Proofs

With one selector per loop of permutation contact, the shape of the loop constraints is the nerve of the holonomies' fixed-point sets, every tie or knot among loops is an atom, free contact binds nothing, and a knot among k loops needs k+1 witness values (Lean). Every simplicial complex arises as such a shape.

Two-witness linear reasoning over a ring R misses an impossible permutation network exactly when R can write 1 from the holonomy-orbit sizes, and over the integers first on five values with orbits 2 and 3. Pinning one witness, or holding three witnesses at once, refutes every impossible network over every ring.

Read by competition, so that a system allows the states breaking the fewest rules, rules on pairs make exactly the four not-all-equal knots on three variables, the frustrated triangles, and no knot on four or more; with hidden variables they make every knot. Read strictly, they make none.

Read on allowed states with erasure as forgetting, a knot is a relation constrained as a whole and free in every part: two loops and a contact have exactly 34, from the single clause ¬A ∨ ¬B ∨ c to parity, which hides one full bit. No 2-CNF with witnesses defines a knot, while three-literal clauses with named intermediates define every knot.

A ring of knots joined through witnesses excludes a middle pattern exactly when every transfer along it is a permutation and an odd number swap, so it composes into a knot or nothing (Lean). Only parity has all-permutation transfers and keeps one full bit at every length; other knots' memory fades with the loop's length.

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, 15 for q = 3, 4, 5 (certified exactly), where its nativity drops discontinuously from 1 to below 0.0011. Planted random regular graphs match the tree at every measured radius, so native knots in random colourings end exactly where typical solutions freeze.

Under the source-aware reading emergence depends on the law and the world: a linear law has native knots at every world or at none (on Z^d exactly when it has positive entropy; Ledrappier's law and Tseitin plus gauge are frozen), while proper 3-colourings have frozen worlds where the star knot 'not all distinct' is not native and dense flexible worlds where it is. Native loops are not sufficient: the hard-core law is flexible everywhere but has no knot.

In the q-colouring law, hidden witnesses with q neighbours each give their visible leaves a K1-K3 knot exactly when the witness graph is a Gallai tree; otherwise they impose nothing. Gluing at a witness fuses two knots into one with multiplicative excluded mass, and a shared visible vertex is a hub, so knots here compose by fusion along Gallai trees, never into networks of distinct knots.

In infinite parity systems an open distinction comes from a finite loop or from the ends; the 3-regular tree has a K1-K3 knot at every vertex without any loop, and no finite part or knot verdict tells the sources apart. The law decides the source: 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.

Every law on bits is pairwise, frozen (only 'not all on' or 'not all off' knots, which never compose) or transmitting, where knots compose exactly when they force a shared distinction oppositely and chains grow. Lasting memory of loops holds exactly when all polymorphisms are affine, exactly when bounded local reasoning fails, and hardness lies only inside transmission with memory.

The bits a relation on bits hides from its pairs vanish exactly at two-literal clauses, reach one bit only for parity among knots, reach about n - log2 log2 n on n bits through covering arrays, and the statistical version vanishes only on least sets of pairwise energies, so not-all-equal is the only silent knot. With hidden joints, a network of knots hides at most one bit per independent cycle, exactly for parity, and frustrating patches add up.

If every edge outside a bounded set lies on a short cycle with short detours avoiding it, Tseitin formulas need size 2^Omega(treewidth) in resolution and in polynomial calculus with or without twin variables over characteristic other than 2. This covers plane graphs with short faces and d-dimensional grids, and through wall topological minors every graph excluding a fixed minor, planar graphs included (resolution and PCR).

On a connected graph of maximum degree at most 3, the toggle width equals cutwidth + 1, so an impossible charge network of three-distinction knots is refuted by linear local reasoning over a field that cannot compute in its group exactly at cutwidth + 1. In general the toggle width lies between cutwidth + 1 and cutwidth + floor(maxdegree/2).

In the complex of partial solutions, every knot that excludes two or more states is a hole at its own scope, so the plain hole criterion certifies only the trivial level 2 on parity, sum-mod-3 and not-all-equal networks. How far local reasoning must look is set by the whole network's cuts, not by the shape of its knots.

Over a field that cannot compute in the charge group, linear local reasoning refutes an impossible charge network exactly once it holds as many distinctions as the graph's toggle width, whatever the charges, while matched rings need one knot. On cubic graphs with up to 10 vertices this depth is cutwidth + 1, and it is not a function of treewidth.

Resolution and PCR size of Tseitin formulas transfers along weak immersions and topological minors, giving 2^Omega(tw^(1/9)/polylog) on every graph and reducing 2^Omega(tw/polylog) to finding a cellular or compact topological minor of nearly full treewidth in every subcubic graph, where immersions add nothing. On walls joined by block-scrambled bundles along an expander, no compact minor keeps the treewidth and cells exist only at scale sqrt(b), a candidate counterexample.

Over any field of characteristic other than 2, the polynomial calculus degree of the Tseitin system of a connected graph with an odd charge, written as balanced plus-minus-one binomials and measured modulo y^2 = 1, lies between ceil(cvw/2) and floor(cvw/2) + 1 for carving width cvw, exactly ceil(cvw/2) when cvw is odd. The standard encodings need the larger of the maximum degree and that value, plus at most one before reduction.

Over any field of characteristic other than 2, every polynomial calculus refutation (0/1 basis, with or without twin variables) of the Tseitin formula of the a x b grid has size 2^Omega(min(a,b)), and size 2^O(min(a,b)) poly(ab) suffices. Since PCR simulates resolution, resolution needs 2^Omega(min(a,b)) on grids as well.

A tangle of order k gives Tseitin formulas a pseudo-expectation up to degree k-1 that defeats Sherali-Adams and sum-of-squares, so SoS degree is the carving width up to 4 and Sherali-Adams degree lies between cvw and 3cvw/2. With cell restrictions, SoS and Sherali-Adams size is 2^Theta(tw) on grids, d-dimensional grids, plane graphs with short faces and walls, the first such bounds beyond expanders that we found.

Refutations of Tseitin formulas that never forget a variable at the region where the hole must lie need size 2^Omega(treewidth), because the hole of a random critical assignment stays uniform there and regions fragment size-biased; Davis-Putnam refutations are of this kind. A smaller refutation must re-read variables and forget at the hole's region, and Urquhart's conjecture is equivalent to the open 2^Omega(tw) problem at bounded degree.

A law with a Mal'tsev polymorphism composes its knots like cycles, so that any two sharing a distinction leave a constraint once it is forgotten and form matroid circuits, exactly when every subalgebra of its polymorphism algebra has a unique maximal congruence; Boolean laws do exactly when affine. Elsewhere knots fuse or meet at hubs, so cycle-space structure among knots marks difference laws whose distinctions do not split.

A law's allowed states carry a graph exactly when its knot matroid is graphic, and a sphere when it is also cographic: they determine a 3-connected graph exactly and a 2-connected one up to Whitney twists. Presence is decided by Tutte's excluded minors, not by how the law was written, so a randomly written local law can carry a sphere while almost every balanced code carries none; the mathematics is classical.

Composing knots like cycles does not force a difference operation: parity beside an unrelated law does it without one. The converse reduces exactly to rectangularity of knots, since rectangular knots compose like cycles precisely when no two meeting at a variable split its range; this is proved on two values, and every tested three-variable knot on up to four values supports it.

