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 $(h_1,\dots ,h_n)\in G^n$.

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 $p,p^{\prime }\in P(n,E)$ are gauge-equivalent if $p^{\prime }=h\cdot p$ 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 $p,p^{\prime }\in P(n,E)$ 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 $△=(i,j,k)$ in the graph: $\Omega (△):=g_{ij}{g}_{jk}{g}_{ki}\in G.$ 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 $[p],[p^{\prime }]\in C(n,E)$ 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 $w_{ij}=\epsilon >0$ (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 $E^{\prime }=E\cup \{(v,j):j\in J\}$ 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 ${\varphi }_t(F)=\theta$, 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 $G_{SPLIT/MERGE}$: crossing of $\{\varphi (F)=\theta \}$ (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.
• $\Delta \Omega :=\{△:{\Omega }_t(△)\ne {\Omega }_{t+1}(△)\}$ — changed holonomies.
• $\Delta \mathfrak{F}:={\mathfrak{F}}_t△{\mathfrak{F}}_{t+1}$ — birth/death of fruits (symmetric difference).
• $\Delta \Sigma :=\{(F,i):i\in {\Sigma }_t(F)△{\Sigma }_{t+1}(F)\}$ — 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 $\mathcal{S}={⨆}_{(n,E)}C(n,E)$ — all possible world states.
6. Observable map $\mathrm{Obs}:\mathcal{S}\to \mathrm{Obs}-\mathrm{Space}$ — 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 ${⨆}_{(n,E)}C(n,E)$ 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.