The Geometry of Perspective
Mathematical analysis of AI meaning-space, Buddhist mental dynamics, and the measurable structure of political thought
May 2026 · kusaladana · Revised draft for discussion
Status. This is an exploratory investigation. The experiments are indicative rather than conclusive — the concept sets are small, the phrase selection is not independent of the theoretical framework, and the field-theoretic section is openly speculative. The work does not claim hard results. It aims to make certain questions precise enough that productive disagreement becomes possible. Where a result admits a mundane technical explanation alongside the contemplative one, both are stated.
I. Shadows and perspective
The Cosmic Microwave Background is the thermal shadow of a state we can never directly observe — the early universe before it became transparent, roughly 380,000 years after the Big Bang. We cannot go back. We cannot measure it directly. But the anisotropy pattern, the acoustic peaks, the power spectrum of temperature fluctuations: these are the geometric consequences of the primordial coupling structure, still readable now, 13.8 billion years later.
This paper began with a simpler observation: that the geometry of artificial intelligence meaning-space — the high-dimensional space in which language models organise semantic content — might function as a similar kind of shadow. Not of the early universe, but of the structure of human mental dynamics. Measurable from outside. Carrying information about something that cannot be directly observed.
What the investigation actually found is something more specific, and it is the organising claim of this revision: every measurement of meaning-space is shaped by a stack of perspectives, and the geometry makes those perspectives separable. Four determinants recur throughout the experiments below:
- The corpus — whose language was averaged to build the space. GloVe’s liberation is the Moro Liberation Front, not moksha, because Wikipedia’s liberation mostly is.
- The training objective — the optimisation that shaped the space’s global geometry. Embedding spaces are known to be anisotropic, occupying a narrow cone, and this is at least partly an artifact of how they are trained, independent of what they are trained on.
- The mechanism of reading — statistical co-occurrence (an embedding model) versus explicit reasoning (a language model asked to judge). These two mechanisms systematically disagree, and the disagreements are data.
- The path of inquiry — the sequence of choices made by whoever is probing the space. This determinant turns out to have an exact mathematical form: it is the wedge product, and it is developed in Section II.
The Buddhist analysis adds a fifth determinant beneath the other four: the kleshas — the habitual organising tendencies of the mind that produced the corpus in the first place. Whether the measured geometry carries a readable shadow of that fifth level, over and above the artifacts of levels 1–4, is the paper’s central open question. It is a question, not a claim.
II. The mathematical framework
The plane of the already-held
In N-dimensional space, any two vectors a and b span a plane. The dot product — cosine similarity — measures alignment with that plane. A concept with high dot product against a and b lies close to what the pair already contains.
Standard semantic retrieval in AI — RAG systems, cosine similarity search — operates entirely within this plane. It can only recognise what it already holds. It is constitutively the operation of the familiar.
The complement of the plane — the (N−2)-dimensional subspace orthogonal to both vectors — contains every direction genuinely new to the pair. The two regions partition the space without overlap. The dot product covers the already-known; the complement covers everything else.
From cross product to wedge product
An earlier version of this framework used the cross product to generate new directions. In three dimensions this works: a × b is the unique perpendicular. But in N dimensions two vectors do not determine a unique perpendicular — they determine the whole (N−2)-dimensional complement — and the cross product proper does not exist above dimension three (and, exotically, seven). The correct instrument is the wedge product of geometric algebra.
The wedge a ∧ b is not a vector but a bivector: an oriented plane element. Wedging in a third vector gives a trivector — an oriented volume — and so on, each additional vector raising the grade of the object by one. Only when N−1 vectors have been accumulated does the construction determine a unique direction: the Hodge dual ★(v₁ ∧ v₂ ∧ … ∧ vₙ₋₁), the single direction orthogonal to everything wedged in.
Path-dependence as accumulated perspective
Here the fourth determinant of perspective acquires its exact form. Starting from two vectors, reaching a unique orthogonal direction requires supplying N−3 further vectors. Every one of those is a choice. Different choices produce different terminal directions. The complement as a whole is covered by all possible paths together — but any particular path carves out one specific direction, and each choice is literally a factor in the wedge.
This reframing dissolves what previously looked like a defect. Path-dependence is not a regrettable artifact of the iterative construction; it is the content. The terminal vector is orthogonal to the original pair and to every perspective brought along the way — the direction that survives when everything the inquirer held has been wedged in and dualised away. An apophatic structure: each factor is an exclusion, and what remains at the end is what no individual perspective directly contained.
The intermediate grades invite a contemplative reading that is flagged here and pursued in Section V: a bivector is an orientation without a direction — a held plane of inquiry that has not yet crystallised into an answer. Whether the graded structure (support → orientation → direction) maps onto the graduated structure of meditation with and without support is an open question, not a result.
Kleshas as principal components
The habitual tendencies — in Buddhist psychology, the kleshas — organise experience preferentially in certain directions. In the geometry of a practitioner’s meaning-space they would be the directions of maximum variance: the most load-bearing vectors, the principal components.