A law composes like cycles exactly when its ranges have tops, its ties are bijections modulo them, and a ternary polymorphism is Mal'tsev modulo the tops of its knot ranges, which proves the series' Conjecture 7 through rectangular top images. The tops of knot ranges are then elementary abelian p-groups with polymorphisms acting affinely, and every system carries a direct sum of matroids representable over finite fields, all of which occur.

In every law whose circuits compose like cycles, each system has exactly the circuits of its image modulo the tops, because each variable and its blurred copy are parallel elements; ties become bijections between tops, and every reflexive pp-definable relation lies below its range's top. The earlier conjecture that such laws are Mal'tsev on all their tops fails for disjoint unions, and holds in every sampled law when restricted to the ranges of knot variables.

Every finite structure is homomorphically equivalent to a law that composes like cycles, and every finite core to a core that does, so composing like cycles alone admits NP-complete constraint problems, even among cores; the core of such a law need not compose like cycles, and it can be non-Taylor. With constants, or for a core with no nontrivial automorphism, composing like cycles does give polynomial time, and both hypotheses are needed.

No statement of the composition characterization turned up in a bounded literature scan; its uses are a composition test for linearity of codes, under which the non-Pappus almost affine code fails through an explicit three-copy system, and a tractability certificate for laws that keep composing like cycles with constants. Counting is not covered, and constants can destroy composing like cycles, so whether every such law is tractable is open.

The local converse and the diagonal lemma are false: a 3-variable knot on four values (x+y+z=0 mod 3 with 3 read as 2, minus (2,2,2)) is non-rectangular, yet its law composes every circuit like cycles because every system keeps the constrained sets of the Z3 law. Composition sees only which sets are constrained, so a dominating difference law hides any fibre pattern inside the components of a fibre graph.

A law whose circuits are all functional modulo common proper pp-definable equivalences composes like cycles, and among 3-nets of finite abelian groups exactly the p-groups do; every overlapping and non-group candidate tested failed. The 3-net of the quaternion group Q8 survived with Latin knot images yet has no polymorphism Mal'tsev modulo its top, so either it fails in larger systems or Mal'tsev structure on the tops is not forced.

A knot link remembers a loop exactly when its transfers permute blocks, for any number of values, and in a law composing like cycles its networks remember twisted first cohomology over a finite field modulo the tops, with frame-matroid free sets. The owner's claim holds for two values and fails beyond, through the Z6 and S3 3-nets, dressed parity, and x + y + z = 0 over F3.

Composing like cycles is incomparable with congruence permutability, n-permutability and modularity: a lift of the Z3 law is 3-permutable and modular without a Mal'tsev operation, while a shadow of it composes like cycles with neither, its fibre clone having only conjunctions and constants. What the known examples share is a polymorphism that is Mal'tsev modulo each range's maximal congruence, proved for difference laws, their lifts and suitable shadows, and conjectured for all.

Single relations carry any matroid as free sets, the Vamos matroid included, but their laws fail; a top lemma shows that group and field laws composing like cycles carry only direct sums of matroids representable over finite fields, one prime at a time, so the Vamos matroid never appears there or in their shadows, lifts and disjoint unions, and every law composing like cycles has tops even without a difference operation.

Two knots sharing one distinction in a law composing like cycles pass the contact-network gate under recodings, erasure, the law's own restrictions and pp-definitions, with pinning and homomorphic or product transports as the deciding conditions. In a declared substrate the allowed states fix contact adjacency, routes that compose into single contacts, native separation and an extended bond distance, but not the drawing, and the law fixes no substrate.

In a law that composes like cycles no circuit overlaps modulo the tops, so composing like cycles is exactly having Latin circuits (multiary quasigroups) modulo proper pp-definable equivalences, while Mal'tsev structure on the tops stays open. The 3-net of the quaternion group Q8 fails through Thomsen's configuration, whose top image is the non-affine relation behind the earlier certificate, and a Latin square's 3-net composes like cycles exactly when the square is isotopic to an abelian p-group.

Given McQuillan's winding theorem, native counting with pins is approximable for the 26 windable knots, because the knot axiom K2 is his terraced condition; with the four topological knots this settles 28 of 34. The six left open are the Fibonacci knot with one or two negated places, which escape winding and holography and freeze native trees, as not-all-equal and parity do.

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 Holant dichotomy. Closed not-all-equal worlds are an antiferromagnetic Ising model, whose count is a determinant on the sphere.

In an unsatisfiable hub-free network of three-place knots, every knot whose removal keeps the network connected allows exactly half of its edge 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. Native contradictions are hard for resolution exactly when the law has a parity knot, while hubs carry NP-hardness.

Rerouting one terminal's arm is injective for each fixed difference pattern, so the arm inequality holds whenever patterns map injectively, as on trees. For gadgets with loops, explicit two- and three-knot gadgets satisfy the inequality but break every local strengthening (per partner, per image pattern, one-sided path flips, the minor form), so any proof must be global.

No native gadget of any knot is an exact hub, and pairs of solutions that do not link all three terminals reroute onto off-diagonal values, which bounds the hub ratio of windable gadgets by the square root of 3 (given McQuillan) and of parity and no-lone gadgets by 1. For the six open knots, tree gadgets are proved to have hub ratio at most 3; gadgets with cycles remain a conjecture.

A trisum of three non-colourable cubic pieces along a square has amplitude rho1 rho2 rho3 - alpha1 alpha2 alpha3 - beta1 beta2 beta3, where (alpha, beta) splits a piece's colourings by the square point its far line leans to. The knot's not-all-equal rule reappears one level up on the pieces' leanings, and pieces glued further down act only as weights on these two numbers.

For every three-fold space the colourings with one line removed are counted in polynomial time without any signing: a product over braces, a planar brace one Berman determinant, any other brace a Robertson-Seymour-Thomas trisum tree evaluated with two weighted determinants per planar leaf. Exact counts reach 1720-vertex instances where variable elimination would need factors on 30 variables.

The wave reproduces itself on a gluing exactly when six signed sums over phase assignments with mixed corner phases vanish, which phase locking and mirror cancellation achieve up to five triangles and genuine identities achieve later. The law selects thousands of gluings and no growth exponent, so it does not draw a space by itself.

At the knot angle a contact of three triangles with odd base count is a not-all-equal knot on their phases, so a gluing of such contacts is the unsigned problem on its dual, which needs seven triangles, and seven suffice. With distinct corner triangles the seven-triangle such gluings are exactly the Fano gadgets, exact for both the signed and the unsigned knot.

On every three-fold space the knot's exactness factor is (-1)^(n-1) times a product over the pieces of the space: 5 for each Fano atom (each handle of its surface), 1 for each doubled square, and for each planar piece of glue Tutte's tree-trinity number, the spanning trees of a map for omnitruncation glue. Across a tight cut the count of colourings factors exactly, through a point lemma.

For every connected cubic triple system that is not 2-colourable, removing any line leaves the same number of colourings, and the transpose gives that number at every point. Counting an Ising weight at half the knot's angle in two orders makes each line's count the average of its points' and vice versa, so every brace has one amplitude, and trisum amplitudes depend on the gluing.

