Philip Gibbs

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Philip Edward Gibbs (b 20 February 1960) is a British independent mathematician and physicist known for his work on number theory, geometric optimisation, lattice gauge theory and quantum gravity. He is the principle founder and administrator of the viXra repository.

Early Life and Education[edit]

Philip Gibbs was born in North London in 1960 to Edward (b 1930) and Shirley Gibbs (b 1934, née Edwards). He has a twin sister and an older brother. His father known as Ted was an electronics engineer who worked for British oscilloscope maker Telequipment and later Tektronix. In 1964 the family moved North to Balerno in Scotland where Gibbs started primary school.

A Tektronix 4052

Gibbs was introduced to computer programming from the age of 10 by his mother who was a lecturer in Computer Science at Napier College. He began by learning to program in the BASIC language using a teletype terminal connected to a Honeywell mainframe at the college, storing his programs on paper tape and punch cards. Later Tektronix invented the self-contained graphic workstation 4050 series and in 1977 his father was able to bring units such as the 4052 home during holidays. Gibbs then perfected his programming technique by writing games including an interactive shooter based on the Star Wars film, and an AI program capable of competing at draughts. He also owned an HP45 programmable calculator from the age of 16.

In 1977 Gibbs was entered into a Scottish Mathematics Challenge competition at Stirling university by his secondary school Currie High, a local comprehensive near Edinburgh. He surprised his teachers by coming second in a strong field mostly from private schools. He went on to compete in the International Mathematics Olympiad as a member of the British team in 1977 (Yugoslavia) and 1978 (Romania). Despite having no access to special training or practice papers he was awarded silver medals on both occasions. In 1978 Gibbs achieved joint first place in the British Mathematical Olympiad, tying with Richard Borcherds who later went on to win the Fields Medal for his proof of the Monstrous Moonshine conjecture.

Gibbs studied for the Maths Tripos at Sidney Sussex College Cambridge, graduating in 1981 as a wrangler. After completing part III he gained his doctorate in Theoretical Physics at the University of Glasgow in 1985. In his thesis, Gibbs demonstrated the first numerical evidence for spontaneous chiral symmetry breaking at high temperature in Lattice QCD[1]. This was confirmed experimentally decades later using Heavy-ion collisions at the RHIC and LHC.

IT Career[edit]

In 1987 after a year's post-doctoral research at Edinburgh University, Gibbs left academia for a career as a software engineer. He worked for three years for the Naval Procurement Executive at Foxhill in Bath. His main assignment was to enhance the GODDESS system for computer aided ship design.

In 1991 he moved to France and spent five years at Eurocontrol, Europe's Air Traffic Control research centre at Bretigny-Sur-Orge south of Paris. He developed the first prototype for the HIPS system of interactive conflict resolution.[2]

In 1996 Gibbs returned to England where he took a year out to write a physics book "Event Symmetric Space-Time."[3]

From 1998 he worked as a developer in the City in London ending his career in 2006 after becoming a Senior Vice President of Lehman Brothers in Fixed Income Research where he headed a development team for the bank's European client research intranet site.

Mathematics Research[edit]

Number Theory[edit]

Philip Gibbs

In his teen years Gibbs developed an interest in recreational mathematics and problem solving. He often spent time solving puzzles posed in specialist computer magazines which his parents subscribed to. His biggest interest was number theory. In 1978 he learnt about a Diophantine problem in the Bulletin of the British Computer Society. Diophantus had sought sets of rational numbers such that the product of any two is one less than a perfect square, finding examples of triples and quadruples of such numbers. Later Fermat found quadruples of integers (e.g. 1, 3, 8, 120) and Euler found quintuples of rational numbers. Gibbs was intrigued by the algebraic relations that he found to exist between the elements of known solutions. Twenty years later he finally worked out that the mysterious equation satisfied by Fermat's Diophantine quadruples is a special case of Cayley's Hyperdeterminant[4]. Gibbs was also able to find the first sextuples of rational numbers with the Diophantus property by computer search[5]. Other equations that generalise the hyperdeterminant exist for some quintuples and sextuples which Gibbs called regular[6]. A community of mathematicians led by Croatian mathematician Andre Dujella has made much further progress on the problem in recent years. The mathematics of hyperdeterminants also connected to Gibbs's research in physics and led to some collaboration with Mike Duff and his students at Imperial College.

