In one paragraph. The curled torus burps is a computational physics paper about a hypothetical sharp change in which flat space emerges from something more curled up. It uses a toy model of space as a network of dots and links. The toy is the paper's own: a close cousin of the published model called combinatorial quantum gravity, with one knob turned above the published value, and not that model itself. In the paper's toy, a tube (a torus curled to four steps around) sits above the flat arrangement in energy and is stable for now. The paper counts what it costs to start the change (12 units at the main setting, for every size of tube tested), predicts how long the tube waits by counting its exits, watches the flat arrangement spread, seals the system to see where the released energy (the burp) goes, and counts what is left behind (about one four-dot scrap). It also reads what the tube turns into: the tight curl is gone and the energy is released, but the result is still a narrow tube, six steps around, and a second large direction does not appear. A toy can show that a mechanism is possible or impossible within a class of models; that is the whole claim.
Everything you have ever seen happened in space with three large directions you can move in. We know its rules well. What we do not know is how it came to be.
This paper tests one piece of one hypothesis about that: a sharp change from something else into space. In a small world of dots and links it asks three questions. What does it take to start such a change? What comes out when it happens? What is left behind?
Who is saying what
The project keeps four kinds of statement apart, and this edition tags them where it matters, so that a sentence is never quietly promoted from one kind to another.
- EXACT Arithmetic, or a complete listing of every possibility: true whatever a run does.
- MEASURED A simulation whose rules and predictions were written down before it ran, with its verdict. MEASURED, EXPLORATORY means a run that was not formally pre-registered, or a closer look at runs already read.
- OUR READING, UNVERIFIED A reading of a result, offered by the author and her AI assistants and checked by no physicist.
- THE AUTHOR'S CLAIM The author’s hypothesis about reality, stated as hers.
- PUBLISHED Other researchers’ published results.
Numbers in square brackets point to the sources listed at the end.
CHAPTER 1 · THE PLACE WE START
We live in space, and we cannot rerun its birth
What was there before the Big Bang?
Nobody knows. This chapter sets up one way to study the question without a time machine: treat space as one arrangement of something deeper, and ask what kind of change could have produced it.
Side by side: the same ground in the technical edition, Sec. I, Introduction »
The author. Emily Smith is a software engineer, not a physicist, and also a startup founder and a foster mom. Her recreational time is limited and the computing bills are her own. Fittingly, the project repository is called RecreationalPhysics. She works, on the side, to map her intuition about how reality is organized onto what humanity knows about the universe, and to share what she learns with other curious people.
Her hypothesis. THE AUTHOR'S CLAIM Space is one settled arrangement of something deeper, which she calls X because nobody knows what it is. X was not chaos. It was a specific arrangement, stable for now, the way a reagent is stable until something gives it a push. The change that made space started at a seed, spread as a front, released a definite amount of energy all at once, which became matter and light, and left a remainder. Across the change the energy books balance. Figure 1 is that paragraph as a picture.
The approach. That is a hypothesis, not a result, and nothing in a computer can prove it about reality. What a computer can do is take a small world whose rules are written out in full, in which space is an arrangement and not a given, and ask whether a change of this kind can happen there at all. If the answer were no, the idea would have no home in that world. If it is yes, the idea gains a laboratory, and every number in it becomes a thing to check. Three habits keep the laboratory honest: published results are reproduced first, each prediction is written down before its run, and a result that goes against the author is reported as prominently as one that goes her way.
The parts of this paper. Chapter 2 is the starting point: a published model, what we reproduced of it, and the one thing we change. Chapters 3 to 8 are the experiment. They take one specific arrangement and ask what it is, what it costs to start changing, how long it waits, what takes shape, where the energy goes, what remains, and whether the answers hold at other settings. Chapters 9 and 10 say where the simulations stop, what held, what failed, and what the next papers take up.
A toy, and why this one. Physicists call a small world of that kind a toy model: simple enough that every consequence of its rules can be counted or simulated, and not to be mistaken for the real thing. Ours starts from a published one, combinatorial quantum gravity [1, 2, 3], for four reasons. Space is not built into it: there are only dots and links, and geometry has to form. Its energy is a version of the curvature in Einstein’s theory of gravity. It has published results to check our code against. And at the smallest sizes every possible arrangement can be listed, which gives exact answers to test the simulations on.
CHAPTER 2 · THE TOY
A published model, reproduced, and the one knob we add
Is space made of something smaller?
Some physicists think so. This chapter describes one published model, combinatorial quantum gravity, in which space is a network of dots and links and nothing else, and then the toy this paper works in: a close cousin of that model with one knob turned up, not that model itself.
Side by side: the same ground in the technical edition, Sec. II, Model and exact results »
PUBLISHED Combinatorial quantum gravity is the work of Carlo Trugenberger, Christy Kelly and Fabio Biancalana [1, 2, 3]. It is a network of dots and links with no positions at all: a dot has no place, only neighbors. Mathematicians call such a thing a graph. Figure 2 is the whole of it.
PUBLISHED With those two prices the energy measures how curved the network is: a graph version of the curvature in Einstein’s equations of gravity. It has no meters or joules in it. We call its amount a unit, and leave the conversion into physical energy for a later paper. The published work also studies a simpler version with the surplus price switched off, where a third square is free.
PUBLISHED A run is a long chain of swaps, each one proposed at random. A swap that lowers the energy is always accepted. A swap that raises it by ΔE units is accepted with a chance of e−ΔE/g, where e is a fixed number, about 2.7, and g plays the part of temperature. We call g the warmth: the bigger the cost, the rarer the swap, and the warmer the network, the more often a costly swap goes through. Picture an endless bath that lends energy for an uphill swap and takes back what a downhill swap gives off. Time is counted in sweeps: one sweep is two proposed swaps for every dot. This is the standard way to sample such a model, not a claim about how anything moved in real time.
Reproduced before changed
PUBLISHED The published work starts from a random tangle of links, cools it, and asks how geometry condenses out of the tangle. Sharply, the way water freezes, giving off heat with ice and water side by side? (Very clean water can even stay liquid below zero, stuck, until a jolt sets it off.) Or smoothly, the way a fridge magnet loses its pull as it warms, with no burst of heat and no moment when half of it has changed? Physicists call the two kinds first order and continuous. Published work reports continuous behavior for curvature-based models [2, 3] and first-order behavior in a related model with no surplus price [4], where the network breaks into closed pieces.
MEASURED We checked published results before extending the model.1 Our simulations match the 2019 square-density curve [2] with a root-mean-square difference of 0.005 at the same size, match exact averages at sizes small enough to list every possible arrangement, and reproduce the sharp change into closed pieces with the surplus price off. Then, with the rules written down first, we measured the jump as the tangle orders. With the surplus price off it is large, but what forms is a pile of closed pieces, not a space. With the full price list we found no clear jump at any size we ran, which settles nothing: a small network can hide a jump a larger one would show.
