October 25th, 2026: updates following the development of the Absolute Frame Theory. Why is the physical action a phase, exp(iS) (oscillatory, norm-preserving, interference-bearing), rather than a real weight, exp(−S) (diffusive, dissipative)? This is the difference between a quantum and a merely statistical world. In Absolute Frame Theory (AFT) it is the difference between the Lorentzian observable manifold ℳ and the Euclidean substratum 𝒜. The imaginary unit is introduced by the ℳ–𝒜 signature transition, the Wick rotation. We show that this transition is the single deepest posit beneath the AFT account of quantum theory. It is the same root from which the complex structure, the Born rule, the arrow of time, and the Tsirelson bound were separately shown to follow. We then reduce it as far as a physical theory can reduce its own ground. The signature transition factorizes into three layers: the existence of a time direction (the sliding of the embedding interface), the orientation (the arrow, fixed by informational irreversibility), and the complex/unitary character (the i and the phase). For the last and only open layer we prove three statements. First, the complex structure is not smuggled in: it acts on each real mode as the real, antisymmetric generator J₀ of the first-order phase-space closure of the mode's oscillation, so that the imaginary unit is an operator of the real phase plane, not an imported ℂ. Real time and the imaginary unit enter together, through the reconstruction. Second, we prove a theorem (for the free sector with uniform sliding) that the reconstructed evolution on ℳ is unitary (the phase exp(iS), with a single complex structure), and that no dissipative exp(−S) sector is admissible. The comoving frame is static, and Osterwalder–Schrader reconstruction delivers a positive Hamiltonian and unitary continuation. Third, that single complex structure acts compatibly across subsystems, J_S ⊗ 𝟙 = 𝟙 ⊗ J_S′ (an operator identity, not a shared scalar sign). Hence the reconstructed theory is locally tomographic, complex rather than real (rebit). The phase is therefore forced by the fact that 𝒜 is Euclidean. That fact is not an arbitrary extra assumption. It is entailed by 𝒜's role as the atemporal, stable, deterministic ground. Atemporality forbids a timelike direction, and a bounded-below action requires positive-definiteness. Hence the two-signature architecture reduces to one posit. Asking why 𝒜 is Euclidean is asking the theory to derive its own ground. That is a foundational question of self-referential, not identifiability, character. The reduction is sharp: it does not derive the posit from its origins, but it shows the posit to be single, non-arbitrary, and necessary given the role. The open residual is thereby narrowed to two sectors, the interacting and the curved/horizon, the latter addressed by modular reflection positivity in the companion dynamical reconstruction.
October 25th, 2026: updates following the development of the Absolute Frame Theory. Why is the physical action a phase, exp(iS) (oscillatory, norm-preserving, interference-bearing), rather than a real weight, exp(−S) (diffusive, dissipative)? This is the difference between a quantum and a merely statistical world. In Absolute Frame Theory (AFT) it is the difference between the Lorentzian observable manifold ℳ and the Euclidean substratum 𝒜. The imaginary unit is introduced by the ℳ–𝒜 signature transition, the Wick rotation. We show that this transition is the single deepest posit beneath the AFT account of quantum theory. It is the same root from which the complex structure, the Born rule, the arrow of time, and the Tsirelson bound were separately shown to follow. We then reduce it as far as a physical theory can reduce its own ground. The signature transition factorizes into three layers: the existence of a time direction (the sliding of the embedding interface), the orientation (the arrow, fixed by informational irreversibility), and the complex/unitary character (the i and the phase). For the last and only open layer we prove three statements. First, the complex structure is not smuggled in: it acts on each real mode as the real, antisymmetric generator J₀ of the first-order phase-space closure of the mode's oscillation, so that the imaginary unit is an operator of the real phase plane, not an imported ℂ. Real time and the imaginary unit enter together, through the reconstruction. Second, we prove a theorem (for the free sector with uniform sliding) that the reconstructed evolution on ℳ is unitary (the phase exp(iS), with a single complex structure), and that no dissipative exp(−S) sector is admissible. The comoving frame is static, and Osterwalder–Schrader reconstruction delivers a positive Hamiltonian and unitary continuation. Third, that single complex structure acts compatibly across subsystems, J_S ⊗ 𝟙 = 𝟙 ⊗ J_S′ (an operator identity, not a shared scalar sign). Hence the reconstructed theory is locally tomographic, complex rather than real (rebit). The phase is therefore forced by the fact that 𝒜 is Euclidean. That fact is not an arbitrary extra assumption. It is entailed by 𝒜's role as the atemporal, stable, deterministic ground. Atemporality forbids a timelike direction, and a bounded-below action requires positive-definiteness. Hence the two-signature architecture reduces to one posit. Asking why 𝒜 is Euclidean is asking the theory to derive its own ground. That is a foundational question of self-referential, not identifiability, character. The reduction is sharp: it does not derive the posit from its origins, but it shows the posit to be single, non-arbitrary, and necessary given the role. The open residual is thereby narrowed to two sectors, the interacting and the curved/horizon, the latter addressed by modular reflection positivity in the companion dynamical reconstruction.