Although his main areas of research have been in physics, Gibbs returned to mathematics more in later life, realising that by combining his skills in problem solving and computing he could sometimes make progress on difficult problems. He has said that while progress in fundamental physics remains frustratingly controversial and hard to verify, nobody can deny the worth of a proven solution to an old mathematics problem.

In 2015 Gibbs saw an opportunity to put this into practice when Stefan Steinerberger published a discovery of a hidden signal in the Ulam number sequence[7]. Gibbs was able to refine Steinerberger's conjecture[8] and exploit the patterns to develop an algorithm that could compute the Ulam sequence in linear time[9]. Don Knuth who had previously used slower methods to compute the sequence checked the algorithm with an implementation in his own language CWEB[10], and cited the new work in his series of textbooks "The Art of Computer programming."[11] The Ulam numbers have now been calculated up to one trillion[12].

Geometric Optimisation[edit]

Another area where Gibbs has been able to combine his mathematics and computer skills to good effect is geometric optimisation. In 2013 he was inspired by a post on John Baez's blog about the Lebesgue Universal Covering Problem posed by Henri Lebesgue nearly 100 years earlier. The problem asks for the covering of smallest area for all planar shapes of diameter one. By developing algorithms to compute optimal solutions he was able to find a variation of previous best coverings that allowed some improvement. Once the improved solutions were understood is was possible to prove new upper bounds for the area without the need for computation. Gibbs published a paper with Baez and his student Bagdasaryan[13] before improving the result further on his own[14]. This work was featured in Quanta Magazine in 2018[15].

Other geometric optimisation problems that Gibbs has contributed to include the moving sofa problem[16] and Bellman's lost in a forest problem[17] which is related to Moser's well known worm covering problem.

Physics Research[edit]

Philip Gibbs left academic research in 1987 after completing his doctorate and two years of post-doc research. In 1990 his interest in physics returned and he sought to use home computing to continue unfinished lines of research in his own time. While working at Eurocontrol in France he had access to the internet from about 1993 including usenet discussion groups and the preprint servers at LANL later known as arXiv.org. He witnessed the birth of the World Wide Web and realised that his scientific research could now continue independently of academia. Most of his papers are available on viXra, arXiv and researchgate. He has submitted essays to most of the FQXi contests and won minor prizes on two occasions.

His physics theories have been largely based on principles of symmetry, but Gibbs believes that consistency rather than mathematical beauty is the main driver of theoretic physics. The combination of consistency with observation and logical self-consistency strongly limits the directions that physicists can take.[18]

A number of Gibbs' discoveries in physics and elsewhere have been used or copied by academic authors without citation. They then received considerable credit for their work. Gibbs has said that such plagiarism seems to be a common fate for independent researchers and demonstrates the need for authenticated publication in a repository in order to prove priority.

Lattice Gauge Theories[edit]

For his doctorate, Gibbs chose to work on Lattice Gauge Theories to make use of his computer programming skills for physics research. With his supervisor Ian M Barbour and collaborators from DESY and Edinburgh University he worked in the computation of chiral symmetry breaking in lattice QCD at high temperature. He also used a combination of analytic and computational methods to understand chiral symmetry breaking a finite density. An innovative feature of his work was a new method based on Lanczos Algorithm for finding the eigenvalues of large sparse matrices.[19][20][21][22][23][24][25]

Energy Conservation in General Relativity[edit]

A number of well known physics bloggers including Lubos Motl, Sean Carroll and Sabine Hossenfelder propagate the idea that energy conservation in general relativity is approximate, trivial or that it only works in special cases. Philip Gibbs is the leader of a small minority of commentators who dispute this. Gibbs himself has described non-conservation of energy as an internet meme supported by numerous specious arguments that are easily refuted yet persistent[26]. For example it is commonly stated even by well known physicists that energy is not conserved in general relativity because Noether's Theorem requires time invariance while in fact the universe is changing. This argument is clearly wrong because the theorem only requires that the equations must be unchanging, not the solutions.