Every such system other than the Fano plane contains at least two Fano planes with one element removed, attached by three-edge cuts; everything else is planar Pfaffian glue (doubled squares, planar braces, and trisums built from them), and every planar cubic bipartite graph occurs as glue. The approved grammar of splices and odd cycles was incomplete: a 43-point system has the cube as glue.

Below 23 points such systems have exactly 7, 13, 19, 21 or 22 points, with 1, 1, 4, 1 and 1 isomorphism classes, each a small core dressed with Fano planes minus an element; none exists at 8 to 12, 14 to 18 or 20 points. With one twisted incidence the sizes are 14, 15, 18, 20, 21 and 22.

Such a gluing is a space for the knot exactly when its dual triple system resists 2-colouring but stops resisting once the corner line is removed, which is the classical even-cycle problem whose one indecomposable exception is the Fano plane. Up to 13 triangles only the Fano plane and its splice occur, and Fano planes splice together without limit.

A Fano plane minus a line forces its three free points to one colour, so lines meeting it twice act as not-equal constraints, and a cycle of an odd number of such blocks resists 2-colouring. The knot therefore has three-fold spaces with 21, 27, 35 and more triangles built from both classical obstructions to 2-colouring, the Fano plane and the odd cycle.

On every minimal map of every three-fold space with planar glue each face has length 2 mod 4, so the knot's own signs satisfy Kasteleyn's condition; the condition survives every tight-cut splice because exactly three crossing faces form, each joining one face from each side. Non-planar trisum glue behaves the same on all its minimal maps.

Every spin structure induced by the knot's signs and a perfect matching is even, with Arf invariant 0 independent of the matching, and even on every Fano handle; the matchings induce exactly 3^h of them, and on the Fano torus exactly the three even ones, eight matchings each. Odd spin structures for the same signs appear only on non-minimal maps, from genus 3 on.

The knot's colouring counts are Arf-signed averages of spin sectors, so its non-colourability is an exact balance of even and odd sectors (8, 8, 8 against 24 on the Fano torus), and its sign class is the only non-colourable one among the surface's Kasteleyn classes. Opening a contact gives 16, 16, 16 and 28, so five per handle does not come from the even spin structures, and at genus 2 the sectors do not factor over handles.

A three-fold space whose glue is planar has genus equal to its number of Fano atoms, because a piece that keeps its genus when a vertex is deleted adds exactly its genus across a three-edge splice, while in general such splices can lose handles. Its minimal maps are exactly connected sums of the chiral Heawood map, one per atom; the Fano plane's own map is the seven-vertex torus.

At a trisum square, a side carrying the knot's own signs acts on the other closed side as a similitude of the matchgate identity with multiplier alpha^2 + alpha beta + beta^2, the norm of its split at the knot's angle pi/3. So counting stays polynomial under every sign class unless some square has twisted sides on all three branches.

With nonnegative counts the junction's not-all-equal square lines can never force a relation between the host's two line-neighbours without pinning them, so no junction is a crossover; its points can carry a wire. Yet twisted truncated-octahedron pieces give soft crossings outside the matchgate class, such as (40000, -1112, 3336, -192), so hardness by interpolation remains open.

For every k >= 4, signed not-all-equal never fails on k-regular k-uniform systems: a minimal failing system would be square with a sign-nonsingular incidence pattern, but its digraph has in- and out-degree at least 3 and is even by Thomassen's theorem. At three places it fails exactly on the knot's Pfaffian spaces, so Thomassen's degree threshold is the two/three boundary for this law.

The 30 value-free four-place knots form four classes: parity's count is polynomial, not-exactly-two and twelve build equality without constants and are NP-complete on 4-regular systems, and not-all-equal never fails. So, unless NP-complete problems are polynomial, mutual selection among value-free laws is special to three places.

A knot's space minus one incidence, placed on an edge, is an exact negation at the knot's law (lam = 0) and multiplies the paths through the edge by a ratio other than +-1 elsewhere (Fano: -141/6079 at lam = 1/2). At the dimer coupling lam = -3 every plain system counts (-2)^n times its perfect matchings, a determinant exactly on the knot's spaces.

Counting 2-colourings of the knot's spaces with weight lam per monochromatic line is polynomial exactly at lam in {0, 1, -1, -3, oo} and #P-hard at every other complex lam, by chains of atoms and interpolation from the hard Pfaffian systems, with torsion points on the curve of two atom ratios closing the complex case. Beyond the affine couplings only the knot's law (count 0) and the dimer law (a determinant) are easy.

Counting parity's coupling (weight lam per line missing its parity) on connected simple cubic systems is polynomial exactly at lam in {0, 1, -1, i, -i, oo} and #P-hard at every other complex lam, since the theta substitution turns it into the knot's coupling at lam + 1/lam - 1. It stays #P-hard on systems where parity fails, because a parity certificate survives grafting into any host, so parity's recognisable failures select nothing.

The 34 three-place knot relations fall into five classes (parity, Fibonacci, not-all-equal, x or (y != z), clause), and only not-all-equal has no preferred value, so only it cannot pin a value. A law that can pin a value, turn a pinned line into an equality and absorb spare occurrences is as hard on cubic systems as with unbounded occurrences, which makes the Fibonacci law and x or (y != z) NP-complete there.

Unless NP-complete problems are polynomial, not-all-equal is the only three-place knot whose failing cubic systems are non-empty and recognisable while its count is hard in general; parity is affine, the clause never fails, and the two pinning knots have coNP-complete failures. On plain systems, unconditionally, not-all-equal is the only knot that can fail.

Ladders of plain cube junctions with rungs act by (NW)^k with characteristic polynomial (x^2 - 19x - 1)(x^2 + 7x + 13), whose roots, a unit of Q(sqrt 365) and the Eisenstein primes above 13, satisfy the lattice condition, so interpolation over odd k yields an exact crossing. K3,3 is counted from plain Pfaffian braces of 20 to 104 vertices.

Weaving a hard source into a prism frame, with every crossing a ladder of plain cube junctions, gives plain Pfaffian braces (a bond lemma, a cube junction lemma and Hall's condition), so counting on plain Pfaffian braces is #P-hard. Among them planarity and the knot's atom, the Heawood graph, are the polynomial cases found.

An unsigned long edge with k capped pairs, read at t = -1 by interpolation over its length, acts as a negation, and lengthening edges inside planar pieces keeps the Pfaffian certificate. So counting proper 2-colourings of 3-regular 3-uniform hypergraphs whose incidence graphs are Pfaffian is #P-hard, and K3,3 is counted end to end from 100 plain Pfaffian instances.

Euler's formula leaves every planar cubic bipartite graph a square face away from any chosen square, where a Pfaffian signing needs an odd sign, so every plain page is twisted and every square of a plain trisum brace has three twisted branches. Since a brace's 4-cycles are nice, the Heawood graph is the only brace whose plain system is not 2-colourable.

Repeated k times with capped plumbing, the junction of two twisted truncated-octahedron pages multiplies its four crossing states by 810000, 7506, -22518 and -432 per step, whose prime exponents at 5, 139 and 3 separate them, so interpolation yields an exact crossover inside Pfaffian graphs. Hence counting colourings of every cubic bipartite signed system reduces in polynomial time to counting on cubic bipartite Pfaffian graphs under signings.

