J.Konstapel,Leiden, 8-11-2026.

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Dit artikel onderzoekt of één eenvoudige regel op elke schaal in de werkelijkheid terugkomt, met knopentheorie en ternaire getallen als testterrein.

De nieuwe knopeninvariant Θ blijkt oneindige families van knopen tot één formule te comprimeren, maar groeit door aanbouw, niet door zelfgelijkvormigheid.

Θ heeft blinde vlekken: de theorie wil richting en spiegeling vastleggen, maar Θ ziet richting niet en geeft alleen bij spiegeling een tekenwisseling.

De echte schaalloze structuur zit niet in Θ, maar in ternaire getallen, met de Lo Shu als meetbaar voorbeeld waarin dezelfde sluitingsregel op elk trit-niveau geldt.

Dit levert een exacte entropielimiet op (≈1,1681) die eerder alleen numeriek bekend was.

De wiskunde is solide, maar de koppeling met de natuurkunde (Hopf-lading versus knoopvorm) blijft een open, goed gedefinieerde vraag.

Can one simple rule describe the world at every size? From particles to galaxies? This question sits at the heart of the Vacuum.Net theory. The original paper tested that idea with mathematics. It asked two things. First, can knot theory turn infinitely many knots into a few rules? Second, do those rules repeat at every scale, like a fractal?

The answer is mixed, and that makes it interesting. A repeating rule does exist, and it can be measured exactly. But it lives in ternary counting, not in the new knot invariant Θ. This essay explains the findings in plain language. Every claim in the original work was either recomputed from scratch or checked against a second source.

What was tested

The starting point was the invariant Θ, defined by Bar-Natan and van der Veen. An invariant is a quantity that stays the same for a knot, however you bend it. Θ consists of two parts. The first is the well-known Alexander polynomial Δ. The second is a new two-variable polynomial θ.

The author rebuilt Θ independently, with computer code. The rebuild reproduced the original paper exactly. It also reproduced every number in an earlier essay, The Knot That Rings. Four small errors in that essay were found and corrected. For example, the phrase “6,758 pairs undistinguished” should read “a deficit of 6,758”, because the paper counts knots, not pairs.

The rebuilt Θ was also stress-tested. It gave the correct answer for the same knot drawn in different ways. It separated 97.84% of all 313,230 prime knots with up to 15 crossings. In short: the tool works.

Result 1: Infinite families become one formula

Knots come in infinite families. The surprise is that θ of many family members follows from a single formula.

For torus knots — knots wound on the surface of a doughnut shape — θ matches a formula published by Ohtsuki in 2004. This was checked coefficient by coefficient, on knots up to T(3,17). Every coefficient matched, with no scaling factor.

A second family was discovered by the author. This is the family 5₂, 7₃, 9₃, and so on. These knots are not torus knots. Their θ also has an exact form, built from concentric hexagonal rings. The formula was first read off a few small cases. Then it was used to predict two larger cases in advance. Both predictions came out exactly right. No published formula for this family was found.

There is a pattern here. For both families, the inner part never changes. Each larger member only adds a ring at the outer edge. So an infinite family is a fixed core plus a growing rim, described by one integer.

Result 2: What Θ cannot see

Θ is powerful, but it has blind spots. Three properties of the Vacuum.Net theory are not visible to it.

Closure. The theory’s closure rule matches Fox three-colouring. This colouring is carried by Kauffman’s bracket polynomial and by the Jones polynomial at a special value. It is not carried by θ. So the theory’s closure rule lives elsewhere in knot theory.

Direction. The theory says the first passage fixes a direction of travel. Knot theory is largely blind to this. Reversing a knot leaves θ unchanged, as it does the classical polynomials. The smallest non-invertible knot, 8₁₇, even has θ = 0.

Mirror image. Here the news is better. A knot’s mirror image gives exactly −θ. Knots that equal their own mirror give θ = 0. This matches the theory’s rule that reversal of an address is negation and zero is rest. But this is agreement, not proof. Knot theory predicts it on its own.