The standard formalism for energy conservation in general relativity uses pseudo-tensors which some physicists find unacceptable. Gibbs says that although pseudo-tensors do work, a better expression for energy and momentum currents is the Komar superpotential. In 1998 he showed that an expression can be derived using Noether's theorem that reduces to the superpotential when the Einstein Field Equations are applied[27]. In this approach the energy and momentum charges take the form of a moment map which is an element of the dual of the adjoint representation of the diffeomorphism group.

Complete Symmetry[edit]

Gibbs' work in physics is strongly influenced by the Noetherian Principle that once the symmetry of a physical theory is known the details can be discovered by the logic of mathematics consistency. In quantum field theory this reduces the standard model to a limited system of possibilities requiring only a finite number of parameters to be determined by experiment. In 1980 while a student at Cambridge, Gibbs attended Stephen Hawking's inaugural lecture as Lucasian professor[28]. Hawking talked about how supergravity theories could lead to the final theory of physics starting only from an assumption that spacetime is supersymmetric. This proved too optimistic but in 1983 while working on his PhD in Glasgow, Gibbs attended a seminar by Michael Green describing the renewed interest in superstring theories which seemed to further such hopes.

Different superstring theories exhibit different gauge groups, yet they are known to by equivalent through duality transforms. The string phenomenon of topology change also implies that spacetime diffeomorphism groups are variable. This has led many string theorists to believe that symmetry is less fundamental than Hawking believed. While Gibbs agrees that symmetry is emergent at some deeper level, he advocates that in fact string theory must possess a much larger hidden universal symmetry structure which may only be apparent in a pregeometric algebraic version of the theory. The observed gauge groups for different superselector sectors must be remnants of this universal symmetry.[29]

The holographic principle proposed by Susskind and t' Hooft to resolve Hawking's black hole information loss paradox requires that the state within any volume of space can be determined by information on its bounding surface. Gibbs has proposed since 2010 that this requires a "complete symmetry" where one dimension of continuous symmetry is present for every degree of freedom in the bulk theory. This would include ordinary symmetry for bosonic states and supersymmetry for any fermionic variables. If this were true then the state within a volume can effectively by transformed away by gauge transformation leaving only holonomic information at the surface corresponding to an infinite number of conserved charges. Complete symmetry would occur when the field state belongs to a representation of the adjoint representation of the universal symmetry group for the theory[30].

Gibbs has found no support for the idea of complete symmetry, yet no alternative mechanism for the holographic principle has been suggested.

Event Symmetric Space-Time[edit]

While working on lattice QCD as a postdoc in 1986, Gibbs heard conference presentations on the first attempts to explore quantum gravity computationally using random triangulation models. In 1990 after leaving academia he considered how the idea could be modified as random graph models with permutation symmetry on discrete space-time events as a precursor to the diffeomorphism symmetry of general relativity. Gibbs was able to do some simulations of random graphs on his home computer. The natural progression from random graphs is to random matrices where the elements of the matrix represent an amplitude for the linkage in a corresponding adjacency matrix. This could be seen as a second quantisation of the random graph. At the time he had no access to academic literature and was unaware of any research in random matrix by mathematicians and physicists.