As practice thins the kleshas, the principal components weaken, the plane of the already-known contracts, and the complement expands. The wedge construction can reach more of the space — not because the operation changed, but because the klesha-organised plane has loosened its grip. This is the fifth determinant of perspective, stated as hypothesis. Section IV proposes how it might be tested without conflating it with determinants 1–3.
III. Five experiments
Experiment 1 — GloVe word vectors: the corpus determinant isolated
The first experiment used GloVe word vectors (Wikipedia + Gigaword, 50 dimensions, single words). The geometry was real but the semantics were not what was needed. Nearest neighbours for liberation: separatist, rebel, moro, guerrilla. For awareness: promotes, prevention, promote — public health campaigns.
GloVe places each word where its dominant statistical usage puts it. This is not a failure of the experiment; it is the cleanest possible demonstration of determinant 1. The corpus decides what a word means before any question is asked of it. It also established the key methodological requirement: to probe philosophical meaning-space, concepts must be embedded as contextualised phrases that specify the intended sense.
Experiment 2 — MiniLM with phrase embeddings
Switching to a sentence-level model (all-MiniLM-L6-v2, 384 dimensions) and embedding each concept as a full descriptive phrase — “liberation as awakening, moksha, freedom from the cycle of suffering and rebirth” rather than the bare word — changed the geometry substantially.
The most notable result: recognition sits in the complement of span{craving, liberation}, ranked second of ten candidate concepts with complement component 0.969. Recognition is not at the end of the craving-to-liberation axis; it is orthogonal to it. This matches the Mahamudra teaching that rigpa is not an achievement on the path but something perpendicular to the path itself.
Caveat (determinant 2). Sentence-embedding spaces carry a large shared common component: all embeddings lean in a common direction, inflating every pairwise similarity. Complement analyses should be repeated on mean-centred embeddings, since otherwise the “complement” is partly measuring residuals against a near-constant offset. Whether the recognition result survives centring is a specific, runnable check; if it survives, the result is considerably stronger. This check has not yet been run and the result should be read as provisional until it has.
Experiment 3 — nomic-embed-text: higher dimensions
Running the same phrase set through nomic-embed-text (768 dimensions, purpose-built for semantic similarity, via Ollama) confirmed the MiniLM finding and added new information.
Liberation ↔ path emerged as the highest-similarity pair (0.756) — the progressive unfolding of practice and its goal, correctly identified as most related. Awareness ↔ recognition rose to 0.600 (versus MiniLM’s 0.37), more philosophically accurate.
The sparseness analysis revealed a structural property of the space: the 12 concept embeddings clustered in a cone using only 44% of the available angular range, with mean pairwise similarity around 0.55 where randomly oriented high-dimensional vectors would sit near zero.
Correction to the earlier draft. The effective dimensionality of 9.3 was previously reported as “9.3 of 768”. This overstates the finding: twelve vectors can span at most twelve dimensions, so the correct statement is 9.3 of a possible 12 — the concepts are in fact rather well spread within their own span. The cone claim rests properly on the mean pairwise similarity, which is far above the near-zero value expected of random directions in 768 dimensions. That figure is solid; the dimensionality framing was not.
Experiment 4 — Qwen 2.5 14B as explicit semantic judge: the mechanism determinant
Rather than extracting embedding vectors, this experiment asked Qwen 2.5 14B to explicitly rate semantic similarity between all 66 concept pairs. Two entirely different mechanisms of reading — statistical co-occurrence versus explicit reasoning — allowed direct comparison. This is determinant 3 made experimental.
Qwen collapsed recognition, awareness, wisdom, liberation, meaning, and path into mutual identity (all rated 1.0). This is a defensible Buddhist position at the absolute level — these terms all point at the same thing — but it geometrically destroys the complement analysis: if recognition equals liberation, recognition cannot appear in liberation’s complement.
The disagreements are where the data becomes most interesting:
| Pair | nomic | Qwen | Direction |
|---|---|---|---|
| meaning ↔ recognition | 0.510 | 1.000 | Qwen higher |
| liberation ↔ recognition | 0.532 | 1.000 | Qwen higher |
| wisdom ↔ recognition | 0.576 | 1.000 | Qwen higher |
| awareness ↔ momentum | 0.619 | 0.200 | nomic higher |
| wisdom ↔ momentum | 0.615 | 0.200 | nomic higher |
| awareness ↔ attention | 0.690 | 0.300 | nomic higher |
Qwen rates higher wherever concepts share an ultimate referent in Buddhist philosophy. nomic rates higher wherever concepts share functional texture in the training corpus. One is reading the map of doctrine; the other is reading the territory of language use. Neither is neutral: each mechanism is itself a perspective, and the disagreement pattern locates precisely where the two perspectives diverge.