The conclusion is practical. An address that serves the theory needs three parts: closure, handedness, and direction. Ordinary polynomial invariants supply only the first two.

Result 3: Two meanings of “knot”

Physics and knot theory use the word “knot” differently. In physics, a knotted field is counted by the Hopf charge Q: one integer, measuring how many times two field lines wind around each other. The Vacuum.Net theory uses exactly this quantity. Knot theory instead studies the shape of a single closed curve.

The two meanings meet in torus knots. Physicists have built light fields whose field lines are torus knots. In the Skyrme–Faddeev model, many torus knots appear as stable shapes. And the one formula that covers all torus knots is the one that describes these physical shapes. Both the theory’s address (Q = p·q) and the invariant Θ can be read from the same two winding numbers.

This is a correspondence, not a demonstrated link. Q does not determine the knot type. Θ does not determine Q. But it makes torus knots the natural meeting point.

Result 4: The fractal is found — in the Lo Shu

The knot families grow at the rim. They are not fractals. But a true fractal structure was found, in an unexpected place: the Lo Shu magic square.

The Lo Shu is the ancient 3×3 square in which every row, column, and diagonal sums to 15. Subtract its centre value and something remarkable appears. Every line sums to zero. Each entry splits into two ternary digits (“trits”). Each trit level obeys the theory’s closure rule: all three different symbols in every line. And all moments up to order 2 vanish.

The square can be grown to orders 9, 27, 81, and beyond. The closure rule holds on every trit level, at every scale tested. The spectrum of the resulting square has a closed form, derived by the author. From it follows an exact entropy limit: H∞ ≈ 1.1681. This explains a number — “about 1.168” — that Loly and Cameron had found numerically but could not derive.

This is the paper’s clearest result. Self-similarity in a measurable quantity exists, exactly computable. But it sits in ternary addressing, not in the knot invariant θ.

What it means for the theory

Six points follow. None of them confirms the theory; they say where the tested results touch it.

1. The theory’s closure rule has a home in knot theory, in the Alexander determinant and the Jones polynomial.
2. The theory’s mirror-sign rule matches θ. This is consistency, not support.
3. Direction is a real gap. If the theory derives direction from knotting, it should produce an invariant that detects the difference in knot 8₁₇. That would be a genuine, testable contribution.
4. The theory’s one-integer address is coarser than the shape. One integer cannot separate 313,230 knot types. Address and shape are complementary.
5. The fractal is in the ternary addressing, shown exactly by the Lo Shu family.
6. Zero-sum ideas — Wheeler’s “boundary of a boundary is zero”, Rowlands’ nilpotency — all point the same way. Whether the universe obeys them, or our descriptions do, remains open.

What remains open

One published formula could not be confirmed: Ohtsuki’s cabling formula did not reproduce θ for cables of the trefoil, despite two dozen attempts with different conventions. This is left open, and nothing else depends on it.

Four questions only the theory can answer: Is direction part of the address? Does “not crossing” mean no crossing or a virtual crossing? Does “knot” mean the Hopf charge or the curve’s shape? Is Kauffman’s bracket the theory’s natural instrument? Four more can be settled by further computation, from proving the ring formula to comparing stable and unstable knotted solitons.

Conclusion

The paper set out to test whether one rule holds at every scale. It found a precise answer with a precise location. Knot invariants compress infinite families into single formulas, but they grow by accretion, not by self-similarity. The true scale-free structure lives in ternary arithmetic, in the Lo Shu and its compound squares, where one closure rule holds on every digit level at every scale — and where the entropy converges to an exact number. The mathematics is solid. The link to physics remains a well-defined open question.

