J. Konstapel, Leiden, 3 October 2026.
© 2026 J. Konstapel, Constable Research, Leiden. Quotation with attribution is permitted; reproduction of the whole requires written permission.
Dit essay past de Vacuum.Net-theorie toe op het klimaat. DHet toont bouwt een schaal met lagen van atomaire tijd en lengte tot aan het waarneembare heelal, en plaatst daaronder klimaat, water, zon en galactisch centrum.
Zo wordt weer een gevolg van grond en water, en CO₂ een uitlezing van het stralingsvenster, niet de knop.
Met een driedelige “+1, 0, –1”-register bouwt dit stuk een vaste ladder van lengten en tijden.
Klimaat, getijden, El Niño, zonnevlekken en de precessie van de aardas blijken op specifieke treden te liggen, in drie families.
Daaruit volgt: het weer is afgeleid van grond en water, CO₂ hoort bij het stralingslaagje, en de theorie is toetsbaar met openbare reeksen.
Abstract
The Vacuum.Net theory writes every place in the net as a depth plus a number in balanced ternary. This essay takes that register at its word and follows it through to its consequences. It derives the full list of layers from the trit, from depth 0 to depth 306. It gives each layer a length and a time from the natural units. It then names what is found at each layer, from a virus at depth 0 to the observable universe at depth 71, so that every rung of the ladder carries a measured quantity beside it and the distance to the layer can be read off at a glance.
The climate is then placed on that list. In space it occupies five consecutive layers, 22 to 26, the shell of the Earth from the ground where people live to the edge of the atmosphere. Its radiation sits on layer 5, the window through which the Earth loses its heat. Its rhythms occupy the layers 43 to 58, from the tide to the precession of the Earth’s axis, a stretch of the ladder that runs in length from the solar system to the centre of the galaxy.
The closure rule of the net is then applied to neighbouring layers. One result follows in three lines: in a closed chain every third layer carries the same state. The layers therefore fall into three families, A, B and C. Two are free. The third follows: $C = -(A+B)$. The weather layer is the one that follows. The ground, the orbit, the inner core, the Sun and the centre of the Milky Way all fall in one family; water falls in the second. Carbon dioxide, on this reading, is not the cause of the state of the weather layer but a reading of the state of the radiation layer, which belongs to the family of water.
The essay ends with three measurements on public data that count how often the closure windows close, and so make the whole chain testable against numbers that anyone can download. Every step is written out with its numbers. No reference needs to be opened to follow it.
1. The image and the question
The image is a fishnet in water. Its strands run everywhere and are always under tension. Where strands wind around each other, knots appear. Between strands lie meshes. A winding persists only if it returns in phase inside its mesh. What does not fit does not stay. This is the whole picture: a net under tension, windings that must close, and meshes that decide what remains.
The climate debate, by contrast, reads one number. The concentration of carbon dioxide is followed month by month, and from its rise a story is told about where the climate is going. The debate says little about where the climate sits in the whole: on which scales it lives, next to which phenomena, under which rule. A single number without a place is a reading without a register.
The question of this essay is plain. On which layers of the net does the climate lie, and what does the net’s own rule say about those layers? The register exists — the theory writes every place in the net as a depth plus a number in balanced ternary — so the question can be answered exactly, layer by layer, with measured values.
The question came with a second one. The cycles of the climate seem to run from the core of the Earth to the black hole at the centre of the galaxy: the tide, the year, El Niño, the sunspot cycle, the precession of the Earth’s axis. Is there a relation between those cycles and the trit, the three-valued state of the net? The essay answers both questions with one list.
2. The register and the rule
The theory rests on five axioms.
- N1, Strand. There is one strand. Every distinction is the strand touching itself.
- N2, Closure. A configuration persists only if it closes on itself.
- N3, Memory. The net keeps the orientation of what was first laid down.
- N4, Context. Constants are state functions of the local vacuum.
- N5, Scale. The same winding appears at every depth.
From N1 and N3 follows a three-valued state at every crossing: the strand passes over, passes under, or does not cross. The three values are written $+1$, $0$, $-1$. This is the trit, the unit of information of the net, as the bit is the unit of the binary register. An address in the net is a depth plus a number in balanced ternary — a number written with the digits $+1$, $0$ and $-1$, in which every integer has exactly one representation. Every integer therefore has exactly one address, and every address names exactly one place.
The closure rule works on a window of three neighbours. The window is closed when the three trits add to $0$, counted modulo 3. Of the 27 possible triples, 9 are closed: six hold all three values (the permutations of $+1, 0, -1$) and three hold the same value three times ($+1}1{1$, $0}0{0$, $-1}1{1$). The rule is local and strict: with two neighbours fixed, exactly one value closes the window. There is no choice, and there is no second-best.
The natural units are those of Larson. The unit of time is one half-period of the Rydberg frequency,
$$T_0 = 1.521 \times 10^{-16}\ \text{s},$$
and the unit of space is that time multiplied by the speed of light,
$$S_0 = 45.6\ \text{nm}.$$
Both units come from two of the best-measured constants in physics — the Rydberg constant and the speed of light — and so the ladder that is built on them is anchored at a point that does not move.
3. The layers, derived
Nine steps lead from the trit to the list. Each step is small, and each brings its own numbers.
- Register. One trit per crossing: $+1$, $0$, $-1$.
- Layer. An address is a depth plus a number. Each depth $n$ is one layer. A layer holds $n$ trits.
