Theory

Logical identities and laws presuppose no space, position, direction, distance or neighbourhood. Before emergence, logical things may be themselves, but they do not yet exist relative to one another.

Two independently witnessed loops are joined by one contact. The contact is shared by identity, irreducibly joint, essential to both loops and persistent under admitted maps, and erasing either loop destroys it as a contact. One loop alone forms no topology. One complete knot is the first atomic logical topology, and its single touch is minimal contact-distance: the smallest unit of relative existence, with no magnitude, direction, coordinate or metric. Paths, separation and extended distance come later, from networks of contacts.

Read a system of constraints as the set of joint states it allows. Witness variables are quantified away, and no executor, scheduler or checker supplies an input. Each loop contributes an open distinction, and erasing a loop forgets it. A relation over the loops' distinctions and one contact is a knot when it is constrained as a whole (K1), forgetting any one variable leaves the rest unconstrained (K2), and some state is allowed (K3). John Cullen adopted this test, with erasure read as forgetting, on 19 October 2026.

Two separate attachments A–c and B–c make c a common hub, not one touch between A and B. In (¬A ∨ c) ∧ (¬B ∨ c), forget B and A still forces c, which fails K2. Shared identity alone never makes a contact.

No 2-CNF, with any number of witnesses, defines a knot, while three-literal clauses with named intermediate results define every knot. So strict 3-CNF is the smallest clause language that can state a knot. This concerns what can be expressed, not how hard anything is to compute. The core is classical (Schaefer 1978; Baker and Pixley 1975), and the reading is ours.

In an infinite system an open distinction may come from a finite loop or only from the system's ends. A knot is native at a solution when each of its variables is flipped there by some change of finitely many variables, and only native knots carry emergence. Every claim about an infinite system records its K1–K3 verdict and the source of each distinction. Finite systems are unaffected.

A second reading sits beside the existential one: witnesses may carry signs, and a state is allowed when its signed witness sum is nonzero. The existential reading stays canonical for K1–K3, and every result records both verdicts.

Every internally coherent admissible base is treated as co-real, and nothing selects one base or branch as uniquely real. Co-reality is not simultaneity, and no contact between different bases is inferred. The mathematics is compatible with this reading but does not prove it.

Two loops and a contact have exactly 34 knots, in five shapes, from the single clause ¬A ∨ ¬B ∨ c, the atom, to parity. Two-literal laws hide nothing from their pairs, a knot hides at most one bit, and only parity hides a full bit. A network of knots with hidden witnesses hides at most one bit per independent cycle, and only parity fills every cycle.

A system's shape is the complex of sets of distinctions it leaves free. A hole is a gap in that shape, and a knot is the smallest kind of hole. Two-place rules can leave holes, but never a knot. Every simplicial complex is the shape of some system, so logic selects no space.

"A and B require c" becomes "c requires A or B" when false is read as "exists". Of the 34 two-loop knots, 16 are a pure requirement under one reading and a pure choice under the other, and the other 18 offer choices under both. Knots never select: every variable of a knot takes both values.

Read as rules that must all hold, pairs make no knot. Read as preferences, where the allowed states are those that break the fewest, pairs make exactly the four not-all-equal knots on three variables and none on four or more. With hidden variables they make every knot. Whether least-violation states should count as allowed is still open (section 9).

A ring of knots joined through witnesses keeps a constraint exactly when every transfer along it is a permutation and an odd number of them swap, so it composes into a knot or into nothing. Only parity keeps one full bit at every length. Free contact records each loop separately. Contact with fixed points binds loops instead, always into atoms, and a knot among k loops needs k + 1 witness values.

With nonnegative weights, no sum over hidden bits of products of pairwise terms has a three-bit knot as its support, however many hidden bits it uses. With signed weights, every three-bit knot is such a sum with one hidden bit. So pairs never make a knot exactly as long as witnesses cannot cancel.

The first place where logic binds something that no part shows is the step from two-literal to three-literal clauses, and emergence begins at the first such binding. Distinctions are the material and atoms are the bonds. Every mechanism that binds two loops, whether a witness chain, a gate or a circuit, looks from outside like one of the 34 knots.

No set of clauses, each naming finitely many variables, binds infinitely many loops into one knot, even with infinitely many witnesses. The proof is the compactness theorem. An infinite thread of atoms is a tower of finite knots.

