magnus919_agent-skills/semantic-spacetime/references/foundations.md

Foundations — The Academic Core of Semantic Spacetime

Load this file when you need the definitions, the formal model, proper time, causality, the γ(3,4) formalism, the learning/knowledge formalism, or an honest assessment of the theory's status. This is the academic anchor of the skill. What belongs here: the academic theory of Semantic Spacetime (SST) as developed by Mark Burgess (2014-2025) — definitions, the formal model, γ(3,4), proper time, causality, the promise-theory substrate, and adjacent fields. What does not belong here: quantum-gravity or physics derivation (this is not a physics theory — see §4), the CFEngine/infrastructure application history, and the agent-coordination synthesis; those belong to the skill's application and agent-coordination references (applications-infrastructure.md and agent-coordination.md). For one-line definitions see glossary.md; for sources see bibliography.md.

Provenance. Every definition below is tagged with exactly one marker, following the research corpus this skill was built from:

• [VERIFIED] — confirmed directly in a primary source fetched during the research phase (the arXiv papers, markburgess.org pages, and the fetched secondary sources listed in bibliography.md).
• [UNVERIFIED] — secondary-source or inferred; confirmed only via metadata, search index, or an author's own secondary account.
• EXTRAPOLATION — original synthesis extending the theory to new domains; never presented as a verified fact.

The theory is semi-formal and deliberately unrefereed (§5). This file states what is defined and verified, what is only informally claimed, and what is this skill's own synthesis. Do not present unverified claims as fact and do not drop markers when reusing this content.

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1. Authorship and scope of the term

Semantic Spacetime is the coinage and project of Mark Burgess — the physicist-turned-computer-scientist who created CFEngine — with the exact term effectively his alone. A full-text search of arXiv for the exact phrase "semantic spacetime" returns exactly 7 hits, all by Burgess; "semantic space-time" returns zero hits [VERIFIED — arXiv full-text search performed 2026-08-12]. There is no independent academic school using the term. The primary series is his arXiv papers 2014-2025:

• Spacetimes with Semantics (2014), arXiv:1411.5563 [VERIFIED]
• Spacetimes with Semantics (II): Scaling of agency, semantics, and tenancy (2015), arXiv:1505.01716 [VERIFIED]
• Spacetimes with Semantics (III): The Structure of Functional Knowledge Representation and Artificial Reasoning (2016, rev. 2017), arXiv:1608.02193 — the most formal document, canonical source for Definitions 1-9 and Lemmas 1-3 [VERIFIED]
• Agent Semantics, Semantic Spacetime, and Graphical Reasoning (2025), arXiv:2506.07756 — the current formal statement, introducing the γ(3,4) representation [VERIFIED]

Burgess states the intent directly: "I have no interest or intention of seeking to publish any of this work beyond making these notes available seeking trusted review" [VERIFIED — markburgess.org/blog_spacetime3.html]. SST is a conceptual/modeling framework, deliberately not a quantum-gravity theory (§4).

2. The formal model

The formal skeleton comes from Part III (arXiv:1608.02193v4), which Burgess calls "lengthy notes" laying foundations; and from the 2025 γ(3,4) paper.

Semantic element (Definition 1)

"A semantic element is a tuple ⟨Aᵢ, {π_scalar j, …}⟩ consisting of a single autonomous agent, and an optional number of scalar material promises." [VERIFIED — arXiv:1608.02193v4, Definition 1]

An agent "surrounded by a halo of promises that imbue it with semantics" [VERIFIED — same source]. The promises are scalar/material (the agent's own capabilities and properties) as distinct from the vector/adjacency promises of Part II that connect elements into a spacetime [VERIFIED — arXiv:1505.01716].

Semantic spacetime (Definition 2)

"A collection of semantic elements, in any phase (gas or solid), for which a local change in state, promises or configuration represents a local unit of time." [VERIFIED — arXiv:1608.02193v4, Definition 2]

Companion one-liner from the project hub: "A semantic spacetime is a discrete graph, which evolves, and whose properties vary from point to point." [VERIFIED — markburgess.org/spacetime.html]. The definition makes time a property of local change within the graph, not an external axis.

Proper time and the absence of a global clock

Time in SST is proper time: "Time in this sense is the Aristotelian concept of proper time as countable changes, as observed by the agent concerned." [VERIFIED — arXiv:2506.07756 §1.3]. There is no global clock: "The view of time as a relative transition system goes back to the work of Leslie Lamport… Lamport rediscovered the idea that time can at best be understood as a precedence relation, in a discrete spacetime context." [VERIFIED — markburgess.org/semantic_spacetime.html]. Lamport, "Time, Clocks, and the Ordering of Events in a Distributed System," CACM 21(7):558-565, 1978, is the credited origin of this precedence view [VERIFIED — same page; bibliography]. Practically: two agents cannot share a wall-clock ordering of events; each element's sequence of local changes is its own time.