- States. Layer $n$ has $3^n$ states. Layer 5 has 243.
- Range. The largest number at depth $n$ is $(3^n – 1)/2$. For $n = 1$ to 6 this gives 1, 4, 13, 40, 121, 364. A larger number forces the next depth; the register does not overflow, it climbs.
- Addresses. An address is a period without a starting point. Rotations of one string are one address. The count is the necklace number$$g(n) = \frac{1}{n} \sum_{d \mid n} \phi(n/d)\cdot 3^{d},$$
summed over the divisors $d$ of $n$, with $\phi$ the Euler totient. For $n = 1$ to 6 this gives 3, 6, 11, 24, 51, 130. - Closed window. Three neighbours close when their sum is $0$ modulo 3.
- Closed ring. Section 3.1 derives which rings close completely.
- Scale. N5 gives a factor 3 per layer. Layer $n$ has the time $3^n \cdot T_0$ and the length $3^n \cdot S_0$.
- Closure depths. Section 3.2 derives where the ladder of three meets the ladder of two.
3.1 Which rings close completely
Take a ring of $n$ trits and ask that every window of the ring is closed. Two neighbouring windows give two equations:
$$T(i-1) + T(i) + T(i+1) = 0 \quad\text{and}\quad T(i) + T(i+1) + T(i+2) = 0.$$
Subtract the first from the second. Everything in the middle cancels, and what remains is
$$T(i+2) = T(i-1).$$
Every trit equals the trit three places back. A fully closed ring has period 3. This three-line derivation is the heart of the essay; everything later depends on it.
Two cases follow. If $n$ is divisible by 3, the period-3 structure fits the ring: the first two trits are free and the third is fixed by them, and that pattern repeats around the ring. That gives $3 \times 3 = 9$ closed rings. If $n$ is not divisible by 3, period 3 and period $n$ together force period 1 — the two rhythms can only agree by standing still — and only the three uniform rings remain: all $+1$, all $0$, all $-1$.
So a ring can close with all three values present only at depths divisible by 3. The count was checked by listing every ring for $n = 3$ to 9. It gives 9, 3, 3, 9, 3, 3, 9 — the pattern of the rule, seen in a brute enumeration.
3.2 Where three meets two
The address ladder triples. The dimensional ladder of the theory doubles: 1, 2, 4, 8. The two never coincide, because $\log_2 3 = 1.58496\ldots$ is irrational — no power of 3 is ever a power of 2. But they come close, where $3^q \approx 2^p$, and those places of closest approach are the convergents of the continued fraction of $\log_2 3$: 2/1, 3/2, 8/5, 19/12, 65/41, 84/53, 485/306.
The denominators of the convergents are the closure depths. The quotient $3^q / 2^p$ is the residue, the part that does not close — the small remainder that the best available compromise still leaves over.
| Depth $q$ | $3^q \approx 2^p$ | Residue | Time $3^q \cdot T_0$ | Length $3^q \cdot S_0$ |
|---|---|---|---|---|
| 1 | $3^1 \approx 2^2$ | 0.75000 | 0.46 fs | 136.8 nm |
| 2 | $3^2 \approx 2^3$ | 1.12500 | 1.37 fs | 410.4 nm |
| 5 | $3^5 \approx 2^8$ | 0.94922 | 37.0 fs | 11.08 µm |
| 12 | $3^{12} \approx 2^{19}$ | 1.01364 | 80.8 ps | 24.23 mm |
| 41 | $3^{41} \approx 2^{65}$ | 0.98860 | 92.4 min | 11.12 AU |
| 53 | $3^{53} \approx 2^{84}$ | 1.00209 | 93.4 y | 93.4 ly |
| 306 | $3^{306} \approx 2^{485}$ | 0.99898 | $4.8 \times 10^{122}$ y | $4.8 \times 10^{122}$ ly |
At depth 12 the residue is 1.01364. That is the Pythagorean comma: the excess of twelve perfect fifths over seven octaves, known to music for two and a half thousand years, standing here at a depth of the net. Two of the seven closure depths are divisible by 3: depth 12 and depth 306. At those two depths both kinds of closure coincide — the ring of period 3 and the approach of the two ladders.
4. The layers, named
Each layer has a length and a time. The next step is to write beside each layer what is found at that size and what lasts that long. Appendix A gives the full table for the layers 0 to 71. Each name carries its measured value, so that the distance to the layer can be read off. A selection:
| $n$ | Length | What has that size | Time | What lasts that long |
|---|---|---|---|---|
| 0 | 45.6 nm | Small virus | 0.15 fs | Orbit of the electron in hydrogen |
| 2 | 410 nm | Violet light; hydrogen line Hδ | 1.4 fs | Oscillation of visible light |
| 5 | 11.1 µm | Heat radiation of the Earth; body cell | 37 fs | Bending vibration of molecules |
| 16 | 1.96 m | The human body | 6.5 ns | – |
| 24 | 12.9 km | Height of the weather layer | 43 µs | – |
| 28 | 1,043 km | Inner core of the Earth | 3.5 ms | – |
| 30 | 9,389 km | The Earth | 31 ms | Fast brain rhythm |
| 33 | 253,493 km | Orbit of the Moon | 0.85 s | Heartbeat |
| 34 | 760,480 km | The Sun | 2.5 s | One breath |
| 41 | 11.1 AU | Orbit of Saturn | 92 min | Sleep cycle |
| 51 | 10.4 ly | Nearest stars | 10.4 y | Sunspot cycle |
| 53 | 93.4 ly | Nearest star cluster | 93.4 y | Long solar cycle; a human life |
| 58 | 22,700 ly | Centre of the Milky Way | 22,700 y | Precession of the Earth’s axis |
| 70 | 12.1 billion ly | Hubble radius | 12.1 billion y | Age of the universe |
| 71 | 36.2 billion ly | Observable universe | 36.2 billion y | – |
Two things stand out at once. The human body sits at depth 16 in length, and its rhythms sit at depths 30 to 34 and 41 in time: the fast brain rhythm at 30, the alpha rhythm at 31, the reaction at 32, the heartbeat at 33, the breath at 34, the sleep cycle at 41. A person is one rung of the ladder, and the whole of the body’s timing is a rung of the ladder too. And the ladder itself is finite in what it can name: it reaches the observable universe at depth 71.
