Formulas Reference

Complete specification of mathematical formulas used in the Babylon simulation engine. All formulas are implemented in babylon.formulas.

Constants

Constant	Value	Description
`LOSS_AVERSION_COEFFICIENT`	2.25	Kahneman-Tversky loss aversion multiplier
`EPSILON`	1e-6	Small value to prevent division by zero

Imperial Rent Formulas

Core Extraction Formula

Calculates value extracted from periphery to core:

\[\Phi(W_p, \Psi_p) = \alpha \times W_p \times (1 - \Psi_p)\]

Where:

\(\Phi\) = Imperial rent extracted
\(\alpha\) = Extraction efficiency coefficient [0, 1]
\(W_p\) = Periphery wage share [0, 1]
\(\Psi_p\) = Periphery consciousness (0 = submissive, 1 = revolutionary)

Implementation:

from babylon.formulas import calculate_imperial_rent

rent = calculate_imperial_rent(
    alpha=0.8,
    periphery_wages=0.3,
    periphery_consciousness=0.2
)  # Returns 0.192

Labor Aristocracy Ratio

Determines if workers receive more than value produced:

\[\text{LA Ratio} = \frac{W_c}{V_c}\]

When ratio > 1, the class is labor aristocracy (benefiting from imperial rent).

Implementation:

from babylon.formulas import (
    calculate_labor_aristocracy_ratio,
    is_labor_aristocracy,
)

ratio = calculate_labor_aristocracy_ratio(
    core_wages=120.0,
    value_produced=100.0
)  # Returns 1.2

is_la = is_labor_aristocracy(
    core_wages=120.0,
    value_produced=100.0
)  # Returns True

Consciousness Formulas

Consciousness Drift

Models ideological change based on material conditions:

\[\frac{d\Psi_c}{dt} = k(1 - \frac{W_c}{V_c}) - \lambda\Psi_c\]

Where:

\(\Psi_c\) = Current consciousness level
\(k\) = Drift sensitivity coefficient
\(W_c\) = Core wages
\(V_c\) = Value produced
\(\lambda\) = Decay coefficient

Fascist Bifurcation Extension:

When wages are falling (wage_change < 0), agitation energy is calculated:

\[E_{agitation} = |W_{change}| \times \lambda_{loss}\]

Where \(\lambda_{loss} = 2.25\) (loss aversion coefficient).

This energy routes based on solidarity:

With solidarity (solidarity_pressure > 0): Energy adds to drift (revolutionary)
Without solidarity (solidarity_pressure = 0): Energy subtracts from drift (fascist)

Implementation:

from babylon.formulas import calculate_consciousness_drift

drift = calculate_consciousness_drift(
    core_wages=80.0,
    value_produced=100.0,
    current_consciousness=0.5,
    sensitivity_k=0.1,
    decay_lambda=0.05,
    solidarity_pressure=0.8,
    wage_change=-10.0
)

Ideological Routing

Multi-dimensional consciousness routing from crisis conditions (George Jackson analysis):

\[E_{agitation} = |W_{change}| \times \lambda_{loss} + |X_{change}| \times \lambda_{loss}\]

Where:

\(W_{change}\) = Wage change (negative = crisis)
\(X_{change}\) = Wealth change (negative = extraction)
\(\lambda_{loss} = 2.25\) (loss aversion coefficient)

Routing based on solidarity infrastructure:

High solidarity + material loss → Class consciousness increases (revolutionary)
Low solidarity + material loss → National identity increases (fascist)

Historical examples: Germany 1933 (crisis + atomization → fascism) vs Russia 1917 (crisis + organization → revolution).

Implementation:

from babylon.formulas import calculate_ideological_routing

new_class, new_nation, new_agitation = calculate_ideological_routing(
    wage_change=-20.0,
    wealth_change=-10.0,  # Wealth extraction compounds crisis
    solidarity_pressure=0.9,
    current_class_consciousness=0.5,
    current_national_identity=0.5,
    current_agitation=0.0,
    agitation_decay=0.1
)
# High solidarity routes agitation to class consciousness

Survival Calculus Formulas

Acquiescence Probability

Probability of survival through compliance with the system:

\[P(S|A) = \frac{1}{1 + e^{-k(W - S_{min})}}\]

Where:

\(W\) = Current wealth
\(S_{min}\) = Subsistence threshold
\(k\) = Curve steepness

At the threshold (W = S_min), probability is exactly 0.5.

