Chapter 0 — Formal State Space

Motivation: This chapter formalizes the state space of RelationWorld—the mathematical object describing all possible world configurations and the structure of the transitions between them. Chapter 8 defines the instantaneous world $\mathfrak{W}_t = (\mathcal{W}_t, \mathfrak{F}_t, \Sigma_t)$; this chapter defines the space in which $\mathfrak{W}_t$ lives as $t$ varies, and how events act on this space.

0.1 The Raw Parameter Space#

Definition 0.1 (Graph topology)#

An admissible graph topology is a pair $(n, E)$ where:

• $n = |V| \ge 1$ is the number of nodes.
• $E \subseteq V \times V$ is a set of ordered edges, with no self-loops: $(i,i) \notin E$.

The collection of all admissible topologies is denoted:

Definition 0.2 (Raw world configuration per topology)#

For fixed graph topology $(n,E)$ with $n \ge 1$ and $E \subseteq V_n \times V_n \setminus \{\text{diag}\}$:

A point $p = (w, g) \in P(n,E)$ is a raw world configuration:

• $w = (w_{ij})_{(i,j) \in E}$ — the edge weights (real numbers $\ge 0$).
• $g = (g_{ij})_{(i,j) \in E}$ — the edge transits (group elements in $G$, a compact Lie group).

Notation#

For clarity:

• $P(n,E)$ is the raw parameter space on topology $(n,E)$.
• A point $p \in P(n,E)$ is a raw configuration.
• The pair $(n,E)$ is the discrete topology index.

0.2 The Gauge Group and Its Action#

Definition 0.3 (Gauge group action)#

The gauge group $G^n$ acts smoothly on $P(n,E)$ by:

for all $(i,j) \in E$, where $(h_1, \ldots, h_n) \in G^n$.

Key properties:

• The action on weights is trivial: $w_{ij}$ is gauge-invariant.
• The action on transits is the conjugation action (natural for principal bundles).
• Two raw configurations $p, p' \in P(n,E)$ are gauge-equivalent if $p' = h \cdot p$ for some $h \in G^n$.

0.3 The Configuration Space (Gauge Quotient)#

Definition 0.4 (Configuration space per topology)#

A point $[p] \in C(n,E)$ is a configuration (or world configuration) on topology $(n,E)$.

Equivalently: two raw configurations $p, p' \in P(n,E)$ represent the same configuration iff they lie in the same $G^n$-orbit.

Topology and structure:

• $C(n,E)$ inherits the quotient topology from $P(n,E)$.
• For generic (non-degenerate) configurations, the $G^n$-action is free, and $C(n,E)$ is a smooth manifold.
• For degenerate (flat/reducible) configurations (all $g_{ij} = e$), the stabilizer may be non-trivial, and $C(n,E)$ may have orbifold or stratified structure.

0.4 The Total State Space (Stratified)#

Definition 0.5 (Total state space)#

The total world state space is the disjoint union of configuration spaces across all admissible topologies.

Discrete topology index: For any state $s \in \mathcal{S}$, there exists a unique topology $(n_s, E_s) \in \mathrm{Top}$ such that $s \in C(n_s, E_s)$. Define:

Stratification: $\mathcal{S}$ is naturally stratified: each stratum is $C(n,E)$ for some $(n,E) \in \mathrm{Top}$.

0.5 Observable Map (Gauge-Invariant Content)#

Definition 0.6 (Observable map)#

The observable map is:

where:

Components:

• Weights: $W_{ij} = (w_{ij} + w_{ji})/2$ (symmetrized, gauge-invariant).
• Holonomies: for each triangle $\triangle = (i,j,k)$ in the graph: $\Omega(\triangle) := g_{ij} \, g_{jk} \, g_{ki} \in G.$ This is gauge-invariant: $(h_i g_{ij} h_j^{-1})(h_j g_{jk} h_k^{-1})(h_k g_{ki} h_i^{-1}) = g_{ij} g_{jk} g_{ki}$ (independent of $h$).

Injectivity: Two configurations $[p], [p'] \in C(n,E)$ are equal iff they have the same observable content (weights and holonomies). Thus $\mathrm{Obs}$ is injective when restricted to a fixed topology.

