Findings

108 = 2² × 3³. Two factors with no common factor must take all of the 2s or none of them, and all of the 3s or none of them. So the coprime splits are exactly 1 × 108 and 4 × 27, and with both factors greater than 1, 4 × 27 is the only one. (In the language of number theory: 108 has four unitary divisors, 1, 4, 27 and 108.) The same count for the other common mala sizes is in Experiment 2, test 4: 27 = 3³ has no such split, 54 has one (2 × 27), and 1008 = 2⁴ × 3² × 7 has three.

Joining each bead k to k + n (mod 108) splits the 108 beads into the cosets of the subgroup generated by n. That subgroup has 108 / gcd(n, 108) elements, so the lines form gcd(n, 108) closed polygons of 108 / gcd(n, 108) beads each: 27 squares for n = 27, 4 rings of 27 for n = 4, 36 triangles for n = 36, 12 nonagons for n = 12, and one loop through all 108 for any n coprime to 108.

The bead address (lap ⌊k/27⌋, turn ⌊k/4⌋, station k mod 27, corner k mod 4) depends on k alone. The thread controls (ring and tube size, petals, rise and fall, cross-section, twist) change only where the torus sits in space, never the order of beads along the thread. So every polygon is defined before any geometry is chosen, and no bending can break it. For n = 27, beads k and k + 27 share a station and differ by one corner, so each square is the tube’s cross-section at one station. For n = 4, the corner stays fixed and the station steps by 4, which visits all 27 stations because 4 and 27 are coprime.

The six weaves in the lab: each is a thread preset, a join step and a flow of beads, with every shape leaving its recent positions behind as cloth. They are drawings of the same arithmetic, not additional results.

The state is a vector ψ ∈ ℂ¹⁰⁸, and |ψₖ|² is the probability of finding the particle at bead k + 1. The Hamiltonian is built from three hopping rules, each a step s with a strength tₛ ≥ 0: thread (s = 1), squares (s = 27) and rings (s = 4), all mod 108:

H = −Σₛ tₛ Σₖ ( |k⟩⟨k+s| + |k+s⟩⟨k| ), i dψ/dt = Hψ (ħ = 1).

With all three rules on, the bead network is the Cayley graph of the cyclic group ℤ₁₀₈ with generator set {±1, ±4, ±27}. Each hopping rule alone is a circulant graph, and so H is real, symmetric and circulant: its eigenvectors are the Fourier modes e^(2πink/108), and its eigenvalues are

E(n) = −2[t₁·cos(2πn/108) + t₂₇·cos(πn/2) + t₄·cos(2πn/27)], n = 0, …, 107,

with t₁, t₂₇, t₄ the three hopping strengths. The 4 × 27 structure shows as the periods 4 and 27 of the second and third terms. The thread’s shape appears nowhere in it. On the page, the lab integrates with fourth-order Runge-Kutta, six substeps per frame, renormalising after each step. The tests use exact diagonalisation (numpy.linalg.eigh). Measure draws a bead with probability |ψₖ|² and collapses the state onto it.

Numbers below are from a fresh run of the replication script, which CI checks against the published answer key on every change. Protocol and code are on the replication page; the original run is experiment-2-four-tests.py with its log quantum-tests-2-log.json.

Test 1, the dark state. Squares only, ψ(0) = (|0⟩ + e^(iφ)|54⟩)/√2. Prediction: P_empty(t) = cos²(φ/2)·sin²(2t) on beads 28 and 82. Maximum over 0 ≤ t ≤ 4:

The first maximum at φ = 0 falls at t = π/4 ≈ 0.785. The state (|1⟩ − |55⟩)/√2 is an eigenvector of the 4-cycle Hamiltonian with eigenvalue 0, so it is stationary; the symmetric combination (|1⟩ + |55⟩)/√2 oscillates with P_empty = sin²(2t). A general phase φ mixes the two with weights cos²(φ/2) and sin²(φ/2), which is the prediction above. At 180° only the stationary part remains, and nothing moves.

Test 2, the factorisation. Squares and rings, strength 1 each, thread off. The network is then the Cartesian product C₄ □ C₂₇, because ℤ₁₀₈ ≅ ℤ₄ × ℤ₂₇ by the Chinese remainder theorem (4 and 27 are coprime). The explicit relabelling is k = (27a + 4b) mod 108, with a ∈ {0, …, 3} and b ∈ {0, …, 26}: a step of 27 moves a by one and leaves b alone, and a step of 4 moves b by one and leaves a alone. Under it, H = H₄ ⊗ I₂₇ + I₄ ⊗ H₂₇ entry by entry, with H₄ and H₂₇ the nearest-neighbour rings; the independent replication verified this exactly. So the return probability factorises, F(t) = F₄(t)·F₂₇(t), with F₄(t) = cos⁴t. Largest gap over 2,000 samples of [0, 4π]: 2.8 × 10⁻¹⁵, which is rounding error. At t = π, 2π, 3π, 4π the 4-ring factor returns to 1 but the 27-ring factor does not, so F stays small: 0.0485, 0.0248, 0.0167, 0.0334.

Test 3, quantum walk and classical walk. Thread only. Spread from bead 1, measured along the ring:

These are the textbook values σ = √2·t for the continuous-time quantum walk and σ = √(2t) for the classical random walk at rate 1 per neighbour: the wave spreads ballistically, the walker diffusively. Konno (2005) gives the limit theorem for the continuous-time walk on the line, where position grows linearly in time; Kempe (2003) sets the quantum and classical scalings side by side, there for the discrete-time walk.

