Hierarchical Concepts Under Finite Resolution: Observation Quotients and Adaptive Symmetry

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Hierarchical Concepts Under Finite Resolution:
Observation Quotients and Adaptive Symmetry
 Brian Theory
29 September 2026
Abstract
An ultrametric hierarchy does not itself make distinct states indistinguishable: its metric
topology is Hausdorff. Indistinguishability instead requires an observation map. This note gives a
self-containedsynthesisofthedistinctionforfiniteballpartitions. Asinglequotientsimultaneously
describes the partition topology, its Kolmogorov quotient, and the metric identification of a
distance-preserving ultrapseudometric collapse. Resolution towers recover a compact space, and
recover the completion when only total boundedness is assumed. For the full regular rooted tree,
we give an exact adaptive refinement index and characterize when the corresponding stabilizer
inclusion is normal. This separates a finite index, which always exists, from a quotient group,
which need not. A short information calculation also separates symmetry index from entropy
gain. The underlying topology, tree theory, and quotient constructions are standard; the purpose
is a precise correction and synthesis, not a claim of a new general theory. The connection with
ultrametric models of cognition is restricted to class-relative observability and does not formalize
Matte-Blanco’s full bi-logic or establish an empirical model of consciousness.
1 The distinction and the main statements
A node of a rooted tree can represent a category by the set of branches passing through it.
This interpretation is established in thep-adic andm-adic mental-space literature. In particular,
Khrennikov’s [10, Section 4] associations are already fixed-prefix equivalence classes. Earlier
subconscious and memory models also use non-Archimedean state spaces [9, 1]. The present note
does not introduce either hierarchy or equivalence classes into that literature.
The point requiring care is the passage fromsimilarity to inability to distinguish. An ultrametric
is a metric, so different points have disjoint sufficiently small neighborhoods. Equal distances from
an external point do not make those points identical or topologically indistinguishable. Lauro-Grotto
[12, pp. 537–539] links clusters with symmetrization and invokes the perception of distant objects.
Here that observational restriction is made explicit by a map; it does not follow from ultrametricity
alone. An inference of indiscreteness from ultrametricity in the author’s thesis [4] is not valid.
Nor is the broad two-layer architecture new. Khrennikov and Yurova [8, Sections 4 and 9]
combine an observational quotient with ap-adic or ultrametric latent space in a protein model.
Kleiner [11, Sections 3.3–3.4] uses automorphism-orbit quotients to describe accessible references
to experience. These are different models, but they preclude treating “latent state, restricted
observations, equivalence classes” as a distinctive invention here.
We isolate two useful precise statements. First, ifP is a finite partition of an ultrametric space
into nonempty clopen balls, then
dP(x,y ) =
{
0, x,y belong to the same cell,
d(x,y ), otherwise (1)
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is an ultrapseudometric. Its zero classes and topological indistinguishability classes coincide with
the classes defined by agreement under all cell-constant observables. The quotient is universal in
the senses made precise in theorem 2.2. This is a synthesis of standard quotient constructions, not
a new separation theorem.
Second, consider the full rootedm-ary tree Tm, with vertices the finite words overΣm =
{0,...,m−1}. A finite complete frontierF is a finite prefix-free set of words whose cylinders
partition the infinite branches. LetI(F) be its proper prefixes, and letGF fix every vertex ofF
individually. WriteF⪯F′when F′refinesF.
Theorem 1.1 (Adaptive index and normality). Form≥2 and finite complete frontiersF⪯F′,
put N =|I(F′)\I(F)|. Then
GF∼=
∏
s∈F
Aut(Tm), [GF :GF ′] = (m!)N. (2)
MoreoverGF ′ is normal inGF if and only if, for everys∈F, there is an integerhs≥0 such that
{t∈F′:s is a prefix oft}=sΣhs
m. (3)
In this case, withTm,h the full tree of heighth,
GF/GF ′∼=
∏
s∈F
Aut(Tm,hs). (4)
The orbits ofGF on the boundary are exactly the cylinders indexed byF.
The theorem is proved in section 4 using standard sections and stabilizers of full tree automor-
phism groups [2, Sections 1.2.2–1.2.3]. Its role is to make the hypotheses of an adaptive “symmetry
loss” statement explicit. Neither the uniform level-stabilizer formula nor the general group-theoretic
mechanism is asserted to be original.
