Seventeen Open Places of the Net, and Where to Meet
Them
J. Konstapel, Leiden, 21 October 2026.
© 2026 J. Konstapel, Constable Research, Leiden. All rights reserved. Quotation with
attribution is permitted; reproduction of the whole requires written permission
Every age draws a line around what it can know. Whatever falls outside the line is called impossible. Or paranormal. Or unscientific. The name suggests something is wrong with the phenomenon. Usually the trouble lies in the line itself.
The book this essay follows makes that move precise. Modern knowledge holds dozens of commitments without marking them as commitments. Only repeatable things count as known. Only waking consciousness is a knowing state. The human ends at the skin. Every event has a doer. Remove these commitments and the category “impossible” does not get explained. It dissolves.
The book applies the same move to a physical theory: the Vacuum.Net theory. This theory pictures the vacuum as a fishnet under tension. Matter is where the net closes on itself. Several of its claims sound impossible at first hearing. Light that rests. A form whose identity is its order of making. A flow that never stops. Two clocks that agree without a master.
The book does not defend these claims. It places them. For each claim it names the line that makes it look impossible. It states what the theory says, with numbers. Then it shows where a reader can meet the same thing elsewhere: in books, films, works of art, exhibitions, and places that can be visited.
That third step matters most. A claim that looks impossible on paper often has a long life in practice. A calligrapher meets the order of making every day. Christiaan Huygens met agreement without a master in 1665. A water court in Valencia has met it every Thursday for centuries. They do not call it physics. They call it the way the work goes.
The theory in brief
The newest form of the theory starts from one statement: the only thing there is, is light. From this it reaches three rules.
- Crossing, place and step. There is one movement that leaves a place through a crossing. The place is light. The movement is bound to it.
- Return gives birth. What arises returns to its source. From that return the next thing arises.
- Mutual support. Two things that have arisen support each other. From that support the next layer arises.
The rules are older than the theory. They already appear in the oldest form of the character 道 and in the oldest Daoist account of origin.
The scale is fixed by one photon. The unit of length is 45.588 nm, derived from the Rydberg constant of hydrogen. The unit of time is that length divided by c. The unit of energy is 27.197 eV. Length, time and energy of the net are together one quantum of light.
A “register” of trits runs the rules. A trit is +1, 0 or −1. A window of three neighbouring trits closes when its sum is 0 modulo 3. Nine of the 27 possible windows close. A ring of trits is rewritten by one simple rule. A ring stays fixed under that rule exactly when every window in it closes. This is proved.
Every claim in the book carries a status word: proved, computed, source, reading, or open. A reading is not a result. Numbers were recomputed. Where the theory interprets, the book says so.
The seventeen open places
Each of the seventeen claims below sounds impossible because of a line. Each has witnesses.
Light and form
Light is a location. The line: light travels, so light moves. The reading: light stays at its location, and the locations progress. Matter is a location that holds itself against that progression. Dewey Larson stated this in 1959: the photon remains stationary while the reference system progresses. James Turrell’s Skyspaces let a visitor meet it. After a few minutes the sky stops being far away. It rests in the opening like a plane of colour. Three such rooms lie near Leiden and The Hague.
Identity lies in the order of making. The line: a thing is its configuration, and its history is forgotten. But the character 道 has a fixed stroke order. A character written in another order counts as wrong, even if it looks the same. The order belongs to the character. Calligraphers meet this daily. The ensō, a circle painted in one stroke, is the pure case: one stroke, and the stroke is the work.
The form reads itself. The line: every rule needs someone to apply it. The answer: the rewrite rule is not applied from outside. Its fixed points are forms that stay equal under self-reference. Louis Kauffman calls these eigenforms, after Heinz von Foerster. M. C. Escher’s Drawing Hands shows it. Two hands, each drawing the other. No drawer outside the picture.
Movement and number
Random, but bound. The line: staying near a starting point needs a pulling force. But Pólya proved in 1921 that a random walk on a line returns to its start with certainty. No force. No binder. In three dimensions the chance of return drops to about 34 percent. Where the third dimension comes from is the theory’s clearest open question.
