Local mathematics assumes the existence of number structures of different types, vector spaces, etc. localized at each space time point. Relations between number structures at different locations are based on two aspects: distinction between two so far conflated concepts, number and number value and the "No information at a distance" principle. This principle forbids the choice of the value of a number at one location to determine the value of the same number at another location. Value changing connections, related to a real valued field, $g,$ move numbers between structures at different locations. The effect of the $g$ field, or its exponential equivalent, $g(y)=e^{α(y)},$ on numbers extends to other mathematical structures, vector spaces, etc. The presence of $α$ affects theoretical descriptions of quantities in physics and geometry. Two examples are described, the effect on the Dirac Lagrangian in gauge theory, and the effect on path lengths and distances in geometry. The gradient field of $α$, $\vec{A},$ appears in the Lagrangian as a spin $0$, real scalar field that couples to the fermion field. Any value for the mass of $\vec{A}$ is possible. The lack of direct experimental evidence for the presence of the $g$ or $α$ field means that the field must be essentially constant within a local region of the cosmological universe. Outside the local region there are no restrictions on the field. Possible physical candidates, (inflaton, dark matter, dark energy) for $α$ are noted.
Local mathematics assumes the existence of number structures of different types, vector spaces, etc. localized at each space time point. Relations between number structures at different locations are based on two aspects: distinction between two so far conflated concepts, number and number value and the "No information at a distance" principle. This principle forbids the choice of the value of a number at one location to determine the value of the same number at another location. Value changing connections, related to a real valued field, g, move numbers between structures at different locations. The effect of the g field, or its exponential equivalent, g(y)=eα(y), on numbers extends to other mathematical structures, vector spaces, etc. The presence of α affects theoretical descriptions of quantities in physics and geometry. Two examples are described, the effect on the Dirac Lagrangian in gauge theory, and the effect on path lengths and distances in geometry. The gradient field of α, A, appears in the Lagrangian as a spin 0, real scalar field that couples to the fermion field. Any value for the mass of A is possible. The lack of direct experimental evidence for the presence of the g or α field means that the field must be essentially constant within a local region of the cosmological universe. Outside the local region there are no restrictions on the field. Possible physical candidates, (inflaton, dark matter, dark energy) for α are noted.
The work is based on two premises: local availability of mathematics to an observer at any space time location, and the observation that number systems, as structures satisfying axioms for the number type being considered, can be scaled by arbitrary, positive real numbers. Local availability leads to the assignment of mathematical universes, Vx, to each point, x, of space time. Vx contains all the mathematics that an observer, Ox, at x, can know. Each Vx contains many types of mathematical systems. These include the different types of numbers (natural numbers, integers, rationals, and real and complex numbers), Hilbert spaces, algebras, and many other types of systems. Space time dependent scaling of number systems is used to define representations, in Vx, of real and complex number systems in Vy. The representations are scaled by a factor ry,x relative to the systems in Vx. For y a neighbor point of x,ry,x is the exponential of the scalar product of a gauge field, A(x), and the vector from x to y. For y distant from x,ry,x is a path integral from x to y. Some consequences of the two premises will be examined. Number scaling has no effect on general comparisons of numbers obtained as computations or as experimental outputs. The effect is limited to mathematical expressions that include space or space time integrals or derivatives. The effect of A on wave packets and canonical momenta in quantum theory, and some properties of A in gauge theories, is described.