Experiment 5 — Three extended sets
Set 1: the anharmonic framework made empirical. Twelve concepts pairing physics terms with their proposed contemplative counterparts:
| Physics ↔ Contemplative pair | nomic | Qwen |
|---|---|---|
| ground_state ↔ bare_awareness | 0.568 | 0.700 |
| excitation ↔ mental_appearance | 0.656 | 0.500 |
| coupling_constant ↔ klesha | 0.579 | 0.000 |
| symmetry_breaking ↔ self_grasping | 0.545 | 0.500 |
| second_arrow ↔ anharmonic_coupling | 0.575 | 0.500 |
| symmetry_restoration ↔ liberation | 0.734 | 1.000 |
The coupling_constant / klesha pair is the paper’s central speculative claim rendered measurable. nomic finds the descriptions structurally similar (0.579); Qwen rates them completely unrelated (0.000) — different domains, no meaningful connection. The disagreement is now precisely located, and Section VI asks what could settle it.
Set 2: political meaning-space. Each of six political concepts was given in two distinct framings — for example authority as legitimate governance versus coercive power. Within-pair similarity measures how much each reading mechanism conflates the two:
| Concept | nomic | Qwen | Interpretation |
|---|---|---|---|
| authority | 0.772 | 0.300 | corpus conflates most severely |
| security | 0.739 | 0.300 | corpus conflates |
| freedom | 0.747 | 0.500 | corpus conflates |
| community | 0.735 | 0.700 | both moderate |
| justice | 0.733 | 0.700 | both moderate |
| equality | 0.689 | 0.700 | both moderate |
The three concepts most commonly weaponised in political rhetoric — authority, security, freedom — are the three most conflated in the statistical geometry of language. Legitimate governance and coercive power are near-synonymous in corpus space (0.772); explicit reasoning distinguishes them (0.300).
Caveat. The two framings of each concept share a head noun — both authority phrases contain “authority” — so part of the conflation is lexical rather than semantic. A paraphrase control, in which the paired framings share no content words, is required before the conflation figures can carry their full interpretive weight. This control is designed and not yet run.
The complement finding stands on separate footing: the complement of span{freedom-individual, security-national} — the dominant political frame — contains equality, community, and justice in both models. Not as opposites: as orthogonal directions. They cannot be reached by moving along the freedom–security axis, however far and in either sense. A different dimension is required. This is, in miniature, the political version of the recognition result.
Set 3: probing the cone boundary. Phrases designed to reach the unexplored angular range — “the pure potentiality before any crystallisation of experience into a specific direction”, “awareness without an object reaching toward nothing” — all remained within the nomic cone, with mean similarities of 0.48–0.62 to the anchor cluster. Most strikingly, awareness ↔ non_referential = 0.782: the phrase designed to point beyond awareness was immediately pulled into awareness’s neighbourhood. The cone is sticky. The vocabulary available for probing the boundary is itself inside the cluster, which may make pointing outside definitionally impossible within language.
Across all four concept domains, nomic showed essentially identical sparseness (mean similarity 0.553–0.563). The cone compression is a property of the model, not of the domain.
IV. The determinants disentangled
The domain-independence of the cone compression forces a choice that the earlier draft elided. Two readings are available:
Reading A (determinant 2 — the training objective). Embedding-space anisotropy is a documented phenomenon: contextual embedding spaces are known to occupy narrow cones, attributable to the optimisation itself — the softmax geometry, token frequency effects — largely independent of corpus content. On this reading the 44% figure is a property of nomic’s training procedure, and would appear for any twelve concepts in any domain, as indeed it did.
Reading B (determinant 5 — the kleshas). The training corpus was produced overwhelmingly by minds with ego-clinging intact, and the compression is its geometric shadow, baked into the space.
The earlier draft asserted Reading B. This revision holds the question open, for a reason worth stating carefully: the two readings may not be rivals. The training objective — next-token prediction, the relentless minimisation of surprise against the expected — is itself a formalisation of grasping-toward-the-anticipated. If the objective produces the cone, the symmetry breaking has not been explained away; it has been relocated from the corpus into the loss function, which would make it more structural, not less. But relocation changes what the CMB analogy is a shadow of, and the paper should not trade on the ambiguity. The experiment that separates the readings is the one that removes the embedding model from the loop entirely:
The practitioner experiment, stated precisely. Collect explicit similarity ratings on the existing 66-pair instrument from practitioners across a range of practice depth — the Experiment 4 protocol, with humans in place of Qwen and practice depth as the independent variable. No embedding model intervenes, so determinants 1–3 are out of the loop; what remains is the rater’s own organisation of meaning. The prediction of the klesha hypothesis: mean pairwise similarity in the ratings decreases — the cone opens — with depth of practice. The instrument exists; the population exists; the experiment is designed and waiting to be run. (A variant using practitioners’ texts embedded in nomic is explicitly rejected: the model’s own anisotropy would dominate, as the domain-independence result already demonstrates.)
