Seedless: Cross-Disciplinary Methods

• Seedless is a cross-disciplinary concept that removes the need for an externally supplied seed, shifting initialization complexity into geometry, statistics, or exhaustive search.
• Its applications span colloidal crystallization, nanocrystal synthesis, gravitational-wave clustering, quantum randomness extraction, and graph matching, each employing field-specific compensatory mechanisms.
• Methodologies include diffusio-electric flows in colloidal assembly, template banks in gravitational-wave analysis, and deterministic maps in cryptography, providing scalable and efficient alternatives.

Seedless is a cross-disciplinary descriptor for methods that eliminate an externally supplied initialization object—such as a crystal seed, a pre-matched node set, a thresholded pixel, a uniformly random extractor seed, a seed magnetic field, or a designated seed assembly—and still obtain controlled growth, inference, or extraction (Niu et al., 2018, Thrane et al., 2014, Foreman et al., 2024, Cannon et al., 2012). This suggests a broad methodological family rather than a single formalism. In the cited literature, seedless methods appear in colloidal crystallization, nanocrystal synthesis, gravitational-wave analysis, quantum randomness extraction, graph matching, tile self-assembly, plasma-field generation, Feynman-integral reduction, NMR pulse design, and jet finding.

1. Terminological scope and recurrent structure

In colloidal materials, seedless refers to transport and nucleation without a rigid seed in the growth center. In nanocrystal chemistry, the “Sodium citrate seedless method” or Turkevich method performs reduction, nucleation, and growth in one pot rather than by adding preformed seeds (Niu et al., 2018, Bai et al., 2022). In gravitational-wave searches, seedless clustering avoids first identifying bright “seed” pixels and instead integrates spectrogram power along hypothesized tracks (Thrane et al., 2013). In graph matching, seedless methods remove the need for a seed set of correctly matched vertex pairs (Hayhoe et al., 2018). In device-independent cryptography, a seedless extractor is a deterministic map $\mathrm{Ext}:\{0,1\}^n\to\{0,1\}^m$ with no additional randomness (Foreman et al., 2024). In the two-handed tile-assembly model, seedless means that there is no distinguished seed tile or seed assembly (Cannon et al., 2012).

Across these usages, the common technical problem is not the absence of all prior structure, but the removal of one privileged initialization resource. The compensating mechanisms differ by field. Colloidal assembly uses diffusio-electric and electro-osmotic transport; gravitational-wave pipelines use large template banks or time-delay trials; quantum extraction replaces an independent seed by Bell-violation promises or by a second entropy source; network alignment uses centrality, percolation, or structural signatures; hierarchical self-assembly permits pairwise supertile combination; and jet algorithms enumerate all stable cones rather than iterating from seeds. A plausible implication is that “seedless” usually shifts complexity away from initialization and into geometry, statistics, or exhaustive coverage.

2. Seedless growth, nucleation, and synthesis

A clear physical realization appears in seedless assembly of colloidal crystals by inverted micro-fluidic pumping. A millimeter-sized cation-exchange resin fixed to the top glass of a closed cell exchanges trace impurity cations for protons, creating a radial pH gradient. Different diffusivities of $\mathrm{H}^+$ and the impurity cations produce an inward pointing diffusio-electric field $E(r)$, and on the negatively charged bottom substrate this field drives a converging electro-osmotic flow. At low ionic strength, colloidal particles are entrained by this quasi-two-dimensional flow and transported to the projected center of the ion-exchange resin without any solid seed. Because there is no rigid seed in the center, nucleation begins uniformly underneath the pump, avoiding voids and seed-crystal incommensurability that typically introduce defects (Niu et al., 2018).

The flow field is quantified by

$v(d)\simeq C(R,H)\,d^{-n},$

with $n\approx 1$ in the quasi-two-dimensional confined regime and $n\approx 2$ when the ion-exchange-resin size approaches the cell height. The prefactor scales approximately as

$C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$

Empirically, at $H=1\,\mathrm{mm}$ and $R\approx 250\,\mu\mathrm{m}$, the measured radial surface flow is $v(d)\simeq (0.3$–$\mathrm{H}^+$0 for $\mathrm{H}^+$1 and obeys $\mathrm{H}^+$2. Varying $\mathrm{H}^+$3 from $\mathrm{H}^+$4 to $\mathrm{H}^+$5 decreases the peak flow by factors $\mathrm{H}^+$6. The crystal radius follows the transport-limited law

$\mathrm{H}^+$7

reaching $\mathrm{H}^+$8–$\mathrm{H}^+$9 in $E(r)$0, while vacancies, dislocations, and grain boundaries anneal to final defect fractions $E(r)$1. With two pumps spaced at $E(r)$2, the average lattice orientation $E(r)$3 aligns within $E(r)$4 of the pump axis, and arrays of pumps generate hexagonal, star-shaped, and “JGU” motifs (Niu et al., 2018).