0.6 Event Maps#

Definition 0.7 (Primitive event maps)#

The six primary events (from Chapter 14) induce maps on the state space:

Event 1: DEFORM(i,j, δ)#

Updates edge parameters within fixed topology:

given by $(w_{ij}, g_{ij}) \mapsto (w_{ij} + \delta w, g_{ij} \cdot \exp(\delta \xi))$ for perturbation vectors $\delta w, \delta \xi$.

Constraint: The perturbation must not drive $w_{ij}$ to zero (i.e., $(i,j)$ must remain in the edge support $E$).

Event 2: CONTACT(i,j)#

Adds a new edge (activates dormant pair):

given by: new edge $(i,j)$ is initialized with $w_{ij} = \varepsilon > 0$ (small) and $g_{ij} = e$ (identity).

Constraint: $(i,j) \notin E$ initially (no duplicate edges).

Event 3: BIRTH(v)#

Introduces a new node and initial edges:

where $E' = E \cup \{(v, j) : j \in J\}$ for some $J \subseteq V_n$ (specified a priori or by initialization rule).

Initialization: New edges from $v$ start with small weight and identity transit.

Event 4: DEATH(v)#

Removes a node and all its incident edges:

where $E_v := \{(i,j) \in E : i = v \text{ or } j = v\}$.

Constraint: $v$ must be a node in the current configuration.

Event 5: SPLIT(F)#

Detected transition (not a primitive operation): When the Cheeger conductance of a fruit $F$ equals the threshold $\phi_t(F) = \theta$, a fruit may split into two. This is a consequence of continuous DEFORM events and is logged in the evidence but does not correspond to a single map.

Event 6: MERGE(F₁, F₂)#

Detected transition (not a primitive operation): When inter-fruit coupling weights exceed a threshold, two fruits unite. Again, this is a detected consequence of DEFORM.

Definition 0.8 (Discrete topology transitions)#

The events BIRTH, DEATH, and CONTACT induce transitions between strata:

for event $e_t \in \{\mathrm{BIRTH}, \mathrm{DEATH}, \mathrm{CONTACT}\}$.

0.7 The Existence Triple#

Definition 0.9 (Existence per fruit)#

Recall from Chapter 7:

where:

• $[A_\infty] \in \mathcal{M}(F, \Sigma_t(F))$ is the canonical (optimal-gauge) connection on the fruit interior.
• $\Sigma_t(F)$ is the door set (Chapter 6).
• $\mathbf{e}_t(F)$ is the door energy.

Gauge-invariance: The existence triple is gauge-invariant by Theorem E (Chapter 7).

0.8 Hybrid Dynamical System Formulation#

Definition 0.10 (World evolution as hybrid system)#

The world evolution system is a hybrid dynamical system $(Q, X, f, G_e, R_e)_{e \in E}$ where:

Discrete events as guards:

• Guard $G_{\mathrm{CONTACT}}$: triggered by external command to activate edge $(i,j)$.
• Guard $G_{\mathrm{BIRTH}}$: triggered by external command to introduce node $v$.
• Guard $G_{\mathrm{DEATH}}$: triggered by external command to remove node $v$.
• Guard $G_{\mathrm{SPLIT/MERGE}}$: crossing of $\{\phi(F) = \theta\}$ (detected threshold).

Reset semantics:

• After DEFORM, the topology $(n,E)$ is unchanged; only parameters update.
• After CONTACT, BIRTH, DEATH, the topology changes; initial conditions for new edges/nodes are specified.

0.9 Evidence Extraction Functional#

Definition 0.11 (Evidence/explanation map)#

Given a state transition from $s_t$ to $s_{t+1}$ triggered by event $e_t$, the evidence extraction functional is:

where:

• $e_t$ — the event type (DEFORM, CONTACT, BIRTH, DEATH, SPLIT, MERGE, or a combination).
• $\Delta W := \{(i,j) : W_t(i,j) \ne W_{t+1}(i,j)\}$ — changed edge weights.
• $\Delta \Omega := \{\triangle : \Omega_t(\triangle) \ne \Omega_{t+1}(\triangle)\}$ — changed holonomies.
• $\Delta \mathfrak{F} := \mathfrak{F}_t \triangle \mathfrak{F}_{t+1}$ — birth/death of fruits (symmetric difference).
• $\Delta \Sigma := \{(F, i) : i \in \Sigma_t(F) \triangle \Sigma_{t+1}(F)\}$ — doors created/removed per fruit.

