A categorical model of the Elementary Process Theory incorporating Special Relativity Marcoen J.T.F. Cabbolet∗ Department of Philosophy, Free University of Brussels Abstract - The purpose of this paper is to show that the Elementary Process Theory (EPT) agrees with the knowledge of the physical world obtained from the successful predictions of Special Relativity (SR). For that matter, a recently developed method is applied: a categorical model of the EPT that incorporates SR is fully specified. Ultimate constituents of the universe of the EPT are modeled as point-particles, γ-rays, or time-like strings, all represented by integrable hyperreal functions on Minkowski space. This proves that the EPT agrees with SR. 1 Introduction The Elementary Process Theory (EPT) is, in a sentence, a collection of seven mathematically abstract formulas that can be interpreted as process-physical principles describing the individual processes at supersmall scale by which interactions have to take place for the gravitational interaction between matter and antimatter to be repulsive [1]. One of its two main issues is that there is no proof that the EPT is consistent with existing knowledge of the fundamental interactions-that is, there is no proof that the interactions as we know them can take place in the individual processes as described by the EPT. Recently a method has been developed for proving that the EPT agrees with a modern interaction theory T : a categorical model C of the EPT has to be specified such that C reduces empirically to T [2]. The purpose of this paper is to demonstrate the method by fully specifying a categorical model CSR of the EPT that reduces empirically to Special Relativity (SR), first published in [3]. This categorical model CSR is thus a category in the sense of category theory as introduced in [4]; it consists of (i) a collection of objects, each of which is a set-theoretic model of the EPT in the reference frame of an inertial observer; (ii) a collection of arrows, each of which corresponds to a Lorentz transformation that transforms one set-theoretic model into another. The specification of the category CSR is straightforward but some elaboration is in place on how the components of the universe of the EPT have been modeled. It has to be taken that the EPT is a mathematically abstract theory that states elementary principles in terms of ultimate components but without reference to any coordinate system of an observer, while each model Mp in {Mi}i∈F1 is a mathematically concrete interpretation of these principles in the reference frame of an inertial observer. Recall that the universe described by the EPT consists of world and antiworld : a component of this universe is designated by a 2× 1 matrix [ φ φ ] , where the abstract set φ designates a constituent of the world and the abstract set φ a constituent of the antiworld-observers who live in "our" forward time-direction thus only observe a manifestation (i.e., a state) of the constituent φ of the world, while a (hypothetical) observer in opposite time-direction would observe a manifestation of the constituent φ of the antiworld. In this study, however, only inertial observers are considered who live in "our" forward time-direction: all models Mp in {Mi}i∈F1 are thus models of the world, not of the antiworld. The outline of this paper is as follows. The next section describes the purely pragmatic approach taken towards specifying a categorical model of the EPT. The section thereafter introduces the main result of this study: the categorical model CSR of the EPT. The final section elaborates on the corresponding world view in terms of particles and events, and states the conclusions. ∗e-mail: Marcoen.Cabbolet@vub.ac.be 1 2 Pragmatic approach: pointillism The EPT describes without reference to any coordinate system how new ultimate building blocks are formed from existing ones by discrete transitions that take place in individual processes. The idea of a set-theoretic model of the EPT is then that the symbols of the EPT that refer to ultimate building blocks are interpreted in a concrete set-theoretical domain D, such that an interpretation I(φ) of a symbol φ mathematically models the state of that building block in the reference frame of an observer. In the model, the discrete transitions of the EPT then become state transitions in the reference frame of an observer: from there, quantitative predictions can be derived. Recall from the introduction that the aim here is to prove that the EPT agrees with SR and nothing more than that. This calls for a purely pragmatic approach: it is enough to specify the simplest categorical model of the EPT that reproduces SR. First of all, the definition of a reference frame of an observer can be taken from SR: Definition 2.1 (IRF) The reference frame of an inertial observer is Minkowski space M = R4 with signature (−,+,+,+). Such an inertial reference frame will henceforth be referred to by the acronym 'IRF'. For a point X = (x0, x1, x2, x3) ∈ M, the real number x0 is the time coordinate, the three real numbers x1, x2, x3 are the spatial coordinates. Planck units are used: both Planck length and Planck time are scaled to 1.  Def. 2.1 thus implies that the present categorical model of the EPT only applies for inertial observers: it is, thus, a presupposition that all observers are inertial observers. Furthermore, for the sake of simplicity we will use rectangular coordinates so that we can use the components ηαβ of the metric tensor η = diag(−1, 1, 1, 1). Secondly, to show agreement with SR it suffices that the set-theoretic models of the EPT in the category CSR are pointillistic. Originally referring to a technique in painting, the term 'pointillism' in physics is defined as the doctrine that a physical theory's fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there; so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts [5]. Thus speaking, a pointillistic model of the EPT is one in which the state of a phase quantum-an ultimate constituent of the universe of the EPT-in the IRF of an observer at every moment of its existence is modeled by a point-particle. Butterfield made a case against pointillism in [5], but it is once more emphasized that we take a purely pragmatic approach in this study: the pointillistic model of the EPT is an idealization that is purely intended to prove agreement with SR-the model needs to be refined to have a wider area of application. Applying hyperreal Dirac delta functions, introduced in [6], to represent the state of a system made up of point-particles, we come to the following pointillistic state postulate: Postulate 2.2 In the categorical model CSR of the EPT, the state of a phase quantum in the IRF of an observer O is represented by a function f :M→ ∗+R for which f : (t, x, y, z) 7→ E * χ(t)δ3(r1,r2,r3)(x, y, z) (1) where E is the energy of the state and χ : M→ ∗+R is a characteristic function having the value 0 at times t when the state doesn't exist, and the value 1 at times t when the state exists. That is, at every time t that the state exists, the energy E of the state is then (i.e. at the time t) distributed over the one point (t, r1(t), r2(t), r3(t), n) ∈M.  