This paper argues that amplitude calculus is not an auxiliary quantum construction but the relational coordinate framework forced into existence when the Lorentzian chart collapses at the null boundary. Within the causal domain accessible to a massive probe, the coordinates (x,t) retain clear geometric meaning: worldlines exist, proper time accumulates, and classical trajectories are well defined. At the null boundary these structures fail simultaneously. The invariant interval vanishes, the Lorentzian chart reaches its degenerate limit, and no geometric observable survives. Beyond this boundary the analytic continuation of (x,t) is not spacetime but the diagrammatic overreach of a coordinate system that has exhausted its operational content. Three structural consequences follow. First, null propagation carries no proper time: the boundary is a zero--interval relation, not a limit of motion. Second, the null boundary is not a worldline and admits no geometric in--between. Third, once (x,t) loses operational meaning, only a relational description is possible. In this regime Feynman's amplitude assignments are not postulates but the unique Lorentz--invariant extension assigning complex weights to admissible relational configurations whose coherent aggregate replaces the classical trajectory. Quantum mechanics is thus the relational mathematics demanded by Lorentzian geometry once its coordinates collapse. The non--spacetime domain is not an extension of spacetime but the region where geometry terminates and amplitude begins. This is version 1.1: This version corrects minor structural issues and removes an obsolete appendix; no scientific conclusions are changed. Authors' note (October 2026). This paper is preserved as an exploratory precursor to our later research, not as a validated derivation of quantum mechanics. Several central geometric claims are incorrect: a null interval does not collapse spacetime coordinates to (0,0) or make the Lorentzian metric degenerate, and spacelike-separated events remain part of ordinary spacetime, not a non-spacetime exterior. The further claim that quantum amplitude calculus and complex amplitudes are uniquely forced by null geometry is not established. In particular, the proposed weight e^(−mℓ/ℏ) is a positive real exponential that does not supply the oscillatory relative phases needed for the claimed interference construction; for massless particles (m = 0), this weight is identically one. The original manuscript is retained unchanged as a historical record of the questions that motivated our subsequent investigations. Our later paper, The Lorentz-Analytic Exterior: A Conditional Mathematical Framework (Version 2, October 2026), revisits the geometric and analytic questions under explicit mathematical assumptions, without relying on the coordinate-collapse argument. https://doi.org/10.5281/zenodo.23400176
This paper argues that amplitude calculus is not an auxiliary quantum construction but the relational coordinate framework forced into existence when the Lorentzian chart collapses at the null boundary. Within the causal domain accessible to a massive probe, the coordinates (x,t) retain clear geometric meaning: worldlines exist, proper time accumulates, and classical trajectories are well defined. At the null boundary these structures fail simultaneously. The invariant interval vanishes, the Lorentzian chart reaches its degenerate limit, and no geometric observable survives. Beyond this boundary the analytic continuation of (x,t) is not spacetime but the diagrammatic overreach of a coordinate system that has exhausted its operational content. Three structural consequences follow. First, null propagation carries no proper time: the boundary is a zero--interval relation, not a limit of motion. Second, the null boundary is not a worldline and admits no geometric in--between. Third, once (x,t) loses operational meaning, only a relational description is possible. In this regime Feynman's amplitude assignments are not postulates but the unique Lorentz--invariant extension assigning complex weights to admissible relational configurations whose coherent aggregate replaces the classical trajectory. Quantum mechanics is thus the relational mathematics demanded by Lorentzian geometry once its coordinates collapse. The non--spacetime domain is not an extension of spacetime but the region where geometry terminates and amplitude begins. This is version 1.1: This version corrects minor structural issues and removes an obsolete appendix; no scientific conclusions are changed. Authors' note (October 2026). This paper is preserved as an exploratory precursor to our later research, not as a validated derivation of quantum mechanics. Several central geometric claims are incorrect: a null interval does not collapse spacetime coordinates to (0,0) or make the Lorentzian metric degenerate, and spacelike-separated events remain part of ordinary spacetime, not a non-spacetime exterior. The further claim that quantum amplitude calculus and complex amplitudes are uniquely forced by null geometry is not established. In particular, the proposed weight e^(−mℓ/ℏ) is a positive real exponential that does not supply the oscillatory relative phases needed for the claimed interference construction; for massless particles (m = 0), this weight is identically one. The original manuscript is retained unchanged as a historical record of the questions that motivated our subsequent investigations. Our later paper, The Lorentz-Analytic Exterior: A Conditional Mathematical Framework (Version 2, October 2026), revisits the geometric and analytic questions under explicit mathematical assumptions, without relying on the coordinate-collapse argument. https://doi.org/10.5281/zenodo.23400176