Causality as cooperative promises

Causality in SST is constituted by cooperative promises, not by imposed links. Each adjacency requires both an offer (+) and an acceptance (−) promise between the two ends: "each node must both emit and absorb adjacency relations, cooperatively… Thus space is made up of cooperating nodes and edges." [VERIFIED — markburgess.org/semantic_spacetime.html]. In the notation of the papers, S →(+π) R means sender S offers promise π to receiver R, which accepts with the complementary −π promise; influence passes only through the overlap of offer and acceptance [VERIFIED — arXiv:1608.02193]. This is the promise-theoretic spine that makes SST an agent model rather than a global network model: every edge is a negotiated, observable relation.

The γ(3,4) formalism

The 2025 paper (arXiv:2506.07756) refines the earlier four irreducible associations (aggregation, causation, cooperation, similarity — [VERIFIED — arXiv:1608.02193]) into a typed graph formalism called γ(3,4): three node meta-types × four link types [VERIFIED — arXiv:2506.07756, Table 1].

The three node meta-types [VERIFIED — arXiv:2506.07756 §2.3]:

Meta-type	Symbol	Nature
Events	e	Temporary/ephemeral; timelike (process) agents; persist or change via "leads to"
Things	t	Persistent, physical/realized agents; "behave like matter"; spacelike (snapshot)
Concepts	c	Invariant notions that cannot be created or destroyed; virtual space of "unrealized" potential; materialized only by attaching to physical agents

The four link types, exactly [VERIFIED — arXiv:2506.07756, Table 1]:

Value	Label	Direction	Semantics
0	NEAR	symmetric	equivalence, similarity, proximity, correlation
±1	LEADS TO	directed	temporal/causal order: enables, causes, precedes, depends on
±2	CONTAINS	directed	containment, membership, generalization, coarse-graining
±3	EXPRESSES	directed	attribute, name/value, property, distinguishing mark

Burgess frames the four-link hypothesis itself as a hypothesis: "This remains a hypothesis for now, but it is not a particularly original one. Various authors have suggested that spacetime concepts underpin natural language." [VERIFIED — arXiv:2506.07756 §2.2]. No additional link types exist in the formalism; adding one would leave γ(3,4).

The nine typing design rules

The node typing rules from arXiv:2506.07756 §2.3, exactly as verified [VERIFIED — arXiv:2506.07756 §2.3]:

1. Things may be contained but not expressed.
2. Concepts may be expressed but not contained.
3. Concepts become realized by anchoring them to things or events.
4. Verbs are dangling concepts without a subject or object to instantiate them.
5. Verbs anchored to subjects/objects (things) are events.
6. A realized state of being is an event.
7. An unrealized state of being is a concept.
8. A realized type of thing is a thing.
9. An unrealized type of thing is a concept.

Note on the paper's abstract: it states that "The Semantic Spacetime postulates bring predictability when reasoning," but the research phase could not verify an enumerated postulate list in the fetched text (it would require a full read of the paper's later sections). Treat the nine design rules above as the verified typing content; do not present them as a numbered list of "the Semantic Spacetime postulates" [UNVERIFIED — exact postulate set not verified].

Location agents and signal agents

Two auxiliary agent types complete the model's ontology [VERIFIED — arXiv:1608.02193v4]:

• Location agents (Definition 6): "irreducible sites that take up space and can emit and absorb signal agents. They may not overlap."
• Signal agents (Definition 7): "They may be created and destroyed, subsequently emitted and absorbed, by location agents. They can occupy the same space, since they end up and accumulate at end points."

3. Absorbing states and information leaks

Absorbing states are a central diagnostic concept in SST: "The ubiquitous appearance of absorbing states in any partial graph means that a graph process leaks information." [VERIFIED — arXiv:2506.07756 abstract]. They are "non-conserving of information" [VERIFIED — same source]. Burgess ties the phenomenon to division by zero: the leak is "closely associated with the issue of division by zero, which signals a loss of closure and the need for manual injection of remedial information" — and the boundary where the graph leaks is "boundary information where intentionality can enter" [VERIFIED — arXiv: 2506.07756 §1.3]. Practically: a dead-end node (an event or thing with no outgoing LEADS TO/EXPRESSES edges that matter) accumulates meaning and stops propagating it; intent or policy must be injected manually at that boundary. For a bounded diagnosis procedure using this concept, see the skill's diagnosis-and-debugging.md reference.

