Vector Substrates in Materials & Computation

• Vector Substrates are decisive platforms that preserve structural information across diverse systems, serving as crystalline membranes, computational algebras, or vector execution layers.
• In oxide integration, a transferred BaTiO3 membrane provides a crystallographic template that promotes high-quality epitaxy and improved ferroelectric performance with PZT.
• In vector-symbolic computation and processor architecture, vector substrates enable efficient binding operations and flexible register management, overcoming conventional system bottlenecks.

The expression “Vector Substrates” (VS) is used in multiple research contexts to denote an underlying substrate that carries decisive structure across otherwise dissimilar systems. In oxide integration, a vector substrate is a transferred crystalline oxide membrane that conveys crystallographic information—especially out-of-plane orientation and in-plane registry—to a film grown on top of it after transfer onto a dissimilar host platform (Haque et al., 7 Sep 2025). In vector-symbolic computation, it denotes a fixed-width, algebraically compositional, high-dimensional vector medium in which binding, bundling, similarity, and inverse operations implement symbolic-style computation (Bazhenov, 2022, Shaw et al., 9 Jan 2025, Alam et al., 2024). In vector computer architecture, it denotes the vector execution substrate exposed to software and compiler, including register count, grouping, and strip-mining behavior (Xu et al., 22 Apr 2025).

1. Terminological scope and acronymal ambiguity

The provided literature uses the term in multiple ways, and it also uses the acronym VS in an entirely different sense in biomolecular screening. In the biomolecular literature, VS usually means virtual screening, including sequence-based virtual screening (SVS) and structure-based hit discovery workflows, rather than vector substrates (Shen et al., 2022, He et al., 10 Nov 2025, Zhang et al., 14 Aug 2025, Haque et al., 7 Sep 2025, Xu et al., 22 Apr 2025).

Context	Meaning of “vector substrate” or “VS”	Representative source
Oxide integration	Transferred crystalline membrane conveying crystallographic information	(Haque et al., 7 Sep 2025)
Vector-symbolic computation	High-dimensional distributed vector substrate for binding, bundling, similarity, and inverse operations	(Bazhenov, 2022, Shaw et al., 9 Jan 2025, Alam et al., 2024)
Vector processor architecture	Vector execution substrate defined by register count, grouping, and strip-mining behavior	(Xu et al., 22 Apr 2025)
Biomolecular screening	VS = virtual screening	(Shen et al., 2022, He et al., 10 Nov 2025, Zhang et al., 14 Aug 2025)

This terminological split matters because the materials-science meaning, the computational meaning, and the processor-architecture meaning all concern an operative substrate, whereas the biomolecular meaning uses the same acronym for a screening task.

2. Vector substrates in oxide integration

In the oxide-integration literature, a vector substrate is explicitly defined as a freestanding or transferred crystalline oxide membrane that conveys crystallographic information—especially out-of-plane orientation and in-plane registry—to a film grown on top of it after transfer onto a dissimilar host platform (Haque et al., 7 Sep 2025). In this formulation, the vector substrate is not merely a physical support, seed layer, or buffer layer. The transferred BaTiO$_3$ membrane provides the overlying Pb(Zr$_{0.52}$Ti$_{0.48}$)O$_3$ film with a dominant $(001)$ / $(00l)$ out-of-plane template and an in-plane cube-on-cube alignment template, with the structural relation described as

$$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$

and approximately

$$[100]_{\mathrm{PZT}} \parallel [100]_{\mathrm{BTO}}, \qquad [010]_{\mathrm{PZT}} \parallel [010]_{\mathrm{BTO}}.$$

The need for this strategy is tied to the longstanding difficulty of integrating high-performance ferroelectric oxides with silicon. The stated obstacles are lattice mismatch, thermal expansion mismatch, interfacial chemical incompatibility, and the need for high-temperature epitaxy directly on Si. The VS approach addresses this by decoupling growth of the crystal template from final integration onto Si: the membrane is grown first on a lattice-matched oxide substrate under epitaxial conditions and only later transferred onto Pt/Ti/SiO$_2$/Si, after which PZT is deposited by chemical solution deposition (CSD) and crystallized at $650^\circ\mathrm{C}$ (Haque et al., 7 Sep 2025).

