A purely mathematical proof of the existence of the …As Exhibit A, consider a recent book, titled...

The road from reality: A purely mathematical proofof the existence of the observable universeAlexandre Harvey-Tremblay 1,2

1 independent researcher2 [email protected]

May 22, 2020

Cartesian Paradigm

My ProposalI prove

I am Pitfall @ Mind-body ProblemI thinkI doubt

Structure of Reality(inherited from reality, at the end of the

road)

Statistical Ensembleof Experiments

Universal Equation of State

Observable UniverseThe difference in entropy is the

mathematical origin of thequantum measurement

Set of RealizedExperiments

Observed Universe

Reality (defined as the totality of what

"I" (the system) proves)

Physics is obtained, first, by creating a purelymathematical model of science, then by

solving it.

(mathematical descriptionthereof)

Observer(mathematical description

thereof)

Physics

Figure 1: This chart describesthe most fundamental descrip-tion of reality I believe to bepossible. Specifically, my pro-posal is sufficiently powerful toinherit the coveted indubitableproperty of the Cartesian uni-versal doubt method then tocarry it forward such that thestructure of reality, in the formof a model of the observableuniverse including the laws ofphysics, is inherited indubitablyfrom pure reason, whilst avoid-ing the pitfall at the mind-bodyproblem. The strategy is to con-struct a purely mathematicalmodel of science, then to solveit for the laws of physics. In themodel, reality is described asa set of realized experiments(the set of what "I" can prove),and the laws of physics arethose that remain universallyvalid for all permutations orrearrangements of experimentsconstrained only by the set ofall possible transformations(which we call nature). Essen-tially, experiments constrainedby nature equals physics, in themost general mathematicalsense imaginable.

a purely mathematical proof of the existence of the observable universe 2

Contents

1 Notation 3

2 Towards a mathematical theory of reality 42.1 What, exactly, is wrong with postulated laws? . . . . . . 5

2.2 What, exactly, is missing from mathematics? . . . . . . . 11

2.3 What is reality contingent upon? . . . . . . . . . . . . . . 15

2.4 Hint: John A. Wheeler . . . . . . . . . . . . . . . . . . . . 19

2.5 Hint: Gregory Chaitin . . . . . . . . . . . . . . . . . . . . 23

I Science 25

3 The Axioms of Universal Physics 25

II Nature 28

4 Technical Introduction 284.1 Recap: Statistical Physics . . . . . . . . . . . . . . . . . . . 29

4.2 Recap: Algorithmic Thermodynamics . . . . . . . . . . . 32

4.3 Attempt 1: Algorithmic Equation of State . . . . . . . . . 34

4.4 Hint: Entropy, Information and Space-time . . . . . . . . 39

4.5 Attempt 2: Designer Ensemble . . . . . . . . . . . . . . . 41

5 Universal Thermodynamics 425.1 Recap: Geometric Algebra . . . . . . . . . . . . . . . . . . 47

5.2 Universal Representation . . . . . . . . . . . . . . . . . . 50

5.3 Universal Metric . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 Universal Ensemble . . . . . . . . . . . . . . . . . . . . . . 63

III Physics 67

6 Results (Spacetime Entropy) 676.1 Law of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Beckenstein-Hawking Entropy . . . . . . . . . . . . . . . 68

6.3 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.4 Fermi-Dirac statistics of events . . . . . . . . . . . . . . . 70

6.5 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 72

6.6 Decoherence at the Time-like/Space-like Boundary . . . 74

6.7 Quantum Field Gravity . . . . . . . . . . . . . . . . . . . 76

7 Results (Cosmology) 777.1 de Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . 77


7.2 Cosmological Pressure . . . . . . . . . . . . . . . . . . . . 78

7.3 Cosmological Inertia . . . . . . . . . . . . . . . . . . . . . 79

7.4 Entropic Derivation of ΛCDM . . . . . . . . . . . . . . . 80

8 Discussion (Quantum Measurements) 818.1 Recap: measurement problem . . . . . . . . . . . . . . . . 81

8.2 The true cause of measurements . . . . . . . . . . . . . . 83

9 Conclusion 85

IV Annex 85

A Universal Interval and Norm of a universal event in four dimen-sions 85

1 Notation

Parentheses will be used to denote the order of operations, andsquare brackets will be used to define valued maps. For instance amap f : X → R will be written as f [x] for x ∈ X. S will denote theentropy, and S the action. Sets, unless a prior convention assigns itanother symbol, will be written using the blackboard bold typogra-phy (ex: L, W, Q, etc.). Matrices will have a hat (ex: A), vectors willbe in bold (ex: a, A) and most other constructions (ex.: scalars) willhave normal typography (ex. a, A). Finally, the identity matrix is 1,the null matrix is 0 and the unit pseudoscalar (of geometric algebra)is I.


2 Towards a mathematical theory of reality

We are all familiar with the current practice of science. Let me firstdo a quick summary of it, then we will discuss my proposition.

Theoretical PhysicsEmpirical Physics

Postulated Laws

PredictedExperimental States

MeasuredExperimental States

Empirical laws

Falsification/Refinements

Figure 2: The current practiceof physics.

Figure 2 summarizes the current practice of physics. In it, boththeoretical and empirical physics work in "tandem" to eventually, andhopefully, converge towards a correct model of reality via an iterativefalsification/refinements process. In the one hand, postulated lawsare compared to empirical laws, and in the other, measured exper-imental states are compared to predicted experimental states. Anydiscrepancy then ought to trigger a modification of the postulatedlaws, and the process begins again.

Figure 3 summarizes a reorganization which I believe is morerepresentative of reality, and mathematically much more powerful:it is in fact able to derive the laws of physics as a theorem, therebyproviding an account for their origins.

Empirical Physics

MeasuredExperimental States

Empirical laws

Equivalence-thesis

PostulatedExperimental States

Universal laws

Universal Physics

unchanged untangled reversed

Figure 3: In my proposal, thefalsification/refinements "algo-rithm" of science is untangledto a mere equivalence-thesis by"flipping" the typical relation-ship between states and lawsprevalent in theoretical physics.

In my proposal we flip the usual relationship between laws andstates employed in theoretical physics to what I call Universal Physics:


a mathematical reproduction of empirical physics. Universal physics,as a formal system, is able to leverage the precision, idealization andclarity that mathematical formalisation often provides, and thus itenjoys the same perks as theoretical physics, but it also acquires the’strong connection to reality’ empirical physics enjoys — It is conse-quently the best of both worlds. Its strength comes from its axiomaticanchor to reality in the form of postulated experimental states. Fur-thermore, the laws derived from Universal Physics are mathemati-cally derived in a manner conceptually identical to their empiricalcounterparts. An empirical law is derived by repeating an experimentover a wide range of (similar) conditions then a general pattern isidentified, and a universal equation of state in Universal Physics isderived as the universal pattern found by permuting over all possiblearrangements and rearrangements of postulated states. Because of itsparallels with the real experimental case, a very strong case can thenbe made that this is the philosophically safest procedure possible toderive/define the law of physics.

I propose, and hope that this manuscript will make abundantlyclear, that having neglected to fully formalize the practice of sciencewithin mathematics such that the corpus of physics is derivable as itstheorem, is one of the biggest missed opportunity in physics.

2.1 What, exactly, is wrong with postulated laws?

As Exhibit A, consider a recent book, titled "The Road to Reality: Acomplete guide to the laws of the universe" by Roger Penrose, span-ning over 1123 pages organized in 32 chapters from natural numbersto complex numbers to manifolds to quantum field theory (and be-yond!). At each step of the way, the author introduces a few moremathematical concepts (in the forms of postulates or definitions) withthe goal to bring us ever closer to "reality". If one’s goal is to postu-late one’s way to reality, then Mr. Penrose is definitely the man tospeak to. His book embodies, in my opinion, the most complete workin line with this methodology. But full stop, near the end on page1033, Penrose ends with the following conclusion:

"I hope that it is clear, [...] our road to the understanding the nature ofthe real world is still a long way from its goal."

then continues with:

"If the ‘road to reality’ eventually reaches its goal, then in my viewthere would have to be a profoundly deep underlying simplicity aboutthat end point. I do not see this in any of the existing proposals."

If even Mr. Penrose does not see a road to reality in any of the ex-isting proposals, then what hope is there for the rest of us. Generally


speaking, is erecting a massive "tower of postulates" for all to see,truly the best way to get to reality?

My earliest memory of thinking about this was on my second dayof school (I promise it is relevant), but before I can explain what hap-pened and why it happened I need to lay out the context leading upto it. My father’s strategy of choice to prepare me for the world was,I would summarize, to "sync my mind to reality" by constantly tryingto transform the environment against my expectations (often whileI wasn’t looking). I believe that he intuitively felt that by permutingover all possible (reasonable) states of the environment was the bestand possibly only way to make sure I would not develop an idea thatis "disconnected from reality". Essentially, he attempted to falsify myexpectations. A specific example that comes to mind was one Easterwhen he brought a large chocolate bunny home and two smaller onesfor myself, himself and my mother, respectively. I ate a tiny little bitaround the ear of my chocolate bunny, then safely placed it in thecupboard. On the next day I woke up to find that half of my bunnywas eaten, and my father is insisting that I am the one who ate ityesterday. Upon my objection, he insisted that I am just confusedabout the quantity I ate because he remembers it clearly, causing meto ponder on whose memories can be trusted more; his or mine, andhow would I know. To cope with these random transformations andconstant requisitioning of the assumptions, I came to the conclusionthat I had to train myself not to inject any of my biases into my ex-pectations of the world and, instead to simply accept that the presentstate of the world is the undeniable foundation to reality; any expec-tations I might have of its future states (and in the extreme case evenmy memory of past states could be questioned) can be no less thanthe set of all ’physically-permissible’ rearrangement of the environ-ment. Consequently, I reasoned that the best case strategy to adopt ina wild environment was to assign a likelihood to each scenario andhave a backup plan for undesirable scenarios so as to manage therisk over time. My intuitive mental foundation to reality was that theinstantaneous state of the system is the only arbiter of truth.

On my first day of school, the teacher taught us that one plus oneequals two (and showed us how to work the symbols out as an equal-ity). I remember being so flabbergasted by the genius of this equationthat I barely slept during the night. Then, on the second day, theteacher extended this concept to all the numbers: "We learned yes-terday that one plus one equals two, but it also works with two plusthree equals five, and with three plus one equals four, and so on".Then at some point she said, "and this is why if you take a rock fromoutside and then grab another rock, you will have two rocks in yourhand". As soon as she say that, my face changed completely. I could


not understand why she seemingly conceived of the relationshipbetween ’rules on a blackboard’ and ’reality’ in the opposite logicaldirection of its true entailment. Of course, at that age I wasn’t ableto articulate that thought using the language that I use in the presenttext — I just had the intense intuition that she misunderstood some-thing fundamental about reality and therefore her statements had tobe verified before they could be trusted. So during the lunch break(for about 1.5 hours) I set out to do just that. I picked up rocks fromthe schoolyard and added them all out, permuting over the differ-ent arrangements I could construct and by so doing, verified a (tiny)subset of arithmetic. Okay, so I have established that it works withrocks, but does it work with... branches? So I went to get branches,and verified it again, and sure it worked for branches too. One ofthe other kids asked me what I was doing and I told him that I wastrying to verify that what the teacher had said was true. He asked,surprised, "oh. You don’t believe her?". I responded along the linesof: "I am almost certain that she is right, but I cannot take the riskto take it on faith". Eventually, the bell rang and I ran out of time.Back in class began a long process of ruminating over what had tran-spired. Before I could continue with the program, I had to somehowgrind away at the claim that all arithmetical permutations of num-bers holds in reality. Clearly, it works with small numbers; I in factjust recently verified it in the schoolyard. For numbers larger thanwhat I could personally verify, I convinced myself of the somewhatreasonable argument that possibly millions of other human beingswhere taught these equations before me and themselves have surelyverified very exhaustively the claims. However, I reckoned that therewas still a limit (a very large one indeed) beyond which arithmeticstatements remain unverified by anybody. And an even a larger limitbeyond which nobody ever could (presuming that the resources ofour universe are finite). I could not rule out the fact that, outsidesome verifiable boundary, arithmetic holds some statement to be truethat are outside the scope of reality. Thus I held as stringly suspecteven something as seemingly banal as the inclusion of ’unbounded’arithmetic within the "tower of postulates" of reality.

During the following school years, I developed and nurtured ahealthy existential angst regarding our willingness to use an un-scoped axiomatic basis (first with arithmetic, but eventually with anymathematical or physical theory) which I know does not connect ex-actly to reality. I tried to express my concerns a number of times withmy teachers, but I do not think I made myself sufficiently clear as Irecall one of the responses to be that I would learn all about rocksin the third year of high school. So I waited to let the program un-fold expecting an eventual deconstruction of the disconnected tower


of postulates in some upcoming more advanced classes, but the de-construction never came; instead the complexity simply piled up;from natural numbers, to classical mechanics to eventually quantumfield theory and everything in between. In my mind each additionallayer takes me further away from reality. It would take me decadesto merely acquire the technical language sufficient to understand theproblem clearly, then to pinpoint exactly what causes it, and finallyhow to cure it.

Returning to the question at hand, I do not share the belief thatbuilding a tower of postulates will ever bring us closer to reality. Itis quite the curiosity why theoretical physics ended up being con-structed in terms of a reversed logical entailment:

Postulated Laws PredictedExperimental States

Instead of the real logical entailment:

PostulatedExperimental States Universal laws

which is simply the mathematical equivalent of its experimentalcounterpart:

MeasuredExperimental States Empirical laws

The inverted construction is present in all formal physical theories;from Newton to quantum field theory. As an example, let us con-sider Newton’s second law of motion, mathematically expressed asfollows:

∑F∈F

F = ma (1)

To come up with such a law, one presumes that Newton at leastreviewed the published experimental data of his time, in additionto have conducted numerous experiments of his own. So clearlythe law is logically entailed by ’something’, and that ’something’ isempirical evidence. How shaky then is the logical foundation of atheory that claims that a law (known to be derived from ’something’)is an axiom (derived from nothing)? What price to we pay when weerase that ’something’ by writing "F = ma" as an axiom instead of asa theorem? Specifically, we create what amounts to a cargo cult, andallow me to explain.

A cargo cult is characterized as a belief, by a technologically lessadvanced culture, that building an airstrip or a tower out of bamboo


sticks will trigger the arrival of modern re-supply transport planesto deliver highly desirable cargo, based on the observation that atechnologically advanced society has previously build a functionalcargo-receiving airstrip nearby. These cults were first reported inMelanesia in the late 19th century following contact with westernsocieties. According to one theory, the belief is held due to a lackof proper understanding of supply chain logistics essential to thedelivery mission, as well as to an unawareness of the necessity ofbuilding the airplanes in some (out of sight) assembly plant. What isthe parallel with modern theoretical physics? In theoretical physics,we construct the largest "tower of postulate" that we can, whilst eras-ing from the formalism all of the logistics that brings us the laws ofphysics from reality (by writing them as axioms instead of as theo-rems), yet we somehow expect reality to be delivered to us on a silverplatter by merely having constructed the tower. Well, it turns out thatthe structure of reality is all in the supply chain, and not in the tower.

I also place a not insignificant part of the blame in the human biasto wish to set the foundation of a mathematical theory to its simplestexpression. It appears that since F = ma is simpler than "100 pagesof experimental data", then it gets to be the axiom and not the data,even though reality is logically entailed in the reverse. Extending theargument to something as complex as the observable universe mayappear as another problem and that may also have something to dowith it. Let us consider a theory which takes the present experimen-tal arrangement of the entire observable universe as its axiomaticbasis. Since it may require upwards 10122 bits of information3 to 3 Seth Lloyd. Computational capacity

of the universe. Physical Review Letters,88(23):237901, 2002

write down its axiom, it could therefore be qualified as intractable.Even if such a theory could logically imply no wrong (by virtue of itsaxiom being an exact description of reality), one might nonethelesswant something simpler (hence the bias part). My proposal howeveris able to derive the laws of physics without having to individuallyinterrogate all 10122 bits of reality, by instead using algorithmic in-formation theory to produce a concise representation of any and allpossible experimental states, and to study the universal equationof state resulting from all arrangements and re-arrangements of ex-perimental states. This preserves the universality of the problem yetmakes it tractable.

I feel I must apologize to Roger Penrose for singling out his bookand for the blunt introduction; In fact, I do have the utmost respectfor the quality of his book as a reference tool of the mathematicalconcepts important to physics, and I have relied upon his work toformalize my own work in this very paper (we do after-all recoverthe laws of physics here, therefore a good portion of the tools remainusable). The clarity, utility and completeness of his book is in my


opinion unmatched by any other modern works. Consequently, myintent is of course not to be attribute fault to Mr. Roger Penrose orto his book, but more to use his book as an illustration of the currentstate of affairs and (incorrect) expectations of the field as a whole, ofwhich Penrose’s book just happens to be the single most completeembodiment of such.

I note that various other more technical problems can be associ-ated, this time, to the practice of science itself. Many of these havebeen discussed ad nauseam, so let us just reiterate them quickly forthe sake of completeness. For instance, we can wonder if the iter-ative falsification/refinements process eventually converges to thetruth. In principle, it is strictly possible to construct a theory whosedomain is immune to the falsification/refinement algorithm. Russellwarned us about the dangers of the teapot, but it was believed (orhoped) by many that we would be sufficiently alert to avoid the trap.However, the largesse of string theory caused many to pause and re-consider (this will be Exhibit B). String theory has avoided confirma-tion/falsification for some 35-plus years. Although thousands haveparticipated in the attempt, thus far, the falsification/refinementsprocess has not been able to restrict the "landscape" of string theory,which, without those checks and balances, has grown largely unim-peded. It may seem that out of concerns for the scientific process weought to at some point stop investigating it, but beaware this couldalso be a mistake; what if ’the truth’ was just a few more years away?To stop or not to stop, that is the dilemma of all non-presently falsifi-able, but possible, leads.

To accommodate long-tail processes such as string theory or oth-ers, it is more convenient to think of the problem of science as ahalting problem instead of a convergence problem. I would in factmark the late 20th century onwards practice of theoretical physics bythe passage from an immediate falsification requirement4 (practic- 4 We recall the 1920-1970 period in

which the theoretical research onquantum field theory/standard-modelwas experimental tested in particleaccelerators oftentimes within just a fewyears of their theoretical publications

ing science as a convergent algorithm) to a long-tail non-convergentapproach (practicing science as a halting problem). In this contest,the problem adopts the least constraining form: will the falsifica-tion/refinement algorithm ever halt on the truth? As a halting prob-lem, we are then free to investigate a given theory for as long as itremains fashionable, safely tucked away from the "dangers" of falsi-fication and still perform science, because the process could, in prin-ciple, eventually just halt on it (without actually converging towardsit).

But with this now less restrictive definition, new dilemmas emerges:are we to investigate every possible non-presently falsifiable, butstrictly possible, leads (of which there are infinitely many) with thesame intensity as we do for string theory? Without convergence, any


possible lead could be the one that happens to halt, and could do sowithout any prior warnings. Clearly we don’t have the financial orphysical resources to investigate everything. To avoid this dilemma,maybe we need to tweak the algorithm some more; but then we arein danger of entering a perpetual tweaking-hell; where tweaking thealgorithm is on par with the difficulty of finding the actual solution.

In comparison, my proposal only requires an equivalence-thesisbetween measured states and postulated states, and thus has none ofthese problems. As long as the thesis holds, the laws derived frommy process cannot diverge from reality.

2.2 What, exactly, is missing from mathematics?

Proving the existence of the observable universe purely mathemat-ically is counter-intuitive and therefore it is often assumed to be animpossibility. However, in contradiction with expectations, a numberof years ago I was able to lay out a precise path able to do so. Here,I will provide a simplified example of my technique, which illus-trates the key concepts, then further in the paper we will thoroughlyinvestigate the technique to produce a mathematical theory of reality.

Specifically, as our starting example, I will create a formal math-ematical theory that has a shelf-life. Wait, "a shelf-life", in a mathe-matical theory... a shelf-life like with milk, or eggs? Yes, a shelf-life;meaning, the mathematical theory is perfectly usable today, but insome amount of "time" it will eventually rot. To the best my knowl-edge, rotting mathematical theories are a novel invention.

The construction is surprisingly simple, yet its philosophical impli-cations are incredibly powerful. To construct such a theory, I simplyobfuscate a statement behind a computationally-intensive algorithmthat I then add as an axiom. For instance, consider the contradictorystatement of arithmetic 1 + 1 = 1 that we assume I have encryptedusing a secure5 perfect6 hash function. 5 A secure hash function can only be

inverted by brute force.6 A perfect hash function is an injectivefunction that maps each input to anhash, with no collisions.

For example, suppose my hash produces the following result:

hash[1 + 1 = 1] = fa1869db4bfbf1767a5446b6a9290243 (2)

Specifically, the hash function takes as input an element of LPA,the set of all valid sentences of arithmetic, and outputs an element ofLhex, the set of all hexadecimal sentences:

hash : LPA −→ Lhex

statement 7−→ hash(3)

I also define the inverse function:


bruteforce : Lhex −→ LPA

hash 7−→ statement(4)

The bruteforce function finds the solution by brute force: it hashesall statements of LPA in shortlex in a loop then halts once it findsthe statement that matches the hash, then it outputs said statement.Reversing the map of a hash function is, by design, computationallyintensive.

Now, let me define a new axiom as follows:

Definition (Axiom of rot).

rot := bruteforce[fa1869db4bfbf1767a5446b6a9290243] (5)

Finally, using the axiom of rot, I define a new formal theory as theunion between the axiom of rot and the Peano’s axioms of arithmetic(PA):

Definition (Rotting arithmetic).

rot∪PA (6)

In the present case since I already revealed the rot statement toyou, it follows that you know that Rotting arithmetic is ultimatelyinconsistent without having to execute the bruteforce function. Butconsider instead the following axiom:

Definition (Axiom-X).

Axiom-X := bruteforce[0cfae383362bc63d7ac429a5755fef05] (7)

Now I ask you, knowing the hash but not the statement, is the for-mal theory comprised of PA∪Axiom-X, consistent or inconsistent?Maybe the original statement I chose was 1 = 0 (inconsistent), ormaybe it was 1 + 1 = 2 (consistent). It may not be so obvious nowwhether Axiom-X causes the theory to rot or not, is it? If you arewilling to work at it, you will eventually find the non-obfuscatedform of the axiom by brute force. In this context, I find it ratherillustrative to employ the terms fresh/rotten (as opposed to con-sistent/inconsistent) to accentuate the timely connection betweenfinitely axiomatic systems and some notion of work. A finitely ax-iomatic system is either fresh (if no contradictions are known) orrotten (if contradictions are known). We note that one who randomlyproves theorems in rotten arithmetic will almost certainly map outa very large portion of standard arithmetic before the axiom of rotbecomes a problem. We also note that no finitely axiomatic systemcan rot without work.