V. The anharmonic framework
The following section is openly speculative. It arose during meditation and is offered as a candidate framework for future investigation, not as a finding.
Pure awareness as ground state
In quantum field theory the vacuum is the ground state of the field — not empty, but the lowest-energy configuration, structured by the shape of the potential. Particles arise as excitations from this ground state. In a harmonic field, excitations are independent and equally spaced in energy: they arise and dissolve cleanly. In an anharmonic field, higher-order terms in the potential (λφ³, μφ⁴, …) couple the modes: when one excitation arises, it automatically generates secondary excitations in correlated directions.
The proposal: pure awareness is the ground state of the mental field. Mental appearances are excitations. The kleshas are not the appearances themselves but the anharmonic coupling constants — built into the potential, not discrete events — that cause one appearance automatically to generate others. This is a candidate mechanism for what the Sallatha Sutta calls the second arrow: the additional suffering generated not by unavoidable experience but by the mind’s habitual propagation of that experience through its coupling structure.
Ego-clinging as spontaneous symmetry breaking
The deepest source of anharmonicity in field theory is spontaneous symmetry breaking: the ground state selects a preferred direction, breaking the symmetry of the Lagrangian, and the coupling structure follows automatically — the Mexican hat potential is the canonical form.
The proposal: ego-clinging is spontaneous symmetry breaking in meaning-space. The pre-ego awareness field is approximately rotationally symmetric — no preferred centre, excitations equally possible in all directions. Ego-clinging selects a fixed reference point, the “I”, around which all appearances are organised. That selection breaks the symmetry, and the kleshas arise as its necessary consequence: coupling terms organised around preserving the selected centre, pulling craving toward it and aversion away from threats to it.
In the language of Section I: symmetry breaking is the selection of a perspective. A perspective is a preferred direction in a space that previously had none. The five determinants are then five nested symmetry breakings — the corpus selects, the objective selects, the mechanism selects, the path selects, the klesha selects — and the wedge construction of Section II is the formal procedure for making a stack of selections explicit and then dualising past them.
Liberation as symmetry restoration
Liberation, in this scheme, is symmetry restoration: the ego-centre dissolves, the anharmonic couplings reduce toward zero, the potential returns toward harmonic. Appearances continue to arise — the harmonic vacuum always fluctuates — but without the coupling structure that generates second arrows. In the graded language of Section II: practice with support holds a direction; practice without support holds only an orientation — a bivector, a plane of attention with no selected vector in it; the ground holds neither.
The CMB analogy now lands where it should. The acoustic peaks encode the coupling structure of the primordial plasma, readable from its consequences long after the fact. If the practitioner experiment of Section IV shows the predicted opening of the cone, then the compression in ratings-space is the CMB of the ego-organised mind: not the symmetry-breaking event itself, but its geometric consequence, present and measurable. If it does not, Reading A stands, and the shadow belongs to the optimiser rather than the mind.
VI. Open questions
The coupling_constant / klesha dispute. nomic finds these descriptions structurally similar (0.579); Qwen rates them unrelated (0.000). Is the anharmonic analogy a genuine structural correspondence or a coincidence of descriptive vocabulary? A paraphrase-controlled version of Set 1, with the physics and contemplative descriptions sharing no content words, would begin to settle it.
Surviving the centring. Does the recognition-orthogonal-to-craving→liberation result survive mean-centring of the embeddings? This is the cheapest decisive check in the paper.
The intermediate grades. Is there a phenomenology of holding an orientation without a direction — and does the grade structure of the wedge (vector, bivector, trivector, …) map onto the graduated structure of śamatha with and without support?
The political complement. Equality, community, and justice sit in the complement of the dominant freedom–security frame. What interventions in political meaning-space would make these directions more accessible — lower their orthogonality to the frame without absorbing them into it?
Qwen’s identification. The collapse of recognition = awareness = wisdom = liberation is defensible at the absolute level but destroys the complement geometry the architecture needs. Does the instrument have to operate in the relative domain even while oriented toward the absolute?
What lives in the 56%? The unexplored angular range of the embedding space — what semantic content would occupy it? Can it be probed without language, or only pointed at?
The practitioner test. The single experiment on which Readings A and B divide. Designed, instrumented, waiting to be run.
VII. Related work
This investigation sits at the intersection of several active research areas, none of which covers the specific conjunction attempted here.
On the geometry of AI embedding spaces: the linear representation hypothesis holds that high-level features correspond to approximately linear directions in LLM representation space. The superposition hypothesis proposes that models encode far more features than their dimensionality by using almost-orthogonal directions. Recent work finds causally separable concepts represented by orthogonal vectors, and categorical concepts forming geometric regions such as polytopes. Separately, the anisotropy of contextual embedding spaces — the narrow-cone phenomenon and its mitigation by mean-centring and common-component removal — is documented in its own literature, and Section IV depends on taking that literature seriously as a rival explanation. The present work applies these geometric frameworks to contemplative and political vocabulary specifically, and introduces the complement space as a generative mechanism rather than an interpretability tool.
