Local Units as Domain Gates: A Multineutral Algebra View on Retrieval-Augmented Generation

Abstract

We show how to construct local units for each semantic cluster in an embedding space and how these units act as domain-specific masks (gates) in Retrieval-Augmented Generation (RAG). The construction is formulated purely algebraically, dispensing with implementation and revealing the direct connection to multineutral structures .

Full text

Local Units as Domain Gates:
A Multineutral Algebra View on
Retrieval-Augmented Generation
Ondˇrej Br˚uha1
1Alpha Codes s.r.o. ondrej.bruha@alphacodes.eu
2025
Abstract
We show how to construct local units [1] for each semantic clus-
ter in an embedding space and how these units act as domain-specific
masks (gates) in Retrieval-Augmented Generation (RAG). The con-
struction is formulated purely algebraically, dispensing with imple-
mentation and revealing the direct connection to multineutral struc-
tures.
1 Preliminaries
Let E∈Rn×dbe a matrix whose rows eiare d–dimensional text embeddings
and let L∈ {−1,0, . . . , C −1}nbe cluster labels produced by any clustering
algorithm (e.g. HDBSCAN), where label −1 denotes noise.
For a fixed cluster index c≥0 define the index set Ic:= {i|Li=c}. We
assume Ic=∅(otherwise the cluster is absent). We denote the cardinality
of Icby k.
Denote by e(c)∈ {0,1}dthe local unit of cluster c; components are defined
as for some ϵ > 0
e(c)
j:= (1 iff1
kPi∈Ic|Eij |> ϵ,
0 otherwise. (1)
1

Thus e(c)
j= 1 precisely when at least one member of the cluster exhibits
non-zero activity in dimension j.
2 Gating with Local Units
Given a query embedding q∈Rd, its cluster-gated variant is
q
c:= q⊙e(c),(2)
where ⊙is the Hadamard (component-wise) product. The operation is idem-
potent and commutative inside the multineutral field determined by the em-
bedding space and the masking vectors.
Relevance test. Let scbe the cosine similarity between qcand the nearest
neighbour vector retrieved inside the cluster c. If scfalls below a threshold
τwe discard the cluster; otherwise, the corresponding passages are included
in the RAG context.
3 Algebraic Properties
1. Idempotence. e(c)⊙e(c)=e(c)by definition.
2. Commutativity. e(c)⊙e(d)=e(d)⊙e(c)for all c, d.
3. Local unity. For every embedding xbelonging to cluster cwe have
x⊙e(c)=x, because every non–zero coordinate of xsurvives the mask
according to (1).
These three facts show that the set {e(c)}c≥0forms a system of local units in
the sense of multineutral algebra [1].
4 Discussion
Local units are extremely cheap to compute and store; a single pass over
the embeddings suffices. Although cruder than centroids or soft masks, they
work well when the dominant goal is to filter away irrelevant dimensions
rather than to re-weight them.
Future work will explore smooth relaxations e(c)
j∈[0,1] that allow differ-
entiable learning of mask values.
2

References
[1] Bruha, Ondrej. (2025). Locally Regular Algebraic Struc-
tures: A Unified Approach to Multineutral Groups and Fields.
10.13140/RG.2.2.32459.50729.
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