical memory; three-axiom reconstruction; Born rule; quantum measurement; weak measurement; quantum trajectories; entanglement; spin and statistics; open quantum systems; Volterra equations; semiclassical limit; quantum topology; numerical certification; experimental falsification.