Counting colourings of cubic bipartite Pfaffian graphs under signings is #P-hard, by reduction from NAE-3SAT counting through merge paths, dead caps and the sharpened soft crossing, and the instances are twisted on all three branches at every junction. Since the knot's own class and every unbranched twist count in polynomial time, the knot's sign class and the branching of twists away from it divide easy from hard.

Of the 20 classes of four-place knots, every class that fails on small 4-regular systems builds a link (directly or by lifting a three-point core that realizes the twelve class) and is NP-complete on 4-regular systems, while parity's count is polynomial and the clause and not-all-equal never fail. So, unless NP-complete problems are polynomial, a mutually selecting four-place knot would have to be one of five weak classes, none of which failed in any search.

The known cycle-composing laws are coset laws with uniform fibres, so they count easily. Doubling a value keeps a law's failures and circuits but makes its count #P-complete, so blurred parity (three values) and blurred Z3 (four values) compose like cycles, fail recognisably and count hard; every Mal'tsev law on three values composes like cycles.

The knot fails exactly on Pfaffian substrates, always globally; parity where perfect matchings are even, never globally; Z3 everywhere, globally exactly at Z3-corank 1. Blurred Z3 on corank-1 substrates both selects (global failures, easy after removing a line, hard in general) and composes like cycles, but its selected spaces are rigid, only ties, while the knot's can keep a knot.

Across every nontrivial 3-edge cut of a knot's space the line side is exactly the knot (not-all-equal) and the point side exactly a hub (equality), and two sides that both accept knots can fail together only as complementary parity cosets. So the knot's spaces compose natively only through hubs, and a hub-free failing composition is a parity one.

Qualification pending: false at 2-edge cuts, where parts meet as two-place parity, and exact only up to signs at 3-edge cuts (see NATIVE-PATHS-20261101-001).

Across every nontrivial 2-edge cut of a knot's space one side accepts exactly equality and the other exactly disequality (complementary two-place parity, no hub), and across every nontrivial 3-edge cut the point side accepts exactly an antipodal pair and the line side its complement (a knot against a hub, up to signs). This corrects KNOT-COMPOSE-20261101-001-D001: the knot's spaces do not compose only through hubs.

Every nontrivial 3-edge cut of a knot's space is a proper splice cut or a shifted 2-edge cut, and in ten spaces proper cuts never cross, so the parts and routes are fixed by the space. A route through atoms carries its length mod 2 and a route through a hub nothing beyond it; cutting both routes separates, and these are routes of a declared substrate, not native paths of contacts.

A knot's space stays non-colourable under a twist pattern exactly when the pattern is a cocycle of its incidence graph (sign-nonsingularity and the uniqueness of Pfaffian signings up to switching). Hence if both sides of a nontrivial cut accept knots on its channels, the cut has exactly two edges: knot meets knot only as two-place parity, and three-place emergence never passes between parts without a hub.

Over a field every global failure of a knot system has corank 1, so removing any line leaves only ties; blurring keeps this, S3 and D_p have only blurred-parity knots on full ranges, and simple non-affine three-value Mal'tsev laws have none. Over Z_(p^k) a failure is global exactly when its class lies in every cyclic subgroup generated by a line's class, and cyclic solution groups leave only ties.

Global failures over Z_(p^k) with k at least 2 keep knots: signed Z4 does after every removal on a 6-point 4-regular substrate, and on a 9-point cubic substrate, the minimum at three places. Over any finite chain ring a global failure leaves, after any removal, solutions whose top layer is one line, so a kept knot has at most one full-range variable and hides at most k - 1 digits.

Blurred signed Z4 on five values composes like cycles, fails globally and recognisably, counts #P-complete and keeps knots inside its selected spaces, below its top layer. At two values a law with knots that composes like cycles is affine and hence rigid, while the knot keeps knots inside its selected spaces.

Each of the five weak four-place classes yields a knot with a forcing place, and two copies sharing that place form a hub, so none composes its knots like cycles. A Boolean law with a knot whose knots compose like cycles is affine, so its count is 2^(n-r) or 0 and it has no mutual selection: selection and cycle-like composition exclude each other.

Across every nontrivial 2- or 3-edge cut of a knot's space each side has one count on its accepted channel values and the two counts multiply to the amplitude, so counted length (its logarithm) is additive over the decomposition: log 5 per Fano block plus the glue's tree-trinity logarithm, independent of the routes' relations and unchanged by transposition. It is a law-intrinsic measure on substrate routes, not a native extended distance.

Interpretation pending: whether counted length is a native magnitude has not been decided; read it as a count, not a distance.

On 24 planar knot spaces beyond the census and the planar census, no removal on the planar side creates a knot across the opened cut (2172 removals). Created knots, found only on non-planar sides with six port patterns, have three points: the removed line's hub, a near point and a far point; the first two pin all three ports and the third separates them.

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 n + 2 faces for n lines force failure and, after a removal, the solutions are the perfect matchings of the other lines to the faces off the removed line. Genus g removes 2g faces and adds 2g homology parities.

Removing a line from a knot's space ties its three points into a hub, exactly the point side of the 3-edge cuts along which the space is spliced from Fano atoms and planar glue. After any removal off a cut's point side the hub holds one bit, and by the knot's symmetry under complementing every value the knots crossing the cut are exactly the 2-sums of a circuit through the hub on each side.

After removing a line on the point side, the interface is what the closed piece allows on its closing line: one complementary pair (the hub re-ties and the 2-sum holds), two pairs (one port released) or three (the hub becomes a knot). With two or three pairs composition fails, and in the 13-point Fano splice 72 crossing knots arise from no chain of eliminations.

Whenever the hub of the removed line and a near point pin the gluing pattern on a plane side, the hub is tied to a non-odd port or forms a near knot with the two non-odd ports one step from the pinned pattern, giving every pinned-and-separated knot an elimination route for any far side. This holds on all 30 plane spaces tested and fails on every pinning of the Fano plane and census 58.

Uniting the knot's near solutions over the four homology sectors of a torus embedding removes all creation for some embeddings and cut positions and not for others, even within one homology class, while dropping one parity removes about nine tenths of the new knots and local point flips remove none. Homology carries most of a handle's creation, but forgetting it is not canonical in point values.

On a plane knot space minus a line, line-face matchings carry integer potentials forming an L-natural convex set whose parities are values relative to the hub, and rounded midpoints show that a hub pinning never selects a hub-tied port's state and otherwise points one step from the pinned pattern. So every pinned three-point knot through the hub is generated on a plane near side, whatever the far side.

In two knot spaces glued along a three-edge cut, a crossing knot whose excluded pattern one port separates is the elimination, through that port, of its near part plus the port and its far part plus the port, or is tie-equivalent to a one-sided circuit. The proof uses only complement symmetry and minimality, and it covers 33,556 of the 58,875 crossing knots tested.

Re-centring the potentials at any point keeps them convex, so on a plane side every pinning pair reaches the ports and every crossing knot pinned by two near points is generated, while on the Fano plane and census 58 no pinning by two non-port points has access. The premise then rests on 12,102 residual knots, all generated on every splice tested but without a proof.

Removing a line holds every cut pointing toward it and opens every cut pointing away; beyond a held cut nothing depends on the removal. The knots crossing only held cuts are exactly the iterated 2-sums, through held hubs, of knots crossing no cut, so new emergence lives only across opened cuts.