The one thing we change
The toy turns one price into a knob. In the toy a surplus square costs 4λ units, where λ (lambda) is a number that can be turned up or down, so we call it the knob. With the knob at exactly λ = 1 the toy is the published model. With the knob at λ = 0 it is the published simpler version. Every other setting is our addition, and there the energy is still a perfectly good rule for a network, but it is no longer a curvature. So two names are kept apart from here on. The published model is theirs. The toy is ours: a close cousin of combinatorial quantum gravity, not the thing itself.
We have not found published work that sets the surplus price to any value above 1 [13]. In print, as far as we have found, are the two settings just described and, in the text of two of the papers [2, 14], a version in which a third square is forbidden outright. So, barring a gap in our search, the approach taken here is new.
Why turn the knob at all? Not to settle the published question: neither this paper nor the author’s idea rests on it. What she asks of the toy is plainer: does a prepared arrangement give way with a jump in energy at the sizes run, and how large is the jump? Chapter 3 shows how the knob makes room for that question.
CHAPTER 3 · THE PHASE BEFORE
What if what came before space was not random?
What existed before space?
The author's hypothesis is that it was not nothing and not chaos, but a specific curled-up arrangement. This chapter builds a stand-in for that arrangement inside the toy model.
Side by side: the same ground in the technical edition, Sec. II, Model and exact results »
The experiment: the arrangement • the cost • the wait • the change • the energy • other settings
The published question starts from a random tangle. This program asks a different one: what if what came before space was not random at all, but a specific arrangement, stable for now, with energy to give? To ask it in the toy, we first need to say what space looks like in a network with no positions, and what the alternative to it could be.
Directions
Space, at its simplest, is a set of directions with room in them. Figure 3 shows how to find a direction in a network that has no positions.
A grid with no edges
A grid drawn on paper has edges, where the dots run out of links. The toy removes the edges.
Curling
If a direction is not large, what else can it be? Small: curled up on itself.
Ring and doughnut: flat is not the same as open
The drawings so far leave something out: the edges are joined. EXACT Join the tube’s two ends and it is a thin ring, like a hula hoop, four steps around the thin way and L steps around the long way. The sheet is a fatter doughnut, long both ways (Figure 8). Nothing in the toy is open-ended, and nothing can be: every dot must have its four links, so there is no edge to stop at, and with a finite number of dots a walk that keeps going must come back around.
Two words are kept apart from here on. Flat is about each dot’s surroundings: every link on exactly two squares, the wiring of a plain grid, at zero energy. Wide is about the whole shape: how many steps it takes to walk around it and come back. A doughnut from a bakery is not flat, since its skin is stretched on the outside and squeezed around the hole, but the toy’s doughnut is the game-screen kind and is not bent anywhere. The sheet is flat and wide. The tube is neither.
Can anyone inside tell a wide doughnut from an endless grid? Suppose something lived in the network and could learn only what it reaches by walking links. Let it count the dots it can reach from home in one step or fewer, then two, then three. EXACT On an endless grid the dots it can reach make a diamond that grows by one ring each step: 5 dots, then 13, then 25, then 41. On a grid that wraps, the diamond grows the same way until it has spread halfway around. Then the walk that set out one way and the walk that set out the other way arrive at the same dot from behind, and that dot is counted once where an endless grid would have two different dots there (Figure 9). The count comes out one short. That is the first thing an inhabitant can notice about the shape it lives in, and it can notice it only at half the way around: at two steps on the tube, at four steps on a grid eight around, at six steps on a grid twelve around.
EXACT So from inside, a doughnut much wider than its inhabitants is open in every way they can check: until they walk halfway around, their counts are an endless grid’s. The tube is the opposite case. Its count falls short at two steps, as early as a count can, because its way around is the smallest loop the rules allow. That direction has no room in it at all, which is what d = 1 records: from inside, the tube is little more than a line.
PUBLISHED Astronomers put the same question to real space. On the largest scales, observations are consistent with zero spatial curvature within the cosmological analysis used to interpret them [12], though gravity still curves spacetime around stars and planets. Whether space is also endless is a separate question. A search of the sky for the repeated patterns that a space closed up like a doughnut would show found none on scales smaller than the part of space we can see [15]. Flat has been measured. Endless has not.
THE AUTHOR'S CLAIM This is where the toy meets the author’s picture. In it, what came before space was curled tight, as the tube is, and the change opened it into the space we live in, with three large directions. OUR READING, UNVERIFIED For that story, flat is not enough. The space we see is also wide, wider than the part of it we can observe, while a change that made every point flat and left the ring narrow would give its inhabitants a world they could walk around in a few steps. So the picture becomes three questions, taken up in order. Does the change make a direction flat, or wide as well? Chapter 6 measures it. If only flat, what would pay for the widening? That is for a later paper. And with three curled directions instead of one, do they open together, and how wide? That needs a network with six links per dot, which is how a grid gets three directions, and a later paper.
The knot, and the count for all three
Curl both ways around to four steps and every dot reads d = 0: a closed sixteen-dot knot called the 4-cube. It is a torus too, curled both ways. Most of the closed pieces of Chapter 2 are these knots. Figure 10 puts the three arrangements side by side.
The ladder
EXACT Put the three arrangements side by side: flat sheet, tube, knot. All three are tori, and we tell them apart by how they are curled: flat, curled, knot. Each curls one more direction than the one before, so we call them the rungs of a ladder. The price list of Chapter 2 gives each rung its bill, per dot:
| flat sheet | tube | knot | |
|---|---|---|---|
| Directions curled | none | one | both |
| Squares per dot | 1 | 1¼ | 1½ |
| saved on squares, at 16 each | 0 | 4 | 8 |
| Surplus squares per dot | 0 | 1 | 2 |
| paid for surplus, at 4λ each | 0 | 4λ | 8λ |
| Bill per dot: paid minus saved | 0 | 4(λ − 1) | 8(λ − 1) |
| at λ = 1.25, where a surplus square costs 5 | 0 | 1 | 2 |
Curling is a trade between the squares it gains and the surplus it pays for, and the knob decides which side wins.
This is what the knob is for. EXACT At the published setting, curled and flat cost the same, so turning one into the other gives off nothing. Equal cost does not make the two equally likely, and it does not forbid a change. Turn the knob above 1, and a tube that ends at zero energy gives off exactly 4(λ − 1) units per dot. That is a reason to study this cousin of the published model. It is not proof of a first-order transition, and it is not a reason why nature should choose it.
That release is what the rest of this paper is about, so it gets a name of its own. The author calls it the burp: the energy given off, and by extension the event that gives it off, a quiet wait, a start, a spreading change and a definite release.
