Step 1 — Defining the geometry of the target system
Human cognition “can observe fragments of these systems, but… cannot hold their full geometry in mind. AI, however, can.” Step 1 formalises what that “geometry” is, so that subsequent representation and inference can be built on a precise, engineered object rather than an informal description.
1. Objective
Goal: Construct a mathematically explicit, machine‑usable geometric model of a complex system that captures its interacting dimensions, structures, and constraints without compressing them into low‑dimensional narratives. Formally, the system is represented as a high‑dimensional state space
where Rn encodes entity‑level variables, Rm×k encodes spatio‑temporal tensors, G is a heterogeneous graph of relationships, and M is a set of domain manifolds. The geometry must support nonlinear update mappings
allowing the system to evolve without collapsing structural complexity.
Outcome: A high‑dimensional state space S with defined variables, relationships, and dynamics, serving as the substrate for Adaptive Representation and Adaptive Inference.
2. System scoping and boundary definition
Identify domains: Enumerate all relevant domains (climate, economics, ecology, technology, geopolitics, etc.) that contribute materially to the behaviour of interest. Represent the system formally as
where D is the domain set, E the entity set, and B the boundary definitions.
Define entities: Specify core entities (countries, firms, ecosystems, infrastructures, populations) and their roles. Each entity ei is represented as an object with attributes and behavioural rules.
Set boundaries: Decide spatial, temporal, and functional limits, e.g. global vs regional, decadal vs annual, macroeconomic vs microeconomic. Spatial boundaries are defined as
and temporal boundaries as
Example: For a global climate–economy system, entities might include nation‑states, energy producers, major ecosystems, financial institutions, and key infrastructures (grids, pipelines, ports).
3. Dimensionality and variable design
State variables: Define a set of variables x1,x2,...,xn that collectively describe the system state at time t. Each entity has a state vector
with variables spanning physical, economic, ecological, technological, and geopolitical dimensions.
Latent dimensions: Reserve capacity for latent variables learned by the AI to capture non‑observed but structurally important factors. These are defined as
where fθ is a neural encoder.
Example: A single state vector might include national CO₂ emissions, energy mix, GDP, trade balances, forest cover, technology diffusion indices, and policy stance indicators for each country.
4. Structural relationships and topology
Graph structure: Represent entities and their interactions as a heterogeneous graph
where V are entities, E are edges encoding relationships, and A is an attribute tensor. Each edge of type k has attributes
representing weight, direction, and volatility.
Manifolds and subspaces: Identify subspaces corresponding to domains (e.g. climate manifold, financial manifold) and define mappings between them
Constraints: Encode physical laws, conservation constraints, and institutional rules as geometric or algebraic constraints on S:
Example: An edge between two countries might carry attributes for trade volume, energy dependency, and shared river basins, forming a multi‑layer graph.
5. Temporal dynamics and update rules
Dynamics model: Define how the state evolves:
where u(t) are control inputs (policies, shocks) and ϵ(t) is noise.
Multi‑scale time: Allow different components to evolve at different temporal resolutions (e.g. daily climate data vs quarterly economic data), with appropriate interpolation or aggregation:
Feedback loops: Explicitly model feedbacks (e.g. climate impacts on agriculture → economic output → policy response → future emissions) using recursive dependencies such as
Example: A policy change in carbon pricing modifies energy investment decisions, which alter future emissions and climate trajectories, feeding back into economic and ecological variables.
6. Data integration and mapping
Data sources: Catalogue all data streams (climate models, satellite observations, economic statistics, trade databases, conflict datasets, technology adoption surveys).
Alignment: Map raw data into the state space S via a transformation
Normalisation and scaling: Apply transformations such as
Spatial alignment:
Temporal alignment:
Entity resolution: Match different datasets to common entities.
Missing data: Define strategies for imputation and uncertainty representation, e.g. graph‑aware imputation
Example: Align satellite‑derived deforestation data with national land‑use statistics and trade data on agricultural exports to populate ecological and economic dimensions.
7. Geometric representation format
State encoding: Choose a representation format such as
for entity‑level variables.
Structured tensors:
for spatio‑temporal fields.
Graph embeddings:
for relational structure.
Coordinate systems: Define coordinate systems for spatial data (lat‑long grids, regional aggregations) and for abstract spaces (e.g. principal components, learned latent coordinates).
Storage and access: Implement data structures and APIs that allow efficient access to slices of S (by entity, domain, time).
Example: Use a combination of a global grid tensor for climate fields, a country‑indexed matrix for economic and policy variables, and a graph structure for cross‑border relationships.
8. Validation of geometric fidelity
Structural tests: Verify that the geometry preserves known relationships:
Economic identities, physical constraints, and ecological dependencies must hold.
Behavioural tests: Simulate simple scenarios and check that the geometry supports plausible trajectories:
Expert review: Have domain experts assess whether the state space captures the essential structures they recognise.
Example: Test whether a simulated global recession propagates through trade and finance edges in ways consistent with historical crises.