Implementation:

from babylon.formulas import calculate_acquiescence_probability

prob = calculate_acquiescence_probability(
    wealth=100.0,
    subsistence_threshold=100.0,
    steepness_k=0.1
)  # Returns 0.5 (at threshold)

Revolution Probability

Probability of survival through collective action:

\[P(S|R) = \frac{O}{R + \epsilon}\]

Where:

\(O\) = Organization/cohesion level [0, 1]
\(R\) = State repression capacity [0, 1]
\(\epsilon\) = Small constant preventing division by zero

Implementation:

from babylon.formulas import calculate_revolution_probability

prob = calculate_revolution_probability(
    cohesion=0.8,
    repression=0.2
)  # Returns 1.0 (clamped)

Rupture Condition

A Rupture Event occurs when:

\[P(S|R) > P(S|A)\]

This is the crossover threshold - the wealth level where revolution becomes a rational survival strategy.

Implementation:

from babylon.formulas import calculate_crossover_threshold

threshold = calculate_crossover_threshold(
    cohesion=0.6,
    repression=0.4,
    subsistence_threshold=100.0,
    steepness_k=0.1
)

Loss Aversion

Applies Kahneman-Tversky loss aversion (losses weighted 2.25x):

\[\begin{split}V_{perceived} = \begin{cases} V & \text{if } V \geq 0 \\ 2.25 \times V & \text{if } V < 0 \end{cases}\end{split}\]

Implementation:

from babylon.formulas import apply_loss_aversion

perceived_gain = apply_loss_aversion(100.0)   # Returns 100.0
perceived_loss = apply_loss_aversion(-100.0)  # Returns -225.0

Unequal Exchange Formulas

Exchange Ratio

Quantifies value extraction via trade:

\[\rho = \frac{L_p}{L_c} \times \frac{W_c}{W_p}\]

Where:

\(\rho\) = Exchange ratio (uses rho to avoid collision with epsilon constant)
\(L_p\) = Periphery labor hours
\(L_c\) = Core labor hours (same product)
\(W_c\) = Core wage rate
\(W_p\) = Periphery wage rate

When \(\rho > 1\), periphery gives more value than it receives.

Implementation:

from babylon.formulas import calculate_exchange_ratio

ratio = calculate_exchange_ratio(
    periphery_labor_hours=100.0,
    core_labor_hours=100.0,
    core_wage=20.0,
    periphery_wage=5.0
)  # Returns 4.0

Exploitation Rate

Converts exchange ratio to percentage:

\[\text{Exploitation Rate} = (\rho - 1) \times 100\%\]

Implementation:

from babylon.formulas import calculate_exploitation_rate

rate = calculate_exploitation_rate(exchange_ratio=4.0)  # Returns 300.0

Value Transfer

Calculates actual value transferred:

\[\text{Transfer} = V_{production} \times (1 - \frac{1}{\rho})\]

Implementation:

from babylon.formulas import calculate_value_transfer

transfer = calculate_value_transfer(
    production_value=1000.0,
    exchange_ratio=4.0
)  # Returns 750.0

Prebisch-Singer Effect

Models terms of trade decline for commodity exporters:

\[P_{new} = P_{initial} \times (1 + \eta \times \Delta Q)\]

Where \(\eta\) is price elasticity (typically negative).

Implementation:

from babylon.formulas import prebisch_singer_effect

new_price = prebisch_singer_effect(
    initial_price=100.0,
    production_increase=0.2,
    elasticity=-0.5
)  # Returns 90.0

Solidarity Transmission

Models consciousness transmission via solidarity edges:

\[\Delta\Psi_{target} = \sigma \times (\Psi_{source} - \Psi_{target})\]

Where:

\(\sigma\) = Solidarity strength on edge [0, 1]
\(\Psi_{source}\) = Source consciousness (periphery)
\(\Psi_{target}\) = Target consciousness (core)

Transmission only occurs if source_consciousness > activation_threshold.