In colloidal chemistry, the same label denotes one-pot citrate reduction. In the Turkevich route, seedless synthesis proceeds by simultaneous reduction, nucleation, and growth of metal atoms in boiling aqueous solution of metal precursor and trisodium citrate. The overall reductions are written as

$E(r)$5

and

$E(r)$6

The mechanism is described by a LaMer–Dinegar sequence: monomer supersaturation, burst nucleation above a critical supersaturation $E(r)$7, diffusion-limited growth, and citrate capping. For Au–Ag composites, two pathways are proposed: “silver attach-on-Au,” in which Ag nucleates on citrate-reduced Au colloids, and “Ag-core / Au-shell via galvanic exchange,” governed by

$E(r)$8

The choice between these mechanisms depends on order and rate of addition, and DLS, UV–Vis, and SEM distinguish dispersed 20–30 nm Au colloids, $E(r)$9 nm Ag colloids, 50–100 nm core–shell composites, and micron-scale aggregates under slow Ag reduction or poor solvation conditions (Bai et al., 2022).

3. Seedless clustering in gravitational-wave searches

In long-duration gravitational-wave searches, the seedless idea emerged from a limitation of seed-based clustering. When a transient is long-lived and narrowband, its energy is spread over many time–frequency pixels, so individual pixels may never exceed a seed threshold. Seedless clustering therefore abandons per-pixel thresholding and integrates excess cross-power along entire trial tracks in spectrograms. In the original “stochtrack” formulation, a track is parameterized as a quadratic Bézier curve through three control points, and the cluster statistic is obtained by summing the cross-power signal-to-noise ratio along the curve. Monte Carlo studies with Gaussian Advanced LIGO design-sensitivity noise found distance gains of 150–200% relative to seed-based algorithms at fixed false-alarm and false-dismissal rates, corresponding to detection-volume gains of 420–740% (Thrane et al., 2013).

The all-sky extension replaced a known sky direction by a trial inter-detector time delay $v(d)\simeq C(R,H)\,d^{-n},$0 and evaluated

$v(d)\simeq C(R,H)\,d^{-n},$1

Templates remained quadratic Bézier tracks, but each was tagged with a random $v(d)\simeq C(R,H)\,d^{-n},$2 drawn uniformly over the light-travel interval between detectors. With sufficiently large banks, true signals are expected to be “hit” by a template with nearly correct shape and delay. In Monte Carlo and recolored real noise, all-sky seedless clustering detected benchmark long-lived waveforms at about $v(d)\simeq C(R,H)\,d^{-n},$3–$v(d)\simeq C(R,H)\,d^{-n},$4 the distance of a seed-based algorithm that knew the true sky location, corresponding to $v(d)\simeq C(R,H)\,d^{-n},$5–$v(d)\simeq C(R,H)\,d^{-n},$6 larger sensitive volume. Because all templates are independent, the method is “embarrassingly parallel”; on a single NVIDIA Kepler GK104 GPU, a deep search with $v(d)\simeq C(R,H)\,d^{-n},$7 templates on a $v(d)\simeq C(R,H)\,d^{-n},$8 spectrogram took about $v(d)\simeq C(R,H)\,d^{-n},$9 versus about $n\approx 1$0 on a CPU, a $n\approx 1$1 speed-up (Thrane et al., 2014).

A central practical issue is background estimation. For seedless clustering, the loudest cluster among many templates is an extreme-value statistic. A semi-analytic background model exploited the near-Gaussianity of the single-template statistic,

$n\approx 1$2

and the independence approximation

$n\approx 1$3

For all-sky Bézier searches with $n\approx 1$4 and $n\approx 1$5 time delays, the analytic $n\approx 1$6 threshold was $n\approx 1$7, while time slides gave $n\approx 1$8; for PN chirps, the analytic and measured thresholds were $n\approx 1$9 and $n\approx 2$0. The paper reported threshold prediction to better than $n\approx 2$1 and a reduction in background-estimation cost from $n\approx 2$2–$n\approx 2$3 CPU-hours to $n\approx 2$4 CPU-hours (Coughlin et al., 2015).