Three years later while working in France Gibbs became connected to the internet and regained access to academic literature via the LANL archive, SPIRES and the BIPR library as Jessieu in Paris. At the time string theory was becoming popular but understanding its fundamental pregeometric nature manifested in the topological phase at high temperatures was considered an important problem. Gibbs realised that his ideas about permutation symmetry on space-time events could be relevant and decided to develop them further. Gibbs discovered that the symmetry in these models could be extended further to supersymmetry and to discrete version of string symmetries developed for string field theory by Michio Kaku. Gibbs also realised that the permutation symmetry over spacetime events could be linked to permutation symmetry of particles. These ideas were published on the LANL archive and in the "International Journal of Theoretical Physics" and also on a website created by Gibbs called "The Cyclotron Notebooks."[31][32][33][34]

The World Wide Web was much smaller at that time so his work attracted much interest. Gibbs was contacted by Leonard Susskind while he was working with t' Hooft on black holes. Susskind said he found the ideas of event-symmetric string theory very interesting. Later Susskind and his collaborators published a supersymmetric matrix model for M-theory[35]. In this model space-time events are replaced by the language of instatons but Gibbs saw this as an equivalent interpretation of his ideas and a partial realisation of event-symmetry in string theory. Lubos Motls who independently discovered similar matrix models at this time also later admitted that he may have been influenced by Gibbs work and the use of permutation symmetry[36]. A Japanese group saw the connection between permutation groups and diffeomorphism symmetry[37]

About ten years later philosopher John Stachel published very similar concept renaming the "principle of event symmetry" as the "principle of maximal permutability"[38]. A similar permutation principle on points in space rather than spacetime events was adopted in the theory of quantum graphity [39] but no other work has taken the concept of space-time symmetry as far as that of Gibbs. Although these later derivative works gathered large numbers of citations, the prior work of Gibbs has rarely been referenced.

Multiple Quantisation and Necklace Lie Algebras[edit]

As the theory of event-symmetric space-time was developed with the addition of further symmetries, Gibbs noticed that multiple levels of quantisation featured in the mathematics. Physicists speak of quantum field theory as a second quantised theory because its classical formulation includes the Dirac equation which is already seen as a quantised field. David Finkelstein who had encouraged Gibbs to submit his work to the International Journal of Theoretical Physics pointed out the work on multiple quantisation by his friend Carl Friedrich von Weizsäcker. Gibbs has often written about the idea that an iterated form of algebraic quantisation is at the root of structures deep in the laws of physics and that the levels of quantisation we see now are just pale shadows of this theory. Philosophically this quantisation is linked to the ensemble of possible universes and how they are selected by information available to our experience of reality. Uncertainty about information can be applied iteratively so that probabilities are themselves uncertain leading to multiple levels of quantisation. However, this is to be formulated in an algebraic description of relationships between possible realities. The most general form of the laws of physics before information selects the vacuum and state that we live in is a fixed point of this iterated process of multiple quantisation.

As Gibbs worked on a version of event-symmetric string field theory in 1995 he discovered that discrete string Lie algebras could be constructed by assembling elements of an underlying Lie algebra or super-Lie algebra in loops and chains. This process could be repeated iteratively to generate higher dimensional structures related to higher category theories. It is this process that Gibbs believes may be the correct pre-geometric formulation of multiple quantisation [40].

Fifteen years later Gibbs learnt that algebras of this type were known as Necklace Lie algebras [41] and the canonical example is simply a free Lie-algebra generated from a vector space. This algebra has dual forms in terms of both chains and loops and can be mapped onto amplitudes for continuous strings and loops in the underlying vector space. Gibbs believes that this is the key to understanding the underlying foundations of string theory. Despite its promise very little attention has been given to the idea by other string theorists.

Theory of Theories and Universality[edit]

In Lattice Gauge Theories quantum physics is computer by using a statistical ensemble of field configuration. As well as being inspired by his doctoral work to consider random graphs and random matrices as models for pregeometric spacetime, Gibbs also considered the more ambitious ensemble of all mathematical theories. He published this idea in 1995 in the Cyclotron Notebooks under the title "The Theory of Theories" Shortly afterwards Max Tegmark published a similar idea calling it the Mathematical Universe Hypothesis.

A common criticism of ultimate multiverse models of this type is that they require a probability measure and there is no unique way to determine one. Gibbs appeals to information theory to resolve this. If you make a random selection from a collection that gives a result that has a probablity then the amount of information you have gained in bits is . Turning this around it means that the probability measure for a universe containing a quantity of information is , The measure in an ultimate multiverse model is therefore determined by information content.