A central mechanistic point is that the immediate growth interface seen by the PZT is BTO, not Pt. The paper attributes the resulting dominant $_{0.52}$0 orientation and cube-on-cube epitaxy to nucleation on a single-crystal perovskite surface, rather than on a metal electrode surface that would ordinarily yield only oriented or polycrystalline growth (Haque et al., 7 Sep 2025).

3. BaTiO$_{0.52}$1 membrane transfer, crystallography, and device performance

The reported donor heterostructure is

$_{0.52}$2

with SrVO$_{0.52}$3 as a sacrificial layer and SrTiO$_{0.52}$4 as the original growth substrate. A PMMA support layer is spin-coated and soft-baked at

$_{0.52}$5

after which the sample is immersed at room temperature in an aqueous KI + HCl etchant. This chemistry dissolves the SVO sacrificial layer in about

$_{0.52}$6

compared with $_{0.52}$7 days in water for SrVO$_{0.52}$8 and approximately 8 days in water for CaVO$_{0.52}$9 and SrMoO$_{0.48}$0 (Haque et al., 7 Sep 2025).

After transfer to Pt/Ti/SiO$_{0.48}$1/Si, the membrane preserves crystalline quality. The paper reports sharp BTO $_{0.48}$2 and $_{0.48}$3 peaks after transfer, disappearance of the SVO peaks, and rocking-curve full width at half maximum values of $_{0.48}$4 for transferred BTO $_{0.48}$5 and $_{0.48}$6 for as-grown BTO $_{0.48}$7. The subsequently deposited PZT shows predominant $_{0.48}$8 orientation, a fourfold $_{0.48}$9-scan of PZT $_3$0, spot-like selected-area diffraction, and a sharp PZT/BTO interface in cross-sectional HAADF-STEM. Surface morphology is also improved, with

$_3$1

for PZT on VS and

$_3$2

for PZT on Pt/Si (Haque et al., 7 Sep 2025).

The electrical and electromechanical results are likewise framed as evidence that the transferred membrane functions as a crystallographically directive substrate. At room temperature and 10 kHz, PZT on VS exhibits remanent polarization

$_3$3

and coercive field

$_3$4

Ferroelectric endurance remains stable up to

$_3$5

on the VS, whereas the PZT on Pt-Si control fatigues beyond about

$_3$6

From piezoelectric butterfly loops, the extracted effective coefficient is

$_3$7

for PZT on VS and

$_3$8

for PZT grown on conventional Pt Si substrates. The final interpretation is that the VS converts an otherwise polycrystalline Pt/Si surface into a crystallographically directive oxide platform (Haque et al., 7 Sep 2025).

4. Vector substrates in vector-symbolic computation

In vector-symbolic computation, the substrate is not a physical membrane but a computational substrate built from high-dimensional distributed vectors and a small algebra of operations. The cited work describes vector-symbolic architectures (VSAs) as methods of computation in which concepts are represented by long hyperdimensional symbols, dimensionality is kept fixed across processing steps, and structure is manipulated by algebraic operations on vectors rather than by changing tensor shape as in convolutional systems (Bazhenov, 2022).

A concrete instance is the Fourier Holographic Reduced Representation (FHRR) substrate. In that formulation, each symbol is an $_3$9-dimensional vector of normalized phases in $(001)$0, with complex representation

$(001)$1

The paper defines similarity as

$(001)$2

bundling as

$(001)$3

and binding as

$(001)$4

with generalized bundling promoted to a trainable neural layer,

$(001)$5

On this basis, residual and attention-based neural architectures are expressed in substrate-native terms. Residual connections use binding rather than additive skips, and attention is reformulated as

$(001)$6

The same symbolic attention architecture is reported for both image classification and molecular toxicity prediction, and the reported FashionMNIST accuracies are 85.8% for a 24-layer residual FHRR network, 88.6% for symbolic self-attention, and 85.5% for symbolic cross-attention (Bazhenov, 2022).