One act, three values. The line: three values need at least two acts. But on the ring of three, one act repeated gives all three values. Once is +1. Twice is −1. Three times is 0. The Setun computer, built in Moscow from 1958, counted this way in balanced ternary. Steve Reich’s Clapping Music makes the idea audible: one pattern set against itself, meeting and parting until both return to unison.
The third value. The line: logic has two values, and paradoxes are errors. In 1975 Francisco Varela added a third: the autonomous value of self-reference. How it maps onto the register’s three values is open. Varela was not only a formalist. He practised meditation and founded the Mind & Life Institute. His work joins logic to lived experience.
Return and memory
Arising by return. The line: arising goes forward and outward. But the Taiyi sheng shui, a bamboo-slip text from around 300 BCE found at Guodian, says otherwise. Water returns to the Great One, and heaven is completed. Heaven returns, and earth is completed. Products arise by return and mutual support. The later, familiar counting cosmogony of chapter 42 does not occur in the older text.
The past is in the present shape. The line: the present form of a thing says what it is, not where it came from. But a tree’s rings record every year’s water and warmth. A hedgerow records the people who laid it. On the register, freezing what has closed raises the closed share to a plateau near 0.62 — ordered, but not frozen. Memory nearly doubles closure. It does not finish it.
A resource is a fit. The line: a resource is a stock under the ground someone owns. In the net, a resource is a fit of shape, phase and depth. This is already governed in practice. Every Thursday at noon, the Tribunal de las Aguas meets at the door of Valencia Cathedral. Its judges are working farmers. The procedure is oral. Nothing is written down. It has held for centuries.
The person and the place
The person is in the mesh. The line: an experiment must not depend on who runs it. But the register shows that a held fraction of sites keeps a direction alive. A hold can be a wall. It can also be a hand, a body, a group. The clearest documented parallel is mainstream: Harry Collins showed that laboratories building the TEA laser succeeded only through contact with someone who had built one. The knowledge lived in persons, not in papers.
Constants differ by place. The line: constants are the same everywhere. But clocks at different heights already tick at different rates; a 2010 NIST experiment saw this over 33 centimetres. The theory treats constants as state functions of local tension. At the human scale the claim is practical: choosing the place is choosing what fits. Newgrange proves the point in stone. Sunlight enters its chamber only at the winter solstice.
Care is closing loops. No clinical claim is made here. The reading: a disorder is a loop that does not close, and care sets the state so the closed configuration stays. Witnesses come from practice: Aaron Antonovsky’s “sense of coherence”, Feldenkrais’s loops of action and attention, taijiquan as taught by its lineage holders in Chenjiagou.
What never rests
Perpetual circulation. The line: every motion runs down. But on the register, 99.86 percent of rings of length 7 circulate forever. Rest is the exception, not the rule. This is not perpetual motion: nothing enters, nothing leaves, no work is taken out. The laboratory echo is exact. Currents in superconducting rings have persisted for years with no driver. Hans Haacke’s Condensation Cube shows a sealed cycle as art.
Poised matter. The line: a small cause gives a small effect. But near a tipping point the lever becomes very large. “Poised matter” is a medium held permanently at that point, answering the smallest touch. Per Bak’s sandpile drives itself to its edge and stays there. Nitinol wire takes any form when cold and returns to its first shape in hot water. Nothing yet shows both at once.
The store that shows nothing. The line: a full store shows its contents. But a winding held as a pair with its mirror sums to zero while both exist. Kenneth Snelson’s Needle Tower is the clearest picture: still aluminium tubes, held only by tension, full of opposed forces in balance. Cut one cable and the stored load is released. Lord Kelvin’s water dropper separates balanced charge from neutral water, and can be built at home.
Agreement and selection
Agreement without a master. The line: agreement needs a reference or a conductor. Couple two rings with a counter-weight, and on lengths that are powers of three they become identical. On any other length they stay two. This is proved. Huygens saw the physical case in 1665: two clocks on one beam settled into exact opposition. Fireflies in the Smoky Mountains flash in unison with no leader. Two metronomes on a board resting on two cans do it on a kitchen table.