9. Interfaces for Adaptive Representation and Inference
Input interface: Define how Adaptive Representation modules ingest the state x(t) and graph G:
Output interface: Specify how Adaptive Inference modules receive updated states, trajectories, and derived structures:
Modularity: Ensure that changes to the geometry (e.g. adding a new domain) can be integrated without breaking downstream components:
Example: Provide a unified API that returns the current state vector for any entity, its local neighbourhood in the graph, and relevant spatio‑temporal fields for climate and ecology.
10. Example: global climate–economy–energy system
Entities: Countries, energy firms, ecosystems, infrastructures.
Variables: Emissions, energy mix, GDP, trade flows, technology adoption, land‑use, policy indices.
Edges: Trade relationships, energy dependencies, ecological couplings, financial exposures.
Dynamics: Policy decisions alter energy investments and emissions; climate impacts affect agriculture and infrastructure; economic changes feed back into policy and technology adoption. For example:
Operational Construction of the System Geometry
To make Step 1 computationally actionable, the conceptual description of the system’s geometry must be translated into an ordered sequence of operations that a program can execute deterministically. The pseudocode below expresses this translation: it shows how the system descriptor is created, how boundaries and manifolds are allocated, how state variables and relationships are instantiated, how dynamics and constraints are attached, and how data is integrated into the resulting geometric structure. Each operation appears in dependency order, ensuring that every mathematical object is constructed only after the components it relies on exist. This produces a fully specified, machine‑usable high‑dimensional geometry that forms the substrate for all subsequent adaptive representation and inference.
Pseudocode for Constructing the System Geometry
###############################################
# STEP 1 — DEFINE GEOMETRY OF TARGET SYSTEM
###############################################
FUNCTION BuildSystemGeometry():
###########################################
# 1. INITIALISE SYSTEM DESCRIPTOR
###########################################
U = NEW SystemDescriptor()
U.domains = DEFINE_DOMAIN_SET()
U.entities = DEFINE_ENTITY_SET()
U.boundaries = EMPTY
###########################################
# 2. DEFINE SPATIAL, TEMPORAL, FUNCTIONAL BOUNDARIES
###########################################
Ω = DEFINE_SPATIAL_BOUNDARIES() # {global, regional, local}
T = DEFINE_TEMPORAL_BOUNDARIES() # {t0, t1, ..., tn}, Δt
U.boundaries.spatial = Ω
U.boundaries.temporal = T
###########################################
# 3. ALLOCATE DOMAIN MANIFOLDS
###########################################
FOR each domain d IN U.domains:
M[d] = INITIALISE_MANIFOLD(d)
###########################################
# 4. DEFINE STATE VARIABLES AND LATENT VARIABLES
###########################################
FOR each entity i IN U.entities:
x[i] = INITIALISE_STATE_VECTOR(i, T)
z[i] = LATENT_ENCODER(x[i]) # z_i(t) = fθ(x_i(t))
###########################################
# 5. BUILD STRUCTURAL RELATIONSHIPS AND TOPOLOGY
###########################################
G = NEW Graph()
G.vertices = U.entities
G.edges = INITIALISE_EDGES(U.entities)
G.attributes = ASSIGN_EDGE_ATTRIBUTES(G.edges)
FOR each domain pair (a, b):
φ[a,b] = DEFINE_MAPPING(M[a], M[b])
C = DEFINE_CONSTRAINTS() # C(X(t)) = 0
###########################################
# 6. DEFINE TEMPORAL DYNAMICS
###########################################
F = DEFINE_UPDATE_RULE() # x(t+1) = F(x(t), u(t), ε(t))
FEEDBACK = DEFINE_FEEDBACK_FUNCTIONS() # e.g., econ ← climate, policy
###########################################
# 7. INTEGRATE DATA INTO GEOMETRY
###########################################
RAW_DATA = LOAD_ALL_DATA_SOURCES()
FOR each dataset d IN RAW_DATA:
X_raw = TRANSFORM_DATA(d) # Ψ: D → X
X_norm = NORMALISE(X_raw) # (x - μ)/σ
X_spatial = SPATIAL_ALIGN(X_norm, Ω)
X_temporal = TEMPORAL_ALIGN(X_spatial, T)
X_imputed = IMPUTE_MISSING(X_temporal, G)
STORE_IN_STATE_SPACE(X_imputed)
###########################################
# 8. ENCODE GEOMETRY INTO MACHINE-USABLE FORMATS
###########################################