Property: The evidence is sufficient to reconstruct the transition (up to gauge): the state $s_{t+1}$ and the diff $\mathrm{Ev}(s_t, s_{t+1}, e_t)$ together determine the path taken.

0.10 Summary and Dependencies#

Key definitions in order:

1. Graph topology $(n, E)$ — indexing the stratification.
2. Raw parameter space $P(n,E) = \mathbb{R}_{\ge 0}^{|E|} \times G^{|E|}$ — the gauge-dependent data.
3. Gauge action $G^n$ on $P(n,E)$ — conjugation on transits, trivial on weights.
4. Configuration space $C(n,E) = P(n,E) / G^n$ — the gauge-invariant semantic space.
5. Total state space $\mathcal{S} = \bigsqcup_{(n,E)} C(n,E)$ — all possible world states.
6. Observable map $\mathrm{Obs} : \mathcal{S} \to \mathrm{Obs-Space}$ — extracting gauge-invariant weights and holonomies.
7. Event maps — transitions within and between strata.
8. Hybrid dynamics — continuous evolution punctuated by discrete topology transitions.
9. Evidence extraction — explaining transitions.

0.11 Relationship to Chapter 8 (World)#

Chapter 8 defines the instantaneous world:

where $\mathcal{W}_t$ is the relational field (a gauge class) at time $t$.

This chapter provides the global structure:

• $\mathcal{W}_t$ is an element of $C(n_t, E_t)$ for the topology $(n_t, E_t)$ active at time $t$.
• The world evolves through the stratified state space $\mathcal{S}$.
• Fruits $\mathfrak{F}_t$ and doors $\Sigma_t$ are derived from $\mathcal{W}_t$ via the procedures in Chapters 4--6.

0.12 Open Questions#

Six foundational questions remain:

Q1 (YM Uniqueness for Non-Abelian G): For $G = \mathrm{SU}(2)$, is the minimizer of the Yang--Mills energy $\mathcal{E}_{F^\circ}(h)$ unique? If not, how does the lack of uniqueness affect the existence triple definition?

Q2 (Event Completeness): Are the six events {DEFORM, CONTACT, BIRTH, DEATH, SPLIT, MERGE} complete and minimal? Can every admissible trajectory be uniquely decomposed into a sequence of these events?

Q3 (Inter-Stratum Topology): Is there a natural (e.g., metric or diffeological) structure on the disjoint union $\bigsqcup_{(n,E)} C(n,E)$ that makes BIRTH, DEATH, CONTACT continuous (or at least measurable in a meaningful sense)?

Q4 (Evidence Completeness): Does the evidence functional $\mathrm{Ev}(\cdot, \cdot, \cdot)$ capture sufficient information to uniquely identify which event occurred and certify that alternative events were impossible?

Q5 (Fruit Structure as Functor): Can the fruit set $\mathfrak{F}_t$ be viewed as a continuous functor on the category of weighted graphs? What are its continuity properties with respect to the topology on $C(n,E)$?

Q6 (Minimal Variables): Is the pair $(w_{ij}, g_{ij})$ the minimal sufficient statistic for the world, or can the system be more parsimoniously described using the existence triples alone?

0.13 Reading Notes#

• This chapter is logically prior to Chapters 1--8 in that it provides the global stage. In practice, readers should:

  1. Read Chapters 1--8 first (they develop intuition).
  2. Return here to understand the formal state space.
  3. Continue to Chapters 9--14 with the state space structure in mind.

• The hybrid dynamical system formulation (Definition 0.10) makes precise the phrase "smooth dynamics punctuated by discrete topological events" from Chapter 14.

• The stratified structure of $\mathcal{S}$ is the mathematical reason why the world evolves as a hybrid system, not as a single smooth flow.