Recall that the EPT is not a quantum theory, so in the present categorical model of the EPT the above state postulate is to be viewed as an equivalent of e.g. the state postulate of standard quantum mechanics, which states that a quantum state is represented by an element ψ of a Hilbert space H with norm ‖ψ‖ = 1-this goes back to Schrödinger's early works, e.g. [7]. Similarly, here we have that the state of a phase quantum is represented by an element f of the function space ∗+RM for which∫ +∞ −∞ ∫ +∞ −∞ ∫ +∞ −∞ f(t, x, y, z)dxdydz = E (2) at any time t with χ(t) = 1. In the next section, set-theoretic models of the EPT are specified in accordance with this pointillistic state postulate. 2 3 Result: the categorical model CSR 3.1 Overview In Sect. 3.2 a generic set-theoretic model MZ,ω,O of the EPT is specified in a number of interpretations that apply the state postulate 2.2. In this generic model, the set of all integer-valued degrees of evolution is modeled by Z, and the number of individual processes from any integer-valued degree of evolution n to the next is ω: this is a generic constant which does not depend on n.1 Correspondingly, the set Sω is the section of positive integers up to and including ω: Sω := {1, 2, . . . , ω} (3) For the constant k in the kth process from the nth to the (n + 1)th degree of evolution we thus have k ∈ Sω. There is, thus, a class of concrete set-theoretic models for each value of ω. That said, the objects of the category CSR can then be defined as follows: Definition 3.1 An object of the category CSR is a concrete set-theoretic model MZ,ω,O of the EPT, that is, a structure 〈|MZ,ω,O|,E, IZ,ω,O(R)〉 consisting of: (i) the set of individuals |MZ,ω,O|, the universe of MZ,ω,O, which is the union of the following sets: • the set Aω,O specified by Def. 3.4; • the set 〈GZ,ω,O〉 specified by Ints. 3.7, 3.9, 3.11, 3.12, 3.13, Def. 3.19, and Rem. 3.26; • the set ΘZ,ω,O = {Θnk | n ∈ Z, k ∈ Sω} made up of the sets Θnk specified by Int. 3.25; • the set φZ,ω,O made up of the choice functions specified by Int. 3.25. (ii) the unary existence relation E specified by Int. 3.24, which can be identified with a subset of 〈GZ,ω,O〉; (iii) the ternary relation IZ,ω,O(R) specified by Ints. 3.14, 3.15, 3.16, 3.17, and Rems. 3.23 and 3.26, which can be identified with a subset of 〈GZ,ω,O〉 × 〈GZ,ω,O〉 × 〈GZ,ω,O〉. In this structure, the axioms of the EPT and the universality of the speed of light are valid.  The collection of objects of CSR is (uncountably) infinite; which model applies to the physical world depends, then, on the system to be modeled. E.g. for a system consisting of a single electron, for any r ∈ R there are models in which the electron at t = 0 is co-moving with the observer at spatial distance r, but there are also models in which the electron at t = 0 is moving relative to the observer; moreover, there are models in which the 4-momentum of the electron remains constant in the time interval (0, 2), and there are models in which the 4-momentum of the electron remains changes in the time interval (0, 2). In Sect. 3.3 the arrows of the category CSR are specified in a number of definitions. If for an inertial observer O a concrete set-theoretic model MZ,ω,O of the EPT applies to a given physical system, then for a different inertial observer O′ a different model MZ,ω,O′ applies to the same physical system. The point is, then, that these models are related by an arrow T in the collection of arrows of CSR. That being said, we can define precisely what such an arrow is. Definition 3.2 Let the objects of the category CSR be structures as in Def. 3.1. Then an arrow of the category CSR is an isomorphism T of a structure MZ,ω,O = 〈|MZ,ω,O|,E, IZ,ω,O(R)〉 and a structure MZ,ω,O′ = 〈|MZ,ω,O′ ,E, IZ,ω,O′(R)〉, which maps |MZ,ω,O| bijectively to |MZ,ω,O′ | such that T (f1) + T (f2) = T (f1 + f2) for any f1, f2 ∈ 〈GZ,ω,O〉 (4) ET (f)⇔ Ef for any f ∈ 〈GZ,ω,O〉 (5) 〈T (f1), T (f2), T (f3)〉 ∈ IZ,ω,O′(R)⇔ 〈f1, f2, f3〉 ∈ IZ,ω,O(R) for any f1, f2, f3 ∈ 〈GZ,ω,O〉 (6) Every arrow of the category CSR corresponds with a Lorentz transformation.  So, once we have a concrete set-theoretic model MZ,ω,O that applies to a given system for inertial observer O, then the arrows of CSR transform this to models MZ,ω,O′ ,MZ,ω,O′′ , . . . that will apply to the same physical system for other inertial observers O′,O′′, . . . That is, the arrows relate the predictions of observer O to those of observers O′,O′′, . . . This way, the categorical model CSR of the EPT reproduces relativity of length and time as in standard SR. 1Here ω is a finite integer, not to be confused with the hyperreal number with the same symbol; in the remainder of this text it is assumed that it will be clear from the context to which number the symbol ω refers. 3 3.2 The objects of the category CSR Agreement 3.3 Greek indices α, β, etc. for the components of vectors and tensors can take all values from 0 to 3, but Roman indices i, j, k, etc. can only take a value from 1 to 3. So xα can be any of the components of the 4-tuple (x0, ..., x4), while xj refers only to x1, x2, or x3. Furthermore, (~x)α denotes the αth component of the 4-vector ~x.  To specify the generic set-theoretic model MZ,ω,O of the EPT, we must begin by defining the set of monads. A 'monad' in the EPT is an abstraction of an indivisible massive particle: in this model, a monadic state is an indivisible building block of the world as seen by observer O-its properties then relate to the properties of the monad defined below. Definition 3.4 (Monads) Let MZ,ω,O be a set-theoretic model of the EPT. The set of all monads in MZ,ω,O is then the set Aω,O = {〈k, σk, χk〉 | k ∈ Sω} (7) For any k ∈ Sω, the three-tuple 〈k, σk, χk〉 ∈ S is the kth monad; the constant σk is the rest mass spectrum of the kth monad; the constant χk ∈ {−1, 1} is the characteristic number of normality of the kth monad. In this model, the rest mass spectrum is a constant function σk : Z→ R, σk : n→ mk (8) that adds the number mk > 0, the rest mass of the k th monad, to a degree of evolution n.  The interpretations of constants and axioms of the EPT make use of the following notation and definition: Notation 3.5 Let MZ,ω,O be a set-theoretic model of the EPT; let IZ,ω,O be the interpretation function that maps any constant φ of the EPT to its interpretation IZ,ω,O(φ) in the language of MZ,ω,O. For a constant φ of the EPT referring to a phase quantum, the expression φ O−→ f (9) is then a notation for IZ,ω,O(φ) = f , and has to be read as: 'the state of the phase quantum, designated by φ, in the coordinate system of the observer O is represented by f '. (This notation is loosely based on a notation used in [8].)  Definition 3.6 Let {t} ⊂ R be any singleton, and let (t, u) ⊂ R be any open real interval. Then the characteristic functions χ{t} : R→ R and χ(t,u) : R→ R are given by χ{t} : x 7→ { 1 iff x = t 0 iff x 6= t (10) χ(t,u) : x 7→ { 1 iff x ∈ (t, u) 0 iff x 6∈ (t, u) (11) The latter equation is to include the cases that (t, u) = (t,∞).  Interpretation 3.7 For integers n ∈ Z and k ∈ Sω, the constant EPμnk of the EPT designates the extended particlelike matter quantum at the nth degree of evolution associated to the kth monad. In the model MZ,ω,O we then have EPμnk O−→ snk (12) snk : (t, x, y, z) 7→ EEPn,k * χ{tn,k}(t)δ 3 (xn,k,yn,k,zn,k) (x, y, z) (13) So, the state of the particlelike matter quantum, designated by the symbol EPμnk in the EPT, in the IRF of the observer O is modeled as a point-particle with energy EEPn,k > 0 at the spatiotemporal position Xn,k = (tn,k, xn,k, yn,k, zn,k), represented by the function s n k :M→ ∗+R.  Note that the support of the function snk is a singleton: supp s n k = {(tn,k, xn,k, yn,k, zn,k)} = {Xn,k}.2 That means that in the IRF of O, the point-particle only exists at the spatiotemporal position Xn,k. 2Recall that, for any nonempty set X and any a vector space V , the support of a function f : X → V is the set denoted by 'supp f ' for which supp f = {x ∈ X | f(x) 6= 0}; see e.g. [9]. 