Matter by selection, and the cut. The line: to make a substance, convert another. In the net an address cannot be pushed. Matter is not converted; it is selected. The method is three steps: soften the medium, impose the mesh, harden again. Roger Hiorns’s Seizure shows it. A flat was flooded with supersaturated copper sulphate solution. As it cooled, only one form remained: blue crystal on every surface. A genuinely new address needs something else: a break in the strand. Rutherford’s 1919 disintegration of nitrogen was such a cut.
How the resources were found
The catalogue behind these meetings was not gathered casually. Predictions were sealed before any search: 85 cells, seventeen places by five kinds of resource. Seven searches ran. Four independent judges re-opened every address and rated the fit. Of 278 candidate resources, none proved invented. Two concert listings were refuted and corrected. About ninety details were fixed. The sealed prediction was right in 68 of 85 cells — 80 percent.
Two rules governed contested material. Accounts of builders such as Keely, Moray and Schauberger are reported as their sources report them, with contrary sources where these exist. Healing practices are described as practices. No clinical claim is made.
The empty cells
Sixty cells were filled. Twenty-five stayed empty. Three patterns stand out.
Mathematics has books but no places. Random-but-bound and one-act-three-values live in papers. No visitor can enter them. The one object that would embody them — a Setun computer or Fowler’s ternary machine — could not be confirmed on display anywhere.
Practice has places but no exhibitions. The water court, the taijiquan class, the solstice at Newgrange: all are met weekly or yearly. No museum shows them.
Three places have almost no resource at all. Where three dimensions come from. Two distant structures held in one state. A builder’s own statement that his device worked for him and not for others. These are not gaps in the search. They are the frontier of the theory.
What it means for people
A theory that sounds impossible often sounds that way because of who is left out of it. The fire age removed the person from the apparatus. That removal made modern technology possible. It also made some knowledge invisible.
The witnesses show that the knowledge is still there. A calligrapher holds the order of making in the hand. A water judge holds centuries of fits in a procedure that is never written down. Such knowledge is fragile. It lives in villages, guilds, temples and lineages. When the village empties, the knowledge goes with it.
The net, if it is right, needs fits held by people who know them. Access then depends on understanding, not on owning the ground above a deposit. Whether that understanding is shared or held is a human choice. The net does not make it for us.
Annotated references
A selection of the strongest sources from the book’s catalogue, with the book’s own reading advice. Fit is the judges’ rating. Inside marks whether the maker met the phenomenon from within.
Light, form and self-reference
Dewey B. Larson. Outline of the Deductive Development of the Theory of the Universe of Motion. Reciprocal System collection, Internet Archive (free). — Statement 17 says the photon “remains stationary in the dimension of motion”. The most direct text for light as a location. Fit: strong.
Lawrence Weschler. Seeing Is Forgetting the Name of the Thing One Sees. — A book-length conversation with Robert Irwin. The clearest account by a maker of what it means to stop seeing light as a carrier of images. Fit: strong · Inside: yes.
Louis H. Kauffman. “EigenForm” (2005). Free on his university pages. — Self-reference as a fixed point: a form that is the result of its own operation. Sections 2 and 3 need little mathematics. Fit: strong.
George Spencer-Brown. Laws of Form (1969). — Begins with one act: draw a distinction. In chapter 11 an expression re-enters its own form and oscillates; time enters the calculus. Fit: strong · Inside: yes.
Douglas Hofstadter. I Am a Strange Loop (2007). — The “I” as a loop that perceives itself, with no observer outside it. Draws on the author’s own life. Fit: strong · Inside: yes.
Chiang Yee. Chinese Calligraphy (1938; 3rd ed. Harvard, 1973). — The classic introduction to rhythm, line and the fixed order of strokes. Written by a practising calligrapher. Fit: strong · Inside: yes.
Taiwanese Ministry of Education stroke-order site. — Animates the standard order of 6,063 characters, including 道. Watching 道 being written is the shortest way into the identity chapter. Free.