Quantum Theory as the Operational Shadow of the M--A Channel: a Substratum Reconstruction in Absolute Frame Theory
The reconstruction program of quantum foundations asks why nature is described by complex Hilbert spaces with the Born rule, and why its correlations stop at the Tsirelson bound rather than saturating the no-signaling limit. Absolute Frame Theory (AFT) embeds the observable four-dimensional manifold M in a continuous Euclidean substratum A carrying a pilot wave Ψ, with an information-bottleneck channel that projects A onto M and traces out an inaccessible fiber. We propose that finite-dimensional quantum theory is the operational shadow of that channel---the channel viewed from within M, restricted to its M-accessible, identifiable part---and we carry out the reconstruction in that reading. This is a program, not a completed theorem: we mark every claim as derived, as a structural correspondence, or as an open frontier, and we state plainly that AFT is not yet formulated as an operational-probabilistic theory in full rigor and that the complex Hilbert structure is here a mechanism (the analytic-signal representation of the channel's sampling), not a posit. Within those limits the reconstruction reduces the why-of-quantum-theory to a single physical fact---the M--A signature transition, the Euclidean-to-Lorentzian Wick rotation---of which the complex structure and the Born rule are the two faces. Concretely, a linear map carries both the complex generator and the conserved ∣Ψ∣2 density if and only if it is unitary. The same signature transition is, in AFT, the root of parity violation and of the arrow of time; quantum theory, parity violation, and time's arrow thus share one origin. The operator-level complex structure that makes the theory locally tomographic is supplied by the Osterwalder--Schrader reconstruction about that single Euclidean time---reflection positivity being a theorem for the free embedding sector and preserved, by a local Feynman--Kac argument, for static configurations---hence complex quantum theory is derived, not posited, in the static sector. The named open frontier is then the time-dependent and interacting reconstruction, while the Born measure ∣ψ∣2 is shown to be the equivariant density of the reconstructed unitary and generic macroscopic matter is derived to be classical, leaving only the strict Dürr--Goldstein--Zanghì uniqueness open. That the embedding yields the Lorentzian signature, and with it emergent local Lorentz invariance, is established in a companion paper on emergent Lorentz invariance.
From the Pilot Wave Ψ to the Schrödinger Wave Function ψ: Dimensional Aliasing as the Bridge Between A and M in the Absolute Frame Theory
Version of October 13th, 2026, following technical developments within the Absolute Frame Theory programme. We establish a bridge between the pilot wave Ψ on the Absolute Frame 𝒜 and the wave function ψ on the observable sub-manifold ℳ within the Absolute Frame Theory. We start from the perpendicular projection of Ψ along the directions transverse to the embedding. The pilot wave is not continued to the Lorentzian signature. The dynamics on ℳ is taken to be the Osterwalder–Schrader reconstruction of the measure of the embedding field. We treat the free field with a declared mass, about a flat background, for the homogeneous plane-wave pilot wave whose wavevector is normal to the image, and at fixed regularization. There the vacuum-to-one-particle amplitude of ψ, defined as an Osterwalder–Schrader pairing, obeys the Klein–Gordon equation and, in the non-relativistic limit, the Schrödinger equation. The relation between ψ and the one-particle states is not established here. We derive the Born rule |ψ(x)|² for the position measure, given the frame-function hypothesis on the full projection lattice, and we exclude the fibre-marginalized density by the empirical observation of quantum interference. The derivation applies Gleason's theorem and assumes the pure-state regime.