The defining property of the substrate in this literature is therefore operational: the same fixed-width vector space and the same algebra implement representation, composition, residual learning, attention, and output decoding. The paper explicitly characterizes this as a common, reusable algebraic medium for computation across domains (Bazhenov, 2022).

5. Formal and linearized substrate theories

Two additional lines of work attempt to formalize or redesign the vector-symbolic substrate itself. A categorical proposal states that a VSA is a (division) rig in a category enriched over a monoid in $(001)$7, the category of Lawvere metric spaces (Shaw et al., 9 Jan 2025). In that formulation, the essential ingredients of the substrate are a representational carrier $(001)$8, a similarity operation

$(001)$9

bundling

$(00l)$0

binding

$(00l)$1

unbinding

$(00l)$2

and, when needed, a braiding or permutation operator

$(00l)$3

Worked examples include function encoding,

$(00l)$4

tuple encoding,

$(00l)$5

and list or tree encodings built from repeated braiding or repeated binding. This line of work is foundational rather than algorithmic, but it makes metric structure and algebraic structure coequal components of the substrate (Shaw et al., 9 Jan 2025).

A more operational redesign is the Hadamard-derived linear Binding (HLB) substrate, which is derived from the Hadamard matrix and then recast into an elementwise real-valued form (Alam et al., 2024). The starting Hadamard-domain binding is

$(00l)$6

with projection

$(00l)$7

and the final operational substrate becomes

$(00l)$8

To stabilize division-based unbinding, the paper introduces a Mixture of Normal Distributions (MiND) initialization,

$(00l)$9

It then derives the retrieval relation

$$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$0

and the norm relation

$$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$1

The reported computational advantage is $$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$2 binding and unbinding, and the empirical claim is that HLB is competitive on classical VSA retrieval while outperforming HRR, VTB, and MAP across the reported CSPS and XML datasets (Alam et al., 2024).

Taken together, these works treat the vector substrate as a mathematically structured carrier that can be formalized abstractly, or simplified into a hardware- and autodiff-friendly binding algebra.

6. Vector execution substrates in processor architecture

In processor architecture, the term shifts from symbolic algebra to the vector execution substrate itself. The Zoozve proposal is explicitly framed as a vector-substrate innovation on top of RISC-V-style vector execution that replaces RVV’s fixed-count, power-of-two-grouped vector register model with a more elastic substrate supporting flexible vector register count, flexible effective register-group size, and direct support for ultra-long vectors without strip-mining (Xu et al., 22 Apr 2025).

The architectural motivation is the mismatch between very long vector workloads and RVV’s substrate. In RVV, long logical vectors are partitioned into strips, and register grouping is limited to power-of-two LMUL choices. Zoozve responds by redefining the register substrate. It introduces a v_head field that names the starting address of the vector register range for each operand, supports access to up to $$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$3 vector registers with further extension through vsetcsr, and carries target vector length explicitly in a scalar register rs_avl. The compiler or hardware derives the occupied range from

$$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$4

and

$$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$5

This yields arbitrary register grouping, including asymmetric source and destination groups for operations such as gather and scatter (Xu et al., 22 Apr 2025).

The compiler implementation is equally substrate-oriented. The workflow uses Clang built-ins, intrinsics, a custom intrinsic splitting pass, modified register allocation that forces grouped virtual registers into consecutive physical vector registers, and a Zoozve assembly coalescing pass that reconstructs a single instruction after allocation. The paper describes this as data-adaptive register allocation, because the required contiguous range depends on the vector value type and target length (Xu et al., 22 Apr 2025).

