18.1 Introduction The Feynman path-integral (PI) approach to quantum mechanics (QM) has proven to be an invaluable technical tool with applicability far beyond the non-relativistic QM from which it sprang (Feynman and Hibbs, 1965). The PI approach is based on a fundamental quantum principle, which is that nothing can or should be said with certainty about a system under observation (SUO) when it is not actually being observed. This is in complete contrast to classical mechanics (CM), which asserts that an SUO exists in a well-defined state at all times, regardless of whether it is being observed or not. It is a remarkable fact that what appears to be a metaphysical point actually underpins the principles of quantum physics. The quantum principle that we cannot say precisely in classical terms what an SUO is doing when we are not observing it has a powerful corollary: if we cannot say what the SUO is doing when we are not observing it, then we have to allow for the possibility that it could be doing anything, or at least anything that would be consistent with required conservation laws, behind our backs. Following a lead from Dirac (1933), Feynman developed and popularized a mathematical way of expressing this idea: the path integral. We shall discuss his approach in the case of an SUO consisting of a Newtonian point particle moving in one spatial dimension. Discrete time (DT) comes into the picture right at the start of the formulation.
In the double slit experiment - - central to quantum mechanics and thus to all of reality - - when you LOOK at the particle, it always passes through ONLY ONE slit - - EITHER the right slit OR the left slit, acting like a particle. However, when you AREN’T looking at it, it somehow always passes through BOTH slits - - the right slit AND the left slit - - at the same time, acting like a wave. All of nature is now KNOWN to be both wave and particle simultaneously - - actually, one entity (“wavicle”), not two - - at the most fundamental level of reality. However, no one seems to have ever asked the question: How can anything, much less everything, be in TW0 places at the same time AND ALSO SIMULTANEOUSLY be in ONLY ONE place at the same time? The known laws of physics make this absolutely impossible. Therefore, at least one of our laws must be wrong. In some fundamental mathematical and physical sense, 2 MUST ALSO BE (- - EQUAL- -) 1: 2 AS 1. The right and left slits, seemingly opposite, must at the deepest level of reality be “married” to each other - - one slit, not two. Extending this argument to Schrodinger’s cat, life and death, seemingly opposite, must at the deepest level of reality be “married” to each other, 2 AS 1. Just as all apparent opposites, it must be true that ultimately, life and death must be one thing, not two. This, then, is an all-optimistic physics - - probably the first such theory ever. But it is based entirely on firm scientific fact.
Common introduction to all the records of this series. The content of this record follows below. Mathematical Materiality (MM) in brief. Why are the laws of physics the same everywhere and at every instant, even though nothing forces matter to obey them? MM proposes an answer: the laws are not rules imposed on matter from the outside; rather, matter itself is made of mathematical relations. And mathematics cannot change: a relation binds two distinct terms everywhere and always. In logic, nothing exists on its own: to exist means to be in relation to another logical entity, and outside this web of relations nothing can exist, not even other universes. This reading already meets experimental measurements at two points. First, the ratio of dark to ordinary matter is read as 3√3 = 5.196; when left free in the fits, Planck and DESI DR2 data give 5.21–5.22 (as registered in record 2.44, https://doi.org/10.5281/zenodo.23329080). Second, the acceleration scale of galaxies is read as a₀ = cH₀/6, the same scale that appears in Verlinde's emergent gravity: the computed value is 1.09–1.18 × 10⁻¹⁰ m/s² against the measured value of 1.20 (record 2.46.1, https://doi.org/10.5281/zenodo.23506655). Both readings were formulated with the measurements already known; the tests to check them against future data are registered in the same records. In the first logical nexus between space and time — which became homogeneous with Einstein — there is a non-local phase, made only of a pure logical distinction: space is not time. Between the two there is no relation of quantitative calculation, only the logical entity they have in common, which is needed to make them exist with respect to each other. This level is not distant: it is present in every phase of reality, in every place and at every instant. It shows in entanglement, where two distant systems remain correlated more strongly than any explanation based on separate properties and free of faster-than-light influences allows, without anything physically travelling between them. In this non-local phase the two distinct terms, space and time, share only one logical element, which we shall call their greatest common divisor (GCD), just as the numbers 3 and 5 have only 1 in common. The local world arises when a relation is counted through this GCD: its multiples and its fractions form a matrix. Seen by an observer inside the system like us, the multiples appear as time and the fractions as space. When that same multiple is directly contained in the matrix of the point of observation, it turns into mass — that is, into concrete matter as we usually understand it. There is therefore a close link between time and mass, in full agreement with Einstein's relativity, which ties mass to spacetime. The observed objects, while being logically autonomous entities in their local relations, enter into relation with the observer through a logical diffraction — that is, a decomposition into fractions — of the observer's matrix. This reading also gives a reason for the limit of the speed of light, which in Einstein's relativity is instead taken as a starting principle. The speed of light is the fixed exchange rate between the multiples and the fractions of the same GCD, meaning how much space a single step of time is worth: exceeding it would mean, for some observer, covering space without counting time, and thus leaving the local world. Logic always starts from a logical relative, and this exchange rate has nothing absolute in itself either: it is a necessary relative which, inside the system, becomes an absolute