This record contains the manuscript Quantum Mechanics as the Unique Lorentzian Continuation of an Osterwalder–Schrader Positive Theory by Chast K. Wolfe. The paper proves that the full mathematical structure of quantum mechanics—complex Hilbert space, linear and unitary time evolution, the Born rule, canonical (anti-)commutation relations, Poincaré invariance, the spin–statistics correspondence, and the existence of a unique vacuum—arises uniquely from the standard Osterwalder–Schrader (OS) axioms: reflection positivity, Euclidean covariance, locality, regularity, and the Markov semigroup property. Starting from a reflection-positive Euclidean field theory, the work establishes a monotone chain of implications showing that OS positivity forces:(1) a unique complex structure on the reconstructed Hilbert space,(2) linear unitary Lorentzian time evolution generated by a self-adjoint Hamiltonian,(3) the Born rule via Gleason’s theorem,(4) CCR/CAR relations in the Gaussian case, and(5) full OS → Wightman reconstruction, including Poincaré covariance, TCP analyticity, spin–statistics, and vacuum uniqueness. The paper also proves that nonlinear, real, or quaternionic variants of quantum theory are incompatible with OS positivity together with Euclidean covariance and locality. Thus, within the class of OS-positive models, standard complex quantum mechanics is the unique Lorentzian continuation. This Zenodo deposit includes:• the full manuscript (PDF),• a license file (CC BY-NC-ND 4.0),• a bibliography file, and• a README describing the archive contents.
Complex Hilbert space is usually assumed at the start of quantum mechanics. This paper formulates a reverse construction for the one-particle and operator-algebraic setting. Starting from a real Hilbert space carrying an orthogonal representation, the admissible imaginary units are identified with the skew-adjoint square roots of − Id in the Real commutant of the complexified representation. The novelty is not the isolated use of complexification, spectral orientation, or polar decomposition, but their order of dependence: first the symmetry representation defines an intrinsic operator-valued phase space; then an oriented internal time flow selects a distinguished point of that phase space; only afterward are the usual complex one-particle, observable, probability, and Weyl–CCR structures invoked under their standard hypotheses. In the terminology used here, this oriented internal flow is the reversible or inner-generative side of a dual-time distinction. It is not a second observer clock and not an additional Lorentzian coordinate: mathematically it is the strongly continuous orthogonal group U ( t ) = exp ( t A ) on the real Hilbert space. Its Stone-dual variable is energy/frequency λ ; canonical conjugation pairs λ with − λ ; positive energy orients this pair and selects the canonical phase on the dynamic sector through A = J | A | = | A | J . Thus inner-generative time is the source of phase orientation, while irreversible record time, projection, memory, and decoherence enter only as later reduced-dynamics layers. No spectral gap at zero is required, and zero modes remain visible as residual phase data rather than being hidden by the construction. The selected phase is invariant under time-orientation-preserving symmetries and changes sign under time reversal, giving the usual antiunitary action after complex reconstruction. The paper also makes explicit the monoidal limitation: two selected local phases do not make the raw real tensor product the standard composite system; the standard composite is recovered only after imposing the J -balanced, equivalently complex, tensor product rule. Thus the result is a reconstruction of compatible complex kinematics on the selected one-particle/operator-algebraic sector, not a derivation of all quantum foundations from purely real data.
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
The M-A Information Channel: A Quantum-Theoretic Characterization of the Information Interface in Higher-Dimensional Embedding Frameworks
July 8th, 2026: Technical updates following the development of the Absolute Frame Theory. We characterize, in the language of quantum channel theory, the information interface between a four-dimensional observable manifold M and a higher-dimensional substratum A proposed in the Absolute Frame Theory. We show that the routing term of the interaction Lagrangian induces a completely positive trace-preserving map Φ: B(H_M) → B(H_A) whose effective output dimension is fixed by the action quantization axiom. We compute the four standard capacities of this channel—quantum capacity Q, classical capacity C, entanglement-assisted classical capacity C_E, and private classical capacity P—and show that each carries distinct physical interpretation: Q bounds the number of coherent pointers that can be sustained simultaneously, C bounds the rate of classical information transmissible from M to A, C_E − C quantifies the structural cost imposed by the thermodynamic isolation axiom, and P relates to the recoverability of information once transferred. We discuss the framework as a quantum-theoretic instantiation of a multidimensional informational substratum—a notion structurally related to the historical concept of aether but free of the empirical constraints that refuted its classical version. We show that the channel framework yields a natural resolution of the black hole information paradox: information transferred through Φ to the substratum is not destroyed by horizon formation, only made operationally inaccessible to observers confined to M. We discuss connections with holographic frameworks, implications for the macroscopic-classical transition, and the experimental signatures accessible through current quantum information platforms.