4 Agreement 3.8 We will henceforth refer to the state represented by the function snk as the 'particle state of the kth monad at the nth degree of evolution in the IRF of the observer O'.  Now that we have the monadic particle states, we are going to let these evolve according to the principles of the EPT, which are formulated in terms of phase quanta: the idea for this model is that the particlelike state of the kth monad at the nth degree of evolution is the initial state at the start of the kth process from the nth to the (n + 1)th degree of evolution. So we first interpret the constants of the EPT referring to phase quanta, and then we interpret the principles of the EPT. Interpretation 3.9 For integers n ∈ Z and k ∈ Sω, the constant EPΦnk of the EPT designates the extended particlelike phase quantum occurring in the kth process from the nth to the (n+ 1)th degree of evolution. In the model MZ,ω,O we then have EPΦnk O−→ EP fnk (14) EP fnk = s n k (15)  Thus speaking, in MZ,ω,O the state of the phase quantum, designated by the symbol EPΦnk in the EPT, in the IRF of the observer O is the particle state of the kth monad at the nth degree of evolution in the IRF of the observer O. Thus speaking, in the IRF of the observer O, the kth process from the nth to the (n + 1)th degree of evolution starts with a point-particle with energy EEPn,k at spatiotemporal position Xn,k. Moreover, Int. 3.9 associates the k th process from the nth to the (n+ 1)th degree of evolution with the kth monad: the properties of the monad defined in Def. 3.4 will thus occur in the said process. Remark 3.10 To emphasize it: in a more elaborate model of the EPT the phase quantum EPΦnk will be modeled as an aggregation of monadic particle states, and these do not have to be point-particles. Thus speaking, Int. 3.9 forces us to treat, for example, a deuterium nucleus as a monadic state-although we already know that it is composed of a neutron and a proton. The crux here is that we are only interested in showing that the EPT agrees with SR: therefore, we keep the internal states of massive particles as simple as possible-that is, all massive particles are modeled as elementary point-particles.  Interpretation 3.11 For integers n ∈ Z and k ∈ Sω, the constant NWΦnk of the EPT designates the non-local wavelike phase quantum occurring in the kth process from the nth to the (n + 1)th degree of evolution. In the model MZ,ω,O we then have NWΦnk O−→ NW fnk (16) NW fnk : { (t, x, y, z) 7→ ENWn,k * ω3 if (t, x, y, z) ∈ ∆Xn,k (t, x, y, z) 7→ 0 if (t, x, y, z) 6∈ ∆Xn,k (17) for a line segment ∆Xn,k in the IRF of the observer O determined by the spatiotemporal position Xn,k of Int. 3.7 and a displacement vector ∆~xn,k = (∆tn,k,∆xn,k,∆yn,k,∆zn,k) in spacetime with ∆tn,k > 0 such that ∆Xn,k = {Xn,k + λ *∆~xn,k ∈M | λ ∈ (0, 1)} (18) η(∆~xn,k,∆~xn,k) = −1 (19) Thus speaking, the state of the phase quantum, designated by the symbol NWΦnk in the EPT, in the IRF of the observer O is a time-like string with energy E = ENWn,k > 0 and spatiotemporal extension ∆Xn,k, represented by the above function NW fnk ∈ ∗+RM. At every point X = X(λ) of its spatiotemporal extension (with the above parametrization), the time-like string is associated with a 4-momentum ~pNWn,k for which ~pNWn,k = mk * ( dx0 dλ , dx1 dλ , dx2 dλ , dx3 dλ ) = (ENWn,k , p 1 n,k, p 2 n,k, p 3 n,k) (20) η(~pNWn,k , ~p NW n,k ) = −(ENWn,k )2 + (p1n,k)2 + (p2n,k)2 + (p3n,k)2 = −(mk)2 (21) where mk in Eq. (20) is the rest mass of the k th monad as given by Def. 3.4.  5 Note that the components pαn,k of ~p NW n,k in Eq. (20) are constants that do not depend on λ, so d2xα dλ2 = 0. We can view the time-like string NW fnk therefore as a wave traveling in a straight line, associated with energy ENWn,k and constant spatial momenta p j n,k. Furthermore, note that the function prescription (17)-in which the symbol ω refers, of course, to the hyperreal number-can be rewritten in the form of Eq. (1) of Post. 2.2. We have NW fnk : (t, x, y, z) 7→ ENWn,k χ(tn,k, tn,k+∆tn,k)(t)δ 3 〈x1(t),x2(t),x3(t)〉(x, y, z) (22) with χ(tn,k, tn,k+∆tn,k) the characteristic function of the interval (tn,k, tn,k + ∆tn,k) as in Def. 3.6, and with  x1(t)x2(t) x3(t)  =  xn,kyn,k zn,k + t− tn,k ∆tn,k *  ∆xn,k∆yn,k ∆zn,k  (23) This gives precisely the same function values of NW fnk . Interpretation 3.12 For integers n ∈ Z and k ∈ Sω, the constant NPΦn+1k of the EPT designates the non-extended particlelike phase quantum occurring in the kth process from the nth to the (n+ 1)th degree of evolution. In the model MZ,ω,O we then have NPΦn+1k O−→ NP fn+1k (24) supp NP fn+1k = {(tn+1,k, xn+1,k, yn+1,k, zn+1,k, 0)} = {Xn+1,k} , tn+1,k = tn,k + ∆tn,k (25) NP fn+1k : (t, x, y, z, u) 7→ E NP n+1,kχ{tn+1,k}(t)δ 3 〈xn+1,k, yn+1,k, zn+1,k〉(x, y, z) (26) Thus speaking, in MZ,ω,O the state of the phase quantum, designated by the symbol NPΦn+1k in the EPT, in the IRF of the observer O is modeled by a point-particle with energy E = ENPn+1,k > 0 represented by the above function NP fn+1k ∈ ∗+RM. Note that the point-particle only exists at the one spatiotemporal position Xn+1,k in the IRF of O, so χ{tn+1,k}(t) = 1 if t) = tn+1,k and χ{tn+1,k}(t) = 0 else.  Interpretation 3.13 For integers n ∈ Z and k ∈ Sω, the constant LWΦn+1k of the EPT designates the local wavelike phase quantum occurring in the kth process from the nth to the (n+1)th degree of evolution. In the model MZ,ω,O we then have LWΦn+1k O−→ γn+1k (27) γn+1k : { (t, x, y, z) 7→ ∆En+1,k * ω3 if (t, x, y, z) ∈ `γn+1,k (t, x, y, z) 7→ 0 if (t, x, y, z) 6∈ `γn+1,k (28) for a line segment `γn+1,k ⊂M in the IRF of the observer O determined by the spatiotemporal position Xn+1,k of Int. 3.12 and a null vector (1, v 1, v2, v3) ∈M: `γn+1,k :  x0 x1 x2 x3  =  tn+1,k xn+1,k yn+1,k zn+1,k + μ *  1 v1 v2 v3  , μ ∈ (0, tend) (29) η((1, v1, v2, v3), (1, v1, v2, v3)) = −1 + (v1)2 + (v2)2 + (v3)2 = 0 (30) Thus speaking, in MZ,ω,O the state of the phase quantum, designated by the symbol LWΦn+1k in the EPT, in the IRF of the observer O is modeled by a γ-ray with spatiotemporal extension `γn+1,k and with energy E = ∆ENPn+1,k > 0, represented by the above function γ n+1 k ∈ ∗+RM. If the γ-ray gets absorbed at a time t > tn+1,k, then tend in Eq. (29) has the finite value t − tn+1,k; if no absorption takes place, then we have (0, tend) = (0,∞). At every point X(μ) of its path (with the above parametrization), the γ-ray is associated with a 4-momentum ~pLWn+1,k for which ~pLWn+1,k = ∆En+1,k * ( dx0 dμ , dx1 dμ , dx2 dμ , dx3 dμ ) = (∆En+1,k,∆p 1 n+1,k,∆p 2 n+1,k,∆p 3 n+1,k) (31) η(~pLWn+1,k, ~p LW n+1,k) = −(∆En+1,k)2 + (∆p1n+1,k)2 + (∆p2n+1,k)2 + (∆p3n+1,k)2 = 0 (32)  6 Given Eq. (33) we here also have d 2xα dμ2 = 0, so we associate the γ-ray with constant spatial momenta ∆pjn+1,k. The idea of the γ-ray implements a ray theory of light in this model, with the front of the ray being a photon. We thus conveniently ignore that phenomena like interference and diffraction require wave theory. But recall that the aim is to show that the EPT agrees with SR: in the framework of SR, photons are point-particles too! Furthermore, similar to the case of the time-like strings, the function prescription (28) can be rewritten in the form of Eq. (1) of Post. 2.2. We get γn+1k : (t, x, y, z) 7→ ∆En+1,k * χ(tn+1,k, tn+1,k+tend)(t)δ 3 〈x1(t),x2(t),x3(t)〉(x, y, z) (33) with χ(tn+1,k, tn+1,k+tend) the characteristic function of the interval (tn+1,k, tn+1,k + tend) as in Def. 3.6, and with x1(t)x2(t) x3(t)  =  xn+1,kyn+1,k zn+1,k + (t− tn+1,k) *  v1v2 v3  (34) where the vj 's are the spatial components of the null vector from Eq. (29). This gives precisely the same function values of γn+1k . Having modeled the objects in the universe of the EPT in terms of point-particles, time-like strings and gamma-rays, we are now ready to model the elementary principles of the EPT. Interpretation 3.14 For integers n ∈ Z and k ∈ Sω, in the model MZ,ω,O the expression |= 0 : EP fnk → NW fnk (35) models the Elementary Principle of Nonlocal Equilibrium, the first of seven axioms of the EPT; here the symbol '0' refers to the function 0 : M→ ∗+R, 0 : X 7→ (0, . . . , 0). Since EP fnk = snk , cf. Int. 3.9, this expression means that in the IRF of the observer O, the particle state of the kth monad at the nth degree of evolution, located at the spatiotemporal position Xn,k, transforms spontaneously into the time-like string NW fnk , which over time occupies the open line segment ∆Xn,k.  