2 A coordinate-free observation quotient
An ultrapseudometric on X is a mape : X×X→[0,∞) with e(x,x ) = 0, symmetry, and the
ultrametric inequality; it may assign zero distance to different points. Throughout,X is nonempty.
For a partitionP let q :X→Psend a point to its cell, and set
τP ={q−1(A) :A⊆P}.
This is thepartition topology, equivalently the initial topology induced byq into the discrete set
P [7, Definition 1]. Its definition requires neither a metric nor a hierarchy. Every cell, with its
subspace topology, is indiscrete.
Lemma 2.1 (Separation of disjoint balls). If B,C are disjoint nonempty balls in an ultrametric
space,d(x,y ) is constant onB×C.
Proof. Forx,x′∈B and y /∈B, the ball inequality givesd(x,x′)< d(x,y ), for either an open or
a closed ball. The strong triangle inequality in both directions impliesd(x′,y ) =d(x,y ). Repeat
inside C. The singleton case is immediate.
Theorem 2.2 (Finite ball quotient). LetP be a finite partition of an ultrametric space(X,d ) into
nonempty clopen balls. DefineδP(B,B ) = 0 and, forB̸= C, defineδP(B,C ) = d(x,y ) for any
x∈B,y ∈C. Then:
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(i) δP is an ultrametric anddP =δP◦(q×q) is the ultrapseudometric in(1). Its topology isτP.
(ii) q realizes both the Kolmogorov quotient of(X,τP) and the metric identification of(X,dP).
For anyT0 spaceY, every continuousf : (X,τP)→Y factors uniquely throughq.
(iii) For an arbitrary setY, the mapsX→Y constant on cells are exactly the maps¯f◦q. If
Y is metric andf : (X,d )→Y is also nonexpansive, its unique factor¯f : (P,δP)→Y is
nonexpansive.
(iv) dP is the greatest pseudometrice≤d that vanishes on each cell.
In particular, two points share a cell if and only if they have zerodP-distance, are topologically
indistinguishable inτP, or agree under every cell-constant observable.
Proof. The preceding lemma proves representative independence. Choose representatives from three
cells and apply the ultrametric inequality inX; repeated cells cause no difficulty. Since the quotient
is finite, its metric topology is discrete. Pullback therefore gives exactlyτP, with zero classes equal
to the cells.
Two points in the same cell belong to precisely the same open sets. Points in different cells
are separated by those cells. A continuous map into a T0 space must identify topologically
indistinguishable points: otherwise a target open set distinguishing their images would pull back to
an open set distinguishing them. The factor exists and is unique becauseq is onto; it is continuous
becauseP is discrete. This also identifies the quotient topology.
The set-theoretic factorization is immediate. For different cellsB,C and representativesx,y,
nonexpansiveness givesdY ( ¯f(B), ¯f(C))≤d(x,y ) = δP(B,C ). Finally, any admissiblee is zero
inside cells and is at mostd =dP across cells. AsdP itself is admissible, it is greatest. Cell indicators
show that equality under all cell-constant observables implies membership in the same cell.
The topological and pseudometric identifications are established theory; see [19, Theorem 5.3].
The collapse is a special case of Ishiki’s pseudo-ultrametric amalgamation [6, Proposition 2.31],
taking every within-cell pseudo-ultrametric to be zero. The useful geometric feature here is retaining
all distances between resolved cells, not the existence of a quotient set.
Example 2.3 (Why the ball condition matters). On binary words of length two used(x,y ) = 2−ℓ(x,y),
whereℓis common-prefix length. Collapse the block{00, 10}while leaving{01}and{11}as singleton
blocks. Formula (1) gives
dP(10, 01) = 1> 0 + 1
2 =dP(10, 00) +dP(00, 01).
An arbitrary partition still defines a partition topology, but this distance-preserving collapse need
not even be a pseudometric. The ball hypothesis is sufficient, not necessary: every partition of an
equilateral finite metric space admits such a collapse.
Observable concepts at a fixed resolution form the finite Boolean algebra of all unions of cells.
Individual balls are generally not closed under union or complement. In particular, introducing this
observation algebra does not remove ordinary Boolean negation.
3 Resolution towers, distinguishing scales, and completion
Let (X,d ) be a totally bounded ultrametric space, and chooser0≥diamX with r0>r 1>···> 0
and rk→0. Define
x∼k y ⇐⇒d(x,y )≤rk, Pk =X/∼k, q k(x) = [x]k.