Consider the case where Axiom-X may have been hashed suchthat more work would be required to brute force the solution thanwhat is available in the universe. Seth Lloyd7 estimates that there 7 Seth Lloyd. Computational capacity


are approximately 10122 bits (and approximately the same amountof operations) available for computations in the universe. What ifour bruteforce function requires, say, 10122 + 1 bits or higher to halt?Such a finitely axiomatic system, although rotten in principle, couldactually never rot in our present day universe. Its shelf-life wouldexceed the age and size of the universe. Rotting arithmetic, with a> 10122 bits bruteforce function would be mathematically rotten, but"physically" fresh.

As per the Gödel incompleteness theorem, we recall that a (suf-ficiently expressive) finitely axiomatic system cannot prove its ownconsistency. It could be the case, hypothetically, that some finitelyaxiomatic system, perhaps believed to be consistent, contain a deeplyhidden contradiction. In fact, since the dept of mathematical proofcomplexity knows no bound8, then a contradiction could be injected, 8 Alan M Turing. On computable

numbers, with an application to theentscheidungsproblem. Proceedings ofthe London mathematical society, 2(1):230–265, 1937; and Gregory J Chaitin. Metamath! the quest for omega. arXivpreprint math/0404335, 2004

accidentally or on purpose, at any level of complexity within a the-ory. My specific example with a bruteforce function shows how topurposefully inject a contradiction at a tunable level of complex-ity, but nonetheless, in principle, all (sufficiently expressive) finitelyaxiomatic systems have the potential to rot. In the general case, math-ematics offers us no tool to rat out rot, other than pure computingpower.

For the present example, I have used a hashing function in orderto make my point obvious; inverting a hashing function is known tobe computationally intensive, thus we immediately notice a connec-tion between work and our knowledge of the non-obfuscated formof the axiom. But do not let the presence of an hashing function dis-tract you; in fact, all mathematical theorems require the consumptionof some, always non-zero, quantity of computing resources to beproven. In essence, all mathematical theorems are hidden behind a"computing pay-wall" which must be paid with computing resourcesto unlock the proof. In many day-to-day cases the price is negligibleand thus goes unnoticed. Ex: prove that 1 + 1 + 1 = 3 is a theoremof PA — the truth of this statement is immediately obvious and so wedo not easily notice the computing cost, but it is there nonetheless.All proofs have a computing cost, whether the proofs are verified bycomputer or by any other devices.

Let us take another example by asking the question: "Is ZFC con-sistent?". An answer along the following lines is often provided:"Although according to Gödel’s incompleteness’ theorem ZFC cannotprove its own consistency, it has been studied for over 80-90 yearsby hundreds of thousands of people. Certainly if there was any (ob-


vious) contradictions it would have been found by now. The factthat no one has so far found any is very convincing evidence (butof course will never be not proof of) that no contradictions will befound". So why do I believe very strongly that ZFC is consistent, butI would have must less faith in whatever some random guy postedonline yesterday, even if I may not be immediately aware of any con-tradictions in either theories? If the consistency of a theory is entirelyimplied by its axioms, why does work (or time spent studying it)have anything to do with our belief in its consistency? More specif-ically, why does it appears to be the case that a person who spends,say, 1020 bits and operations of computation on a theory is seeminglycloser to identify rot (if rot is present) than a person who spendsonly, say, 105 bits and operations of computation, and therefore toquantify why one may logically have more faith in the former thanthe later.

So now comes the problem of actually constructing a frameworkable to formalise this intuition. To do so, we will rely on algorithmicinformation theory, initially on the works of Gregory Chaitin9, but 9 Gregory J. Chaitin. A theory of

program size formally identical toinformation theory. J. ACM, 22(3):329–340, July 1975

also on the more recent works of Baez and Stay regarding algorithmicthermodynamics10 which imports the tools of statistical physics into

10 John Baez and Mike Stay. Algorith-mic thermodynamics. Mathematical.Structures in Comp. Sci., 22(5):771–787,September 2012

algorithmic information theory. Using these tools, we will create astatistical ensemble of programs comprised of a manifest, a domainand a ground state. The elements of the trio are related to each otheras shown on Figure 4.

Axioms(ground state) Domain∞

AlgorithmicEquation of State

(states)

Statistical Ensemble of Programs

Manifest(exited state)

(mathematical work)

Figure 4: The algorithmic equa-tion of state quantifies theamount of mathematical workrequired to excite the system toa given manifest.

On the left, we have the set of axioms of the finitely axiomaticsystem. In the middle, we have the manifest. The manifest is theinstantaneous state of the system; specifically, it is the set of all state-ments that are proven and verified to be true by the system. In thissense, the manifest is the mathematical description of the proven stateof the system. Then, on the far right we have the domain spannedby the axioms. In the general case, the domain is a non-computableinfinite set comprising all statements provable from the axioms. Themanifest is always sandwiched between the axioms and the domain,and is related to them as follows:


Axioms ⊆ Manifest ⊂ Dom[Axioms] (8)

In the ground state, the manifest is equal to the axioms. To leavethe ground state (and thereby prove any theorems), the system mustconsume mathematical work. It is the algorithmic equation of statethat quantifies the amount of mathematical work required to excitethe system as the manifest is moved further along the axis.

2.3 What is reality contingent upon?

Philosophy allows us to understand why the this framework pro-duces a purely mathematical description of reality. Specifically, itsboils down to the logical relationship between manifest and mathe-matical work:

Definition 1 (The fundamental structure of reality). The existence ofthe manifest is contingent on the existence of mathematical work. Conse-quently, since the manifest exist indubitably (as we will argue in a moment),then so does mathematical work.

To fundamentally understand the power of this statement, wehave to start at the very beginning of the rational inquiry. Allowme first to lay out the groundwork using classical philosophy, thenwe will use our tools to modernize the argument. We will recall thephilosophy of René Descartes (1596–1650), the famous 17th-centuryfrench philosopher most directly responsible for the mind-bodydualism ever so present in western philosophy. Descartes’ main ideawas to come up with a test that every statement must pass beforeit will be accepted as true. The test will be the strictest imaginable.Any reason to doubt a statement will be a sufficient reason to rejectit. Then, any statement which survives the test will be consideredirrefutable. Using this test and for a few years Descartes rejectedevery statement he considered. The laws and customs of society, asthey have dubious logical justifications, are obviously amongst thefirst to be rejected. Then, he rejects any information that he collectswith his senses (vision, taste, hearing, etc) because a “demon” couldtrick his senses without him knowing. He also rejects the theoremsof mathematics, because axioms are required to derive them, andsuch axioms could be false. For a while, his efforts were fruitlessand he doubted if he would ever find an irrefutable statement. But,eureka! He finally found one which he published in 1641. He doubtsof things! The logic goes that if he doubts of everything, then it mustbe true that he doubts. Furthermore, to doubt he must think and tothink, he must exist (at least as a thinking being). Hence, ’cogito ergo


sum’, or ’I think, therefore I am’. This quite remarkable argument is,almost by itself, responsible for the mind-body dualism of westernphilosophy.

How did Descartes addressed the mind-body problem? In theTreatise of man (written before 1637, but published posthumously)and later in the The passions of the soul (1649), Descartes presents hispicture of man as composed of both a body and a soul. The bodyis a ’machine made of earth’ and the soul is the center of thoughts.Communication between the two parts would be handled by the’infamous’ pineal gland, whose inner working he covers in greatdetail in almost a hundred pages. Specifically, he states "[The] mech-anism of our body is so constructed that simply by this gland’s beingmoved in any way by the soul or by any other cause, it drives thesurrounding spirits towards the pores of the brain, which direct themthrough the nerves to the muscles; and in this way the gland makesthe spirits move the limbs"11. We note that bio-electrical signals were 11 Gert-Jan Lokhorst. Descartes and

the pineal gland. In Edward N. Zalta,editor, The Stanford Encyclopedia ofPhilosophy. Metaphysics ResearchLab, Stanford University, winter 2018

edition, 2018

characterized in 1791 by Luigi Galvani, as bioelectromagnetics some≈150 years later. Of course, the Cartesian pineal gland theory wasincorrect, but the problem on how to connect the mind and the bodyremains.

The proof of the ’cogito ergo sum’ by Descartes is the only proofI have encountered that I feel exceeds the absolute genius that oneplus one equals two is (that is quite an achievement). Furthermore,it is the only proof of the existence (of something/anything) that Ifind to be completely satisfactory within the body of philosophy; allother attempts fall short in one way or another. However, as good asthe first part is, appealing to the biological structures of the brain toattempt a connection from mind to body violates the standards of theproof for the simple and obvious reason that the existence of thesebiological structures do not survive universal doubt and thus oughtnot to be used in the proof. How can this pitfall be avoided?

It may be subtle, but one may nonetheless have already noticedthat the construction of the statistical ensemble of programs I haveproposed, such that proving statements from a finitely axiomaticsystem is contingent on the consumption of mathematical work is arepeat, in disguise, of the universal doubt method of Descartes (in-cluding its conclusion), but applied to the whole structure of reality.Essentially, Descartes may have described the very first universalproof checker, and applied it to the set of universal mathemati-cal statements (although in an informal manner, and consequentlymissed out on a massive opportunity).

Let us now do an exercise that will reveal this missed opportu-nity. We will attempt to produce a description of reality constructedentirely and exclusively out of indubitable statements, of the kind


that Descartes would not rule out by his universal doubt method,justified on the simple idea that we will eventually want to claim thatreality exist indubitably and that those statements will be key for thispurpose.

Let us start by defining what a language is:

Definition 2 (Language). A language L, with alphabet Σ, is the set of allsentences (s1, s2, . . . ) that can be constructed from the elements of Σ and itincludes the empty sentence ∅:

L := ∅, s1, s2, . . . (9)

For instance, the sentences of the binary language are:

Lb := ∅, 0, 1, 00, 01, 10, 11, 000, . . . (10)

and its alphabet is:

Σb = 0, 1 (11)

Definition 3 (Shortlex ordering). A shortlex ordering is a list of thesentences of L, first ordered by length from shortest to longest, then alpha-betically.

For instance, the shortlex ordering of Lb is:

(∅, 0, 1, 00, 01, 10, 11, 000, . . . ) (12)

Let us now define a "Cartesian-Turing machine" which works asfollows:

Definition 4 (Minimal proof checker).

MPC : L −→ 1,@sentence 7−→ result

(13)

Under the hood, the machine works as a brute force automatic theoremprover. Specifically, the machine contains a set of internal rules of inference(logical axioms) which it uses to attempt to formulate a proof that the inputsentence is true within the rules. If a proof is found, then MPC[sentence]outputs 1 otherwise it never halts. The machine may take a very long timeto find a proof, but time is not our concern as we only require that shoulda proof exists, it eventually finds it. For example, once given a sentence asinput, the machine could analyse every sentences of L in shortlex one by oneand by scheduling its work according to a dovetailing algorithm, until one isfound to be the proof, then outputs 1 and halts.


Let us now investigate how MPC behaves using an example. SayDescartes feeds the sentence (1 + 1 = 2) to the MPC. Will the ma-chine find a proof for it? Well lets see. To prove (1 + 1 = 2), onerequires PA (or an equivalent). However, since the machine only con-tains logical axioms, it will never halt because no proof will ever befound. On the surface, it may seem that the conclusion of Descartesregarding the idea that one can doubt of all mathematical theoremsbecause the axioms they rely upon need not be true, is sound.

However, once in a while something quite interesting happens.Let’s say we feed all sentences of L in shortlex to MPC using dove-tailing scheduling. Eventually this statement will be feed to MPC:

PA ` (1 + 1 = 2) (14)

The statement states: PA proves that one plus one equals two. Aswe did with the previous example, we also ask here will MPC[PA `(1 + 1 = 2)] eventually halt? In this case, the answer is yes. Indeed,PA just so happens to be the missing part required for the machineto prove the statement. The statement supplied the missing set ofaxioms required to prove itself.

The second step of the exercise will be to construct a manifestexclusively using such statements. There are of course infinitelymany statements of the type A ` B, and such statements include allpossible mathematical proofs for all possible mathematical theories.Their significance rely on the fact that they are the means to describereality to any level of complexity or expressivity desired or requiredwithout having to construct a preliminary "tower of postulates" to doso.

Let us define a set M, which we call a manifest, of n statementsprovable by MPC:

M := A1 ` B1, A2 ` B2, . . . , An ` Bn (15)

The Bs are the theorems, the As are their axiomatic basis, and eachpair forms an individual fact of the system. In the case of a real sys-tem such as that of the observable universe, any associated manifestswould of course be extremely large and as such n is also very big12. 12 Seth Lloyd. Computational capacity


The union of all As forms the axiomatic basis of a "computationaltheory of everything" the Bs. Although it is not the typical theory ofeverything referenced in physics, it does provide the means to proveall facts of the system, but it does so in a patchy manner, it is suscep-tible to initial graining effects and to the possibility of future rotting.In the case where two or more facts have the same axiomatic basis,say A4 = A9 = A3, the theory forms a "logical grain" such that the


theorem B4, B9, B3 are entailed from the same axiomatic basis. In theground state, all facts simply imply themselves B1 ` B1, B2 ` B2, . . .(e.x "there is milk in the fridge because there is milk in the fridge (noinsight)"), but to leave the ground state, mathematical work must beproduced. Outside the ground state, there exists a "spread" betweenthe axioms and the facts, whose magnitude is quantified in terms ofmathematical work. The larger the spread, the more "algorithmically-rich" the system is and the more insightful the system has the poten-tial to be.

The set of all scientific theories for the system are the variouspatches or grains that are formed as the mathematical work is ex-panded to increase the spread. Scientific theories are subject to re-finements as grains are fused or reorganized. Describing reality asa manifest allows one to produce a plurality of logically indepen-dent scientific theories, each valid within their own domain of ap-plicability and each corresponding to a grain. In this sense, we maysuspect that all scientific theories are subject to possible falsifica-tion/refinements as more mathematical work is produced. One mayalso expect knowledge of the facts themselves to be essential.

But one formulation, qualified as universal, rises above all others;it is a universal equation of state which remains unfalsified withrespect to all possible configurations of facts, grains and state ofmathematical work. This universal equation of state is the origin ofthe laws of physics in the system; it is the theory of everystate and ofeverytransformation.

We are almost there, there are just a few things missing. In fact,mathematical algorithms admit a slightly more general formula-tion than what we have used so far and it just so happens to be re-quired in order to derive the universal equation of state as the lawsof physics. The language A ` B is not entirely universal as, amongstother things, it leaves out all theories that may not admit the ` sym-bol. Additionally, any specific choices of logical axioms (to define theproof solver) eliminates unduly all other choices. Consequently, tobe truly universal we will think of statements as arbitrary programsthat halt, instead of as sentences with specific structures. Let us nowinvestigate how mathematical algorithms or programs truly connectto the physical world, as we enter the next hint.

2.4 Hint: John A. Wheeler

So, what is the right way to think of reality in terms of information?Information, physics and entropy have, of course, a long and rich

history. As an example, let us recall the Landauer13 limit, an expres- 13 Rolf Landauer. Irreversibility and heatgeneration in the computing process.IBM journal of research and development,5(3):183–191, 1961; and Rolf Landaueret al. Information is physical. PhysicsToday, 44(5):23–29, 1991

sion for the minimum amount of energy required to erase one bit of


information for a system at thermodynamic equilibrium:

E ≥ TkB ln 2 (16)

where E is the energy (in Joules), T is the temperature (in Kelvin)and kB is Boltzmann’s constant. Such relation, although consideredextremely fundamental, cannot in our case be used as the startingequivalence. Indeed, since we are not yet at the stage where energyor temperature are defined, we thus cannot use a relation whichrefers to them in order to connect mathematical information to physi-cal reality.

What about other connections found in the literature? A strongcontender relying upon modern notions such as black-hole entropy(or more generally entropy-bearing horizons) and the holographicprinciple suggests a connection between information (on the surfaceof an horizon) and physical states (in the bulk of the enclosed vol-ume). Lets hypothesise how and if we could use this contender inour case. Perhaps we are to map our manifest to the surface of aninformation-bearing horizon then use the holographic principle torecover the bulk? Alas, no - the same problem as before also occurshere but instead of energy/temperature, we have geometry/surface-gravity as the presumed pre-existing physical concepts. Unfortu-nately, any pre-existing physical quantities needed to formulate arelation between information and some element of physical real-ity, precludes those quantities from been given an origin within thisinformation.

So we ask, is such a connection possible - can all physical conceptsbe reduced to information, or are there irreducible physical concepts?As we work our way to the proposed solution, let us review the bestcontender I have thus far identified in the literature. We will nowinvestigate two hints; the first by John A. Wheeler regarding the’participatory-universe’ hypothesis, the second by Gregory Chaitinregarding the undecidability of mathematical formalism and thelink between mathematics and science. Together these two hintswill allow us to identify a universal relation entirely free of physicalbaggage.

We summarize John A. Wheeler’s participatory universe hypoth-esis as follows. First, for any experiments, regardless of their sim-plicity or complexity, the registration of counts (in the form of binaryyes-or-no alternatives, the bit) is taken as a common book-keepingtool, unifying the practice of science. Further to that, John A. Wheelersuggests (in the aphorism "it from bit" 14) that what we consider to 14 John Archibald Wheeler. Frontiers

of time. In Problems in the Foundationsof Physics. North-Holland for the Soci-eta italiana di fisica, 1978; and John AWheeler. Information, physics, quan-tum: The search for links. Complexity,entropy, and the physics of information, 8,1990

be the "it" is simply one out of many possible mixtures of theoreticalglue that binds the "bits" together. Essentially, the ’bit’ is real and the


’it’ is derived. John A. Wheeler states;

"It from bit symbolizes the idea that every item of the physical worldhas at bottom — at a very deep bottom, in most instances — an im-material source and explanation; that what we call reality arises in thelast analysis from the posing of yes-no questions and the registeringof equipment-evoked responses; in short, that all things physical areinformation-theoretic in origin and this is a participatory universe"

The bit is the anchor to reality. The bit would come into being inthe final act, so to speak, and then constrains the possible "it"s, whosetheoretical formulation must, of course, be consistent with all bitsgenerated (and not erased) thus far. Furthermore, he mentions thatthe bit is registered following an equipment-evoked response. Tofurther illustrate his point of view, John A. Wheeler gives the photonas an example of the theme:

"With polarizer over the distant source and analyzer of polarizationover the photodetector under watch, we ask the yes or no question,"Did the counter register a click during the specified second?" If yes,we often say, "A photon did it." We know perfectly well that the photonexisted neither before the emission nor after the detection. However,we also have to recognize that any talk of the photon "existing" duringthe intermediate period is only a blown-up version of the raw fact, acount."

For John A. Wheeler, it makes little sense to speak of the pho-ton existing (or not existing) until a detector registers a count. Buthe goes further and suggests that even after the registration of acount, deducing that the photon existed in-between the counts is a"blown-up version of the raw fact, a count". Here, John A. Wheelerimplies that the counts are what is real, not the theory that explainsthe counts. The theory is one hypothesis among many alternativesand is, at best, a mathematical tool to make some sense of the counts,which by themselves define the world irrespectively of the theory.

In "Frontiers of time" (about a decade before ’it from bit’), John A.Wheeler lays out multiple attempts to derive some form of physicalbehavior/law from the study of experimentally-derived bits, but hisapproaches suffer from introducing physical baggage to get themstarted. Taking a specific example, on page 150, he reasons that timeshould emerge out of entropy. So far so good, but then he argues thatbecause the universe goes from Big Bang to Big Stop, to Big Crunch,the statistics of entropy must be time-symmetric. Therefore, he con-cludes that the acceptable rules of statistics to describe the dynamicsof this entropy are those that he calls "double-ended statistics" whichworks in both directions of time (pages 150-155). The argument has,of course, an obvious fatal error: if time is derived from the bits,


then so should the cosmos — why would one not be allowed to re-fer to time apriori (it must be derived from entropy), but be allowedto refer to the cosmos’ hypothetical future time-reversal to justifysome properties on the bits? Thirty-nine years later, the results of thePlanck Collaboration15 indicate a critical density consistent with flat 15 N Aghanim, Y Akrami, M Ashdown,

J Aumont, C Baccigalupi, M Ballardini,AJ Banday, RB Barreiro, N Bartolo,S Basak, et al. Planck 2018 results. vi.cosmological parameters. arXiv preprintarXiv:1807.06209, 2018

topology and eternal expansion, possibly contradicting Wheeler’sargument relying upon the necessity of some upcoming future cos-mological reversal. Obviously, the eventual correct approach is onlyappealing if all physical statements (the ’its’) follow from the bitssuch that the future time reversal, if any, ought to be derived fromthe ’bits’. John A. Wheeler’s book presents a myriad of similarly con-structed arguments. John A. Wheeler does understand this to be aproblem, and in his defense, he does present "double-ended statis-tics" only as an example of what might be done. Some 11 years laterhe corrects his approach to the participatory-universe hypothesis.

In "Information, physics, quantum: The Search For Links", he pro-vides general guidance on how to rectify this. It is there that he in-troduces the core idea that the bits are the result of the registeringof equipment-evoked responses. With this John A. Wheeler discardsthe idea of referring to the cosmos at all to enforce any kind of prop-erties on the distribution of the bits and instead refers to equipmentevoked responses exclusively. After-all, evidence for both time andthe cosmos are derived from the information provided to us by ex-perimental devices (including the biological senses).

This completes our summary of the core concepts of John A.Wheeler’s participatory universe hypothesis16. 16 John A Wheeler. Information, physics,

quantum: The search for links. Complex-ity, entropy, and the physics of information,8, 1990; and John Archibald Wheeler.Frontiers of time. In Problems in theFoundations of Physics. North-Hollandfor the Societa italiana di fisica, 1978

So why this brief mention by John A. Wheeler of associating bitsto an equipment-evoke response, essential — why can’t bits juststand on their own merits? To understand this, we have to first recog-nize that the bits only have meaning if they are associated with somelogical structure and that bits without it are meaningless. Let’s seewhy with the following example.

Let’s say that we were to provide someone with a list of bits:

111010110001001110101010101 (17)

How valuable would this person find this information? Probablynot much —why? As a hint, imagine if we were to tell this personthat these bits represent the winning numbers of the next lotterydraw. Then, all of sudden and although the sequence of bits stays thesame, the bits are much more valuable.

Alternatively, we could have said that these bits are the resultsof random spin measurements. The bits once again stay the same,but their meaning is now completely different. Thus, some form ofa logical structure must be associated with any bits that we acquire


about the world otherwise they are without context or sense. Thisis why the pairing of experimental results (in the form of bits) andthe experimental setup (under which the bits are acquired) are bothequally crucial for a meaningful description.