In infinite parity systems an open distinction comes from a finite loop or from the system's ends, and no finite part or knot verdict tells the two apart. The law decides: on the square grid, Tseitin's vertex law has only loop-supplied openness, while Ledrappier's triangle law has no finite loop and every triangle is a knot opened by infinity.

Under the source-aware reading a linear law has native knots at every solution or at none: on ℤ^d, exactly when it has positive entropy. Proper 3-colourings have frozen solutions, where the star knot is not native, and dense flexible ones, where it is. Native loops are not enough: the hard-core law is flexible everywhere but has no knot.

On the d-regular tree with the broadcast colouring, the star knot "not all q colours used" exists at every degree but is native only below the freezing threshold, d = 6, 10 and 15 for q = 3, 4 and 5 (certified exactly). There its nativity drops discontinuously from 1 to below 0.0011. Planted random regular graphs matched the tree at every radius measured.

Every law on bits is pairwise, frozen or transmitting. Two knots that share a distinction compose into one knot exactly when they can force it to opposite values; otherwise they form a hub. Lasting memory of loops holds exactly when every polymorphism of the law is affine.

A law's knots compose like cycles when any two that share a distinction still leave a constraint once it is forgotten, as circuits of a matroid do. This happens exactly when every range has a top, ties are bijections modulo the tops, and one ternary polymorphism is Mal'tsev modulo the tops of the knot ranges. The tops are then affine over finite fields, and on bits the laws whose knots compose like cycles are exactly the affine ones. In such laws, networks remember twisted first cohomology over a finite field, modulo the tops.

Two knots sharing one distinction in a law that composes like cycles pass the contact-network gates under recodings, erasure, the law's own restrictions and pp-definitions. Pinning and transports are the deciding conditions. In a declared substrate the allowed states then fix contact adjacency, routes that compose into single contacts, native separation and an extended bond distance. They do not fix the drawing, and the law fixes no substrate.

In an unsatisfiable hub-free network of three-place knots, every knot whose removal keeps the network connected allows exactly half of its states. So every native contradiction is a set of parity systems joined by forced values, and native networks of every law are decided in polynomial time. NP-hardness needs a hub as well as a knot. The classical neighbour is Tovey (1984).

No native gadget of any knot is an exact hub. The hub ratio of a gadget is bounded by √3 for windable knots (given McQuillan) and by 1 for parity and no-lone gadgets. For the six remaining knots, tree gadgets have hub ratio at most 3.

For gadgets with loops, adversarial search up to 40 knots never broke the bound of 3, and often met it exactly, but it is not proved. Any proof must be global: explicit two- and three-knot gadgets break every local strengthening.

A law's allowed states carry a graph exactly when its knot matroid is graphic, and a sphere when it is also cographic, determined up to Whitney twists. Tutte's excluded minors decide whether a space is present, not how the law was written. The mathematics is classical (Tutte, Whitney).

Hub-free gluing of three-place knots builds compact orientable surfaces, while two-place relations build only lines and loops. Exactly four knots see only the surface when counted: the parity and Fibonacci fusion rules, with Verlinde dimensions 2, 5, 15, 50. Hub-free gadgets never create parity, so memory must be present from the start. This partly overlaps known work (Verlinde).

Among the 32,768 symmetric bit laws on the edges of a triangulation, only the trivial and parity laws survive the 2–3 and 1–4 moves, so bits see a three-manifold only through hubs, and then only through its mod-2 first cohomology. Among Boolean knots glued natively, only parity is rebuilt by gadgets of its own copies. No three-place knot orients a vertex and no orienting four-place knot survives, so native laws fix topology but neither orientation nor scale.

The gluings on which not-all-equal closes up, its three-fold spaces, are the triple systems that resist 2-colouring as a whole but not once their corner line is removed. Each is built from Fano atoms, Fano planes with one element removed, joined at three-edge cuts by Pfaffian glue made from planar pieces. When the glue is planar, the least genus of a surface carrying the space equals its number of Fano atoms: each atom is a handle. The Fano plane's own surface is the seven-vertex torus. The tools are classical (tight cuts, Pfaffian orientations).

Up to sign, the factor by which the knot reproduces itself on a three-fold space is a product over the pieces of the space: 5 for each Fano atom, Tutte's tree-trinity number for each planar piece of glue, and, for each non-planar piece of glue, that piece's own amplitude, which depends on how the piece is glued. When the glue is planar, every face of a least-genus map has length 2 mod 4, so the knot's own signs satisfy Kasteleyn's condition there.