Local availability of mathematics and number scaling provide an approach to a coherent theory of physics and mathematics. Local availability of mathematics assigns separate mathematical universes, x, to each space time point, x.. The mathematics available to an observer, Ox, at x is contained in x . Number scaling is based on extending the choice freedom of vector space bases in gauge theories to choice freedom of underlying number systems. Scaling arises in the description, in x, of mathematical systems in y . If ay or ψy is a number or a quantum state in y, then the corresponding number or state in x is ry,xax or ry,xψx. Here ax and ψx are the same number and state in x as ay and ψy are in y . If y = x+ μdx is a neighbor point of x, then the scaling factor is ry,x = exp( A(x) • μdx) where A is the gradient of a scalar field. The effects of scaling and local availability of mathematics on quantum theory show that scaling has two components, external and internal. External scaling is shown above for ay and ψy. Internal scaling occurs in expressions with integrals or derivatives over space time. An example is the replacement of the position expectation value, ∫ψ*(y)yψ(y)dy, by ∫x ry,ψx* x(yx)yxψx(yx)dyx. This is an integral in x . The good agreement between quantum theory and experiment shows that scaling is negligible in a space region, L, in which experiments and calculations can be done, and results compared. L includes the solar system, but the speed of light limits the size of L to a few light years. For observers in L and events outside L, at cosmological distances, scaling is not limited by theory experiment agreement requirements.
This record. Twelve phenomena, entanglement among them, read through the same non-local phase: «non-local» as not located in space and time, not as action at a distance. Reading path: Foundations: why the laws hold, 5 of 5 · before: Is mass a contained rhythm? · all themes. 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. This record develops one aspect of the theoretical framework. 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. In physics «non-locality» mostly names one fact: the correlations of entanglement. In Mathematical Materiality (MM) it is the foundation: a non-local phase, not located in space and time and present everywhere, of which the local world is the count made from a point. Here «non-local» means not located in space and time; it does not mean action at a distance. This record collects, from MM's published records and the author's premises, twelve phenomena in which MM's reading uses the same non-local phase as the reason, and shows that entanglement is the case in which it rests most directly on a measured datum, not the only one. There are no new calculations: the result is the unity of the reading. The twelve phenomena: the correlations of entanglement beyond Bell's limit, and their Tsirelson ceiling, set by the local reading (MM 2.48, 2.48.1); the same laws everywhere and always, as copies of the same nexus (MM 2.68); identical particles, as the same unit counted twice (a proposal of the AI); the objects two observers see alike, as two counts of one non-local relation (MM 2.62.1); the boundary of the local as a number of copies, not a distance (August status report, reread); probability and the Born rule (MM 2.48.1); the mass below which an object stops being at one point (MM 2.66); the present, and the loop in which the future chooses what is written (MM 2.48, 2.68.1); space and time themselves: a clock counting events without pre-assigned places runs at Einstein's proper time (MM 2.40.2); the limit of light, the same for everyone; dark matter as the ordinary matter of a second mirror, with no reciprocal non-local link to us (MM 2.44); the boundary within which a gravitating system reads only its own field, set by the Galaxy (MM 2.70). How each is set out. For every phenomenon: what is observed, with sources; how known physics reads it; what the non-local phase does; where the link comes from (already written in a deposit, a premise of the author, or a proposal of the AI); what rests on calculations already deposited; what remains open. In ten phenomena the link is already in the deposits or in the author's premises; in one it comes partly from the AI, and in one from the AI alone. Three ways organise them: copies without distance, existence through a link, and a reading that is always local and partial. What is not claimed. The non-local phase transmits no signals and no influences outside light cones. Today it has no distinct experimental signature: where MM gives numbers, they coincide with known physics (the Tsirelson bound, decoherence, wave optics) or come from readings with declared choices. The record also lists the other senses in which «non-local» appears in MM's records, so that they are not confused with this one. Where the reading is tested. Gaia DR4 in December 2026 (rules registered in MM 2.70) and the mass threshold, with a prediction dated 31 December 2035 (MM 2.66); five open questions from MM 2.68. Contents: REA
This paper is a small step towards the goal of constructing a coherent theory of physic and mathematics together. It is based on two ideas, the localization of mathematical systems in space or space time, and the separation of the concepts of number from number value. The separation of number from number value along with the freedom of choice of number values at different points of space or space time enables the introduction of a space or space time dependent number valuation field. The presence of a location dependent number value field affects theoretical descriptions of many physical and geometric quantities. A simple geometric example is worked out in detail, that of the length of a path. The localization of mathematical systems and the separation of number from number value or meaning both emphasize the role of observers. The separation of number from number value shows the role of observers in that value or meaning are conscious observer related concepts. Nothing, including numbers, has value or meaning to an unconscious observer. The localization of mathematical systems also shows the role of observers in that the mathematics that is potentially available to an observer is that at the same position as is the observer. This represents the mathematical knowledge that can reside in an observers brain. As an observer moves in space or space time, the mathematical knowledge potentially available to the observer is the collection of mathematical systems at the same location as the observer. It is hoped that this work, which was begun in 2010, will lead to a better understanding of the relation between the foundations of mathematics and physics, and the role that observers play in this relation.
