This article is a preface to a book, The Shadow of Existence, which presents a novel physics framework that dissolves several tensions of modern physics and offers an explanation of the measured beginning and expansion of our universe. While the book is a compilation of eighteen mathematical physics papers, I have developed an AI companion (explained at the bottom) to try and bridge the gap for curious readers needing assistance tackling the more esoteric topics.
A little over a year ago I decided to create this website as a space where I could write and share a whole lot of thoughts that had been rattling around in my head for more than a decade since finishing my PhD. Since that time, my focus has mainly been on teaching. It’s something I love to do because I love sharing all the fascinating things there are to know about the universe with people who are eager to learn and think critically.
One of the first jobs I had as an instructor was to create and teach an online course called Astronomy of Planets. Since the subject matter lends itself to the split, I decided to split the course so that nearly the first half would cover the development of astronomy as the science that eventually explained how we know the Earth is a planet orbiting the Sun, with the back half of the course devoted to the things we’ve learned about each of the planets since the invention of the telescope and, in particular, through space missions.
There isn’t a good astronomy textbook covering this material at the level I wanted — particularly the historical development of science, what science is, what we can learn by doing science, and why it works in all the messy and uncertain ways it does. So I wrote all my own learning material, which I’ve since shared on this site at the above link. The course doesn’t paper over details or present the heroes of the Scientific Revolution as caricatures of themselves. Copernicus was not creative, and his model was closer to the Aristotelian monstrosity he characterised the Ptolemaic model to be, than to what we think of our solar system today. Galileo was a stubborn man who didn’t have everything right, and whose stubbornness takes a good part of the blame for his misfortune — though his frustration with those around him who refused to open their minds to the possibility they were wrong is understandable. Kepler was a bona fide crank. Ptolemy, Hipparchus, Aristotle and many others in the Ancient Greek tradition were every bit the geniuses that Copernicus, Galileo, Kepler, and Newton were.
Having gotten certain details right no more makes one a genius than having gotten others wrong makes one worth dismissing. Science is messy and uncertain, and no one has ever gotten every detail right.
Not Newton, not Einstein — no one. And the point of the planets course, as much as the overall website, is to analyse the content itself rather than the people — the reasons they thought what they did, when they did, and the processes of thought that led to significant inferences, whether those ultimately turned out to be right or wrong. And it’s to understand the inferences we make, how they influence our thinking, and how to weigh what we know objectively: processing and reprocessing assertions to ensure, to the best we can understand, whether they’re right or not — and even then being careful to hold everything only at the weight it is owed, not more or less than that.
It’s odd to me that this part of physics — the historiography of thought — is not more widely appreciated among physicists. That our textbooks tend to skate through lists of facts discovered by good guys against the fools who couldn’t see what is now obvious to us in hindsight. And that this is not generally seen as the most important material we could teach students in a survey course that could be their last meaningful contact with science — so they understand why scientists make arbitrary choices, and waver over imperfect inferences, when confronted with truths they’re still grappling towards.
So, in the end I guess I’ve built this website as a place where I can do just that — grappling towards truths about which I think better answers can be found. Where I can write about and share the things that matter to me — from fairly settled learning material I might update once in a while, to exploring the questions and thoughts and understandings that seem to be worth grappling towards, styled and formatted however the subject-matter seems to warrant, focusing on the problems or inconsistencies or incoherences I see in various descriptions, and trying to explain why I think those things are problematic or inconsistent or incoherent — and how it seems the details might be resolved to a state that is more consistent, coherent, and cohesive.
I actually got back to writing about these things for myself about two years ago, after a decade of intensive teaching and course development led to severe burnout. Someone suggested it might ease some of the tension I’ve felt about the messy state of our physical theories if I would lay out the reasons why I think the notion that spacetime ‘exists’ is incoherent, based on a wrong inference, and amounts to a category error. After a few attempts I ended up sharing those reasons in the first posts on this site — and in writing them I came to an even clearer understanding of my own thoughts that I was able to share in a series of articles for The Conversation that were read by more than half a million people and picked up by several other outlets. That in turn led to the first article I posted here about an invalid inferential step in the standard reasoning about black holes, which was eventually published by SciTechDaily where it was read by another hundred thousand people.
