Revised: Going in Circles – Part II

In Going in Circles – Part I we presented the basic reasoning underlying Q-Orders by analyzing the configurations of elements on 1 Circle and the combination of 2 circles. Already in the discussion of Axiom 4 we said informally that the combination of circles should create new ones and specified which would be permitted based on pictures.

Yet it would be a fruitless and hence useless task, to continue that way, since very rapidly we would be defeated by combinatorial explosion, as 4 elements on 1 circle already allow 3 configurations according to Axiom 1 and Lemma A 1. We need urgently rules of interference! For more than 4 elements on 1 Circle, Lemma A 2 solved the problem, as shown by Lemma A 5.

Yet for the combination of more than 2 circles –at least 3^3 = 27 possibilities- we have no rules yet and analyzing even in this case the configurations one by one more would be very clumsy, and literally impossible for only countable many elements.

A part of the problem is solved by Axiom 3.

Axiom 3  Regular

It claims that

1. whenever all pairs of 3 points can be found on some circle –regardless their configurations there-, there should be a 4th element to give a complete circle, where all 4 may be positioned.
2. whenever all triplets of 4 points have their circle, the 4 points themselves may be positioned on 1 circle.

As this is a global claim, Axiom 3 is truly an Axiom i.e. a Thinking – Hypothesis that may be false in the real world (as the Parallel Axiom in non-Euclidian Geometries). Actually I’ve done some investigation on the consequences, if it were false in the frame work of Q-Orders. Due to its hypothetical character we will flag further on any proof that uses it.

Sorry, no pictures for Axiom 3! (it would have been 91 for 3.1 and 271 for 3.2) …. but whoever has the time and resources to draw them up all, including the configuration for the resulting circle, is kindly invited to provide them, or even better a small animation applet …).

Axiom 3 and 4 combined express a second Thinking Hypothesis as a rule of interference to allow later on for consistent orientation.

Lemma A.6

Whenever a pair {c,d} is separated by another pair {a,b}, and this pair appears on the same side of some circle , then {a,b} separate {c,d} also on this circle. Or more loosely, if {a,b} separates {c,d}, then it does so anywhere or even more general: {{a,b},{c,d}} is a universal relation.

This Lemma has far reaching consequences as it’s implied by the concept of transitivity of partial orders and likewise by the concept of continuous functions in Real Analysis.

Then Lemma 7 shows that a pair {u,v} not only separates all circles that run trough {u,v} but cuts them into two equivalence classes of points- say inner-outer, left-right, upper-lower - without defining which is which.

Lemma A 7 

Q-Orders sometimes have adjacent points:
Lemma A 8 Adjacent Points 

Adjacent are those points which can not be separated by Q. It’s a local property, i.e. something that can’t be separated by one interval, can’t be separated by any other.

With the above, we are done with all local Axioms, remaining only the Axioms 5 and 6  that will generalize our findings about orientation.

Revised: Going in Circles – Part I

Strange as is may sound, to me Axioms are not just formal statements –obviously if we like to do Mathematics later on, they have to be also formal statements in some formal language-, but rather the intent to express as precisely as possible a concept found in reality, Plato would have said an idea.

In this post we’re going to show for 3 of the first 5 Axioms how they have been found and what was the essential input for the remaining 2 of them. Our basic idea are the circle and relations among 4 distinct points on that circle, expressed by an unordered pair of 2 unordered pairs.

Axiom I Q-Relation

Why this relation? Let’s have a look on a circle and just any 4 distinct points and their possible configurations.

To avoid any premature symbolic interpretation, we use 4 small hearts in 4 different colors to mark the 4 distinct points. Then the simplest way to express the only invariant of these configurations is by stating that the set {red heart, yellow heart} separates the set {blue heart, green heart}, a statement that remains true from which ever side we look on the circle; from before, from behind, from top, from bottom, from left, from right … and anywhere in-between.

The remaining definitions just introduce short notations for the relation itself and to denote that 4 points occur actually jointly on one Circle.