Interpretation 3.15 For integers n ∈ Z and k ∈ Sω, in the model MZ,ω,O the expression |= NW fnk : EP fnk → NP fn+1k (36) models the Elementary Principle of Nonlocal Mediation, the second of seven axioms of the EPT. Since we have EP fnk = s n k , cf. Int. 3.9, this expression means that in the IRF the observer O, the time-like string NW fnk effects a transition from the particle state of the k th monad at the nth degree of evolution, located at the spatiotemporal position Xn,k in the IRF of the observer O, to the point-particle NP fn+1k located at the spatiotemporal position Xn+1,k in the IRF of O. This has to be taken that at t = tn+1,k, the time-like string "collapses" into, i.e. transforms into, the point-particle NP fn+1k .  Interpretation 3.16 For integers n ∈ Z and k ∈ Sω, in the model MZ,ω,O the expression |= 0 : NP fn+1k → γ n+1 k (37) models the Elementary Principle of Local Equilibrium, the third of seven axioms of the EPT; here '0' has the same meaning as in Int. 3.14. This expression means that in IRF of the observer O, the point-particle NP fn+1k spontaneously emits a γ-ray γ n+1 k .  Interpretation 3.17 For integers n ∈ Z and k ∈ Sω, in the model MZ,ω,O the expression |= γn+1k : NP fn+1k → s n+1 k (38) models the Elementary Principle of Local Mediation, the fourth of seven axioms of the EPT. This expression means that in the IRF of the observer O, the emitted γ-ray γn+1k causes the transition of the point-particle NP fn+1k to the particle state of the k th monad at the (n + 1)th degree of evolution. Note that supp NP fn+1k = supp EP fn+1k = {Xn+1,k}, cf. Ints. 3.7 and 3.12, so the discrete transition NP fn+1k → EP f n+1 k involves no spatiotemporal displacement. The particle state of the k th monad at the (n + 1)th degree of evolution is then the starting point of the kth process from the (n + 1)th to the (n+ 2)th degree of evolution.  7 At the level of abstractness of the EPT, the phase quanta in terms of which the elementary principles are stated are abstracted from their properties. In the present model, however, we have endowed the phase quanta with properties, in particular (spatiotemporal) position, energy and spatial momentum. To exclude inapplicability to the physical world the formulation of conservation laws is required; this has the status of an additional postulate. Postulate 3.18 (Conservation of 4-momentum) Upon the collapse of the time-like string NW fnk with 4-momentum ~pNWn,k to the point-particle NP fn+1k the momenta are conserved, so we associate NP fn+1k with a 4-momentum ~pNPn+1,k := ~p NW n,k = (E NW n,k , p 1 n,k, p 2 n,k, p 3 n,k) (39) The γ-ray γn+1k with associated 4-momentum ~p LW n+1,k emitted by the point-particle NP fn+1k then causes the latter to transform to the point-particle EP fn+1k , so we associate EP fn+1k with a 4-momentum ~p EP n+1,k for which ~p EPn+1,k := ~p NP n+1,k − ~pLWn+1,k (40) η(~p EPn+1,k, ~p EP n+1,k) = −mn+1 (41) By a discrete state transition, the point-particle EP fn+1k subsequently transforms into the time-like string NW fn+1k with 4-momentum ~p NW n+1,k. If a γ-ray γ p m with associated 4-momentum ~p LW p,m is absorbed, that is, if a γ-ray γpm has a path {X(t) | t ∈ (0, tend} ⊂ M such that lim t→tend X(t) = Xn+1,k (42) then 4-momentum is conserved according to ~pNWn+1,k = ~p EP n+1,k + ~p LW p,m (43) If no γ-ray is absorbed, then Eq. (43) holds with ~pLWp,m = 0.  Definition 3.19 Let GZ,ω,O = {EP fnk , NW fnk , NP f n+1 k , γ n+1 k | n ∈ Z, k ∈ Sω}; then 〈GZ,ω,O〉 is the commutative monoid generated by the set GZ,ω,O under function addition, for which f + g : X 7→ f(X) + g(X) (44) Note that snk ∈ 〈GZ,ω,O〉 since snk = EP fnk .  Interpretation 3.20 For integers n ∈ Z and k ∈ Sω, the constant ψnk of the EPT designates the state of the kth monad from the nth to the (n+ 1)th degree of evolution. In the model MZ,ω,O we then have ψnk O−→ tnk (45) tnk :M→ ∗+R (46) such that the expression |= tnk = EP fnk + NW fnk (47) models the Elementary Principle of Binad Composition, the fifth of seven axioms of the EPT. Recall that in the EPT the constant βnk ≡ EPΦnk + NWΦnk designates the binad occurring in the kth process from the nth to the (n+ 1)th degree of evolution; the expression (47), thus, means that the state of the binad βnk in the IRF of the observer O is modeled by the monadic state tnk which is made up of the point-particle EP fnk and the time-like string NW fnk .  In a more advanced model of the EPT the state of the binad βnk = EPΦnk + NWΦnk may be identified with an aggregation of monadic states. The next two examples will formalize electrons and positrons in the present framework, but it works the same way for neutrons, antineutrons, protons, antiprotons, and all other massive particles and their antimatter counterparts. 8 Example 3.21 Suppose that the kth monad, introduced in Def. 3.4, is an electronic monad: then the rest mass spectrum σk maps any degree of evolution n to the rest mass σk(n) = mk = me of an electron; the characteristic number of normality χk has then the value +1. The particle state s n k of the k th monad at the nth degree of evolution in the IRF of the observer O, introduced in Int. 3.7, is then a particle state of an electron: the lowest possible value of its energy EEPn,k is the rest mass of an electron me, which is thus predetermined by the rest mass spectrum σk, and it is a normal particle state as indicated by the value +1 of the characteristic number of normality χk. The time-like string NW fnk , created from the particle state of the electron on account of the principle stated in Int. 3.14, can be viewed as a wave state of an electron. Together, the particle state of the electron and the wave state of the electron form the state tnk , which is the (temporally extended) state of the electron from the n th to the (n+1)th degree of evolution-see Int. 3.20.  Example 3.22 Suppose that the jth monad is a positronic monad, then the rest mass spectrum σj is the same as that of an electronic monad: σj maps any degree of evolution n to the rest mass of an electron, so σj(n) = mj = me = σk(n). However, the characteristic number of normality χj has now the value −1. The particle state snj of the jth monad at the nth degree of evolution in the IRF of the observer O is then a positron in a particle state: the lowest possible value of its energy EEPn,j is the rest mass of an electron me, which is thus predetermined by the rest mass spectrum σj , and it is an abnormal particle state as indicated by the value −1 of the characteristic number of normality χj . The state tnk is then the (temporally extended) state of the positron from the nth to the (n + 1)th degree of evolution. In this as well as in the previous example, the characteristic number of normality has the same value as the lepton quantum number in quantum theory.  