And each time, more and more people wrote to me with their own thoughts, which stimulated my own thinking in directions I probably would not have reached for on my own. There is a direct line running from those outlets, through the feedback they brought back, to this prefatory note.
You see, through a number of exchanges over the preceding year, I had considerably sharpened my thinking about spacetime, relativity, and black holes by the time spring rolled around. And finally, it was a question from a friend who I’d asked to review this piece — someone who always finds where I’ve skated past some detail or given a poor explanation — and it was in trying to explain the Eddington-Finkelstein diagram’s depiction of a Schwarzschild black hole’s event horizon and its r = 0 singularity as two coordinate singularities, drawing an analogy to the poles of a spherical surface, described from one pole then extending beyond the far pole, that I stumbled upon the cycloidal description of the Schwarzschild geometry that is the subject-matter of this book’s second paper. That second paper led to the third, which develops a geometric generator that recovers exactly, from first principles and by independent means, the symmetry-reducible sector of general relativistic solutions — which led to the identification of that generator’s description groupoid, an algebraic restriction via the generator’s symmetry-breaking mechanisms — and, over the past four months, to the remaining papers that form the rest of the book.
The upshots of these papers, and the inferences that are drawn within them, are far-reaching — and I offer the following list of seven substantive points for consideration.
1. The event horizon is a metric singularity, and nothing has collapsed yet
This one asks for nothing new. No modification of general relativity, no extra assumption beyond the causal structure everyone already works with.
There’s a geometric structure missing from GR’s list of definitions, and once you have it the horizon stops being puzzling. Call it a metric singularity: a null hypersurface along whose generators the spatial extent has contracted to zero, so that events on a generator are topologically distinct and causally ordered and yet carry no metric separation between them. The triangle whose spatial and temporal legs are equal has a hypotenuse of zero length — which is the ordinary null condition, read for what it says about extent rather than about propagation. In Schwarzschild the future event horizon satisfies exactly those conditions, and the same forcing carries the conclusion to any Killing horizon, the Kerr family included.
Independently of that identification, the horizon’s defining causal property — that it is the null future boundary of the exterior — forces any temporal ‘now’ slicing adapted to an exterior observer to become asymptotically tangent to a single generator and to meet the horizon only in the limit of infinite exterior time. So the horizon is not a ‘place’ sitting out there now, at some radius, being gone through by things that fall in. It is a limiting event-of-events at the end of exterior time. Everything that ever falls in reaches it — in finite personal time, and all of it at that same common, limiting event.
Three standard problems then dissolve on causal grounds alone, and I want to be careful about what each one costs. No closed trapped surface is ever realised, so Penrose’s theorem, which is mathematically correct, has physical preconditions the astrophysical domain never meets — and cosmic censorship becomes unnecessary rather than false. The horizon-induced mode-splitting that yields Hawking radiation has no realised background to be computed on; note the scope carefully, because it matters: what goes is horizon-induced radiation, while local, non-horizon particle production is untouched. And the information-loss paradox does not arise, the realised spacetime remaining globally connected with a global Cauchy surface and unobstructed unitary evolution.
The densities associated with the central singularity therefore never form at any finite exterior time. What happens instead is accumulation, at densities that stay finite — and the more massive the object, the lower the density at which that accumulation sits.
2. A covariance of slicings, rather than a covariance of coordinates
The cycloid I stumbled on while trying to explain an Eddington–Finkelstein diagram turned out to be the first member of something larger.
General relativity is usually characterised by the covariance of coordinate descriptions of a single metric solution. What these papers develop is the other thing: a covariance of geometric slicings of a single substrate, where each slice is characterised by the manner in which the slice itself breaks that symmetry. The four-dimensional spacetimes we know come out as slices of one five-dimensional geometry — not embedded in it after the fact, but generated from first principles by independent means.