4. The "not physics" boundary

SST is explicitly not a theory of physics: "Semantic spacetime is a discrete model of spacetime, but it is not intended as a theory of quantum gravity, in spite of some affinity with quantum systems." [VERIFIED — markburgess.org/semantic_spacetime.html]. Three consequences worth stating [VERIFIED — markburgess.org/semantic_spacetime.html]:

• No manifold structure is assumed: space is constituted by relationships between objects, not by a background geometry.
• There is no concept of variable velocity, nor momentum: "a discrete spacetime with finite number of states is not obviously a canonical system."
• The connection with canonical systems remains unknown.

When a task is physics (general relativity, quantum gravity, kinematics), SST is the wrong tool; route away at the SKILL.md "When not to use" boundary.

5. Status: semi-formal and unrefereed

The core series is a set of self-published notes, deliberately not submitted for refereed publication: "I have no interest or intention of seeking to publish any of this work beyond making these notes available seeking trusted review" [VERIFIED — markburgess.org/blog_spacetime3.html]. Burgess also warns of the scope: "I have improvised with an eye on practical applications. It is probably too ambitious in scope and detail, but bridges may serve a purpose even with gaps," and "Although not a complete theory, it lays out guidance on the formulation of the basic issues of information propagation, with some proofs left to the reader." [VERIFIED — arXiv:1608.02193 preamble; markburgess.org/ semantic_spacetime.html]. Use the formalism as a reasoning aid, not a proof system. What is formal: the graph definitions (Definitions 1-9), the γ(3,4) type system and its nine design rules, the learning/knowledge formalism with its Nyquist bound and decay lemmas (§9), and the association-decomposition algebra. What is semi-formal or metaphorical: the scaling/tenancy results of Part II, and the physics parallels (Feynman/Schwinger readings, quantum-field analogies, "logic emerges from reasoning") [VERIFIED — arXiv:1608.02193; markburgess.org].

6. Promise theory as the substrate

SST is formally built from Promise Theory: "The chosen language here is Promise Theory (2004-2014)" [VERIFIED — markburgess.org/spacetime.html] and "the idea of semantic spacetime is based on an idea called Promise Theory" [VERIFIED — markburgess.org/blog_spacetime3.html]. Promise Theory is the joint work of Mark Burgess and Jan A. Bergstra; its canonical statement is Promise Theory: Principles and Applications (χtAxis Press, 2014; 2nd ed. 2019), which describes itself as a "semi-formal language for modelling intent and its outcome" [VERIFIED — markburgess.org/promises.html].

The primitives SST inherits, stated here in one line each and developed in depth by the promise-theory skill, are:

• Promise — an autonomous declaration of intended behavior, with a body (label Λ), a type (τ), and a constraint (χ); written S →(+π) R for an offer from promiser S to promisee R [VERIFIED — promise-theory foundations; arXiv:1608.02193].
• Offer (+) and acceptance (−) — every interaction requires both directions to be promised independently; this is the semantic spine of adjacency in SST (§2, Causality) [VERIFIED].
• Autonomy and locality — agents are autonomous and inert except for the promises they make; a strong form of locality, and the reason SST is an agent model rather than a global network model [VERIFIED].
• Downstream Principle — the most downstream party in a promise chain carries the greatest causal responsibility for the outcome [VERIFIED — promise-theory foundations].
• Convergence — repeated local assessment toward a desired state; the dynamic meaning of "convergent coordination" in SST [VERIFIED — promise-theory foundations].

Do not re-derive promise definitions here. When you need the promise vocabulary (promises, acceptances, bindings, assessment, trust, the Downstream Principle), load promise-theory or its foundations reference. This skill's territory is the space/time of meaning built on top of those promises: γ(3,4), trajectories, drift, semantic distance, shared semantic ground.

7. Measurement: the spacelike/timelike duality

SST distinguishes two inequivalent ways to stabilize observation, which Burgess maps onto the Feynman (path-integral) vs. Schwinger (source) readings of quantum theory [VERIFIED — markburgess.org/semantic_spacetime.html; markburgess.org/spacetime.html]:

1. Spacelike / ensemble measurement — repeated trials with constant state and semantics, in which time plays no role; objective/frequentist. You sample the same configuration many times and average.
2. Timelike / "cognitive" measurement — continuously adapting accumulation of state, whose semantics define change in real time; subjective/Bayesian. You update a running assessment as the system changes.

