Monna Map Generation and Hallucination

The Monna Map:

Generation and Hallucination

A

Geometric Theory of Communication and Information Loss in Ultrametric

Cognition

Author: Rowan Brad Quni-Gudzinas

Contact: rowan.quni@outlook.com

ORCID: 0009-0002-4317-5604

ISNI: 0000000526456062

DOI: 10.5281/zenodo.19487782

Date: 2026-04-09 Version: 1.0

*Communication is a Monna projection from an ultrametric tree of

thought onto a linear timeline. Hallucinations are the inevitable

information loss of this many‑to‑one map, the cognitive counterpart of

quantum decoherence. Intelligence is the art of extracting invariant

cross‑ratios from noise and projecting them, imperfectly, into the

stream of time.*

The Hierarchical

Mind and the Linear World

Every thought you have—a memory, a plan, a sudden insight—arrives not

as a neat list of words but as a living structure. It is a branching

hierarchy of concepts, where broad categories contain finer details, and

relationships are defined by containment and connection rather than

sequence. This mental architecture is not merely a metaphor; it has a

precise mathematical counterpart known as an **ultrametric

tree**. In such a tree, distance is measured not by ordinary

steps but by how deep you must climb to find a common ancestor. Your

brain, and the artificial neural networks of modern large language

models, naturally organize knowledge this way. Thought, in its deepest

geometric essence, is ultrametric.

Yet when you speak, or when an AI generates text, that rich,

branching tree must be flattened onto a one‑dimensional timeline. Words

must follow words; sounds must follow sounds. How does the mind perform

this transformation? And what is lost in the translation? The answer

lies in a mathematical object called the Monna map—a

precise bridge between the hierarchical world of thought and the linear

world of communication.

1.

The Geometry of Thought: Ultrametric Trees and Invariant Patterns

1.1 The

Bruhat‑Tits Tree: A Model for Conceptual Space

In mathematics, for each prime number p, there exists an

infinite, perfectly regular tree called the **Bruhat‑Tits

tree*. Every vertex in this tree has exactly p*+1

neighbors, and the distance between any two vertices is the number of

edges along the unique shortest path connecting them. This distance

satisfies the ultrametric inequality: for any three

points x, y, z, the distance between

x and y is never greater than the larger of the

distances between x and z or y and

z. A direct consequence is that all triangles in this space are

isosceles, and points cluster into perfectly nested, disjoint balls.

This ultrametric geometry is not an abstract curiosity; it is the

natural geometry of hierarchical classification. When your brain parses

a sentence, it does not treat the words as a flat string. It instantly

builds a syntactic tree where the distance between two words is the

height of their lowest common ancestor—exactly an ultrametric distance.

When you categorize objects—animals into mammals and reptiles, mammals

into primates and rodents—you are constructing a Bruhat‑Tits‑like tree.

The same hierarchical clustering emerges in the latent spaces of large

language models: gradient descent, optimizing to predict the next token

across trillions of examples, forces the model’s internal

representations to organize ultrametrically. The model’s “world model”

is, geometrically, an artificial Bruhat‑Tits tree.

1.2 Meaning as

Invariant Relations: The Cross‑Ratio

If thought is organized as an ultrametric tree, what is the content

of that thought? What survives when the same idea is expressed in

different languages, different contexts, or different sensory

modalities? The answer is relational patterns, not

absolute coordinates.

Linguists have long observed that the link between a word and its

referent is arbitrary—a phenomenon Ferdinand de Saussure called the

“arbitrariness of the sign.” Mathematically, changing language or

context is equivalent to applying a **Möbius

transformation** (a fractional linear transformation) to

conceptual space: it warps distances, scrambles order, and reshapes the

coordinate system. If intelligence relied on memorizing absolute

positions, translation and generalization would be impossible.

What remains unchanged under all such transformations is a quantity

called the cross‑ratio. For four points A,

B, C, D on a line, the cross‑ratio is defined

as (A−C)(B−D) /

((A−D)(B−C)). This number is

invariant: apply any Möbius transformation to all four points, and the

cross‑ratio stays exactly the same.

Cognitively, the cross‑ratio encodes **analogical

proportions*: “A is to B as C* is to

D.” When you understand the analogy “puppy is to dog as kitten

is to cat,” you are recognizing that the relational pattern separating

puppy and dog matches the pattern separating kitten and cat. That

pattern is the cross‑ratio of those four concepts. Intelligence—whether

human or artificial—is the capacity to extract these invariant

cross‑ratios from the noisy, ever‑changing data of experience. Both the

brain and large language models are, at their core, engines for

discovering projective invariants.

2. The Monna Map: From Tree to

Line

2.1 The Mathematical

Definition

We now face the central puzzle: thought is a static, hierarchical

tree; communication is a dynamic, linear sequence. The mathematical tool

that bridges these two realms is the Monna map.