Introduction Our concern in this chapter is locality in quantum mechanics (QM). Locality is a heuristic physics principle based on the following propositions. No Action-at-a-Distance All the evidence points to the principle that physical actions, taken within restricted (localized) regions of space and time by observers or other agencies such as systems under observation (SUOs), do not cause instantly observable effects on other SUOs at large distances. This does not apply to mathematical/ metaphysical concepts such as quantum wave functions or correlations, as these are conceptual objects (Scarani et al., 2000). Statements about instantaneous wave function collapse are vacuous (have no empirical significance) and are therefore not an issue of significance in physics. Such statements are an issue to theorists who objectivize wave functions, as in Hidden Variables (HV) theory. Action-at-a-distance is generally regarded as anathema by most physicists. For example, Newton's law of universal gravitation is well known for mathematically encoding action-at-a-distance. There is direct evidence, however, in the form of a letter written by Newton to Bentley, that Newton believed that gravity acting “at a distance through a vacuum without the mediation of anything else” was an absurdity (Newton, 2006). The no action-at-a-distance principle is encoded in quantized detector networks (QDN) by the requirement that labstate preparation and consequent signal detection never occur at the same stage. Causal Transmission All physically observable consequences of local actions taken by an observer or SUO are transmitted by identifiable physical processes, such as electromagnetic waves or neutrinos. There is no such thing as magic or action-at-a-distance . In QDN this proposition is taken into account implicitly in the labstate outcome amplitudes at each stage, as these model how information is propagated from stage to stage. When necessary, the information void can be modeled as if there were fields and/or particles propagating through it, giving scope for different mathematical models, such as Euclidean space, curved spacetime, noncommuting spacetimes, and so on. The structure of the information void is essentially a discussion of whatever modules have to be taken into account between labstate preparation and signal detection.
Introduction My presentation aims at answering a part of the questions posted in the invitation to the session. The questions arose in and originated from the FIS discussion on Physical informatics … in October 2014. No doubt, they were formulated in provocative manner. The goal was to challenge discussion. I plan to illustrate my personal answers with a few examples quoted from the history of 20th c. physics. My answers to the questions are not intended to be enunciations and to provide final solutions, rather they serve as arguments and indicate that nothing is closed, the discussion is open. Methods 1 What do we consider physical information? Can one speak about physical information when there is no live percipient to accept, evaluate and use it? Can one speak about physical information (e.g., signal exchange) between inanimate physical objects (cf., e.g., Feynman diagrams)? And if so, what is it for? Is (physical) information a passive phenomenon, or its existence presumes activity? Interpretation of ‘activity’ plays important role in the possible answers. One of the interpretations says that activity is an antropomorph phenomenon. It is a privilege of the human mind, which is able to perceive and process information, able to teleologically evaluate its possible consequences and (re)act accordingly. Another interpretation says that there is inanimate activity, that means, reception of information between physical agents and their reaction to it. I argue for the latter concept. Let us see the example of the interaction between two (electric) charges. There were two approaches in the classical age of developing these theories (1928-33). When two electric charges interact, there appear two types of interaction. One is a Coulomb-type repulsion/attraction (according to their mutual signs) governed by their scalar potential, and the other is a Lorentz-type one governed by their vector potential depending on their relative velocity to each other. One type of the theories considered first the interaction between the scalar potentials and calculated the effect of the vector potentials as perturbation. It considered two charges approaching to each other from the infinity, when in first approximation they have got information on the amount of the Coulomb charge of the other, but not its velocity and the caused Lorentz force. The latter was considered in the perturbation process. The other type of the theories considered first the effect of the Lorentz force of the approaching charges, and took into consideration the