EXACT The changed rule is the published curvature energy plus 4(λ − 1) times the surplus square count. The extra term vanishes when that count is zero. THE AUTHOR'S CLAIM The author proposes reading it as a curling cost: energy has to be paid to curl space tight, and is given back when it uncurls. She holds that as something to test. EXACT In the toy it is plain bookkeeping: a curled tube sits 4(λ − 1) per dot above the flat arrangement, so uncurling gives that much back and curling again would have to pay it. OUR READING, UNVERIFIED What is not supplied is a derivation. At λ = 1 the rule is the published graph version of the curvature in Einstein’s gravity; nothing known requires the extra term or fixes its size, so its meaning for nature is unverified.
Stable for now: is there a window?
Energy to give is not enough. The clean water of Chapter 2, still liquid below zero, has energy to give and sits there. For the tube to be a stand-in for X it must be stuck: leaving its equal-energy family must cost energy. EXACT At 18 dots the toy is small enough to list every allowed arrangement: there are 26, counting arrangements that differ only in the names of their dots as one. Among them, arrangements above the flat sheet from which every single swap costs energy exist for knob settings between 1 and 1.6, and for no larger setting. So the knob has a window, and a narrow one.
The stand-in
Our stand-in for X is therefore the tube. At λ = 1.25 it sits exactly one unit per dot above the flat sheet. It is a stand-in and not X: it has one direction with room in it where space has three, its curled direction is always four dots around, and it lives in a network of a few hundred dots. What it shares with X is the shape of the idea under test: a specific, curled, stable-for-now arrangement with energy to give.
The running example
Take a tube of 96 dots: four around, 24 along. At λ = 1.25 it holds 96 units above the flat sheet. That is the burp on offer. We will follow this tube through the rest of the paper so that the amounts stay concrete. It is a running example, not an invented history: the evidence comes from hundreds of repeated runs and, where possible, from counting every allowed move.
Now the question that decides whether any of this is a story at all. Ninety-six units are waiting to come out. What does it cost to start?
CHAPTER 4 · THE OBSTACLE
What does it cost to start?
What could have set off the Big Bang?
Unknown. In the toy model a change of this kind needs a push, and the push can be counted exactly: twelve units at the paper's main setting of the knob (λ = 1.25), the same for every size of tube tested.
Side by side: the same ground in the technical edition, Sec. III, The cost of leaving »
The experiment: the arrangement • the cost • the wait • the change • the energy • other settings
Before any number, consider what the number could mean. If the push needed were tiny, a few units, then any warmth at all would set the tube off, and it would not be stable for now. If the push were as large as the burp itself, 96 units for our tube and more for a bigger one, then starting the change would cost what the change gives back, and the cost would grow with the size of the tube. Nothing small could ever start something large. A sharp change of the kind pictured at the start of this paper needs a middle answer: a push that is small, and that does not grow with the size of the thing being pushed, so that a bigger tube releases a bigger burp from the same spark. A fixed push and a growing release is the condition for a runaway.
So: is the cheapest exit from a perfect tube small, fixed, or neither?
Pricing the cheapest exit
EXACT Start from a perfect tube and try every swap the rules allow, the one move of Chapter 2. Each one breaks some squares and removes some surplus, and the price list gives its cost. Most are expensive. Some cost nothing: they twist the tube around itself without changing any dot’s neighborhood, so they do not count as leaving. Among the swaps that do leave, two kinds are cheapest. Call them route A and route B. Here are their bills at λ = 1.25, where a surplus square costs 5:
| Route A | Route B | |
|---|---|---|
| Squares broken | 2 | 4 |
| charged, at 16 each | + 32 | + 64 |
| Surplus squares removed | 4 | 10 |
| refunded, at 5 each | −20 | −50 |
| The bill | 12 | 14 |
Route A is the cheaper at this setting; the two tie at λ = 4/3, and B is the cheaper beyond. (For readers who want every exit listed with its formula, the technical paper [11] has them in Section II.)
The cheapest way out of the tube costs 12 units.
Twelve. Not 96, and not 2. For our 96-dot tube the push is an eighth of what comes out; for a 288-dot tube, which holds 288 units, the same 12 is a twenty-fourth. EXACT The cost of the cheapest single swap does not depend on how long the tube is, because a swap only touches a handful of neighboring dots; the burp does, because every dot contributes one unit. That is the middle answer the change needed, and it is arithmetic.
Is 12 really the floor?
One cheap exit from the perfect tube is not the same as no cheaper exit anywhere nearby. The free twists lead to other arrangements with the same energy, and one of those might have a cheaper door. EXACT So the calculation followed every free twist, at six sizes from 48 to 288 dots, until no new arrangement appeared. At each size there are exactly three reachable arrangements, counting arrangements that differ only in the names of their dots as one; every one has the same menu of exits, and the cheapest costs 12. That is a complete listing at those sizes, not a proof for every size: at 32 dots the pattern is different.
The test from the other side
Arithmetic says nothing below 12 can leave. Does 12 suffice? To find out, take the bath away and hand the tube an exact budget and nothing more (Figure 13), a standard way to simulate at a fixed total energy [7].
MEASURED, EXPLORATORY Try budgets of 4, 8, 10, 11, 11.5, 12, 13 and 16 units, eight runs at each budget at each of five sizes from 48 to 192 dots: 320 runs in all, each 12,000 sweeps long.
EXACT The lower half of this staircase is a theorem of the toy at these sizes: with no exit cheaper than 12 and a store that cannot go negative, a budget below 12 can only pay for free twists. The upper half is a measurement. MEASURED, EXPLORATORY Together they say that the wall is local, exactly 12 units high at λ = 1.25, and the same at every size tried, while what waits behind it grows with the system.
Why we went here. This is the pivot of the whole paper. Suppose the push had grown with the size of the tube, so that a tube twice as long needed a push twice as large. Then no single spot could start the change: something would have to push the whole tube at once, and the picture of a seed and a spreading front would fail in this toy. We would have had to look for this kind of change in another one. A fixed, cheap push against a growing release is the condition under which something small can start something large.
CHAPTER 5 · THE WAIT
How long before the door opens, and what kind of waiting is it?
What is a metastable state, or false vacuum?
A state that is stable for now: it lasts until chance hands it enough energy to leave. This chapter predicts how long the toy's version waits, and checks the prediction.
Side by side: the same ground in the technical edition, Sec. IV, The first exit »
The experiment: the arrangement • the cost • the wait • the change • the energy • other settings
A rim of 12 says what a push must be. It does not say when a tube left to itself will go. For that we put it back in the bath of Chapter 2 and let the network jiggle at warmth g. A swap that climbs the rim, costing 12, is accepted with chance e−12/g: warmer, and the rim is cleared more often.
A lottery with counted tickets
Now the wait becomes a lottery, and every ticket can be counted. EXACT In one sweep the computer proposes two swaps for every dot, 2N in all for a tube of N dots, each one picked at random from every swap there is. How many of those proposals are exits?