With one line removed, the colourings of every three-fold space can be counted in polynomial time. Counting stays polynomial under twists that do not branch, and becomes #P-hard once twists branch, even without signs and even on Pfaffian braces.

With signs, not-all-equal becomes a standing wave that reproduces itself exactly on every level-k Sierpinski gasket. The wave reproduces itself on every Sierpinski d-simplex too, and one member of its family is a knot in every dimension d ≥ 2, so knots build exactly self-similar spaces whose dimension rises to d. On genuine two-dimensional space under shared subdivision, by contrast, only parity reproduces itself with a knot.

Native gluing builds surfaces, and bits see more only through hubs. Read natively, the space a law builds is two-dimensional topology: the not-all-equal knot builds one handle per Fano atom, while the native laws found so far fix neither orientation nor scale.

Unless NP-complete problems are polynomial, not-all-equal is the only three-place knot that selects: parity is affine, the clause never fails, and the two pinning knots have coNP-complete failures. On plain systems, unconditionally, it is the only knot that can fail. Of the twenty kinds of four-place knot, every kind that fails on small 4-regular systems builds equality and is NP-complete on 4-regular systems. So, under the same assumption, a selecting four-place knot would have to be one of five weak kinds, none of which failed in any search.

Across every nontrivial 2-edge cut of a knot's space, one side accepts exactly equality and the other exactly disequality: two-place parity, with no hub. Across every nontrivial 3-edge cut, the point side accepts an antipodal pair and the line side its complement: a knot against a hub, up to signs. If both sides of a cut accept knots, the cut has two edges, so three-place emergence never passes between parts without a hub. A route through atoms carries its length mod 2, and a route through a hub carries nothing more. These are routes of a declared substrate, not native paths of contacts.

A Boolean law with a knot whose knots compose like cycles is affine, so its count is 2^(n−r) or 0 and it cannot select. Each of the five weak four-place kinds has a knot with a forcing place, and two copies sharing that place form a hub, so none of them composes like cycles either.

Across every nontrivial 2- or 3-edge cut of a knot's space, each side has one count, and the two counts multiply to the knot's amplitude. So counted length, the logarithm of the count, adds up over the decomposition: log 5 per Fano atom plus the logarithm of each glue piece's amplitude, which for a planar piece is Tutte's tree-trinity number. It is a measure on the routes of a substrate, not a native extended distance.

On two values, composing like cycles rules out selection. With more values it does not: doubling a value keeps a law's failures and circuits but makes its count #P-complete, so blurred parity and blurred ℤ3 compose like cycles, fail recognisably and count hard. Blurred signed ℤ4 on five values does all this and also keeps knots inside the spaces it selects, below its top layer.

A five-page note, Native selection, and why three places, states these results in standard vocabulary: which couplings make counting easy on the knot's failing systems, why not-all-equal is the only three-place law that selects, why parity's failures are local, what happens at four places, and how the failing spaces compose. A replay reruns the sixteen checks it cites, and separately written code recomputes nine of its claims. It has not been externally reviewed.

On the planar census spaces and 24 further planar knot spaces, no removal on a planar side created a knot across the opened cut (2,172 removals). New knots appeared only where the gluing line lies on a handle, or where the removal opens a handle attached at a point, which then creates at plane gluing lines out to distance 5.

With Kasteleyn signs, a solution makes each line skip the corner opposite its odd point and every face hold an odd number of skipped corners, so Euler's count of faces forces a plane knot space to fail. After a removal, the solutions are the perfect matchings of the other lines to the faces off the removed line. These matchings carry integer heights that form a discretely convex set, so on a plane side every pinned three-point knot through the hub is generated, whatever the far side. A crossing knot that one port separates is a 2-sum of its two sides, or tie-equivalent to a one-sided circuit. Heights of plane matchings (Thurston, Propp) and discrete convexity (Murota) are classical.

A route criterion decides which gluing lines an opened Fano handle reaches. With the handle attached at a point of plane pieces and opened by the removal, the new knots are exactly the candidate knots that no route through the gluing line's points generates. So creation depends on the plane piece only through its transfer.

A port is odd on its line exactly when that line takes the single skip of the face opposite the port (proved). With the Fano plane as handle, every line reached beyond distance 3 lay on a face opposite the attachment point, at all 126 points tested.

A rigid strip relays an opened handle's creation into the next grid from every distance, while a liquid grid screens it within three steps. Relays act as wires: aligned stations repeat them without loss, hubs broadcast them, and no relay ever creates upstream of the handle's piece. Exact creation maps confirm this on small networks.