Remark 3.23 Formulas (35), (36), (37), and (38) describe all individual processes in the IRF of the observer O: there are no other processes (but see Rem. 3.26). In the EPT, the corresponding four elementary principles all use expressions of the form [ a a ] : [ x x ] → ← [ y y ] , which are notations for 〈 [ a a ] , [ x x ] , [ y y ] 〉 ∈ R (48) where R is a ternary relation on a finitely generated commutative monoid (〈g1, g2, g3, . . . , gΩ〉,+); an individual [ a a ] , [ x x ] , or [ y y ] in an expression (48) can, thus, be a sum of generators gj . In the present model MZ,ω,O, however, by these formulas (35), (36), (37), and (38) this relation R is interpreted as a ternary relation IZ,ω,O(R) on the set 〈GZ,ω,O〉.  Having described the elementary processes in this model, we can now interpret the unary existence relation ME of the EPT, which is straightforward. Interpretation 3.24 For any generator f ∈ GZ,ω,O and for any finite sum f1 + . . .+ fn ∈ 〈GZ,ω,O〉 of such generators, the expressions |= Ef ⇔ f 6= 0 (49) |= Ef1 + . . .+ fn ⇔ Ef1 + . . .+ fn−1 ∧ ((Efn ∧ f1 6= fn ∧ f2 6= fn ∧ . . . ∧ fn−1 6= fn) ∨ fn = 0) (50) model the existence relation for the objects in the IRF of the observer O, where 'Ef ' denotes f ∈ E with E = IZ,ω,O(ME).  So, in the model MZ,ω,O we have E EP fnk for any n ∈ Z, k ∈ Sω, but we do not necessarily have Eγ n+1 k for any n ∈ Z, k ∈ Sω. The point is that there may be elementary processes in which no γ-ray is emitted: in that case γn+1k = 0, and thus ¬Eγ n+1 k ; formula (37) is then trivially true. It remains to be established that the present model is a deterministic model of the EPT, which contains an elementary principle of choice. In the IRF of the observer O, a choice takes place at every event that a time-like string NW fnk with spatiotemporal extension ∆Xn,k transforms into a point-particle NP fn+1k at spatiotemporal position Xn+1,k. The time-like string corresponds to a displacement vector ∆~xn,k = (∆tn,k,∆xn,k,∆yn,k,∆zn,k) inM, but although we have from Eq. (25) for the time coordinate that tn+1,k = tn,k+∆tn,k it does not follow from the foregoing that Xn+1,k = Xn,k+∆~xn,k. It is, thus, the principle of choice that guarantees continuity. That is to say: the point-particle NP fn+1k is chosen from a set of possibilities Θn+1k . 9 Interpretation 3.25 Let Θn+1k be the set of all functions NPhn+1k :M→ ∗+R for which NPhn+1k : (t, x, y, z) 7→ E NW n.k * χ{tn+1,k} * δ 3 〈,x1,x2,x3〉(x, y, z) (51) so that NPhn+1k (tn+1,k, x 1, x2, x3) = NP fn+1k (Xn+1,k): the support is a singleton {X} whose element X = (tn+1,k, x 1, x2, x3) differs only with respect to the spatial coordinates xj from Xn+1,k. Let, for Y = (y0, y1, y2, y3) with y0 = tn+1,k, the choice function φY : {Θn+1k } → Θ n+1 k be given by φY (Θ n+1 k ) = NPhn+1k ⇔ supp NPhn+1k = {Y } (52) Let n ∈ Z, k ∈ Sω, and X(t) ∈ ∆Xn,k with x0 = t; then in the model MZ,ω,O the expression |= NP fn+1k = φY (Θ n+1 k ) ∧ Y = limt→tn+1,k X(t) = Xn+1,k (53) models the Elementary Principle of Choice, the sixth of seven axioms of the EPT. This expression means that in the IRF of the observer O, the point-particle NP fn+1k is a choice from a set of possibilities Θ n+1 k strictly determined by the spatiotemporal extension ∆Xn,k of the time-like string NW fnk . See Fig. 1 for an illustration in a spacetime diagram.  Figure 1: Spacetime diagram illustrating the elementary principle of choice, Eq. (53). The two black dots represent the positions Xn,k and Xn+1,k as indicated: these are the positions of the particle states EP fnk and NP fn+1k , respectively. The two diagonal line segments represent the line segments ∆Xn,k and ∆Xn+1,k as indicated: these are the spatiotemporal extensions of the time-like strings NW fnk and NW fn+1k , respectively (cf. Int. 3.11). The spacetime diagram shows a discontinuity: without the principle of choice there is no guarantee that Xn+1,k = Xn,k+∆~xn,k, so the transition from the time-like string NW fnk to the point-particle NP fn+1k at the position Xn+1,k could then involve a discontinuity as shown in the diagram. But the principle of choice, as given by Int. 3.25, guarantees that we have Xn+1,k = Xn,k + ∆~xn,k and thus that no such discontinuity occurs. So in the IRF of the observer O, the particle state sn+1k is located where the spatiotemporal extension of the time-like string NW fnk ends. (In MZ,ω,O, the higher black dot thus continues the lower line segment). Remark 3.26 We leave constants SΦn+2k , which designate the spatial phase quanta that occur in the universe of the EPT, uninterpreted; the same then goes for the Elementary Principle of Formation of Space, the last of seven axioms of the EPT. The reason for this omission is that these interpretations are not needed for showing that the EPT agrees with SR-after all, in SR spacetime is not a substance. For those who find this omission unacceptable, we can interpret an individual constant SΦn+2k as a function Sfn+2k :M→ ∗+R for which Sf n+2 k (X) = γ n+1 k (X − E1) where E1 = (1, 0, 0, 0) ∈ M. The Elementary Principle of Formation of Space, which involves a continuous process, then becomes the expression |= Eγn+1k ⇒ E Sfn+2k (54) (with 'E' as in Int. 3.24, and with the assumption that the set GZ,ω,O now also contains the functions Sfn+2k ) meaning that in the IRF of the observer O, an existing γ-ray leaves a (vanishing) trace of substantial space. To emphasize it: this is just to trivially complete the model.  10 3.3 The arrows of the category CSR There are, then, three kinds of special arrows ('ur-arrows') in the collection of arrows of CSR: • permutation arrows that correspond to a permutation of counting numbers; • translation arrows that correspond to a translation in spacetime; • Lorentz arrows that correspond to a Lorentz transformation. Below these ur-arrows will be defined precisely; all other arrows are then compositions of these ur-arrows. To define such an ur-arrow, it suffices to define how the individuals in the set Aω,O and the individuals in the set GZ,ω,O of generators of 〈GZ,ω,O〉 transform: that determines everything else. To see that, let T be an arrow T : MZ,ω,O → MZ,ω,O′ ; if T (NW fnk ) and T (NP f n+1 k ) are known for all n ∈ Z, k ∈ Sω, then ΘZ,ω,O′ and φZ,ω,O′ are determined by Int. 3.25. Definition 3.27 Let MZ,ω,O be a concrete set-theoretic model of the EPT, and let Σω be the set of all permutations on the section of positive integers Sω. Then for every π ∈ Σω there is a permutation arrow TZ,ω,O,π and a concrete set-theoretic model MZ,ω,O′ of the EPT given by TZ,ω,O,π : MZ,ω,O →MZ,ω,O′ (55) TZ,ω,O,π : 〈k, σk, χk〉 7→ 〈π(k), σπ(k), χπ(k)〉 ∧ σπ(k) = σk ∧ χπ(k) = χk (56) TZ,ω,O,π : αfnk 7→ αf ′ nπ(k) ∧ αfnk = αf ′ nπ(k) (57) (here α denotes EP,NP,NW,LW,S).  