Why do local actions and exponential Euclidean weights arise so universally in classical, statistical, and quantum theories? We offer a structural explanation from minimal constraints on finite descriptions of admissible histories. Assume that histories admit finite, self-delimiting (prefix-free) generative codes that can be decoded sequentially in a single forward pass. These purely syntactic requirements define a minimal descriptive cost, interpretable as a smoothed minimal program length, that is additive over local segments. First, any continuous local additive cost whose stationary sector coincides with the empirically identified classical variational sector is forced into a unique Euler--Lagrange equivalence class. Hence the universal form of an action is fixed by descriptional structure alone, while the specific microscopic Lagrangian and couplings remain system-dependent semantic input. Second, independently of microscopic stochasticity, finite prefix-free languages exhibit exponential redundancy: many distinct programs encode the same coarse history, and this redundancy induces a universal exponential multiplicity weight on histories. Requiring this weight to be real and bounded below selects a real Euclidean representative for stable local bosonic systems, yielding the standard Euclidean path-integral form. When Osterwalder--Schrader reflection positivity holds, the Euclidean measure reconstructs a unitary Lorentzian amplitude.
Why do local actions and exponential Euclidean weights arise so universally in classical, statistical, and quantum theories? We offer a structural explanation from minimal constraints on finite descriptions of admissible histories. Assume that histories admit finite, self-delimiting (prefix-free) generative codes that can be decoded sequentially in a single forward pass. These purely syntactic requirements define a minimal descriptive cost, interpretable as a smoothed minimal program length, that is additive over local segments. First, any continuous local additive cost whose stationary sector coincides with the empirically identified classical variational sector is forced into a unique Euler--Lagrange equivalence class. Hence the universal form of an action is fixed by descriptional structure alone, while the specific microscopic Lagrangian and couplings remain system-dependent semantic input. Second, independently of microscopic stochasticity, finite prefix-free languages exhibit exponential redundancy: many distinct programs encode the same coarse history, and this redundancy induces a universal exponential multiplicity weight on histories. Requiring this weight to be real and bounded below selects a real Euclidean representative for stable local bosonic systems, yielding the standard Euclidean path-integral form. When Osterwalder--Schrader reflection positivity holds, the Euclidean measure reconstructs a unitary Lorentzian amplitude.
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.
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.
Abstract Event-State Theory (EST) proposes a pre-geometric, cost-based explanatory framework for why variational structures arise so broadly in physics. EST does not claim to replace General Relativity (GR) or Quantum Mechanics (QM). Instead, it frames both as effective descriptions emerging from different uses of a single additive, real-valued history cost functional. The classical (GR-compatible) regime is characterized by dominance of extremal-cost histories (minimization), while the quantum regime is characterized by interference between many admissible histories, which requires a unitary representation of the additive cost as a complex phase. This paper is a programmatic bridge: it cleanly separates core postulates from auxiliary assumptions, identifies the precise boundary where purely positive weighting fails to generate interference, and states the minimal conditions under which complex amplitudes of the form exp(iS/ℏ) are forced. We outline a research program: (I) mathematical sharpening of the cost functional on discrete causal substrates, (II) derivation of continuum variational limits (GR-side), (III) derivation of unitary amplitude propagation (QM-side), and (IV) empirical diagnostics where deviations from standard models may arise.
Quantum theory universally represents a physical history by a phase of the form exp(iS/ℏ), adds amplitudes across unresolved alternatives, and assigns probabilities through the Born rule. These structures are empirically successful but are ordinarily introduced only after a quantum state space has been assumed. This article develops a conditional reconstructional account of their common origin. The premetric reconstruction domain contains neither fields, action, amplitudes, nor quantum probabilities; a quantum platform arises only through constitutive read-out. On a connected platform whose persistent sectors descend from one neutral parent, Indefinite Reconstruction Stability requires locally readable primitive returns to agree on overlaps. Under compact connected scalar return, faithful overlap comparison, common orientation, and smooth-holonomy bridges, the local return data form one global U(1) phase structure. A continuous composition-preserving representation of additive post-read-out action then has the form Θ[γ]=exp(iS[γ]/a0). If the platform supplies one total action for joint histories and no additional action-calibration field, common-parent and overlap compatibility require the same a0 in every physical sector; empirical calibration identifies it with ℏ. The resulting invariant is an action period 2πℏ, not a minimum positive action. IRSP saturation retains every inequivalent admissible alternative, while an explicitly declared linear-composition bridge produces coherent history amplitudes. Positivity, normalization, orthogonal additivity, and a Hilbert-space event read-out then give the Born representation conditionally. The construction yields parameter-free falsification relations: equal action-to-phase slopes across sectors, one total-action phase for mixed-sector histories, invariance under observationally equivalent history refinements, and vanishing genuine third-order interference. A nonzero residual identifies which bridge fails. Previously reported constants, particle ratios, and law-selection results provide cross-sector motivation but are not premises. The construction does not derive Hilbert space or linear composition from IRSP alone; its bridges, inherited ingredients, and failure conditions are separately audited.
We develop an information-theoretic reconstruction of quantum dynamics based on inference over action space. The fundamental object is a density of action states encoding the multiplicity of dynamical alternatives between configurations. Maximum-entropy inference introduces a finite resolution scale in action, implying that sufficiently close action contributions are operationally indistinguishable. We show that this indistinguishability, together with probability normalization and action additivity, selects complex amplitudes and unitary evolution as the minimal continuous representation compatible with action additivity, probability normalization, and inference under finite resolution. Quantum interference and unitarity therefore emerge as consequences of these assumptions rather than independent postulates. From the resulting propagator, the Lagrangian, Hilbert-space structure, and Schrödinger equation follow as derived consequences. In the infinitesimal-time limit, action differences universally fall below the resolution scale, making coherent summation the minimal consistent description at every step. The numerical value of the action scale is fixed empirically and identified with ℏ.
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.