effect of the Coulomb force in course of the perturbation. Representatives of both types of theories agreed that the roles of the interacting electric charges must be symmetrical, but Christian Møller. Møller [7], who belonged to the latter type of theoreticians, applied scattering matrices (1931). He showed that there appeared a component among the matrix elements that was asymmetric in respect to the two interacting charges. H. Bethe (that time a doctoral student of E. Fermi, 1932) could not accept this asymmetry and ‘corrected’ Møller’s theory. He ‘symmetrised’ those matrix elements artificially [1]. That was a rough and unjustified involvement in Møller’s equation, but due to the later attained high authority of both Bethe and Fermi, the physics community accepted the apparently convenient symmetrisation of the theory without discussion and has not treated it until the recent years. Thus any possible distinction between (roles and properties of) interacting charges were unrevealed until the past decade. However, the question can be formulated so: how do the charges, ready to interact, get information from each other? All theories agree that interaction between two charges take place by the exchange of a boson. In case of electromagnetic interaction this boson is a photon. Which of them emits the first photon towards the other? Did this emitter receive any information from the direction of the partner charge prior to the photon emission? What else is this if not an asymmetry between the roles, and possibly between properties of the two interacting charges? The distinction between the interacting charges was introduced by the isotopic field-charge theory, and the notion of ‘isotopic field-charge’ made the distinction between the properties of the two charges [2],[3],[4],[5]. The latter led to the proof of the gauge invariance under rotation of a newly introduced property, the isotopic field-charge spin, in an abstract field, and its conservation. This conservation is a result of a symmetry. Neverthe-less, this mechanism argues for activity between physical objects without an animated (human) agent. 2 What are the limits between (closed and open) systems, from the aspects of information and of symmetries? Further, if so, how wide can we extend the meaning of activity to be still accepted for generating information? What are the roles of different appearances of symmetries in taking a stand in the mentioned questions? What kinds of symmetry (or their absence) may play a role in making decision in the listed problems? The ‘classical’ (20th c.) relativity theories demanded that all physical laws were invariant under the Lorentz transformation. This was established first in the special theory of relativity that was formulated for electromagnetic interactions [6]. Lorentz invariance of physical laws was in fact a symmetry principle (conservation of the form of the laws during reference frame change) [9]. This invariance proved to hold for many other physical laws, so later also the symmetry principle was extended to other physical laws as well. The Lorentz invariant relativity theory included another consequence: there is no distinct (odd) reference frame in nature. In other words, all reference frames are equivalent. Aren’t they? Based on Noether’s theorems [8], one can show that conservation laws hold in all reference frames. However, the quantity of the conserved property (e.g., mass, charge, etc.) may change in the different reference frames. E.g., the amount of mass of matter in a closed system depends on the velocity of the observer relative to that system where the mass is to be measured. One can always find a reference frame from which the amount of the measured mass is minimal. This fact contradicts to the absence of a distinct (odd) reference frame. And there are more. Is there any invariance that compensates this lost equivalence of all reference frames? The observer can be not only a human agent who measures with instruments and reads the records. It can be another inanimate mass that perceives information about the mass from the observed system. Its ‘activity’ is that this information can be obtained by experiencing a force. This force can be gravitational or inertial. According to the general theory of relativity the ‘observer mass’ is unable to make distinction whether the experienced force is of ‘inertial’ or ‘gravitational’ origin. At the same time, the inertial mass changes its value according to its relative velocity to the observer, while the gravitational mass does not. Thus the ‘observer mass’ in different reference frames will experience different inertial forces originating from the ‘observed mass’. Something similar distinction can be made between the Coulomb charges and the Lorentz-type (current) charges. They are sources of the Coulomb force and the Lorentz force, respectively, and similar to the