The layers 72 to 306 carry no known phenomenon. They lie beyond what has been observed. Closure depth 306 lies in that range — the last and best meeting of the two ladders stands in a part of the register that no measurement has yet entered.
One identity is exact, and it deserves to be written out. The hydrogen line Hδ, the jump from level 6 to level 2, has the wavelength $9/(2R)$, with $R$ the Rydberg constant. Since $S_0 = 1/(2R)$, that is $3^2 \cdot S_0$. It follows from the arithmetic of the levels alone:
$$\frac{1}{4} – \frac{1}{36} = \frac{8}{36} = \frac{2}{9},$$
and $9/(2R) = 3^2/(2R)$. Closure depth 2 is a spectral line. Of all the correspondences in the appendix, this one is not a rounding: it is an identity, and it fixes the bottom of the ladder to the spectrum of hydrogen.
5. Where the climate sits
The position of a phenomenon on the ladder is $\log_3$ of its size over $S_0$, or of its duration over $T_0$. The nearest whole number is its layer. The climate is placed on the ladder in three ways: in space, in radiation, and in time.
5.1 In space
| Phenomenon | Size | Position | Layer |
|---|---|---|---|
| Lowest air layer, the ground where people live | 1.5 km | 22.04 | 22 |
| Mean depth of the ocean | 3.7 km | 22.86 | 23 |
| Top of the weather layer | 12 km | 23.94 | 24 |
| Continental crust; ozone layer | 35 km | 24.91 | 25 |
| Edge of the atmosphere | 100 km | 25.87 | 26 |
The climate is the shell of the Earth, and it occupies five consecutive layers. The Earth itself lies at layer 30, its outer core at 29, its inner core at 28. The weather — the part of the climate that moves — is a single layer: layer 24, the twelve kilometres between the ground and the top of the weather layer.
5.2 In radiation
Layer 5 has the length 11.08 µm. That is the centre of the window through which the Earth loses its heat, 8 to 13 µm. The ozone band at 9.6 µm lies at position 4.87. The carbon-dioxide band at 15 µm lies at position 5.28. Both belong to layer 5. Layer 5 is a closure depth: the window of the Earth’s heat radiation sits where the ladder of three meets the ladder of two.
5.3 In time
| Phenomenon | Duration | Position | Layer |
|---|---|---|---|
| Tide | 12.42 h | 42.90 | 43 |
| Weather system | 5 d | 44.96 | 45 |
| Spring tide | 14.77 d | 45.95 | 46 |
| Tropical oscillation | 45 d | 46.96 | 47 |
| Solar rhythm | 154 d | 48.08 | 48 |
| Year | 365.24 d | 48.87 | 49 |
| El Niño | 3.5 y | 50.01 | 50 |
| Sunspot cycle | 11.0 y | 51.05 | 51 |
| Orbit of Saturn | 29.46 y | 51.95 | 52 |
| Long solar cycle | 88 y | 52.95 | 53 |
| Solar cycle of 208 y | 208 y | 53.73 | 54 |
| Solar cycle of 1,000 y | 1,000 y | 55.16 | 55 |
| Solar cycle of 2,300 y | 2,300 y | 55.92 | 56 |
| Precession | 25,772 y | 58.12 | 58 |
In space the climate lies on layers 22 to 26. In time it lies on layers 43 to 58. Those are two different stretches of the ladder, and the second is the answer to the second question of Section 1. The layers 43 to 58 have, in length, the size of the solar system up to the centre of the galaxy. Layer 58 holds both the precession of the Earth’s axis and the distance to the galactic centre. The rhythms of the climate do run from the Earth to the central black hole — not as a metaphor, but as a fact of the register: the climate’s slowest rhythm and the galaxy’s centre share one rung of one ladder.
6. The closure step: three families
Now the closure rule is applied to the layers themselves. Each layer carries one trit, its state. Three neighbouring layers form a window. The derivation is that of Section 3.1, read along the ladder instead of around a ring.
- Window $n$ closes: $T(n-1) + T(n) + T(n+1) = 0$.
- Window $n+1$ closes: $T(n) + T(n+1) + T(n+2) = 0$.
- Subtract: $T(n+2) = T(n-1)$.
In a closed chain every third layer carries the same state. The layers therefore fall into three families, by their number modulo 3.
- Family A: $n = 1$ modulo 3.
- Family B: $n = 2$ modulo 3.
- Family C: $n = 0$ modulo 3.