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These are the closed-threshold quotients of Mémoli, Smith and Wan [15, Section 2.1, Equation (5)
and Definition 10], with thresholdrk. The strong triangle inequality makes∼k transitive. Its classes
are closedrk-balls, which are also open becauserk > 0. A finite cover by balls of radius less thanrk
proves thatPk is finite. No regular branching assumption is needed.
Proposition 3.1 (A calibrated tower). Letdk and τk arise fromPk as in theorem 2.2. Then
0≤d−dk≤rk, d k≤dk+1, τ k⊆τk+1. (5)
The topology generated by⋃
kτk is thed-topology. Forx̸=y, ifκ(x,y ) = min{k≥1 :qk(x)̸=qk(y)},
then
rκ(x,y)<d (x,y )≤rκ(x,y)−1. (6)
The natural mapX→lim←−kPk is injective and has dense image. The inverse limit is canonically
the completion ofX; whenX is compact, it isX itself.
Proof. The collapse formula gives(5); every fine cell lies in a coarse cell. Everyτk-open set isd-open.
Conversely, for any open neighborhoodU of x, chooserk smaller than a radius witnessingx∈U.
Then [x]k⊆U. The definition of first distinction gives (6).
The quotient maps give bonding mapsPk+1→Pk. A compatible sequence is a nested sequence
of nonempty cells of diameter at mostrk. In the completionˆX, their closures remain nested and
nonempty. Choosing one point in each cell gives a Cauchy sequence. Its limit belongs to all the
closures, and uniqueness follows fromrk→0.
For completeness of the identification, at each fixed level there are finitely many cells and the
distances between distinct cells are positive constants. Their closures therefore form a disjoint finite
partition of ˆX. These closures remain positively separated and hence are clopen, so the coordinate
maps from ˆX toPk are continuous. Every point ofˆX determines exactly one compatible sequence.
A basic inverse-limit neighborhood specifies finitely many coordinates and thus a nonempty cell at
the deepest specified level; it meetsX. This proves density and a continuous bijection fromˆX to the
inverse limit. It is a homeomorphism sinceˆX is compact and the inverse limit is Hausdorff. Equip
the inverse limit withD(a,b ) = limkδPk(ak,bk). For distinct sequences, these distances are positive
and constant after their first differing coordinate. Passing to completion limits of representatives
shows thatD equals the distance inˆX, proving the isometry.
This is the finite clopen partition description of profinite spaces [20, Lemma 5.22.2], specialized to
a calibrated ultrametric. Tree/end-space representations are much older and more general; Hughes
[5] establishes a categorical equivalence with specified morphisms, while Dovgoshey [3, Theorem
10.25 and Proposition 10.29] relates representing trees to isometric completions of totally bounded
spaces and realizes the completion by maximal chains. These results should not be confused with a
claim that every hierarchy is a full regular tree.
Example 3.2 (Why completion cannot be omitted). The eventually zero binary sequences are
totally bounded under the prefix ultrametric. Every finite binary word occurs as a prefix. Their
inverse limit is therefore the entire binary boundary, including the all-ones sequence, which was not
in the original space.
3.1 The regular prefix and prime-base specializations
On X = Σ N
m, number coordinates from zero and letℓ(x,y ) be the number of initial coordinates that
agree. For 0< λ <1 set dλ(x,x ) = 0 and dλ(x,y ) = λℓ(x,y) for x̸=y. Common prefixes satisfy
ℓ(x,z )≥min{ℓ(x,y ),ℓ(y,z )}, proving the ultrametric inequality. A length-k cylinder is a closed
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ball of radiusλk; fork≥1 it is an open ball of radiusλk−1. An open ball of radiusλk instead fixes
k + 1 digits. Positive-radius balls are clopen and intersecting balls are nested, not necessarily equal.
Here qk records the firstk digits and
κ(x,y ) =ℓ(x,y ) + 1, d λ(x,y ) =λκ(x,y)−1.
The exact equality depends on this scale calibration; the general statement is(6). The union of
the finite observable Boolean algebras generates the Borel sigma-algebra; it is not itself generally a
topology or a sigma-algebra.
When m =p is prime andλ=p−1, the least-significant-digit-first map
(x0,x 1,...)↦−→
∑
j≥0
xjpj
is an isometry ontoZp. The first differing digit is the valuation of the difference, and compatible
residues uniquely recover the digits. Consequently
qk : Zp−→Z/pkZ, Zp∼= lim←−k
Z/pkZ.