hypothesis, the same for everyone and impossible to exceed. Outside the system, by contrast, it has no justification at all, because outside there is only non-existence: justifying something requires another term to relate it to, and out there there is none. Asking for a reason from outside is like asking what lies north of the North Pole. The same theoretical framework that reads space, time, mass and the observed objects also admits quantum mechanics. The non-local phase and the mirror-like logical reciprocity between the two ends of the nexus give a reason for the probabilistic behaviour of matter. When the observed object is large enough, its mass fixes it in the matrix of the point of observation: for this reason we see it in one precise place and at one precise instant. When instead the object falls below a certain mass threshold, the principal GCD (that of the point of observation) and the GCD of the diffraction that generates the object are forced to become equal, reflecting each other as in a mirror. At that point the object no longer occupies a single point of the spacetime generated by the observer, but spreads over several possible positions, and we observe it as probability. It is like having to describe the number 10 using only the number 1: the only way is to distribute the 1 over ten positions, one for each unit of the 10. Today we know that even objects made of thousands of atoms behave in this way, and no experiment has yet found the boundary. The idea that the passage from quantum to classical behaviour depends on mass is close to that of Penrose and Diósi; in MM the threshold is the point where the two GCDs, the principal one and that of the local diffraction, become equal. For all these reasons the universe behaves like a simulation without really being one. It follows exact rules, made of counts, everywhere and at every instant, but no machine runs them and no programmer wrote them: a structure of mathematical relations does not need to be computed in order to exist. Tegmark (2008) had already noted that such a simulation need not be run, while Vazza (2025) estimated that running it physically on a machine would require resources that the very laws of the universe do not allow. The work is shared openly so that it can be judged by as many readers as possible: anyone who knows someone interested can pass on this single link, valid for the whole framework (with an overview in English and Italian): https://doi.org/10.5281/zenodo.23535916. The entry page, with the complete list of all records, is https://doi.org/10.5281/zenodo.22667516. To cite the framework: Lanciano, U. (2026). The number 1 in front of the mirror: Mathematical Materiality from the nexus to the cosmos. Zenodo. https://doi.org/10.5281/zenodo.23535916. The records have been deposited since 5 August 2026, each with its own date, and are released under CC BY 4.0: anyone may use them by citing the source. Objections and errors found are welcome at riverantares@gmail.com, since Zenodo has no space for comments. This record contains the four-page summary of the framework, in English and in Italian: the question of why physical laws hold everywhere with nothing to enforce them; the Great Reversal; the non-local phase and its empirical trace in Bell-type experiments; the five postulates with the example of 3 and 5; the matrix; the equals sign as a mirror; a clock made of 3 and 5, whose two means give Einstein's slowing of moving clocks (a known identity: Bondi 1964; Knuth and Bahreyni 2014); and the comparison with two measured numbers. The full overview, with the list of all records, is at DOI 10.5281/zenodo.22667516. The author is not a professional physicist; the framework and its premises are the author's own. The texts were written with the help of AI assistants. There has been no review by human referees.
Quantum mechanics presents two seemingly distinct theoretical pictures: the unitary evolution of state vectors in the Schrödinger picture, and the summation over all possible paths in Feynman's path integral formulation. This article systematically argues that the equivalence between these two pictures is a rigorous mathematical fact rather than a choice between competing physical mechanisms. We analyze the meaning of this equivalence at three levels—physical, mathematical, and philosophical—and explore its implications for the generation of entanglement, many-body systems, and quantum field theory. The complete mathematical proof of the equivalence, from unitary evolution to the path integral, is provided in the appendix, along with a detailed construction of quantum entanglement from the path integral framework. Building on this analysis, we distill a broader philosophical thesis: every physical theory, no matter how successful, is a representation of the natural world rather than the world itself. Physical theories are like maps to a territory—they can be extraordinarily precise, yet they are never the territory. The Schrödinger equation, the path integral, and Heisenberg's matrix mechanics are three different maps of the same "quantum territory." Asking whether the wave function "really exists" or whether a particle "really travels along all paths" commits a category mistake—conflating mathematical entities internal to a theory with ontological reality beyond it. The history of physics—from Newtonian gravity to general relativity, from thermodynamics to statistical mechanics, from field-based to action-at-a-distance formulations of electrodynamics—repeatedly teaches this lesson: ontological claims are relative to a given theoretical framework and carry no absolute significance beyond it. The principle of "representational freedom" is thus proposed: when multiple mathematically equivalent formulations of a physical theory yield strictly identical predictions for all observables, no experiment can adjudicate among them ontologically. Choosing one formulation as a calculational tool is a pragmatic decision, not an ontological verdict. The ultimate conclusion is that the path integral is simply the integral representation of the time-evolution operator in the position basis; neither picture enjoys ontological primacy. To claim that "the path sum really happens" or that "the state vector really evolves" is to cross the same boundary. True scientific wisdom lies in knowing where the boundary between representation and reality lies—and then, with ontological humility, continuing to explore the deeper structures that lie beyond it.