As easily seen, any 4 distinct points may be grouped into exactly into 1 set of 2 sets. As candidate for an axiom we note therefore (it will be a consequence of our final Q-Order Axiom 4):
Lemma A.1 

Any additional statement would need additional information from either some external frame of reference –like the blue heart is on the left- or an additional convention to describe a way how one heart after another may be visited, i.e. an inner orientation. We note that apparently there are just 2 inner orientations.

Axiom 2 Reduced
 
Axiom 2 states then that all shall be expressed in terms of points and circles, such that 2 points which appear anywhere –at any circle- in identical configurations shall be considered identical. And for the sake of completeness, we eliminate the all-isolated and the all-connect points.

What happens if a 5th heart enters the game? Let’s see: The 4 purple hearts with 5 show just the 4 possible positions of the 5th point. Fortunately we may describe its position exactly using only the already defined relation Q, as the next axiom candidate show (again as Lemma a consequence of the later Axiom 4)

Lemma A.2 

In words: If 5 points are on a circle and for 4 of them their configuration is known, then the 5th falls in exactly 1 of 4 alternatives. As a consequence, the configuration of any set of distinct points on a circle may be defined using only the relation Q.

In the finite case, there are (n-1)! circle-configurations for n elements. As stated already elsewhere, the Sumerian used 5, 6 and 7 elements and their configurations as the base for their mathematics and geometry.

Looking on the above configurations, we find a triplet of relations of 5 points that never occurs:

Lemma A.3 

It just expresses that {a,b} splits the circle in 2 halves.

What if we have two circles and not only one? Axiom 4 deals with these cases and the above at the same time.

Axiom 4 Ordered

Let’s picture the first case. Obviously there should be some new circles formed by elements of both and the new circles will share the elements a and b. Yet not all combinations of 4 new circles would be mutually compatible (lemma 3). Axiom 4.1 offers the only two choices possible: (an inner {{a,b},{d,e}} and an outer circle {{a,b},{c,f}}) or (a left {{a,b},{c,e}} and a right circle {{a,b},{d,f}}).

Now the second case:

Again not all possible combinations would be compatible. Axiom 4.2 offers the only two choices possible: Either an inner circle with 5 elements {a,b,d,e,f} or an outer circle with again 5 elements {a,b,c,e,f}.

Please note that the terms like inner, outer, left and right have no intrinsic meaning yet. They just may help in seeing the circles.

Finally the third possible case:  Here the original circles are {{a,c},{d,b}} and {{a,f}},{{e,b}} and almost any additional circle is possible, as that there is none at all. Hence Axiom 4 makes no claims for this situation.

Lemma 1 and 2 were initial working hypotheses (candidates for axioms). We leave it to the reader that now - as intended -they are consequences of Axioms 1, 2, 4.1 and 4.2.

We will go back for a moment to a single circle and show how implicitly there is an order defined as soon as one decides, which points should be the beginning and the end respectively.

With Lemma 2 already holds the following:
Lemma A.4  
This may be used, once fixed a and b to define a partial order.

Lemma A.5 

Have a look on the following picture

and you easily verify that {{b,n},{m,e}} as defined for the circle, will order the whole set {1,2,3,4,5,6} as one would expect. Note that inverting the order of (b,e) to (e,b) inverts the order of the set, again as one should expected.

The dual orientation of circles is the underlying principle. The remaining parts of the first 6 Axioms will insure that there is a consistent orientation –up to duality- for the whole Q and in all of its admitted circles, by virtue of which in any Interval we may operate as usual with partial orders.

Revised Axioms for Q-Spaces: Q-Orders

There have been month of silence in this BLOG, as I was running into serious troubles beyond the elementary Axioms for Q-Orders, i.e. those that equip Q-Spaces with a suitable topology. More over it turned out that –at least for the moment- an additional axiomatic relation may be needed: while the relation Q clearly models the conformal invariant structure –the Topology- of Space-Time, it misses the projective invariant part, which only jointly as shown by Weyl long time ago fix also the metric. The work isn’t done yet. Nevertheless I would like to present the current stage with its stabilized Axioms for Q-Orders and the first intents to manage the additional relation.