Take route A. A tube of N dots has 3N different route-A swaps: three for every dot, so a longer tube has more ways out. But a longer tube also has more swaps of every other kind, and those grow faster, as N times N. So the chance that any one proposal lands on a route-A exit is 3 in 2N, smaller for a longer tube. A sweep makes 2N proposals. Multiply the two and the length cancels: \[2N \;\times\; \frac{3}{2N} \;=\; 3 .\] A sweep offers three route-A exits on average, whatever the size of the tube. A longer tube has more exits, each one is harder to hit, and with two proposals per dot in a sweep the two effects balance exactly. The same count for route B gives two.
Each offer of A is accepted with chance e−12/g, each offer of B with chance e−14/g. So the chance of leaving in any one sweep is 3 e−12/g + 2 e−14/g, and the mean wait is one divided by that, with nothing fitted: \[\text{mean wait} \;=\; \frac{1}{3\,e^{-12/g} + 2\,e^{-14/g}} \quad\text{sweeps.}\] At g = 1.5, the warmth used for most runs below, that is 845 sweeps. Chemists know this shape as an Arrhenius law. Here both of its ingredients, the height of the rim and the number of tries per sweep, are counted from the arrangement rather than fitted to the data (the technical paper [11], Section II, gives the formula for any λ).
Does the tube keep the appointment?
MEASURED, EXPLORATORY A first scan at λ = 1.25, six warmths, two sizes and sixteen tubes each, over a ninety-fold range of waits, came within 2 percent of the count on average.
MEASURED A stricter, pre-registered test then counted every single accepted swap in 240 further runs across six settings of λ and size, and compared the first exits with the full formula, both routes. All 240 first exits were observed, and their mean arrived when the formula said it would in every cell, within the scatter expected from that many runs.
What kind of waiting
If the chance of escape is the same on every attempt, the waits are memoryless. It is like rolling a die until a six comes up: twenty rolls without one do not make a six any likelier on the next roll, and a tube that has waited a long time is no nearer to leaving. One sign of a memoryless wait is that the waits spread out by about as much as their average. MEASURED That check was pre-registered, and it passed at every size in the main run. MEASURED, EXPLORATORY Checked afterwards, the whole pattern of first exits fits that picture too.
MEASURED One more thing the move-by-move count revealed: the first swap out does not always lead anywhere. Think of that first swap as a door opening. About a third of the time the door closes again, and the tube goes back to being a perfect tube. Only 56 to 68 percent of first exits went on to a real change, with no trend in λ. The formula predicts when the door opens, not the departure.
Two clocks
EXACT The three tickets a sweep come from a cancellation, and the cancellation belongs to the clock. Every proposal is picked from the whole network, so in a tube twice as long any one neighborhood is offered its swaps half as often per sweep, while there are twice as many neighborhoods. A clock can instead give every neighborhood the same chances whatever the size of the network. We call it the fair clock: one fair sweep gives each neighborhood the chances it gets in one sweep of a 96-dot network, which takes N/96 sweeps. On the fair clock every 32 dots of tube offer one route-A exit per fair sweep, so a tube twice as long offers twice as many and waits half as long.
| The choice of clock changes | It does not change |
|---|---|
| how a wait compares across sizes: the same at every size in sweeps, half as long for a tube twice as long in fair sweeps | the 12-unit push, the release, the energy books, what the tube turns into, and the count of scraps a cold box keeps |
| how the speed of the change compares across sizes (Chapter 9) | any comparison at one size, including the match between the measured wait and the lottery, which counts both in sweeps |
OUR READING, UNVERIFIED The toy has no time in it, so neither clock is the “right” one. The fair clock is the natural one for comparing sizes, and the next paper in the series uses it. It also carries a caution: on the fair clock, the longer a stuck tube, the sooner it goes, so “stable for now” describes the sizes that were run.
Why we went here. If the measured wait had disagreed with the counted lottery, something larger than a single swap would be holding the tube shut, and 12 would have been the wrong number. It agrees. The rim is the single swap, and the lottery is the whole explanation of the first exit.
CHAPTER 6 · THE CHANGE
What takes shape on the other side
Can space change phase, the way water freezes?
In the toy model it can: a patch of the new arrangement appears and grows while the old one is still there beside it. This chapter watches that happen, says what the pictures do and do not prove, and reads what the tube turns into: a tube six steps around, not a wide sheet.
Side by side: the same ground in the technical edition, Sec. V, The conversion »
The experiment: the arrangement • the cost • the wait • the change • the energy • other settings
Chapter 5 was about the first swap out. This chapter follows the change through the tube and asks what the tube turns into.
A zipper, or several?
The natural picture is a zipper: one spot of the tube opens, and the opening runs along the tube the way a zipper runs along a jacket. A moving boundary like the zipper’s slider, with opened tube behind it and curled tube ahead, is called a front. EXACT The instrument is the direction count d of Chapter 3: dots still in the tube read 1, dots on a flat sheet read 2, and a disordered dot reads something else. A progress meter runs from 0 (all tube) to 1 (all flat), and the program photographs the whole network when the meter first reads a quarter, a half and three-quarters. These are stages of the change, fixed in advance, and nothing is special about them.
MEASURED At λ = 1.25 and g = 1.5, thirty tubes at each of four sizes were run twice, with three predictions written down first: a memoryless wait; at least 80 percent of dots in one of the two signatures at the half mark; and at least 70 percent of the flat-like dots in one connected region, on average. All three held at every size in both runs. At every stage photographed, 98.5 to 99.8 percent of dots read as tube or sheet, with almost nothing in between.
MEASURED So one large open region is weaker evidence than one zipper. Single runs did not always pass the 70-percent rule (as many as 9 of 30 at the largest size), and a separate registered test of a simple one-front law came out NOT ESTABLISHED: 21 of 59 tubes converted in several patches, mostly in the warmer bath. One connected region usually dominates by the half mark, and the change often starts in more than one place.
What the tube turns into
EXACT A count of d = 2 at a dot means only that the four-step curl is gone there: two buildings can have the same number of rooms and different floor plans. So the finished networks were rebuilt from their saved wiring. MEASURED, EXPLORATORY The ones that finished at zero energy, with nothing left over, are rings again: one surface, flat at every dot, six steps around where the tube was four, and as long as the tube was. That held for all 120 read at λ = 1.05, and for 3,446 of the 3,521 zero-energy finishes of the map in Chapter 8 (the others came apart into separate pieces, or closed up with a twist).
How does a ring four around become six around? Its seam is re-glued two dots along (Figures 17 and 18). A walk four steps around no longer comes back to where it began. It lands two dots along the tube, so each ring runs on into the next, and the rings join into spirals that wind around the whole doughnut. The shortest walk back to the start is now six steps: four around and two back.