Opened cuts do not act separately: new knots of one are carried across others, and two or three create knots of three or four points jointly. On 96 chains of K4^(3) and Fano pieces, neither touching hubs, nor a shared port on the removed line, nor the removed line's hub decides this, and chains of K4^(3) pieces create nothing.

In every Mal'tsev law a knot is pulled back from abelian sections of its ranges, so a knot system with declared domains is linear over the abelianized domains, and if every abelian section is simple every global failure on a substrate is rigid. Hence all laws on three values that compose like cycles (all 1020 non-simple ones checked at three places) and all non-affine laws on four values are rigid, while T2(F2) shows two layers are necessary but not sufficient.

A group's coset law composes like cycles exactly when every subgroup has a unique maximal normal subgroup (18 non-abelian such groups up to order 60), and its knots are pulled back from cyclic p-group relations. For systems with declared domains the top image after any removal from a global failure is at most one line; the law is rigid at prime-order sections, and C5:C4 and D9 keep knots below the top.

A simple Mal'tsev law on five values containing Z4 composes like cycles and keeps knots in its top, so the statement fails as phrased; the rigid layer is the top of each point's own section, proved when the sections come from one chain ring. A law that composes like cycles thus keeps emergence only below the top of a two-layer abelian section, whereas the knot keeps it on two values in its only layer.

Around an attachment point the own arc between two ports is barred exactly when they disagree, and a frozen line claims an own arc from one port pattern but an opposite face from two. A handle creates at a frozen line exactly when a pair of its points freezes it and neither point alone excludes either claim (one direction proved, one computed), so the Fano plane, whose single points remove one pattern per other class, never uses own arcs.

Across 305 handles from 24 non-planar knot spaces, every port order on a ladder, the faces a handle creates through are decided by its port relation alone: whether a pair of its points pins the face's barred state while each single point keeps the face's claims. Nine signatures occur; census 58 uses its own arc exactly when the bottom port is a third point of its two lines sharing two points.

On a ladder arm spliced onto a grid, at a junction point the frozen region of a port class runs along the arm's far long face with two-pattern claims, and the Fano plane and census 58 create there, off the faces around the attachment point; exact creation confirms this on a 21-point piece. So the halo lies on frozen lines whose claims survive the handle's single points, refuting PHASE-HALO's law of the faces around the point.

Qualification pending: the law holds after reduction (see the FACE-CLAIM-20261103-001 closure).

A three-point crossing knot at a gluing line is generated when one of the line's points separates it or a pair of the line's points, forced by the near pair and covered by the far point's excluded block, carries a two-step elimination. With a Fano handle attached at a point of plane pieces and opened by the removal, the new knots are exactly the candidates without such a route, so creation depends on the plane piece only through its transfer.

In a point split of a plane knot space the solutions are corner matchings, and which pairs of port patterns and gluing-line patterns occur together is decided exactly by heights: fix the line's corner, project the difference constraints onto six regions by shortest paths, and test parities by halving. This makes the halo computable on plane pieces of hundreds of points, and it matched enumeration on all 1,221 small cases.

An opened Fano handle attached at a point creates at the farthest gluing line of every ladder tested (prisms to 80 points) but never beyond distance 3 on grids (subdivided-octahedron duals to 256 points). Fixing the handle's point leaves every ladder line with two patterns at any distance while every grid line at distance 7 or more is exactly independent of it, so the halo follows frozen against liquid forcing.

Plane signed systems with as many lines as odd faces have perfect matchings as solutions and the access lemma for every pinning, and in a splice an intact side acts on the cut as the point or line it replaced. So every pinning by two points of a plane piece has access wherever the removal is, and only pinnings from inside an opened handle can fail.

No new knot appeared unless the gluing line lies on a handle or the removal opens a handle attached at a point, which then creates at plane gluing lines out to distance 5; when the cut carries one bit, a screening lemma reduces creation to a plane knot space. Near opened handles 1,092 of 2,496 new knots are unpinned three-point knots, and four-point new knots appear inside handles, so the pinned three-point premise fails on composite spaces.

A port is odd on its line exactly when that line takes the single skip of the face opposite the port, so every line of that face feels it. With the Fano plane as handle, every halo line beyond distance 3 lay on a face opposite the attachment point at all 126 points tested, so the reach is set by those faces' size, not by a dimer phase; census 58 can also use the point's own face.

Qualification pending: the face law fails at junctions and holds after reduction (see FACE-CLAIM-20261103-001).

The prism is the cyclic chain of consecutive triples, whose transfer matrix has powers of period 2, so every far line's transfer depends only on its parity and the halo of an opened Fano handle runs the full opposite face on every prism. On grids, far lines have the full transfer and are independent of the handle, as height ranges grow with distance while on ladders they stay zero.

A port pattern freezes exactly the points whose height difference to the ports cannot vary, and a side beyond a 3-edge cut that holds neither the handle nor the gluing line can be collapsed to the vertex it lost without changing creation. In the collapsed space the halo again lies on the faces around the attachment point, so a ring spliced onto a grid shows the full ladder halo along its far face.

In a grid, strip and grid joined through two 3-edge cuts, an opened handle in the strip or the far grid creates in the first grid near its cut, from every distance along a rigid strip and at the same lines for every length, while a liquid grid screens it within three steps. The relay enters only a side split at a point there, the direction of a handle's own splice.

In a space made of plane glue and atoms at 3-edge cuts, whether a removal creates at a gluing line depends only on the atoms holding the two lines, since every other atom acts as the vertex it replaced (proved for atoms attached at a point, checked exactly for atoms attached at a line on 13 designs). So each removal's halo is its own atom's, and atoms attached at a line have no halo on plane glue.

The lines outside a largest creation-free set fall into one piece per atom, each piece an atom, when plane glue separates the atoms (416 of 420 separated spaces with two or three atoms), while touching atoms merge. On the 32 three-fold spaces up to 22 points, where atoms mostly touch, this count equals the genus in 10, and exploratory blocks of mutual creation equal it in 29.

On small plane networks, a rigid strip or a hub lands one atom's halo on a second atom's cut lines in every placement tested (13 exact on the strip, 24 at the hub). Yet each halo stays its own atom's and the native count is unchanged: one piece per atom in all 11 separated strip placements, since a halo changes neither the quiet glue nor the lines the glue creates at.

After a frozen hop a relay is an arc signal, two ports fixed and disagreeing, and an arc signal freezes exactly the lines with a corner on the own arc between them. So a station passes it on exactly when its out-line lies on that arc: aligned chains repeat it without loss and carry relays four cuts deep, misaligned ones stop it, and at separated cuts every creation k cuts away passed cut k as a live signal.

A hub receiving an opened handle's relay copies it into every receiver its frozen region reaches, so one handle is read in several grids at once, and no relay ever creates upstream of the handle's piece. Stations with adjacent cut vertices are bypassed without loss; on three small networks exact creation confirms the broadcast, a three-cut chain and a two-cut relay.

A Fano atom's lines create at one another in both directions, as in the complete Fano plane (proved for atoms attached at a line by the screening lemma, computed for atoms attached at a point). Across a cut between two atoms, the receiver's cut lines and the emitter's lines are mutual while its intact lines create one way at the emitter's (676 of 676 pairs), and lines sharing two points are silent.