We are looking for a combinatorial framework that, in an essential way, includes the structure of Space-Time as a continuous model on one side and the structure of of Petri-Nets as a finite (countable) model on the other.

Essential means that physically different Space-Times and logically different Petri-Nets shall have different models and that different models produce different Space-Times and different Petri-Nets respectively.

For Space-Times a seminal contribution of S. W. Hawking¹ introduced a unique combinatorial structure –a partial-order– attached to Lorentzian Manifolds with some additional restrictions, that up to conformal mappings defines the manifold (review² on Causal Space-times). David Malament³ showed how this combinatorial structure alone, under suitable conditions, is sufficient to reconstruct Space-time up to conformalty.

For Petri-Nets since 1973 there as been a systematic effort⁴ to detect the underlying combinatorial structures, specifically in the theory of Concurrency or causal structures defined by event-occurrence systems.

The problems

1. the mentioned Space-Time models in their definition make a heavy use of concepts typical for the continuous world, like Hausdorff-spaces as basic model-domain or using properties borrowed from Linear Algebra, all which as such can not be transported into the finite/countable domain.

2. the mentioned Petri-Net models -namely concurrency-theory- require countable models to work and therefore as such a not suited to express all the technical concepts as used in continuous models. On structural level, there is no Linear Algebra, hence appears on first sight impossible to express concepts like convex let alone tensors or more complicated constructs.

3. Both models depend on Global Partial Orders even when expressing purely local concepts, a slight contradiction with the basic idea of General Relativity as something locally defined.

The ideas for solution

1. Both model domains use Paths respectively Curves as a basic building block, where Curves in both domains model trajectories -world-lines- of particles (more precisely potential trajectories see⁵). All expressed relations and properties can be re-written using only curves and the relations among points as defined by curves.

2. As Petri pointed out quite early⁶, on partial orders there exists a generalization for the concept of Dedekind-continuity and -completeness that allows for countable models, yet if applied to full-orders produces the known results. Crucial are two types of points, closed and open, while retaining the idea that the emerging topologies should be connected.

3. A little bit later Petri proposed the separation relation {{a,b},{c,d}} -an unordered pair of unordered pairs- as the basic order-producing relation. This relation expresses the separation of 4 points on a line, and is well defined on any Jordan-Curve, open or closed, i.e. there is no difference between a line, may be with suitable compactification, and a circle.

4. A careful analysis of the original article from Hawking, specifically analyzing the relation between local time-like cones, which form the base for the topology, the definition of regular paths in that topology and their relation to time-like curves, allowed to eliminate the reference to linear concepts like convex and to define local time-like cones and their properties using only combinatorial concepts.

5. This revision in turn demanded a revision of concepts in Petri-Nets. While in the original model the open elements are conditions and the closed elements are the events, and likewise sets border by conditions considered open while sets bordered by events closed, we need exactly the dual: events and event-bordered sets are open, conditions and condition-bordered sets are closed. It should be noted that for countable structures -Petri-nets are normally assumed to be countable- both sets -open and closed- define the same dual Alexandrov⁷ Topology. However already the comparison of Dedekind continuity between total orders and half-orders alas Occurrence-Nets shows that the common type of elements in both -the non-branching conditions- must be closed.

6. In a Hawking-space all points are closed. Therefore it was necessary to overcome the initial interpretation of Einstein as if world points would correspond to physical events. They do not! If the Hawking-space models the loci -the geometry-, then a physical event can not have an exact place as Quantum-Mechanics tells us. A similar observation made decades ago Pauli⁸. Curiously enough, in this interpretation nothing ever happens in Hawking-Space as there are no events. To have events we must coarse grain first.
   