Movement, number and the third value
G. Pólya (1921), original paper; Jonathan Novak, “Pólya’s Random Walk Theorem” (arXiv 1301.3916). — Novak gives the proof in a few pages: recurrent in one and two dimensions, transient in three. Fit: strong.
Scientific American. “This mathematician proved a brilliant theorem to justify his social awkwardness.” — The theorem grew out of Pólya’s own repeated walks near Zurich. Look for Kakutani’s line: “a drunk bird may get lost forever.” Fit: strong · Inside: partly.
Peter Doyle and J. Laurie Snell. Random Walks and Electric Networks (1984). Free on arXiv. — The recurrence theorem by asking how easily current flows to infinity. Fit: strong.
N. P. Brusentsov et al. “Development of ternary computers at Moscow State University.” Virtual Computer Museum. — The builders’ own account of the balanced-ternary Setun, and of why production stopped in 1965. Fit: strong · Inside: yes.
Brian Hayes. “Third Base.” American Scientist (Nov–Dec 2001). — The classic readable essay on balanced ternary, the weighing puzzle, and Setun. Fit: strong.
Francisco J. Varela. “A Calculus for Self-Reference” (1975); Principles of Biological Autonomy (1979). — The autonomous value: a third logical value for forms that re-enter themselves. Fit: strong.
Varela, Thompson and Rosch. The Embodied Mind (1991; rev. 2017). — Joins mindfulness practice to cognitive science. Logic studied by people who do it. Fit: strong · Inside: yes.
Return, memory and the commons
Scott Cook. The Bamboo Texts of Guodian (2012). — The complete annotated English translation of the 1993 find, including the Taiyi sheng shui: the oldest known account of arising by return. Fit: strong.
Sarah Allan. “The Great One, Water, and the Laozi: New Light from Guodian.” T’oung Pao (2003). — The central scholarly reading of the water cosmogony itself. Fit: strong.
Oliver Rackham. The History of the Countryside (1986). — Fields, hedgerows and woods as the archive of the people, climate and soils that shaped them. The landscape is the record. Fit: strong · Inside: yes.
Valerie Trouet. Tree Story (2020). — World history read from tree rings, by a working dendrochronologist. The Hohenheim oak-pine chronology spans 12,460 years. Fit: strong · Inside: yes.
Lynne Kelly. The Memory Code (2016). — Ancient monuments read as memory devices for oral cultures. The author uses the techniques herself. Fit: strong · Inside: yes.
Elinor Ostrom. Governing the Commons (1990). — Nobel-recognized study of commons that lasted for centuries without privatisation or state takeover. Fit: strong · Inside: partly.
J. Stephen Lansing. Priests and Programmers (1991). — Bali’s water temples schedule irrigation across whole watersheds, with no designer above. Fit: strong.
The person, the place and care
Harry Collins. “The TEA Set: Tacit Knowledge and Scientific Networks.” — Laboratories built the TEA laser only after contact with someone who had built one. Knowledge lives in persons, not papers. Fit: strong.
Michael Polanyi. Personal Knowledge (1958). — All scientific knowing rests on personal skill and judgment, much of it tacit. Written by a working chemist. Fit: strong · Inside: yes.
Richard Sennett. The Craftsman (2008). — How craft knowledge lives in the maker’s body and practice. Fit: strong · Inside: yes.
Aaron Antonovsky. Unraveling the Mystery of Health (1987). — Salutogenesis: why some people stay well under great strain. The study began with survivors of the concentration camps. Fit: strong.
Christian Norberg-Schulz. Genius Loci (1979). — The basic work on why a place has its own character, which building must answer. Fit: strong.
What never rests
Yoseph Imry. “Tireless electrons in mesoscopic gold rings.” Physics (2009). — Persistent currents in ordinary, non-superconducting gold rings, measured in 2009. What returns in phase does not stop. Fit: strong.
Arthur Ord-Hume. Perpetual Motion: The History of an Obsession (1977). — The standard history. Separates the dream of work from nothing from the plain fact of a self-sustaining flow. Fit: strong.