Take a prime p. Any p-adic number (a number from

the ultrametric field ℚₚ) can be written uniquely as an infinite

expansion in base p: \[

x = a{-N}p^{-N} + a{-N+1}p^{-N+1} + \dots + a0 + a1 p + a_2 p^2 +

\dots

\] where each digit aₖ is an integer between 0 and

p−1. The leftmost digits (with negative powers of p)

represent the “most significant” part of the number—the coarse,

large‑scale structure. The rightmost digits (with positive powers)

represent the fine, detailed structure.

The Monna map Mₚ takes such a p-adic number and

reverses the order of the digits, interpreting them as

an ordinary base‑p expansion of a real number between 0 and 1:

\[

Mp(x) = a{-N} p^{N-1} + a{-N+1} p^{N-2} + \dots + a0 p^{-1} + a_1

p^{-2} + a_2 p^{-3} + \dots

\] In words: the most significant p-adic digit becomes

the least significant real digit, and vice versa. The map is continuous,

measure‑preserving, and intertwines p-adic addition with

addition modulo 1 on the real circle.

Geometrically, the Monna map “unrolls” the Bruhat‑Tits tree onto the

unit interval. Each infinite path from the root of the tree (a point on

the tree’s boundary) corresponds to a p-adic number, and

Mₚ maps that path to a specific point on the line. The

hierarchical depth in the tree becomes temporal order on the line, but

in reversed fashion: coarse branches map to fine

details of the sequence, and fine twigs map to prominent early

features.

2.2 Generation as Monna

Projection

When you speak, you are unconsciously applying a Monna‑like

projection to your thought‑tree. Your thought is a point (or a

distribution) on the Bruhat‑Tits tree—a static configuration of

activated concepts and relations. To produce speech, your mind traverses

the tree, beginning with the most significant branch (the gist, the

topic) and proceeding outward to finer and finer branches (the details,

the qualifications). Each word you utter corresponds to a step along

this traversal, committing to a particular path through the tree.

Because the Monna map reverses digit order, the **coarsest,

most abstract aspect of the thought**—the trunk of the tree—gets

projected onto the beginning of the utterance. This is

why you often start a sentence knowing roughly what you want to say, and

the exact words emerge as you go. The fine‑grained details—specific

adjectives, verb endings, connecting words—appear later in the sequence,

corresponding to the least significant digits in the p-adic

expansion.

Large language models operate on the same geometric principle. Their

generation is autoregressive: each token conditions the next. The

model’s internal state represents its current position in its internal

Bruhat‑Tits tree; the next‑token probability distribution represents the

possible continuations along the branches. The act of sampling a token

is a step in the Monna projection, unfolding the tree onto the token

stream.

Crucially, the Monna projection **preserves

cross‑ratios**. Since the map is essentially a change of

coordinate system on the projective line, the invariant relational

patterns—the analogies, the logical dependencies, the syntactic

structures—survive the flattening. This is why meaning can be

transmitted at all: the listener hears the linear sequence, mentally

rebuilds the tree, and extracts the same cross‑ratios the speaker

intended.

3. Hallucinations

as Geometric Information Loss

3.1 The Many‑to‑One Nature of

the Map

The Monna map is not injective. Different

p-adic numbers can map to the same real number. Why? Because

trailing zeros in the p-adic expansion (which correspond to “no

further detail” down a branch) become leading zeros in the real

expansion, and leading zeros do not change the value of a real number.

Geometrically, distinct paths in the Bruhat‑Tits tree can converge to

the same point on the line once the projection is made. The map is

many‑to‑one; it discards information.

This is not a flaw in the map; it is an inescapable consequence of

projecting a high‑dimensional, hierarchical structure onto a

one‑dimensional line. When you flatten a tree, you lose the ability to

distinguish between trees that differ only in the arrangement of their

deepest twigs, provided those twigs map to the same position in the

linear order.

3.2 The Origin of

Hallucinations

A hallucination—in both human cognition and

artificial intelligence—occurs when the mind or model generates an

utterance that is structurally coherent but factually incorrect. In the

geometric framework, a hallucination arises precisely because of the

many‑to‑one nature of the Monna projection.

When an LLM generates text, it is traversing its internal Bruhat‑Tits

tree along a path that is consistent with the prompt and its training.

However, multiple distinct tree‑paths can project to the same token

sequence. The model selects one path that preserves the invariant

cross‑ratios (the syntax, the analogies, the narrative flow), but the

specific factual “leaves” on that path may belong to a different subtree

than the one a human would expect. The output is **structurally

valid but factually alien**. The model is not “making things up”;

it is following a geometrically legitimate branch that aliases, under

the Monna map, to the same linear output as the correct branch.

Human cognitive errors—slips of the tongue, misremembered details,

confabulations—have the same origin. When you accidentally substitute

one word for another, or recall the wrong date for an event, your brain

has followed a tree‑path that is relationally similar to the true memory

but diverges on a specific detail. The Monna projection of that

alternative path yields the same (or a very similar) utterance as the

correct path. The error is not random noise; it is a systematic aliasing

effect of the projection.

