Preface: Two People, One Loop

Note

Please read this first. This book introduces attachment theory and research on couple interaction, and uses simplified mathematical models and programs to make its arguments. It is not marital or family therapy, couples counselling or psychotherapy, and it cannot be used to diagnose anyone. The models capture only a very small part of real relationships, and unless a source is given their parameters are illustrative.

If your relationship involves physical or sexual violence, threats, stalking, financial control or any other form of coercive control, or if you fear for your own or your children’s safety, none of the exercises in this book apply. Put safety first: call your local emergency number, or contact local domestic-violence services, a shelter, a doctor, the police or legal aid. In a relationship with violence or control, advice such as “communicate more”, “make repair attempts” or “take a break to calm down” can be misused and can even increase the danger. If you are in serious psychological distress, or have thoughts of harming yourself, please contact a doctor, a mental-health professional or a local crisis service as soon as you can.

Attachment anxiety and attachment avoidance are continuous dimensions, not types, still less diagnoses; everyone sits somewhere on both, and where they sit changes with relationships and circumstances. The “pursue–withdraw loop” in this book describes a pattern between two people. It is not either partner’s fault, nor the “illness” of any attachment style.

Many couples know a certain kind of argument. One wants to talk, the other wants to get away; the more one presses, the quieter the other becomes; the quieter the other becomes, the more anxious the first. A few rounds later they are no longer talking about the small thing that started it, but about “you don’t care about me at all” and “you never give me room to breathe”.

This book asks three questions. First, how does such a loop form, and why does it often spin faster and faster? Second, how far do relational experiences in childhood determine this loop in adulthood? Third, are there measurable, testable ways to slow the loop down before it escalates?

What the book covers

The book discusses couple interaction at five levels:

Internal working models. Bowlby’s and Ainsworth’s attachment theory, and the two dimensions of anxiety and avoidance in adult attachment research; written as a Bayesian prior about “whether others will be there for me”, to see how different priors and ways of updating shape belief.
The pursue–withdraw loop. Two coupled rate equations describe “one pursues, one withdraws”, and show under what conditions the loop sustains itself and under what conditions it erupts periodically.
Gottman and Murray’s marriage equations. A full implementation of the difference equations in The Mathematics of Marriage, with their steady states and basins of attraction; Gottman’s descriptive observation about stable couples (about 5 positive to 1 negative in conflict discussions) is kept separate from the debunked “Losada ratio”; and why “predicting divorce” requires cross-validation.
Physiological co-regulation. “Linkage” between partners’ heart rates and other physiological signals, and the difference between co-regulation and escalating co-dysregulation; how to measure linkage, and how not to mistake a shared situation for mutual influence.
Change and repair. How stable attachment is from infancy to adulthood (the pooled average correlation is about 0.39, and over intervals longer than 15 years it is no longer significant); how relationship experience might revise old expectations; why repair attempts act like a damper; and the physiology of taking a break.

Chapter 7 reads this modern research beside the Book of Changes hexagrams Xian and Jiaren, the Book of Songs poem “Beating the Drum”, the Great Learning and Mencius, and states the fundamental boundary between hierarchical classical family ethics and modern equal partnerships. Chapter 8 offers a four-step interaction-loop checklist and a pre-registered randomised controlled trial.

This book complements 5D Library volume 15, The Sandbox, Determinism and Active Inference: that book discusses how a pessimistic prior can lead one person into the same trap again and again; this one discusses how two people’s priors and reactions interlock into a loop.

Claims corrected on checking

Several popular claims needed correction once they were checked. First, “your family of origin determines your whole life” does not hold up statistically: a meta-analysis pooling 127 studies found an average stability correlation of about 0.39 for attachment security from infancy to early adulthood, so the earlier measure explains only about 15 % of the variance of the later one, and over intervals longer than 15 years stability is no longer significant; a large study that followed children from birth to age 18 found that early caregiving quality and other factors predicted adult attachment, but with small effects. Second, the “positivity ratio of 5.6 to 1” is not Gottman’s finding: 5.614 is the value Losada reported for high-performing business teams, and 2.9013 is the “critical positivity ratio” proposed by Fredrickson and Losada, whose mathematical derivation Brown, Sokal and Friedman showed in 2013 to be invalid, after which the journal published a correction withdrawing that modelling; Gottman’s “about 5 to 1” is a descriptive observation of stable couples’ conflict discussions, not a threshold that must be reached. Third, “attachment types” behave more like continuous dimensions than separate categories. Fourth, in the Book of Songs, “in death or life, near or far, I made my pledge with you” was read by the Han commentators as an oath between soldiers; Zhu Xi read it as a soldier recalling his marriage vow to his wife, and today’s familiar use as a lovers’ vow is the later reading. Each chapter says so where it arises, and the Editor’s Notes published with the book files collect these points.

How to read the boxes

As in other 5D Library books, different kinds of statement are marked separately.

Evidence

Evidence boxes report published findings, with sources.

Calculation

Calculation boxes give formulas and short Python programs. Every program runs as printed in a standard Python 3 installation using only the standard library; its actual output is shown beneath it. The “couples”, “diaries” and data in the book are fictional or synthesised by the programs, not any real person.

Hypothesis · not established

Hypothesis boxes hold the author’s ideas, marked “not yet established”, with a statement of what result would show them wrong.

Source text

Source boxes quote classical texts, with the author’s plain-language reading.

Note

Note boxes give cautions and explain choices.

When the book applies concepts from physics and dynamical systems (coupling, limit cycles, damping, basins of attraction) to relationships, they are analogies and simplified models: they help us see structure, but people are not oscillators and relationships are not equations.

About this edition

This is version 1.0.0, the first complete edition: a preface and eight chapters, in Chinese and English. Corrections and missing references are welcome through the 5D Library website.

Disclosure

The author uses the works of 5D Library, including this book, as the methodological basis of his community courses, the 5D Ascension Cards (Dimensional Shift Card Decks) and the Yafu AI knowledge base, and therefore has a commercial interest in the subject of this book. The book promotes no course, card deck or service; the Chapter 8 trial is designed to be run and analysed by researchers independent of the author’s business and does not recruit from the author’s community, students or customers; all the code and the study protocol are published openly for anyone to check.

Fan, Y. (2026). Computational Attachment Theory and Couple Interaction Dynamics: From Anxious-Avoidant Coupled Oscillators and Gottman's Nonlinear Difference Equations to Updating Early Relational Priors [依恋理论的计算模型与伴侣互动动力学] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23754715 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).