Implementation:

from babylon.formulas import calculate_solidarity_transmission

delta = calculate_solidarity_transmission(
    source_consciousness=0.8,
    target_consciousness=0.2,
    solidarity_strength=0.5,
    activation_threshold=0.3
)  # Returns 0.3

Territory Heat Formulas

Heat Accumulation

Territories accumulate heat from high-profile activities:

\[H_{t+1} = H_t + \Delta H_{activity} - \Delta H_{decay}\]

Heat Thresholds
Heat Level	Consequence
< 0.4	Normal operations
0.4 - 0.8	Increased surveillance
>= 0.8	Eviction trigger - classes must relocate

Configuration Parameters

Parameter	Default	Effect
`territory.heat_threshold`	0.8	Heat level triggering eviction
`territory.heat_decay`	0.1	Heat reduction per tick
`territory.spillover_coefficient`	0.2	Heat transferred on displacement

Dynamic Balance (Bourgeoisie Decision)

Models bourgeoisie policy decisions based on imperial rent pool and tension:

Decision Matrix
Condition	Decision	Effect
pool >= 0.7, tension < 0.3	BRIBERY	wages +5%
pool < 0.1	CRISIS	wages -15%, repression +20%
pool < 0.3, tension > 0.5	IRON_FIST	repression +10%
pool < 0.3, tension <= 0.5	AUSTERITY	wages -5%
else	NO_CHANGE	status quo

Implementation:

from babylon.formulas import calculate_bourgeoisie_decision

decision, wage_delta, repression_delta = calculate_bourgeoisie_decision(
    pool_ratio=0.8,
    aggregate_tension=0.2
)  # Returns ("bribery", 0.05, 0.0)

Metabolic Rift Formulas

Ecological limits on capital accumulation (Slice 1.4).

Biocapacity Delta

Models change in ecological carrying capacity:

\[\Delta B = R - (E \times \eta)\]

Where:

\(R\) = Regeneration (fraction of max restored per tick)
\(E\) = Extraction intensity × current biocapacity
\(\eta\) = Entropy factor (waste multiplier, typically 1.2)

When \(\Delta B < 0\), the system is depleting faster than regenerating.

Implementation:

from babylon.formulas import calculate_biocapacity_delta

delta = calculate_biocapacity_delta(
    regeneration_rate=0.02,   # 2% max restored per tick
    max_biocapacity=100.0,
    extraction_intensity=0.05,
    current_biocapacity=50.0,
    entropy_factor=1.2
)  # Returns -1.0 (depletion)

Overshoot Ratio

Measures ecological overshoot:

\[O = \frac{C}{B}\]

Where:

\(C\) = Total consumption across all entities
\(B\) = Total biocapacity available

When \(O > 1.0\), the system is in ecological overshoot (consuming more than the planet can regenerate).

Implementation:

from babylon.formulas import calculate_overshoot_ratio

ratio = calculate_overshoot_ratio(
    total_consumption=200.0,
    total_biocapacity=100.0
)  # Returns 2.0 (2x overshoot)

Vitality Formulas

Mortality rate calculations for population attrition (Mass Line Refactor Phase 3).

Mortality Rate (Coverage Ratio Threshold)

Calculates population attrition based on wealth coverage and inequality:

\[ \begin{align}\begin{aligned}\text{coverage\_ratio} = \frac{W_{pc}}{S}\\\text{threshold} = 1 + I\\\begin{split}\text{attrition\_rate} = \begin{cases} 0 & \text{if coverage\_ratio} \geq \text{threshold} \\ (\text{threshold} - \text{coverage\_ratio}) \times (0.5 + I) & \text{otherwise} \end{cases}\end{split}\end{aligned}\end{align} \]

Where:

\(W_{pc}\) = Wealth per capita
\(S\) = Subsistence needs (s_bio + s_class)
\(I\) = Inequality coefficient [0, 1]

Mechanic: High inequality raises the survival floor, causing mortality even when average wealth appears sufficient. This models how wealth concentration exposes marginal workers to attrition.

