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 $\mathcal{M}$ in a continuous Euclidean substratum $\mathcal{A}$ carrying a pilot wave $\Psi$, with an information-bottleneck channel that projects $\mathcal{A}$ onto $\mathcal{M}$ 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 $\mathcal{M}$, restricted to its $\mathcal{M}$-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 $\mathcal{M}$--$\mathcal{A}$ 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 $|\Psi|^2$ 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 $|\psi|^2$ 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.
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.
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.
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.
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.
A no-signaling theory may, in principle, be more nonlocal than quantum mechanics: the Popescu--Rohrlich box reaches the algebraic Clauser--Horne--Shimony--Holt value 4, whereas nature stops at the Tsirelson value 22. Absolute Frame Theory (AFT) produces nonlocal correlations---two systems entangled in M share a single point of the substratum A (Axiom II)---and forbids superluminal signaling (the channel is Bremermann-bounded). We ask where those correlations fall, and we establish, by two independent routes, that they cannot exceed the Tsirelson bound: AFT admits no super-quantum correlations. The operator route proves, inside the reconstructed Hilbert theory, the Tsirelson operator identity, with the local commutativity [Ai,Bj]=0 itself derived from AFT structure---the tensor factorization that local tomography supplies and the field microcausality of the constrained embedding---rather than assumed. The information route reduces the bound to a physical principle: AFT's shared resource is read out through a completely positive, trace-preserving channel of finite classical capacity (a theorem of the channel, founded on action quantization and the trace over the inaccessible fiber), this channel obeys the data-processing inequality, and the combination places AFT in the class that satisfies Information Causality, which singles out 22 among the no-signaling theories. We state plainly what this does and does not achieve. It does not close the question of why the bound is 22 rather than some other value: both routes, chased to their foundations, terminate at the same posited fact---the M--A signature transition (the Wick rotation) that installs the complex-Hilbert structure---and finite capacity alone, stripped of that structure, does not exclude a Popescu--Rohrlich box. The two routes are thus the operator-algebraic and information-theoretic faces of one ceiling, and they reduce the selection of 22 to a single substratum fact, with the residue an instance of the deepest open question of the programme rather than a problem specific to nonlocal correlations.
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.
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.