Another feature of Gibbs version of the Theory of Theories is universality. He believes that there must be an emergent meta-law that arises from a principle of universality in the complex system of the ensemble of mathematical theories. This would be an algebraic meta-law from which real universe are derived as geometric solutions.

In Gibbs's view the multiverse ensemble should be considered a representation of logical possibilities rather than a platonic realm of existence that would then require a higher explanation. Reality is relative to the observer and limited to the scope of what they can experience[42][43][44].

Higgs Boson[edit]

Unofficial Higgs Combination Plot by Philip Gibbs

Gibbs covered the build up to the discovery of the Higgs Boson on his blog viXra log.[45][46][47][48]

The ATLAS and CMS detector teams at the Large Hadron Collider at CERN published frequent updates on the Higgs boson search in the form of Brazil charts that would show a bump at the mass of the boson when enough data was collected to make it apparent. Although the two teams of experimentalists combined their own channels they were very slow to combine the results of the two experiments. Gibbs found a way to combine them very quickly. The result was approximate but accurate enough for the needs of theorists. As well as combining the two LHC experiments, Gibbs was able to include published results from the Tevatron to further improve the statistics. These were published on viXra log and an applet was provided to allow other theorists to combine channels and results. The experimentalists were not happy with this interference in their results but top theorists including David Gross, John Ellis and Nima Arkani-Hamed all showed results from Gibbs' combinations in their conference talks. Other theorists applied the Gibbs combination method in published papers but did not cite the originator.

Because of the large teams of thousands of physicists involved in the Higgs search there were frequent rumours and leaks on the progress which were reported on blogs such as viXra log. On 2nd December 2011 a commenter named Alex leaked the first news via viXra log that a 2-3 sigma signal for a Higgs boson at 125 GeV had been seen. Two days later Gibbs became the first physicist to point out that this value for the Higgs placed it on the borderline for stability of the vacuum at high energy for a standalone standard model.

When the official announcement was made on 13th December Gibbs was able to combine all the published results giving a clear 3 sigma signal. To make a discovery announcement CERN physicists would need a 5 sigma signal. They were not happy that there was a strong possibility that both experiments would have a four sigma signal by the summer and Gibbs would then be the one to combine them and demonstrate the discovery. During the next few months the experiment teams gathered more data and went to extraordinary lengths to add smaller signals to increase their statistics. They were competing against each other and also Gibbs to claim the discovery. The analysis of the experiments is supposed to be done double blind but with frequent updates the team were able to unblind and adjust their analysis to improve the signal before reblinding. When the results were finally announced on 4th July 2012 they had saved the day. Both teams had somehow just managed to achieve a 5 sigma signal independently allowing them to share credit for the discovery of the Higgs boson and avoid the embarrassment of having a blogger confirm the discovery with an unofficial combination. In fact the signals exceeded standard model expectations by a significant margin leading Gibbs to suggest that the double blind analysis may have been more than a little compromised allowing room for experimental bias. When more data was gathered over the next year the signal did indeed return to the standard model prediction.

Philip Gibbs was featured in Episode 10 of Colliding Particles, a series of video's about the Higgs Discovery produced by Mike Paterson.

Debunking[edit]

Gibbs believes that open science and open review is important to ensure the correct course of science. He has said that while peer-review is important, science is not a tick-box exercise. Reviews can help highlight potential errors but discoveries should be judged on the merits of the reasoning and experiments behind them, not on the authority or expertise of those who support them. In his blog viXra log Gibbs compiled a collection of true stories about people who were called "crackpots" who later turned out to be right. Such cases are rare but they need to be looked out for. This is one reason why he keeps viXra open to as many submissions as possible. According to Gibbs it is important to keep an open mind until evidence is overwhelming for a conclusion. It is important to be ready to change your mind if knew evidence is presented that does not support your initial leanings.