On 50 spaces, the native relation between two atom lines is a function of their place in the decomposition wherever the atoms are the same, adjacent or joined through atoms only (21 places, 10,074 pairs), with fixed rules for chains, siblings and co-receivers, while through plane glue it can depend on more. The source rule fixed from these rules counts the Heawood braces, and the genus, on all 47 spaces without a non-Fano brace.

On 111 fresh spaces (every three-fold class of 23 to 26 points and new trees of atoms), all 32,016 pairs at the dictionary's places and all 163 cuts between atoms behaved as predicted, and the source rule counted the Heawood braces, and the genus, on all 110 spaces without a non-Fano brace. The glue rule was refuted: across a Fano atom, glue lines can create (12 pairs, one class).

Native networks of a triangulation see only its dual graph, and among the 32,768 symmetric bit laws on edges only 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. The Fibonacci rule, which sees surfaces exactly, needs a negative weight to see three-manifolds, so no count of hidden witnesses realizes it.

Hub-free gluing of three-place knots builds compact orientable surfaces, while two-place relations build only lines and loops, and exactly four knots see only the surface with counts: the parity and Fibonacci fusion rules, with one hidden bit per handle and Verlinde dimensions 2, 5, 15, 50. Hub-free gadgets never create parity, so memory must be present from the start.

On ordered triangulations, a counting law whose long edge is a function of the two short edges survives the moves exactly when its working triangles form a finite groupoid with equal component values; the objects are the idempotent labels. Its count is a sum of |Hom(pi1(M), G)|/|G|, so such laws see a three-manifold only through the finite quotients of its fundamental group.

On ordered triangulations with one bit per edge, twists and orientations add nothing: exactly the seven symmetric counting laws survive the ordered moves, so counts see a three-manifold only through its mod-2 first cohomology. Positive Turaev-Viro weights for the Fibonacci and twisted Fibonacci rules cannot have full support and are pure gauge on the parity class: the sign survives the twist.

Up to six copies, the three-bit knots that reproduce themselves with uniform counts are parity, twisted not-all-equal and two three-excluded classes; not-all-equal does so at seven. Their towers keep the corner law at every scale with growth exponents log2 3, 2 and log3 7, but are finitely ramified, and softening is repelled at least threefold per level.

Among Boolean knots glued natively, only parity is rebuilt by gadgets of its own copies: other three-place knots' gadgets exclude fewer tuples, so towers of patterns of patterns go free within three levels, and at four places (up to three copies) only parity and one chained implication between pair parities survive. No three-place knot orients a vertex and no orienting four-place knot survives, so native laws fix topology but neither orientation nor scale.

With hubs, not-all-equal is rebuilt by its copies on the seven lines of the Fano plane with one point split into the legs, and no smaller hub gadget rebuilds it, so its smallest pattern of patterns is a finite projective plane; every allowed tuple has exactly five completions, an exact counted fixed point. Within six copies, hub gadgets rebuild exactly seven of the sixteen three-bit knot classes, all through copies in parallel.

With nonnegative values no such sum has a three-bit knot as its support, however many hidden bits it uses; with signed values every one of the 16 classes of three-bit knots is exactly such a sum with one hidden bit, and a clause of any width is one hidden bit (inclusion-exclusion). So the program's rule that two-variable constraints never make a knot holds exactly as long as witnesses cannot cancel.

Where nonnegative counting reproduces only parity, signed laws of 14 of the 16 knot classes are rebuilt and 12 are reproduced exactly, 10 of them by the native triangle that generates the Sierpinski gasket, so signed not-all-equal persists at every scale. The two Fibonacci fusion knots are never rebuilt, and on closed spheres the shared-boundary substitution still keeps only parity.

On the triangular lattice with shared subdivision, only 22 of 122 knot rules reproduce themselves exactly at scale 2 and 6 at scale 3, all with affine solution sets and zero entropy density: parity (Newman-Moore, Baxter-Wu), rules equal to parity on tori, and twisted not-all-equal collapsed to equality lines. Exactness on genuine two-dimensional space keeps a knot only through parity; not-all-equal never reproduces itself.

Every signed one-bit edge law with weights in {-1, 0, 1} invariant under the ordered 2-3 and 1-4 moves has a counting support, and up to sign changes it is a counting law, the twisted Dijkgraaf-Witten law (-1)^(x01 x12 x23) on a cocycle support, or a mix of these. No twisted law is symmetric, so signs need the vertex order, as order alone added nothing to counting.

The twisted laws compute the Gauss sum of the cubic form a -> a^3 on H^1(M; Z/2), which is linear, so they see exactly whether the linking form gives 1/2 on an element of order 2. That bit is not a function of homology: it separates RP^3 # RP^3 from S^3/Q_8 and sampled triangulations that every counting law identifies.

Signed not-all-equal, (-1)^a times the Legendre symbol of a+b+c mod 3, reproduces itself exactly on every level-k Sierpinski gasket, with a factor congruent to ±1 mod 3, because its two Eisenstein phase sectors stay locked along the gasket's edges. It therefore builds exact self-similar spaces of dimension log(k(k+1)/2)/log k for every k, approaching two, while shared subdivision of the plane admits no such knot.

Of 1,168 signed three-bit laws, 68 are exact on some gasket up to level 12, and every one that persists is a rotation, relabeling or field twist of the Eisenstein standing wave, or linear parity at powers of two. The other 42 are single-level coincidences at levels 3 and 5.

Under shared subdivision of the triangular lattice, weights ±1 make no shape outside the 22 affine exact ones exact, even on allowed states, and real weights add none wherever decidable: positive weights keep the counted supports, a sign-parity obstruction excludes 54 of the 100 non-affine shapes for every weighting, and the affine shapes' exact real weightings are gauge copies of the ±1 laws. Exact self-reproduction on genuine two-dimensional space stays affine; 46 dense shapes remain open.

The standing wave (-1)^(x_0) sin(theta E)/sin(theta) reproduces itself exactly on every level-k Sierpinski d-simplex, with a factor that is a monic integer polynomial in 2 cos(theta), equal to 2^N at theta = 0 and never zero at odd prime-power angles. Its Legendre member is an exact knot in every dimension d >= 2, and in odd d the whole family is, so knots build exact self-similar spaces whose dimension rises to d in every dimension.

The cosine wave (-1)^(x_0) cos(theta E) reproduces itself exactly when the number of simplices is odd and otherwise turns into the sine wave, which explains the period-4 family of the plane. In exact censuses on Sierpinski tetrahedra and four-simplices the only knots exact at every computed level are standing waves, and the wave's rational exceptional angles are only pi/2 and, at even levels, pi/4.

A three-bit law symmetric in its two non-base corners reproduces itself on the Sierpinski gaskets of levels 2 to 5 exactly when it is a standing wave, a product, an equal-corner law or one of two frozen laws, and then it does so at every level. Fewer levels admit impostors: a whole curve at levels 2 and 3, and 16 complex laws at levels 2 to 4.

Besides rotations of the symmetric classes, general three-bit laws reproducing themselves on gasket levels 2 to 5 include the implication laws, two-sector laws whose support is a one-way implication; they compose like a chain and build every gasket. In the classified strata the only knots that do so are signed not-all-equal and its rotations.