7. Likewise a too naive interpretation by Net-Theory of GRT had to be abandoned, as if each world-point branches into infinite many world-lines. Actually a world-point summarizes the whole time-like pre- respectively post-cones as such and not individual lines. This is the essence of the construction of regular paths by Hawking and the distinguishing conditions from Malament.

Based on the above I obtained the Axiom-sets shown in the picture below.

Some models for Q-Orders are:

(1) Occurrence-Nets (with the above change and some additional requirements) as subclass of Petri-Nets
(2) The Real Numbers (but not Rationals nor Integers) (Q-order is derived from classical order)
(3) The Unit-Circle S1 (and the Circle Group) (but not n-cyclic Groups) and the Real Line (Q-order is derived from the relation among four points)
(4) The Minkowski-Space and the Quaterions (Q-order is derived from Q-Topology)
(5) The Causal structure of a Lorentzian manifold as defined by Hawking and others (Q-order is derived from the relation among four points on a time-like curve)

1 S. W. Hawking A.R. King and P. J. McCarthy, A new topology for curved space-time which incorporates the causal, differential and conformal structures Journal of Mathematical Physics Vol. 17, No 2, February 1976
2Alfonso García-Parrado, José M. M. Senovilla, Causal structures and causal boundaries, arXiv:gr-qc/0501069v2
3 D. Malament, The class of continuous timelike curves determines the topology of spacetime Journal of Mathematical Physics, July 1977, Volume 18, Issue 7, pp. 1399-1404
4 Olaf Kummer, Mark-Oliver Stehr: Petri's Axioms of Concurrency - A Selection of Recent Results , Proceedings of the 18th International Conference on Application and Theory of Petri Nets, Toulouse, June 23-27, 1997, Lecture Notes in Computer Science 1248, © Springer-Verlag , 1997
5 David B. Malament, Classical Relativity Theory, arxiv.org/abs/gr-qc/0506065v2
6 Petri, C.A., Concurrency. Lecture Notes in Computer Science Vol. 84: Net Theory and Applications, Proc. of the Advanced Course on General Net Theory of Processes and Systems, Hamburg, 1979 / Brauer, W. (ed.) --- Berlin, Heidelberg, New York: Springer-Verlag, 1980, Pages: 251-260
7Not to be confused with the Alexandrov Topology used by Hawking
8Pauli, Vorlesungen in Turin über nichtlokale Feldtheorien in Google-Books http://books.google.com/books?id=NU9OUj-f8cYC&hl=es Page 34 ff.

The Great Simplification – Part II

After slashing the original definition of Axiom 6, here comes a similar reduction of Axiom 7, originally introduced in The trapped Arrow of Time – Part III.

The new
Axiom 7 Q-Loops
    
Axiom 7.1 are technical definitions:  completely Q-ordered sets and the closed hull of a set. 

Axiom 7.2-4 introduce a substitute for Jordan-Curves. 7.2 defines the property of being connect for a set in terms of topology. 7.3 defines the relation of being topologically separated for two points.  Please note that we don’t require that the whole space is T1. 7.4 defines J-Curves as sets that contain for each point at least one separated partner and fall apart exactly if a separated pair is removed. This definition requires implicitly the Axiom of Choice, therefore it’s flagged.

For the moment let’s assume that J-Curves exist -later on we will claim the existence of rather specific ones- and see whether the definition meets our expectations. Axiom 7.5 introduces the set of connected subsets into which a separated pair splits the J-Curve. By definition for J-Curves, there must be at least 2 of them.    
Lemma S 2 Segments  
 
The lemma shows that –as intended- a separated pair splits a J-Curve in just 2 segments. And these 2 segments contain at least each a point for a separated pair, that is we’ve got two pairs that mutually separate each other.

Sounds  familiar? Well, be aware that the term J-Curve was introduced and some of it’s properties shown without reference to the initial Q-Order, yet –as intended- on a J-Curve there exists a natural Q-Order, as defined by Axiom 7.6.
Lemma S 3 J-Order 

Be aware that by no means all J-Curves correspond to q-ordered sets, as shown below. 