Later work compared seedless clustering with hidden Markov model tracking on cross-power spectrograms. In that implementation, a total of $n\approx 2$5 monotonic quadratic Bézier templates were drawn per spectrogram, and the cluster statistic was

$n\approx 2$6

At false-alarm probability $n\approx 2$7, the hidden Markov model recovered 90% of magnetar-spin-down injections at roughly half the $n\approx 2$8 amplitude required by seedless clustering in Gaussian O1-recolored noise, and the relative loss of seedless sensitivity persisted in time-shifted real O2 data. For accretion-disk-instability waveforms, which are well approximated by quadratic Bézier curves, the two methods performed comparably. Measured runtimes were $n\approx 2$9 for intermediate-duration seedless searches and $C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$0 for long-duration searches, compared with $C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$1 and $C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$2 for the HMM tracker, a speedup of $C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$3–35 (Banagiri et al., 2019).

A specialized variant targeted low-mass compact binary coalescences with post-Newtonian-inspired chirp tracks instead of generic Béziers. For low-mass systems, chirp-based seedless templates extended the sensitive distance by $C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$4 relative to generic Bézier templates, corresponding to roughly 240% in volume, and there was no significant sensitivity loss in going from targeted to all-sky searches. This established seedless clustering as a complement rather than a replacement for matched filtering: less model-specific, but generally less sensitive when accurate waveform models exist (Coughlin et al., 2014).

4. Seedless extraction, condensing, and quantum randomness

In randomness extraction and privacy amplification, “seedless” has a literal formal meaning: the extractor is deterministic and uses no independent random seed. In device-independent cryptography, the core proposal is to replace the usual min-entropy promise by a Bell-violation promise. In the memoryless-device model, Alice applies a deterministic function

$C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$5

to her raw outcomes, and security is certified by shifted-CHSH operator bounds rather than by a separate private seed. The simplest extractor is the global XOR,

$C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$6

for which the output distance from uniform is bounded by the expectation of a product of single-round shifted CHSH witnesses. The paper states that this is the first deterministic seedless extractor secure against unbounded quantum adversaries in a device-independent setting, while also emphasizing a limitation: fresh randomness is still needed for setting choices and spot-checking, so the seed is removed only at the extraction step (Foreman et al., 2024).

A subsequent development introduced a truncation-based proof technique that estimates Bell parameters using an asymptotically vanishing fraction of rounds and proves the optimal asymptotic key rate of one key bit per singlet. The protocol sets the estimation fraction to $C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$7, chooses the key as a deterministic linear map $C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$8, and derives a composable security theorem with efficient implementation. In this formulation, seedless extraction is compatible with $C(R,H)\propto R/H^m,\qquad m\approx 1\text{–}2.$9 post-processing and with a class of binary matrices $H=1\,\mathrm{mm}$0 satisfying a combinatorial weight-distribution bound. The resulting asymptotic rate

$H=1\,\mathrm{mm}$1

reaches $H=1\,\mathrm{mm}$2 at maximal CHSH violation $H=1\,\mathrm{mm}$3 (Lin et al., 29 May 2026).

Outside the device-independent setting, seedless extraction has also been developed as a two-source primitive. A real-time two-source quantum-proof extractor treats each QRNG output as a forward block source. If

$H=1\,\mathrm{mm}$4

is a $H=1\,\mathrm{mm}$5–forward block source, then every suffix block tuple has conditional min-entropy at least $H=1\,\mathrm{mm}$6 given any earlier fixed prefix and quantum side information. The extractor applies an inner-product gadget over $H=1\,\mathrm{mm}$7 blockwise,

$H=1\,\mathrm{mm}$8

and produces a constant fraction of the total min-entropy from two infinitely long independent forward block sources. The construction is block-wise parallelizable, and the paper reports a simulated extraction speed as high as $H=1\,\mathrm{mm}$9 on a Xilinx Kintex-7 FPGA (Li et al., 2024).

A related proposal addresses “the randomness loop” in seedless QRNG bootstrapping. Two independent QRNG devices provide raw strings $R\approx 250\,\mu\mathrm{m}$0 and $R\approx 250\,\mu\mathrm{m}$1; one source serves as the extractor input and the other as the seed for a modified Toeplitz hash family. If $R\approx 250\,\mu\mathrm{m}$2 and $R\approx 250\,\mu\mathrm{m}$3, then it suffices that

$R\approx 250\,\mu\mathrm{m}$4

to produce an $R\approx 250\,\mu\mathrm{m}$5-bit output $R\approx 250\,\mu\mathrm{m}$6-close to uniform. The security proof is based on the Quantum Leftover Hash Lemma, and the same framework is extended to hybrid PQC-QKD key combination by modeling PQC keys with HILL entropy (Giestinhas et al., 27 Mar 2026).