Per Bak. How Nature Works (1996). — Self-organised criticality: a sandpile drives itself to its tipping point and stays there. The closest known thing to poised matter. Fit: strong.
Frank Close. The Void (2007). — Empty space as full of pairs of particles that appear and vanish. By a working particle physicist. The only book found on a pair that sums to nothing. Fit: strong · Inside: yes.
Agreement and selection
M. Bennett et al. “Huygens’s clocks” (2002). — A rebuilt 1665 set-up. Lock happens only in a narrow window: near-equal frequencies, real coupling, heavy cases. Huygens used about 40 kg of lead. Fit: strong.
Steven Strogatz. Sync (2003). — The standard account of locking without a conductor: fireflies, clocks, heart cells. Fit: strong · Inside: yes.
James Pantaleone. “Synchronization of metronomes.” American Journal of Physics (2002). — The recipe for agreement on a kitchen table: two metronomes, one board, two drink cans. Fit: strong.
Lawrence Principe. The Secrets of Alchemy (2012). — Alchemical recipes read as coded laboratory procedures, rerun by the author at the bench. Fit: strong · Inside: yes.
Roger Hiorns. Seizure (2008). Yorkshire Sculpture Park. — A flat flooded with supersaturated copper sulphate; only the crystal remained. Selection made visible. Visitable in timed slots from 24 October to 31 December 2026. Fit: strong.
Stuart Kauffman. Investigations (2000). — The “adjacent possible”: what can appear next is one step from what exists. The map of the net can only grow by being walked. Fit: strong.
What the Lectures Leave Out: One Rule, Three States, and the Vacuum as Net
An essay on J. Konstapel’s “What the Lectures Leave Out” (Constable Research, Leiden, 2026)
Introduction
Physics curricula share a quiet habit: the vacuum is taught as a background, a stage with fixed constants on which particles and fields perform. Mathematics curricula share another: modular arithmetic, random walks, and coupled oscillators live in separate chapters with separate exams. In the four-page paper “What the Lectures Leave Out,” J. Konstapel (2026) argues that both habits hide a single structure. His Vacuum.Net theory treats the vacuum as the only thing there is, and everything else as a pattern within it that closes on itself. The paper’s claim is not that lectures are wrong, but that they are fragmentary: results proven in one discipline reappear, unrecognised, as selection rules in another. This essay summarises the theory’s central picture, examines how it unifies three “chapters” of mathematics into one rule, considers its most striking physical and historical claims, and weighs what the text leaves for the reader to do.
The Picture: A Fishnet Under Tension
The theory begins with an image rather than an equation: a fishnet under tension, in which a single strand crosses itself. At each crossing the strand goes over, under, or does not cross at all — recorded as +1, −1 and 0. One crossing is a trit; the register of the theory is balanced ternary. From this picture Konstapel derives five starting rules, of which the most heterodox is N4: the response of the medium is a state function. Constants such as c and h are not inputs but readings — a constant that appears fixed is simply a state that does not change on the scale at which it is measured. This is the paper’s titular point. No lecture makes it, because the standard frame must assume its constants before it can compute anything at all.
One Rule from Three Chapters
The paper’s most persuasive move is to show that the closure rule of the net is a rule the reader already knows. Three crossings in a row form a “window”; a window closes when its trits sum to a multiple of three. Of the 27 possible windows, exactly 9 close: three uniform ones, (+1,+1,+1), (0,0,0), (-1,-1,-1), and six mixed ones, the permutations of (-1,0,+1). In \mathbb{Z}_3, three elements sum to zero exactly when they are all equal or all different. This is the rule of the card game SET, where a valid triple must be all-equal or all-different on every attribute — an affine line in \mathbb{Z}_3^n. A combinatorial curiosity taught in recreational mathematics is thus reframed as a selection rule for what persists in a physical medium. The 9 closing windows split into two kinds with distinct behaviours: uniform closures hold without turning (“the form of a crystal”), while mixed closures keep turning while staying closed (“the form of living matter”). Where the two kinds meet, the window on the seam stays open and carries tension.