The next figure zooms in on a short stretch of the new tube to show the dots and the walk.
EXACT Shift the seam by two more and the shortest way around is eight; each further shift adds two, and none of them is rewarded or charged. Every swap out of a flat arrangement costs energy, at least 20 units from six around, so in a cool bath the network stays at the first flat arrangement it reaches. From the tube, that is the nearest one: six around.
MEASURED, EXPLORATORY That is the clean finish, and it is the common one. Of the 11,880 runs in the map of Chapter 8, across all its settings, 9,865 finished clean, about five in six, and nearly all of those are the six-around ring. The other 2,014, about one in six, kept a scrap, and those can hold more than one width: about half have a stretch wider than six somewhere, and in every one that holds both a stretch six around and a wider one, a scrap sits between them. This paper follows the six-around ring.
The scraps themselves are the subject of the next paper, What the change leaves behind. Read from their wiring they come, so far, in three kinds, and some scraps sit where a stretch six around meets a wider one: a kind of seam. The next chapter meets the most common kind.
The energy comes out exactly
MEASURED When a tube finishes at zero energy, the energy given off is exactly one unit per dot at λ = 1.25: the arithmetic of Chapter 3, now measured. In about 26 runs of every 30 the release was within one percent of that. The rest kept small defects, 8 to 45 units’ worth read from their wiring, and held back that much. The ideal gap is exact; how much of it comes out depends on where the network stops.
Why we went here. The hypothesis pictures a seed and a spreading front. The photographs test what a toy can check: whether the two arrangements sit side by side with almost nothing between, and whether one connected region dominates. They do not by themselves establish a single front or a wide space.
CHAPTER 7 · CLOSE THE LID
Where does the energy go when it has nowhere to go?
Where did the energy of the Big Bang come from?
In the author's hypothesis the change itself released it, the way freezing water gives off heat. In the toy model that release, the burp, can be counted exactly and followed when nothing is allowed to leave, and the change leaves a small scrap of the old arrangement behind.
Side by side: the same ground in the technical edition, Sec. VI, Sealed system: the reservoir and the relic »
The experiment: the arrangement • the cost • the wait • the change • the energy • other settings
Apart from the budget test of Chapter 4, every run so far sat in the bath, and the bath carried the released energy away as fast as it appeared. If a change like this made all of space, there was no outside for its energy to go to. So seal the system again, as in Chapter 4, and watch what the burp does to its own surroundings.
EXACT The sealed system of Chapter 4 had one store. Here there are C stores beside the network, each holding a non-negative number of units. Every proposed swap picks one store at random: an uphill swap is refused unless that store can pay; a downhill swap deposits its release there. Nothing enters or leaves, so network plus stores is constant to the unit, and it was checked to be constant in every run.
Return to our 96-dot tube, holding 96 units. Put 12 more in a store and seal the lid. The account reads 108. Suppose the change runs and leaves one 14-unit scrap in the sheet, as it often does in a cold box. The network has gone from 96 to 14; the stores have gone from 12 to 94.
The surroundings can spoil the ending
Here is the twist. The released energy is now in the stores, and the stores pay for uphill swaps. Energy that has nowhere to go can buy swaps that break the very squares the change has just made. How much damage depends on how many stores there are. PUBLISHED Many stores share the release thinly, like a large cold room; few stores concentrate it, like a small hot one.
MEASURED Pre-registered: tubes of 64, 96 and 192 dots, seven choices of C from a single store to two per dot, twenty runs at each setting, 420 runs. The outcomes were classified by rules fixed beforehand: sheet-like if at least 90 percent of dots read d = 2 with almost no melted dots; melted if 15 percent or more read as disorder; stalled if both orders sat side by side with nothing melting; in between otherwise.
MEASURED A bonfire with a threshold, as the pre-registered verdict put it. Where the heat has room, the change feeds itself to completion; where it does not, the heat damages what the change made (“melted” in this classification is a label: at least 15 percent of the dots have lost their neat two directions). The switch-over moves in step with size, as a release of one unit per dot shared among the stores predicts. OUR READING, UNVERIFIED A rough energy budget puts it near one store per three dots; the measured switch-over sits between one per four and one per two.
MEASURED No run ended stalled, with tube and sheet side by side and nothing melting, in 420 runs of 30,000 sweeps. That is an observation about these runs. It does not show that such a standoff is impossible, which would take evidence these sealed runs do not supply.
What “melted” means here
EXACT A sealed box holds only the energy it started with: at λ = 1.25, one unit per dot, plus the push. A missing square costs 16 units. So the burp’s own heat can break at most about one square in sixteen, however few stores there are. MEASURED, EXPLORATORY Every one of the 154 networks classed as melted still had 94 to 100 percent of its squares. That is damage: warm fabric with flaws in it. It is not the random tangle the published model starts from, which has about an eighth as many squares as flat fabric. The burp cannot melt its own product.
MEASURED A later registered test asked whether heat kept in the stores makes the new ring wider. It does not. With plenty of room the rings ended six around; with little room they ended as patchworks, six around in one stretch and eight or more in the next, with flaws frozen in where the stretches meet (the joints of the next paper). None of 720 ended as one clean ring wider than six.
What a cold box leaves
MEASURED Eighty fresh runs in the coldest box, two stores per dot, at 64, 96, 192 and 288 dots. Almost every finished ring kept one scrap of the old arrangement: four dots in a loop that still closes a square, holding 14 units. The count was about one per box whatever the size of the tube (0.90, 0.85, 1.05 and 0.95 per finished ring). The prediction written down beforehand was that it would grow with the space, and it did not. A second test found the scrap anywhere along the tube, not where the change began or met itself. What scraps are, how many kinds there are, and which of them last is the next paper’s subject.
Why we went here. What the burp turns into, and whether the energy books balance across the change, only have meaning in a sealed system. In a bath the burp leaves and nothing can be asked of it. The sealed box is where the balance becomes a measurement.
CHAPTER 8 · THE MAP
Turning the knob
How do you test an idea about the universe with a simulation?
By turning one knob at a time with the rules written down first, and publishing every setting, including the ones that failed. This chapter is that map.
Side by side: the same ground in the technical edition, Sec. VII, Dependence on λ »
The experiment: the arrangement • the cost • the wait • the change • the energy • other settings
Everything so far was at one setting of the knob, λ = 1.25, chosen in advance. A single setting proves little: a result that only holds at one carefully chosen number is the kind of thing the project’s rules forbid. So the whole stretch of the knob was run, from 1.05 to 1.45 in steps of 0.05, at four sizes, with the rules fixed first.
What the map shows
MEASURED Three things.
- Where the tube is stuck. At every setting from 1.05 to 1.35, and at none from 1.40 up. Wherever it is stuck the change keeps the character of Chapter 6: 95 to 100 percent of dots in one signature or the other at the half mark, and one connected region holding most of the flat-like dots.