The above yellow Rhombus  is a  J-Curve, yet it’s not a Q-ordered set but rather built by of 2 Q-ordered sets, the left and right side.

It appears as if we might start out just with some topology with some nice properties … and [re-]construct the Q-Order. For the moment we ask only –consistent with our whole approach- that every totally q-ordered set shall be consistently embeddable into some J-Curve, where consistency means that original Q-Order and derived Q-order of the curve are the same.

Axiom A 7.7 Consistent Embedding 

Axiom A 7.7 has backward consequences for the Q-topology.
Lemma S 4 Connected Space 
  
As immediate consequence of Axiom 7.6, J-Curves finally do exist. J-Curves connect the whole space (S 4.2), which is hence a connected topological space (S 4.3).

Finally
Axiom 7.8 Local Orientation

This Axiom establishes an intrinsic relation between J-Curves –remember they are closed- and the underlying Q-Topology. For each point –respectively its closed hull- there shall exist at least one neighborhood, sufficiently large that it can be split into two subsets –sometimes called local future and local past, or local input and local output- such that  each of these subsets is J-convex – i.e. any two points can be connected by a J-Curves, but sufficiently small that any J-Curve that connects between the sets contains at least one external point.

The picture below illustrates the concept.

The two sets are
{ {(0,0),(-1,0)}, (-1,0), {(-1,0),(-1,1)}, (-1,1), {(-1,1),(0,0)}, {(-1,1),(0,1)}, (0,1),{(0,1),(0,0)} }
{ {(0,0),(0,-1)}, (0,-1), {(0,-1),(1,-1)}, (1,-1), {(1,-1),(0,0)} {(1,-1),(1,0)}, (1,0),{(1,0),(0,0)} }

Axiom 7.8 requires at least two dimensions (or two J-Curves). As to be shown, it captures the  underlying  essence of the Hawking-construction  for regular curves, which in the original text is scattered between local properties of the manifold –existence of local convex neighborhoods in terms of the Manifold-Topology, global causality-conditions –strong causality-, all needed to effectively define Regular Curves, and finally the properties  defined by the construction as such.

This ends our preliminary presentation of the new Axiom 7 Q-Loops.

The Great Simplification - Part I (corrections)

As stated clearly in the presentation, this BLOG documents Work in Progress, not at all final results. Some decades ago, when I started to study seriously Mathematics, I always wondered: how the hell were those powerful initial axioms and definitions found, which then gave origin to such powerful theories? Most of my teachers (and most Text-Books) presented only the final results in the sequence Axiom, Axiom, Definition, Definition, Lemma, Lemma, Lemma .. and out of box jumps a wonderful theorem. Actually it took me some time to discover that the way the final results were presented had little, if any, to do with how they had been constructed. Similar I suspect most traditional papers as published by Journals are –may be due to the harsh space-limits of Journals and time-limits of potential readers- rather an intent to impress than to explain making understand how. (I do doubt however that it truly saves time for the really interested reader, because unless he or she knew already, they have to reconstruct a living body of knowledge by analyzing only its bar bones). 

So here goes a new round in my construction process: While working on a new post about Dedekind-Completeness and its extension to partial orders and Q-orders I got stuck in some little, tiny detail: from Net-Topology I know that there should be never adjacent open or adjacent close points. And Hawking-Topology requires that pieces of world-lines have to be continuous, hence connected, images of [0,1], which boils down to exactly the same requirement as for Net-Topology, yet it turned out to be impossible to deduce this simple property from the Axiom VI in its former form. Reluctant to dump not only already written pages but some 20 Lemmata or so and their proofs, I tried first –what I suspect many do- to patch the initial Axioms –in this case Axiom 6   -the results can be seen in The trapped Arrow of Time  Part II- but finally decided to start over again.