Gibbs has himself been responsible for debunking some firmly held beliefs. For example in 2014 he was surprised how easily many people accepted an incident where a sky diver filmed what appeared to be a meteoroid falling passed him after he opened his parachute. Other blogger scientists including Sabine Hossenfelder and Phil Plait were convinced that the claim was correct, mainly because an expert on meteoroids who had studied it was very certain. Gibbs was suspicious because their had been virtually no video recordings of a meteorid visible as a falling rock so the chances of this happening to a sky diver with a camera was vanishingly small. Closer examination showed that it was more likely that a small rock trapped in the parachute canape had fallen out and that it looked much larger and faster due to a lack of perspective. After Gibbs explained this on his blog it was quickly accepted as the most likely explanation.[49]

Gibbs was also an early sceptic of the stories of faster than light neutrinos at CERN and of the BICEP observation of primordial gravitational waves. In the latter case he was initially excited by the doscovery but grew suspicious when he saw that the claimed 5 sigma confidence level fell to only three sigma when dust backgrounds are considered. It later transpired that the dust background had been underestimated and was the signal[50][51]

Another area where Gibbs has yet to be vindicated is Energy Conservation in General Relativity. He is against the consensus that energy conservation in cosmology is not robust. Although more people are now taking up his point of view a clear conclusion has not yet been reached.

Despite his reputation for successful debunking, Gibbs is not a contrarian and avoids populist anti-science views. He is not a climate sceptic and has even suggested that increased levels of carbon dioxide in the air could be directly responsible to the worldwide crash in insect populations. The popular view is that this is caused by overuse of pesticides, loss of habitat and/or global warming. Gibbs has also argued in favour of string theory which he says is supported by arguments of consistency rather than mathematical beauty. He has also taken the unpopular view that string theory and loop quantum gravity are different aspects of the same underlying theory because they share many mathematical features and grew out of the same physical roots in dual models.

viXra[edit]

Philip Gibbs is the principle founder and administrator of viXra.org. The preprint repository was created on 9th July 2009 after a science blog discussion concerning the openness of repositories such as arXiv.org.[52][53][54][55][56][57][58][59]

The reaction from academics to viXra have been very mixed. Some have accused the site of being a magnet for cranks because it does not filter submissions for a minimum level of quality. Gibbs says that this is not a problem. Unlike a peer-reviewed journal, the repository does not claim to lend any credibility to its contents. It just provides a place for people to save their research in a fixed place where it can be openly accessed and referred too. There is no reason why the presence of low quality work should have any bearing on the judgement of any other work in the repository. Other scientists complain that they receive unsolicited email from "crackpots" and blame viXra. Some have launched personal attacks against Gibbs on social media. Gibbs points out that viXra discourages sending such email because it is unsafe. He says that authors should not discuss their work in private with people they do not know well because there is a risk of plagiarism. Gibbs has also pointed out that it is arXiv that sends authors in search of academics to provide endorsements for submissions.

Despite its detractors, viXra receives more than 5000 new articles each year are from those who appreciate the service viXra provides. This includes academics and gifted amateurs who find that the freedom viXra provides is valuable.[60] Gibbs' view is that all the work on viXra is valuable in its one way regardless of quality and that free speech is as important in science as any other endeavour.

Although Gibbs has committed to keep viXra online permanently, the site now faces an existential threat from the European Union who want to force sites such as viXra that upload user content to filter for copyright violations. This would not be possible because it would require an expensive subscription to a service that provides copyright checks on the fly. viXra runs on a budget of £300 per year with no external source of funding so it could not afford such a service. Gibbs has also said that there is a concern that such filtering could be used by governments to censor content and believes that governments are seeking stealth means to control the internet.

Trivia[edit]

A note on FRACTRAN from John Horton Conway sent to Gibbs in 1980 as an undergraduate at Cambridge University.

[61].

In the biography of John H Conway by Siobhan Roberts it is mentioned that Conway was stopped by a student in the supermarket and shown an improved solution to his minimal algorithm for computing prime number. Philip Gibbs was that student.

Gibbs was beaten for priority on his concept of Event-Symmetric Space-Time in a science fiction story written by Greg Egan. This appeared in the well known novel "Permutation City" but had originally been published as a short story called "The Dust Theory"[62].