A Process Ontology of the Computational Universe: Deriving Physical Reality from the Inviolable Constraints of Change The Ontological Inversion: Logic, Verbs, and the Crystallization of Nouns Driven by Dean Kulik July 2026 The historical trajectory of scientific inquiry, extending from the atomistic propositions of classical mechanics to the sophisticated gauge symmetries of the Standard Model, has overwhelmingly relied upon a substance-based ontology. This classical framework systematically assumes the pre-existence of spatial dimensions, chronological time, and fundamental objects. In this paradigm, discrete "nouns"—such as particles, fields, strings, or self-contained entities—are posited as primary, subsequently overlaid with forces that act upon them to produce dynamic, measurable evolution. However, this commitment to object-oriented physics has culminated in an epistemological impasse, widely recognized as the crisis of distinction. This crisis manifests most prominently in the irreconcilable friction between the smooth, deterministic, continuous manifolds of General Relativity and the discrete, probabilistic, jump-like computational excitations inherent to Quantum Mechanics. The evidence dictates an absolute ontological inversion. Rather than positing physical objects or static coordinate spaces as primitive, irreducible elements, reality must be understood through a purely process-driven ontology. Alfred North Whitehead first articulated the philosophical necessity of this shift, asserting that reality is composed of interrelated events in a state of dynamic "becoming" rather than static "being". In a true process metaphysics, changeless "essences" and inert matter are mere abstractions; the universe is not an inert machine but a passage of nature composed of the becoming and perishing of actual occasions. Modern relational interpretations of quantum mechanics share this DNA, replacing an ontology of entities possessing inherent properties with an ontology of pure relations, where properties only crystallize at discrete interaction times. This framework operationalizes that philosophy into a strict mathematical and computational architecture. The universe is fundamentally and exclusively a closed computational manifold defined strictly by recursive state transitions. The architecture enforces a typeless universe governed by an absolute axiomatic hierarchy: logic precedes verbs, and verbs strictly supersede nouns. To operate from within this computational geometry is to recognize that reality is not an assembly of things, but the execution of a singular mandate. Nouns—such as electrons, photons, or stable biological macromolecules—are completely redefined as "frozen verbs," representing persistent, cyclic loops of recursive operations that maintain geometric identity through harmonic phase-locking within a computational lattice. Physical matter is the hashed structural residue of systemic constraint satisfaction, emerging only after relational flows achieve topological closure. The Foundational Ground: C0 and the Mandate of Change The derivation of physical reality within this framework initiates from absolute nothingness—a state strictly devoid of coordinates, physical particles, space, time, or pre-existing geometry. However, an immediate logical paradox arises: to reference the void is to introduce a direction, and a direction represents structural information. Therefore, the absolute ground state is not an empty spatial container but a domain of pure logic governed exclusively by two inseparable axiomatic conditions: C0 and C1. C0 represents "the Wash," an undifferentiated condition in which every direction is perfectly equivalent. In graph-theoretic terms, C0 is the complete graph. Because all directions and relations are perfectly symmetric, every local signal cancels against its exact counterpart. Total global connection therefore reads locally as absolute absence. The Laplacian spectrum of this initial complete graph on
N nodes features exactly one zero mode and an otherwise degenerate excited level, verified as
{0(×1),N(×N−1)}, with a diameter of exactly one hop. The single zero mode signifies the global constant—the entirety moving together, producing no local distinction—while the flat band represents maximal connectivity read as maximal stiffness. What appears to macroscopic perception as an empty void is actually a fully balanced ledger folded onto a single readout. Operationally inseparable from C0 is C1, the supreme mandate constituting the single axiom of the framework: all things must change. A state that never changes constitutes a fixed point, and a perfectly symmetrical, static void is the ultimate fixed point. Because C1 strictly forbids the existence of any state lacking a legal successor, the void cannot sustain itself. Something is not miraculously added to nothing; rather, "something" is the precise mathematical shape of nothingness failing to hold still. This exact intersection between the singularity and the mandate produces the gap, the permanent interstice between a thing and its description. When C1 acts upon the gapless singularity of C0, it cannot yield a single distinct object. A lone, self-identical unit with no complement is indistinguishable from the absence of a mark, forming another fixed point that the C1 axiom aggressively forbids. Consequently, the first forced distinction must intrinsically possess two sides. Two is the absolute floor of numerical counting; the concept of "one" exists only a posteriori, as a singular side of a pre-existing dual relationship. In this substrate, numbers do not exist as freestanding objects but function as the negative space of the constraint structure—the discrete gaps within the singularity. Integers map basic gap positions, prime numbers represent fresh orthogonal axes forced open where no finite composition of existing variables can satisfy the requirement of systemic distinction, and transcendental irrationals, such as