Two families are free. The third follows: $C = -(A + B)$. Whatever the two free families carry, the third has no vote of its own.
6.1 The climate in space
| Layer | What lies there | Family |
|---|---|---|
| 22 | Lowest air layer, the ground | A |
| 23 | Water, the depth of the ocean | B |
| 24 | Weather layer | C |
| 25 | Crust, ozone layer | A |
| 26 | Edge of the atmosphere | B |
The weather layer is the dependent one: $T(24) = -(T(22) + T(23))$. The state of the weather follows from the state of the ground and the state of the water. In the shell of the Earth, the moving part is the part that follows.
6.2 The climate in time
| Family | Layers | What lies there |
|---|---|---|
| A | 43, 46, 49, 52, 55, 58 | Tide, spring tide, year, Saturn, 1,000-year cycle, precession |
| B | 44, 47, 50, 53, 56 | Tropical oscillation, El Niño, long solar cycle, 2,300-year cycle |
| C | 45, 48, 51, 54, 57 | Weather system, 154-day solar rhythm, sunspots and Jupiter, 208-year cycle |
The division comes from the layer number alone — nothing else was used to sort them. Yet the families sort themselves. Family A turns out to hold orbit and axis: the tide, the year, the orbit of Saturn, the precession. Family B holds the slow swings of ocean and Sun: El Niño, the long solar cycle, the cycle of 2,300 years. Family C holds weather and sunspots: the five-day weather system, the 154-day solar rhythm, the eleven-year sunspot cycle. Again the weather is in the family that follows.
6.3 The three centres
| Centre | Size | Position | Layer | Family |
|---|---|---|---|---|
| Inner core of the Earth | 1,220 km | 28.14 | 28 | A |
| The Sun | 696,340 km | 33.92 | 34 | A |
| Centre of the Milky Way | 26,670 ly | 58.15 | 58 | A |
The three centres fall in one family. They share it with the ground at layer 22 and with the year and the precession. The outer core at layer 29 falls in B, with the water at layer 23. The Earth as a whole at layer 30 and the orbit of the Moon at layer 33 fall in C. The radiation layer 5 falls in B — the family of water, as Section 7 will use.
7. What the families say
The weather is not a control. It is the layer that follows. In the closed chain it has no freedom of its own. It is what remains when the two other families have their state. Any account that treats the weather layer as the driver has the arrow pointing the wrong way: the driver cannot be the layer whose state is fixed by the two beside it.
Two things set it, and one is within reach. Family A is orbit, axis, core, Sun and galactic centre. That state is given; no village can move the orbit of Saturn. Family B is water: the ocean, El Niño, the slow swings. At the scale of a village that is ditch, water level, soil and groundwater. There the state can be set. Restoring balance is a matter of water. This is not a preference, and not a policy argument read into the numbers. It is the place of water in the chain: layer 23, family B, one of the two free states.
Sun and weather correspond. Sunspots (layer 51) and weather systems (layer 45) lie in the same family and carry the same state. Neither sets the other. It is the same winding at two depths — one register entry, read at two rungs of the ladder.
The centres and the ground belong together. Inner core, Sun, galactic centre and the ground on which people live lie in one family. The core is the beginning, not a detail.
Carbon dioxide is a reading, not a lever. Its band at 15 µm lies inside the radiation layer, beside the ozone band at 9.6 µm, and layer 5 falls in family B, the family of water. Water vapour sets the short-wave edge of that window, below 8 µm. The concentration of carbon dioxide is then a reading of the state of that layer — the number the debate watches is the output of a gauge, not the position of a handle. That is the sharpest conclusion of the essay, and it is stated here as the register states it; Section 10 gives the measurements by which it can be tested.
8. The half-step family
One group of cycles does not lie on a layer. It lies exactly between two. The positions below use the current value of the Rydberg constant, $T_0 = 1.51983 \times 10^{-16}$ s.
| Cycle | Duration | Position |
|---|---|---|
| Day | 86,400 s | 43.4999 |
| Rotation of Venus | 243.0226 d | 48.5000 |
| Beat of the two lunar months | 2,190.35 d | 50.5013 |
| Saros | 6,585.32 d | 51.5033 |
| Triple Saros | 19,755.96 d | 52.5033 |
Among themselves these five lie whole powers of 3 apart: $3^0$, $3^5$, $3^7$, $3^8$ and $3^9$ days. The residues against the day are 0.009% for Venus, 0.15% for the lunar beat and 0.37% for the Saros. As a group they lie one half layer, a factor $\sqrt{3}$, off the ladder that starts at $T_0$. The day lies 0.013% from $3^{43.5} \cdot T_0$.
The lunar beat follows from two lunar cycles. The line of apsides turns in 8.85 years, the line of nodes in 18.61 years. Their beat is
$$\frac{1}{8.85} + \frac{1}{18.61} = 0.1667\ \text{per year},$$
a period of 6.00 years. The six-year oscillation in the length of day, read as a signal of the core, has a measured period of 5.9 years. Core and Moon share one clock.
The ladder as derived does not yet give the half step. It is recorded here as measured: five cycles, all at $n + \tfrac{1}{2}$, all whole powers of 3 apart. A theory that derives the half step from the axioms would turn this table from a curiosity into a prediction; that work is open.