The seedless viewpoint also appears in the theory of condensers and in classical variable-length extraction. For $R\approx 250\,\mu\mathrm{m}$7 source models, recent work shows a sharp distinction between online NOSF and fully non-oblivious NOSF sources. When $R\approx 250\,\mu\mathrm{m}$8, condensing with error $R\approx 250\,\mu\mathrm{m}$9 above rate $v(d)\simeq (0.3$0 is impossible for oNOSF sources, and this bound is tight; when $v(d)\simeq (0.3$1, explicit condensers exist for uniform oNOSF sources with output length $v(d)\simeq (0.3$2 and error $v(d)\simeq (0.3$3. By contrast, for NOSF sources, constant-error condensing above rate $v(d)\simeq (0.3$4 is impossible, which rules out non-trivial condensing when $v(d)\simeq (0.3$5 (Chattopadhyay et al., 2023). In a different classical line, variable-length extraction from imperfect stochastic processes defines efficiency by

$v(d)\simeq (0.3$6

For uncertainty parameter $v(d)\simeq (0.3$7, no extractor can exceed efficiency $v(d)\simeq (0.3$8 uniformly over all sources within that uncertainty class, while a seedless cascade using a small extractor-generated seed from independent source fragments achieves the same asymptotic efficiency (Zhou et al., 2012).

5. Seedless network alignment and graph matching

In network alignment, the seedless problem is to recover a structure-preserving correspondence between two graphs without a seed set of known matched pairs. SPECTRE addresses this problem through spectral centralities and percolation. It first ranks vertices by eigenvector centrality, constructs a noisy seed set by pairing top-ranked nodes within a window, then uses bootstrap percolation on the product graph to extract a confident partial matching and expand it. The phases are EstimateSeeds, SafeExpand, and LooseExpand with Backtracking. The point of the method is that the initial seed set may contain a majority of incorrect pairs, yet the subsequent token-spreading dynamics suppress incoherent matches and amplify coherent neighborhoods. The paper states that SPECTRE is the first truly seedless, scalable network alignment method of this kind (Hayhoe et al., 2018).

Formally, for a matching $v(d)\simeq (0.3$9, one objective is to maximize edge correctness,

$\mathrm{H}^+$00

SPECTRE empirically maintains high performance even when graph correlation is not high. On synthetic correlated networks, precision and recall stay above 90% until 80% correlation and remain above 75% at 60% correlation; edge correctness is about 60–70% down to 80% correlation and about 30–40% even at 60% correlation. On protein–protein interaction networks, SPECTRE achieves $\mathrm{H}^+$01, $\mathrm{H}^+$02, precision about $\mathrm{H}^+$03, and recall about $\mathrm{H}^+$04 on degree-$\mathrm{H}^+$05 nodes. Runtime scales nearly linearly with the number of edges and ranges from less than $\mathrm{H}^+$06 to about $\mathrm{H}^+$07 on a 4-core 2.2 GHz machine with 16 GB RAM for graphs with $\mathrm{H}^+$08 to $\mathrm{H}^+$09 edges (Hayhoe et al., 2018).

A different seedless graph-matching strategy uses the tail of the degree distribution of neighboring nodes. For each vertex, the Tail Degree Signature computes breadth-first neighborhoods up to depth $\mathrm{H}^+$10, forms the multiset of degrees at each level, and concatenates the $\mathrm{H}^+$11 smallest and $\mathrm{H}^+$12 largest degree values into a feature vector. Similarities between graphs are then matched by either the Hungarian algorithm or a greedy rule. Under the correlated Erdős–Rényi model, this method is evaluated in the sparse regimes $\mathrm{H}^+$13 and $\mathrm{H}^+$14, where exact recovery is feasible if and only if

$\mathrm{H}^+$15

For $\mathrm{H}^+$16 and $\mathrm{H}^+$17, the TDS methods achieve about 80% accuracy at $\mathrm{H}^+$18 and about 45% at $\mathrm{H}^+$19, while baselines remain markedly lower; on noisy real networks including Bitcoin-OTC, GR-QC, and Facebook, mean accuracy is about 60% for TDS-h and TDS-g, about 30% for DP, and below 20% for other seedless baselines (Bozorg et al., 2019).