The reasons: Though very common, in my feeling for esthetics these patches damage any beauty of a true Axiom-System. Second the former versions used a concept –Lines- that has its own flaws already on conceptual level: it requires the Axiom of Choice twice for its definition, which makes it very cumbersome to use later on and its in a way unphysical, as a physical process can never correspond to a geometric world-line, again a consequence of the uncertainty-principle, both considerations mentioned already in The trapped Arrow of Time - Part I.

Yet as both Hawking and Petri use Lines very intensively, I gave them at least a try, but finally decided to drop the concept Line as something fundamental and tried to rewrite Axiom 6 and Axiom 7 without using it … and received as gratification a great simplification without –as to be shown- loosing essence in modeling, that is if there are lines, then they behave as before.

Here the new Axiom 6 corrected
Axiom 6 Q-Topology            
    
The Axiom starts with the definition of U-neighborhoods of a point using the Q-Relation itself. Each element of neighborhood U shall by member of some limiting pair (A 6.1.1), for each member of a pair there shall be a representative (A 6.1.2), and finally a point of the same interval closer to the point than some already included, shall be likewise included (A 6.1.3). For the notion closer see Going Backward, Going Forward - Part I and Going in Circles - Part II.

The picture below may be helpful:

A blue neighborhood { (0,0), {(0,0),(-1,1)}, {(0,0),(-1,-1)},)}, {(0,0),(1,-1)},)}, {(0,0),(1,1)} }, a red neighborhood { {(0,0),(0,1)}, (0,1), (0,0) }.
The color-families green and purple represent similar as before related pairs w.r.t. the point (0,0).  

Lemma S 1 shows that arbitrary unions and meets of two U –neighborhoods are U –neighborhoods.    corrected 
   
Axiom 6.2-6 Q-Topology    
   
As before, we introduce closed points as all those which can be distinguished from any other point by using the Q-Relation. An open set of the Q-topology includes for each closed point  a U-neighborhood. Lemma S 1 essentially proves already that O is a topology. We claim consistency of concepts: what is closed as Q-Relation shall be closed as Q-Topology. The complete cone of a point shall be open.

Here ends the presentation of the simplified Axiom 6.

A Glimpse of the Big Picture

I’ve been asked whether there is a single text, that comprises the most essential of Q-Orders. Well – there is not, or not yet. To get an idea of what is and what not yet, may be the below picture helps.

The Big Picture

The left side shows in a very simplified manner the tower of mathematics beneath contemporary, classical General Relativity Theory. The right side, as far as I’m aware  less solidified and standardized yet, the tower of mathematics beneath contemporary usage of Petri-Nets in Informatics.

I had this picture already in my office about 30 years ago. Now, I thought, if one would like to relate seriously the truly interesting part on top of the left tower with something may be interesting on top of the right tower –meaning by proofs and not by analogies- then one would need a mathematical bridge between both towers, starting already on some quite low-level of both towers.  This is, where the work on Q-orders started. They should permit both types of domains –Real and Countable- and should produce one single category of a topology to relate both towers.

To complicate the issue, I discarded partial orders as the funding concepts, as both from Physics and Net-Theory we knew that it are the cycles, that produce basic invariants, on the very end even enable measurement: while we can measure our time as cycles, we can’t measure space without using time. [I know Carlo Rovelli will most strongly disagree].

I did know already the red elements towards the center, they were developed while I was still a GMD. 10 years later, Olaf Kummer and  Mark-Oliver Stehr (1) give a quite complete résumé of what has been found out. Yet –though published already in 1976- I was not aware of the proposal of Hawking et. al (1) for a New Topology for Space-Time, i.e. the blue elements towards the center.

May be if I’d known, my life’s history would have been different. Yet I did know already then that the basic invariant of embedding Petri-Nets into (1+n)-Vector Spaces seemed to be the group of conformal transformations. So after leaving GMD in 1985 I accumulated notes and proofs on predecessors for Q-Orders, yet without any serious break-through, still I succeeded giving Axioms 1-5 for Q-orders their current form. (See Going in Circles Part I to III).