9. The human layer
Layer 22 is where people live: the ground and the first kilometre of air above it. Layer 23 is the water beneath and beside them. A farmer, a water board and a village can set the state of layer 23 on their own land. They can screen a ditch, raise a water level, keep water on the soil. The two free states of the weather window are ground and water, and the second of them is small enough to hold.
Peat shows what that does. Drained peat releases carbon and sinks. Raising the water table by 10 cm lowers the emission by about 3 tonnes of carbon dioxide per hectare per year (Evans and others, 2021). The same act holds the water, keeps the soil, and cools the air above the field in summer. People who live there see it within a season. They see the ground stop sinking.
The chain says the same in its own terms. The weather layer follows from ground and water: $T(24) = -(T(22) + T(23))$. Whoever sets the water sets one of the two free states. The essay began with the observation that the climate debate reads one number; it ends at the place where a person with a shovel can set the number that counts.
10. Three measurements
The closure rule can be counted. Each layer gets a measured series. Each month becomes a trit: $+1$ when the series rises, $0$ when it turns, $-1$ when it falls. Per month the three neighbouring trits are added. The count is how often the sum is $0$ modulo 3. By chance that is one third. The comparison is the same series, shuffled in time — a closure count above the shuffled count is the signature of the chain. Claude Code runs all three.
Measurement 1, the time chain. Layer 49 is the seasonal course of temperature. Layer 50 is the El Niño index of the Pacific. Layer 51 is the monthly sunspot number. Claude Code counts the closed months of the window (49, 50, 51) since 1870. If the three families are real, this window should close more often than chance gives.
Measurement 2, ground, water and weather at one place. Layer 22 is the temperature just above the ground. Layer 23 is the groundwater level in a monitoring well. Layer 24 is rainfall or air pressure at the nearest station. Claude Code counts how often weather $= -$ (ground $+$ water), for one well in a peat-meadow area. This is the measurement that people on the spot can follow themselves — a water board can read its own ditch in the register.
Measurement 3, the same family. Layers three apart carry the same state. Claude Code compares El Niño (50) with the long solar cycle (53), and sunspots (51) with the solar rhythm of 154 days (48). Same family, same state, different depth: the third measurement tests the identity of the winding across the ladder.
11. Status
- Taken from the sources. The five axioms, the trit, the unique address, the closure rule on a window, the natural units, the closure depths.
- Derived. The list of layers with states, range, addresses, time and length. The period-3 theorem for closed rings and the count of 9 or 3. The three families and the rule $C = -(A + B)$. The identity $H\delta = 3^2 \cdot S_0$.
- Corresponded. The name beside each layer. A phenomenon is placed on the nearest layer; its measured value stands beside it. The membership of the three centres, of ground, water and weather, and of the climate rhythms follows from that placement alone.
- Checked against measurement. The positions of day, Venus, lunar beat and Saros at $n + \tfrac{1}{2}$. The lunar beat of 6.00 years against the 5.9 years of the core.
- Open. The half step of $\sqrt{3}$. The states of the layers today, and with them which windows are open. The three measurements of Section 10. The layers 72 to 306.
The boundary is drawn sharply on purpose. What is taken, what is derived, what is corresponded and what is checked are not mixed; the reader can see at every step which kind of step is being taken.
12. Conclusion
The essay asked two questions: on which layers of the net does the climate lie, and is there a relation between the climate’s cycles and the trit. Both are answered by one list.
The climate lies on five layers in space, 22 to 26, from the ground to the edge of the atmosphere; on one layer in radiation, layer 5, the closure depth of the Earth’s heat window; and on sixteen layers in time, 43 to 58, from the tide to the precession, a stretch whose lengths run from the solar system to the galactic centre.
The closure rule then does the rest. Every third layer carries the same state; the layers fall into three families; two are free and the third follows. The weather layer is the third. Ground, core, Sun and galactic centre fall in one family; water falls in the other; and the water is the free state that is within reach of the people who live on the ground. The carbon-dioxide reading belongs to the radiation layer, in the family of water — a gauge, not a lever.
One number was never going to be enough, because a number is not a place. The register gives the place, the rule gives the relation, and three countable measurements on public data give the test. What does not fit does not stay.
Appendix A. The layers 0 to 71
A star marks a closure depth. A dash means that no phenomenon of that duration is known.