But how do we describe the very complex world of experimentalequipment without invoking physical baggage? I have the impressionthat this may have been a primary roadblock encountered by John A.Wheeler: formalizing equipment-evoked response seems to requiresome physical description of said equipment, and as this wouldcontain physical baggage, then the fundamentality of the theorywould be compromised.

The solution that I retained was to define an experiment not bythe physical devices that are used in it, but instead by the proto-col that must be followed to realize it. This is how the connectionto the algorithmic formulation is made. As shown on Figure 1, in-stead of connecting information to some complex pre-existing phys-ical quantity, we here connect the algorithm to an experiment. The’it’ of Wheeler is a consequence of protocol-evoked responses, notequipment-evoked responses — a very important but subtle differ-ence. As we will see with the next hint, shifting the description fromequipment to protocol is the key to make the endeavor mathemati-cally precise.

2.5 Hint: Gregory Chaitin

If we are to construct physical reality as a statistical ensemble of ex-periments (Figure 1), then there is likely one better way to recover thelaws of physics; one must simulate the practice of science within theensemble. But before we can formalize science within mathematics,we first need to identify a mathematical structure that behaves asscience does.

Gregory Chaitin summarizes his work on the halting probability17, 17 Gregory J. Chaitin. A theory ofprogram size formally identical toinformation theory. J. ACM, 22(3):329–340, July 1975

the Ω construction, in the book "Meta Math!"18. Let U be the set of all

18 Gregory J Chaitin. Meta math!the quest for omega. arXiv preprintmath/0404335, 2004

universal Turing machines, then:

Ω : U −→ [0, 1]UTM 7−→ ∑p∈Dom[UTM] 2−|p|

(18)

The image of Ω is a set of real numbers that are normal, incom-pressible and provably algorithmically random due to their connec-tion to the halting problem in computer science. The reader maywish to read the first few paragraphs of our technical introduction(Section 4.2) on algorithmic information theory for a more detailedprimer on Ω, and then come back to this section.


In the book "Meta Math!" Gregory Chaitin states that the followingis his ’strongest’ incompleteness theorem:

"A finitely axiomatic system (FAS) can only determine as many bits ofΩ as its complexity.

As we showed in Chapter V, there is (another) constant c such that aformal axiomatic system FAS with program-size complexity H[FAS]can never determine more than H[FAS] + c bits of the value for Ω."

where H[p] is the Kolmogorov complexity of p.

This result essentially quantifies the general incompleteness inmathematics (originally identified/proved by Gödel for a specificcase: the Gödel sentences in Peano’s axioms) and equates it to theKolmogorov complexity, measured in quantities of bits, of the ax-iomatic basis of the finitely axiomatic system.

Gregory Chaitin dedicated a considerable amount of time to con-sider the implication of his Ω construction regarding the philosophyof mathematics. What does such widespread incompleteness meanfor mathematics? He concludes the following:

"I, therefore, believe that we cannot stick with a single finitely ax-iomatic system, as Hilbert wanted, we’ve got to keep adding new ax-ioms, new rules of inference, or some other kind of new mathematicalinformation to the foundations of our theory. And where can we getnew stuff that cannot be deduced from what we already know? Well,I’m not sure, but I think that it may come from the same place thatphysicists get their new equations: based on inspiration, imaginationand on — in the case of math, computer, not laboratory-experiments."

Finally, Gregory Chaitin further suggests:

"So this is a “quasi-empirical” view of how to do mathematics, whichis a term coined by Lakatos in an article in Thomas Tymoczko’s inter-esting collection New Directions in the Philosophy of Mathematics.And this is closely connected with the idea of so-called “experimentalmathematics”, which uses computational evidence rather than con-ventional proof to “establish” new truths. This research methodology,whose benefits are argued for in a two-volume work by Borwein, Bai-ley, and Girgensohn, may not only sometimes be extremely convenient,as they argue, but in fact, it may sometimes even be absolutely nec-essary in order for mathematics to be able to progress in spite of theincompleteness phenomenon..."

In another more recent article19, Gregory Chaitin provides con- 19 Gregory Chaitin. Doing math-ematics differently. https:

//inference-review.com/article/

doing-mathematics-differently, 02

2019. Accessed: 2019-12-04

crete examples of how the incompleteness phenomenon can entersome fields of mathematics. Specifically, he states:

"In theoretical computer science, there are cases where people behavelike physicists; they use unproved hypotheses. P 6= NP is one example;

https://inference-review.com/article/doing-mathematics-differently




it is unproved but widely believed by people who study time complex-ity. Another example: in axiomatic set theory, the axiom of projectivedeterminacy is now being added to the usual axioms. And in theo-retical mathematical cryptography, the use of unproved hypotheses isrife. Cryptosystems are of immense practical importance, but as far asI know it has never been possible to prove that a system is secure with-out employing unproved hypotheses. Proofs are based on unprovedhypotheses that the community currently agrees on, but which could,theoretically, be refuted at any moment. These vary as a function oftime, just as in physics."

Finally, we note Gregory Chaitin’s Meta-biological theory pro-posed in20, "Proving Darwin: making biology mathematical", which refer- 20 Gregory Chaitin. Proving Darwin:

making biology mathematical. Vintage,2012

ences many of these concepts.

Part I

Science3 The Axioms of Universal Physics

Universal Physics is very likely to be both the least-biased and themost universal framework of reality possible. It can be applied toany and all physical systems. In the case were it is applied to thesum-total of all reality, both the microscopic description and theequation of state are universal; the first in the mathematical sense (itspawns the domain of a universal Turing machine) and the secondin a physical sense (it accounts for all possible transformations of thesystem).

The fundamental object of study of formal science is not the elec-tron, the quark or even the microscopic super-strings, but the experi-ment. An experiment represents an ’atom’ of verifiable knowledge.

Definition 5 (Experiment). An experiment p is a tuple comprising twosentences of L. The first sentence, h, is called the hypothesis. The secondsentence, TM, is called the protocol. Let UTM: L×L→ L∪@ be a univer-sal Turing machine, then we say that the experiment holds if UTM[TM, h]halts, and fails otherwise:

UTM[TM, h]

= r halts =⇒ p holds

@ ¬halts =⇒ p fails(19)

If p holds, we say that the protocol verifies the hypothesis. Finally, r, alsoa sentence of L, is the result. Of course, in the general case, there exists nocomputable function which can decide if an experiment holds or doesn’t.


An experiment, so defined, is formally reproducible. Indeed, forthe protocol TM to be a Turing machine, the protocol must specifyall steps of the experiment including the complete inner workings ofany instrumentation used for the experiment. The protocol must bedescribed as an effective method equivalent to an abstract computerprogram. Should the protocol fail to verify the hypothesis, the entireexperiment (that is the group comprising the hypothesis, the protocoland including its complete description of all instrumentation) isfalsified.

The set of all experiments that hold are the programs that halt.The set includes all provable mathematical statements and it is uni-versal in the computer theoretic sense.

Definition 6 (Domain of science). Let D be the domain (Dom) of formalscience. We can define D in reference to a universal Turing machine UTMas:

D := Dom[UTM] (20)

Thus, for all sentences s in L, if UTM[s] halts, then s ∈ D.

Definition 7 (Manifest). A manifest M is a subset of D:

M ⊂ D (21)

Assumption 1 (The fundamental assumption of science). The stateof affairs of the world is describable as a set of reproducible experiments.Therefore, the state of affairs is describable as a manifest. Furthermore, toeach state of affairs corresponds a manifest, and finally, the manifest is acomplete description of the state of affairs.

Definition 8 (States — set of all manifests). The set of all manifestscomprises all possible state of affairs of the world, and is defined as:

S := P [D] (22)

where P [D] is the powerset of D. Of course, M ∈ S.

Axiom 1 (Existence of the reference manifest). As the world is in agiven state of affairs, then there exists, as a brute fact, a manifest M whichcorresponds to its state:

∃!M (23)


• M is called the ’reference manifest’.

• The symbol M will denote any manifest in S, whereas M specificallydenotes the reference manifest corresponding to the present state of affairs.

• We consider the overhead ring symbol to be the designator of ontologicalexistence and to be distinct from mathematical existence referenced bythe symbol ∃. For instance, in set theory, all manifests M exists (∃),but in formal science, only the state of affairs described by M existsontologically.

• Unique to M, its elements are verified.

Intuition: The reference manifest is how the world presents itselfto us in the most direct, unmodelled, uninterpreted and uncom-pressed manner. Brutely knowing the manifest is how one perceivesthe world without understanding any patterns and without knowingany laws of physics.

As stated in the previous philosophical section, the fact that theexperiments of the reference manifest are verified implies, implicitly,the existence of mathematical work in quantities exactly sufficient toverify them, which in the present context, we call nature.

Definition 9 (Nature). Nature is a system of mathematical work used toverify experiments. Thus, experiments are verified in nature. Precisely,nature is the set of all functions mapping manifests in S to computingtransformations in T (usually taken to be a complete set of matrices) —g : S→ T, and is noted as N such that:

N := ST (24)

Note: We will investigate nature in more detail (including many exam-ples) in the next section.

As infinitely many manifests M can be constructed from the ele-ments of D, one may wonder why it is the reference manifest M thatis actual and not any other. This brings us to the next assumption.

Assumption 2 (The fundamental assumption of ’nature’). We adoptthe Bayesian principle of insufficient reason: The reference manifest is ran-domly selected from the set of all possible manifests S according to a proba-bility measure ρ[M].

With this assumption, we abandon all hope, as difficult to copewith as it may be, of there being a model which tells us why M andnot M is actual. Essentially, this is a side-effect of the notion that thestate of affairs represents the axiomatic basis of the model.


However, as dreadful as this may be, it is the key to recover thecorpus of physics. The first step is to associate knowledge of M toinformation, and it is precisely because M is randomly selected froma larger set that this is possible. We briefly recall the mathematicaltheory of information of Claude Shannon: Specifically, M will beinterpreted as a message randomly selected from the set S. Usingρ[M], we will be able to quantify the amount of natural informationin the message M.

Definition 10 (Natural Information). We define natural information asthe information one gains by knowing which manifest is randomly selectedfrom S, according to the probability distribution ρ[M] that maximizes theentropy under the constraint of nature N .

Those familiar with statistical physics will no doubt have identi-fied that the setup of Universal Physics is fundamentally that of aensemble of statistical physics. Indeed, in the present case, the lawsof physics will be derived as a universal equation of state of naturevalid over all possible permutations or rearrangements of manifestsor of experiments, and resulting from maximizing the entropy in theusual sense of statistical physics. Nature, defined as a uncountableset of functions, here acts as the universal constraints of the system.In this context, we refer to the laws of physics as the ’structure’ ofreality so as to contrast them to typical theories of physics in whichthe laws of physics are postulated, in which case we call them the’model’ of reality. Universal Physics through its connection to en-tropy reveals that the laws of physics is the structure that maximizesthe amount of information associated to the message M under theconstraint of nature: intuitively, they are the laws that make realitymaximally informative, which is why they are "unbeatable". In thenext section, we will produce an exact mathematical expression fornatural information.

Part II

Nature4 Technical Introduction

To precisely quantify the relationship between natural information,entropy, mathematical work and how this implies the laws of physicsas the bulk description of system of verified experiments, we willeventually introduce universal thermodynamics, but first, we willprovide a recap of statistical physics, and then of algorithmic thermo-dynamics.


4.1 Recap: Statistical Physics

The applicability of statistical physics to a given physical systemrelies primarily upon two assumptions21. 21 Jos Uffink. Compendium of the

foundations of classical statisticalphysics. Philosophy of Physics, 03 20061. The average of all experimental measurements of a given observ-

able in a macroscopic system converges to a well defined value,called a constraint.

2. "Any macroscopic system at equilibrium is described by the max-imum entropy ensemble, subject to constraints that define themacroscopic system."22 22 Victor S. Batista. Postulates of statis-

tical mechanics. http://xbeams.chem.

yale.edu/~batista/vaa/node20.html.Accessed: 2019-12-17

The first assumption is responsible for implying a number of fixedmacroscopic quantities, known as the constraints. Let Q be a set ofmicro-states and N be a set of constraints, then set of all probabilitymeasures compatible with the constraints is:

P :=

ρ : Q→ [0, 1]

∣∣∣∣∣∣ ∑q∈Q

ρ[q] = 1

∣∣∣∣∣∣N (25)

The observables, in general, are n functions defined as:

Oi : P −→ R

ρ 7−→ ∑q∈Q ρ[q]Oi[q](26)

where Oi : Q → R. All constraints are observables, but not allobservables are constraints. Typical thermodynamic quantities areshown in Table 1.

Symbol Name Units Type

E[q] energy Joule extensive1/T = kBβ temperature 1/ Kelvin intensiveE average energy Joule macroscopic

V[q] volume meter3 extensivep/T = kBγ pressure Joule /(Kelvin-meter3) intensiveV average volume meter3 macroscopic

N[q] number of particles kg extensive−µ/T = kBδ chemical potential Joule/(Kelvin-kg) intensiveN average number of particles kg macroscopic

Table 1: Typical thermodynamicquantities

The second assumption is responsible for implying the probabilitymeasure which maximizes the entropy:

S : P −→ [0, ∞[

ρ 7−→ −kB ∑(q∈Q) ρ[q] ln ρ[q](27)

http://xbeams.chem.yale.edu/~batista/vaa/node20.html



under said constraints. This probability measure, which can beobtained from the method of the Lagrange multipliers by maximizingthe entropy under the constraints, is the Gibbs ensemble:

ρ : Q×Rn −→ [0, 1](q, α1, . . . , αn) 7−→ Z−1 exp

(−α1O1[q]− · · · − αnOn[q]

)(28)

where α1, . . . , αn are Lagrange multipliers. The partition function Zis:

Z : Rn −→ R

(α1, . . . , αn) 7−→ ∑(q∈Q) exp(−α1O1[q]− · · · − αnOn[q]

)(29)

The observables form a set of n constraints that we call the set ofobservables, or the thermodynamic bulk state:

Oi = Z−1 ∑(q∈Q)

Oi[q] exp(−α1O1[q]− · · · − αnOn[q]

)(30)

The thermodynamic bulk quantities are also given by the follow-ing n relations:

∂ ln Z[α1, . . . , αn]

∂αi= Oi (31)

And the variance by the following n relations:

∂2 ln Z[α1, . . . , αn]

∂α2i

= (∆Oi)2 (32)

The entropy for this ensemble is:

S[α1, . . . , αn] = kB(ln Z + α1O1 + · · ·+ αnOn) (33)

Taking the total derivative of the entropy, we obtain:

dS[α1, . . . , αn] = kB(α1 dO1 + · · ·+ αn dOn) (34)

which is called the equation of the state of the system.Thermodynamics is derived from statistical physics which is

concerned primarily by the equation of state (34). Thermodynamicchanges (and cycles) can be realized by changing the quantities


α1, . . . , αn and/or by modifications of Q. Under modification ofQ, usually by cross product: Q×Q1 = Q2, or by set complementQ \Q3 = Q4, quantities which are invariant α1, . . . , αn are calledintensive, and quantities which are variant A1, . . . , An are calledextensive.

As an example, replacing the generalized quantities by the typicalthermodynamic quantities, in Table 1:

α1 := β (35)

α2 := γ (36)

α3 := δ (37)

A1[q] := E[q] (38)

A2[q] := V[q] (39)

A3[q] := N[q] (40)

the partition function would be:

Z[Q, β, γ, δ] = ∑q∈Q

exp(−βE[q] + γV[q] + δN[q])

)(41)

The Gibbs measure is:

ρ(q, β, γ, δ) =1Z

exp(−βE[q]− γV[q]− δN[q]

)(42)

The observables are:

E =1Z ∑

q∈Q

E[q] exp(−βE[q]− γV[q]− δN[q]

)(43)

V =1Z ∑

q∈Q

V[q] exp(−βE[q]− γV[q]− δN[q]

)(44)

N =1Z ∑

q∈Q

N[q] exp(−βE[q]− γV[q]− δN[q]

)(45)

The entropy is:

S[β, γ, δ] = kB(ln Z + βE + γV + δN) (46)

and the equation of state is:

dS[β, γ, δ] = kB(β dE + γ dV + δ dN) (47)


4.2 Recap: Algorithmic Thermodynamics

Many authors23 have discussed the similarity between the Gibbs en- 23 C. H. Bennett, P. Gacs, Ming Li,P. M. B. Vitanyi, and W. H. Zurek. In-formation distance. IEEE Transactionson Information Theory, 44(4):1407–1423,July 1998; Gregory J. Chaitin. A theoryof program size formally identical toinformation theory. J. ACM, 22(3):329–340, July 1975; Edward Fredkin andTommaso Toffoli. Conservative logic.International Journal of Theoretical Physics,21(3):219–253, Apr 1982; Andrei Niko-laevich Kolmogorov. Three approachesto the definition of the concept “quan-tity of information”. Problemy peredachiinformatsii, 1(1):3–11, 1965; A K Zvonkinand L A Levin. The complexity offinite objects and the developmentof the concepts of information andrandomness by means of the theoryof algorithms. Russian MathematicalSurveys, 25(6):83, 1970; R.J. Solomonoff.A formal theory of inductive inference.part i. Information and Control, 7(1):1 –22, 1964; Leo Szilard. On the decreaseof entropy in a thermodynamic systemby the intervention of intelligent beings.Behavioral Science, 9(4):301–310, 1964;Kohtaro Tadaki. A generalization ofchaitin’s halting probability omegaand halting self-similar sets. HokkaidoMath. J., 31(1):219–253, 02 2002; andKohtaro Tadaki. A statistical mechanicalinterpretation of algorithmic informa-tion theory. In Local Proceedings of theComputability in Europe 2008 (CiE 2008),pages 425–434. University of Athens,Greece, Jun 2008

tropy S = −kB ∑q∈Q ρ[q] ln ρ[q] and the entropy in information theoryH = −∑q∈Q ρ[q] log2 ρ[q]. Furthermore, the similarity between thehalting probability Ω and the Gibbs ensemble of statistical physicshas also been studied24. First let us introduce Ω. Let U be the set of 24 Ming Li and Paul M.B. Vitanyi. An

Introduction to Kolmogorov Complexityand Its Applications. Springer PublishingCompany, Incorporated, 3 edition, 2008;Cristian S. Calude and Michael A. Stay.Natural halting probabilities, partialrandomness, and zeta functions. Inf.Comput., 204(11):1718–1739, November2006; John Baez and Mike Stay. Algo-rithmic thermodynamics. Mathematical.Structures in Comp. Sci., 22(5):771–787,September 2012; and Kohtaro Tadaki. Ageneralization of chaitin’s halting prob-ability omega and halting self-similarsets. Hokkaido Math. J., 31(1):219–253, 02

2002

all universal Turing machines, then:

Ω : U −→ [0, 1]UTM 7−→ ∑p∈Dom[UTM] 2−|p|

(48)

Here, |p| denotes the length of p, a computer program. The do-main, Dom[UTM], is the domain of the universal Turing machine (theset of all programs that halt for it). The sum represents the proba-bility that a random program will halt on UTM. Chaitin’s construc-tion25 (a.k.a. Ω, halting probability, Chaitin’s constant) is defined

25 Gregory J. Chaitin. A theory ofprogram size formally identical toinformation theory. J. ACM, 22(3):329–340, July 1975

for a universal Turing machine as a sum over its domain (the set ofprograms that halts for it) where the term 2−|p| acts as a special prob-ability distribution which guarantees that the value of the sum, Ω,is between zero and one (The Kraft inequality 26). As the sum does

26 L. G. Kraft. A device for quanitiz-ing, grouping and coding amplitudemodulated pulses. Master’s thesis,Mater’s Thesis, Department of ElectricalEngineering, MIT, Cambridge, MA,1949

not erase halting information, knowing Ω is enough to know theprograms that halt and those that do not on UTM. Since the haltingproblem is unsolvable, Ω must, therefore, be non-computable. Ω’sconnection to the halting problem guarantees that it is algorithmi-cally random, normal and incompressible.

It is possible to calculate some small quantity of bits of Ω. Assuch, Calude27 calculated the first 64 bits of Ω for some well-defined 27 Cristian S Calude, Michael J Dinneen,

Chi-Kou Shu, et al. Computing aglimpse of randomness. ExperimentalMathematics, 11(3):361–370, 2002

universal Turing machine utm1 ∈ U as:

Ω[utm1] = 0.0000001000000100000110...2 (49)

Running the calculation for a handful of bits is certainly possible,however, any finitely axiomatic systems will eventually run out ofsteam and hit a wall. Calculating the digits of π, for instance, willnot hit this kind of limitation. For π, the axioms of arithmetic aresufficiently powerful to compute as many bits as we wish to calcu-late, limited only by the physical resources of the computers at ourdisposal. To understand why this is not the case for Ω, we have torealize that solving Ω requires solving problems of arbitrarily highercomplexity, the complexity of which always eventually outclasses thepower of any finitely axiomatic system.

In 2002, Tadaki28 suggested augmenting Ω with a multiplication 28 Kohtaro Tadaki. A generalization ofchaitin’s halting probability omega andhalting self-similar sets. Hokkaido Math.J., 31(1):219–253, 02 2002

constant D, which acts as an ’algorithmic decompression’ term on Ω.


Chaitin construction → Tadaki ensemble

Ω[UTM] = ∑p∈Dom[UTM]

2−|p| → Ω[UTM, D] = ∑p∈Dom[UTM]

2−D|p|

(50)

With this change, Tadaki argued that the Gibbs ensemble com-pares to the Tadaki ensemble as follows:

Gibbs ensemble Tadaki ensemble

Z[β] = ∑q∈Q

e−βE[q] Ω[UTM, D] = ∑p∈Dom[UTM]

2−D|p| (51)

Interpreted as a Gibbs ensemble, the Tadaki construction formsa statistical ensemble where each program corresponds to one of itsmicro-state. The Tadaki ensemble admits the following quantities— the prefix code of length |q| conjugated with D. As a result, itdescribes the partition function of a system which maximizes theentropy subject to the constraint that the average length of the codesis some quantity |p|;

|p| = ∑p∈Dom[UTM]

|p|2−D|p| (52)

The entropy of the Tadaki ensemble is proportional to the averagelength of prefix-free codes available to encode programs:

S[UTM, D] = ln Ω + D|p| ln 2 (53)

The constant ln 2 comes from the base 2 of the halting probabilityfunction instead of base e of the Gibbs ensemble.