- When it leaves. The mean wait follows the counted lottery of Chapter 5 over a hundred-fold range of waiting times, with nothing fitted.
- How clean the ending is. Cleanest near the published setting, and messier as the knob turns. At 1.05 every one of 120 tubes ended as a perfect flat torus, at 1.25 about 88 percent did, at 1.35 fewer than half, and beyond the edge the change starts in several places at once in most runs and ends mostly on a sheet with defects.
OUR READING, UNVERIFIED There is a trade here. A larger λ means a larger burp per dot but a less patient X and a messier birth; a smaller λ means a cleaner change that releases less. The ideal release is 4(λ − 1) per dot, reached only when the change ends at zero energy, and it vanishes at the published setting, so this family of rules has no setting with a large release and a quiet, clean X at once.
Does it hold up?
MEASURED The map was scored against rules fixed before it ran, and it passes: in all 24 cells from λ = 1.05 to 1.30 and 64 to 192 dots, the waits spread as a memoryless wait should, the dots sit in one signature or the other, and one connected patch dominates. A larger size, 288 dots, was predicted before it was read, and all six settings landed inside their predicted ranges. At 1.35 the connected-patch test fails at 192 and 288 dots, so the stretch where everything holds is 1.05 to 1.30.
Two things keep that pass in proportion. It took four tries: three earlier maps each failed one of their rules and stay inconclusive, for reasons that mostly traced to the measuring and not to the tube, and the technical paper [11] has that story. And the single patch weakens toward the edge: at 288 dots and λ = 1.30 the largest patch holds 75 percent of the flat-like dots on average, against a cutoff of 70, and only 65 of 120 single runs clear it.
Why we went here. The project’s first success condition is no cherry-picking: a family of rules may be explored, but only along knobs fixed in advance, and every setting run is published. The map is that rule applied, and it also locates where the stand-in for X stops being stable for now.
CHAPTER 9 · THE EDGES
Where the simulations stop, and what lies past each edge
Can a computer simulate the beginning of the universe?
Not the real one. This chapter lists exactly how far these simulations go, where each stops, and what lies past each edge.
Side by side: the same ground in the technical edition, Sec. VIII, Discussion »
Every knob the toy has was turned only so far. This chapter is the honest map of the box: for each knob, what was run, why it stopped there, and what is on the other side. “Unknown” means no one has looked.
You can turn the knob, the warmth, the size, the push and the stores yourself in the interactive companion to this figure, The Tube’s Curling Ladder: its exact numbers follow you to any setting, and its measured results appear only inside the teal bars.
The knob λ: 1.05 to 1.45
Below 1.05 the burp gets smaller and cleaner, not longer-awaited. EXACT The wait levels off as the knob approaches 1 (the counted lottery gives about 8,300 sweeps at 1.05 and about 14,000 at 1.001), while the release shrinks to nothing at exactly 1, where curled and flat tie. Below 1 the ladder flips. MEASURED Above the range, at 1.5, the tube falls apart within 50 sweeps in every run. EXACT At 1.6 route B costs nothing, and at 18 dots no stuck state above flat exists; by a setting of 5 to 10 the toy behaves as one with a third square simply forbidden. Nothing curled is stable for now out there.
The warmth g: 1.4 to 2.5, with the maps at 1.5
EXACT Colder, the wait grows like e12/g: about 50,000 sweeps at g = 1 and nearly three million at g = 0.75. MEASURED, EXPLORATORY At g = 1, two of four tubes were still tubes after 30,000 sweeps, which is why the range stops there. Hotter, the tube opens almost at once and from many places, and near g ≈ 3.5 the flat sheet itself melts into the random tangle; the reservoir estimate of Chapter 7 used that melting point. How it moves with the size of the sheet is taken up in a later paper, and until then every warm result carries a size caveat.
The size N: 48 to 288 dots
The upper limit was computing time, spent on repeats rather than size. EXACT The exit count was certified at 48 through 288; a control at 32 dots breaks the pattern, and 16 dots is the knot. MEASURED Counted in sweeps, the front’s speed fell roughly as the size to the power −0.7. EXACT Counted on the fair clock of Chapter 5 the same numbers give a speed that does not fall with size, and if anything rises a little, as the size to the power 0.3. Like the wait, the speed compares across sizes only once a clock is chosen. The technical paper claims nothing beyond 288; networks into the thousands of dots are the subject of a later paper on the scrap.
The push and the stores
EXACT With a push below 12 nothing can leave by single swaps at the certified sizes, and a tube with nothing to spare waits for ever. MEASURED, EXPLORATORY Pushes of 12 to 160 units at 64 dots all left, and the hotter the box the more disordered the sheet it ended with, as predicted from the toy’s equilibrium curve before the run: 0.97 squares per dot measured against 0.95 predicted at the smallest push, 0.81 against 0.81 at the largest. Larger pushes were not run. MEASURED With a single store every tube ends damaged; two stores per dot leave clean sheets with one scrap. Colder boxes than that were not tried.
CHAPTER 10 · WHAT WE HAVE LEARNED, AND WHAT COMES NEXT
A mechanism we can price, and two questions it opens
Is this a theory of where the universe came from?
It is a hypothesis, with one mechanism tested in a toy model and no physicist's review yet. This chapter says what was learned and what would have to come next.
Side by side: the same ground in the technical edition, Sec. VIII, Discussion »
Return one last time to the 96-dot tube. Here is everything this paper asked of it, with the answer and the kind of answer it is.
| The question | The answer, in the toy | Kind |
|---|---|---|
| What is it holding? | 96 units above flat: one per dot at λ = 1.25. | EXACT |
| What does it cost to start? | 12 units, the same at every size certified, from 48 to 288 dots. | EXACT |
| When does it first leave? | When the counted lottery says: over a ninety-fold range of waits in a first scan, and in every cell of a stricter registered test. | MEASURED |
| What takes shape? | Halfway through, 99 dots in 100 belong to one arrangement or the other. | MEASURED |
| What does it turn into? | Finishing clean, a ring six around where it was four: flat at every dot, and still narrow. | MEASURED, EXPLORATORY |
| Where does the energy go? | Finishing flat, it gives off exactly one unit per dot. Sealed, every unit is followed into the stores. | MEASURED |
| What remains? | In a cold box, about one four-dot scrap, 14 units, whatever the size of the tube. | MEASURED |
| Does it hold at other settings? | The fourth map passes from 1.05 to 1.30. Three earlier maps stay inconclusive. | MEASURED |
The rows come from different tests and different ranges of settings; the sealed ones are at λ = 1.25 only.