Quantum phenomena are traditionally described as states evolving in spacetime, yet the zero–proper–time behavior of massless systems points to a different architecture—one in which propagation is not fundamental. Motivated by this observation, we develop a relational framework in which physical processes are represented not as trajectories but as relations between contexts and outcomes. The central object is a relational amplitude function R:B×O→C, organized as a fiber bundle whose slices R_b are selected by experimental contexts. Expressed entirely in standard mathematical terms—probability theory, quantum amplitudes, and fiber‑bundle structure—this relational architecture is precise and unambiguous. Interference, entanglement, and measurement emerge as geometric properties of slice selection rather than as dynamical evolution of a spacetime system. Interference arises when a slice contains multiple coherent contributions; entanglement arises from the non‑factorization of the relational object; and measurement corresponds to a context update rather than a physical collapse. The geometry of zero proper time motivates a shift from a spacetime ontology to a relational one, yielding a unified, non‑spatiotemporal account of quantum phenomena. The result is a structurally simple and conceptually coherent reformulation of quantum mechanics that dissolves long‑standing paradoxes and clarifies the relational foundations from which spacetime events arise. This is version 1.0 of the manuscript. Authors' note (October 2026). This is the original March 2026 manuscript, preserved unchanged as the starting point of an ongoing research program. It is a conceptual proposal, not a set of established results: the framework reformulates the standard quantum formalism in relational language and does not derive it. The proofs of Theorems 5.3 (entanglement) and 5.5 (no-signaling) require additional assumptions that the paper does not state, and claims that the framework “dissolves” the measurement problem or is “complete” should be read as aspirations. The original motivation subsequently led to investigations of geometric, analytic, and measurement-theoretic questions under more explicit assumptions. One such development is The Lorentz-Analytic Exterior: A Conditional Mathematical Framework (Version 2, October 2026), which examines related geometric and analytic structures without relying on the original paper’s zero-proper-time argument as a derivation of quantum mechanics. This later work does not validate the original manuscript’s unsupported claims. https://doi.org/10.5281/zenodo.23400176
Standard quantum descriptions treat interference, entanglement, measurement, and no signaling as separate operational rules, yet their shared structural features point to a deeper geometric origin. In this work we identify that origin by developing the analytic exterior: a non‑spatiotemporal analytic domain specified intrinsically by a single analytic direction, a compatibility condition, and a singularity–residue geometry. A relational bundle then supplies the curvature and boundary projections that generate the operational structure of the amplitude calculus. This intrinsic specification requires neither explicit analytic functions nor global coordinates. Within this analytic framework, familiar quantum structures emerge naturally as boundary effects. The Hilbert‑space inner product appears as the boundary pairing induced by residue geometry; analytic flow along the distinguished direction reveals the Schrödinger‑type evolution already encoded in analytic displacement; and boundary evaluation combined with normalization invariance yields the Born rule as the unique probability assignment compatible with the analytic exterior. Interference, entanglement, measurement, and no signaling thus arise not as independent postulates but as geometric consequences of singularities, curvature, and contextual transport. The relational object introduced in [6] is recovered as the boundary value of an analytic‑exterior amplitude kernel, exposing the analytic origin of its interference and measurement properties. The analytic exterior therefore provides a unified, non‑spatiotemporal foundation in which amplitudes, probabilities, and correlations are boundary manifestations of a single underlying analytic geometry. Quantum mechanics emerges as the boundary theory of this analytic structure, offering a unified and conceptually coherent reformulation parallel in spirit to the relational architecture developed in our earlier work. Version 1.1. This revision adds a new subsection clarifying the ontological divergence between the analytic–exterior framework and the standard Lorentzian manifold ontology. The update explains why the analytic exterior cannot arise within the classical spacetime model and introduces the notion of mathematical ontology as a representational constraint. No mathematical results or definitions have been changed.