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Annotated references

Knot theory and invariants

• Bar-Natan, D. & van der Veen, R., A Fast, Strong, Topologically Meaningful and Fun Knot Invariant (arXiv 2509.18456). Annotation: The paper that defines Θ. Its trefoil example, mirror-sign conjecture, and knot counts were reproduced exactly by the author’s independent implementation.
• Ohtsuki, T., A cabling formula for the 2-loop polynomial of knots (Publ. RIMS 40, 2004). Annotation: Source of the closed torus-knot formula that θ matches coefficient by coefficient. The published cabling formula could not be reproduced for cables of the trefoil; this discrepancy is left open.
• Kauffman, L., Virtual Knot Theory (1999). Annotation: Introduces the virtual crossing (“not really there”) and the bracket polynomial. The bracket carries the theory’s closure rule (three-colouring) and matches its crossing alphabet more closely than Θ.
• Przytycki, J., 3-coloring and other elementary invariants of knots; Lickorish & Millett (1986). Annotation: Cited for the formula linking three-colourings to the Jones polynomial at e^{iπ/3}. Verified numerically on 12 knots.
• Knot Atlas (entries 7₃, 9₃); OEIS A002863. Annotation: Used to confirm braid words, Alexander polynomials, and the count of prime knots up to 15 crossings (313,230).

Physics

• Cho, Y., Oh, P. & Zhang, H., Knots in Physics (arXiv 1803.08952, 2018). Annotation: Defines physical knots by Hopf charge Q ∈ π₃(S²) = ℤ — the same quantity the Vacuum.Net theory uses as its address.
• Arrayás, M. & Trueba, J., Torus-knotted electromagnetic fields (2011). Annotation: Shows light fields whose field lines are torus knots, connecting the theory’s Hopf charge to concrete knot shapes.
• Sutcliffe, P., Knots in the Skyrme–Faddeev model (2007). Annotation: Computes which knot shapes are stable up to Hopf charge 16; many torus knots appear, motivating the comparison of Θ for stable versus unstable solitons.
• Rowlands, P., Are there alternatives to our present theories of physical reality? Annotation: Source of the nilpotency principle (“square root of 0”) and the “zero totality”, placed by the Ninth Edition under its closure axiom.
• Does General Relativity Highlight Necessary Connections in Nature? (on Wheeler and Misner–Thorne–Wheeler, ch. 15–17). Annotation: Source for ∂∂ = 0, “the boundary of a boundary is zero”, read as the geometric form of the theory’s loop rule N1.

Magic squares

• Loly, P. & Cameron, A., Frierson’s 1907 Parameterization of Compound Magic Squares Extended to Orders 3^l (arXiv 2008.11020). Annotation: Builds compound squares by Kronecker products, reports rank 2l+1 and entropy “about 1.168”. The author derived this limit exactly: H∞ ≈ 1.1681.
• Loly, P., The Invariance of the Moment of Inertia of Magic Squares (Mathematical Gazette, 2004). Annotation: Shows the moment of inertia is arrangement-independent; the value 60 for the Lo Shu was confirmed by computation.
• Navas, L., Khajuraho’s magic square is an hypercube (arXiv 2608.06740); Wikipedia: Lo Shu Square, Magic square, Constructions of magic squares. Annotation: Background on the Lo Shu, its uniqueness among 3×3 magic squares, and historical compound constructions going back to Yang Hui.

The theory

• The Vacuum.Net Theory, Ninth Edition (Foundations 9.2 / Physics 9.0, 27 October 2026). Annotation: The primary source. Its closure (Fox colouring, T4/T19), negation (T12.1), direction (N3), and scale (N5) rules were compared one by one with knot-theoretic results.
• The History of the Derivation as It Happened. Annotation: Establishes that the trit, closure, and orientation were meant to follow from the knotting (programme of 9 July 2026); two references in the earlier essay were corrected against this history.
• The Knot That Rings (earlier essay) and Exploring Knot Theory: Kauffman’s Vision and Applications (constable.blog, 13 January 2026). Annotation: The first relates Θ to the theory; four of its numerical statements were corrected. The second provides background on Kauffman’s programme.

Filling the early universe with knots can explain why the world is three-dimensional

One Rule at Every Scale: Knots, Ternary Closure and the Lo Shu

DOI: 10.13140/RG.2.2.17102.42568