Dialectical Field Topology Reference

Technical reference for the dialectical field topology systems implemented in Feature 002. Three systems (positions 16–18 in _DEFAULT_SYSTEMS) compute contradiction fields, spatial/temporal derivatives, and edge mode transitions on the simulation graph.

Field Registry 

The field registry is an extensible container of named contradiction fields. Core computation logic (gradient, Laplacian, temporal derivatives, principal contradiction) is field-name-agnostic — new fields can be registered without modifying computation code.

Default Contradiction Fields
Field	Computation	Raw Range	Normalization
`exploitation`	`(subsistence - wealth) / subsistence`	[0, 5.0+]	Linear ×10, clamp [0, 10]
`immiseration`	`(prev_wealth - wealth) / prev_wealth`	[0, 1.0]	Linear ×10, clamp [0, 10]
`imperial_rent`	Direct from `unearned_increment`	[0, ∞]	Saturating: `10(1 - e^{-x/10})`
`displacement`	`(prev_population - population) / prev_population`	[0, 1.0]	Linear ×10, clamp [0, 10]

All field values are normalized to [0.0, 10.0] for cross-field comparability. Normalization preserves relative ordering and derivative signs.

Implementation: babylon.engine.field_registry.DefaultFieldRegistry

Protocol: babylon.engine.field_registry.FieldRegistryProtocol

Spatial Derivatives 

Gradient (per edge)

The gradient along edge (i, j) is the signed difference:

\[\nabla f(i, j) = f(j) - f(i)\]

Positive gradient means the field increases from source to target. Stored in edge attribute field_gradients: dict[str, float].

Graph Laplacian (per node)

The unweighted graph Laplacian at node i is:

\[\Delta f(i) = \sum_{j \in N(i)} [f(j) - f(i)]\]

Where N(i) is the set of all neighbors (undirected — both in-edges and out-edges). Stored in node attribute field_derivatives[field]["laplacian"].

Laplacian Interpretation
Sign	Meaning
Negative	Pressure peak — node has higher field value than neighbors
Positive	Pressure trough — node has lower field value than neighbors
Zero	Equilibrium with neighbors, or isolated node (EC-002)

Temporal Derivatives 

Computed from tick-keyed history using backward finite differences:

\[\frac{df}{dt}(i, t) = f(i, t) - f(i, t-1)\]

\[\frac{d^2f}{dt^2}(i, t) = f(i, t) - 2f(i, t-1) + f(i, t-2)\]

History Requirements
Derivative	Min Ticks	If Insufficient
df/dt	2	Returns `None` (not 0.0)
d²f/dt²	3	Returns `None` (not 0.0)

Stored in node attribute field_derivatives[field]["df_dt"] and field_derivatives[field]["d2f_dt2"].

Predicates referencing undefined derivatives evaluate to False (EC-001).

Principal Contradiction 

At each tick, the system identifies the principal contradiction: the field with the largest maximum absolute first derivative across all nodes.

Selection criteria:

Field with max(\|df/dt\|) across all nodes
Tie-break: total magnitude Σ\|df/dt\| descending
Tie-break: exploitation preferred (structural primacy)

Output: Graph-level attribute principal_contradiction:

{
    "field_name": str | None,
    "max_abs_df_dt": float,
    "changed": bool,
}

Event PRINCIPAL_CONTRADICTION_SHIFT is emitted when the principal field changes between ticks.

Ollivier-Ricci Curvature 

Discrete curvature computed per edge using optimal transport. Structural property — recomputed only when topology changes.

\[\kappa(u, v) = 1 - \frac{W_1(\mu_u, \mu_v)}{d(u, v)}\]

Where:

\(\mu_u, \mu_v\) = probability measures over neighborhoods
\(\alpha\) = self-loop weight (default 0.5)
\(1 - \alpha\) = uniform distribution over neighbors
\(W_1\) = Wasserstein-1 (Earth Mover’s) distance
\(d(u,v)\) = shortest-path distance

Curvature Interpretation
Sign	Meaning
κ > 0	Well-connected, clustered (redundant topology)
κ < 0	Bottleneck (bridge between sparse regions)
κ ≈ 0	Balanced, flat topology

Validated Topologies
Graph	Expected κ	Note
K₄ (complete)	Positive	High redundancy
P₄ (path)	Negative at bridges	Bridge edges are bottlenecks
S₄ (star)	Known hub value	Hub-spoke topology

Implementation: babylon.formulas.curvature.compute_ollivier_ricci()

Dependencies: scipy (linear programming), numpy, networkx

Edge Mode Transition State Machine 

Five Modes 

Mode	Direction	Value Flow	Political Content	Stability
EXTRACTIVE	Unidirectional	Exploited → exploiter	Produces resistance	Unstable
TRANSACTIONAL	Symmetric	Market exchange	Neutral	Stable until disrupted
SOLIDARISTIC	Mutual	Shared reproduction	Builds collective power	Stable under pressure
ANTAGONISTIC	Oppositional	Contested/destroyed	Open conflict	Unstable
CO_OPTIVE	Asymmetric	Concessions for quiescence	Prevents resistance	Fragile to crisis