What the toy shows, and where it stops
OUR READING, UNVERIFIED The toy gives the larger hypothesis a concrete first question: can a prepared curled arrangement wait, escape and release an amount of energy that can be calculated? Under the stated rules, it can. But its ending limits the comparison with the picture that opens this paper: the runs stop at the nearest flat arrangement, the narrow one, and their own trapped heat does not widen it. Two cautions go with the result. The fixed cost is the cost of opening the door, not of committing to the change. And an energy gap between two ideal arrangements is not a test of a thermodynamic first-order transition.
THE AUTHOR'S CLAIM The author’s wider proposal is that an organized arrangement came before space. This paper is a controlled test of part of that proposal in a toy, not a test against observations of the universe. What it hands to the next model: the local energy can be priced exactly, the first exit can be predicted, and flatness dot by dot is not enough to make a shape wide. What did not go the way it might have: the remainder did not grow, a single front was not established, and three registered maps stayed inconclusive before a fourth passed.
What comes next
The question that began the work survives its first test. Can an organized arrangement give way sharply, release energy and leave a remainder? In this small laboratory it can, and the first step can be priced to the unit. Two things were left standing at the edge of what this paper measured, and each has work of its own coming, with its predictions written down first.
The scrap. A scrap is the only thing in the new space that still carries the old arrangement, so the next paper, What the change leaves behind, asks what it takes to keep one. MEASURED One result: the scrap counted here is not permanent. Cool the new sheet slowly enough and it heals away, while another kind of leftover survives every cooling we could run.
Three directions. The tube has one long direction, and space has three, so the paper after that, How curled directions open, runs the same rules with six links per dot. MEASURED One result: escape from the four-link tube does not settle the question for more directions. Later runs find a separate barrier for each curled direction. Some end in wider shapes, but no fixed budget has been shown to open every direction.
Twelve units start an escape. What opens a space?
Continuing into the technical paper
This edition follows the technical paper [11], which holds the equations, tables, uncertainty calculations and full references. Every number above is taken from it or from the project’s record, and the kinds-of-statement tags follow the project’s own rule that no statement is promoted from one kind to another without a dated note saying why. Its sections run in the same order as the chapters here: the toy and its exact results (Chapters 2 and 3), the cost of leaving (4), the first exit (5), the conversion (6), the sealed system and the scrap (7) and the dependence on λ (8).
A few translations for the next step
| In the paper | What to keep in mind |
|---|---|
| “The model” | In the technical paper, our toy: the published model with the knob λ. Only λ = 1 is the published model, combinatorial quantum gravity. |
| N | Number of dots. The running example uses N = 96. |
| Torus | Any grid that wraps both ways: the surface of a doughnut. The flat torus is the sheet (a fat doughnut), the curled torus is the tube (a thin ring), and the smallest one, curled both ways, is the knot. |
| Flat, wide | Flat is about each dot’s surroundings: every link on exactly two squares, at zero energy. Wide is about the whole shape: many steps around both ways. A torus six around is flat and narrow. |
| 4 × L | Four steps around the short wrap and L around the other, L even and above 4. 4 × 4 is the knot. |
| λ and g | Lambda sets the price of a surplus square and so the whole ladder; g is the bath’s warmth, which sets how often an uphill swap is accepted. |
| H = 16(N − S) + 4λX | The price list: S squares, X surplus squares. Zero on the flat sheet; 4(λ − 1) per dot on the tube. |
| ΔE‡ = 32 − 16λ | The rim: the cost of route A, 12 at λ = 1.25. Route B costs 64 − 40λ. |
| d | The square-count signature: pairs of links at a dot that close no square. Sheet 2, tube 1, knot 0. Not a certified geometry. |
| f | The progress meter, 0 for the tube and 1 for the sheet. Disorder raises it too. |
| Arrhenius law | Uphill events becoming much rarer as the bath cools, as e−cost/g. Here its two ingredients are counted, not fitted. |
| Memoryless | The chance of leaving is the same on every attempt, like rolling a die until a six comes up; a long wait so far makes leaving no nearer. |
| Metastable | Stable for now: escaping the tube’s neutral family costs energy. The map’s edge near 1.37 is where the wait becomes too short to see, not where that stops being true. |
| Microcanonical | Sealed: total energy fixed. Inferring temperature from the stores also requires a checked equilibrium distribution. |
| Pre-registered | The prediction and the rules of the test were committed before the run. Exploratory runs say so. |
| 95% interval | A range from a statistical procedure that, under its assumptions, covers the truth 95 times in 100. |
Where the evidence is recorded
The technical manuscript and its methods supplement are in the recorded repository version, and the supplement maps every claim to code, configurations and result files. Experiment labels in that record: T7 and T8 (the decay and the map), T9 (the sealed reservoir), T10 and T11 (the scrap and its position), T22 (exits counted move by move), T38 (the long-wait test). The questions teased above, what the scrap does afterwards, larger networks and more directions, have their own labels in the record and their own papers. The push threshold covers 48 to 192 dots and the exit enumeration 48 to 288; neither extends to other sizes.
Disclosure
The author used Claude (Anthropic) and ChatGPT/Codex (OpenAI) to assist with code, analysis and writing, and is responsible for the result. This edition was drafted with Claude from the technical manuscript and the project’s record. Its analogies and pictures introduce no new experiments or evidence. No physicist has reviewed any of it.
Sources
- C. A. Trugenberger, Combinatorial quantum gravity: geometry from random bits, J. High Energy Phys. 09 (2017) 045, arXiv:1610.05934. Cited as in the technical paper.
- C. Kelly, C. A. Trugenberger, and F. Biancalana, Self-assembly of geometric space from random graphs, Class. Quantum Grav. 36, 125012 (2019), arXiv:1901.09870.
- C. A. Trugenberger, Networks as the fundamental constituents of the universe, J. Phys. Complex. 6, 042001 (2025), arXiv:2512.17676.
- A. Gorsky and O. Valba, Interacting thermofield doubles and critical behavior in random regular graphs, Phys. Rev. D 103, 106013 (2021), arXiv:2101.04072.
- C. Kelly, Emergent geometry and discrete curvature in random graphs, Ph.D. thesis, Heriot-Watt University (2022), https://www.ros.hw.ac.uk/handle/10399/4591.
- C. A. Trugenberger, Dark matter and dark energy in combinatorial quantum gravity, arXiv:2409.09385 (2024).
- M. Creutz, Microcanonical Monte Carlo simulation, Phys. Rev. Lett. 50, 1411 (1983). Cited as in the technical paper; we have not read it in full.
- S. Coleman, Fate of the false vacuum: semiclassical theory, Phys. Rev. D 15, 2929 (1977). Described here from general knowledge of the paper; we have not read it in full.
- H. Jang and G.-W. Chern, Suppressed coarsening after an interaction quench in the Holstein chain, arXiv:2602.05815 (2026). Cited in the technical paper only, for the vocabulary of a neighboring field.