John C. Baez and Mike Stay29 took the analogy further by suggest- 29 John Baez and Mike Stay. Algorith-mic thermodynamics. Mathematical.Structures in Comp. Sci., 22(5):771–787,September 2012

ing a connection between algorithmic information theory and ther-modynamics, where the characteristics of the ensemble of programsare equivalent to thermodynamic constraints. A stated goal was toimport tools of statistical physics into algorithmic information theoryto facilitate its study. In algorithmic thermodynamics, one extends Ωwith algorithmic quantities to obtain the Baez-Stay ensemble:

Ω : U×R3 −→ R

(UTM, β, γ, δ) 7−→ ∑p∈Dom[UTM] 2−βE[p]−γV[p]−δN[p] (54)

Noting its similarities to the Gibbs ensemble of statistical physics,these authors suggest an interpretation where E[p] is the expected


value of the logarithm of the program’s runtime, V[p] is the expectedvalue of the length of the program, and N[p] is the expected valueof the program’s output. Furthermore, they interpret the conjugatevariables as (quoted verbatim from their paper):

"

1. T = 1/β is the algorithmic temperature (analogous to temperature).Roughly speaking, this counts how many times you must doublethe runtime in order to double the number of programs in theensemble while holding their mean length and output fixed.

2. p = γ/β is the algorithmic pressure (analogous to pressure). Thismeasures the trade-off between runtime and length. Roughly speak-ing, it counts how much you need to decrease the mean length toincrease the mean log runtime by a specified amount while holdingthe number of programs in the ensemble and their mean outputfixed.

3. µ = −δ/β is the algorithmic potential (analogous to chemical po-tential). Roughly speaking, this counts how much the mean logruntime increases when you increase the mean output while hold-ing the number of programs in the ensemble and their mean lengthfixed.

"

–John C. Baez and Mike Stay

From (Equation 54), they derive analogs of Maxwell’s relations andconsider thermodynamic cycles, such as the Carnot cycle or Stoddardcycle. For this, they introduce the concepts of algorithmic heat andalgorithmic work. Finally, we note that other authors have suggestedother alternative mappings in other but related contexts30. 30 Ming Li and Paul M.B. Vitanyi. An

Introduction to Kolmogorov Complexityand Its Applications. Springer PublishingCompany, Incorporated, 3 edition,2008; and Kohtaro Tadaki. A statisticalmechanical interpretation of algorithmicinformation theory. In Local Proceedingsof the Computability in Europe 2008 (CiE2008), pages 425–434. University ofAthens, Greece, Jun 2008

4.3 Attempt 1: Algorithmic Equation of State

The Journal of Natural Computing defines the subject as:

"Natural Computing refers to computational processes observed innature, and human-designed computing inspired by nature."31

31 https://link.springer.com/

journal/11047We are interested in how systems of algorithmic thermodynamicsrelate to the first part of this definition: how and under what con-ditions are such systems realized/realizable in nature? A relatedquestion is how much of nature can be described as natural com-puting — is it all of it or is it only part of it? One (naive) applicationof algorithmic thermodynamics could be as follows: consider thearchetypal ensemble of statistical physics — the classical system ofa perfect gas in a box of constant volume. One can surely interpretthe changing distribution of the gas molecules within the box as a

https://link.springer.com/journal/11047

https://link.springer.com/journal/11047


computation that, over time, maps out the space of solutions for thedynamical equations for the perfect gas. How insightful is that appli-cation likely to be? Well, this application amounts to just plasteringa computing description on top of an already satisfactory physicaldescription of the system. Why hinder ourselves with the additionaloverhead? Instead, we will be looking for a much more fundamen-tal description, we want the computing system to stand on its ownmerits. Specifically, our goal is not to describe the laws of physicsas analogous to performing a computation, but to instead find theproper statistical description under which the equation of state of thecomputation gives us the laws of physics.

The first step to connect algorithmic thermodynamics to nature isto not shy away from the computer-theoretic origins of algorithmicthermodynamics and to use quantities consistent with this origin.Therefore, instead of arbitrarily mapping, say the runtime to the en-ergy and the program length to the volume (or permutations of such)we will ground said quantities within the terminology of computerscience.

We will introduce two partition functions. The first is a canonicalensemble over the domain of a universal Turing machine. The quan-tities of this partition function are listed in Table 2. They are ok, thecomputing repetency conjugated with Ox[p] the program length, and o f

the computing frequency conjugated with Ot[p] the program time. Thepartition function is:

Z : U×R2 −→ R

(UTM, ok, o f ) 7−→ ∑p∈Dom[UTM] 2−okOx [p]−o f Ot [p] (55)


Ox[p] program length [bit] extensiveok computing repetency [1/bit] intensiveOx average tape size [bit] macroscopic

Ot[p] program time [operation] extensiveo f computing frequency [1/operation] intensiveOt average clock time [operation] macroscopic

Table 2: Algorithmic quantitiesof the canonical ensemble ofprograms

The second partition function is a grand canonical ensemble. It issimilar to the previous case, but the sum is over the finite elements ofthe power set of the domain of UTM, or P [D]. Executing a manifestM ∈ P [D] of programs on a universal Turing machine refers to aspecific computation involving multiple programs. In this ensemble,we add the quantity oµ, the computing overhead conjugated to On[M],


the quantity of programs in the manifest. The quantities of this ensembleare shown in Table 3 and its partition function are:

Z : R3 −→ R

(ok, o f , oµ) 7−→ ∑M∈P [D] 2−okOx [M]−o f Ot [M]−oµOn [M] (56)


Ox[M] length of programs in the manifest [bit] extensiveok computing repetency [1/bit] intensiveOx average tape usage [bit] macroscopic

Ot[M] running time of programs in the manifest [operation] extensiveo f computing frequency [1/operation] intensiveOt average clock time [operation] macroscopic

On[M] quantity of programs in the manifest [program] extensiveoµ computing overhead [1/program] intensiveOn average concurrency [program] macroscopic

Table 3: Algorithmic quantitiesof the grand canonical ensem-ble of programs

The corresponding probability measure is:

ρ : W×R3 −→ R

(M, ok, o f , oµ) 7−→ Z−12−okOx [M]−o f Ot [M]−oµOn [M] (57)

The probability measure maximizes the entropy subject to thefollowing bulk constraints:

Ox = ∑M∈P [D]

Ox[M]2−okOx [M]−o f Ot [M]−oµOn [M] (58)

Ot = ∑M∈P [D]

Ot[M]2−okOx [M]−o f Ot [M]−oµOn [M] (59)

On = ∑M∈P [D]

On[M]2−okOx [M]−o f Ot [M]−oµOn [M] (60)

The Lagrange multipliers (ok, o f and oµ) are interpreted, in thestyle of Baez and Stay, as:

• The computing repetency ok counts how many times the averagetape usage Ox must be doubled to double the entropy of the en-semble while holding the average clock time Ot and the averageconcurrency On fixed.

• The computing frequency o f counts how many times the averageclock time Ot must be doubled to double the entropy of the en-semble while holding the average tape usage Or and the averageconcurrency On fixed.


• The computing overhead oµ counts how many times the averageconcurrency On must be doubled to double the entropy of theensemble while holding the average clock time Ot and the averagetape usage Or fixed.

Various systems of natural computing can be produced usingother resources. Let us give a few examples.

1. Computing time to program frequency formulation:

Z′ : U×R2 −→ R

(UTM, ok, ot) 7−→ ∑p∈Dom[UTM] 2−okOx [p]−otO f [p] (61)

To formulate this relation, we introduce the program frequencyO f [p] as the inverse of the program time Ot[p], thus O f [p] :=1/Ot[p]. This formulation fixes an average clock frequency O f byhaving the programs executed under a constant computing timeot:

• The Computing time ot counts how many times the averageclock frequency O f must be doubled to double the entropy ofthe ensemble while holding the average tape usage Ox and theaverage concurrency On fixed.

2. Size-cutoff formulation:

Z′′ : U×R2 −→ R

(UTM, ok, x) 7−→ ∑p∈q:Dom[UTM]|Ox [q]<x 2−okOx [p]

(62)

The sum Z′′ only includes programs with length less than or equalto x. Ω is recovered in the limit when x → ∞ (and when ok = 1).Z′′ represents the first n bits of Ω up to a cutoff proportional to x.

3. Time-cutoff formulation:

Z′′′ : U×R3 −→ R

(UTM, ok, o f , t) 7−→ ∑p∈q:Dom[UTM]|Ot [q]<t 2−okOx [p]−o f Ot [p]

(63)

The sum Z′′′ only includes programs that halt within a time cutofft. Thus, Z′′′ contains no "non-halting information" and is com-putable. Ω is recovered in the limit when t → ∞ (and whenok = 1).


4. Computational-complexity cutoff formulation:

Z′′′′ : U×R× G : f → h −→ R

(UTM, ok, g) 7−→ ∑p∈q:Dom[UTM]|O[UTM[q]]≤g 2−okOx [p]

(64)

The sum only includes programs that halt and whose computa-tional complexity is less than or equal to a term g.

Interpretation:

1. Feasible computing complexity:

The use of the letter O to identify thermodynamic observablesis coincidentally very convenient here. Recall the Big O notationused in computational complexity theory to denote program com-plexity. Unlike in algorithmic thermodynamics, computationalcomplexity theory has no need for physical resource indicators(clock speed, time-cutoffs, etc.) to define the computational com-plexity of programs because said difficulty is defined as the re-lation between the size of the input and the number of steps re-quired to solve the problem (a definition independent of physicalresource availability). For example, in complexity theory, a pro-gram with input n which takes 109999n steps to halt would likelytake longer to run than the age of the universe on any physicalcomputer (even for n = 1), but computational complexity theoryconsiders this intractable problem to be an easier problem thanone requiring n2 steps. Consequently, computational complex-ity theory based on Big O notation does not quite connect to thephysical reality of computation with limited available resources.

However, using an ensemble of algorithmic thermodynamics, acost-to-compute, measured in entropy, can be attributed to carry-ing out a computation using finite resources.

2. Entropy as a measure of computational ’distance’

Consider an equation of state based on computing resources. Apartition function of algorithmic thermodynamics (such as the oneof Equation 57), has the following equation of state:

dS = ok dOx + o f dOt + oµ dOn (65)

Using this equation of state, we can quantify the computing ’dis-tance’ between two states of the system using the difference in


entropy as the ’meter’. This equation forms a specific type of met-ric, known as a taxi-cab metric and represents a typical equation ofstate of thermodynamics.

3. Reservoirs of computing resources:

It is common in statistical physics to appeal to various reservoirssuch as a thermal reservoir or a particle reservoir, etc. The typicalGibbs ensemble in physics is Z(β) = ∑q∈Q exp

(−βE[q]

). It’s av-

erage energy is given by E = −∂ln Z/∂β and its fluctuations are(∆E)2 = ∂2ln Z/∂β2. To justify that fluctuations are possible andcompatible with the laws of conservation of energy, the system isclaimed/idealized to be in contact with a thermal reservoir. In thisidealized case, both the system and the reservoir have the sametemperature and they can exchange energy. The reservoir is con-sidered large enough that the fluctuations of the smaller systemare negligible to its description. Mathematically, the reservoir hasinfinite heat capacity. Thus, the reservoir abstractly represents aninfinitely deep pool of energy at a given, constant temperature.

A similar analogy can be supported for a system of natural com-puting, in which the computing resources are provided to thesystem in the form of reservoirs. For instance, instead of a thermalreservoir, we may have runtime and tape reservoirs. These reser-voirs have mathematically infinite runtime and tape capacities andthus act as infinitely deep pools of computing resources. Com-puting is made possible by the interaction of the reservoirs withthe system and the intensity of the exchanges is calibrated by thecomputing repetency and the computing frequency, instead of bythe temperature.

By considering that the group of reservoirs is the representation ofan idealized ’supercomputer’, the analogy is completed and algo-rithmic thermodynamics describes the dynamics of computationin equilibrium with the resources made available by a ’supercom-puter’.

So far so good; but why not a quantum computation? Where isquantum mechanics, the qubit, the geometry of space-time... etc.?

4.4 Hint: Entropy, Information and Space-time

In 2002, Lloyd32 calculated the total number of bits available for com- 32 Seth Lloyd. Computational capacityof the universe. Physical Review Letters,88(23):237901, 2002

putation in the universe, as well as the total number of operationsthat could have occurred since the universe’s beginning.

For both quantities (the quantity of bits stored in the universeand the quantity of operations made on those bits), Lloyd obtains


the number ≈ 10122kB[bit]. This number is consistent with other ap-proaches; for instance, the Bekenstein-Hawking entropy33 of a ’holo- 33 Stephen W Hawking. Black hole

explosions? Nature, 248(5443):30, 1974;and Damien A. Easson, Paul H. Framp-ton, and George F. Smoot. Entropicinflation. International Journal of ModernPhysics A, 27(12):1250066, 2012

graphic surface’ at the cosmological horizon34 (also ≈ 10122kB[bit]).

34 Leonard Susskind. The world asa hologram. Journal of MathematicalPhysics, 36(11):6377–6396, 1995

How did Lloyd derive these numbers? First, he calculated thevalue for these quantities while ignoring the contribution of gravityand he obtained ≈ 1090kB[bit]. It is only by including the degreesof freedom of gravity that the number ≈ 10122kB[bit] is obtained,which he does in the second part of his paper. As we are interestedin the totals, we will go directly to the calculations that include thecontribution of gravity. We state Lloyd’s main result and note that thedetails of the calculation can be reviewed in his paper. Lloyd obtainsa relation between time and number of operations for the universe:

#ops ≈ ρcc5t4

h≈ t2c5

Gh=

1t2

pt2 (66)

where ρc is the critical density and tp is the Planck time and t isthe age of the universe. With present-day values of t, the result is≈ 10122kB[bit]. He states:

"Applying the Bekenstein bound and the holographic principle to theuniverse as a whole implies that the maximum number of bits thatcould be registered by the universe using matter, energy, and gravity is≈ c2t2

l2p

= t2

t2p."

A particularly interesting consequence of this result is that theserelations appear to imply conservation of both information and op-erations in space-time (the numerical quantity of 10122 is obtainedby summing over all available degrees of freedom in space-time). Sowith this hint, we are now looking for a fundamental relationshipbetween entropy, information, operations, and... space-time.

A relation between entropy and space-time has been anticipated(or at least hinted at) since probably the better part of four decades.The first hints were provided by the work of Bekenstein35 regarding 35 Jacob D Bekenstein. Black holes and

entropy. Physical Review D, 7(8):2333,1973; Jacob D Bekenstein. General-ized second law of thermodynamics inblack-hole physics. Physical Review D,9(12):3292, 1974; and Jacob D Beken-stein. Black-hole thermodynamics.Physics Today, 33(1):24–31, 1980

the similarities between black holes and thermodynamics, culminat-ing in the four laws of black hole thermodynamics. The temperature,originally introduced by analogy, was soon augmented to a real no-tion by Hawking36 with the discovery of the Hawking temperature

36 Stephen W Hawking. Black holeexplosions? Nature, 248(5443):30, 1974

derived from quantum field theory on curved space-time. We notethe discovery of the Bekenstein-Hawking entropy, connecting the areaof the surface of a horizon to be proportional to one fourth the num-ber of elements with Planck area that can be fitted on the surface:S = kBc3/(4hG)A.

We mention Ted Jacobson37 and his derivation of the Einstein field 37 Ted Jacobson. Thermodynamics ofspacetime: The einstein equation ofstate. Phys. Rev. Lett., 75:1260–1263, Aug1995

equation as an equation of state of a suitable thermodynamic system.


To justify the emergence of general relativity from entropy, Jacobsonfirst postulated that the energy flowing out of horizons becomeshidden from observers. Next, he attributed the role of heat to thisenergy for the same reason that heat is energy that is inaccessiblefor work. In this case, its effects are felt, not as "warmth", but asgravity originating from the horizon. Finally, with the assumptionthat the heat is proportional to the area A of the system under someproportionality constant η, and some legwork, the Einstein fieldequations are eventually recovered.

Recently, Erik Verlinde38 proposed an entropic derivation of the 38 Erik P. Verlinde. On the origin ofgravity and the laws of newton. Journalof High Energy Physics, 2011(4):29, Apr2011

classical law of inertia and those of classical gravity. He comparedthe emergence of such laws to that of an entropic force, such as apolymer in a warm bath. Each law is emergent from the equationT dS = F dx, under the appropriate temperature and a posited en-tropy relation. His proposal has encouraged a plurality of attempts toreformulate known laws of physics using the framework of statisticalphysics. Visser39 provides, in the introduction to his paper, a good 39 Matt Visser. Conservative entropic

forces. Journal of High Energy Physics,2011(10):140, 2011

summary of the literature on the subject. The ideas of Verlinde havebeen applied to loop quantum gravity (40), the Coulomb force (41), 40 Lee Smolin. Newtonian gravity in

loop quantum gravity. arXiv preprintarXiv:1001.3668, 2010

41 Tower Wang. Coulomb force asan entropic force. Physical Review D,81(10):104045, 2010

Yang-Mills gauge fields (42), and cosmology (43). Some criticism has,

42 Peter GO Freund. Emergent gaugefields. arXiv preprint arXiv:1008.4147,2010

43 Rong-Gen Cai, Li-Ming Cao, andNobuyoshi Ohta. Friedmann equationsfrom entropic force. Physical ReviewD, 81(6):061501, 2010; Miao Li andYi Wang. Quantum uv/ir relationsand holographic dark energy fromentropic force. Physics Letters B, 687(2-3):243–247, 2010; and Damien A Easson,Paul H Frampton, and George F Smoot.Entropic accelerating universe. PhysicsLetters B, 696(3):273–277, 2011

however, been voiced44, including by Visser45.

44 Sabine Hossenfelder. Commentson and comments on comments onverlinde’s paper" on the origin ofgravity and the laws of newton". arXivpreprint arXiv:1003.1015, 2010; ArchilKobakhidze. Gravity is not an entropicforce. Physical Review D, 83(2):021502,2011; Shan Gao. Is gravity an entropicforce? Entropy, 13(5):936–948, 2011;BL Hu. Gravity and nonequilibriumthermodynamics of classical matter.International Journal of Modern PhysicsD, 20(05):697–716, 2011; and ArchilKobakhidze. Once more: gravity isnot an entropic force. arXiv preprintarXiv:1108.4161, 2011

45 Matt Visser. Conservative entropicforces. Journal of High Energy Physics,2011(10):140, 2011

Even more recently, a connection between entanglement entropyand general relativity has been supported by multiple publications46.

46 Brian Swingle. Entanglement renor-malization and holography. PhysicalReview D, 86(6):065007, 2012; MarkVan Raamsdonk. Building up spacetimewith quantum entanglement. GeneralRelativity and Gravitation, 42(10):2323–2329, 2010; Glen Evenbly and GuifréVidal. Tensor network states and ge-ometry. Journal of Statistical Physics,145(4):891–918, 2011; Thomas Faulkner,Monica Guica, Thomas Hartman,Robert C Myers, and Mark Van Raams-donk. Gravitation from entanglementin holographic cfts. Journal of HighEnergy Physics, 2014(3):51, 2014; ThomasFaulkner, Felix M Haehl, Eliot Hijano,Onkar Parrikar, Charles Rabideau, andMark Van Raamsdonk. Nonlineargravity from entanglement in conformalfield theories. Journal of High EnergyPhysics, 2017(8):57, 2017; BartlomiejCzech, Lampros Lamprou, SamuelMcCandlish, and James Sully. Tensornetworks from kinematic space. Jour-nal of High Energy Physics, 2016(7):100,2016; Juan Maldacena. The large-nlimit of superconformal field theo-ries and supergravity. Internationaljournal of theoretical physics, 38(4):1113–1133, 1999; Brian Swingle and MarkVan Raamsdonk. Universality of grav-ity from entanglement. arXiv preprintarXiv:1405.2933, 2014; Juan Maldacenaand Leonard Susskind. Cool horizonsfor entangled black holes. Fortschritteder Physik, 61(9):781–811, 2013; FabioSanches and Sean J Weinberg. Holo-graphic entanglement entropy conjec-ture for general spacetimes. PhysicalReview D, 94(8):084034, 2016; LeonardSusskind. Entanglement is not enough.Fortschritte der Physik, 64(1):49–71,2016; ChunJun Cao, Sean M Carroll,and Spyridon Michalakis. Space fromhilbert space: recovering geometry frombulk entanglement. Physical Review D,95(2):024031, 2017; Ning Bao, ChunJunCao, Sean M Carroll, and Liam McAl-lister. Quantum circuit cosmology: Theexpansion of the universe since the firstqubit. arXiv preprint arXiv:1702.06959,2017; Ning Bao, ChunJun Cao, Sean MCarroll, and Aidan Chatwin-Davies.de sitter space as a tensor network:Cosmic no-hair, complementarity,and complexity. Physical Review D,96(12):123536, 2017; Leonard Susskind.Dear qubitzers, gr= qm. arXiv preprintarXiv:1708.03040, 2017; Thanu Pad-manabhan. Thermodynamical aspectsof gravity: new insights. Reports onProgress in Physics, 73(4):046901, 2010;Ted Jacobson. Entanglement equilib-rium and the einstein equation. Physicalreview letters, 116(20):201101, 2016; andSean M Carroll and Grant N Remmen.What is the entropy in entropic gravity?Physical Review D, 93(12):124052, 2016

Finally, we mention the body of work of George Ellis regardingthe evolving block universe hypothesis detailed in 47 and the con-

47 George FR Ellis. Physics in the realuniverse: Time and spacetime. Generalrelativity and gravitation, 38(12):1797–1824, 2006; George FR Ellis and Ritu-parno Goswami. Spacetime and thepassage of time. In Springer Handbookof Spacetime, pages 243–264. Springer,2014; and George FR Ellis. The evolv-ing block universe and the meshingtogether of times. Annals of the New YorkAcademy of Sciences, 1326(1):26–41, 2014

nection between space-time events, general relativity and quantummechanics.

We are now ready to investigate our second attempt at a solution.

4.5 Attempt 2: Designer Ensemble

Our second series of attempts could be grouped under a simple con-cept: we attempted to construct a specific system of statistical physicshaving a double interpretation; one, as a system of algorithmic ther-modynamics admitting an equation of state involving bits and oper-ations, and second, that said equation of state be interpretable as aphysical system of space-time.

Finding a specific system of statistical physics means attribut-ing an implementation to the thermodynamic observable functions(Ox[p], Ot[p], etc.) used in the partition function. A similar approachwas used by Ted Jacobson and Erik Verlinde in the context of con-necting general relativity and classical gravity, respectively, to en-tropy. In each of their papers, the degrees of freedom of space are as-sumed to be quadratic (i.e. they grow as an area law). Consequently,


the thermodynamic observables are quadratic degrees of freedom.Attempting to expand upon these ideas, we have investigated theemergence of many physical laws, including a toy model of a cosmol-ogy emergent from quadratic degrees of freedom. However, in theend, we felt that there was a general problem with this approach.

The problem with this approach, even if it successfully lead tosome set of laws, is that any results would be specific to the con-structed ensemble. A "tower of postulates" would be sneaked into thedefinition of a postulated ensemble.