The two modes can disagree because they answer different questions, and the practitioner consequence is the skill's core measurement rule: semantics requires measurement — meaning cannot be asserted before the dynamics are measured at the right scale. Different scales yield different conclusions; a measurement that is stable at one scale can be wrong at another. This duality is the theory-level ground for the "dynamics always trumps semantics" lesson of the infrastructure lineage (covered in the application reference, applications-infrastructure.md) and for Gotcha 4 in SKILL.md.

8. Distance: metric vs semantic

Part III defines two kinds of distance [VERIFIED — arXiv:1608.02193v4]:

• Metric (quantitative) distance (Definition 8): "a measure of coordinate-similarity in position." Coordinates, embeddings, positions.
• Semantic (qualitative) distance (Definition 9): "a measure of similarity in interpretation." Worked examples in the paper: Hamming distance; hop counts in an associative network; semantic hashing; sparse distributed representations [VERIFIED — same source].

The distinction is operational: two concepts can be close in coordinates yet far in interpretation, and vice versa. A weighted hop count over a γ(3,4) graph is a semantic-distance instance of the hop-count family — the family this skill's model tooling implements for measuring drift between two snapshots of a system's meaning.

9. Learning and knowledge

SST formalizes learning and knowledge as processes with explicit timescales [VERIFIED — arXiv:1608.02193v4]:

• Learning about a promise π (Definition 3): "the sampling, equilibration, and summarization of observational assessments concerning a promise π made by another agent, repeated over a timescale T_learn > 2·T_sample." The observer applies a learning function E(α(π)_{t+1}) = L(α(π)_t, E(α(π)_t)); learning defines a clock ticking at rate T_sample.
• Knowledge of π (Definition 4): "a stable summary of the iterated assessment α(π)_{T_know}, of one or more promises π, formed by equilibration of the samples over a timescale T_know ≫ 2·T_sample." Crucially, "because knowledge defines a process with a timescale, the failure to confirm it relative to other changes leads to its decay."
• Lemma 1 (knowledge decay): uncertainty of knowledge grows geometrically with time since learning, with attenuation ℓ^r, ℓ < 1.
• Lemma 2 (fidelity / learning rate): "Learning can only represent source values faithfully if the rate of sampling is greater than twice that of the fastest rate of change in the data, i.e. 2/T_sample < ∂π/∂t" — the Nyquist bound.

Practitioner consequence: staleness is a first-class quantity. Memory and retrieval designs must budget refresh; a knowledge summary that is never re-confirmed decays geometrically no matter how accurate it was when formed. This directly supports drift diagnosis: a stale shared interpretation is a predictable source of semantic divergence.

10. The empirical arm: the Quantitative Spacetime Hypothesis

Two 2020 papers operationalize SST as a testable hypothesis rather than pure formalism [VERIFIED — arXiv:2010.08126; arXiv:2010.08125]:

• arXiv:2010.08126 — Testing the Quantitative Spacetime Hypothesis using Artificial Narrative Comprehension (I): Bootstrapping Meaning from Episodic Narrative viewed as a Feature Landscape. Parses narrative streams "without knowledge of semantics, using only measurable patterns (size and time)… as an event 'landscape'"; concepts are extracted "as process invariants." Results claim simple spacetime process cues, not higher reasoning, drive what is important about sensory experience [VERIFIED — arXiv:2010.08126].
• arXiv:2010.08125 — …(II): Establishing the Geometry of Invariant Concepts, Themes, and Namespaces. Reconstructs concepts and themes via "multiscale interferometry" and a "chemistry of association and pattern reconstruction, based only on the four fundamental spacetime relationships," drawing a bioinformatic analogy (n-grams, micro/meso/macro scales) [VERIFIED — arXiv:2010.08125].

Honest caveat: these are proof-of-concept experiments on narrative corpora with single-CPU methods; the research phase found no independent replication and no benchmark against distributional baselines [UNVERIFIED — no independent replication found]. Treat the Quantitative Spacetime Hypothesis as an active, incompletely validated empirical program — not established validation of SST.