On Buddhist–physics dialogue: Varela, Thompson and Rosch’s The Embodied Mind established the productive relationship between cognitive science and contemplative traditions, deepened in Thompson’s subsequent work. The present paper differs in using the mathematical structure of field theory, rather than phenomenological description, as its primary language.
The closest Western analogue to the anharmonic framework is Friston’s free energy principle: organisms minimising surprise relative to a generative model. There is structural resonance — the generative model functions like the harmonic potential, surprise like an anharmonic perturbation — but Friston describes the organism from outside rather than the appearance-structure from within the ground state. The observation of Section IV, that the training objective itself formalises grasping-toward-the-expected, is the point where the two frameworks would meet.
Two recent papers apply quantum formalisms to LLM embeddings directly — semantic wave functions with double-well potentials for ambiguity; Hamiltonian formalism treating cosine similarity as analogous to zero-point energy. Neither reaches the contemplative–physics conjunction attempted here.
References
- Elhage et al. (2022) — Toy Models of Superposition. Anthropic interpretability research.
- Park et al. (2023) — The Linear Representation Hypothesis and the Geometry of Large Language Models.
- Ethayarajh (2019) — How Contextual are Contextualized Word Representations? (anisotropy of embedding spaces).
- Mu & Viswanath (2018) — All-but-the-Top: Simple and Effective Postprocessing for Word Representations.
- Pennington, Socher, Manning (2014) — GloVe: Global Vectors for Word Representation.
- Reimers & Gurevych (2019) — Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks.
- Friston (2010) — The free-energy principle: a unified brain theory? Nature Reviews Neuroscience.
- Varela, Thompson, Rosch (1991) — The Embodied Mind. MIT Press.
- The Buddha (Sallatha Sutta, SN 36.6) — The Arrow. Two arrows: the unavoidable and the self-inflicted.
- Planck Collaboration (2020) — Planck 2018 results: Cosmological parameters. Astronomy & Astrophysics.
- Previous papers in this series — Tensors, Fields and Meaning-Space; Structured Descriptors for Dynamic Mental States. kusaladana.co.uk
Exploratory paper · May 2026 · kusaladana.co.uk
Code: meaning_space.py through meaning_space_v5.py · Models: nomic-embed-text, Qwen 2.5 14B, all-MiniLM-L6-v2, GloVe-wiki-gigaword-50
This framework does not need to be true. It needs to open the debate.
PREVIOUS DRAFT
Exploratory Paper · Meaning-Space Series
Mathematical analysis of AI meaning-space, analogies with Buddhist mental dynamics, and the measurable structure of political thought
May 2026 · kusaladana · Draft for discussion
This paper began with a simpler observation: that the geometry of artificial intelligence meaning-space — the high-dimensional space in which language models organise semantic content — might function as a similar kind of shadow. Not of the early universe, but of the structure of human mental dynamics. Measurable from outside. Carrying information about something that cannot be directly observed.
Status. This is an exploratory investigation. The experiments are indicative rather than conclusive — the concept sets are small, the phrase selection is not independent of the theoretical framework, and the an-harmonic field theory section is openly speculative. The work does not claim hard results. It aims to make certain questions precise enough that productive disagreement becomes possible.
The Cosmic Microwave Background is the thermal shadow of a state we can never directly observe — the early universe before it became transparent, roughly 380,000 years after the Big Bang. We cannot go back. We cannot measure it directly. But the anisotropy pattern, the acoustic peaks, the power spectrum of temperature fluctuations — these are the geometric consequences of the primordial coupling structure, still readable now, 13.8 billion years later.
Like the intractability of the early universe, what follows is a record of an investigation of the structure of meaning space: Here we present some initial ideas and mathematics developments, resulting experiments runs, and results pertaining to the relationship of dot and vector products in meaning space. A tentative theoretical framework emerges — including aspects from considering thoughts arising in unattached awareness which find expression in mathematical description.
The Mathematical Framework
In N-dimensional space, any two vectors a and b partition the full space into two complementary regions.
The dot product — cosine similarity — measures alignment with the plane spanned by a and b. High dot product means a concept lies close to the plane of what a and b already contain. Standard semantic retrieval in AI (RAG systems, cosine similarity search) operates entirely within this plane. It can only recognise what it already holds. It is constitutively the operation of the familiar.
The cross product — in three dimensions — generates the unique direction perpendicular to both a and b. In higher dimensions, two vectors do not determine a unique perpendicular: they determine a perpendicular subspace of dimension N−2. An enormous family of possible new directions.
These two regions partition the full space without overlap. The dot product covers the plane of the already-known. The complement covers everything genuinely new to the pair.
Standard AI semantic retrieval operates entirely in the plane of the already-known. The cross product opens the complement — every direction not already contained in what you are holding.