This adds canonical ring quotients and arithmetic compatibility to the tree model. No arithmetic
dynamics is used in this note, andZp is a ring, not the fieldQp. The prime-base specialization is an
example, not a prerequisite for the results.
4 Adaptive frontiers and symmetry
Finite complete frontiers are the finite maximal prefix codes of the regular tree; their expansions are
standard objects [13]. Their internal setsI(F ) are finite prefix-closed sets, and conversely every such
set determines a frontier by stopping at the first vertex outside it. Counting edges in the resulting
full finite tree gives
|F|= 1 + (m−1)|I(F )|. (7)
For a frontierF, writeCs for the cylinder indexed bys∈F and apply theorem 2.2 to{Cs :s∈F}.
This yields adaptive, rather than uniform-depth, observation quotients.
4.1 Proof of the adaptive theorem
Proof of theorem 1.1.An automorphism fixing every frontier vertex fixes its ancestors and has
arbitrary, independent actions on the descendant trees rooted at frontier vertices. Taking these
sections gives the product in(2), with inverse given by extension by the identity on the finite upper
tree.
The full automorphism group of a regular rooted tree is transitive on its infinite branches: along
two specified rays choose, at each vertex, a permutation sending the next digit of the first ray
to that of the second; extend at all other vertices arbitrarily. This defines compatible finite-level
automorphisms and hence an automorphism of the infinite tree. Applied inside each frontier cylinder,
it proves the orbit assertion.
A refinement can be obtained by replacing one current leaf at a time by itsm children. For one
replacement at a leafs, the homomorphism from the current stabilizer toSym(m) records its action
on those children. It is onto and its kernel is the refined stabilizer. Thus each replacement has index
m!. There are exactlyN replacements, and multiplication of subgroup indices proves(2), without
any assertion that the final subgroup is normal.
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Suppose GF ′ is normal inGF. Its boundary orbit partition, already proved to be{Ct :t∈F′},
must be invariant underGF: conjugation by an element ofGF permutes these orbits. Fixs∈F
and take any descendantt∈F′at relative depthh. The full section group ats acts transitively on
the words of lengthh. Invariance therefore places every vertex ofsΣh
m in F′. Prefix freeness and
completeness exclude any other relative depth in that subtree. This is(3). Here distinct vertices
have distinct cylinders becausem≥2, so permuting frontier cylinders also permutes their indexing
vertices.
Conversely, under(3), the subgroup in each factorAut(Tm) is the kernel of restriction to the
finite treeTm,hs. It is normal, the restriction is onto, and its quotient isAut(Tm,hs). Taking the
product proves normality and (4).
Corollary 4.1 (Uniform and one-step refinement). ForGk =GΣ km,
Gk∼= Aut(Tm)mk
, G k/Gk+1∼= Sym(m)mk
, G k+1 ⊊Gk.
More generally, simultaneously splitting a subsetS⊆F of current leaves gives the normal quotient
Sym(m)|S|.
Example 4.2 (Index without a quotient group). Form = 2, takeF ={ε}and F′={00, 01, 1}.
There are two splits, so[GF : GF ′] = 4. But the root interchange sends the orbit partition to
{C0,C 10,C 11}, a different partition. ThusGF ′ is not normal inGF and GF/GF ′ is only a coset
set, not a quotient group. The successive local steps each have a normal kernel; normality is not
transitive along their composition.
4.2 What is specific to the full regular tree?
The product and transitivity arguments useall rooted automorphisms and identical descendant
trees. They do not hold for an arbitrary chosen subgroup. For a nonregular rooted tree, section
factors are the potentially different descendant-tree automorphism groups; child permutations must
preserve rooted subtree isomorphism types. Some levels can have no nontrivial permutations.
A three-point ultrametric already shows that geometry alone is insufficient for the orbit clause.
Putd(a,b ) = 1/2 andd(a,c ) =d(b,c ) = 1. At the coarsest observation all three points are equivalent,
but the full isometry group has orbits{a,b}and{c}. The pointc is geometrically distinguished.
Observation equivalence is therefore primary; an orbit representation requires additional symmetry
hypotheses.