Implementation:

from babylon.formulas import calculate_mortality_rate

attrition = calculate_mortality_rate(
    wealth_per_capita=0.5,
    subsistence_needs=0.4,
    inequality=0.3
)  # Returns attrition rate [0, 1]

TRPF Formulas (Tendency of the Rate of Profit to Fall)

Marx’s rate of profit formulas (Capital Vol. 3). Currently used as surrogate for organic composition via extraction efficiency decay.

TRPF Multiplier (Surrogate)

Models declining extraction efficiency over time:

\[\text{multiplier} = \max(\text{floor}, 1 - (\text{coefficient} \times \text{tick}))\]

Where:

\(\text{coefficient}\) = TRPF decay rate (default 0.0005)
\(\text{floor}\) = Minimum efficiency (default 0.1)

Implementation:

from babylon.formulas import calculate_trpf_multiplier

mult = calculate_trpf_multiplier(
    tick=1000,
    trpf_coefficient=0.0005,
    floor=0.1
)  # Returns 0.5

Rent Pool Decay

Models imperial rent pool depletion:

\[\text{new\_pool} = \text{current\_pool} \times (1 - \text{decay\_rate})\]

Implementation:

from babylon.formulas import calculate_rent_pool_decay

new_pool = calculate_rent_pool_decay(
    current_pool=100.0,
    decay_rate=0.002
)  # Returns 99.8

Rate of Profit (Epoch 2 Placeholder)

Full Marx formula for rate of profit:

\[r = \frac{s}{c + v}\]

Where:

\(s\) = Surplus value
\(c\) = Constant capital (machinery, materials)
\(v\) = Variable capital (wages)

Implementation:

from babylon.formulas import calculate_rate_of_profit

rate = calculate_rate_of_profit(
    surplus_value=100.0,
    constant_capital=200.0,
    variable_capital=100.0
)  # Returns 0.333

Organic Composition (Epoch 2 Placeholder)

Ratio of constant to variable capital:

\[\text{OCC} = \frac{c}{v}\]

Rising OCC is the mechanism behind the tendential fall in profit rate.

Formula-to-System Mapping

Formula	System	Module
Mortality Rate (Coverage Ratio)	VitalitySystem	`engine/systems/vitality.py`
Production (Labor × Biocapacity)	ProductionSystem	`engine/systems/production.py`
Imperial Rent	ImperialRentSystem	`engine/systems/economic.py`
TRPF Multiplier	ImperialRentSystem	`engine/systems/economic.py`
Rent Pool Decay	ImperialRentSystem	`engine/systems/economic.py`
Consciousness Drift	ConsciousnessSystem	`engine/systems/ideology.py`
Ideological Routing	ConsciousnessSystem	`engine/systems/ideology.py`
Survival Calculus	SurvivalSystem	`engine/systems/survival.py`
Solidarity Transmission	SolidaritySystem	`engine/systems/solidarity.py`
Territory Heat	TerritorySystem	`engine/systems/territory.py`
Bourgeoisie Decision	ContradictionSystem	`engine/systems/contradiction.py`
LA Decomposition (15%/85% split)	DecompositionSystem	`engine/systems/decomposition.py`
Control Ratio (Guard:Prisoner)	ControlRatioSystem	`engine/systems/control_ratio.py`
Metabolic Rift	MetabolismSystem	`engine/systems/metabolism.py`

Formulas Reference

Constants

Imperial Rent Formulas

Core Extraction Formula

Labor Aristocracy Ratio

Consciousness Formulas

Consciousness Drift

Ideological Routing

Survival Calculus Formulas

Acquiescence Probability

Revolution Probability

Rupture Condition

Loss Aversion

Unequal Exchange Formulas

Exchange Ratio

Exploitation Rate

Value Transfer

Prebisch-Singer Effect

Solidarity Transmission

Territory Heat Formulas

Heat Accumulation

Configuration Parameters

Dynamic Balance (Bourgeoisie Decision)

Metabolic Rift Formulas

Biocapacity Delta

Overshoot Ratio

Vitality Formulas

Mortality Rate (Coverage Ratio Threshold)

TRPF Formulas (Tendency of the Rate of Profit to Fall)

TRPF Multiplier (Surrogate)

Rent Pool Decay

Rate of Profit (Epoch 2 Placeholder)

Organic Composition (Epoch 2 Placeholder)

Formula-to-System Mapping

See Also