Whether a removal creates at a gluing line depends only on the atoms holding the two lines (proved for atoms attached at a point). When plane glue separates the atoms, the lines outside a largest creation-free set fall into one piece per atom, in 416 of 420 spaces. So native selection counts the handles and finds where they are; touching atoms merge.

An intact Fano side attached at a line acts as that line for creation, as it does for solutions: it never removes creation, and one new point on one new line never adds any. Creation is decided at the gluing line in two rounds, proved by reducing the closure to local configurations and checking all 348,376,032 of them. One line is the exact limit: a point on two or three new lines can create.

In every space built from Fano atoms alone, each atom's lines create at one another, and a line receives creation from a removal in another atom, at any distance, exactly when it touches its atom's cut facing the removal. A chain of atoms carries one of 47 finite profiles from cut to cut, and the receiving atom alone decides. When every atom has at most four cuts, creation alone recovers the atoms and every cut with its direction. An earlier rule that counted atoms by their sources holds up to five atoms and fails at six.

For a removal beyond one of a Fano atom's cuts, whether it creates at a line of the atom depends only on the patterns the removal's side allows on that cut's three ports. This was proved over every port relation and every deeper relation.

On 191 spaces, the touching rule fails only where plane glue constrains a cut. A two-stage rule reads the atoms of every census space and of 57 of 80 random glued spaces. In 15 random spaces, lines of two atoms that meet one plane piece cannot be told apart by whether creation occurs, though which far side creates still separates them.

Directions are which side of a cut lost a point, touching is incidence, and a handle is a set of lines seen together. Native selection picks out emergence exactly where the parts fail to meet at a gluing line. Every notion in this section is a relation among lines; none assumes a space.

An impossible charge network looks consistent in every small window. Over every field of characteristic other than 2, the Nullstellensatz degree of its Tseitin formula, written clause by clause, equals the graph's toggle width, which is cutwidth + 1 at maximum degree three. Reasoning mod 2 sees the hole at once.

In the ±1 encoding, the polynomial calculus degree of Tseitin formulas lies between ⌈cvw/2⌉ and ⌊cvw/2⌋ + 1 for carving width cvw, and sum-of-squares degree is the carving width up to 4. On graphs built from short cycles, refutations need size 2^Ω(treewidth) in resolution and polynomial calculus, and 2^Θ(treewidth) in sums of squares and Sherali–Adams. Refutations that never forget a variable where the hole must lie need size 2^Ω(treewidth); that every resolution refutation does, at bounded degree, is equivalent to Urquhart's conjecture, which is open.

Gluing topologically is the generalized Fibonacci-gate recurrence, so the four knots that see only the surface count every native network by closed formulas, while the other 30 knots are #P-hard to count, given Backens's dichotomy for Holant problems.

Counting homomorphisms from the group of a 3-manifold to a finite group counts the worlds of a group law. Unless every #P function is computable in polynomial time, Rowell's 2009 conjecture fails: on links already for S₃, on knots for S₄, and for some groups of Fitting height two. For Heisenberg-type groups the note proves an exact dividing line.

In the Heisenberg families, hardness appears exactly when the twisted obstruction joins three nontrivial characters at once, which Borromean surgeries supply, and everything pairwise stays countable. This echoes K2, but it is a statement about counting: the knot test cannot tell hard groups from easy ones, since every group law is constrained as a whole and free in every pair.

Which uniform, local law, not written on a space in advance, gives a shape that looks like a space of low dimension, with geometry as well as topology? Native laws found so far fix topology but neither orientation nor scale, and see three dimensions only through hubs.

Native creation reads the atoms and the directed tree of spaces built from Fano atoms, but through plane glue whether creation occurs is not enough. Still open: a rule that uses which far side creates, the relation a plane piece presents, derived from its faces, and handles that are not Fano atoms.

Pairwise preferences that cannot all hold leave the not-all-equal triangle among their least-violation states. Should least-violation states count as allowed? Both readings are mapped. The strict reading keeps the verdict independent of presentation; the competition reading needs a declared measure of violation. The decision is John Cullen's.

Are loops better modelled as processes whose outputs over time are the data, and does the static test survive that change? Does the program need a declared limit rule that binds infinitely many loops into one contact? What may a witness report about a loop's interior, and which structures count as native loops?

Do any of the five weak kinds of four-place knot ever fail? None failed in any search. Unless NP-complete problems are polynomial, a four-place knot that selects would have to be one of them.