| $n$ | Family | Length | What has that size | Time | What lasts that long |
|---|---|---|---|---|---|
| 0 | C | 45.6 nm | Small virus (hepatitis B, 42 nm) | 0.15 fs | Orbit of the electron in hydrogen (0.152 fs) |
| 1* | A | 136.8 nm | Far ultraviolet, stopped by oxygen (130–175 nm); large virus (120 nm) | 0.46 fs | Oscillation of far-ultraviolet light |
| 2* | B | 410.4 nm | Violet light; hydrogen line Hδ (410.3 nm); smallest bacterium (300 nm) | 1.37 fs | Oscillation of visible light |
| 3 | C | 1.23 µm | Near infrared; bacterium (1–2 µm) | 4.11 fs | Oscillation of near-infrared light |
| 4 | A | 3.69 µm | Infrared window 3–4 µm; cell | 12.32 fs | Stretch vibration of molecules (O–H) |
| 5* | B | 11.08 µm | Heat radiation of the Earth (window 8–13 µm; ozone 9.6 µm, CO₂ 15 µm) | 36.96 fs | Bending vibration of molecules (ozone 32 fs, CO₂ 50 fs) |
| 6 | C | 33.2 µm | Far infrared; pollen grain (30 µm) | 110.8 fs | Vibration of hydrogen bonds in water (about 170 fs) |
| 7 | A | 99.7 µm | Terahertz radiation; human egg cell (100–120 µm); width of a hair | 332.64 fs | Rotation of small molecules |
| 8 | B | 299.2 µm | Submillimetre radiation; dust mite | 997.92 fs | Lifetime of a hydrogen bond in water |
| 9 | C | 897.5 µm | Background radiation of the universe (peak 1.06 mm); grain of sand | 2.99 ps | Rearrangement of water |
| 10 | A | 2.69 mm | Millimetre waves; head of an ant | 8.98 ps | Relaxation of water (8.3 ps) |
| 11 | B | 8.07 mm | Microwaves 37 GHz; raindrop (2–5 mm) | 26.94 ps | – |
| 12* | C | 24.23 mm | Microwaves 12.4 GHz; width of a thumb | 80.83 ps | – |
| 13 | A | 72.7 mm | Microwaves 4.1 GHz; hen’s egg (60 mm) | 242.5 ps | – |
| 14 | B | 218.1 mm | Hydrogen line of 21.1 cm; human | 727.4 ps | Oscillation of the hydrogen line (0.704 ns) |
| 15 | C | 654.3 mm | Radio waves 458 MHz; arm length | 2.18 ns | Afterglow of molecules (1–10 ns) |
| 16 | A | 1.96 m | Height of the human body (1.6–2.0 m) | 6.55 ns | – |
| 17 | B | 5.89 m | Giraffe (5.5 m); a room | 19.64 ns | Lifetime of the pion (26 ns) |
| 18 | C | 17.67 m | Large tree; blue whale (25 m) | 58.93 ns | – |
| 19 | A | 53.0 m | Short-wave radio; tall tree (50 m) | 176.7 ns | – |
| 20 | B | 159 m | Tallest trees (115 m); Great Pyramid | 530.3 ns | – |
| 21 | C | 476.9 m | Medium-wave radio; a hill | 1.59 µs | Lifetime of the muon (2.2 µs) |
| 22 | A | 1.43 km | Lowest air layer, cloud base (1–2 km) | 4.77 µs | – |
| 23 | B | 4.29 km | Mean depth of the ocean (3.7 km); a large wood | 14.32 µs | Fastest folding of a protein |
| 24 | C | 12.9 km | Height of the weather layer | 42.96 µs | – |
| 25 | A | 38.6 km | Ozone layer (20–50 km); continental crust (35 km) | 128.87 µs | – |
| 26 | B | 115.9 km | Edge of the atmosphere (100 km); lower ionosphere | 386.62 µs | – |
| 27 | C | 347.7 km | Upper ionosphere (250–400 km) | 1.16 ms | Nerve pulse (about 1 ms) |
| 28 | A | 1,043 km | Inner core of the Earth (radius 1,220 km) | 3.49 ms | – |
| 29 | B | 3,129 km | Outer core of the Earth (radius 3,480 km); mantle (2,890 km thick); Moon | 10.46 ms | – |
| 30 | C | 9,389 km | The Earth (radius 6,371 km, diameter 12,742 km) | 31.3 ms | Fast brain rhythm (30–40 Hz); Crab pulsar (33 ms) |
| 31 | A | 28,166 km | Circumference of the Earth (40,075 km); radiation belts | 93.9 ms | Alpha rhythm of the brain (8–12 Hz); ground tone of the cavity between the Earth and the ionosphere |
| 32 | B | 84,498 km | Magnetic field of the Earth on the day side (64,000 km); Jupiter | 281.8 ms | Reaction time; eye movement |
| 33 | C | 253,493 km | Orbit of the Moon (384,400 km) | 845.5 ms | Heartbeat (71 per minute) |
| 34 | A | 760,480 km | The Sun (radius 696,340 km) | 2.54 s | One breath (3–5 s) |
| 35 | B | 0.02 AU | Two to three solar diameters | 7.61 s | Ocean swell (7–10 s); blood-pressure wave |
| 36 | C | 0.05 AU | Corona of the Sun (10–20 solar radii) | 22.83 s | Long earthquake waves (20 s) |
| 37 | A | 0.14 AU | Inner solar wind | 1.14 min | Pulsations of the Earth’s magnetic field |
| 38 | B | 0.41 AU | Orbit of Mercury (0.39 AU) | 3.4 min | Three-minute oscillation of the Sun |
| 39 | C | 1.24 AU | Orbits of Earth and Mars (1.0 and 1.5 AU) | 10.3 min | Granulation of the solar surface (8–10 min); light from Sun to Earth (8.3 min) |
| 40 | A | 3.71 AU | Asteroid belt to Jupiter (2.1–5.2 AU) | 30.8 min | Innermost orbit around the central black hole (32.6 min); tsunami (10–60 min) |
| 41* | B | 11.12 AU | Orbit of Saturn (9.6 AU) | 1.54 h | Human sleep cycle (90 min); orbit just above the Earth (84–93 min) |