Recoloured, fair visible distinctions orient the hidden implications of a not-all-equal world by coin flips, reach through them is bond percolation at exactly 1/2, and a world without a directed crossing of its cut torus in one direction is consistent. So global emergence vanishes below a lattice's bond threshold, stays bounded on the square lattice, where self-duality puts the threshold at 1/2, and diverges when two-way contacts percolate.

Among 204 placements of three-variable knots as fair contacts on square-lattice tori, six classes have a vacuum and hold nothing, parity and swap-or-order frustrate locally, and only not-all-equal and 'x or (h ≠ h′)' hold purely global emergence. The latter is consistent exactly when its disequality edges, bond percolation at 1/2, are bipartite, so its emergence is bounded on both sides and is a torus winding quantity that Pinson's conformal formula predicts.

A fair not-all-equal world is inconsistent exactly when the swap-reversed random orientation of its bipartite double cover has a swap-invariant strongly connected component, that is when its cyclic edges form a cross cluster or an odd winding. On twisted tori primal and dual cyclic edges are exactly complementary, so P(all contractible) = P(cross) and the emergence lies between -log2(1 - P0) and -log2 P0.

On any square-lattice torus, the primitive classes of an orientation's directed closed walks are exactly those in a closed convex cone, and the rotated dual orientation realizes exactly those in its polar; the uniform orientation's cone has a polar-invariant law. So the conjecture that strongly-connected percolation wraps in one direction with probability 3/4 is equivalent to: with probability 1/4, no directed closed walk winds horizontally.

Flipping the sign of a path through degree-2 vertices whose ends are also joined by links of both signs keeps a signed graph's chromatic polynomial, and so its number of acyclic orientations, though not its Tutte polynomial. One such flip explains each of the 48 skeletons that isomorphic pieces left unexplained on L(3,1,3,1), so every exact twist relation found on the genus-2 surfaces now holds skeleton by skeleton.

In every two-literal tiling world on a triangular torus, shared coins included, a configuration contradicts along no class, one class, or every class its twist makes odd, so three cylinders certify a lone line there. Three-place worlds can contradict along some odd classes and not others, so their lone lines need not be lines, which inflated the dual-pair ratios previously reported, although lines pure among eight classes still differ.

On every closed orientable surface, the probability that the twisted strong components wind along a class c and nothing else is the same for a twist and for that twist plus the dual of c, under fair orientations, by a wall involution along the loop. Seams next to a lone loop carry only that dual twist (Lefschetz duality), and in three dimensions no wall exists around a loop.

On every square-lattice torus, at every shear and for every primitive class c, the two twists odd on c give exactly the same probability that the twisted strong components wind along c and nothing else, for fair orientations. An explicit involution proves it: sliding the second twist's seam onto the upper boundary of the odd loop, with bit flips on edges entered from the free side, changes no directed cycle.

In twisted fair worlds on tori, a lone contradiction line has the same count in the two twists odd on its class whenever each coin decides a contact on two places, a theorem on every torus. Exact enumeration of the smallest tori finds every pair not related by symmetry unequal once one coin decides three places, through a three-place relation or two clauses sharing it: the line falls at a coin's reach, not clause width.

Superseded by LINE-DETECTOR-20261023-001-D001.

For fair contacts with three hidden places, a twist is a cohomology class exactly when flipping all three places swaps the contact's two relations. When the doubly allowed patterns tile the plane, no planar patch and no fitting twist contradicts, and every minimal contradiction on a torus winds, along classes on which the twist differs from every fitting twist.

On every closed surface, the number of fair orientations whose cyclic edges form a given skeleton is, in each twist, the number of orientations of the skeleton with every edge cyclic times the number of acyclic orientations of the graph with each cluster shrunk to a point. Walls are odd clusters, and for any rank they leave the counts unchanged under the duals of the radical of the cluster's lattice.

On square-lattice tori, shearing by the width relabels the four twist sectors and the mirror maps each sector to itself, for every orientation, so the effect of the second twist splits into a part with whole harmonics of the shear and a part with only half-integer harmonics that vanishes at half shear. Additivity of P(all strong components contractible) across the sectors is therefore a relation of one basis, dephased by any shear, and it is not exact at finite size.

At three places, rigidity in direction is decided by the geometry of walls, not by algebra: all three-place tiling contacts have trivial polymorphisms, and renaming the places of one relation can change its kind. Contacts with free stripes or straight pinned walls are rigid on every torus, with an exact count; the twelve whose walls can turn are selective, and when their walls have no choices a direction contradicts exactly where every wall collides with its own copy.

For the six directed three-place tiling contacts, whether a direction contradicts is decided in polynomial time: their walls are monotone paths whose good walls form a lattice, Aubry's ordered maximum and a split shorten any good wall to one that is simple on the cover undoing the twist, and max-closed constraints decide each period. Among the bending contacts only the four honeycomb ones with a free place now lack a polynomial rule.

For every three-place tiling contact with a constant frame, a direction has an assignment exactly when some periodic wall crosses its cylinder without meeting its own copies; the rule needs no determinism, because a cylinder with an assignment has a periodic one. Planted staircases make eight of the twelve bending contacts selective on every torus with sides at least 3; for the other four, good walls must sometimes repeat a state, and selectivity on every torus is open.

Counting homomorphisms from knot groups (first homology Z) to S4 that send meridians to transpositions is #P-hard: on tripled alternating diagrams of plane graphs the S3 colourings are free potentials, and the S4 lifts follow the left-right cycles of each monochromatic subgraph, giving a twelve-state model equal to the 12-state Potts model at v = 1/2 for links. Every group with an abelian normal subgroup and nilpotent quotient stays polynomial with bounded mod-p homology.

Every S4 transposition colouring of a knot has trivial longitude, and over each S3 colouring the twisted lifting cohomology meets the boundary torus in a line over the meridian (half-lives-half-dies), so transposition and 4-cycle lifts are equinumerous. Hence the connected quandles C[6,1] and C[6,2] colour every classical knot equally, proving the 2013 conjecture of Clark, Elhamdadi, Hou, Saito and Yeatman for this pair; a virtual knot shows the 3-manifold input is needed.

Superseded by CORRECTIONS-20261026-001-D002.

For every knot, the S4 counts of 2- and 4-surgery satisfy C2 + C4 = 8 + 3T, and 3 C2(K) + C2(K # trefoil) = 16 + 12T, where T counts transposition colourings. With the independently replayed knot hardness, counting homomorphisms to S4 is #P-hard on closed 3-manifolds with first homology Z/2, where metabelian targets stay polynomial.

Dehn-filling parallel copies of the edge curves by square roots kills the reflection channels of the order-16 dihedral group, so interpolation recovers 4^m 8 P(G,4)/2^|V|, and an index-p theorem gives explicit chromatic coefficients for every group with an abelian normal subgroup of prime index acting nontrivially on its p-th powers. Hardness for every non-abelian group already follows from Galindo's 2026 Turaev-Viro dichotomy, which also predates HARD-3D-20261025-001's novelty claims.

Counting homomorphisms from 3-manifold groups into S3 is #P-hard: handlebodies with tori attached along a planar graph turn it into a spin model that recovers 3-colourings, and their closed doubles count pairs of colourings, 2 P(G,9)/3^|V|. The same holds on closed manifolds for most dihedral groups, Q8 and A4, so for class-2 targets bounded H2 is needed.