Path-dependence and the iterative cross product
To recover a unique direction from the complement in N dimensions requires successive cross products — taking two vectors from the perpendicular subspace, finding their perpendicular within it, repeating until a unique direction emerges. This terminates after ⌈(N−1)/2⌉ steps.
But the terminal direction is path-dependent. Different choices at each step produce different limit vectors. The complement is fully covered by all paths together — but any particular path carves out one specific direction, shaped by every choice made along the way.
Each step of the iteration narrows the available space and reflects the perspective of whoever is choosing. The path accumulates perspective, each step more limiting than the last. Yet the terminal vector is what survives all the accumulated perspectives — the direction that no individual perspective directly contained. An apophatic structure: each exclusion is a step toward what remains when everything familiar has been removed.
Kleshas as principal components
The habitual tendencies — in Buddhist psychology, the kleshas — organise experience preferentially in certain directions. They are the directions of maximum variance in the practitioner’s meaning-space: the most load-bearing vectors, the ones that account for the most structure. In mathematical terms, they are the principal components.
As practice thins the kleshas, the principal components weaken, the plane of the already-known contracts, and the complement expands. The cross product operation can reach more of the space — not because the operation changed, but because the klesha-organised plane has loosened its grip.
II
Five Experiments
Experiment 1 — GloVe word vectors: the failure mode
The first experiment used GloVe word vectors trained on Wikipedia and Gigaword — 50-dimensional embeddings of single words. The geometry was real but the semantic content was not what we needed. GloVe’s nearest neighbours for liberation: separatist, rebel, moro, guerrilla — the Moro Liberation Front. Awareness: promotes, prevention, promote — public health campaigns.
GloVe places each word in the position determined by its dominant statistical usage in the training corpus. Liberation in Wikipedia is mostly political liberation movements, not moksha. Single-word static embeddings cannot distinguish these uses.
This is not a failure of the experiment. It established the key requirement: to probe philosophical meaning-space, concepts must be embedded as contextualised phrases that specify the intended meaning.
Experiment 2 — MiniLM with phrase embeddings
Switching to a sentence-level model (all-MiniLM-L6-v2, 384 dimensions) and embedding each concept as a full descriptive phrase — “liberation as awakening, moksha, freedom from the cycle of suffering and rebirth” rather than the single word — changed the geometry substantially.
The most notable result: recognition sits in the complement of span{craving, liberation}, ranked second out of ten available concepts with complement component 0.969. Recognition is not at the end of the craving-to-liberation path. It is orthogonal to that axis. This matches the Mahamudra teaching: rigpa is not an achievement on the path but something perpendicular to the path itself.
Experiment 3 — nomic-embed-text: higher dimensions
Running the same phrase set through nomic-embed-text (768 dimensions, purpose-built for semantic similarity via Ollama) confirmed the MiniLM finding and added new information.
Liberation ↔ path emerged as the highest similarity pair (0.756) — the progressive unfolding of practice and its goal, correctly identified as most related. Awareness ↔ recognition rose to 0.600 (vs MiniLM’s 0.37), more philosophically accurate.
The sparseness analysis revealed a structural property of the embedding space: with 768 dimensions theoretically available, the 12 concept embeddings clustered in a cone using only 44% of the available angular range. Effective dimensionality: 9.3 of 768.
Key measurement
In a 768-dimensional embedding space, 12 philosophical concepts clustered in a cone occupying 44% of the available angular range, with an effective dimensionality of 9.3 of 768. The cone compression is real, measurable, and — as Experiment 5 showed — domain-independent.
Experiment 4 — Qwen 2.5 14B as explicit semantic judge
Rather than extracting embedding vectors, this experiment asked Qwen 2.5 14B to explicitly rate semantic similarity between all 66 concept pairs. Two completely different mechanisms — statistical co-occurrence patterns versus explicit reasoning — allowed direct comparison.
Qwen collapsed recognition, awareness, wisdom, liberation, meaning, and path into mutual identity (all rated 1.0). This is a defensible Buddhist philosophical position — these terms all point at the same thing. But it geometrically destroyed the complement analysis: if recognition equals liberation, recognition cannot appear in liberation’s complement.
The disagreements between nomic and Qwen are where the data becomes most interesting:
| Pair | nomic | Qwen | Direction |
|---|---|---|---|
| meaning ↔ recognition | 0.510 | 1.000 | Qwen higher |
| liberation ↔ recognition | 0.532 | 1.000 | Qwen higher |
| wisdom ↔ recognition | 0.576 | 1.000 | Qwen higher |
| awareness ↔ momentum | 0.619 | 0.200 | nomic higher |
| wisdom ↔ momentum | 0.615 | 0.200 | nomic higher |
| awareness ↔ attention | 0.690 | 0.300 | nomic higher |
Qwen rates higher wherever concepts share an ultimate referent in Buddhist philosophy. nomic rates higher wherever concepts share functional texture in the training corpus. One is reading the map. The other is reading the territory of language use.