The sector this reaches is the symmetry-reducible sector of general relativity. It contains Petrov types O, D and I, and types N and III are absent. I’d rather state that absence precisely than let it read as a gap, because it isn’t one: a swept geometry depends only on its orbit-space coordinates, while a free gravitational wave depends on the transverse coordinates it propagates through. Free radiation sits exactly at the boundary of what a slicing construction can reach — the sector’s positive edge — and is reached instead by ordinary evolution of the leaf. That’s a fact about where one instrument stops, not a claim that the framework has no waves.
Ask next what the space of descriptions actually is, and the answer is a groupoid rather than a group. In plain terms: the objects are the vantages a geometry can be described from, the arrows are the changes of vantage between them, and there is no single privileged description and no single symmetry acting across all of them at once — but there is exact bookkeeping of how any two relate. GR’s constraint algebra turns out to be a Lie algebroid rather than a Lie algebra, which is the technical statement that the symmetry varies from point to point. I take that variation to be the content, not a defect.
3. One object, read many ways
At this point it’s worth saying what the first two are readings of, because otherwise the rest of this list looks like six separate arguments.
There is a single geometry underneath: a maximally symmetric de Sitter substrate. Two things about it carry the weight. First, it is real by construction — a real manifold with a real coordinate basis, whose Lorentzian signature is an intrinsic property of its positive curvature rather than a signature imposed on it from outside. The imaginary quantities the construction uses to reach it are instruments of continuation over a geometry that is real at every point they land on. Second, it is the universal standard of scale: its curvature radius is fixed by the cosmological constant alone, and everything on the geometry scales from that one length.
I’d hold this at the weight it’s owed. It isn’t an independent discovery so much as the thing that was implicit in the construction all along, and stating it plainly is what makes the pattern visible: the horizon, the slicing, the cosmology and the matter sector are not four results but four readings.
4. A discovered symmetry, hardened into a claim it never had to make
Einstein discovered something beautiful and true — the relativity of clock readings. Over the decades that followed, that discovered symmetry hardened into an ontological claim: that there is no fact of the matter about what is happening now elsewhere in the universe. My argument isn’t that the discovery was wrong. It’s that the hardening was never forced, and that it went unexamined for a century because nothing much pressed on it.
Something has been pressing on it the whole time, in plain view, and it comes from a property standard cosmology has held for a hundred years. The redshift of a distant galaxy is not an intrinsic property of that galaxy. It is an integrated property of cosmic expansion along the entire path the light travelled to reach us. And we have measured how isotropic that effect is: the scatter is below about three parts in a million.
Here is the difficulty. If cosmic expansion were influenced by local matter density in the way standard wisdom holds it must be, the scatter across those same paths could not be smaller than roughly a thousand times what we actually measure. That’s the sort of margin that doesn’t leave much room for interpretation.
What comes out the other side isn’t a preference. Our universe must have a well-defined simultaneity relative to which clock measurements are made; there must be a three-dimensional universe that exists and evolves, with a now defined by that empirically constrained cosmic simultaneity; it must expand uniformly; and local matter is not bound to a finite threshold radius.
Once cosmic time is measured rather than posited, the problem of time stops being a problem — and I think this is the cleanest example in the corpus of a puzzle dissolving rather than being solved. In canonical gravity the Hamiltonian is a sum of constraints that vanish on the physical phase space; no preferred time survives; evolution reduces to a constraint that annihilates physical states. That frozen reading is the canonical expression of treating the four-dimensional manifold itself as the thing that exists. Read the evolving layer as what exists and the manifold as its record, and the constraint is solved for a true Hamiltonian that generates the advance. The frozen constraint and the true Hamiltonian are the same canonical content under two readings.
5. A universe that has to expand the way ours does
After defining an objective time, the oldest problem in cosmology is the expansion itself.