Loosely speaking, for every inertial observer O there is an equivalent inertial observer O′ such that the kth process from the nth to the (n+ 1)th degree of evolution in the IRF of O is the π(k)th process from the nth to the (n+ 1)th degree of evolution in the IRF of O′. The point is that the numerical value that an observer gives to the label k is trivial: it is only important that the same value is maintained for its successor and its predecessor, and for the events (i.e. the state transitions) in that process. Definition 3.28 Let MZ,ω,O be a concrete set-theoretic model of the EPT. Then for every function τ for which τ : Sω ×Z→ Z, τ : (k, n) 7→ n+ j(k), there is a permutation arrow TZ,ω,O,τ and a concrete set-theoretic model MZ,ω,O′ of the EPT given by TZ,ω,O,τ : MZ,ω,O →MZ,ω,O′ (58) TZ,ω,O,τ : 〈k, σk, χk〉 7→ 〈k, σk, χk〉 (59) TZ,ω,O,τ : αfnk 7→ αf ′ τ(n,k) k ∧ αfnk = αf ′ n+j(k) k (60) (here α denotes EP,NP,NW,LW,S).  Loosely speaking, for every inertial observer O there is an equivalent inertial observer O′ such that the kth process from the nth to the (n+ 1)th degree of evolution in the IRF of O is the kth process from the (n+ j(k))th to the (n+ j(k) + 1)th degree of evolution in the IRF of O′. The point is that the numerical value that an observer gives to the degree of evolution n is trivial in this categorical model : only the displacement in degrees of evolution matters (vide infra). Definition 3.29 Let MZ,ω,O be a concrete set-theoretic model of the EPT. Then for every ∆X ∈ M with (∆X)4 = 0 there is a translation arrow TZ,ω,O,∆X and a concrete set-theoretic model MZ,ω,O′′ of the EPT given by TZ,ω,O,κ : MZ,ω,O →MZ,ω,O′′ (61) TZ,ω,O,κ : 〈k, σk, χk〉 7→ 〈k, σk, χk〉 (62) TZ,ω,O,κ : αfnk ∧ 7→ αf ′′ nk ∧ αf ′′ nk (X) = αfnk (X + ∆X) (63) (here α denotes EP,NP,NW,LW,S).  Loosely speaking, for every inertial observer O there is an equivalent inertial observer O′′ who does not move relative to O, such that the constituents of the IRF of O′′ are the constituents of the IRF of O shifted by ∆X. The set of monads Aω,O is thus invariant under translation. 11 Definition 3.30 Let MZ,ω,O be a concrete set-theoretic model of the EPT. Then for every Lorentz transformation Λ there is a Lorentz arrow TZ,ω,O,Λ and a concrete set-theoretic model MZ,ω,O′′′ of the EPT given by TZ,ω,O,Λ : MZ,ω,O →MZ,ω,O′′′ (64) TZ,ω,O,Λ : 〈k, σk, χk〉 7→ 〈k, σk, χk〉 (65) TZ,ω,O,Λ : αfnk 7→ αf ′′′ nk ∧ supp αf ′′′ nk = Λ[supp αfnk ] (66) TZ,ω,O,Λ : ~p(X) 7→ Λ(~p(X)) (67) where α denotes EP,NP,NW,LW,S, and ~p(X) is any 4-momentum of any object at the point X in the IRF of the observer O.  Loosely speaking, for every inertial observer O there is an equivalent inertial observer O′′′ who moves relative to O with constant speed, such that the origins of the IRFs of O and O′′′ coincide, and such that the support of the individuals in GZ,ω,O and GZ,ω,O′′′ , as well as the 4-momenta at any point in the support, are related by a Lorentz transformation Λ. In other words, an object that has 4-momentum ~p at position X in the IRF of O has 4-momentum Λ(~p) at position Λ(X) in the IRF of O′′′. The collection of arrows of the categorical model is then generated by the ur-arrows defined above under arrow composition; for any arrows T : dom T → cod T and T ′ : dom T ′ → cod T ′ with cod T ′ = dom T there is thus an arrow T ◦ T ′ : dom T ′ → cod T . See Fig. 2 for a diagrammatic illustration. 4 Discussion and conclusions 4.1 Worldview In this section we want to establish a firm contact with the world view of standard SR by formalizing notions of 'events', 'massive particles', and 'massless particles' in the language of CSR. Definition 4.1 (Events) In the IRF of an inertial observer O, an event E is the manifestation of a discrete transition g1 → g2 at a spatiotemporal position X in the IRF of O; we formalize an event E as a three-tuple 〈α1, α2, α3〉 for which E = 〈X, IZ,ω,O(g1), IZ,ω,O(g2)〉 (68) An event E in the IRF of an inertial observer O and an event E ′ in the IRF of an equivalent inertial observer O′ are equivalent, notation: E ∼ E ′, if and only if E and E ′ are manifestations of the same discrete transition in the IRFs of O adn O′, respectively.  Notation 4.2 An expression 'E O−→ X', meaning: 'for the observer O the event E takes place at spatiotemporal position X', is a notation for 'MZ,ω,O |= (E)1 = X', that is, the first component of the three-tuple E is X. (This notation is based on a notation used in [8].)  Thus speaking, for any n ∈ Z and for any k ∈ Sω, the following events take place in the kth process from the nth to the (n+ 1)th degree of evolution in the IRF of an observer O: • the initial event EIn,k: this is the discrete transition EP fnk → NW fnk at the spatiotemporal position Xn,k, so that EIn,k O−→ Xn,k; • the collapse event ECn,k: this is the discrete transition NW fnk → NP f n+1 k at the spatiotemporal position Xn+1,k, so that ECn,k O−→ Xn+1,k; • the emission event EEn,k: this is the discrete transition NP f n+1 k → γ n+1 k at the spatiotemporal position Xn+1,k, so that EEn,k O−→ Xn+1,k; • the final event EFn,k: this is the discrete transitions NP f n+1 k → EP f n+1 k at the spatiotemporal position Xn+1,k, so that EFn,k O−→ Xn+1,k. The point here is that in particular the absorption and emission of a γ-ray is an event: if γ-rays are absorbed, it is at these events EIn,k; if γ-rays are emitted, it is at these events EEn,k. 12 Figure 2: Diagram illustrating various ur-arrows, identity arrows and composite arrows in the categorical model. The four dots represent the models MZ,ω,O, MZ,ω,O′ , MZ,ω,O′′ , MZ,ω,O′′′ in the collection of objects as indicated. The vertical arrows between MZ,ω,O and MZ,ω,O′ represent two permutation arrows TZ,ω,O,π : MZ,ω,O → MZ,ω,O′ and TZ,ω,O′,π−1 : MZ,ω,O′ → MZ,ω,O as defined by Def. 3.27. The circular arrow at the top middle is the identity arrow TZ,ω,O′,I : MZ,ω,O′ → MZ,ω,O′ corresponding with the identity permutation I : Sω → Sω , I : k 7→ k. Permutation arrows as defined by Def. 3.28 are not shown. The diagonal arrows between MZ,ω,O and MZ,ω,O′′ represent two translation arrows TZ,ω,O,∆X : MZ,ω,O → MZ,ω,O′′ and TZ,ω,O′′,−∆X : MZ,ω,O′′ → MZ,ω,O as defined by Def. 3.29. The circular arrow at the lower right is the identity arrow TZ,ω,O′′,0 : MZ,ω,O′′ →MZ,ω,O′′ corresponding with the zero displacement in M. The diagonal arrows between MZ,ω,O and MZ,ω,O′′′ represent two Lorentz arrows TZ,ω,O,Λ : MZ,ω,O → MZ,ω,O′′′ and TZ,ω,O′′′,Λ−1 : MZ,ω,O′′′ → MZ,ω,O as defined by Def. 3.30. The circular arrow at the lower left is the identity arrow TZ,ω,O′′′,I : MZ,ω,O′′′ →MZ,ω,O′′′ corresponding with the identity transformation I :M→M , I : X 7→ X. The bent arrow at the bottom represents the composite arrow TZ,ω,O,Λ ◦ TZ,ω,O′′,−∆X : MZ,ω,O′′ →MZ,ω,O′′′ ; for the sake of clarity other (composite) arrows are omitted in the diagram. The diagram commutes. Definition 4.3 (Massive particles) Let MZ,ω,O be a set-theoretic model of the EPT that is an object of CSR. Then for any k ∈ Sω, the function tk, for which tk :M→∗+ R , tk = ∞∑ n=−∞ tnk (69) represents the kth massive particle-i.e. an ultimate constituent of matter having rest mass-in the IRF of O, moving on a world line `k for which `k = supp tk = ⋃ {{Xn,k},∆Xn,k | n ∈ Z} (for some k ∈ Sω) (70) (Recall that tnk = EP fnk + NW fnk .) At any X ∈ `k where `k is differentiable, the 4-velocity ~u(X) is given by ~u(X) = 1 mk * ~p(X) = (u0, u1, u2, u3) (71) where ~p(X) is the 4-momentum at X and mk the rest mass as given by Def. 3.4.  13 Definition 4.4 (Massless particles) Let MZ,ω,O be a set-theoretic model of the EPT that is an object of CSR. Then any function γ n+1 k ∈ GZ,ω,O for which Eγ n+1 k represents a massless particle-i.e. an ultimate constituent of matter having no rest mass-in the IRF of the inertial observer O, moving on a world line `γk,n+1 for which `γk,n+1 = supp γ n+1 k (72) The notion of a 4-velocity, as given by Eq. (71), does not apply to massless particles.  