Superseded by CORRECTIONS-20261026-001-D003.

SL(2,3) is polynomial on closed 3-manifolds with bounded first homology: its maps are A4 maps forming affine twisted-cocycle spaces over boundedly many maps to Z3, and the lift through the centre is obstructed by a quadratic function into the bounded group H^2(M; F2), whose zeros Gauss sums count. Since SL(2,3) is not abelian-by-nilpotent while S4 and its double covers are #P-hard, the bounded-homology line separates Fitting heights 2 and 3 as far as tested.

Superseded by CORRECTIONS-20261026-001-D001.

For Heisenberg groups over odd prime fields with a coprime cyclic action, counting homomorphisms from 3-manifold groups with bounded first homology is #P-hard exactly when the action's three characters are nontrivial (else polynomial), already on knot surgeries and knot groups, because Borromean moves steer a three-way eigen-triple form into a Potts gadget. Hence Rowell's 2009 conjecture of polynomial exact counting on link groups for solvable targets fails, also for S4, unless #P functions are polynomial-time computable.

SL(2,3) is polynomial on closed 3-manifolds with bounded first homology: its maps are A4 maps forming affine twisted-cocycle spaces, and the central lift is obstructed by a quadratic function into the bounded group H^2(M; F2), counted by Gauss sums. The bounded-homology line is not the Fitting height: UT3(F4) x| C3, also of Fitting height 2, is #P-hard because its twisted obstruction is a free three-way form.

Every S4 transposition colouring of a knot has trivial longitude, and over each S3 colouring the transposition and 4-cycle lifts are equinumerous, re-proving Livingston's 1995 Theorems 8 and 9 by twisted lifting cohomology. Hence the connected quandles C[6,1] and C[6,2] colour every knot equally; the 2013 conjecture of Clark, Elhamdadi, Hou, Saito and Yeatman for this pair was already implied by Livingston.

Handlebodies with tori attached along a planar graph turn counting maps into S3 into a spin model that recovers 3-colourings, and their closed doubles count pairs of colourings, 2 P(G,9)/3^|V|, with analogous chromatic formulas for most dihedral groups, Q8 and A4. The resulting #P-hardness is a case of Galindo's 2026 Turaev-Viro dichotomy; the explicit chromatic constructions are the contribution.

Counting homomorphisms from 3-manifold groups with H1 = Z/3, and from knot groups, into UT3(F4) x| C3, a solvable group of Fitting height 2, is #P-hard: the twisted obstruction is one cubic form whose alternating part Borromean moves steer freely, and an edge gadget turns its kernel count into a Potts partition function. So the bounded-homology line is not the Fitting height, and Rowell's 2009 conjecture fails for this group unless #P functions are polynomial-time computable.

Counting homomorphisms from 3-manifold groups with H1 = Z/3 into UT3(F4) x| C3, a solvable group of Fitting height 2, is #P-hard: the twisted obstruction is one cubic form whose alternating part Borromean moves steer freely, and an edge gadget turns its kernel count into a Potts partition function. So with bounded homology the easy/hard line is not the Fitting height but pairwise versus three-way twisted obstructions.

Superseded by CORRECTIONS-20261026-001-D004.

For coprime p-group-by-cyclic targets on closed 3-manifolds with bounded first homology, hardness past class 2 does not need a three-way pair: a cross-layer hard pair already gives #P-hardness, and at r = 2, where no hard pair exists and class 2 is always polynomial, counting maps from 2-surgeries on knots into U4(F_p) x| <diag(1,-1,1,-1)> (p >= 5) is #P-hard, carried by a twisted triple Massey product read against a fourth class.

For coprime class-2-by-cyclic groups of class (K) over odd prime fields with r | p - 1, counting homomorphisms from closed 3-manifold groups with bounded first homology, and from knot groups, is #P-hard exactly when a commutator joins two nontrivial characters with nontrivial product (else polynomial), via sector partition functions isolated over disjoint unions and Bulatov-Grohe. Hence Rowell's 2009 conjecture fails for every such group unless every #P function is polynomial-time computable.

For p-groups of exponent p with a coprime cyclic action, counting maps from closed 3-manifolds with H1(M; F_p) = 0 is polynomial at a sector order whenever nontrivial characters bracket only with their inverses, collapsing the action to an involution, and that involution's Lie triple system is abelian; for graded class at most 4 it is #P-hard when pairs hold and a triple survives, at every sector order. Pair failures in classes 3-4 remain open.

On closed 3-manifolds without F_p-homology, every map over a sector of order d lands in the subgroup generated by the nonzero-character part, so trivial-character generators drop out and the earlier cross-layer example is explained. Hardness passes from any quotient with a pinned surgery construction whose knots avoid the kernel's characters, which decides 153 of 329 residual census algebras.

Gadget knots whose blocks return exactly to the meridian value make the count an edge-local integer partition function whose edge count is maximal on the diagonal, where the Borromean insertion is trivial, and strictly smaller across a pair-failing difference. So higher layers cannot cancel hardness, and with a character-0 refinement this decides 55 interacting census algebras.

When a gadget vertex reads the bracket of an edge's own leaves, its equations form a flow conserved at every vertex, and Fourier duality turns the count into a spin system on vertex states and potentials. If the flow vanishes off the diagonal and is bilinear on it, the weights are non-negative and the Brunnian rank witness gives #P-hardness, deciding 8 more census algebras.

For p-groups of exponent p with an involution, counting maps from closed 3-manifold groups with H1(M; F_p) = 0 is polynomial whenever the Lie triple system of the minus part is abelian, in every class, since every homomorphism then lies in a class-2 subgroup; and for graded algebras of class at most 4 it is #P-hard otherwise, already on 2-surgeries of knots. So at r = 2 brackets of two minus elements never matter and brackets of three decide.

When a class-3 edge fails only the vertex-root condition and its dual slot carries no degree-1 data, the conserved vertex flow and the dual slot's constraints are linear in that slot's data, so every Fourier-dual weight counts solutions of linear equations and is non-negative. With the Brunnian rank witness this gives #P-hardness, deciding 8 more census algebras and, with conserved flows, the whole vertex-root family.

If zero-jump gadget counts are local up to a relabeling of degree-3 data, the indeterminacy of a Massey-type obstruction, then linear flows on single edges keep the Fourier dual non-negative even with independent potentials at the two ends of an edge, so counting is #P-hard. Six class-4 census algebras satisfy the single-edge conditions, so they are #P-hard if that locality holds.

The Lean statement behind Braun's claimed exponential lower bound for the bit pigeonhole principle in DAG-like Res(⊕) covers every standard rule set (any sound inference from two earlier lines, reuse unrestricted, every line counted) and exactly the usual clauses with 2^l+1 pigeons. It is not vacuous: the clause set is unsatisfiable for every l, and the system refutes l = 1.

For l >= 32, every DAG-like Res(⊕) refutation of bit PHP with 2^l holes and m pigeons, for 2^l < m with 8192 m l^2 (2l+5) <= 4^l, has more than exp(4^l/(8192 m l^2)) lines, checked in Lean against Braun's replayed development. A proper colouring with m classes turns every bit-colouring refutation into one of bit PHP with m pigeons of the same size, so the bound covers every graph whose chromatic number lies in that range.