Gibbs' ancestral family tree includes the family of William Britain who was famous for his invention of hollow-cast toy soldiers in 1893. His great great grandfather William Gibbs had worked as a brass founder in London with the Britain family and married Willaim Britain's sister Eliza Britain.

Philip Gibbs was the creator of the usenet newsgroup sci.physics.relativity and served as the editor of the Physics FAQ during which time he wrote a number of FAQ articles[63]

Gibbs has an official Erdős number of 4 from his work with John Baez, but if you count a joint article written for the Physics FAQ with Terrance Tao about how to add relativistic velocities, then his Erdős number is 3[64].

In 2015 as a campaigner against development of the green belt Gibbs stood as a candidate local councillor in the Borough of Basildon. He missed out by just two votes after 5 recounts.

Gibbs was a SCUBA diver with the Glasgow and Edinburgh University Sub-Aqua Clubs from 1982 to 1987. He did over 500 dives in Scottish waters including Cape Wrath, Bo fascadale Rock, the Oberon Bank, the Falls of Lora, the Corryvreckan Whirlpool, St Kilda and Foula.

To avoid any confusion this Philip Gibbs has never been a war correspondent, fund manager, football referee, free runner or sky diver.

References[edit]

  1. Barbour IM.; Gibbs P.; Gilchrist JP.; Schneider H.; Schierholz G.; Teper M. (1984). "Strong evidence for spontaneous chiral symmetry breaking in (quenched) QCD". Physics Letters B. 136 (1-2): 80-86. 
  2. Meckiff C.; Gibbs P. (1994). "PHARE highly interactive problem solver". Eurocontrol Report. 273. 
  3. Gibbs P. (1998). "Event-Symmetric Space-Time". viXra. 0911.0042. 
  4. Gibbs P. (2001). "Diophantine quadruples and Cayley's hyperdeterminant". arXiv. math/0107203. 
  5. Gibbs P. (2006). "Some rational Diophantine sextuples". Glas. Mat. Ser. III. 41 (61): 195–203. 
  6. Gibbs P. (2016). "Regular Rational Diophantine Sextuples". viXra. 1609.0425. 
  7. Steinerberger S. (2017). "A Hidden Signal in the Ulam Sequence". Experimental Mathematics. 26 (4): 460-467. 
  8. Gibbs P. (2016). "A Conjecture for Ulam Sequences". viXra. 1508.0045. 
  9. Gibbs P. (2016). "An Efficient Method for Computing Ulam Numbers". viXra. 1508.0085. 
  10. Knuth D. (2017). "Documentation of program Ulam-Gibbs". 
  11. Knuth D. (2019). The Art of Computer Programming, Volume 4A, Section 7.1.3. 
  12. Gibbs P.; McCranie J. (2017). "The Ulam Numbers up to One Trillion". viXra. 1711.0134. 
  13. Baez JC.; Bagdasaryan K.; Gibbs P. (2015). "The Lebesgue Universal Covering Problem". Journal of Computational Geometry. 6 (1): 288–299. 
  14. Gibbs P. (2018). "An Upper Bound for Lebesgue's Covering Problem". arXiv. 1810.10089. 
  15. Hartnett K. (2018). "Amateur Mathematician Finds Smallest Universal Cover". 
  16. Gibbs P. (2014). "A Computational Study of Sofas and Cars". viXra. 1411.0038. 
  17. Gibbs P. (2016). "Bellman's Escape Problem for Convex Polygons". viXra. 1606.0050. 
  18. Gibbs P. (2012). "The Higgs Boson and the Power of Consistency". Scientific GOD Journal. 3 (6): 543-549. 
  19. IM Barbour; P Gibbs; JP Gilchrist; H Schneider; G Schierholz; M Teper (1984). "Strong evidence for spontaneous chiral symmetry breaking in (quenched) QCD". Physics Letters. B136 (1-2): 80–86. 