| 42 | C | 33.35 AU | Neptune and the Kuiper belt (30–50 AU) | 4.62 h | – |
| 43 | A | 100 AU | Edge of the solar wind (85–120 AU) | 13.87 h | Half a day; tide (12.42 h) |
| 44 | B | 300 AU | Scattered disc (several hundred AU) | 1.7 d | Orbit of Io (1.77 d) |
| 45 | C | 901 AU | Farthest point of Sedna (about 940 AU) | 5.2 d | Five-day wave of the atmosphere |
| 46 | A | 2,702 AU | Inner Oort cloud | 15.6 d | Half a month; spring tide (14.8 d) |
| 47 | B | 0.1 ly | Oort cloud | 46.8 d | Tropical oscillation of 30–60 days |
| 48 | C | 0.4 ly | Outer Oort cloud | 140.4 d | Solar rhythm of 154 days |
| 49 | A | 1.2 ly | Edge of the Sun’s reach (1–2 ly) | 1.2 y | Year (365.24 d); wobble of the Earth’s axis |
| 50 | B | 3.5 ly | Distance to the nearest star (4.25 ly) | 3.5 y | El Niño (2–7 y) |
| 51 | C | 10.4 ly | Nearest stars (Sirius 8.6 ly) | 10.4 y | Sunspot cycle (11 y); orbit of Jupiter (11.9 y) |
| 52 | A | 31.1 ly | Nearby stars (Vega 25 ly, Arcturus 37 ly) | 31.1 y | Orbit of Saturn (29.5 y); a generation |
| 53* | B | 93.4 ly | Nearest star cluster (Hyades, 153 ly) | 93.4 y | Long solar cycle (80–100 y); a human life |
| 54 | C | 280.3 ly | Local Bubble in the gas between the stars (radius 300–500 ly) | 280.3 y | Orbit of Pluto (248 y); solar cycle of 208 y |
| 55 | A | 840.8 ly | Nearest star-forming regions (Orion, 1,244 ly) | 840.8 y | Solar cycle of about 1,000 y |
| 56 | B | 2,500 ly | Thickness of the disc of the Milky Way (2,000–2,600 ly) | 2,500 y | Solar cycle of 2,300–2,500 y |
| 57 | C | 7,600 ly | Distance to the next spiral arm (6,400 ly) | 7,600 y | – |
| 58 | A | 22,700 ly | Distance to the centre of the Milky Way (26,670 ly) | 22,700 y | Precession of the Earth’s axis (25,772 y; climate 19,000–23,000 y) |
| 59 | B | 68,100 ly | Disc of the Milky Way (radius 50,000 ly) | 68,100 y | – |
| 60 | C | 204,300 ly | Magellanic Clouds (160,000–200,000 ly) | 204,300 y | Reversal of the Earth’s magnetic field (mean 200,000–300,000 y) |
| 61 | A | 613,000 ly | Halo of the Milky Way | 613,000 y | Shape of the Earth’s orbit (405,000 y) |
| 62 | B | 1.84 million ly | Distance to Andromeda (2.5 million ly) | 1.84 million y | Ice-age period (2.6 million y) |
| 63 | C | 5.5 million ly | Local Group (radius 5 million ly) | 5.5 million y | – |
| 64 | A | 16.6 million ly | Nearest groups of galaxies (12 million ly) | 16.5 million y | – |
| 65 | B | 49.7 million ly | Distance to the Virgo cluster (54 million ly) | 49.6 million y | Passage of the Sun through the plane of the Milky Way (30–45 million y) |
| 66 | C | 149 million ly | Local supercluster (110 million ly) | 148.9 million y | In-and-out swing of the Sun in the disc (150–170 million y) |
| 67 | A | 446.9 million ly | Laniakea (520 million ly); fixed scale in the distribution of galaxies (490 million ly) | 446.8 million y | Cycle of the supercontinents (300–500 million y) |
| 68 | B | 1.34 billion ly | Great Wall of galaxies (1.37 billion ly) | 1.34 billion y | – |
| 69 | C | 4.02 billion ly | Largest groups of quasars (4 billion ly) | 4.02 billion y | Age of Earth and Sun (4.6 billion y) |
| 70 | A | 12.07 billion ly | Hubble radius (14.4 billion ly) | 12.06 billion y | Age of the universe (13.8 billion y) |
| 71 | B | 36.20 billion ly | Radius of the observable universe (46.5 billion ly) | 36.19 billion y | – |
Appendix B. Symbols and numbers
| Symbol | Meaning | Value |
|---|---|---|
| $T_0$ | natural unit of time | $1.521 \times 10^{-16}$ s |
| $S_0$ | natural unit of space | 45.6 nm |
| $n$ | depth, the number of the layer | 0 to 306 |
| $3^n$ | states at depth $n$ | – |
| $(3^n – 1)/2$ | range at depth $n$ | 1, 4, 13, 40, 121, 364, … |
| $g(n)$ | addresses at depth $n$ | 3, 6, 11, 24, 51, 130, … |
| $T(n)$ | state of layer $n$ | +1, 0, −1 |
| A, B, C | families $n = 1, 2, 0$ modulo 3 | $C = -(A + B)$ |
| $q$ | closure depth | 1, 2, 5, 12, 41, 53, 306 |
Annotated references
The list below gives, for each source, what it is, why it is read, and where it enters this essay. The first four are the papers of the theory itself; the rest are the measured anchors.
Konstapel, J. (2026). The Vacuum.Net Theory: Complete Derivation, from Axioms to the Two Roads of Energy. Constable Research, Leiden, 28 September 2026.
Why read? This is the source of the five axioms (N1–N5), the register, the ladder strand–winding–knot–mesh, and the natural units $T_0$ and $S_0$ used throughout Sections 2 and 3. Every length and time in the essay is $3^n$ times these units. Reading advice: Sections 2, 3 and 5.