We are now finally in a position to reap the fruits of all the definitions and interpretations by establishing contact between the language of this model of the EPT and existing physical language. For that matter, a description will be given of the kth process from the nth to the (n+ 1)th degree of evolution: (i) the initial state of the process is the particle state EP fnk of the k th massive particle, having position Xn,k and 4-momentum ~p EP n,k -its rest mass mk is predetermined by the rest mass spectrum σk; (ii) the initial event of the process is the event EIn,k-by the state transition EP fnk → NW fnk the kth massive particle gets in the wave state NW fnk with 4-momentum ~p NW n,k ; (iii) the law of conservation of 4-momentum applies: the 4-momentum of the wave state is identical to the 4-momentum of the initial particle state plus 4-momentum of a possibly observed γ-ray; (iv) the collapse event of the process is the next event ECn,k-by the state transition NW fnk → NP f n+1 k an intermediate particle state NP fn+1k with momentum ~p NP n+1,k is produced at the position Xn+1,k from the wave state of the kth massive particle; (v) the law of conservation of 4-momentum applies: the 4-momentum of the intermediate particle state is identical to the 4-momentum of the preceding wave state; (vi) upon the collapse event, we thus have E NP fn+1k ; (vii) the emission event of the process is the next event EEn,k-by the state transition NP f n+1 k → γ n+1 k the γ-ray γn+1k is emitted from the spatiotemporal position Xn+1,k; (viii) the final event of the process is the event EFn,k-upon the emission of the the γ-ray γ n+1 k , by the state transition NP fn+1k → EP f n+1 k the intermediate particle state turns into the next particle state EP fn+1k of the k th massive particle, having position Xn+1,k and 4-momentum ~p EP n+1,k (ix) the law of conservation of 4-momentum applies: the 4-momentum of the new particle state of the kth massive particle is identical to the 4-momentum of the intermediate particle state minus the 4-momentum of the emitted γ-ray; (x) the spatiotemporal separation, i.e. the invariant interval, between the spatiotemporal positions Xn,k and Xn+1,k of initial and final event is always unity: ∆s = 1. This holds for any n ∈ Z, k ∈ Sω and in the IRF of any observer O: for any observer, all individual processes are essentially the same. By these processes, massive particles alternate between a particle state and a wave state. It doesn't matter whether the massive particle concerns an ultimate constituent of matter or an ultimate constituent of antimatter: the course of events is the same, regardless of the value of the particle's characteristic number of normality. That is a feature that will remain the same also in a more elaborate model of the EPT that includes interactions, but of course then the displacement that takes place will become a function of the particle's properties. That said, below some lemma's are stated without proof, as well as some remarks: these contribute to an understanding of the categorical model CSR in terms of particles and events. Lemma 4.5 For any inertial observer O, any massive particle moves on a continuous, piecewise differentiable world line (i.e., path) in the IRF of O, so that we have η(~u(X), ~u(X)) = 1 (73) for the 4-velocity ~u(X) at any spatiotemporal position X on the particle's world line ` in the IRF of O (provided ` is differentiable at X).(See [10] for a definition of a continuous piecewise differentiable function.)  14 Lemma 4.6 For any inertial observer O, any massive particle moves piecewise unaccelerated; that is, at any point X of any massive particle's world line ` we have for the 4-acceleration ~a(X) = d dτ ~u(X) = (0, 0, 0, 0, 0) (74) provided ` is differentiable at X; here τ is the proper time.  Remark 4.7 One should realize, however, that the fact that the motion of massive particles is piecewise unaccelerated as defined in Lemma 4.6 does not imply that there is no accelerated motion. It is merely the case that if we want to speak about a '4-acceleration' in the present context, then this has to be understood in terms of a change in the 4-velocity of a particle on subsequent pieces of its world line. A formal definition is omitted here, since it is not important for the aim of this paper.  Lemma 4.8 (Universality of light speed) For any inertial observer O, any massless particle moves with the speed of light c = 1 through space. That is, at any point X on its world line ` we have( dx1 dt )2 + ( dx2 dt )2 + ( dx3 dt )2 = 1 (75)  Remark 4.9 (Degrees of evolution vs. invariant interval) The numerical 'degrees of evolution', which occur in the EPT, are a numbering of states in the direction of evolution: every individual process by which a massive particle alternates once between a particle state and a wave state then corresponds to a 'jump' in degrees of evolution of precisely one. In this categorical model CSR of the EPT, every such jump thus effects a displacement in spacetime with unit Minkowski measure: the difference in degrees of evolution between consecutive particle states of any massive particle is identical to the spatiotemporal separation between their (spatiotemporal) positions in the IRF of any inertial observer O. Likewise, photons do not evolve: they remain at the same degree of evolution, and correspondingly the Minkowski measure of any displacement of any photon is zero.  Remark 4.10 (Reality of Planck time) The unit spatiotemporal displacement between initial and final events of the elementary processes, to which massive particles are subjected, means that there a minimum time quantum: for any inertial observerO, this is the time difference between initial and final events of the elementary processes by which a co-moving massive particle evolves. In this model, this minimum time quantum has been identified with the Planck time: it is, thus, implicitly postulated that the individual processes take place at Planck scale. This identification of the minimum 'process-physical time unit' (pptu) with Planck time is somewhat arbitrary: this has the status of a conjecture-it may very well be that the pptu is orders of magnitude larger than Planck time. But nevertheless, a minimum time quantum is real in this model-its identification with Planck time gives reality to the Planck scale, and leads to verifiable predictions.  4.2 Kinematics of some physical processes The objective of this section is to describe three kinds of processes-inertial motion of massive particles, Bremsstrahlung, and laser cooling-in the language of CSR. The statements are purely descriptive: there is no 'why' to the inertial motion or to the Bremsstrahlung. 4.2.1 Inertial motion of massive particles Definition 4.11 (Inertial motion in CSR) For integers n ∈ Z and k ∈ Sω, in the model MZ,ω,O the kth process from the nth to the (n+ 1)th degree of evolution is a process of inertial motion if and only if (i) no γ-ray is absorbed at the initial event EIn,k, that is, at the discrete transition EP fnk → NW fnk at the position Xn,k: we thus have ~p NW n,k = ~p EP n,k as in Eq. (43) with ~p LW p,m = 0; (ii) no γ-ray is emitted at the emission event EEn,k upon the discrete transition NW fnk → NP f n+1 k at the position Xn+1,k: we thus have ¬Eγn+1k and, from Eqs. (39) and (41), ~p NPn+1,k = ~p EPn+1,k.  15 Translated into terms of particles and events, this means for an inertial observer O that if a particle exhibits inertial motion between the events E1 O−→ (t1, x1, y1, z1, n1) and E2 O−→ (t2, x2, y2, z2, n2), t2 > t1 on its world line `, then the 4-momentum of the particle is a constant, and there is no event E3 O−→ (t3, x3, y3, z3, n3) on ` with t2 > t3 > t1 where a massless particle is emitted or absorbed. See Fig. 3 for an illustration with a spacetime diagram. Figure 3: Spacetime diagram of a sequence of processes of inertial motion. Horizontally the spatial coordinates x of the IRF of an inertial observer O, vertically the time coordinates t. The five dots represent subsequent point-particle snk = EP fnk , the line segments connected by the dots represent subsequent time-like strings NW fnk . Together this represents the k th massive particle on its world line `k; the constant slope of `k reflects the constant 4-momentum. 