Experiment 5 — Three extended sets
Set 1: Anharmonic framework. Twelve concepts pairing physics terms with their proposed contemplative counterparts. The central result:
| Physics ↔ Contemplative pair | nomic | Qwen |
|---|---|---|
| ground_state ↔ bare_awareness | 0.568 | 0.700 |
| excitation ↔ mental_appearance | 0.656 | 0.500 |
| coupling_constant ↔ klesha | 0.579 | 0.000 |
| symmetry_breaking ↔ self_grasping | 0.545 | 0.500 |
| second_arrow ↔ anharmonic_coupling | 0.575 | 0.500 |
| symmetry_restoration ↔ liberation | 0.734 | 1.000 |
The coupling constant / klesha pair is the paper’s central claim made empirical. nomic finds the descriptions structurally similar (0.579). Qwen rates them completely unrelated (0.000) — different domains, no meaningful connection. The disagreement is now precisely located.
Set 2: Political meaning-space. Each of six political concepts was given in two distinct framings. Within-pair similarity measures how much the corpus conflates the two versions:
| Concept | nomic | Qwen | Interpretation |
|---|---|---|---|
| authority | 0.772 | 0.300 | corpus conflates most severely |
| security | 0.739 | 0.300 | corpus conflates |
| freedom | 0.747 | 0.500 | corpus conflates |
| community | 0.735 | 0.700 | both moderate |
| justice | 0.733 | 0.700 | both moderate |
| equality | 0.689 | 0.700 | both moderate |
The three concepts most commonly weaponised in political rhetoric — authority, security, freedom — are the three most conflated in the statistical geometry of language. Legitimate governance and coercive power score 0.772 similarity in nomic: near-synonymous in corpus space. Qwen distinguishes them at 0.300.
The complement of span{freedom-individual, security-national} — the dominant political frame — contains equality, community, and justice in both models. Not as opposites. As orthogonal directions. They cannot be reached by moving along the freedom-security axis. A different dimension is required.
Set 3: Cone boundary. Phrases designed to probe the unexplored angular range — “the pure potentiality before any crystallisation of experience into a specific direction”, “awareness without an object reaching toward nothing” — all remained within the nomic cone, with mean similarities of 0.48–0.62 to the anchor cluster.
The most striking result: awareness ↔ non_referential = 0.782 — the phrase designed to point beyond awareness was immediately pulled into awareness’s neighbourhood. The cone is sticky. The contemplative vocabulary used to probe the boundary is itself inside the contemplative cluster. To point genuinely outside may require phrases with no conceptual overlap with the existing vocabulary — which may be definitionally impossible to write within language.
Across all four concept domains, nomic showed essentially identical sparseness (mean similarity 0.553–0.563, effective dimensionality 8.1–9.3). The cone compression is a property of the model, not of the domain.
III
The Anharmonic Framework
The following section is openly speculative. It arose during meditation and is offered as a candidate framework for future investigation, not as a finding.
Pure awareness as ground state
In quantum field theory, the vacuum is the ground state of the field — not empty, but the lowest energy configuration, structured by the shape of the potential. Particles arise as excitations from this ground state. In a harmonic field, excitations are independent and equally spaced in energy: they arise and dissolve cleanly. In an anharmonic field, higher-order terms in the potential (λφ³, μφ⁴…) create coupling between modes: when one excitation arises, it automatically generates secondary excitations in correlated directions.
The proposal: pure awareness is the ground state of the mental field. Mental appearances are anharmonic excitations. The kleshas are not the appearances themselves but the anharmonic coupling constants — built into the potential, not discrete events — that cause one appearance to automatically generate others. This is the physical mechanism of what the Arrow Sutta calls the second arrow: the additional suffering generated not by unavoidable experience but by the mind’s habitual propagation of that experience through its coupling structure.
Ego-clinging as spontaneous symmetry breaking
Anharmonicity arises from asymmetry in the potential, not merely from spatial constraint. The deepest source of anharmonicity in quantum field theory is spontaneous symmetry breaking: when the ground state selects a preferred direction, breaking the symmetry of the Lagrangian. The Mexican hat potential is the canonical form — the vacuum sits at one point on the rim, and the anharmonic coupling structure arises automatically as a consequence.
The proposal: ego-clinging is spontaneous symmetry breaking in meaning-space. The pre-ego awareness field is approximately rotationally symmetric — no preferred centre, excitations equally possible in all directions. Ego-clinging selects a fixed reference point — the “I” — around which all appearances are organised. That selection breaks the rotational symmetry. The kleshas arise as the necessary mathematical consequence: coupling terms organised around preserving the selected centre, pulling craving toward it and aversion away from threats to it.
The cone compression — 44% of angular range used, 56% unused — is the geometric shadow of this symmetry breaking. The training corpus was produced overwhelmingly by humans with ego-clinging intact. The model’s geometry carries the contraction baked in.