These two lines illustrate two distinct meanings of seedlessness in graph inference. SPECTRE manufactures a noisy surrogate seed set from centrality and then bootstraps it, whereas Tail Degree Signature dispenses with seed construction entirely and relies on structurally distinctive node features. The distinction is substantive: one method is seedless at the input level but still internally uses synthetic seeds, while the other remains seedless throughout the matching stage.

6. Formal models, software systems, and additional specialized meanings

Several other literatures use “seedless” in tightly defined ways. In the two-handed model of tile self-assembly, there is no distinguished seed tile or seed assembly; any two producible supertiles may combine if the resulting assembly is $\mathrm{H}^+$20-stable. This seedless model can simulate any seeded aTAM system with only constant blowup, exhibits finite shapes with busy-beaver separation in tile complexity, contains an infinite shape that can be constructed two-handed but not seeded, and renders unique-assembly verification co-NP-complete, whereas the corresponding seeded problem is polynomial-time solvable (Cannon et al., 2012).

In plasma physics, seedless can denote the absence of an externally applied magnetic seed field. Two-dimensional EPOCH simulations of an aluminum tetrafoil target show multi-megatesla magnetic field generation with no seed field at all. The seeded microtube implosion benchmark with a $\mathrm{H}^+$21 seed reaches $\mathrm{H}^+$22 at $\mathrm{H}^+$23, whereas a seedless tetrafoil reaches $\mathrm{H}^+$24 at $\mathrm{H}^+$25 and a mirrored tetrafoil reaches $\mathrm{H}^+$26 with opposite polarity. Hybrid tetrafoil–microtube targets sustain higher flux, about $\mathrm{H}^+$27–$\mathrm{H}^+$28, and keep fields above $\mathrm{H}^+$29 for up to $\mathrm{H}^+$30–$\mathrm{H}^+$31. Mirroring the target reverses the sign of $\mathrm{H}^+$32, so the geometry functions as a passive magnetic-polarity switch (Miranda et al., 9 May 2025).

In perturbative quantum field theory, seedless reduction of Feynman integrals removes the need to preselect a finite set of seed integrals in a Laporta-style IBP reduction. Instead, one constructs a complete set of lowering operators for generic propagator and numerator indices using IBP-generating vectors

$\mathrm{H}^+$33

builds level-by-level systems of relations, and solves for operators that reduce any target integral to simpler integrals and ultimately to master integrals. The resulting method is generic in the exponents and avoids heuristic seed choice, at the cost of a more intricate symbolic construction (Cruz et al., 25 Feb 2026).

In magnetic-resonance pulse design, Seedless is the name of a stand-alone GRAPE-based engine for weakly coupled spin systems that uses RF pulses optimized for a single spin-$\mathrm{H}^+$34 to control multi-spin dynamics. The core single-spin Hamiltonian is

$\mathrm{H}^+$35

and the software analyzes pulses through the schematic decomposition

$\mathrm{H}^+$36

By enforcing this “band-schematic” form uniformly across offsets, Seedless controls chemical-shift and scalar-coupling evolution of the entire weakly coupled system. The paper demonstrates a continuously irradiated joint INEPT that achieves band-selective transfer $\mathrm{H}^+$37 and reports that a single-band schematic pulse with 96 offsets and 3 $\mathrm{H}^+$38 values converges in $\mathrm{H}^+$39 on a modern laptop (Lim et al., 11 Feb 2026).

In collider phenomenology, SISCone is a seedless infrared-safe cone jet algorithm. Instead of starting from energetic seed particles, it finds all stable cones. For low particle multiplicities, an implementation optimized for subset enumeration has stable-cone complexity $\mathrm{H}^+$40 and split–merge complexity $\mathrm{H}^+$41. The low-multiplicity code yields speedups up to about $\mathrm{H}^+$42 for the cylindrical version and up to about $\mathrm{H}^+$43 for the spherical version at $\mathrm{H}^+$44, with crossover around $\mathrm{H}^+$45–10 relative to the standard large-$\mathrm{H}^+$46 implementation (Weinzierl, 2011).

Taken together, these specialized meanings show that seedlessness is neither purely algorithmic nor purely physical. It can refer to hierarchical assembly without a privileged nucleator, spectrogram integration without thresholded pixels, deterministic extraction without private randomness, exhaustive geometric search without seed objects, or symbolic reduction without seed integrals. The unifying pattern is structural: a seedless construction removes a designated starting point and replaces it with a mechanism that is distributed, exhaustive, or certified by an alternative invariant.