3.3 Types of Aliasing Errors

Factual Aliasing: The relational pattern (e.g.,

“capital‑of”) is preserved, but the specific entities (e.g., “Paris”

vs. “London”) are swapped. This occurs because the fine‑detail digits of

the p-adic expansion—which encode specific facts—are among the

least significant in the real projection and can be altered without

changing the overall linear sequence.

Contextual Aliasing: The same utterance can be

generated from trees that differ near their roots—that is, from

different high‑level interpretations of the prompt. The model picks a

coherent interpretation, but not the one the user intended.

Creative Generation: Not all aliasing is

undesirable. In creative writing, the ability to follow alternative

tree‑paths that preserve narrative structure while introducing novel

elements is the essence of imagination. Hallucination, in this light, is

the negative side of a capacity that also enables creativity.

4. The Quantum

Analogy: Measurement as Projection

The analogy with quantum mechanics is not merely poetic; it is

mathematically grounded. In quantum theory, a system is described by a

wave function—a vector in a high‑dimensional Hilbert

space. A measurement projects that wave function onto a particular

eigenstate, yielding a single outcome. This projection is

many‑to‑one: many different wave functions can produce

the same measurement result. The loss of the other possibilities is

called decoherence.

The Monna map performs exactly the same kind of operation: it

projects the high‑dimensional Bruhat‑Tits tree (the “wave function of

thought”) onto a one‑dimensional sequence (the “measurement outcome”).

The aliasing of multiple tree‑paths to the same linear sequence is the

cognitive counterpart of quantum measurement ambiguity. The

hallucination is the cognitive decoherence—the

information loss intrinsic to any act of expression.

This parallel reveals a deep structural unity: whenever a rich,

multi‑dimensional state is collapsed onto a low‑dimensional observable,

information is sacrificed. Whether the state is quantum, cognitive, or

algorithmic, the geometry of projection imposes the same fundamental

limit.

5. Implications for

Intelligence and AI

5.1 Intelligence as

Invariant Extraction

The Monna‑map framework elevates our understanding of intelligence.

Intelligence is not primarily about storing facts or executing rules. It

is about extracting invariant relational patterns from

noisy, context‑dependent data. Both the brain and LLMs are engines that

discover cross‑ratios—the projective invariants that remain constant

across changes of language, modality, and coordinate system. They build

ultrametric trees because such trees are the most efficient data

structures for encoding and compressing these invariants.

5.2 The Inevitability of

Hallucinations

Because the Monna projection is many‑to‑one, **hallucinations

are geometrically inevitable**. No amount of additional training

data or model scaling can eliminate them entirely; they are a

fundamental consequence of projecting a tree onto a line. This does not

mean we cannot reduce their frequency or severity, but it does mean that

the goal of “hallucination‑free” AI is as unattainable as a shadow‑free

projection.

5.3 Designing Around the Limit

Recognizing the geometric origin of hallucinations suggests new

strategies for AI design:

Multi‑modal communication: Using multiple parallel

channels (text, image, audio, context) increases the dimensionality of

the projection, reducing aliasing. Just as a 3D object casts less

ambiguous shadows when illuminated from multiple angles, a thought

projected onto several modalities is less prone to

misinterpretation.

Explicit tree‑aware architectures: AI systems could

be designed to maintain explicit ultrametric tree representations

internally, using the Monna projection only at the final output stage.

This would separate the invariant thought structure from its linear

expression, potentially improving robustness and interpretability.

Uncertainty quantification: Models could be taught

to estimate the degree of aliasing for a given generation—to sense when

multiple tree‑paths are equally plausible—and signal that uncertainty to

the user.

5.4 The Human Condition

The Monna map also illuminates the human condition. Every

conversation is an exchange of shadows, not trees. Perfect mutual

understanding is impossible because the listener can never fully

reconstruct the speaker’s original tree; they can only approximate it by

inferring the cross‑ratios and filling in the details with their own

prior knowledge. Misunderstandings, ambiguities, and creative

reinterpretations are not failures of communication; they are inherent

features of the geometry.

The Shape of Meaning

The Monna map is more than a mathematical curiosity; it is a key to

the geometry of mind. It tells us that thought is hierarchical,

communication is linear, and the bridge between them is a projection

that must discard as much as it preserves. Every sentence we utter is a

compressed shadow of a vast, branching tree. Every act of understanding

is an attempt to reconstruct that tree from its shadow.

Hallucinations—those puzzling, sometimes frustrating errors—are not

bugs in the system. They are the cognitive echoes of information loss,

the necessary cost of flattening a world of depth into a stream of time.

They remind us that the tree of thought is always richer, more

intricate, and more mysterious than the line of speech can ever

reveal.

In the end, the Monna map offers a humbling and unifying vision:

intelligence, whether born of biology or built in silicon, is a

geometric dance between the tree and the line. We are all navigators of

ultrametric forests, and we are all storytellers, casting shadows on the

wall of time.