Resources & Open Science

GitHub Repository

In the spirit of fostering collaboration, reproducibility, and collective advancement, we are committed to open science. All original code used for simulations, data analyses, and generating figures in this book, along with relevant data references (where permissible), are available in our public repository:

Visit github.com/threshold-dialectics

We encourage readers to explore, utilize, critique, and contribute to these resources.

Symbol Glossary

The following table provides definitions and interpretations for key symbols used throughout "Threshold Dialectics."

Symbol Type Definition / Interpretation
Observables
PE vector Raw prediction–error signal, o-g(μ), measuring instantaneous sensory mismatch.
ΔP=Π PE vector Precision–weighted prediction error that drives free–energy descent.
<ΔP>τ scalar Running mean of ΔP over a horizon of τ time steps.
τ integer Averaging–window width (number of samples).
u=(L,S,E,P) vector Context inputs: legitimacy L, social influence S, collective affect E, power asymmetry P.
Latent levers
Π scalar Expected sensory precision (dominant eigenvalue of the noise–covariance inverse).
g scalar Perception–gain lever: scales every component of PE, controlling vigilance.
β scalar Policy precision: soft–max inverse temperature that sharpens or loosens action selection.
Fcrit scalar Energetic slack: residual buffer (ATP, cash, battery head-room) before failure.
ΘT scalar Tolerance sheet: gw1βw2Fcritw3—the breach ceiling separating viability from collapse.
w1,w2,w3 scalars Constant elasticities (w1+w2+w3=1) that allot the collapse margin to each lever.
g(u), hβ(u), hF(u) functions Pathways mapping context u to the three levers.
Diagnostics
ḣ=(β̇,Ḟcrit) vector Instantaneous velocity of the two tolerance levers.
S=‖ḣ‖2 scalar Speed index: joint rate at which rigidity (β) and slack (Fcrit) drift.
C=corr(β̇,Ḟcrit) scalar Couple index: synchrony of lever tightening (−1≤C≤1).
TΘ time First–passage time until <ΔP>τ>ΘT.
Σḣ matrix Covariance of lever velocities; diagonal gives σβ2, σF2, off–diagonal σβF.
Energetics & information
ψ(·) function Generative mapping μ↦οˆ that produces the predicted observation used in the prediction–error signal o−ψ(μ).
κ scalar Proportionality constant in the concave information law I(g)=κ gφ1 linking precision gain to bit-rate.
φ1 scalar Concavity exponent in the information-cost law I(g)=κ gφ1; with 0<φ1≤1 it captures how sharply energy demand tapers as perception gain g rises.
Pbit scalar Marginal energy cost per processed bit (pJ bit−1 in typical substrates).
I(g) scalar Information throughput of the sensing channel; scales as κgφ1.
Auxiliary symbols
Φ(·) function Positive, homogeneous aggregator mapping lever vector to ΘT.
ρβ=g/hβ, ρF=g/hF scalars Gain ratios defining approach direction in (g,β) and (g,Fcrit) planes.
J vector Compact notation for the triplet (g,β,Fcrit).