- B. B. Le, S.-H. Lee, and G.-W. Chern, Energy landscape of the kagome antiferromagnet: Characterization of multiple energy scales, arXiv:2604.18902 (2026). Cited in the technical paper only; we have read it in part.
- E. Smith, The curled torus burps: activated escape and conversion in a graph model of emergent geometry, technical manuscript, 14 October 2026, in the project repository.
- Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), arXiv:1807.06209.
- The written search behind this sentence, 10 October 2026: where we looked, with what terms, and what came closest. File
docs/reading/notes/2026-10-10_prior_work_lambda_above_one.mdin the project repository. - C. Kelly and C. A. Trugenberger, Combinatorial quantum gravity: emergence of geometric space from random graphs, arXiv:1811.12905 (2018).
- Planck Collaboration, Planck 2015 results. XVIII. Background geometry and topology of the Universe, Astron. Astrophys. 594, A18 (2016), arXiv:1502.01593. Described here from its abstract; we have not read it in full.
Sources 1 to 10 carry the same numbers as in the technical paper [11]; 11 to 15 are this edition’s own.
Notes
- The published simulations are not identical experiments: the 2019 simulation forbids surplus squares, the later curvature action permits them, and the study with no surplus price does not use the same fixed two-group networks. Our square-density curve differs from a 2025 figure [3] by up to 0.41 over part of its range. Given the different stated protocols, agreement with the 2019 curve does not validate our whole curve, and this reproduction issue remains open. Kelly’s doctoral thesis [5] adds a hint for four links per dot: cooled and then warmed again, the network does not quite retrace its path, though the thesis declines to call the evidence definitive. ↩
The big questions, and what this work can honestly say
What was there before the Big Bang?
Nobody knows, and this work does not settle it. It tests one hypothesis, the author's: that space is one settled arrangement of something deeper, and that before the change there was a different, specific arrangement, curled up and stable for now. A toy model shows that a curled arrangement can give way sharply in a network of dots and links, with a countable push and a countable release. What it turns into there is a slightly wider tube, not a wide space. That is part of a possibility shown in a toy, not evidence about the real universe.
Where did the Big Bang come from?
One hypothesis, tested here only in a toy model: the hot early universe is the energy released when an earlier arrangement gave way and space opened out, the way water gives off heat when it freezes. In the toy the release can be counted exactly and grows with the size of the system, while the push that starts it stays small and fixed at every size tested.
What was there before time?
This paper cannot say. The toy model has no time in it: its clock counts attempted moves, not seconds. In the author's wider hypothesis, which this paper does not test, what came before space had order but not time as we experience it.
Is space made of something?
Possibly. In the published model this work builds on, space is a network of dots and links with no positions at all, and flat space is one arrangement of that network. Whether real space is like that is an open research question.
Is this an accepted theory?
No. It is a hobbyist's hypothesis, tested on a toy model with the predictions written down first, written with AI assistance and not yet reviewed by a physicist. The paper says exactly what was and was not shown.
Quick answers
What is “The curled torus burps” about?
It is a plain-language edition of a computational physics paper by Emily Smith. It asks what a sharp change from one specific arrangement into flat space would look like, and tests each part of that picture in a toy model made of dots and links: what it costs to start, how long the wait is, what takes shape, where the released energy goes, and what remains.
What is combinatorial quantum gravity, and is this it?
Combinatorial quantum gravity is a published model, built by Carlo Trugenberger, Christy Kelly and Fabio Biancalana, in which space is a network with no positions at all and the energy is a count of squares that acts as a curvature. This paper's toy is a close cousin of that model, not that model itself: it turns one knob, the price of a surplus square (λ), above the published value of 1.
What is a curled torus?
A torus is a grid that wraps around in both directions, the surface of a doughnut, and every shape in the paper is one: flat (the sheet, a fat doughnut), curled (the tube, a thin ring) or knot. Curl one direction of the flat one until it is only four steps around and you have a tube. In the toy the tube has one long direction where the flat sheet has two, and it sits 4(λ − 1) units of energy per dot above the flat sheet: one unit per dot at λ = 1.25.
Is flat space the same as endless space?
No. Flat is about each place's surroundings; whether space goes on for ever is a separate question. In the toy every shape closes on itself, like a doughnut, and a count made from inside matches an endless grid's until half the way around, so a doughnut much wider than its inhabitants looks endless to them. Real space is measured to be flat on the largest scales; whether it is endless has not been measured.
What is the burp?
The burp is the energy given off when the tube's tight curl comes undone. For a clean change it is exactly 4(λ − 1) per dot, so it grows with the size of the tube, while the push needed to start it does not.
Does the tube open into a wide space?
Not in this toy as it stands. Read from the saved wiring, the tube ends as a tube six steps around where it was four: flat at every dot, and still narrow. The toy's rule charges for loops of four and for nothing longer, so nothing pays for widening further.
How much energy does it take to start the change?
Twelve units at λ = 1.25: the cost of the cheapest partner swap out of a perfect tube, counted exactly, the same at every size from 48 to 288 dots. In 320 sealed runs, no tube left with a budget below 12 and every tube left with 12 or more.
How long does the tube wait before it starts?
Count the exits a sweep offers, multiply by the chance each is taken, and you predict the mean wait with nothing fitted. Across twelve conditions the measured mean wait divided by the predicted one averaged 0.98. The waiting is memoryless, like rolling a die until a six comes up.
What is left behind after the change?
About one four-dot scrap of the old arrangement per sheet, holding 14 units, whatever the size of the tube, in the sealed runs with the most room. What the scrap does afterwards is the subject of the next paper.
Does this show how the universe began?
No. A toy can show that a mechanism is possible or impossible within a class of models; that is the whole claim. The clock in the simulation is a sampling rule, not physical time, and no physicist has reviewed the work.
Was AI used to write this?
Yes. The author used Claude (Anthropic) and ChatGPT/Codex (OpenAI) to assist with code, analysis and writing, and is responsible for the result. The predictions for the main tests were written down before the runs.
What this page is about
what came before the Big Bang, emergent spacetime, combinatorial quantum gravity, graph model of space, curled torus, metastable state, false vacuum decay, activation energy, first exit time, memoryless waiting time, nucleation and front, sealed system (microcanonical), Creutz reservoir, relic defect, pre-registration, Markov chain Monte Carlo, decompactification.
Sign the guestbook
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How to cite this page
Emily Smith, The curled torus burps: A small push, a large release, and the question of what survives (plain-language edition), 10 October 2026. https://sidenerdapps.com/mainnerd/01thecurledtorusburpsPL/
@misc{smith2026curledtorus,
author = {Smith, Emily},
title = {The curled torus burps: A small push, a large release, and the question of what survives},
year = {2026},
month = oct,
note = {plain-language edition},
url = {https://sidenerdapps.com/mainnerd/01thecurledtorusburpsPL/}
}