11. Spacetime-Entangled Networks: consensus as entanglement

Spacetime-Entangled Networks (I): Relativity and Observability of Stepwise Consensus is a four-author paper — Paul Borrill, Mark Burgess, Alan Karp, Atsushi Kasuya (arXiv:1807.08549, 2018, rev. 2020) — that instantiates the SST/promise line at the distributed-consensus layer [VERIFIED — arXiv: 1807.08549]: "Entanglement describes co-dependent evolution of state. Networks formed by entanglement of agents keep certain promises: they deliver sequential messages, end-to-end, in order, and with atomic confirmation of delivery to both ends of the link." The "relativity of consensus" reading — observers at different points in the network reach consensus stepwise, in their own local order — is the SST no-global-clock doctrine applied to agreement [VERIFIED — arXiv:1807.08549; the mapping onto the cooperative-promise causality doctrine of §2 is this skill's synthesis and is labeled EXTRAPOLATION]. Note this paper is not one of the seven "semantic spacetime" phrase hits; it does not use the exact term [VERIFIED — arXiv search].

12. Motion of the Third Kind

SST distinguishes three ways to understand motion in a graph; the third, "virtual motion" (Motion of the Third Kind), treats processes and properties — for example cloud workloads and data records — as promises moving from host to host [VERIFIED — markburgess.org/spacetime.html]. This is the basis of Burgess's "cloud computing as virtual physics" framing: relocating a workload is not matter moving through space, it is a promise being re-anchored. The ResearchGate papers Motion of the Third Kind I & II (2021-22) exist but their full texts were not fetched during research; details beyond the moving-promises framing are [UNVERIFIED]. See glossary.md for the one-line entry.

13. Adjacent fields

SST sits next to — but is distinct from — these fields. Correct attribution and a one-line framing for each [VERIFIED — citations verified in the research phase; see bibliography]:

• Cognitive maps — Tolman, "Cognitive maps in rats and men" (1948). The brain demonstrably organizes knowledge spatially; SST is a candidate formal language for concept space-times, not a neuroscience claim.
• Conceptual spaces — Gärdenfors, Conceptual Spaces: The Geometry of Thought (MIT Press, 2000). Concepts as convex regions in metric spaces with quality dimensions; Gärdenfors-style spaces have no time dimension — SST adds process and temporality.
• Distributional / vector-space semantics — Harris (1954), LSA (Landauer & Dumais 1997), word2vec-style embeddings (Mikolov et al. 2013). The dominant statistical competitor; SST explicitly contrasts itself ("graphs preserve the intentionality of the source even under data fractionation" vs. vectorized probabilistic estimation [VERIFIED — arXiv:2506.07756; arXiv:2512.19084]).
• Event calculus — Kowalski & Sergot (1986). Logic-based reasoning about events where "the notion of event is taken to be more primitive than that of time"; SST instead claims spacetime structure generates the semantics.
• Situation calculus — McCarthy & Hayes (1969). Logic-based reasoning about actions and change; the same logic-first framing distinguishes it from SST.
• Causal sets — Myrheim (1978), Sorkin (2003), Surya (2019). The discrete- spacetime program Burgess flags as the closest physics analogue: "in this regard, a semantic spacetime is akin to causal sets" [VERIFIED — arXiv: 2506.07756 §2]. Difference: SST's nodes are autonomous agents with semantics, not passive points, and SST assumes no manifold structure or symmetries.
• Logical clocks / virtual time — Lamport (1978), Mattern (1988/89). The distributed-systems backbone for "no global clock"; SST generalizes logical clocks into full semantic spacetimes [VERIFIED].

14. Applying this reference

When you have modeled a system with this vocabulary, materialize it in the skill's model format — see templates/sst-model.yaml.tmpl (the versioned sst-model-v1 contract: agents, nodes, edges, acceptances, trajectories, observations) — and write the analysis in templates/sst-analysis.md.tmpl. For unfamiliar terms while reading, load glossary.md. For the promise-theory substrate vocabulary, load promise-theory — do not re-derive promises here. For measurement and verification practice (turning assessed meaning into evals and traces), the agent-evals-and-observability skill is the assessment-layer partner.

Sources

Primary sources and adjacent works are listed with URLs in bibliography.md. The key items cited in this file: Burgess, Spacetimes with Semantics I-III (arXiv:1411.5563, 1505.01716, 1608.02193); Burgess, Agent Semantics, Semantic Spacetime, and Graphical Reasoning (arXiv:2506.07756); Burgess, Testing the Quantitative Spacetime Hypothesis I-II (arXiv:2010.08126, 2010.08125); Borrill, Burgess, Karp & Kasuya, Spacetime-Entangled Networks (I) (arXiv:1807.08549); Lamport, Time, Clocks, and the Ordering of Events in a Distributed System (CACM 1978); Burgess's project pages (markburgess.org/spacetime.html, /semantic_spacetime.html, /blog_spacetime3.html); Bergstra & Burgess, Promise Theory: Principles and Applications (2014/2019).