The same rule regenerates the theory’s states from a single act. Since \mathbb{Z}_3 is cyclic, giving one act the value +1 yields +1, then 1+1 \equiv -1, then 1+1+1 \equiv 0. The mirror is not a second thing but the same act taken twice; and there are two roads to zero — an act meeting its mirror, or the same act going round three times — matching the two kinds of closure.
The third chapter is probability. A movement from a point, stepping +1, 0 or −1 with equal chance, returns to its start with certainty, by Pólya’s (1921) theorem on the recurrence of one-dimensional random walks. Exact returns are counted by the central trinomial coefficients, 1, 1, 3, 7, 19, \ldots; counted modulo 3 the number rises from 7 to 9 of 27, and the two extra strings, (+1,+1,+1) and (-1,-1,-1), “do not come back along the line. They go once round a ring of three.” The difference between coming back and going round separates two kinds of binding — and, with Watson’s (1939) return probability of about 0.3405 in three dimensions, seeds one of the paper’s open problems.
From Oscillators to Units to Guodian
Closure reappears a third time as phase locking: a winding that closes on itself does so in phase, which is Huygens’ observation of 1665 in the language of the Kuramoto (1984) model. The paper’s inversion is characteristic: locking is not one phenomenon among others but the persistence rule itself — what does not lock does not stay.
More ambitious is the unit derivation. The net fixes length, time and energy from the reduced-mass Rydberg constant, so that one photon of wavelength S_0 = 1/(2R_H) = 45.588 nm carries all three. Each step deeper multiplies by three, and the resulting ladder lands, by identity rather than fit, on the Balmer Hδ line at 9 S_0; by Wien’s law near the thermal peak of about 262 K at 243 S_0; and within 0.52 % of the lowest ideal mode of the Earth–ionosphere cavity at 3^{31} S_0. The paper is careful about what is computed, what is an identity, and what is a 0.52 % coincidence — but the pattern is intriguing precisely because it links scales no single lecture connects.
The most unusual section places the structure beside the Taiyi sheng shui (“The Great One gave birth to water”), a bamboo-slip text from about 300 BCE found at Guodian in 1993. The text describes arising by return and mutual support, from which Konstapel condenses three rules — a movement bound to the light it leaves through, return as the engine of birth, and mutual support building the ladder — that replace the five starting rules. Whether or not one accepts the correspondence, the methodological point stands: no physics course places a 2,300-year-old cosmology next to Pólya’s theorem, yet here they can be read as descriptions of one structure.
Open Problems as an Invitation
The paper closes with six problems that are explicitly “not exercises.” They range from algebra (what quantity records the order of a closing window, since (+1,-1) and (-1,+1) close differently?) through dimension (if all binding is one-dimensional, where do three dimensions come from?), to a striking physical question: if c is a state function, which measurement would show its local variation first, and at which depth? The cap-set problem of Ellenberg and Gijswijt (2017) becomes a question about the largest tension-carrying configuration of the net, and the Kuramoto model a question about locking on a balanced-ternary lattice. Each is sized, the author claims, for a bachelor or master project — an unusual and welcome gesture in a foundational text.
Assessment
The paper’s virtues are compression and connection. In four pages it links additive combinatorics, random walks, synchronisation, atomic spectroscopy, and Warring States cosmology under one ternary rule, and it is honest about status: each claim is labelled computed, proved, or reading. Its risks are equally clear. The identifications — SET with a selection rule, Guodian with the net, the 9-with-9 of windows and cosmological products — are readings rather than derivations, and the 0.52 % cavity agreement is suggestive, not conclusive. But the text does not pretend otherwise; it ends by handing the reader six problems and a blog. What the lectures leave out, on this showing, is not a missing proof but a missing picture — and the paper’s contribution is to supply the picture and let the reader test it.
Annotated References
Konstapel, J. (2026). “What the Lectures Leave Out: One Rule, Three States, and an Open Problem Set from the Vacuum.Net Theory.” Constable Research, Leiden, v1.0, 21 October 2026.