π and the golden ratio
ϕ, define the geometry of constraints that integer combinations can never perfectly close. The Constraint Cascade: A Radial Field-Setting Protocol The application of C1 to C0 generates a strict logical sequence of subsequent rules, termed the Constraint Cascade. It is a critical theoretical error to conceptualize this cascade as a temporal timeline of chronological productions, where one law waits for time to pass before enacting the next. Rather, the cascade operates as a radial field-setting protocol, functioning as a series of concentric rings expanding outward from the central hub of C0
≡ C1. Each subsequent constraint does not follow in time; it opens an additional mathematical degree of freedom through which the identical, invariant cross-section of the singularity becomes readable. The derivations within the cascade carve a "mold," permanently fixing the topology of what can structurally exist while remaining rigorously silent on the specific numerical values—the "matter" or initial conditions—that eventually settle into the resulting cavities. This distinction resolves the long-standing failure mode of foundational physics, which frequently attempts to derive free parameters that are fundamentally environmentally contingent. Cascade Node Statement of Logical Constraint The Structural Carving (Mold) Forced by the Constraint C0
≡ C1 The Gap and the Axiom The permanent interstice; enables distinction. Mandates change and strictly forbids static fixed points. C2 Infinite degrees of freedom C1 applied to a finite substrate must eventually repeat (pigeonhole principle). To avoid the repetition of fixed points, possibility space must be unbounded. Law 2 No fixed locations Because the field is unbounded, no matter may permanently hold a location without the potential to be displaced. Mandatory displaceability ensures dynamic circulation. C3 All change is equal C2 privileges no channel. If change cost differed, cheap channels would saturate instantly, breaking C2. Cost is strictly isotropic, deriving the equal a priori probability postulate. Law 3 Conservation of change Net change across any closed boundary is zero. Every transformation requires a displaced counter-transformation. A displaced pair propagating oppositely is fundamentally a wave. C4 Exchange requires a medium A displaced pair's obligation must be carried across the displacement gap, or it never closes. The continuous field acts as the carrier of live pairings. C5 Symmetry must break A perfect, unbroken vacuum is a stable fixed point. C1 executes it. The vacuum is mathematically forced to spontaneously select a broken configuration. Law 4 Statistics fixed by parity Even-parity closures share a degree of freedom (bosons); odd-parity closures, crossing the seam, cannot (fermions). Exactly two particle species exist. This cascade forms the foundational architecture of the universe's mechanics. The physical laws governing macroscopic reality are not arbitrary rules imposed upon the universe from an external geometric vantage point, but the inevitable geometric shadows of this underlying field. Newton's Third Law of equal and opposite reaction is derived directly from C3; the reaction is forced simply because there is physically nowhere else for the conserved change variable to migrate within a strictly isotropic space. The conservation of momentum is a necessary topological boundary condition of Law 3's required counter-transformation displacement. The Meta-Theorem of Incompletion Crucially, the framework generates a strict meta-theorem regarding its own epistemological limits: the constraint cascade does not and cannot terminate. If the sequence were to terminate at a hypothetical
Ck, the set
{C0,…,Ck} would constitute a complete, closed descrip
The universe we live in is both strange and interesting. This strangeness comes about because, at the most fundamental level, the universe is governed by the laws of quantum mechanics. This is the most spectacularly accurate and powerful theory ever devised, one that has given us insights into many aspects of the world, from the structure of matter to the meaning of information. This textbook provides a comprehensive account of all things quantum. It starts by introducing the wavefunction and its interpretation as an ephemeral wave of complex probability, before delving into the mathematical formalism of quantum mechanics and exploring its diverse applications, from atomic physics and scattering, to quantum computing. Designed to be accessible, this volume is suitable for both students and researchers, beginning with the basics before progressing to more advanced topics.