Furthermore, we were missing out on the full potential of statis-tical physics as a general framework. Indeed, statistical physics canproduce conservation equations on the broadest of scales. As a typi-cal example, we refer to the fundamental relation of thermodynamicsinvolving the conservation of energy over a change in thermody-namic observables:

dE = T dS + p dV − µ dN (67)

To capture this generality, our retained solution was not to de-fine a specific system of statistical physics, but instead to increasethe generality of thermodynamics; in the present case, with a ge-ometric algebra applied to the thermodynamic observables suchthat the equation of state forms a complete basis of the n× n matrixrepresentation of all possible transformations of the system. In thisgeneralization, which we call universal thermodynamics, the generalconservation relation above becomes a special case of an even moregeneral conservation relation that, remarkably, has the suitable prop-erties, and the equation of state is able to carry the system from anystate to any state.

5 Universal Thermodynamics

I initially identified the potential to generalize statistical physics asI attempted to create thermodynamic cycles that are consistent withthe symmetries of space-time. By doing so, I realized that such cyclescould be produced if the relevant thermodynamic constraints obeyeda non-commutative algebra. By comparison, in normal thermody-namics, the constraints are commuting scalars.

My initial goal was to recover the structure of space-time (Lorentzinvariance, general invariance, speed of light, metric interval, etc.)strictly using the facilities of statistical physics (entropy, observables,conjugate variables, cycles/transformations, etc.). To achieve that, wesimply interpret the speed of light as a tool to hide information inspace-time. Specifically, the speed of light hides information regard-


ing events whose interval to the observer exceeds the speed of light.Interpreted as such, we can then use the entropy in statistical physicsto achieve the same purpose as the speed of light (hide information),provided that we "place" this entropy at the appropriate position inthe system.

We note that attributing an entropy to events separated by a hori-zon to connect to thermodynamics has been done since at least 1973

by J.D. Bekenstein48. Furthermore, from G. W. Gibbons and S. W. 48 Jacob D Bekenstein. Black holes andentropy. Physical Review D, 7(8):2333,1973

Hawking’s 1977 article49, I quote:49 Gary W Gibbons and Stephen WHawking. Cosmological event horizons,thermodynamics, and particle creation.Physical Review D, 15(10):2738, 1977

"An observer in these models will have an event horizon whose areacan be interpreted as the entropy or lack of information of the observerabout the regions which he cannot see."

The part missing to complete a full entropic picture of space-time,I suggest, is to apply the same line of reasoning to configurationsof time-like and space-like separated events. For instance, we canimagine an observer O whose "visibility" is defined by the usuallight cone of special relativity. We can describe this light cone en-tirely using notions of statistical physics by analyzing the number ofconfigurations of events outside the light-cone and associating it toan entropy. Indeed, to prevent faster-than-light communication, allpossible configurations of events outside the light-cone must be ofmaximal entropy to be void of information from the perspective of O.This entropy thus hides events that O cannot see.

The same reasoning can be applied to the future of O. Indeed,to prevent O from knowing its future, future events must also bevoid of information from O’s perspective and thus be at maximal en-tropy. Using this strategy we can construct an ensemble of statisticalphysics that recovers the structure of space-time in the bulk, simplyby attributing an entropy to the microscopic states which comprisesthe random events spread in space-time. We will now describe thephysical quantity relevant to universal thermodynamics.

Definition 11 (Specific repetency). As we construct an equation of state,a physical quantity will be introduced as the Lagrange multiplier: the spe-cific repetency k with units m−1. The specific repentency is a proportionallyfactor that quantifies the entropy associated to the interval between twoevents in space-time. It replaces the role normally assumed by the tempera-ture in thermodynamics.

This specific repetency is the conjugated variable to a distanceX1, X2, X3 and/or time cX0. These quantities are summarized in Ta-ble 4. By convention, we will prefix the Lagrange multiplier withthe word "specific" and its averaged conjugated quantity will be pre-fixed with the word "entropic". The specific repetency is an intensive



X1[q], X2[q], X3[q] space [meter] microscopick specific repetency [1/meter] intensiveX1, X2, X3 entropic space [meter] extensive

X0[q] time [second] microscopicck specific frequency [1/second] intensiveX0 entropic time [second] extensive

Table 4: The physical quantitiesof the geometric ensemble

quantity, whereas entropic space X1, X2, X3 and entropic time cX0

are extensive quantities. Indeed, a process taking 1 min followedby a process taking 2 min takes a total of 3 min (extensive). For theX1, X2, X3 quantities: walking 1 meter followed by walking 2 me-ters implies one has walked a total of 3 meters (extensive). Addingor removing clocks from a group of clocks ticking at a frequency ck(say once per second) has no impact on the frequency of the otherelements of the group (intensive). The same argument applies to thespecific repetency (intensive). The units of k are m−1, the units ofX1, X2, X3 are the meters, the units of X0 are the seconds, and theunits of ck are s−1.

We note that the temperature (kBT = 1/β) has no central role inuniversal statistical physics. In fact, unlike the speed of light, space-time in general (excluding horizons) does not have a constant tem-perature and therefore describing space-time as a thermodynamicsystem (using temperature, energy, and entropy) would be inappro-priate as the system would be outside equilibrium. However, withour strategy, it is precisely because faster-than-light communication isimpossible that a quantity, the specific repetency — closely connectedto the speed of light, can take the role normally assumed by the tem-perature as a Lagrange multiplier of the ensemble. As we will see, byusing this quantity as a Lagrange multiplier instead of the temper-ature, the ensemble applies an entropy to all of space-time, with orwithout horizons, and thus determines its complete structure.

To understand in more detail, let us investigate a hypotheticalthermodynamic transformation involving the observables X1, X2 andX3. The equation of state of such a system would be:

k−1B dS = k(dX1 + dX2 + dX3) (68)

For a change over the quantities X1, X2 and X3 to be consistentwith the symmetries of Euclidean space, one would expect that thechange in entropy along two paths of equal distance, say a path go-ing in a straight line from (0, 0, 0) to (0, 5, 0) and a path going in astraight line from (0, 0, 0) to (3, 4, 0), to be equal. Indeed, the Eu-


clidean distance along either path is the same: in this case, 5 meters.Since the paths are related to one another via rotation of the frame ofreference, the entropic cost of the transformation should only dependon the Euclidean length of the path, and not on the orientation of theframe of reference.

One can enforce this property by demanding that the thermody-namic observables obey a suitable non-commutative algebra. Let’ssee with an example. As the first step, we add the generators of analgebra, say we name them σ1, σ2, σ3, to each quantity. We get:

k−1B dS = k(σ1 dX1 + σ2 dX2 + σ3 dX3) (69)

We note that in this expression, S becomes an entropy vector, andthis will be addressed rigorously in the next section. But for now, wewill see that this entropy will become a real number by squaring itand we will see that the entropy conforms to the Euclidean distance.By squaring, we obtain:

k−2B k−2(dS)2 =σ2

1 (dX1)2 + σ2

2 (dX2)2 + σ2

3 (dX3)2

+ (σ1σ2 + σ2σ1)dX1 dX2 + (σ1σ3 + σ3σ1)dX1dX3 + (σ2σ3 + σ3σ2)dX2 dX3

(70)

In the case where σ1, σ2 and σ3 are commutative, the cross termsσ1σ2 + σ2σ1, σ1σ3 + σ3σ1 and σ2σ3 + σ3σ2 do not cancel, but if they are,say matrices, that obey the following relations:

σ21 = 1 (71)

σ22 = 1 (72)

σ23 = 1 (73)

σ1σ2 + σ2σ1 = 0 (74)

σ1σ3 + σ3σ1 = 0 (75)

σ2σ3 + σ3σ2 = 0 (76)

Then, the cross-terms cancel and we obtain:

k−2B k−2(dS)2 =(dX1)

2 + (dX2)2 + (dX3)

2 (77)

The entropy, here, is a real number again.The resulting equation of state has the mathematical form of the

Euclidean distance. The entropy, as demanded, is invariant under ro-tation of the Euclidean frame of reference. As we will see, if one usesthe flexibility of geometric algebra, one can generalize this argument


to space-times of any dimensions, any signature, and even includingarbitrarily curved space-times.

For instance, a thermodynamic system of special relativity wouldhave X1, X2, X3 and cX0 as its thermodynamic quantities. The equa-tion of state, using the generators γ0, γ1, γ2, γ3 is:

k−1B k−1 dS = γ0c dX0 + γ1 dX1 + γ2 dX2 + γ3 dX3 (78)

Here, X0 is an extensive quantity with units s and it is conjugatedwith ck having units s−1. Squaring the equation of state gives:

k−2B k−2(dS)2 =γ2

0c2(dX0)2 + γ2

1(dX1)2 + γ2

2(dX2)2 + γ2

3(dX3)2

+ c(γ0γ1 + γ1γ0)dX0 dX1 + c(γ0γ2 + γ2γ0)dX0 dX2 + c(γ0γ3 + γ3γ0)dX0 dX3

+ (γ1γ2 + γ2γ1)dX1 dX2 + (γ1γ3 + γ3γ1)dX1 dX3

+ (γ2γ3 + γ3γ2)dX2 dX3 (79)

The cross-terms cancel, provided that the generators obey thefollowing relations:

γ20 = 1 (80)

γ21 = −1 (81)

γ22 = −1 (82)

γ23 = −1 (83)

γ0γ1 + γ1γ0 = 0 (84)

γ0γ2 + γ2γ0 = 0 (85)

γ0γ3 + γ3γ0 = 0 (86)

γ1γ2 + γ2γ1 = 0 (87)

γ1γ3 + γ3γ1 = 0 (88)

γ2γ3 + γ3γ2 = 0 (89)

(90)

Then, the equation of state is:

k−2B k−2(dS)2 = c2(dX0)

2 − (dX1)2 − (dX2)

2 − (dX3)2 (91)

The square of the entropy is a real number again.In the general case, one begins by defining an arbitrary non-

commutative basis as follows:

eµ · eν =12(eµeν + eνeµ) = gµν (92)


To define geometric thermodynamics as a system of statisticalphysics, one first defines n thermodynamic observables using thegeometric basis. The statistical priors, such as E = ∑q∈Q E[q]ρ[q], arenow simply multiplied with a generator ei of the geometric algebra,yielding n equations:

eiXi = ∑q∈Q

eiXi[q]ρ[q] (93)

Then, by maximizing the entropy with these priors as the con-straints and by using the method of the Lagrange multipliers, onewill obtain a generalized non-commutative thermodynamics conser-vation relation instead of equation (34):

k−1B k−1 dS = e0 dX0 + · · ·+ en−1 dXn−1 (94)

We note that had we instead selected a geometric algebra suchthat the generators are commutative, then one would recover, as aspecial case, the traditional conservation relation of energy found instatistical physics. Explicitly, posing the properties of the generatorse1, ..., en to be commutative:

e2i = 1 (95)

eiej = ejei (96)

one obtains the relation k−1B dS = k dX0 + · · · + k dXn−1, which

is of the same mathematical form as equation (34) and describes ataxi-cab metric. Therefore, geometric thermodynamics is indeed ageneralization of thermodynamics; a fact quite important to what weare trying to achieve. Indeed, statistical physics has long been con-sidered by many to be our physical theory least likely to be falsifiedwithin its domain of applicability. The robustness associated with sta-tistical physics will thus be inherited by the laws of physics derivedas a consequence of this generalization.

5.1 Recap: Geometric Algebra

A geometric algebra G is spawned by a generator relation:

eµ · eν =12(eµeν + eνeµ) = gµν (97)

The generators form a basis that includes the generators them-selves and all arrangements of their wedge products. For instance, analgebra of four generators e0, e1, e2, e3 form the complete basis:


basis elements grade

1, grade-0 (98)

e0, e1, e2, e3, grade-1 (99)

e0e1, e0e2, e0e3, e1e3, e2e3, grade-2 (100)

e0e1e2, e0e1e3, e0e2e3, e1e2e3, grade-3 (101)

e0e1e2e3 grade-4 (102)

Multivectors of G can be constructed as a linear combination ofthese basis elements. For instance:

Vectors and multivectors:

example name

v := 1 0-vector, or scalar (103)

v := 3e0 + 4e1 1-vector, or vector (104)

v := 3e0e3 + 2e2e1 2-vector, or bivector (105)

v := 5e0e1e2 3-vector, or trivector (106)

......

v := 2e0e1 . . . ek k-vector (107)

......

v := 1 + 2e0 + 5e2e2 multivector (108)

We note that the k-vectors are a linear combination of basis el-ements of the same grade, whereas a multivector mixes differentgrades.

If the scalars multiplying the basis elements of the multivectorsare elements of the reals, then the algebra is called a real geometricalgebra G(R), and if they are complex then the algebra is called acomplex geometric algebra G(C). For instance:

v := r + r0e0 + r1e1 + r2e2 + r01e0e1 + . . . where r, r0, r1, r1, r01, · · · ∈ R

(109)

is a real algebra G(R), and

v := z + z0e0 + . . . where z, z0, · · · ∈ C (110)

is a complex algebra G(C).We use numbered indices to denote the number of generators of

G. For instance if G has four generators e0, e1, e2, e3 we denote the


algebra by G4 generally, or G4(C) if the algebra is complex with fourgenerators, or G4(R) if the algebra is real with four generators.

Furthermore, if the generator relation is orthogonal;

γi · γj =12(γiγj + γjγi) = ηij (111)

where ηij is the signature of the generator relation, for instance:

ηij = diag(+,−,−,−) (112)

then,

γ0γ0 = 1 (113)

γ1γ1 = −1 (114)

γ2γ2 = −1 (115)

γ3γ3 = −1 (116)

γ0γ1 + γ1γ0 = 0 (117)

γ0γ2 + γ2γ0 = 0 (118)

γ0γ3 + γ3γ0 = 0 (119)

γ1γ2 + γ2γ1 = 0 (120)

γ1γ3 + γ3γ1 = 0 (121)

γ2γ3 + γ3γ2 = 0 (122)

For real algebras, we add an additional indice Cln,m(R), where nis the number of generators squaring to 1, and m is the number ofgenerators squaring to −1. In the case of signature diag(+,−,−,−),the algebra is Cl1,3(R).

The geometric product of two multivectors v and u is:

vu = v · u + v ∧ u (123)

It can be calculated quite simply by expanding the product andapplying the generator relation to simplify the expression. For in-stance, consider the following 1-vectors:

v := ae0 + be1 (124)

u := ce0 + de1 (125)

Then, the geometric product is


vu = (ae0 + be1)(ce0 + de1) (126)

= ae0ce0 + ae0de1 + be1ce0 + be1de1 (127)

= ac + ade0e1 − dbe0e1 + bd (128)

= ac + bd + (ad− db)e0e1 (129)

= v · u + v ∧ u (130)

One can construct higher grades of the basis using the antisym-metrization. Using the gamma matrices γ0, γ1, γ2, γ3 as a represen-tation, the complete basis contains:

1. The identity matrix: 1

2. 4 matrices: γi

3. 6 matrices σµν = 12 [γ

ν, γµ]

4. 4 matrices σµνρ = 16 [γ

µ, γν, γρ]

5. 1 matrix σµνρδ = 124 [γ

µ, γν, γρ, γδ]

The most important fact for us is that a complex linear combina-tion using all 16 elements of Cl4(C) forms a complete basis of 4× 4complex matrices.

5.2 Universal Representation

In mathematics it is common to think of matrices as a representationof the geometric algebra. But if one has an affinity with matrices,can one not instead think of geometric algebra as a representationof such matrices? This may not be possible for all matrices, butfor at least the complex 4 × 4 matrices and Cl4(C), each is a com-plete representation of the other. Since our starting point is algo-rithms/computation/transformation, then as the first and foremostdescription, we can think of a ensemble of programs such that itspossible transformations are described by a complete set of matri-ces. Then, changes in the system are governed by an equation ofstate able to cycle over all states attainable by these transformations.In this case, we will then think of the geometric algebra, forming acomplete basis of such matrices, as a representation of these transfor-mations. Consequently, our interpretation herein of space-time willbe that it is a representation (a very visualizable one indeed) of suchuniversal transformations.


5.3 Universal Metric

Definition 12 (Spacetime Event). A spacetime event s is a 1-vector of thespace generated by Cl3,1(R):

s = X0e0 + X1e1 + X2e2 + X3e3 (131)

Definition 13 (Universal Event). A universal event s is a multivector ofthe space generated by Cl4(C) that uses its complete basis:

s :=a

+ X0e0 + X1e1 + X2e2 + X3e3

+ E1e0e1 + E2e0e2 + E3e0e3 + B1e2e3 + B2e1e3 + B3e1e2

+ V0e1e2e3 + V1e0e2e3 + V2e0e1e3 + V3e0e1e2

+ be0e1e2e3 (132)

This definition generalizes to n dimensions and to arbitrary geometricalgebras Gn.

It is well-known that the metric of space-time events is (in theorthonormal case / special relativity):

(ds)2 = −(c dX0)2 + (dX1)

2 + (dX2)2 + (dX3)

2 (133)

= ηµν dXµ dXν (134)

or (in the general case / general relativity):

(ds)2 = gµν dXµ dXν (135)

But how does a metric generalizes to universal events? Anotherway of asking the question is: how can we define the distance be-tween two arbitrary multivectors of the same geometric algebra?First, let me state my solution, then we will investigate it:

Definition 14 (Universal Interval). Let ∆s be the matrix representationof ∆s ∈ Gn. Then, the universal interval ∆s ∈ C for a given matrixrepresentation of s is given by this relation:

det[∆s1− ∆s

]= 0 (136)

Solving it for ∆s yields the universal norm of s.

We will now investigate how this definition compares to the usualdefinition of the interval. Let us recall that for a 1-vector of Euclideanspace, its norm is defined as the inner product:


‖v‖2 := v · v = x2 + y2 + z2 (137)

It may be tempting to define the interval of a multivector as asimple extension of the usual inner product, but alterations are of-ten required on the usual inner product to create an interval thatis relevant for physics, and specifically which alterations to use ornot to use is not always easy to guess right. My definition assigns aphysically relevant notion of distance automatically to all multivec-tors even in non-obvious cases. In fact, the universal interval is thephysically-relevant quantity describing the ’length’ of vectors andmultivectors in all geometric spaces. To help us understand how itworks, let us first compare various kinds of established norms andintervals to our definition, and then we will look at novel cases. Un-less otherwise stated, we will be working with Cl4(C) represented bythe gamma matrices γ0, γ1, γ2, γ3 (which we note as Clγ

4 (C)).

1. scalar: The event v := a has the following matrix representation:

v :=

a 0 0 00 a 0 00 0 a 00 0 0 a

(138)

The universal interval is obtained as follows:

0 = det

∆s

1 0 0 00 1 0 00 0 1 00 0 0 1

−

∆a 0 0 00 ∆a 0 00 0 ∆a 00 0 0 ∆a

=⇒ 0 = (∆s− ∆a)4 =⇒ ∆s = ∆a

(139)

The universal interval compares to the conventional definition asfollows:

∆s = ∆a Conventional (140)

∆s = ∆a Universal Interval (141)

In this case, both intervals are a.

2. Pythagorean norm (2D): In Cl2(R), the event v := xe1 + ye2 is:


v = x

(1 00 −1

)+ y

(0 11 0

)=

(x yy −x

)(142)

Evaluating the universal interval produces this relation:

0 = (∆s)2 − (∆x)2 − (∆y)2 =⇒ (∆s)2 = (∆x)2 + (∆y)2 (143)

Which is the Pythagorean norm.

3. Euclidean norm (3D): In Clσ3 (R), the event v := xσx + yσy + zσz is:

v =

(z x− iy

x + iy −z

)(144)


0 = (∆s)2 − (∆x)2 − (∆y)2 − (∆z)2 =⇒ (∆s)2 = (∆x)2 + (∆y)2 + (∆z)2

(145)

which is the Euclidean norm.

4. spacetime event: The event v := tγ0 + xγ1 + yγ2 + zγ3 has thefollowing matrix representation:

v =

t 0 z x− iy0 t x + iy −z−z −x + iy −t 0

−x− iy z 0 −t

(146)


0 = ((∆s)2 − (∆t)2 + (∆x)2 + (∆y)2 + (∆z)2)2 (147)

Compared to the conventional approach, the intervals are:

ηµν∆Xµ∆Xν = −(∆x)2 − (∆y)2 − (∆z)2 + (∆t)2 conventional(148)

(∆s)2 = −(∆x)2 − (∆y)2 − (∆z)2 + (∆t)2 our definition(149)

We note that the conventional definition and our definition givethe same interval.


5. bi-vectors: Let us now apply my definition in a geometric contextwhere no intervals are currently known. The event b := E1γ0γ1 +

E2γ0γ2 + E3γ0γ3 + B1γ2γ3 + B2γ1γ3 + B3γ1γ2, a bi-vector spanningall six 2-basis elements, has the following matrix representation:

b :=

−iB3 −iB1 + B2 E3 E1 − iE2

−iB1 − B2 iB3 E1 + iE2 −E3

E3 E1 − iE2 −iB3 −iB1 + B2

E1 + iE2 −E3 −iB1 − B2 iB3

(150)

The conventional approach is speechless with respect to its in-terval, but by using my definition an interval can nonetheless beidentified with physical relevance. The universal interval for thisbivector is verbose:

0 =B41 + 2B2

1B22 + B4

2 + 2B21B2

3 + 2B22B2

3 + B43

+ 2B21E2

1 − 2B22E2

1 − 2B23E2

1 + E41 + 8B1B2E1E2

− 2B21E2

2 + 2B22E2

2 − 2B23E2

2 + 2E21E2

2 + E42

+ 8B1B3E1E3 + 8B2B3E2E3 − 2B21E2

3 − 2B22E2

3

+ 2B23E2

3 + 2E21E2

3 + 2E22E2

3 + E43 + 2B2

1‖s‖2

+ 2B22‖s‖

2 + 2B23‖s‖

2 − 2E21 +‖s‖

2 − 2E22‖s‖

2

− 2E23‖s‖

2 +‖s‖4 (151)

but its four roots are more tractable:

−√−B2

1 − B22 − B2

3 + E21 + E2

2 + E23 − 2i(B1E1 + B2E2 + B3E3)

(152)√−B2

1 − B22 − B2

2 + E21 + E2

2 + E23 − 2i(B1E1 + B2E2 + B3E3)

(153)

−√−B2

1 − B22 − B2

3 + E21 + E2

2 + E23 + 2i(B1E1 + B2E2 + B3E3)

(154)√−B2

1 − B22 − B2

3 + E21 + E2

2 + E23 + 2i(B1E1 + B2E2 + B3E3)

(155)

These terms are not unknown to physics. Let us rewrite themusing dot product notation, to make it pop out more:


−√−||B||2 + ||E||2 − 2iB · E (156)√−||B||2 + ||E||2 − 2iB · E (157)

−√−||B||2 + ||E||2 + 2iB · E (158)√−||B||2 + ||E||2 + 2iB · E (159)

The term −||B||2 + ||E||2 and the term B · E are simply the twofundamental Lorentz invariants of electromagnetism, here groupedas a complex number.