two types of masses, originate from the scalar potential and from the vector potential of the Hamiltonian of the charged object, respectively. The two types of masses and the two types of electric charges are called isotopic field-charge pairs, respectively. They are subject of the same gauge invariance. As such, they can exchange their roles (switch into each other) by the exchange of a gauge boson (in addition to the graviton and the photon, respectively), called delta bosons in the theory. As a consequence of the additional invariance and the corresponding additional mediating gauge boson, the respective systems of the two interacting isotopic field-charges (masses or electric charges) are not subjects of the Lorentz invariance alone. They are subjects of a convolution of the Lorentz- and this additional invariance. One can conclude two things. First, along with the development of physics, there is no more enough to demand invariance under the Lorentz transformation. At extended conditions, one should demand the invariance under a combination of the Lorentz invariance and an additional invariance. In short, we demand invariance under (the applicable) transformations. Secondly, when two charges (let they be either gravitational, electric, or other field-charges) interact, they make a distinction between each other. The system, composed of the interacting two field-charges, follow the Pauli principle. That means, the two interacting field-charges must be in different quantum states. Since this state in which they differ cannot be characterized by any of the earlier known properties, it must be a characteristic of the newly introduced property. It must be one of the two stable positions of the isotopic field-charge that are rotated into each other by the SU(2) symmetry group in the isotopic field-charge field. These two stable positions are called, by an analogically given name, the isotopic field-charge spin (not identical either with the angular momentum spin or the isotopic spin). According to its proven invariance, the isotopic field-charge spin is a conserved property. When two field-charges interact, they must be in the opposite isotopic field-charge spin states. The information that they exchange about each other is about this state: they check whether the partner is in the opposite state. Otherwise they are ‘not allowed’ to interact (Pauli’s exclusion principle). The information exchange takes place by the exchange of a delta boson (called also dion) between them, in addition to the exchange of the traditional mediating bosons (like graviton, photon, weak charged and neutral bosons, or gluons). That delta boson switches the emitting charge from inertial to potential state, and the absorbing charge from potential to inertial state. Conclusions The asymmetry of the interacting charges has been explained. It was subject of information exchange between the interacting particle partners. In order to meet the Pauli principle, physical objects should exchange information about the (opposite) states of each other before getting into active interaction. The explanation led to the loss of an invariance property. However, this loss has been restored by introducing a new physical property (isotopic field-charge spin), by proving its conservation, and completing the Lorentz invariance with the respective invariance attributed to the newly proven conservation. References Bethe, H., Fermi, E., Über die Wechselwirkung von zwei elektronen, Zeitschrift für Physik, 77 (1932), 5-6, 296-306. Darvas, G., The Isotopic Field Charge Spin Assumption, International Journal of Theoretical Physics, 50 (2011), 10, 2961-2991.DOI: 10.1007/s10773-011-0796-9 Darvas, G., The Isotopic Field-Charge Assumption Applied to the Electromagnetic Interaction, Int J Theor Phys, 52 (2013a), 11, 3853-3869, DOI 10.1007/s10773-013-1693-1. Darvas, G., A symmetric adventure beyond the Standard Model – Isotopic field-charge spin conservation in the electromagnetic interaction, Symmetry: Culture and Science, 24 (2013b), 1-4, 17-40. Darvas, G., Electromagnetic Interaction in the Presence of Isotopic Field-Charges and a Kinetic Field, Int J Theor Phys, 53 (2014), 1, 39-51, DOI 10.1007/s10773-013-1781-2. Einstein, A., Zur Elektrodynamik bewegter Körper, Annalen der Physik, 17:891 (1905); English translation: On the electrodynamics of moving bodies, in: The Principle of Relativity, (1923), London: Methuen and Co. Møller, C., Über den Stoß zweier Teilchen unter Berücksichtigung der Retardation der Kräfte, Zeitschrift für Physik, 70 (1931), 11-12, 786-795. Noether, E. A., Invariante Variationsprobleme, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen: Mathematisch-physikalische Klasse, (1918) 235-257. Wigner, E.P. , Events, laws of nature and invariance principles, pp. 38-50, in: Wigner, Symmetries and Reflections, Bloomington: Indiana University Press (1967).