4.2.2 Bremsstrahlung Definition 4.12 (Bremsstrahlung in CSR) For integers n ∈ Z and k ∈ Sω, in the model MZ,ω,O the kth process from the nth to the (n+ 1)th degree of evolution is a process with Bremsstrahlung if and only if (i) no γ-ray is absorbed at the initial event EIn,k, that is, at the discrete transition EP fnk → NW fnk at the position Xn,k: we thus have ~p NW n,k = ~p EP n,k as in Eq. (43) with ~p LW p,m = 0; (ii) a γ-ray is emitted at the emission event EEn,k, that is, at the discrete transition NP f n+1 k → γ n+1 k at the position Xn+1,k: we thus have Eγn+1k and, from Eqs. (39) and (41), ~p EPn+1,k := ~p NPn+1,k − ~p LWn+1,k.  So as a simple example, consider that the point-particle EP fnk has 4-momentum (E, px, 0, 0), such that px > 0 and −E2 + (px)2 = −m2. At its transition to the time-like string NW fnk , this 4-momentum is conserved, so at any point on the line segment occupied by the time-like string NW fnk , the 4-momentum is also (E, px, 0, 0). Upon the transition of the time-like string NW fnk to the intermediate point-particle NP fn+1k , the latter emits a γ-ray with 4-momentum (∆E,∆px, 0, 0) with px > ∆px > 0 and ∆E = ∆px. Upon emission, the point-particle NP fn+1k then transforms into the new point-particle EP fn+1k : its 4-momentum is then (E′, px −∆px, 0, 0) for which −(E′)2 + (px −∆px)2 = −m2 so E′ < E. 16 Translated into terms of particles and events, this means for an inertial observer O that if a particle emits Bremsstrahlung between the events E1 O−→ (t1, x1, y1, z1, n1) and E2 O−→ (t2, x2, y2, z2, n2) on its world line `, t2 > t1, then the energy and spatial momentum of the particle decrease stepwise through the emission of massless particles (photons). See Fig. 4 for an illustration with a spacetime diagram. Figure 4: Spacetime diagram of subsequent processes with Bremsstrahlung. Horizontally the spatial coordinates x of the IRF of an inertial observer O, vertically the time coordinates t. The two dots represent subsequent point-particles snk = EP fnk and s n+1 k = EP fn+1k , the line segments connected by the dots represent subsequent time-like strings NW fn−1k , NW fnk , and NW fn+1k . The wavy blue lines represent emitted γ-rays γnk and γ n+1 k . Together this represents the k th massive particle on its world line `k, plus two emitted photons; the increasing slope of `k reflects the stepwise deceleration. 4.2.3 Laser cooling Definition 4.13 (Laser cooling in CSR) For integers n ∈ Z and k ∈ Sω, in the model MZ,ω,O the kth process from the nth to the (n+ 1)th degree of evolution is a process with laser cooling if and only if (i) a γ-ray γpm from a laser source is absorbed at the initial event EIn,k, that is, at the discrete transition EP fnk → NW fnk at the position Xn,k: for some p ∈ Z and m ∈ Sω we thus have ~pNWn,k = ~p EPn,k +~p LWp,m as in Eq. (43), but in particular with ENWn,k < E EP n,k (decreasing energy); (ii) no γ-ray is emitted at the emission event EEn,k upon the discrete transition NW fnk → NP f n+1 k at the position Xn+1,k: we thus have ¬Eγn+1k and, from Eqs. (39) and (41), ~p NPn+1,k = ~p EPn+1,k.  So as a simple example, consider that the point-particle EP fnk has 4-momentum (E EP n,k , px, 0, 0), such that px > 0 and −(EEPn,k )2 + (px)2 = −m2. At its transition to the time-like string NW fnk , a γ-ray is absorbed with 4-momentum (∆E,−∆px, 0, 0) with −∆px < 0 and ∆E = ∆px. Then at any point on the line segment occupied by the time-like string NW fnk , the 4-momentum is (E NW n,k , px −∆px, 0, 0,m) for which −(ENWn,k )2 + (px −∆px)2 = −m2 so that ENWn,k < EEPn,k . Translated into terms of particles and events, this means for an inertial observer O that if a particle is laser cooled between the events E1 O−→ (t1, x1, y1, z1, n1) and E2 O−→ (t2, x2, y2, z2, n2) on its world line `, t2 > t1, then the energy and spatial momentum of the particle decrease stepwise through the absorption of massless particles (photons) emitted by a laser tube. See Fig. 5 for an illustration with a spacetime diagram. 17 Figure 5: Spacetime diagram of subsequent processes with laser cooling. Horizontally the spatial coordinates x of the IRF of an inertial observer O, vertically the time coordinates t. The two dots represent subsequent point-particles snk = EP fnk and s n+1 k = EP fn+1k , the line segments connected by the dots represent subsequent time-like strings NW fn−1k , NW fnk , and NW fn+1k . The wavy blue lines represent γ-rays γ1 and γ2 from a laser source that are absorbed at the points Xn,k and Xn+1,k, respectively. Together this represents the kth massive particle on its world line `k, plus two absorbed photons; the increasing slope of `k reflects the stepwise deceleration by laser cooling. 4.3 Conclusions In this paper a categorical model of the EPT incorporating SR has been specified: the main conclusion of this tedious exercise is that this proves that the EPT agrees with SR. This result renders the EPT consistent with the outcome of real-world experiments and observations that can be described as predictions of SR-examples are the null result of the Michelson-Morley experiment [11], and the observed prolonged lifetime of fast muons [12]. In addition, it has been shown that laser cooling and Bremsstrahlung can be described in the language of the categorical model CSR. A main outcome is that an individual process effects a unit jump in space-time: for any individual process and for any observer, the spatiotemporal separation ∆s between the spatiotemporal positions of the initial particle state and final particle state of the process is always unit-that is, always satisfies ∆s = √ ∆t2 −∆x2 −∆y2 −∆z2 = 1. This directly relates the process-physical principles of the EPT to observable motion of massive (anti)particles. The present study doesn't purport to yield an advancement in relativity theory. In addition, a limitation of this study is that it has been focused purely at demonstrating the agreement of the EPT with SR, and with SR alone. Further research is therefore required to establish whether or not the EPT agrees with the knowledge of the physical world obtained from the experimentally confirmed predictions of modern, relativistic interaction theories. Acknowledgements This study has been facilitated by the Foundation Liberalitas. References [1] M.J.T.F. Cabbolet, Ann. Phys. 522, 699-738 (2010); 523, 990-994 (2011); 528, 626-627 (2016) [2] M.J.T.F. Cabbolet, A methodolocal note on proving agreement between the Elementary Process Theory and modern interaction theories, Preprint: philpapers.org/rec/MARAGN-2 (2018) 18 [3] A. Einstein, Ann. Phys. (Leipzig) 17, 891-921 (1905) [4] S. Eilenberg, S. MacLane, Trans. Amer. Math. Soc. 58, 231-294 (1945) [5] J. Butterfield, Br. J. Philos. Sci. 57(4), 709-753 (2006) [6] M.J.T.F. 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