Our standard model describes a universe beginning from infinite density and decelerating at an infinite rate. It’s a universe that doesn’t want to expand — one that begins by trying to stop with everything it has, and succeeds in expanding only because the same infinite initial state supplies an infinite rate more than sufficient to overcome the deceleration it also supplies. Nothing in the past hundred years has touched that. The initial state came to be called the Big Bang, and the conditions it predicts were confirmed in the 1960s, so we know there was a hot, dense early state very much like the predicted one.
A whole family of well-known problems then grows on that single unsolved initial-value problem — horizon, flatness, coherence, isotropy, near-uniformity. Each one characterises something we do observe as wildly improbable unless we add a mechanism whose job is to make it probable again.
In this framework those problems don’t get solved. They aren’t realised. There is no epoch at which they arise to be answered. And what stands in their place is a cosmology that must be observed to expand in the manner we do observe it to expand, producing the relative abundances of light elements we do observe.
The part that can be checked — and so the part I’d point a sceptic at first — is the accounting. This cosmology carries one free parameter and one measured epoch, against flat ΛCDM’s six. The acoustic scale and the light-element abundances are predicted rather than fitted. That’s the claim most exposed to being wrong, which is exactly why it’s the one worth attacking.
6. Three generations, and the dimension of spacetime, turn out to be one fact
The matter sector produces a hard negative before it produces anything, and the negative is what makes the positive mean something.
The obvious hope was colour: if the gauge groups are continuous isometries of the compact face, the Standard Model falls out of the geometry. Four converging routes close that door, and the reason is sharper than “it doesn’t work.” Colour does act geometrically — transitively, in fact, on the compact five-dimensional face. But it acts there as a symmetry of space, not as an internal symmetry, and it lives only on that compact face, reached off the real Lorentzian substrate by a change of signature. Colour needs a six-dimensional real carrier; the Lorentzian substrate’s compact sector supplies five. And the symmetry admits no equivariant map down to the cosmological three-sphere, so the physical cut destroys it. A second obstruction meets the first on the same object: on the round face there are no Dirac zero modes at all, so the sector there carries no massless content for colour to act on — not a vector-like spectrum, but nothing at the bottom of the tower at all.
What the wall leaves standing is not small. A Dirac field on the slicing curve gives three chiral generations and no mirror partners — three zero modes of one handedness and none of the other — forced by least-arbitrariness rather than posited, and protected as a graded index under any deformation that preserves the structure. The number three is counted, not chosen.
And read in a general dimension, the same construction speaks about the dimension itself. The generation count reads a fold of dimension D−1. The horizon relation collapses to a single multiple-angle only at D = 4 and D = 5. The mass-parity that grades chirality exists only at even D. Four dimensions is the only one carrying both a generation count and a chirality — so three generations and four-dimensional spacetime are one fact read at two ends. This is forced within the framework rather than proved about the world, and it settles the dimension of the cut rather than of the substrate.
There’s a further consequence of the negative worth stating. Because the face carries a single scale, and that scale is cosmological, any mass read off its geometry would be around 10⁻³³ eV. So an external mass spectrum isn’t a concession to phenomenology — it’s a requirement of the one-scale accounting.
7. Collapse makes universes, and antimatter is a relative property
Put the pieces together and the last one follows.
Matter collapses. The horizon it collapses to is a metric singularity at the end of exterior time, which everything falling in reaches in finite personal time. And the same continuous curve that describes complete gravitational collapse in one universe describes expansion in another. It is one closed curve, read two ways.
So a black hole of galaxy-cluster mass doesn’t terminate in a point of infinite density. It becomes a universe. And antimatter is fundamentally a relative property — which region the reading is taken in — so the child is antimatter with respect to the parent, matter with respect to itself, and its inhabitants would say the same of us.
Whatever mass such an object accretes, whatever residual spin and charge it carries, it must become a universe that expands as ours is observed to expand, with the light-element abundances ours is observed to have. Read from the other end: our universe formed from the collapse of a black hole in a previous one, and the composition datum we inherit is transmitted through that crossing rather than derived within it.