After getting back more seriously, about in 2007 or so, and using the resource Internet (+some additional dollars, unfortunately many seminal papers are still sold, while they should be free for humanity), I got across the cited paper from Hawking and a later companion by David Malament (3). Suddenly there was a correspondence already worked-out: a structure on both sides, whose geometrical invariant is the conformal group.

So the only task remaining was to find an axiomatic definition for the Q-Topology, that covered both sides –the Hawking Topology and the Net Topology-. Yet Hawking and Malament use for definitions and proofs many features intrinsically related with lower parts of their tower, starting with the standard definition of paths, which carries automatically the Hausdorff-properties of Reals into the Topology to be defined, over concepts like locally convex, which make sense in a Real-Linear-Vector-Space setting, yet not in the right tower etc. etc. etc. So it took some time to get to the current axioms of Q-Topology (Axiom 6) and Q-Loop-Topology (Axiom 7), which both do only rely on concepts available on both sides of the Big Picture. (The trapped arrow of Time Part I-III).

So what is finished –at least in my electronic scrap-book- is the basic bridge. And I will continue to present its definition and related results during the next weeks. Specifically I will introduce a rich set of models, all by themselves important, for Q-Orders, which will construct something may be close to the fundaments for middle-tower that might be of some use by itself. Done, I’ll proceed to recompile the essays into one single paper, to be published may be through my arXiv account.

Yet I’m fully aware that there is still neither Informatics nor Physics in the picture, which both start on top of their respective towers. The fundamental problem: we have no tools yet to formulate equations or even quantify invariants. Though it’s known that the Hawking-Topology allows to reconstruct the metric and it’s known that the conformal group of transformation corresponds to a single central source of gravity, but as far as I know –and found googleing the Internet- nobody has investigate yet the full way back: i.e. given a Einstein (or Einstein-Cartan) field-equation and posing some reasonable constraints on its right –Energy-Tensor- side, what are the effects on the underlying Casual Structure? (Though Alfonso García-Parrado  and Miguel Sánchez (4) may give some hints).

As these tools are still missing, it doesn’t make much of sense either to speculate about the formal relations between the big tower underlying contemporary Quantum-Mechanics and the modest elements presented so far.

There is however the sketch of a work-program to complete and solidify the middle-tower, once finished the above presentation.  

1. Going downwards, it appears attractive to introduce the concept of a Q-Manifold. The Q-Manifold will be defined using the Complex or Quaternion (alas 2 and 4 dimensional Minkowski-Spaces) as base-space, using q-continuous functions instead of the usual Euclidian ones. The advantage: it appears as if a Q-Manifold is automatically smooth.
2. As defining measures on S1 is quite standard and the q-continuous images of S1 correspond to Q-Loops, it should be possible to get some notion of distance for Q-Loop-Spaces.
3. Similar remembering that the Tangent-Space may be defined as local equivalence classes of paths (alas Q-loops in our model), it should be possible to have a sort of Tangent-Space for Q-Loop-Spaces.

My only hope: my conditions of work will permit to continue … and the readers of this BLOG don’t get too impatient too soon as truly seminal posts are still month away.

The Axioms 
   

(1) Olaf Kummer, Mark-Oliver Stehr: Petri's Axioms of Concurrency - A Selection of Recent Results , Proceedings of the 18th International Conference on Application and Theory of Petri Nets, Toulouse, June 23-27, 1997, Lecture Notes in Computer Science 1248, © Springer-Verlag , 1997
(2) S. W. Hawking A.R. King and P. J. McCarthy, A new topology for curved space-time which incorporates the causal, differential and conformal structures Journal of Mathematical Physics Vol. 17, No 2, February 1976
(3) D. Malament, The class of continuous timelike curves determines the topology of spacetime Journal of Mathematical Physics, July 1977, Volume 18, Issue 7, pp. 1399-1404
(4) Alfonso García-Parrado, Miguel Sánchez,  Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples, arXiv:math-ph/0507014v2