This note explores what the null interval does and does not imply for single-photon events. The proper time and proper length between a photon’s emission and detection both vanish, but these are one geometric fact, not two. A null interval does not remove the events along the worldline; it removes the proper-time parameter along which a massive system evolves. Motivated by this, and by the standard amplitude rules of quantum mechanics, the note proposes a Photon Draw Principle: each single-photon detection can be described as a draw from a fixed amplitude distribution set by the preparation and the experimental context, with no trajectory needed for the prediction. This is an interpretive proposal, not a consequence derived from relativity. Massive particles show the same interference, and describing an outcome as a draw does not explain why that outcome occurs. Version 2.1 (October 2026). Revises version 2.0, which presented its central statement as a theorem forced by relativity. That framing is withdrawn. The statement is now a Photon Draw Principle within an interpretation, resting on the standard amplitude rules plus one explicitly interpretive assumption (Axiom 4). Other corrections: amplitude addition applies to indistinguishable alternatives, and the claims about gravity and about a link between general relativity and quantum mechanics are removed.
Modern theoretical physics rests on two highly successful but conceptually distinct frameworks: quantum mechanics and general relativity. While both theories are experimentally confirmed to high precision within their respective domains, they rely on fundamentally different starting points. General relativity takes spacetime as a primitive geometric structure, whereas quantum mechanics is formulated in terms of abstract state spaces without intrinsic spacetime meaning. Most approaches to quantum gravity attempt to reconcile this tension by modifying, quantizing, or discretizing spacetime itself. Even in approaches where spacetime is said to be “emergent,” it typically remains the primary conceptual backdrop on which physical processes are defined. Despite extensive efforts, there is no consensus on whether spacetime should be regarded as fundamental or derivative. In this work, a more radical starting point is explored. Rather than modifying spacetime or its geometry, we ask whether spacetime can be excluded entirely from the fundamental layer of the theory and recovered only at a later, operational stage. The central question is not how spacetime behaves, but whether the notion of “what lies between physical objects” must be spatial or temporal at the most fundamental level. We propose a framework—referred to here as Super Relativity (SR)—based on a single pre-geometric postulate: the existence of a monotonically expanding set of possible quantum configurations, referred to as the flux. Formally, this flux is represented as a monotone, σ-finite family of capacity measures on a common configuration space (introduced in Section 2). This postulate deliberately refrains from assuming spacetime, time ordering, metric structure, causality, or dynamics. Within this pre-geometric core, two derived, dimensionless notions are introduced: utilization, describing the realization of configurations, and saturation, defined as the ratio between utilized and available configurations. A central structural element of SR is the strict separation between a core layer and an interface layer. In the core layer, all quantities are pre-geometric and dimensionless, and no reference is made to space, time, motion, energy, or interaction. Only at the interface layer are operational notions introduced, including clocks, rulers, and signal propagation. Spacetime, in this construction, is not a dynamical entity but an effective bookkeeping structure that encodes relative rates of change once a calibration to physical units is performed. Within this framework, gravitational phenomena do not originate from spacetime curvature taken as fundamental. Instead, they arise from variations in saturation that, at the interface level, manifest as uniform rescalings of physical processes. When expressed in geometric language, these rescalings reproduce the standard predictions of general relativity, including gravitational time dilation and the equivalence between gravitation and acceleration. The invariant local speed of light is not postulated but follows as a derived consequence, since both clock rates and signal propagation are affected identically by the same underlying saturation mechanism. The aim of this paper is not to introduce new empirical predictions beyond those of general relativity, nor to present a complete theory of quantum gravity. Rather, the objective is foundational: to demonstrate that the mathematical structure of general relativity can be reconstructed from a single pre-geometric postulate without assuming spacetime as part of the fundamental ontology. In this sense, SR should be understood as a reconstruction and reinterpretation of known physics, rather than as a competing dynamical theory. An extended and fully explicit formulation of the framework, including all intermediate steps, is available in a companion preprint (version V12) deposited on Zenodo. The present paper restricts itself to the minimal conceptual and mathematical structure required to establish logical consistency and to demonstrate the emergence of general-relativistic behavior at the interface level.