Definition 15 (Electromagnetism Interval). The electromagnetisminterval is given by applying the universal interval to a general bivectorof the Clγ

4 (C) algebra, as given by Equation 151.

Using this norm, we now have the opportunity to treat both elec-tromagnetism and general relativity as mere features of a singleuniversal interval. Unlike other proposals in the literature, we notethe benefit that here we do not need to add additional dimensionsbeyond 3+1 spacetime to achieve it.

So, what about the other forces — are they in the universal inter-val? Before we can answer that, let us first show the difficulty.

6. complete multivector: The matrix representation of a completemultivector such as:

s :=a

+ X0e0 + X1e1 + X2e2 + X3e3

+ E1e0e1 + E2e0e2 + E3e0e3 + B1e2e3 + B2e1e3 + B3e1e2

+ V0e1e2e3 + V1e0e2e3 + V2e0e1e3 + V3e0e1e2

+ be0e1e2e3 (160)

is:

s =

a + X0 − iB3 − iV3 B2 − iB1 + V2 − iV1 −ib + X3 + E3 − iV0 X1 − iX2 + E1 − iE2

−B2 − iB1 −V2 − iV1 a + X0 + iB3 + iV3 X1 + iX2 + E1 + iE2 −ib− X3 − E3 − iV0

−ib− X3 + E3 + iV0 −X1 + iX2 + E1 − iE2 a− X0 − iB3 + iV3 B2 − iB1 −V2 + iV1

−X1 − iX2 + E1 + iE2 −ib + X3 − E3 + iV0 −B2 − iB1 + V2 + iV1 a− X0 + iB3 − iV3

(161)

We note that the characteristic polynomial required to solve thismatrix for its eigenvalues is of degree 4. Although an arbitraryquartic polynomial contains an exact algebraic solution, finding a


simplified form of this expression is challenging. To illustrate thedifficulty, we attempted a symbolic computation using Mathemat-ica:

y0=KroneckerProduct[PauliMatrix[3],IdentityMatrix[2]];

y1=KroneckerProduct[I PauliMatrix[2],PauliMatrix[1]];



o01=1/2 (y0.y1-y1.y0);

o02=1/2 (y0.y2-y2.y0);

o03=1/2 (y0.y3-y3.y0);

o12=1/2 (y1.y2-y2.y1);

o31=1/2 (y3.y1-y1.y3);

o23=1/2 (y2.y3-y3.y2);

y5=I y0.y1.y2.y3;

v0=y5.y0;

v1=y5.y1;

v2=y5.y2;

v3=y5.y3;

M=a IdentityMatrix[4]

+X0 y0+X1 y1+X2 y2+X3 y3

+E1 o01+E2 o02+E3 o03+B3 o12+B2 o31+B1 o23

+V0 v0+V1 v1+V2 v2+V3 v3

+b y5;M//MatrixForm

S1=Part[Eigenvalues[M],1]




ToRadicals[S1]

Simplify[%]

The difficulty is that Part[Eigenvalues[M],1] produces a Root[]

expression (specifically, a general polynomial equation of degree4) whose LeafCount is 1703, and ToRadicals[S] produces an al-gebraic expression with a LeafCount of 52007. Using Simplify[%]

on an expression with such a high LeafCount far exceeds the com-puting resources I have at my disposal. The solution containsthousands and thousands of terms. Even then, the simplified re-sult produced by Mathematica is not guaranteed to yield an easilyunderstandable result.

Due to its size, the universal interval of a universal event has beenpushed to Annex A.

Consequently, the universal norm embeds quite a lot of complex-ity. Upon manual inspection, many of the terms appears as related


to unitary groups up to U(3) and to interference between suchterms, and therefore *may* embed a theory of particle physics.However, before we attack this complexity manually (using insightinstead of raw computation), we must develop a better under-standing the universal interval.

Let me here state that there is an alternative way to define the uni-versal interval independently of its matrix representation, that is pos-sibly even better. However I was not able to work out the Cl4(C) casegenerally and thus I elected to use the matrix representation, whichis friendlier to symbolic computations (using the tools/software I amfamiliar with). To achieve this representation-free definition, it suf-fices to construct a polynomial using the geometric product and otheralgebraic and involutive operations to eliminate all basis elements.The resulting polynomial is a characteristic polynomial associated tothe multivector and is analogous to the matrix characteristic polyno-mials. Let me give a few simple examples on how to construct such acharacteristic polynomial from a multivector.

7. In the case of a 1-vector of Cl3(C) one can find the characteristicpolynomial easily by simply taking the geometric product once:

v := xσx + yσy + zσz (162)

=⇒ v2 = x2 + y2 + z2 (163)

where v2 is the geometric product. The roots of v2 are scalars andcorresponds to the eigenvalue of the matrix representation of v.

8. Sometimes the polynomial contains both elements of degree twoand of degree one:

v : = a + xσx (164)

=⇒ v2 = a2 + x2 + 2axσx (165)

= a2 + x2 + 2a(xσx + a)− 2a2 (166)

= x2 − a2 + 2av (167)

=⇒ v2 − 2av = x2 − a2 (168)

The roots on this equation are scalars and are analogous to theeigenvalues of its matrix representation.

9. Finally, in the case of a universal event of Cl4(C), a polynomial ofdegree 4 would be required to produce 4 roots (since the matrixrepresentation has 4 eigenvalues).

Let us now investigate some intervals that are "outside the norm".


10. Euclid/taxicab norm: Let us consider the universal interval of thefollowing multivector: u := a + b + c + σxx + σyy + σzz (of theCl3(R) algebra). It has the following matrix representation:

u =

(z + a + b + c x− iy

x + iy −z + a + b + c

)(169)

Evaluating the universal interval, we obtain this relation:

0 = (∆a + ∆b + ∆c)2 − 2(∆a + ∆b + ∆c)∆s + (∆s)2 − (∆x)2 − (∆y)2 − (∆z)2

(170)

Solving for the roots, we get:

∆s = ∆a + ∆b + ∆c︸︷︷︸taxi-cab elements

±√(∆x)2 + (∆y)2 + (∆z)2︸︷︷︸

Euclidean elements

(171)

Here the departure from familiar norms is quite remarkable: wehave taxi-cab contributions to the interval by the scalar elements.To understand why the scalars find themselves out of the squareroot, let us contrast two ’practical’ examples:

• Say one draws a Cartesian graph with two axes: x and y. Onethen places a token at the origin (0, 0). Then, say one movesthe token 3 units on the x-axis, followed by 4 units on the y-axis. After these translations, one will find the token at point(3, 4). The total distance the token has moved along the path is3 + 4 = 7 units. This is not the shortest path to (3, 4) however.Indeed, since the axes are orthogonal one could have insteadmoved the token in a straight line to (3, 4). In this case, thetoken would have moved 5 units along this path.

• Say one draws a Cartesian graph with two axes: the x-axisdenotes the quantity of apples, and the y-axis denotes thenumber of oranges. Say one wishes to procure 3 apples and4 oranges. Question: can one appeal to the Pythagorean the-orem to negotiate down the price of 3 apples and 4 orangesfor the low price of 5 fruits? The answer is obviously no, andthe reason is that the appropriate metric for this situation is,contrary to a graph drawn in the 2d-plane, not the Euclideanmetric, but instead the taxi-cab metric. Explicitly, the distance —perhaps measured in fruits— between (0, 0) and (3, 4) is givenby d = ∆x + ∆y = 3 + 4 and not d =

√(∆x)2 + (∆y)2.


The universal interval is able to construct a norm which is a com-bination of taxi-cab terms and Euclidean terms, as appropriate forthe situation.

11. Let us now apply a similar logic as the Euclid/taxicab norm, butfor the complex numbers. Here we will use the representation-freeprocedure to eliminate the geometric element I and to find theinterval:

z : = a + bI (172)

z2 = a2 − b2 + 2abI (173)

= a2 − b2 + 2a(a + bI)− 2a2 (174)

= −a2 − b2 + 2az (175)

Consequently the universal interval of a complex number is:

(∆s)2 − 2∆a∆s + (∆a)2 + (∆b)2 = 0 (176)

The roots are:

∆a + ∆bI, ∆a− ∆bI (177)

Why are we not getting the usual complex norm√(∆a)2 + (∆b)2

as the universal interval for the complex numbers? To understandthe discrepancy, consider the following purely hypothetical scenar-ios:

• Say someone asks your help to find a lost friend. If they say, "wewalked 3km east, then 4km north, then I lost him", then clearlythe last known location is 5km away (Euclidean norm).

• Now, say that someone knocks on your door with the followingstory: "My friend and I walked 3 km to the east in a straight line.Remarkably, there was a portal to an imaginary dimension at thispoint. We entered the portal and we walked 4 km inside of it also in astraight line, then I lost my friend. Can you help me find him?" Nowbefore you go rushing to the portal, you suddenly rememberyour complex analysis class in which the complex norm, forsay a + ib is given by

√a2 + b2. Then blindly believing this

equation, you realize there is a faster way to the lost friend. Youcan simply walk 4 km in a straight line to get to the person’slast known position, saving yourself a total of 2 km by skipping


the portal. Will you reach the lost friend’s last known locationfaster than the another person who rushed to the portal thenwalk towards the endpoint? Well, if one believes the norm ofa complex number to be |z| =

√a2 + b2 then one obtains the

invalid conclusion that one can skip the portal to get therefaster, and if one believes the norm to be a complex taxi-cabnorm such as a ± bI, then one concludes that one must enterthe portal to get there. Specifically, with the tax-cab norm forcomplex numbers, one will have to walk 3 km on the real lineand 4 km on the imaginary line for a total of (3+4i) km (not 5)to get to the correct position.

The take home message is that the universal interval does not givethe norm of a complex number as measured within the complexplane (as

√a2 + b2 does) but as measured within physical space. In-

deed, since position/movement in real space presumably does notinfluence position/movement in imaginary space, the norm retainsthe imaginary component and relates it to the real component viaa complex taxi-cab norm.

12. quaternions: In Cl3(R), the event q := a + bσyz + cσxz + dσxy is:

q =

(a + ib ic + dic− d a− ib

)(178)

The universal interval of the quaternion is:

(∆a)2 + (∆b)2 + (∆c)2 + (∆d)2 − 2∆a∆s + (∆s)2 = 0 (179)

And the roots are:

∆s = ∆a± i√(∆b)2 + (∆b)2 + (∆c)2 (180)

Once one enters the quaternionic portal, one is free to use thePythagorean theorem to move around.

We may wonder, is there nonetheless a way to get√

a2 + b2 forcomplex numbers, and

√a2 + b2 + c2 + d2 for quaternions? Interest-

ingly, a universal norm can be constructed using similar techniques.The universal norm of a multivector, unlike its universal interval,maps to a real instead of a complex number, and even recovers andextends the concept of the complex norm.


Definition 16 (Universal Norm). Let ∆s be the matrix representation of∆s ∈ Gn. Then, the universal norm‖s‖ ∈ R for a given matrix representa-tion of s with dimension k2 is given by this relation:

‖s‖ = k√det ∆s (181)

Let us give a few examples.

13. complex number: In Cl2(R), the event z := a + bI has the follow-ing matrix representation:

z :=

(a b−b a

)(182)

The universal norm is:

‖z‖ =

√√√√det

(a b−b a

)=√

a2 + b2 (183)

14. quaternions: In Cl3(R), the event q := a + bσyz + cσxz + dσxy is:

q =


)(184)

The universal norm of the quaternion is:

∥∥q∥∥ =

√√√√det


)=√

a2 + b2 + c2 + d2 (185)

15. Electromagnetism norm: As shown earlier, the matrix represen-tation of the event b := E1γ0γ1 + E2γ0γ2 + E3γ0γ3 + B1γ2γ3 +

B2γ1γ3 + B3γ1γ2 has the following eigenvalues:

−√−||B||2 + ||E||2 − 2iB · E (186)√−||B||2 + ||E||2 − 2iB · E (187)

−√−||B||2 + ||E||2 + 2iB · E (188)√−||B||2 + ||E||2 + 2iB · E (189)


Definition 17 (Electromagnetism norm). The norm of electromag-netism is given by multiplying the eigenvalues (to obtain the determi-nant):

‖s‖2 = (||B||2 + ||E||2)2 + 4(B · E)2 (190)

It groups the imaginary and the real part of the eigenvalue of theinterval of electromagnetism using the complex norm.

16. complete multivector: Due to its size, the universal norm of auniversal event has been pushed to Annex A.

Result 1 (Dimensionality of spacetime limited to 4). We note that thecharacteristic polynomial of the matrix representation of a universal eventof Cl4(C) is of degree 4. Algebraic solutions (in terms of additions, subtrac-tion, multiplication, division and exponentiation by rational numbers) existsfor all polynomials of degrees 4 or lower, however, as per the Abel–Ruffinitheorem, an algebraic solution does not exist for polynomial of degrees 5 orhigher. Consequently, an algebraic solution of a universal metric is onlypossible for the complete basis of a geometric algebras of dimensions 4 orlower. This places the upper bound for a universal metric to no more than4. Beyond four dimensions, no general expression for the universal metric ispossible, and the algebraic form of each metric is (potentially) specific to eachnumerical value of the entry of the matrix.

Let us now turn our attention towards expressing the universalnorm as a metric. When I showed to a few mathematicians that Ihad a new ’cool’ metric and that it is a polynomial of degree 4, theimmediate reaction was skeptical. But it turns out that a polynomialmetric of degree four can be made to follow all the axioms of norms(or pseudo-norms), metric (or pseudo-metric) and distance functions(or pseudo distance functions). There is an already known groupof obscure norms, called p-norms, extending Euclidean norms todegrees p. I think the most evidently valid degree four metric issimply this one:

(ds)4 = (dx)4 + (dy)4 + 2(dx)2(dy)2 (191)

What is this unusual degree 4 metric? Well, if one takes the squareroot on each sides, one gets (ds)2 = (dx)2 + (dy)2 which is the plainold simple Euclidean metric. So at least one metric of degree four isperfectly fine. Lets us now consider more interesting options.

17. Sticking with degree two but adding a bit of degree one spice, forexample we could have:


(ds)2 = 2 da ds− (da)2 + (db)2 (192)

Which, if we solve as a polynomial, we get two roots for the metricas:

ds = da± i db (193)

which assigns the norm of a complex number to either a complextaxi-cab norm or its conjugate.

18. Let us now state a metric of degree 4 that does not have a repre-sentation as degree two:

(ds)4 = (dx)4 + (dy)4 (194)

And another one:

(ds)4 = (dx)4 + (dx)3 dy (195)

19. A general metric of degree 4 is:

(ds)4 = gµναβ dXµ dXν dXα dXβ (196)

where gµναβ is a 4 × 4 × 4 × 4 tensor with the usual symmetryrequirement and is defined in the usual sense as a function whichtakes two multivectors from the tangent space of a manifold andoutputs a scalar. Because it is of degree 4, the metric has extradegrees of freedom compared to a usual degree 2 metric.

In the following section, we will introduce universal thermody-namics as the framework able to tackle a physical system undergoingarbitrary changes described, and we will use the universal metric todo so.

5.4 Universal Ensemble

We are now ready to produce a microscopic description of universalthermodynamics by using the full flexibility of geometric algebra,including that of multivectors. A typical thermodynamic system isdefined by a set of constraints E, V, · · · . Changes in the state of the


system governed by an equation of state such as dS = T dE + p dV,are:

state-a︷︸︸︷(Ea, Va)→

state-b︷︸︸︷(Eb, Vb) (197)

If interpreted as a metric, the equation of state of such a sys-tem is a taxi-cab metric because the constraints are scalar: (dS)2 =

(T dE)2 + (p dV)2 + (2Tp dE dV).In universal thermodynamics, we wish to define a thermodynamic

system able to describe a cycle/transformation between states thatare described by general matrices, just as:

state-a︷︸︸︷a00 . . . a0n...

. . ....

an0 . . . ann

→state-b︷︸︸︷

b00 . . . b0n...

. . ....

bn0 . . . bnn

(198)

As stated earlier, for n ≥ 4 there are no algebraic universal metric,but specifically in the case n = 4 the equation of state can be ex-pressed equivalently as a universal metric, giving the system a purelygeometric interpretation.

Using the definition of nature N , as the set of a maps betweenmanifests and transformations, as the constraints, we can define apartition function of the probability measure that maximizes theentropy subject to said constraints:

Definition 18 (Universal partition function). The partition functionassociated to the probability measure that maximizes the entropy of the spaceof all possible function g ∈ N is a system of functional integrals that max-imizes the entropy for each eigenvalues of the universal metric. Specificallyin 4 dimensions the system contains 4 equations. Let g1, g2, g3, g4 be theeigenvalues of the universal metric, then the universal partition function (in4 dimensions) is the following system of equations:

Z :=

Z1 0 0 00 Z2 0 00 0 Z3 00 0 0 Z4

=

∫D[g1] exp

(−kS [g1]

)0 0 0

0∫

D[g1] exp(−kS [g1]

)0 0

0 0∫


)0

0 0 0∫


)

(199)

Definition 19 (Universal equation of state). The entropy associated tothe ensemble with partition function Z is:


S1 0 0 00 S2 0 00 0 S3 00 0 0 S4

=

kB(k〈S[g1]〉+ ln Z1) 0 0 0

0 kB(k〈S[g2]〉+ ln Z2) 0 00 0 kB k(〈S[g3]〉+ ln Z3) 00 0 0 kB(k〈S[g4]〉+ ln Z4)

(200)

And the equation of state is its total derivative:

dS1 0 0 0

0 dS2 0 00 0 dS3 00 0 0 dS4

=

kB k d〈S[g1]〉 0 0 0

0 kB k d〈S[g2]〉 0 00 0 kB k d〈S[g3]〉 00 0 0 kB k d〈S[g4]〉

(201)

Finally, one can structure the equation of state as a norm by multiplyingthe eigenvalues:

dS1 dS2 dS3 dS4 = (kB k)4 d〈S[g1]〉d〈S[g2]〉d〈S[g3]〉d〈S[g4]〉 (202)

Let us start with a simple example. Assume a 1-vector valuedfunction over Cl4(C) such as g := X0γ0 + X1γ1 + X2γ2 + X3γ3. Thenthe system of equations of state is:

dS

−1 0 0 00 −1 0 00 0 1 00 0 0 1

= kB k d〈√

X20 − X2

1 − X22 − X2

3〉

−1 0 0 00 −1 0 00 0 1 00 0 0 1

(203)

We will rename 〈√

X20 − X2

1 − X22 − X2

3〉 to√

X20 − X2

1 − X22 − X2

3.Let us now write this system as a norm by multiplying the eigenval-ues:

(−dS)2(dS)2 = (−1)(−1)(1)(1)

(kB k d

√X2

0 − X21 − X2

2 − X23

)4

(204)

Then by simplifying:

dS = ±kB k d√

X20 − X2

1 − X22 − X2

3 (205)

= ±kB k1

2√

X20 − X2

1 − X22 − X2

3

(2X0 dX0 − 2X1 dX1 − 2X2 dX2 − 2X3 dX3)

(206)


By noticing that the square root is simply the entropy (within a con-

stant) S ∝ kB k√

X20 − X2

1 − X22 − X2

3, we can rewrite the equationas:

S dS = ±(kB k)2(X0 dX0 − X1 dX1 − X2 dX2 − X3 dX3)

(207)

In normal statistical physics, one simply needs to take the totalderivative once and one gets the equation of state. However, in uni-versal statistical physics one must take the total derivative over andover until all elements are infinitesimals. With 4 eigenvalues, onemay need to take the total derivative up to 4 times. In the present ex-ample however, we will only need to take it once more. We note thatwe take the derivative of an infinitesimal to be 0.

d(S dS) = ±(kB k)2 d(X0 dX0 − X1 dX1 − X2 dX2 − X3 dX3)

(208)

(dS)2 + d(dS) = ±(kB k)2((dX0)

2 + X0 d(dX0)− (dX1)2 + X1 d(dX1)− . . .

)(209)

±(kB k)−2(dS)2 = (dX0)2 + (dX1)

2 + (dX2)2 + (dX3)

2 (210)

which is the interval of special relativity, here expressed in termsof entropy. The above derivation can be repeated with the universalinterval, and we will investigate a few more examples in the resultssection. We now generalize the example to the following results:

Result 2 (Entropic interval). We are already familiar with the relationshipbetween ds = c dτ connecting the interval to the proper time. Now, we haveidentified that the interval of space-time connects to its entropy as follows:

(ds)4 = (kB k)−4(dS)4 (211)

We can further connect the entropy directly to the proper time as follows:

(c dτ)4 = (kB k)−4(dS)4 (212)

Consequently, we identified that any change in proper time is accompa-nied by a change in entropy.

Result 3 (Arrow of time). Any statistical system of physics will evolvetowards the direction of increased entropy as fluctuations occurs. Space-timeis no different. Since both the interval and the entropy have the same signand the same magnitude (related by a multiplication constant), then theentropy of future events is such that the system is powered to evolve towardsits own future.


Part III

Physics6 Results (Spacetime Entropy)

6.1 Law of Inertia

We will now attribute an exact expression for the specific repetency,in terms of existing physical constants. As one may recall, in usualstatistical physics the Lagrange multiplier β is eventually associatedwith the temperature by connecting it to well-known thermodynamicequations having the same mathematical form as those derived fromstatistical physics. Specifically, the following equation (of statisticalphysics):

∂S∂E

= βkB (213)

is equivalent to the following equation (of thermodynamics):

∂S∂E

= T−1 (214)

provided that we pose β := 1/(kBT).To find an expression for k, here a similar strategy is adopted.

We will manipulate our equations until they are mathematically ofthe same form as some familiar laws of physics and then we use theequivalence to assign the correct expression to k such that the two areequated. Let us begin with the 1-vector:

g = X1γ1 + X2γ2 + X3γ3 (215)

The equation of state of this metric is:

(dS)2 = (kB k)2((dX1)2 + (dX2)

2 + (dX3)2) (216)

We now wish to investigate this equation as an entropic forceemergent from the equation of state, connecting the entropy to thedistance quantified by the metric. We recall the definition of an en-tropic force:

dS =FT

dx (217)


As a specific example of an entropic force, we can think of thetension within the chain of a polymer in a warm bath. With ourequation of state, we are not very far from the general definition ofan entropic force. In fact, the units of k are the same as those of F/T.