5 Information gain is not a symmetry index
Let µbe a probability measure on the boundary. For a finite frontier put
Hµ(F ) =−
∑
s∈F
µ(Cs) log2µ(Cs),
using 0 log2 0 = 0. For a vertex of positive mass definep(a|v) =µ(Cva)/µ(Cv), and give zero-mass
vertices zero contribution. ForF⪯F′, successively applying the finite entropy chain rule gives
Hµ(F′)−Hµ(F) =
∑
v∈I(F ′)\I(F)
µ(Cv)H
(
p(·|v)
)
. (8)
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Indeed one split replaces the term−µ(Cv) log2µ(Cv) by the sum over its children; subtraction gives
the displayed summand. This is standard hierarchical information accounting, not a new entropy
definition; compare [18, Equations (7) and (10)].
Under uniform independent digits, the summand ism−|v|log2m, whereas
log2[GF :GF ′] =N log2(m!)
counts split vertices without probability or depth weights. For a binary uniform source, both
{00, 01, 10, 11}and{000, 001, 01, 1}have three internal vertices, so their stabilizers have index eight
in G{ε}. Their entropies are respectively2 and 7/4 bits. There is no general identification of the
index with information gain. Neither quantity is a count of conscious states.
6 Interpretation and limits
The motivating psychological analogy must be narrower than the mathematics. Matte Blanco’s
generalization principle concerns nested class membership, while his symmetry principle concerns
treatment of a relation and its converse [14, pp. 38–39]. The relationq(x) =q(y) directly represents
class-relative indistinguishability. It does not by itself symmetrize an independently given binary
relation.
It is also inaccurate to attribute to Matte Blanco merely a modern equivalence-versus-identity
distinction. His discussion includes stronger treatment of class members and parts as identical
[14, p. 39]. He also distinguishes this treatment from the external observer’s description of parts
and wholes [14, pp. 148–149]. Our quotient is a deliberate restriction:x and y remain different
latent points, andq(x) is a class label, not an assertion that a member equals its containing set.
Likewise the primary discussions of negation, infinite sets, and conscious unfolding [14, pp. 45–46,
146–149, 107–108] are not consequences of this construction. The observable algebra is Boolean, no
part–whole bijection is inferred, and a resolution index is not a psychological time variable.
The wider interpretation of ultrametric structure in bi-logic has been developed by Lauro-Grotto
and Murtagh [12, 16, 17]. The note supplies a precise conditional mathematical model, not an
endorsement of all claims made in that literature. A model of cognitive data would still need justified
features, their hierarchical ordering, an observation mechanism, and tests against nonhierarchical
alternatives. No such validation is supplied here.
Several other boundaries matter. Deterministic cell equivalence does not model measurement
error: equality of stochastic observation laws is a different relation, and distances between laws need
not be ultrametric. A finite observation map is not automatically a factor of a dynamical system; a
transition must respect its fibers. Refinement is not a quantum collapse, and it is not simplicial
decomposition or gluing. Those operations, reserved for a separate project, are not used in any
theorem above.
The mathematical conclusion is consequently limited but definite. Ultrametricity supplies a
coherent hierarchy and a distance-preserving ball collapse. Observation supplies indistinguishability.
Fullregular-treesymmetrysuppliesanoptionalexactorbitdescription, withaquantitativerefinement
index and a nontrivial normality condition. These roles should not be conflated.
A Two elementary frontier facts
For clarity about observation design, finite frontiers have a distributive refinement lattice: under
F ↦→I(F), meet and join correspond to intersection and union of finite prefix-closed sets. A
bottom frontier exists, namely{ε}, but no finite top frontier exists on the infinite full tree. For the
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uniform boundary measure, disjoint covering by cylinders gives the Kraft equality∑
s∈Fm−|s|= 1.
Conversely a finite prefix-free set satisfying this equality is complete, since a nonempty complement
would contain a cylinder of positive measure.
For a finite setA of distinct infinite branches, define
IA ={v :|A∩Cv|≥2}.
This is a finite prefix-closed set: every such vertex is a common prefix of a pair from the finite set
A. Its frontier is the unique least refinement that separates all members ofA. Necessity follows
because a stopping cylinder containing two sample points must be split; sufficiency follows because
every resulting leaf contains at most one. Thus the minimum number of leaves is1 + (m−1)|IA|.
This is the familiar uncompressed trie principle, not a new optimal decision-tree theorem; arbitrary
query choices and probabilistic stopping rules are different problems.
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