Quantum structure in this framework arises from a single analytic kernel defined on the analytic exterior. Boundary wavefunctions are not physical fields but coordinate charts of this kernel, and the familiar canonical operators—Hamiltonian, momentum, position, and angular momentum—appear as boundary projections of one intrinsic analytic generator. Schrödinger evolution becomes the time‑chart expression of analytic flow, while measurement amplitudes correspond to residue components selected by boundary projection. Probability is not an added postulate but the squared residue norm of the projected component, supplied by the invariant residue pairing. A further refinement comes from recognizing that each admissible boundary chart carries its own symmetry group, and each symmetry group forces a unique invariant pairing. The historical operator formalism mixes Hilbert symmetry, Lorentz symmetry, and analytic symmetry, creating the appearance of incompatible dynamical regimes. In this framework, measurement is not merely chart selection but symmetry selection: a boundary projection is well defined only when the symmetry group of the chart aligns with the symmetry of the underlying analytic flow. Symmetry‑compatible charts yield separable boundary expressions, while incompatible charts produce analytic cross‑terms—the geometric counterpart of entanglement. The analytic interval unifies these regimes. When the interval is nonzero, interior analytic structure persists and the boundary receives unitary evolution. When the interval collapses to a null boundary, interior analytic structure is lost and the boundary state becomes static. Because both evolution and collapse are boundary limits of the same analytic flow, the measurement problem dissolves: the apparent duality between unitarity and collapse reflects symmetry misalignment rather than a physical conflict. In this view, quantum mechanics becomes boundary geometry, and its operational content is the shadow cast by the interior analytic kernel. Dynamics, measurement, and probability emerge as coordinated manifestations of a single geometric structure. This is version 1.0 of the manuscript.
Light is never observed in transit. We see it only at the moment it is emitted and at the moment it is absorbed, with no evidence of anything occurring in between. This simple fact, combined with the constancy of the speed of light, implies that a photon experiences zero proper time between emission and detection. Once this is taken seriously, the classical picture of a photon occupying intermediate positions or following definite paths becomes unnecessary. This paper presents a clear and accessible account of the double‑slit experiment based on this geometric insight. The interference pattern is not produced by a photon “exploring” multiple routes, but by the structure of the experimental setup itself. Each detection event is one sample drawn from a distribution fixed by the context—just as a single arrow on a target reflects, but does not define, the underlying distribution of possible hits. Using Feynman’s amplitude rules and a relational interpretation developed in earlier work, we show how the observed pattern arises without invoking motion through space or physical waves passing through both slits. The result is a simple, coherent picture that remains faithful to both relativity and quantum mechanics while being understandable to motivated high‑school students. Version 1.2 (October 2026). Corrects the Lorentz-transformation derivation in Appendix A. A sign error in the left-moving light-pulse condition (Eq. 8) had produced a degenerate transformation. The corrected equations give t′ = αt + βx and x′ = c²βt + αx; reciprocity and spatial isotropy fix α² − c²β² = 1, establishing the invariance of c²dt² − dx². The standard proper-time formula and double-slit amplitude calculations are unchanged. A new note to the reader clarifies that zero proper time motivates, but does not derive, the relational interpretation: photons have no rest frame, and massive particles such as electrons also exhibit interference.