So which value of F and T to use? The natural choices, proposedby Erik Verlinde50 are to take T as the Unruh temperature and F as 50 W. G. Unruh. Notes on black-hole

evaporation. Phys. Rev. D, 14:870–892,Aug 1976

the law of inertia. Let us solve for k using:

TUnruh =ha

2πckB(218)

F = ma (219)

Then:

kB k = FT−1 (220)

= ma2πckB

ha(221)

= 2πkBmch

(222)

We recognize the term mc/h as the inverse of the reduced Comp-ton wavelength. Here, the Compton wavelength is revealed as aproportionality constant between distance and entropy. Intuitively, anobject with a larger Compton wavelength requires less information tospecify its position than an object with a small Compton wavelength.

Result 4 (Expression for the specific repetency).

k = 2πmch

(223)

6.2 Beckenstein-Hawking Entropy

Let us now explore another means to derive/confirm the expressionfor k. Starting from the entropy of inertia, we now consider that thedistance is fixed to the Schwarzschild radius 2Gm/c2.

S = 2πkBmch

√(X1)2 + (X2)2 + (X3)2 (224)

= 2πkBmch

(2Gm

c2

)(225)

= 4πkBGm2

hc(226)

For a black hole, the mass relates to the area as:

A = 4πr2 = 4π

(2Gm

c2

)2= 4π

4G2m2

c4 =⇒ m2 = Ac4

16πG2 (227)


Replacing m2 in our result for S, we get:

S = 4πkBGhc

Ac4

G216π(228)

= kB14

c3

hGA (229)

which is the Bekenstein-Hawking entropy.

6.3 Action

We have established in an earlier result that the arrow of time ispowered by entropy and points towards the future of the observer,but we only worked out the example for a flat spacetime. In the caseof generally curved space-times, the direction of motion in space-timeis the geodesic. In this case, the observer experiences entropic forcesalong their paths in space-time which "tilt" the direction of maximalentropy.

The equation of state corresponding to the movement of a testparticle in a gravitational field is:

dS = ±2πkBmch

√gµν dXµ dXν (230)

Let us parametrize the metric equation of state over a path τ andthen integrate:

∫τ

∂∂τ S[τ]dτ = ±2πkB

mch

∫τ

√gµν

∂Xµ

∂τ∂Xν

∂τ dτ (231)

We rearrange as follows:

h2πkB

∫τ

∂∂τ S[τ]dτ = ±mc

∫τ

√gµν

∂Xµ

∂τ∂Xν

∂τ dτ (232)

To recover the dynamics one merely needs to investigate thechange of entropy under an infinitesimal variation of δ.

h2πkB

∫τ

δ ∂∂τ S[τ]dτ = ±mc

∫τ

δ

√gµν

∂Xµ

∂τ∂Xν

∂τ dτ (233)

The path along τ that extremalize the production of entropy isgiven in the stationary regime by posing:

δ ∂∂τ S[τ] = 0 (234)

and the corresponding equations of motion are:


δ

√gµν

∂Xµ

∂τ∂Xν

∂τ = 0 (235)

This equation is the Euler-Lagrange equation of motion for a testparticle in curved space-time. Expanding it yields the equations ofgeodesic motion. Consequently, system evolution along the geodesicis revealed as the path for which the production of entropy is ex-tremal in space-time.

We have now identified a relation between the action and theentropy. We will use S to denote the action (as we already use S forthe entropy).

Result 5. The action relates to a change of entropy as follows:

S ≡ ± h2πkB

∫τ

∂∂τ S[τ]dτ (236)

6.4 Fermi-Dirac statistics of events

We now attack the microscopic description. We consider that anevent can occur at most once (whatever happens to Schrödinger’scat, for sure, it doesn’t die twice), and thus we will use Fermi-Diracstatistics to study the occupancy distribution of events in space-time.We start with:

dS = ±kB k√(dX0)2 − (dX1)2 − (dX2)2 − (dX3)2 (237)

But, for simplicity, we will consider the 1+1 space-time case:

dS = ±kB k√(dX0)2 − (dX1)2 (238)

We will now attribute each eigenvalue to a different directionof time. The positive eigenvalue points towards the future, and thenegative eigenvalue towards the past. Consequently, the Fermi-Diracdistributions for this equation of state are:

〈n〉future =1

exp(

kB k√

X20 − X2

1

)+ 1

(239)

〈n〉past =1

exp(−kB k

√X2

0 − X21

)+ 1

(240)

To attribute the correct eigenvalue to the direction of time overthe whole interval, we combine 〈n〉future and 〈n〉past as a piecewisefunction:


〈n〉 =

1

exp(−kB k

√X2

0−X21

)+1

X0 < 0

12 X0 = 0

1

exp(

kB k√

X20−X2

1

)+1

X0 > 0

(241)

Figure 5: Graph of the Fermi-Dirac statistics over the occu-pancy of space-time events. Redmeans an occupancy rate of100%, whereas blue means 0%(and with rainbow colors forintermediate values).

〈n〉 is shown in Figure (5) using a 2-dimensional heat map. As wecan see, 〈n〉 has the shape of a light cone in Minkowski space withthe observer at the origin (0, 0). Remarkably it achieves the correctshape of the light cone only by using event occupancy information.The usual description of a light cone is thus augmented, using oc-cupancy statistics, with a different description for the past than forthe future. For the future, the occupancy rate of events is depletedat 0% (future events are upcoming and have not occurred). For thepast, the occupancy rate of events is saturated at 100% (past eventshave occurred and will not reoccur). Interestingly, the occupancyprobability of events nonetheless describes a probability space (theoccupancy rate varies between 0% to 100% within the light-cone). Anobserver O at point (0, 0) evolving towards its future will experiencea transfer in the depleted occupancy of future events to a saturationin the occupancy of past events (Figure 5a and 5b). In other words,the observer evolves forward in time by filling its past with events. To


illustrate, we introduce the analogy of a tide flooding the past withevents as the present advances in space-time. Along with O, the tideadvances in space-time at the speed of light towards the directionof the future (Figure 5c). The observer rides the tide. Three distinctregimes of time are described; the past (100% event occupancy), thepresent (at the inflection point in the occupancy of events) and thefuture (0% event occupancy).

Of specific interest, we also note that events outside the light-cone of the observer (the white region in the figure) have a complexoccupancy rate. We will now explore the significance in the nextsection.

6.5 Quantum Mechanics

The probability of occupancy of an event is obtained by applyingthe Fermi-Dirac statistics to the ensemble. By doing so, we recoverreal-valued occupancy rates but also complex-valued occupancyrates. Both the real probabilities and the complex occupancy ratesplay a role in the same ensemble and, notably, are dependant uponwhich region in space-time (with respect to the observer) the systemis described.

The white region outside the light cone described in Figure 5a),corresponds to a complex-valued occupancy rate for events. Wecan see this as a consequence that the metric contains a square rootand thus calculating the interval from the observer to a space-likeseparated event yields an imaginary value. For instance the space-time interval between (0, 0) and, say, (0, 5) will be imaginary:

√c2(∆t)2 − (∆x)2

∣∣∣∣∆t=0,∆x=5

=√

c2(0)2 − 52 = i5 (242)

We will make use of this imaginary number to recover a formu-lation of quantum mechanics using the path integral approach.Consider the partition function corresponding to the metric g =

X0e0 + X1e1 + X2e2 + X3e3. Using the universal metric method, weassociate the following partition function to the eigenvalues:

Z =∫

D[g] exp(±k

∫τ

√gµν

∂Xν

∂τ∂Xµ

∂τ dτ

)(243)

In the general case, a path in space-time may cross the light-coneboundary of the observer. For example look at of Figure 5a) thenimagine a path τ that may start in the white region, enter the redzone then end at the observer. The part of this path that are in thewhite region will acquire the imaginary term i in the action whereas


the parts that are are in the red zone will not. Paths of this nature areshown on Figure 6. To account for the entire path, we may thus splitthe action into two parts; the real part of the action as the time-likepart and the imaginary part as the space-like part.

Figure 6: a) We investigatethe path integral from P to Oentirely in a space-like regionwith complex occupancy. As anapproximation, we consider Pand O to be so far away fromthe observer that we can con-sider all paths to be space-likeseparated from O (≈ we ne-glect the contribution of anypaths which may cross into thetime-like regions). In this case,we obtain the usual Feynmanpath integral. b) We investi-gate a path from R to O closeto the boundary of the lightcone of O. The path starts asspace-like separated and endswithin the light cone of O astime-like separated. In this case,the path begins as a quantumsystem capable of interfer-ence, and decoherence/loss-of-interference is experienced as itenters the light cone.

Z =∫

D[g] exp

±Re[

k∫

τ

√gµν

∂Xµ

∂τ∂Xν

∂τ dτ

]︸︷︷︸

time-like part of τ

± i Im[

k∫

τ

√gµν

∂Xµ

∂τ∂Xν

∂τ dτ

]︸︷︷︸

space-like part of τ

(244)

When O describes purely space-like paths, the real part is elimi-nated and we recover a formulation very close to the Feynman pathintegral, including the presence of the imaginary term i multiplyingthe action:

Z =∫

D[g] exp

[±i Im

[k∫

τ

√gµν

∂Xµ

∂τ∂Xν

∂τ dτ

]](245)

Furthermore, when O describes purely time-like path, it is theimaginary part that is eliminated:

Z =∫

D[g] exp

[±Re

[k∫

τ

√gµν

∂Xµ

∂τ∂Xν

∂τ dτ

]](246)


6.6 Decoherence at the Time-like/Space-like Boundary

Let us investigate the role of each part of the path in more detailsand explain why we think that having both a real and an imaginarypart leads to a more complete description of the system. Paths may,in the general case, have both a time-like (real) part and a space-like(imaginary) part. As we will see, the part of the path that is space-like gives the normal Feynman path integral, and the part of the paththat is time-like gives a decoherent version of the path integral.

In the space-like separated region, the system experiences complexinterference and it is described by the usual Feynman path integraland with complex amplitudes. However, as the observer advancesin time and the paths connecting two events gradually penetrate thelight cone of the observer, the probability distribution with complexinterference terms abruptly switches to a distribution using onlyreal-valued probabilities. This process occurs continually as the ob-server advances in time and larger and larger parts of the space-likeseparated region are integrated within the time-like region of theobserver.

To see the process in more details, it suffices to replace the La-grange multipliers k by the previously obtained coefficient 2πmc/h.Let AE[q] be the time-like part (the energy part) of the functionalintegral and let AS [q] be its space-like part (the action part):

Z =∫

D[g] exp(

AS [g])

exp(

AE[g])

(247)

where the space-like part is:

AS [g] =ih

Im

−2πmc∫

τ

√gµν

∂Xµ

∂τ∂Xν

∂τ︸︷︷︸Lagrangian

dτ

︸︷︷︸

Action︸︷︷︸Interference states: iS[g]/h

(248)

The factor 2π is attributed to the connection between action andentropy, but otherwise has no impact on the equations of motion.Then for the time-like part of the path, we first multiply the coeffi-


cient with a/a = 1 (and with a 6= 0), then we get:

AE[q] = −2πcha

Re

ma∫

q

√gµν

∂Xµ

∂τ∂Xν

∂τ︸︷︷︸Lagrangian

dτ

︸︷︷︸

Energy︸︷︷︸Thermal states: −βE[q]

(249)

where the term 2πc/ha is the Unruh temperature.A similar process would occur at any horizons, even in the case

of curved spacetimes. From O’s point of view, the statistical weightof paths inside the horizon is described by complex amplitudes in aFeynman path integral until they cross the horizon, at which pointthe system is described by a decoherent sum (in this case thermal)over its energy levels. In curved spacetimes, this temperature is theHawking temperature. Specifically, if we replace a → c4/(4MG) weget:

1kBβ

=hc3

8πkB MG= THawking (250)

Thus, a quantum system will enter the light cone of an observerwith an energy spectrum at the Unruh temperature (horizon re-sulting from uniform acceleration) or at the Hawking temperature(horizon resulting from gravity), or even at the cosmological horizontemperature (horizon resulting from the metric expansion of space— Section 7.4). In the case of a horizon, as no information can leaveit, the time-like radiation of the space-like quantum system is at ther-modynamic equilibrium. We call this an unprepared quantum system.However, this need not be the case for an unaccelerated observer ad-vancing into the future and capturing a larger sector of the space-likeregion within its light cone over time. In flat space-time, the space-like region is not hidden by a horizon, thus information from theregion can eventually enter the light cone of the observer.

A prepared quantum system has additional observables Oi[g] eachwith expectation value 〈Oi〉 but unlike for its unprepared counter-part, the probability measure over the paths is maximized under theadded constraints of the observables’ expectation values. In suchcase, information about the preparation will be available to the ob-server under repeated measurements of multiple copy of the quan-tum system. Outside the light cone, preparation information is givenin the forms of expectation values by the usual quantum equation:


〈Oi〉 =

∫D[g]Oi[g] exp

[ihS [g]

]∫

D[g] exp[

ihS [g]

] (251)

and inside the light, the same information is available but as’information-bearing thermal’ radiation / decohered quantum states:

〈Oi〉 =∫

D[g]Oi[g] exp[−βE[g]

]∫D[g] exp

[−βE[g]

] (252)

In the first case, and using the Von Neumann formalism, prepa-ration information is available in the form of pure states, but in thesecond decoherent case, it is available as a mixture, such as:

ρ =

Ptime-like-1 . . . 0

.... . .

...0 . . . Ptime-like-n

(253)

The absence of off-diagonal terms indicates that the system hasdecohered and that no probability interference will be observed. Thesum obeys a classical sum of real-valued probabilities. An observerwill, therefore, interpret ’coming into causal contact with a quantumsystem’ as performing a measurement on the system.

6.7 Quantum Field Gravity

Let us take a multivector comprised only of the pseudoscalar basis ofCl4(C):

g = R[X0, X1, X2, X3]e0 ∧ e1 ∧ e2 ∧ e3 (254)

This represents the 4-volume element. We can rewrite g using thebasis γ0, γ1, γ2, γ3 instead of e0, e1, e2, e3, and we get:

g = R[X0, X1, X2, X3]√|g|γ0γ1γ2γ3 (255)

Using the universal metric, we obtain the eigenvalues as the rootsof this metric polynomial:

s4 = (R[X0, X1, X2, X3]√|g|)4 (256)

s = ±R[X0, X1, X2, X3]√|g| (257)


When we insert the eigenvalues into the universal partition func-tion as functionals, we get:

Z =∫

D[g] exp(±k

∫R[X0, X1, X2, X3]

√|g|dX0 dX1 dX2 dX3

)(258)

Here, the exponent is the Hilbert-Einstein action. We recall thatR[X0, X1, X2, X3] is a complex number, and consequently its imagi-nary part is the well known path integral over the possible metrics ofthe gravitational fields. The presence of the real as well as the imag-inary part are very important to the universal formulation; they areresponsible for decoherence at the surface of horizons.

7 Results (Cosmology)

Having a system of equations for our partition function means thatthe geometry can be entirely emergent from the entropy of the sys-tem as an equation of state. Here, we will study this concept intogreater detail by constructing an entropy model of ΛCDM.

7.1 de Sitter Space

We recall that de Sitter space is a hyperboloid defined in 3+1 Minkowskispace-time by the relation:

α2 = −c2X20 + X2

1 + X22 + X2

3 (259)

with a cosmological horizon at r = α. Let us now compare it to the1-vector metric entropy:

S = kB ln Z + kB k√−X2

0 + X21 + X2

2 + X23 (260)

First, let us consider the ground state of the system by posingS = 0:

kB ln Z = −kB k√−X2

0 + X21 + X2

2 + X23 (261)

Now, we inject the coefficient 2πkBmc/h previously obtained forthe equation of state of inertia as the Lagrange multiplier, we get:

kB ln Z = ±2πkBmch

√−X2

0 + X21 + X2

2 + X23 (262)


Since Z is a constant, we recover the same mathematical formas the hyperboloid equation in 3+1 space-time characteristic of deSitter space, except that now that the quantities on each side of theexpression have the units of the entropy. The entropy of de Sitterspace, where α is the radius to the horizon, is therefore, in the groundstate, quantified by the relation:

kB ln Z = ±2πkBmch

α (263)

Noting that in de Sitter α is the Hubble radius at c/H, we willderive, using entropy, the cosmological pressure and the cosmologicalinertia.

7.2 Cosmological Pressure

We consider that the cosmological horizon (at r = α) bears an en-tropy for the same reason that the black hole apparent horizon bearsan entropy (it delimits a boundary of information inaccessible to theobserver). As we here consider empty de Sitter space, the cosmolog-ical horizon and the Hubble radius will be the same. Specifically, wedescribe the cosmological horizon using the Hubble radius r = c/Hwhere H is the Hubble constant and we replace a by cH in the Un-ruh temperature51. With these replacements, the Unruh temperature 51 Damien A Easson, Paul H Framp-

ton, and George F Smoot. Entropicaccelerating universe. Physics Let-ters B, 696(3):273–277, 2011; andDamien A. Easson, Paul H. Framp-ton, and George F. Smoot. Entropicinflation. International Journal of ModernPhysics A, 27(12):1250066, 2012

becomes the cosmological horizon temperature52:

52 Damien A Easson, Paul H Framp-ton, and George F Smoot. Entropicaccelerating universe. Physics Letters B,696(3):273–277, 2011

Tde-Sitter-horizon :=hH

2πkB(264)

Starting with the Bekenstein-Hawking entropy with the minussign as a starting point (we are inside the cosmological horizon thuswe flip the sign), we then multiply each side by Tde-Sitter-horizon as aproportionality constant and we get:

T dS = − hH2πkB

kBc3

hG4dA (265)

Let us now write this equation in terms of volume by replacingA with V. With this replacement, the equation will be formally thesame as before, but the coefficient now has the units of pressure.Using A = 4πr2 and V = 4/3πr3 therefore dA = 2r−1 dV, we get:

T dS = − hH2πkB

kBc3

hG42r−1︸︷︷︸

negative pressure

dV (266)


Simplifying (and using the radius replacement r → c/H), we get:

T dS = − hH2πkB

kBc3

hG42

Hc

dV (267)

= − H2

4πGc2 dV (268)

Finally, we rewrite the expression in terms of the critical cosmo-logical density ρ = 3H2/(8πG), and we obtain a negative entropicpressure corresponding to 66% of the total energy of the universe53: 53 Damien A Easson, Paul H Framp-

ton, and George F Smoot. Entropicaccelerating universe. Physics Letters B,696(3):273–277, 2011

T dS = −23

ρc2dV (269)

This result was obtained by Easson54, where it was suggested as a 54 Damien A Easson, Paul H Framp-ton, and George F Smoot. Entropicaccelerating universe. Physics Letters B,696(3):273–277, 2011

candidate explanation of the accelerated expansion of the universe.

7.3 Cosmological Inertia

We repeat the same process as was used to derive the cosmologicalpressure, but instead of rewriting the relation from area to volume,we go from dA to radius dr, using: dA = d(4πr2) = 8πr dr. Usingthis replacement as well as the cosmological horizon temperature andthe Bekenstein-Hawking entropy, we get:

T dS = − hH2πkB

kBc3

hG4(8πr︸︷︷︸

entropic force

dr) (270)

Then, with r → c/H, we get:

T dS = − hH2πkB

kBc3

hG48π

cH

dr (271)

= − c4

Gdr (272)

Since the surface gravity of a horizon is equal to a = c4/(4G), wecan rewrite this expression in terms of acceleration, and we get:

T dS = −4Madr (273)

Using these results, we assign to the energy of de Sitter universe a66% negative pressure component (obtained in the previous section)and 25% inertial matter component (the inertial mass of the cosmos isweighted at one fourth its energy content).


7.4 Entropic Derivation of ΛCDM

In the previous case, we have considered that the cosmological hori-zon is at the Hubble radius (de Sitter space). However, accordingto present observations, this is not quite the case. The cosmologicalhorizon is slightly beyond the Hubble horizon (≈ 5 giga-parsec and≈ 4.1 giga-parsec, respectively).

To account for this difference, we increase the entropy of the sys-tem such that the position of the Hubble horizon with respect tothe cosmological horizon ends up slightly further. For generality wecan in fact include any number of scalar thermodynamic quantitiesµ1N1, . . . , µnNn in the equation, so as to offset the entropy. Theequation becomes:

S− kB ln Z = ±(2πkBmc/h)α︸︷︷︸de Sitter entropy

−µ1N1 − · · · − µnNn︸︷︷︸scalar thermodynamic quantities︸︷︷︸

deformed de Sitter entropy

(274)

This entropy is determined by two competing processes. First, werecall that the entropy is information inaccessible to the observer.Working in the direction contributing to the entropy, ln Z is usuallyinterpreted to be the intrinsic degeneracy of the system (at T → 0it is, in fact, the degeneracy of the ground state), whereas, α repre-sents, as always, the distance to the Hubble horizon. The contribu-tion from these terms works in the opposite direction to that of thescalar terms µ1N1, . . . , µnNn. Finally, we recall that the Hubblehorizon meets the cosmological horizon asymptotically at t → ∞when all matter has exited the cosmological horizon. Thus, for theuniverse to be asymptotically de Sitter at t → ∞, it follows thatlimt→∞ α→ (2πkBmc/h)−1S and for all i, limt→∞ Ni → 0.

By attributing the role of bookkeeper (of the matter and energy yetto leave the horizon) to µ1N1, . . . , µnNn, and by adopting the lawof conservation of energy, then the sum-total of all matter and energyleaving the horizon can be summarized as the continuity equation:

dE = (ρc2 + p)dV (275)

To equate this relation to the entropy we must now introducethe temperature at the real cosmological horizon, using the Unruhtemperature as the starting point with a → c2/r. We use the relationsderived by Easson55, first for the radius: 55 Damien A Easson, Paul H Framp-

ton, and George F Smoot. Entropicaccelerating universe. Physics Letters B,696(3):273–277, 2011rcosmological-horizon :=

c√H2 + k/a2

(276)


where a is a scaling factor. Then for the temperature:

Tcosmological-horizon :=h(c2/rcosmological-horizon)

2πckB(277)

As before, here we use the temperature as a proportionality con-stant, and we rewrite the entropy as follows:

Tcosmological-horizon dS = (ρc2 + p)dV (278)

or, more specifically, as:

Tcosmological-horizonkBc3

4hGdA = (ρc2 + p)dV (279)

This is the sufficient starting point used by Easson56 (in Annex A 56 Damien A Easson, Paul H Framp-ton, and George F Smoot. Entropicaccelerating universe. Physics Letters B,696(3):273–277, 2011

of his paper) to recover the Friedman equations of cosmology:

H − ka2 = −4πG

(ρ +

pc2

)(280)

H2 =8πG

3ρ− kc2

a2 +Λc2

3(281)

as an equivalent representation of (279).It would thus appear that ΛCDM is to universal statistical physics

what the ideal gas law is to statistical physics.