X_tensor = BUILD_STATE_TENSOR(x) # R^{q × n}
C_tensor = BUILD_SPATIOTEMPORAL_TENSOR() # R^{H × W × K}
FOR each entity i:
h[i] = GRAPH_EMBEDDING(G, x[i]) # GNN(G, x_i)
###########################################
# 9. VALIDATE GEOMETRIC FIDELITY
###########################################
IF NOT APPROX_EQUAL(C(X_tensor), 0):
RAISE_WARNING("Constraint violation detected")
X_sim = SIMULATE(F, X_tensor)
VALIDATE_BEHAVIOUR(X_sim)
EXPERT_REVIEW(U, M, G, X_tensor)
###########################################
# 10. BUILD INTERFACES FOR DOWNSTREAM MODULES
###########################################
I_in = { X_tensor, G, C_tensor }
I_out = { NEXT_STATE(F, X_tensor),
FUTURE_STATES(F, X_tensor),
GRAPH_UPDATES(G),
CONSTRAINT_UPDATES(C) }
RETURN {
"SystemDescriptor": U,
"Manifolds": M,
"StateVectors": x,
"LatentVectors": z,
"Graph": G,
"Constraints": C,
"Dynamics": F,
"StateTensor": X_tensor,
"SpatioTemporalTensor": C_tensor,
"Embeddings": h,
"InputInterface": I_in,
"OutputInterface": I_out
}
###############################################
# STEP 1 — DEFINE GEOMETRY OF TARGET SYSTEM
###############################################
FUNCTION BuildSystemGeometry():
###########################################
# 1. INITIALISE SYSTEM DESCRIPTOR
###########################################
U = NEW SystemDescriptor()
U.domains = DEFINE_DOMAIN_SET()
U.entities = DEFINE_ENTITY_SET()
U.boundaries = EMPTY
###########################################
# 2. DEFINE SPATIAL, TEMPORAL, FUNCTIONAL BOUNDARIES
###########################################
Ω = DEFINE_SPATIAL_BOUNDARIES() # {global, regional, local}
T = DEFINE_TEMPORAL_BOUNDARIES() # {t0, t1, ..., tn}, Δt
U.boundaries.spatial = Ω
U.boundaries.temporal = T
###########################################
# 3. ALLOCATE DOMAIN MANIFOLDS
###########################################
FOR each domain d IN U.domains:
M[d] = INITIALISE_MANIFOLD(d)
###########################################
# 4. DEFINE STATE VARIABLES AND LATENT VARIABLES
###########################################
FOR each entity i IN U.entities:
x[i] = INITIALISE_STATE_VECTOR(i, T)
z[i] = LATENT_ENCODER(x[i]) # z_i(t) = fθ(x_i(t))
###########################################
# 5. BUILD STRUCTURAL RELATIONSHIPS AND TOPOLOGY
###########################################
G = NEW Graph()
G.vertices = U.entities
G.edges = INITIALISE_EDGES(U.entities)
G.attributes = ASSIGN_EDGE_ATTRIBUTES(G.edges)
FOR each domain pair (a, b):
φ[a,b] = DEFINE_MAPPING(M[a], M[b])
C = DEFINE_CONSTRAINTS() # C(X(t)) = 0
###########################################
# 6. DEFINE TEMPORAL DYNAMICS
###########################################
F = DEFINE_UPDATE_RULE() # x(t+1) = F(x(t), u(t), ε(t))
FEEDBACK = DEFINE_FEEDBACK_FUNCTIONS() # e.g., econ ← climate, policy
###########################################
# 7. INTEGRATE DATA INTO GEOMETRY
###########################################
RAW_DATA = LOAD_ALL_DATA_SOURCES()
FOR each dataset d IN RAW_DATA:
X_raw = TRANSFORM_DATA(d) # Ψ: D → X
X_norm = NORMALISE(X_raw) # (x - μ)/σ
X_spatial = SPATIAL_ALIGN(X_norm, Ω)
X_temporal = TEMPORAL_ALIGN(X_spatial, T)
X_imputed = IMPUTE_MISSING(X_temporal, G)
STORE_IN_STATE_SPACE(X_imputed)
###########################################
# 8. ENCODE GEOMETRY INTO MACHINE-USABLE FORMATS
###########################################
X_tensor = BUILD_STATE_TENSOR(x) # R^{q × n}
C_tensor = BUILD_SPATIOTEMPORAL_TENSOR() # R^{H × W × K}
FOR each entity i:
h[i] = GRAPH_EMBEDDING(G, x[i]) # GNN(G, x_i)
###########################################
# 9. VALIDATE GEOMETRIC FIDELITY
###########################################
IF NOT APPROX_EQUAL(C(X_tensor), 0):
RAISE_WARNING("Constraint violation detected")
X_sim = SIMULATE(F, X_tensor)
VALIDATE_BEHAVIOUR(X_sim)
EXPERT_REVIEW(U, M, G, X_tensor)
###########################################
# 10. BUILD INTERFACES FOR DOWNSTREAM MODULES
###########################################
I_in = { X_tensor, G, C_tensor }
I_out = { NEXT_STATE(F, X_tensor),
FUTURE_STATES(F, X_tensor),
GRAPH_UPDATES(G),
CONSTRAINT_UPDATES(C) }
RETURN {
"SystemDescriptor": U,
"Manifolds": M,
"StateVectors": x,
"LatentVectors": z,
"Graph": G,
"Constraints": C,
"Dynamics": F,
"StateTensor": X_tensor,
"SpatioTemporalTensor": C_tensor,
"Embeddings": h,
"InputInterface": I_in,
"OutputInterface": I_out
}