  20. IM Barbour; P Gibbs; KC Bowler; D Roweth (1985). "High statistics study of the chiral condensate in quenched lattice QCD". Physics Letters. B158 (1): 61–65. 
  21. Gibbs P. (1986). "The fermion propagator matrix in lattice QCD". Physics Letters. B172 (1): 53–61. 
  22. Gibbs P. (1986). "Lattice Monte Carlo simulations of QCD at finite baryonic density". Physics Letters. B182 (3-4): 369–372. 
  23. IM Barbour; NE Behilil; PE Gibbs; G Schierholz; M Teper (1987). "The Lanczos method in lattice gauge theories". The recursion method and its applications. 58: 149–164. 
  24. IM Barbour; NE Behilil; PE Gibbs; M Rafiq; KJM Moriarty; G Schierholz (1987). "Updating fermions with the Lanczos methods". Journal of Computational Physics. 68 (1): 227–236. 
  25. CF Baillie; KC Bowler; PE Gibbs; IM Barbour; M Rafique (1987). "The chiral condensate in SU (2) QCD at finite density". Physics Letters. B197 (1-2): 195–199. 
  26. Gibbs P. (2010). "The Energy Is Conserved in the Classical Theory of General Relativity". Prespacetime Journal. 1 (6): 1072-1084. 
  27. Gibbs P. (2010). "Covariant Energy-Momentum Conservation in General Relativity with Cosmological Constant". Prespacetime Journal. 1 (6): 899-907. 
  28. Hawking S. (1980). "Is the End in Sight for Theoretical Physics?". Black Holes and Baby Universes and Other Essays. Bantam. 
  29. Gibbs P. (1996). "The Cyclotron Note Books". viXra. 1007.0056. 
  30. Gibbs P. (2010). "A Universe Programmed with Strings of Qubits". FQXi. 
  31. Gibbs P. (1994). "Models on event symmetric space-time". arXiv. hep-th/9404139. 
  32. Gibbs P. (1994). "Event-symmetric physics". arXiv. hep-th/9505089. 
  33. Gibbs P. (1996). "Principle of event symmetry". International Journal of Theoretical Physics. 35 (6): 1037–1062. 
  34. Gibbs P. (1998). "Event-symmetry for Superstrings". International Journal of Theoretical Physics. 37 (4): 1243–1252. 
  35. T. Banks.; W. Fischler; S.H. Shenker; L. Susskind (1996). "M Theory As A Matrix Model: A Conjecture". Phys.Rev. D55: 5112–5128. 
  36. L.Motls. (2006). "The Reference Frame: Quantum Graphity". 
  37. Iso S.; Kawai H. (1999). "Space-Time and Matter in IIB Matrix Model - gauge symmetry and diffeomorphism". Int.J.Mod.Phys. A15: 651–666. 
  38. J Stachel, "`The Relations Between Things' versus `The Things Between Relations': The Deeper Meaning of the Hole Argument", in Reading Natural Philosophy/ Essays in the History and Philosophy of Science and Mathematics, ed. David Malament Chicago and LaSalle, IL, Open Court pp 231-266 (2002)
  39. T Konopka, F Markopoulou, L Smolin, "Quantum Graphity" , (2006), arxiv:hep-th/0611197
  40. Gibbs P. (2013). "An Acataleptic Universe". FQXi. 
  41. Gibbs P. (2009). "This Time - What a Strange Turn of Events". FQXi. 
  42. P. Gibbs (2015). "A Metaphorical Chart of Our Mathematical Ontology". FQXi. 
  43. P. Gibbs (2016). "Consciousness Bootstrapped". FQXi. 
  44. P. Gibbs (2017). "A Universe Made of Stories". FQXi. 
  45. Alexander Abad-Santos (2012). "The Best Physics Gossips You Should Be Reading". The Altantic. 
  46. Adam Mann (2012). "Physics Community Afire With Rumors of Higgs Boson Discovery". Wired. 
  47. Michael Slezak (2013). "Rumour points to completely boring Higgs boson". New Scientist. 
  48. Hamish Johnston (2012). "Fermilab chips in on the Higgs mass". Physics World. 
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