8 Discussion (Quantum Measurements)

8.1 Recap: measurement problem

We are all familiar with the measurement problem. In quantumphysics the unitary evolution of the wave function ψ[r, t] is deter-ministic, but the notion breaks down if measurements are made bythe observer.

In the Von Neumann scheme, a measurement of the second kind,for a quantum object with wave function |ψ〉 and a quantum appara-tus with wave function |φ〉, is defined as:

|ψ〉|φ〉 →∑n

cn|χn〉|φn〉 (282)

After the measurement the system is in one of eigenstates |χn〉with probability |cn|2. Why it is that the deterministic unitary evolv-ing system adopts the "un-deterministic" initiative to collapse itself


randomly in one of multiple states after measurement is quite themystery. Nothing in theoretical quantum physics predicts that sucha thing would occur. Consequently, the notion of the measurement isintroduced into quantum mechanics as a full-blown axiom not deriv-able from the Schrodinger equation itself, or any of the other axiomsof quantum mechanics.

The theory of quantum decoherence is a modern take on VonNeumann measurements. Decoherence from the environment |e〉 isintroduced as follows:

|ψ〉|φ〉|e〉 →∑n

cn|χn〉|φn〉|en〉 (283)

Under contact with an environment having multiple degrees offreedom, any interference pattern normally observable from |ψ〉 willbe smudged by the environment beyond the ability of instrumentsto detect it. Decoherence is a possible mechanism to account forwhy a quantum superposition of eigenstates is unlikely to be ob-served macroscopically because interaction with the environmentvery quickly causes the system to evolve towards a classical proba-bility distribution. However, decoherence is ultimately of no help inregards to explaining why the system is in a definite state after themeasurement.

The problem remains: how does the system go from a mixture ofstates to a definite state, post-measurement? The measurement postu-late, as a law, is derived empirically and it is introduced so that quan-tum physics predicts a single macroscopic world (not a superpositionof many worlds), consistent with observations. The two primarycompeting interpretations of this behavior are a) the Copenhagen in-terpretation and b) the Everett many-worlds interpretation, but thereexists at least half a dozen others. None of these interpretations are,however, considered satisfactory by mainstream physics and thus thequestion remains unsettled: in the first case the collapse is simplypostulated, but no mechanisms are generally accepted for it, and inthe second case it is postulated that the observer becomes coupledwith a specific result of the measurement causing the appearance ofcollapse, but no mechanism to account for this coupling is generallyaccepted either. The interpretational problem is retained in all exten-sions of quantum theory from the Dirac equation to quantum fieldtheory, etc. As one is generally free to apply any of the compatibleinterpretations to quantum theory, deciding which one is correct, ifany, is considered non-falsifiable.


8.2 The true cause of measurements

Rather than taking some arbitrary set of laws as postulates, UniversalPhysics addresses the problem from the other direction by taking asits sole axiom the ontological existence of the state of affairs refer-enced by M. An interpretation of quantum mechanics is availablewithin this setup as a direct consequence of applying the conceptsof statistical physics in terms of microstate/macrostate. The termsmicrostate and macrostate are somewhat misleading in the presentcontext because we are of course not dealing with difference in sizesas we typically do in usual statistical physics. Here, microscopicrefers to a description of the system in terms of a set of experimentsin the form of a reference manifest M, whereas macroscopic refers toa description in terms of the constraints of nature N neutral to anyspecific manifests and invariant with respect to any rearrangementof manifest or experiments within a manifest. Both the microstateand the macrostate description have the ability to spawn the wholeobservable universe.

This is essentially an inversion on the typical place of the ob-server in ordinary statistical physics, where it is associated to a large(macroscopic) device measuring things like the volume or the energy,etc. In Universal Physics, due to this inversion, the observer, a fancyword for microscopic description, is aware of the measurement re-sult, whereas the macroscopic description is a logico-deductive modelformulated by the observer under the principle of insufficient reasonto bound the states the system could involve into. Finally, as theyare derived from statistical physics, the laws of physics acquire theprinciple of maximum entropy with respect to the space of possiblesolutions, which accounts for the quantum measurement problemwhen projected outside the scope of the reference manifest. Simplyput, the laws of quantum physics cannot tell us the result of a mea-surement, for the same reason that pV = nRT cannot tell us where inthe box the molecules of air are.

The quantum measurement problem appears as unexplained onlywhen one postulates the laws of physics because such models erasesthe supply chain, and thus can no longer deliver the laws of physicsat the end of the road. When one postulates the laws, then solvesthem for solutions, one obtains a plurality of manifests as possiblesolutions, only one of which is the reference manifest. Since oneintuitively expected that the laws of physics ought to produce thereference manifest, one may then become baffled as to why the lawsof physics produce a plurality of manifests as their solutions insteadof just the reference manifest.

The fundamental motivation in maximizing the entropy of natural


information is to release the constraints imposed by the microscopicdescription so as to facilitate formulating the broadest possible pat-tern about nature, such that the pattern survives all possible rear-rangements of experiments or permutations of manifests. However,one cannot form a pattern from a single existing candidate (there isonly one reference manifest), unless one invents hypothetical alter-natives (the set of all manifests). For example, one can say "I am aphysicist, but I could have been a doctor instead", or one could say"I measured the spin up, but it could have been down". Althoughneither violates the laws of physics, in reality, one happened andthe other didn’t. It is precisely because natural information is erasedfrom the laws of physics that the claim ’both alternatives (even theone that didn’t happen) are compatible with the laws of physics’ canbe made. Unavoidably, the laws of physics will recover both alterna-tives as possible solutions, but would be unable to determine whichof the two occurred without access to natural information. Considerif one would have instead said: "I am a physicist, but I could havebeen superman". How credible is that claim? Supposedly, we mayadmit that being superman violates the laws of physics, whereas be-ing a doctor doesn’t. Do we then want our laws of physics to ruleout superman, but not the doctor, even though in reality we got thephysicist? Remarkably, we want our laws of physics to permit notonly the reference manifest but also all other possible manifests, wiltsruling out only the "truly" impossible.

In the case of Universal Physics, this concept is taken to its maxi-mum. The description of the state of affairs as a manifest is the mostgeneral description possible (universal in the mathematical sense),and the entropy of natural information is maximized to generate thebroadest possible rules using nature as the constraints, itself com-prising no less than the set of all possible transformations (universalin the physical sense), thereby producing the laws of physics as auniversal equation of state.

Universal Physics states that there is no collapse (thus it rejects theCopenhagen interpretation), and also that the system was never in asuperposition of many-worlds to begin with (thus it rejects the many-world interpretation). Universal Physics states that all alternativemanifests are mathematical creations used to facilitate the formula-tion of the laws of physics as patterns, and thus, have no ontologicalproperties. Universal Physics predicts the discrepancy between whatis observed and what the laws of physics offers as solutions, withoutthe introduction of ad hoc postulates, and quantifies the discrepancyusing the entropy of natural information. The observer is associatedto the recursive enumeration of the domain of science and thus isattributed to the role of constructing the micro-state of the statistical


physics system. Universal Physics admits sufficient mathematicalwork to verify only the reference manifest, consequently all othermanifests (although required to formulate the probability measure)must therefore be hypothetical.

9 Conclusion

As we surely all suspected, the laws of physics, at their most funda-mental level, describes the states of all possible set of experiments asconstrained only by nature, the set of all possible transformations,but now with exact mathematical definitions for each term in italic.An explanation of why the laws of physics are inviolable, and why"God" had no choice in their making, follows trivially. Finally, aninvestigation of the universal metric/norm and how/if it ties to thestandard model (beyond electromagnetism) will be investigated infuture papers.

Part IV

AnnexA Universal Interval and Norm of a universal event in four di-

mensions

I have executed the following code in Mathematica 12:

y0 = KroneckerProduct[PauliMatrix[3], IdentityMatrix[2]];

y1 = KroneckerProduct[I PauliMatrix[2], PauliMatrix[1]];



o01 = 1/2 (y0.y1 - y1.y0); o01 // MatrixForm;







y5 = y0.y1.y2.y3 ;

v0 = y5.y0; v0 // MatrixForm;




M = a IdentityMatrix[4] + X0 y0 + X1 y1 + X2 y2 + X3 y3 + E1 o01 +

E2 o02 + E3 o03 + B3 o12 + B2 o31 + B1 o23 + V0 v0 + V1 v1 +


V2 v2 + V3 v3 + b y5;

Expand[Det[s IdentityMatrix[4] - M]]

Expand[Det[M]]

The output for the universal interval is:

a^4 + 2 a^2 b^2 + b^4 + 2 a^2 B1^2 - 2 b^2 B1^2 + B1^4 + 2 a^2 B2^2 -

2 b^2 B2^2 + 2 B1^2 B2^2 + B2^4 + 2 a^2 B3^2 - 2 b^2 B3^2 +

2 B1^2 B3^2 + 2 B2^2 B3^2 + B3^4 - 8 a b B1 E1 - 2 a^2 E1^2 +

2 b^2 E1^2 + 2 B1^2 E1^2 - 2 B2^2 E1^2 - 2 B3^2 E1^2 + E1^4 -

8 a b B2 E2 + 8 B1 B2 E1 E2 - 2 a^2 E2^2 + 2 b^2 E2^2 -

2 B1^2 E2^2 + 2 B2^2 E2^2 - 2 B3^2 E2^2 + 2 E1^2 E2^2 + E2^4 -

8 a b B3 E3 + 8 B1 B3 E1 E3 + 8 B2 B3 E2 E3 - 2 a^2 E3^2 +

2 b^2 E3^2 - 2 B1^2 E3^2 - 2 B2^2 E3^2 + 2 B3^2 E3^2 + 2 E1^2 E3^2 +

2 E2^2 E3^2 + E3^4 - 4 a^3 s - 4 a b^2 s - 4 a B1^2 s - 4 a B2^2 s -

4 a B3^2 s + 8 b B1 E1 s + 4 a E1^2 s + 8 b B2 E2 s + 4 a E2^2 s +

8 b B3 E3 s + 4 a E3^2 s + 6 a^2 s^2 + 2 b^2 s^2 + 2 B1^2 s^2 +

2 B2^2 s^2 + 2 B3^2 s^2 - 2 E1^2 s^2 - 2 E2^2 s^2 - 2 E3^2 s^2 -

4 a s^3 + s^4 - 2 a^2 V0^2 - 2 b^2 V0^2 + 2 B1^2 V0^2 +

2 B2^2 V0^2 + 2 B3^2 V0^2 + 2 E1^2 V0^2 + 2 E2^2 V0^2 +

2 E3^2 V0^2 + 4 a s V0^2 - 2 s^2 V0^2 + V0^4 - 8 B3 E2 V0 V1 +

8 B2 E3 V0 V1 + 2 a^2 V1^2 + 2 b^2 V1^2 - 2 B1^2 V1^2 +

2 B2^2 V1^2 + 2 B3^2 V1^2 - 2 E1^2 V1^2 + 2 E2^2 V1^2 +

2 E3^2 V1^2 - 4 a s V1^2 + 2 s^2 V1^2 - 2 V0^2 V1^2 + V1^4 +

8 B3 E1 V0 V2 - 8 B1 E3 V0 V2 - 8 B1 B2 V1 V2 - 8 E1 E2 V1 V2 +

2 a^2 V2^2 + 2 b^2 V2^2 + 2 B1^2 V2^2 - 2 B2^2 V2^2 + 2 B3^2 V2^2 +

2 E1^2 V2^2 - 2 E2^2 V2^2 + 2 E3^2 V2^2 - 4 a s V2^2 + 2 s^2 V2^2 -

2 V0^2 V2^2 + 2 V1^2 V2^2 + V2^4 - 8 B2 E1 V0 V3 + 8 B1 E2 V0 V3 -

8 B1 B3 V1 V3 - 8 E1 E3 V1 V3 - 8 B2 B3 V2 V3 - 8 E2 E3 V2 V3 +

2 a^2 V3^2 + 2 b^2 V3^2 + 2 B1^2 V3^2 + 2 B2^2 V3^2 - 2 B3^2 V3^2 +

2 E1^2 V3^2 + 2 E2^2 V3^2 - 2 E3^2 V3^2 - 4 a s V3^2 + 2 s^2 V3^2 -

2 V0^2 V3^2 + 2 V1^2 V3^2 + 2 V2^2 V3^2 + V3^4 + 8 a B1 V1 X0 -

8 b E1 V1 X0 - 8 B1 s V1 X0 + 8 a B2 V2 X0 - 8 b E2 V2 X0 -

8 B2 s V2 X0 + 8 a B3 V3 X0 - 8 b E3 V3 X0 - 8 B3 s V3 X0 -

2 a^2 X0^2 - 2 b^2 X0^2 + 2 B1^2 X0^2 + 2 B2^2 X0^2 + 2 B3^2 X0^2 +

2 E1^2 X0^2 + 2 E2^2 X0^2 + 2 E3^2 X0^2 + 4 a s X0^2 - 2 s^2 X0^2 +

2 V0^2 X0^2 + 2 V1^2 X0^2 + 2 V2^2 X0^2 + 2 V3^2 X0^2 + X0^4 -

8 a B1 V0 X1 + 8 b E1 V0 X1 + 8 B1 s V0 X1 + 8 b B3 V2 X1 +

8 a E3 V2 X1 - 8 E3 s V2 X1 - 8 b B2 V3 X1 - 8 a E2 V3 X1 +

8 E2 s V3 X1 - 8 B3 E2 X0 X1 + 8 B2 E3 X0 X1 - 8 V0 V1 X0 X1 +

2 a^2 X1^2 + 2 b^2 X1^2 - 2 B1^2 X1^2 + 2 B2^2 X1^2 + 2 B3^2 X1^2 -

2 E1^2 X1^2 + 2 E2^2 X1^2 + 2 E3^2 X1^2 - 4 a s X1^2 + 2 s^2 X1^2 +

2 V0^2 X1^2 + 2 V1^2 X1^2 - 2 V2^2 X1^2 - 2 V3^2 X1^2 -

2 X0^2 X1^2 + X1^4 - 8 a B2 V0 X2 + 8 b E2 V0 X2 + 8 B2 s V0 X2 -

8 b B3 V1 X2 - 8 a E3 V1 X2 + 8 E3 s V1 X2 + 8 b B1 V3 X2 +


8 a E1 V3 X2 - 8 E1 s V3 X2 + 8 B3 E1 X0 X2 - 8 B1 E3 X0 X2 -

8 V0 V2 X0 X2 - 8 B1 B2 X1 X2 - 8 E1 E2 X1 X2 + 8 V1 V2 X1 X2 +

2 a^2 X2^2 + 2 b^2 X2^2 + 2 B1^2 X2^2 - 2 B2^2 X2^2 + 2 B3^2 X2^2 +

2 E1^2 X2^2 - 2 E2^2 X2^2 + 2 E3^2 X2^2 - 4 a s X2^2 + 2 s^2 X2^2 +

2 V0^2 X2^2 - 2 V1^2 X2^2 + 2 V2^2 X2^2 - 2 V3^2 X2^2 -

2 X0^2 X2^2 + 2 X1^2 X2^2 + X2^4 - 8 a B3 V0 X3 + 8 b E3 V0 X3 +

8 B3 s V0 X3 + 8 b B2 V1 X3 + 8 a E2 V1 X3 - 8 E2 s V1 X3 -

8 b B1 V2 X3 - 8 a E1 V2 X3 + 8 E1 s V2 X3 - 8 B2 E1 X0 X3 +

8 B1 E2 X0 X3 - 8 V0 V3 X0 X3 - 8 B1 B3 X1 X3 - 8 E1 E3 X1 X3 +

8 V1 V3 X1 X3 - 8 B2 B3 X2 X3 - 8 E2 E3 X2 X3 + 8 V2 V3 X2 X3 +

2 a^2 X3^2 + 2 b^2 X3^2 + 2 B1^2 X3^2 + 2 B2^2 X3^2 - 2 B3^2 X3^2 +

2 E1^2 X3^2 + 2 E2^2 X3^2 - 2 E3^2 X3^2 - 4 a s X3^2 + 2 s^2 X3^2 +

2 V0^2 X3^2 - 2 V1^2 X3^2 - 2 V2^2 X3^2 + 2 V3^2 X3^2 -

2 X0^2 X3^2 + 2 X1^2 X3^2 + 2 X2^2 X3^2 + X3^4

and the universal norm is:

a^4 + 2 a^2 b^2 + b^4 + 2 a^2 B1^2 - 2 b^2 B1^2 + B1^4 + 2 a^2 B2^2 -

2 b^2 B2^2 + 2 B1^2 B2^2 + B2^4 + 2 a^2 B3^2 - 2 b^2 B3^2 +

2 B1^2 B3^2 + 2 B2^2 B3^2 + B3^4 - 8 a b B1 E1 - 2 a^2 E1^2 +

2 b^2 E1^2 + 2 B1^2 E1^2 - 2 B2^2 E1^2 - 2 B3^2 E1^2 + E1^4 -

8 a b B2 E2 + 8 B1 B2 E1 E2 - 2 a^2 E2^2 + 2 b^2 E2^2 -

2 B1^2 E2^2 + 2 B2^2 E2^2 - 2 B3^2 E2^2 + 2 E1^2 E2^2 + E2^4 -

8 a b B3 E3 + 8 B1 B3 E1 E3 + 8 B2 B3 E2 E3 - 2 a^2 E3^2 +

2 b^2 E3^2 - 2 B1^2 E3^2 - 2 B2^2 E3^2 + 2 B3^2 E3^2 + 2 E1^2 E3^2 +

2 E2^2 E3^2 + E3^4 - 2 a^2 V0^2 - 2 b^2 V0^2 + 2 B1^2 V0^2 +

2 B2^2 V0^2 + 2 B3^2 V0^2 + 2 E1^2 V0^2 + 2 E2^2 V0^2 +

2 E3^2 V0^2 + V0^4 - 8 B3 E2 V0 V1 + 8 B2 E3 V0 V1 + 2 a^2 V1^2 +

2 b^2 V1^2 - 2 B1^2 V1^2 + 2 B2^2 V1^2 + 2 B3^2 V1^2 - 2 E1^2 V1^2 +

2 E2^2 V1^2 + 2 E3^2 V1^2 - 2 V0^2 V1^2 + V1^4 + 8 B3 E1 V0 V2 -

8 B1 E3 V0 V2 - 8 B1 B2 V1 V2 - 8 E1 E2 V1 V2 + 2 a^2 V2^2 +

2 b^2 V2^2 + 2 B1^2 V2^2 - 2 B2^2 V2^2 + 2 B3^2 V2^2 + 2 E1^2 V2^2 -

2 E2^2 V2^2 + 2 E3^2 V2^2 - 2 V0^2 V2^2 + 2 V1^2 V2^2 + V2^4 -

8 B2 E1 V0 V3 + 8 B1 E2 V0 V3 - 8 B1 B3 V1 V3 - 8 E1 E3 V1 V3 -

8 B2 B3 V2 V3 - 8 E2 E3 V2 V3 + 2 a^2 V3^2 + 2 b^2 V3^2 +

2 B1^2 V3^2 + 2 B2^2 V3^2 - 2 B3^2 V3^2 + 2 E1^2 V3^2 +

2 E2^2 V3^2 - 2 E3^2 V3^2 - 2 V0^2 V3^2 + 2 V1^2 V3^2 +

2 V2^2 V3^2 + V3^4 + 8 a B1 V1 X0 - 8 b E1 V1 X0 + 8 a B2 V2 X0 -

8 b E2 V2 X0 + 8 a B3 V3 X0 - 8 b E3 V3 X0 - 2 a^2 X0^2 -

2 b^2 X0^2 + 2 B1^2 X0^2 + 2 B2^2 X0^2 + 2 B3^2 X0^2 + 2 E1^2 X0^2 +

2 E2^2 X0^2 + 2 E3^2 X0^2 + 2 V0^2 X0^2 + 2 V1^2 X0^2 +

2 V2^2 X0^2 + 2 V3^2 X0^2 + X0^4 - 8 a B1 V0 X1 + 8 b E1 V0 X1 +

8 b B3 V2 X1 + 8 a E3 V2 X1 - 8 b B2 V3 X1 - 8 a E2 V3 X1 -

8 B3 E2 X0 X1 + 8 B2 E3 X0 X1 - 8 V0 V1 X0 X1 + 2 a^2 X1^2 +

2 b^2 X1^2 - 2 B1^2 X1^2 + 2 B2^2 X1^2 + 2 B3^2 X1^2 - 2 E1^2 X1^2 +


2 E2^2 X1^2 + 2 E3^2 X1^2 + 2 V0^2 X1^2 + 2 V1^2 X1^2 -

2 V2^2 X1^2 - 2 V3^2 X1^2 - 2 X0^2 X1^2 + X1^4 - 8 a B2 V0 X2 +

8 b E2 V0 X2 - 8 b B3 V1 X2 - 8 a E3 V1 X2 + 8 b B1 V3 X2 +

8 a E1 V3 X2 + 8 B3 E1 X0 X2 - 8 B1 E3 X0 X2 - 8 V0 V2 X0 X2 -

8 B1 B2 X1 X2 - 8 E1 E2 X1 X2 + 8 V1 V2 X1 X2 + 2 a^2 X2^2 +

2 b^2 X2^2 + 2 B1^2 X2^2 - 2 B2^2 X2^2 + 2 B3^2 X2^2 + 2 E1^2 X2^2 -

2 E2^2 X2^2 + 2 E3^2 X2^2 + 2 V0^2 X2^2 - 2 V1^2 X2^2 +

2 V2^2 X2^2 - 2 V3^2 X2^2 - 2 X0^2 X2^2 + 2 X1^2 X2^2 + X2^4 -

8 a B3 V0 X3 + 8 b E3 V0 X3 + 8 b B2 V1 X3 + 8 a E2 V1 X3 -

8 b B1 V2 X3 - 8 a E1 V2 X3 - 8 B2 E1 X0 X3 + 8 B1 E2 X0 X3 -

8 V0 V3 X0 X3 - 8 B1 B3 X1 X3 - 8 E1 E3 X1 X3 + 8 V1 V3 X1 X3 -

8 B2 B3 X2 X3 - 8 E2 E3 X2 X3 + 8 V2 V3 X2 X3 + 2 a^2 X3^2 +

2 b^2 X3^2 + 2 B1^2 X3^2 + 2 B2^2 X3^2 - 2 B3^2 X3^2 + 2 E1^2 X3^2 +

2 E2^2 X3^2 - 2 E3^2 X3^2 + 2 V0^2 X3^2 - 2 V1^2 X3^2 -

2 V2^2 X3^2 + 2 V3^2 X3^2 - 2 X0^2 X3^2 + 2 X1^2 X3^2 +

2 X2^2 X3^2 + X3^4

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A purely mathematical proof of the existence of the …As Exhibit A, consider a recent book, titled...

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