Test 4, other bead counts. For each count N and each coprime split N = p × q with 2 ≤ p ≤ q, one hopping rule of step q joins the beads into p-cycles. The particle starts on bead 1 and the bead ⌊p/2⌋·q places on, and we record the lowest probability left on that starting pair over 0 ≤ t ≤ 12, and how many distinct frequencies carry more than 1% of its oscillation.

Among the common mala counts (27, 54, 108, 1008), 108 is the only one whose coprime split produces a cycle with perfect state transfer: its four-bead squares (section 8). The split of 54 joins the beads in pairs, and a pair transfers perfectly too, but trivially, since a pair has nowhere else to go (the two-site case in Christandl and colleagues, 2004): from bead 1 alone the particle reaches bead 28 with certainty at t = π/4. (With the protocol’s start on both beads of the pair, it never moves, which is the row above.) The cycles of 7, 9 and 16 beads from 1008 have no perfect state transfer.

The earlier guess, that only 108 empties its starting pair completely, was wrong: 1008 empties its pair almost completely too, the 16-cycle to within 10⁻⁶, but through six or more frequencies at irregular times. That emptying is a weaker and different thing from perfect state transfer. The guess is kept here, marked wrong, rather than deleted.

Bead k and beads k ± 27 share a station (k mod 27) and sit on adjacent corners of the tube (27 ≡ −1 mod 4): the circulant chord of length 27 is realised as a spatially local coupling across the tube. The graph it makes, C₁₀₈(27) = 27 · C₄, is standard: 27 disconnected copies of the 4-cycle, whose perfect state transfer is section 8’s. (Bašić, 2011, characterises perfect state transfer on connected integral circulant graphs; this graph is disconnected, so it sits outside that theorem and needs only the 4-cycle result.) Models of helical molecules already let sites couple to their neighbours on the next turn (Guo and Sun, 2014, for electron transport). What we did not find, as far as we could find, is a torus-knot winding that makes the chord of 27 spatially local and reads it as the 27 four-cycles that carry perfect state transfer. Other nearest work: Biswas and Ghosh (2019), a continuous particle on a torus knot, with no sites; Gualdi, Kostak, Marzoli and Tombesi (2008), near-perfect transfer in a long-range, dipolar-like chain by removing the sender’s and receiver’s nearest neighbours.

Which bead is nearest depends on the tube. Mean distances between beads, measured on the lab’s own torus with ring radius 1:

Bead k ± 28 is the same corner at the next station. It is the nearest bead of all at the lab’s default proportions, and the square’s corners are every bead’s nearest neighbours only when the tube radius is below 0.141 of the ring radius. Bead k ± 26 (the opposite corner at the next station) is always farther than k ± 27.

Our check, in one simple model. This is the only place on this page where geometry enters the equations; the lab’s hopping modes and Experiments 1 and 2 use the three hopping rules alone. The lab’s Couplings from distance mode (v5) runs the every-pair model below live, and its Wound chain preset shows the transfer appear as the tube thins. Let every pair of beads couple with strength (d₂₇/d)^α, normalised to the square’s edge, for distances below 2.5 bead spacings, with α = 3 (a dipolar fall-off) or α = 6 (Rydberg-type), and evolve for 0 ≤ t ≤ 40:

At the default proportions the rings of k ± 28 dominate and neither effect appears. In a thin winding the square qubit comes back almost fully. The one-corner transfer is slowed by the coupling across the square’s diagonal (k ± 54), which is (√2/2)^α of the edge at every tube size: 0.354 under 1/d³, 0.125 under 1/d⁶. On a lone square that diagonal does not stop the transfer but delays it. The first pass, near t = π/2, reaches 0.738 under 1/d³ and cos²(π/16) = 0.962 under 1/d⁶; the first arrival above 0.999 comes at t ≈ 26.7 and t = 4π ≈ 12.6. On the knot, the weak couplings along the ring act during that wait, so the transfer completes only on the thinnest tubes. That diagonal leaves the square qubit alone: it shifts the two symmetric states (|1⟩ + |55⟩)/√2 and (|28⟩ + |82⟩)/√2 by the same amount. These are our computations in a model we chose, not a result from the literature. Script and output: wound-chain-check.py.

Every pair, no cutoff. The figure repeats the check with all couplings J = d^(−α), no cutoff, normalised to the bead 1 to bead 28 coupling: start on bead 1, record the best P(55) for 0 ≤ t ≤ 40. It agrees with the table to within 0.012; the only difference is the cutoff. The geometry behind it: each step along the thread is a quarter turn of the tube, so a square’s edge is a chord of √2·r ≈ 1.41 r; stations are 2π/27 ≈ 0.23 of the ring radius apart on average, more on the outside of the tube, where bead 1 sits, and less on the inside; bead k + 28 is the same corner at the next station. The switch falls between tube radii of 0.12 and 0.15; at the lab default of 0.4, P(55) < 10⁻³ under both laws. The r → 0 limit is not ideal square hopping, because the diagonal stays; it delays the transfer, as above. A first-pass run to t = 8 gives 0.962 and 0.879 at a tube of 0.02 and is kept as a file. Script and data: experiment-3-leak-geometry.py, leak-test-geometry.json, leak-test-geometry-first-pass.json.

Open. For two isotropic coupling laws, the threshold is computed above. Whether the transfer survives a physical winding with realistic couplings (anisotropic dipoles, exchange that falls off exponentially, disorder in the bead positions) is unanswered. It would make a good student project.