Primary source. The four-page paper this essay is based on: the fishnet picture, the five rules N1–N5, the window register, the depth ladder of units, the Guodian reading, and six open problems. Read sections 2–3 first; they carry the whole argument.
Konstapel, J. (2026). The Vacuum.Net Theory: Foundational Paper, ed. 8.1. Constable Research, Leiden.
The full theory behind the paper: the five starting rules, the window register and the unit derivation. Reading advice: Part 0 first, then Parts III.4 and VII.2. Essential for depth; the four-page text is a distillation of it.
Konstapel, J. (2026). “The Way Before the Net.” constable.blog, 20 October 2026.
The companion piece that rewrites the five rules into three (R1–R3), develops the trit walk, and presents the Guodian material. Read together with sections 4, 5 and 8 of the primary source.
Davis, B. L. & Maclagan, D. (2003). “The card game SET.” The Mathematical Intelligencer 25(3), 33–40.
SET as affine geometry over \mathbb{Z}_3: the cleanest entry to the paper’s section 3, showing why a “SET” is an affine line and why the all-equal-or-all-different rule is the zero-sum rule of \mathbb{Z}_3.
Ellenberg, J. S. & Gijswijt, D. (2017). “On large subsets of \mathbb{F}_q^n with no three-term arithmetic progression.” Annals of Mathematics 185(1), 339–343.
The polynomial-method bound on cap sets, the current best upper bound on SET-free collections. In the paper’s terms, a cap is a configuration where no window closes and everything carries tension — the starting point for open problem 4. Four pages; read it whole.
Pólya, G. (1921). “Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz.” Mathematische Annalen 84, 149–160.
The classical proof that random walks are recurrent in one and two dimensions and transient in three. The mathematical ground of the paper’s section 5 and of open problem 2 on where three dimensions come from.
Watson, G. N. (1939). “Three triple integrals.” Quarterly Journal of Mathematics 10, 266–276.
The evaluation of the return probability of the three-dimensional walk (≈ 0.3405). Supplies the number that makes the dimension problem sharp: binding is certain in one dimension, but far from certain in three.
OEIS A002426. Central trinomial coefficients.
The sequence 1, 1, 3, 7, 19, 51, 141, … counting exact returns of the trit walk, with asymptotics. The bridge between the walk’s combinatorics and the paper’s counts of 7 versus 9 returning strings.
Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence. Springer, Berlin.
The origin of the Kuramoto model of coupled phase oscillators: natural frequencies, coupling strength K, critical coupling K_c, and the order parameter r. The model behind the paper’s reading of closure as phase locking.
Strogatz, S. H. (2000). “From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators.” Physica D 143, 1–20.
The best review of the locking transition and the order parameter, connecting the original model to later developments. Reading advice: sections 1–3 suffice for the paper’s purposes.
Knuth, D. E. (1997). The Art of Computer Programming, Vol. 2, 3rd ed. Addison-Wesley.
Balanced ternary and its symmetries (pp. 195–213): why +1/0/−1 is a natural register, the round-off-free arithmetic the net’s trit register relies on.
Larson, D. B. (1959). The Structure of the Physical Universe. North Pacific Publishers, Portland.
Light as location and matter as rotation against the progression — a precursor physical intuition for the paper’s rule R1, in which a movement is bound to the place it leaves through.
Allan, S. & Williams, C. (eds.) (2000). The Guodian Laozi. Society for the Study of Early China, Berkeley.
The first wide English treatment of the Taiyi sheng shui, the bamboo-slip cosmology from Guodian (unearthed 1993, c. 300 BCE) that supplies the paper’s section 8 and open problem 3: arising by return, and mutual support as the builder of layers.
Williamson, J. G. & van der Mark, M. B. (1997). “Is the electron a photon with toroidal topology?” Annales de la Fondation Louis de Broglie 22(2), 133–160.
Matter as light closed on itself in a worked physical model — a bridge between the paper’s section 7 (units from hydrogen) and the closure rule that governs everything else in the net.