17 Permissible Transitions 

From	To	Condition (summary)	Priority
EXTRACTIVE	ANTAGONISTIC	Exploitation >5.0 AND rising	10
EXTRACTIVE	TRANSACTIONAL	Exploitation <2.0	5
EXTRACTIVE	CO_OPTIVE	Exploitation >3.0, rent >2.0	8
TRANSACTIONAL	SOLIDARISTIC	Exploitation <2.0 both sides	5
TRANSACTIONAL	ANTAGONISTIC	Immiseration rising >1.0	10
TRANSACTIONAL	EXTRACTIVE	Exploitation >5.0	7
TRANSACTIONAL	CO_OPTIVE	Imperial rent >3.0	6
SOLIDARISTIC	TRANSACTIONAL	Immiseration >6.0	5
SOLIDARISTIC	ANTAGONISTIC	Exploitation spike >3.0	10
ANTAGONISTIC	TRANSACTIONAL	Exploitation falling AND <3.0	5
ANTAGONISTIC	SOLIDARISTIC	Exploitation >7.0 both sides	8
ANTAGONISTIC	CO_OPTIVE	Rent >3.0, not rising	6
CO_OPTIVE	TRANSACTIONAL	Exploitation <2.0, not rising	5
CO_OPTIVE	ANTAGONISTIC	Exploitation rising >1.0	10
CO_OPTIVE	SOLIDARISTIC	Exploitation >5.0 both sides	3
CO_OPTIVE	EXTRACTIVE	Imperial rent <1.0	7
ANTAGONISTIC	ANTAGONISTIC	Persistence (no resolution)	—

Transitions not listed are prohibited — they require passing through intermediate states.

When multiple transitions are eligible simultaneously, the highest-priority transition fires (EC-003).

Compound Predicates 

Each transition is governed by a compound predicate — a conjunction of threshold conditions:

PredicateCondition(
    field="exploitation",
    metric="df_dt",        # or "value", "d2f_dt2", "laplacian"
    operator="gt",         # "lt", "gte", "lte"
    threshold=0.0,
    scope="source",        # or "target"
)

A compound predicate evaluates to True only when all conditions are satisfied. Conditions referencing undefined derivatives (insufficient history) evaluate to False.

CO-OPTIVE Mechanics 

Suppression 

Each CO-OPTIVE edge declares which contradiction fields it suppresses (per-edge configurable via co_optive_suppressed_fields). The suppression applies to df/dt at the co-opted node:

\[\text{suppressed} = \frac{df}{dt} \times r_s\]

Where \(r_s\) = co_optive_suppression_rate (default 1.0, meaning full suppression).

Suppressed df/dt accumulates as latent contradiction — tracked per-source-node per-field in persistent data.

Breakdown and Latent Release 

When a CO-OPTIVE edge transitions away (typically to ANTAGONISTIC), the accumulated latent contradiction is released:

\[\text{released} = \text{latent} \times m_r\]

Where \(m_r\) = latent_release_multiplier (default 1.5).

This produces a spike in df/dt that may cause exploitation to reassert itself as the principal contradiction — the “return of the repressed” dynamic.

George Jackson Bifurcation 

When CO-OPTIVE breakdown produces ANTAGONISTIC edges, the direction of antagonism is determined by solidarity topology:

Upward (revolutionary): total solidarity strength across the colonial divide > within-group solidarity
Lateral (fascist): within-group solidarity > cross-divide solidarity

ContradictionFieldDefines
Parameter	Default	Range	Description
`field_min`	0.0	[0, ∞)	Minimum normalized field value
`field_max`	10.0	(0, ∞)	Maximum normalized field value
`history_window`	3	[2, 10]	Rolling tick window for temporal derivatives
`curvature_alpha`	0.5	(0, 1]	Self-loop weight for Ricci probability measures
`co_optive_suppression_rate`	1.0	[0, 1]	Fraction of df/dt suppressed by CO-OPTIVE edges
`latent_release_multiplier`	1.5	[1, 5]	Multiplier on released latent contradictions
`default_transition_priority`	0	[0, ∞)	Default priority for transitions

Event Types 

Event Type	Description
`EDGE_MODE_TRANSITION`	Edge qualitative mode changed
`PRINCIPAL_CONTRADICTION_SHIFT`	Principal field (max \|df/dt\|) changed
`CO_OPTIVE_BREAKDOWN`	CO-OPTIVE edge failed, latent released
`LATENT_CONTRADICTION_RELEASE`	Suppressed df/dt spike after CO-OPTIVE loss
`ASPECT_REVERSAL`	Dominant party switched on directed edge

System Execution Order 

#	System	Purpose
16	ContradictionFieldSystem	Compute normalized field values at social-class nodes
17	FieldDerivativeSystem	Gradients, Laplacian, df/dt, d²f/dt², principal contradiction
18	EdgeTransitionSystem	CO-OPTIVE suppression, predicate evaluation, mode transitions

All three systems are no-ops when services.field_registry is None (backward compatible).