Konstapel, J. (2026). The Vacuum.Net Theory: Foundational Paper, sixth edition. Constable Research, Leiden.
Why read? Theorem T7, the unique address — every integer has exactly one depth and one balanced ternary number — with its proof, and the reason why the register has three values and not two or ten. The uniqueness is what makes the ladder a ladder and not a fog: each phenomenon has one nearest layer. Reading advice: Part II.
Konstapel, J. (2026). Closure Without Alignment. Constable Research, Leiden, 23 September 2026.
Why read? The closure rule on a window of three trits, the enumeration of the 9 closed triples out of 27, and the measure of closure against a shuffled series. Section 10 of this essay uses exactly that measure: a closure count is only meaningful next to the same series shuffled in time. Reading advice: Theorems 1 to 4.
Konstapel, J. (2026). The Current State of the Vacuum.Net Theory. Constable Research, Leiden, September 2026.
Why read? The two ladders — the address ladder of three and the dimensional ladder of two — the closure depths on the convergents of $\log_2 3$, and the comma as the residue that never closes. Section 3.2 of this essay stands on that ground. Reading advice: the part on the thread of three.
Larson, D. B. (1959). The Structure of the Physical Universe. Portland: North Pacific Publishers.
Why read? The natural units used here are Larson’s: the unit of time as half the period of the Rydberg frequency, and the unit of space as that time multiplied by the speed of light. Whatever one makes of the rest of that book, these two units anchor the whole ladder at two of the sharpest constants known. Reading advice: Chapter 13.
Knuth, D. E. (1997). The Art of Computer Programming, Vol. 2, §4.1. Reading: Addison-Wesley.
Why read? Balanced ternary: the uniqueness of the representation and the range $(3^n-1)/2$ at depth $n$. Knuth also notes why balanced ternary was once called the most economical notation; the essay needs only uniqueness and range, both proved here. Reading advice: a few pages; the examples are worth doing by hand.
Hardy, G. H., and Wright, E. M. (1938). An Introduction to the Theory of Numbers. Oxford: Clarendon Press.
Why read? Continued fractions, and why their convergents are the best possible approximations. This is the ground under the closure depths: since $\log_2 3$ is irrational, the depths are exactly the denominators of its convergents, and the residues in Section 3.2 are what those best approximations still leave over. Reading advice: Chapter X.
Fox, R. H. (1970). “Metacyclic invariants of knots and links.” Canadian Journal of Mathematics 22.
Why read? The colouring condition at a crossing of a knot diagram — three colours, and at each crossing either one colour or all three — is the closure rule of the net in its original home. The rule of Section 2 is Fox’s condition read as arithmetic: three values, and a window closes when they sum to zero modulo 3. Reading advice: the definitions suffice.
Margot, J.-L., and others (2021). “Spin state and moment of inertia of Venus.” Nature Astronomy 5, 676–683.
Why read? The measured rotation of Venus, 243.0226 days. It is the sharpest entry of Section 8: a whole planet’s spin sitting at position 48.5000, a half step off the ladder, to four decimal places. If the half-step family is an artefact, this is the number that would have to be an accident.
Meeus, J. (1991). Astronomical Algorithms. Richmond: Willmann-Bell.
Why read? The lunar months, the Saros, and the periods of the line of apsides (8.85 years) and the line of nodes (18.61 years) used in Section 8. The beat of 6.00 years that the essay derives from those two lines — and matches to the 5.9-year signal of the core — is computed with Meeus’s periods. Reading advice: the chapters on the Moon and on eclipses.
Holme, R., and de Viron, O. (2013). “Characterization and implications of intradecadal variations in length of day.” Nature 499, 202–204.
Why read? The oscillation of 5.9 years in the length of day, read as a signal of the Earth’s core. The essay’s lunar beat of 6.00 years stands against it: core and Moon on one clock, to within the width of the measurement. Reading advice: the figure with the filtered series.
GRAVITY Collaboration (2019). “A geometric distance measurement to the Galactic center black hole with 0.3% uncertainty.” Astronomy & Astrophysics 625, L10.
Why read? The distance to the centre of the Milky Way: 8,178 parsec, or 26,670 light years, with an uncertainty of three parts in a thousand. This is the number that places the galactic centre on layer 58, beside the precession of the Earth’s axis — the coincidence that closes the second question of Section 1.
Evans, C. D., and others (2021). “Overriding water table control on managed peatland greenhouse gas emissions.” Nature 593, 548–552.
Why read? The measured relation between water table and emission in peat: about 3 tonnes of carbon dioxide per hectare per year for every 10 cm of water table. This is the number that makes the human layer of Section 9 a practical matter rather than a picture. Reading advice: the main figure; it is the human layer in one graph.
SILSO, Royal Observatory of Belgium. Sunspot Number, monthly mean total, from 1749.
Why read? The observed series for layer 51, used in Measurement 1 (the window 49–50–51) and Measurement 3 (sunspots against the 154-day rhythm). It is the longest continuous solar series there is, and it is public.
NOAA Physical Sciences Laboratory. Niño 3.4 sea-surface temperature index, monthly, from 1870.
Why read? The observed series for layer 50, used in Measurement 1 and in the comparison of Measurement 3 (El Niño against the long solar cycle of layer 53). Together with the SILSO series it carries the time chain of the essay.