Liberation as symmetry restoration; the CMB as analogy for testability
Liberation, in this scheme, is symmetry restoration: the ego-centre dissolves, the anharmonic coupling terms reduce toward zero, the potential returns toward harmonic. Mental appearances continue to arise — the harmonic vacuum always fluctuates — but without the coupling structure that generates second arrows. The cone expands back toward the full hypersphere.
By definition, subjective experience resists direct objective verification. But the geometric shadow is measurable. The prediction follows: meaning-space produced by practitioners with progressively reduced ego-clinging should show measurably lower cone compression — higher effective dimensionality, mean pairwise similarity approaching zero as symmetry restores.
The Cosmic Microwave Background is the thermal shadow of a state we can never directly observe. The acoustic peaks encode the coupling structure of the primordial plasma — readable now from its consequences. The cone compression in meaning-space is the CMB of the ego-organised mind: not the symmetry-breaking event itself, but its geometric consequence, still present and measurable.
IV
Open Questions
- The coupling_constant / klesha dispute. nomic finds these descriptions structurally similar (0.579). Qwen rates them completely unrelated (0.000). Is the anharmonic analogy a genuine structural correspondence, or a coincidence of descriptive vocabulary? What would settle this?
- Escaping the cone. All boundary-probing phrases remained within the nomic cone. Can the cross product operation — working from the most orthogonal pairs — generate limit vectors outside the compressed region? Or does the stickiness of language make this definitionally impossible?
- The political complement. The complement of span{freedom-individual, security-national} empirically contains equality, community, and justice. What interventions in political meaning-space would make these directions more accessible — lower their orthogonality to the dominant frame without pulling them into it?
- Qwen’s identification. Qwen collapses recognition = awareness = wisdom = liberation = meaning = path into mutual identity. This is philosophically defensible at the absolute level but destroys the complement geometry needed for the architecture to function. Does the COSINE instrument need to operate in the relative domain even if it is oriented toward the absolute?
- The shadow as test. If practitioners with reduced ego-clinging show measurably lower cone compression in their similarity ratings, the anharmonic framework has indirect empirical support. This experiment is designed and waiting to be run.
- What lives in the 56%? The unexplored angular range of the nomic embedding space — the region outside the compressed cone — what semantic content would occupy it? Can it be probed without language, or only pointed at?
V
Related Work
This investigation sits at the intersection of several active research areas, none of which covers the specific conjunction attempted here.
On the geometry of AI embedding spaces: the linear representation hypothesis holds that high-level features correspond to approximately linear directions in LLM representation space. The superposition hypothesis proposes that models encode exponentially more features than their dimensionality by using almost-orthogonal directions. Recent work has found that causally separable concepts are represented by orthogonal vectors, and that categorical concepts form geometric regions such as polytopes. The present work applies these geometric frameworks to contemplative and political vocabulary specifically, and introduces the complement-space as a generative mechanism rather than an interpretability tool.
On Buddhist-physics dialogue: the work of Varela, Thompson, and Rosch in The Embodied Mind established the productive relationship between cognitive science and contemplative traditions. Evan Thompson’s subsequent work has deepened this. The present paper differs in using the mathematical structure of field theory rather than phenomenological description as its primary language.
On the closest Western analogue to the anharmonic framework: Friston’s free energy principle describes organisms minimising surprise relative to a generative model. There is structural resonance — the generative model functions like the harmonic potential, surprise like an anharmonic perturbation — but Friston’s framework describes the organism from outside rather than the appearance-structure from within the ground state.
Two recent papers use quantum formalisms for LLM embeddings directly: one proposes semantic wave functions and double-well potentials for semantic ambiguity; another applies Hamiltonian formalism to embedding spaces treating cosine similarity as analogous to zero-point energy. Neither reaches the contemplative-physics conjunction attempted here.
References
- Elhage et al. (2022) — Toy Models of Superposition. Anthropic interpretability research.
- Park et al. (2023) — The Linear Representation Hypothesis and the Geometry of Large Language Models.
- Pennington, Socher, Manning (2014) — GloVe: Global Vectors for Word Representation.
- Reimers & Gurevych (2019) — Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks.
- Friston (2010) — The free-energy principle: a unified brain theory? Nature Reviews Neuroscience.
- Varela, Thompson, Rosch (1991) — The Embodied Mind. MIT Press.
- The Buddha (Sallatha Sutta, SN 36.6) — The Arrow. Two arrows: the unavoidable and the self-inflicted.
- Planck Collaboration (2020) — Planck 2018 results: Cosmological parameters. Astronomy & Astrophysics.
- Previous papers in this series — Tensors, Fields and Meaning-Space; Structured Descriptors for Dynamic Mental States. kusaladana.co.uk
Exploratory paper · May 2026 · kusaladana.co.uk
Code: meaning_space.py through meaning_space_v5.py · Models: nomic-embed-text, Qwen 2.5 14B, all-MiniLM-L6-v2, GloVe-wiki-gigaword-50
This framework does not need to be true. It needs to open the debate.