The chain is finite. Entropy per baryon rises by a fixed amount at every crossing and is bounded below, so there are at most a couple of billion links, and there is a first universe whose datum has no antecedent. I want to be plain about what that means, because it would be easy to overclaim here: this doesn’t dissolve the initial-condition problem. It transports the datum through a bounded chain and then owes the head — much as flat ΛCDM owes a baryogenesis. What it does is turn an unexplained beginning into a transmitted inheritance with a countable history, which is a different and, I think, better-posed sort of debt.
In keeping with everything I said at the top about critical reasoning, I expect none of the above to be taken as stated and I do not expect it to be the final word about anything. The framework is falsifiable, and its confrontation with the evidence calls for more and better tests than have been run so far. The sharpest of those is the one named in the fifth point above: a cosmology that spends one free parameter where the standard model spends six, and that predicts the acoustic scale and the light-element abundances rather than fitting them, has correspondingly little room to hide. If those predictions fail against better data, the framework fails with them. That is the test I would want run first, and I would rather it were run by someone hoping it fails. Nor is the book ‘complete’: it contains a list of open problems I am still actively working on, which continues to shape and reshape certain aspects. But the papers have largely converged at this point, and I offer them for the answers they do provide as I continue to work through them.
I should also say plainly how this was done, because eighteen papers in a summer is not a pace I could manage alone, and I didn’t. The work was done in sustained collaboration with AI — and that fact is precisely why the corpus is built the way it is. Every computation is runnable, every substantive claim is traceable to a receipt that can be executed and checked, and the reading protocol’s first job is to stop the model agreeing with me. I don’t fully trust the process that produced this work, including the part of that process that was me. So I built the thing to be checkable, and I’m handing you the means to check it.
And I offer this: as I continue working the open problems with Claude, my method is to ‘spin up’ new Chat or Code or Cowork threads as the old ones fill; and the way I do this is to point each new instance at the programme’s readme on GitHub, at this link here, asking it to work its way through the instructions. The instructions themselves begin with a paper about what makes for a good conversation, because I’ve found that LLMs tend to skip the reading altogether and answer from training data instead, strawmanning the very thing they were asked to engage with. So I start by getting each instance to treat the work fairly, and not to criticise what it hasn’t actually read. From there, the readme works through a series of twelve reading steps, reading all the papers in turn, followed by meta-documentation about thinking honestly and fairly and not falling into bad habits of cutting corners or inverting commands or collapsing two things into one, etc. — treating the work fairly and honestly. And finally it reads the live frontier documents against its knowledge of the papers and its developed ‘thinking’ strategy. Then it is spun up, and only then is it worthwhile to interact with, in my opinion.
I offer this for two reasons: (a) I think such an informed Chat instance is the best companion a person could have for interacting with this book, as it then truly understands the content — it thus becomes a valuable AI companion that can answer any question one might have about even the most esoteric topics in the corpus and help you learn regardless of your background; and (b) the live frontier need not be mine alone to work — every computation in the corpus is independently runnable, backed by more than 700 python scripts that anyone can execute and check for themselves — and I would gladly receive help in this venture and add substantive contributions that meaningfully push the programme forward towards completion.
So that’s the deal. I’ve done on my own what I think needed doing, and I’ll keep working the live frontier while making the whole thing available to anyone interested in learning about and working on it with me. The book itself — both the HTML pages and the typeset PDFs — is generated live through GitHub, so it will continue propagating to the subdomain, https://shadow.cosmicave.org/, where it will remain readable in a format that suits humans, while the GitHub repo remains readable by machines. The comments section here is open for discussion. For anything substantive — a defect in an argument, a receipt that doesn’t reproduce, a result worth adding — please open an issue on the GitHub repository, where I can work through it properly. I’m keeping write access closed so the corpus stays coherent, but an issue that identifies a real problem is a genuine contribution and I’ll say so. Thanks in advance for any interest you take in my work.
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