The proof-of-concept hardware is implemented in SystemVerilog on an Ara-like vector architecture. Hazard detection is performed over register ranges $$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$6, and the datapath adds a shuffle engine with a crossbar and multiple processing elements to support inter-lane asymmetric operations. For a 64-lane, 1024-register configuration in SMIC 40nm at 400 MHz, the reported costs are 7.2 mm$$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$7 synthesis area, 11.9 mm$$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$8 layout area, and 5.2% area overhead relative to the baseline implementation. The reported dynamic-instruction-count results are a minimum 10.10$$(001)_{\mathrm{PZT}} \parallel (001)_{\mathrm{BTO}},$$9 reduction for FFT, constant 17 instructions for dotproduct versus 52 to 1292 for RVV with up to 76$$[100]_{\mathrm{PZT}} \parallel [100]_{\mathrm{BTO}}, \qquad [010]_{\mathrm{PZT}} \parallel [010]_{\mathrm{BTO}}.$$0 speedup, and constant 12 instructions for axpy versus 25 to 707 for RVV with 58.92$$[100]_{\mathrm{PZT}} \parallel [100]_{\mathrm{BTO}}, \qquad [010]_{\mathrm{PZT}} \parallel [010]_{\mathrm{BTO}}.$$1 speedup (Xu et al., 22 Apr 2025).

Here, the substrate is not a data structure or algebraic carrier but the machine-visible register-and-execution medium on which long-vector semantics are realized.

7. Cross-domain interpretation, misconceptions, and limitations

The provided literature suggests a common pattern: a vector substrate is the operative layer that preserves the structure that matters while relaxing a bottleneck imposed by the host system (Haque et al., 7 Sep 2025, Bazhenov, 2022, Shaw et al., 9 Jan 2025, Alam et al., 2024, Xu et al., 22 Apr 2025). In oxide integration, the transferred BTO membrane preserves crystallographic registry while the host platform becomes Pt/Ti/SiO$$[100]_{\mathrm{PZT}} \parallel [100]_{\mathrm{BTO}}, \qquad [010]_{\mathrm{PZT}} \parallel [010]_{\mathrm{BTO}}.$$2/Si. In vector-symbolic systems, the fixed-width vector algebra preserves compositionality while allowing the same architecture to process images, molecular graphs, functions, tuples, lists, and trees. In Zoozve, the elastic register substrate preserves vector semantics while removing RVV’s fixed-count, power-of-two grouping and the resulting strip-mining overhead.

Several misconceptions are addressed directly by the sources. In the oxide context, a vector substrate is not merely a physical support, seed layer, or buffer layer (Haque et al., 7 Sep 2025). In the computational context, a vector substrate is not simply “using vector features”; it is a substrate in which the core computations are themselves built from binding, bundling, similarity, and inverse operations (Bazhenov, 2022, Shaw et al., 9 Jan 2025, Alam et al., 2024). In biomolecular research, VS generally abbreviates virtual screening, as in SVSBI, S$$[100]_{\mathrm{PZT}} \parallel [100]_{\mathrm{BTO}}, \qquad [010]_{\mathrm{PZT}} \parallel [010]_{\mathrm{BTO}}.$$3Drug, or HelixVS, and should not be conflated with vector substrates (Shen et al., 2022, He et al., 10 Nov 2025, Zhang et al., 14 Aug 2025).

The limitations are equally domain-specific. The oxide work explicitly notes transfer-induced angular misalignment, some non-templated growth outside the membrane area, no large-area scaling demonstration yet, and no detailed strain/mismatch quantification (Haque et al., 7 Sep 2025). The categorical account is presented as a first attempt, with no complete axiomatization of a category $$[100]_{\mathrm{PZT}} \parallel [100]_{\mathrm{BTO}}, \qquad [010]_{\mathrm{PZT}} \parallel [010]_{\mathrm{BTO}}.$$4 and no finished account of how all VSA families instantiate the proposal (Shaw et al., 9 Jan 2025). The FHRR residual and attention architectures are promising but the authors explicitly state that the molecular-toxicity results are not state of the art (Bazhenov, 2022). The HLB work provides strong empirical evidence but no detailed gradient-theory analysis (Alam et al., 2024). Zoozve is marked WIP, and the paper lists no full formal ISA specification, no detailed power analysis, no cycle-accurate performance model, and limited discussion of memory-system, spill, and OS implications (Xu et al., 22 Apr 2025).

Across these literatures, the term therefore does not denote a single object class. It denotes a structurally privileged substrate—crystallographic, algebraic, or architectural—that determines how vectors carry order, interaction, or execution.