A theory of pair bonding, written as an operating system
Everything settles at 0.618
It sounds absurd. Regulus takes the maths an engineer uses to stop a chemical plant blowing up, and points it at a couple. Then you follow it, and the awkward part is how much of it you can measure.
Thomas, who wrote the framework, named this version himself: "R4R." His name, for his own work.
This is a theory of what happens between two people. It is written the way an engineer writes about a machine.
It says love changes your body, and that it has to change both of you, not one.
So it hands you a single question to ask of any bond: who here has not been changed?
If only one of you is changing, that is not closeness. Someone is using you — and for about thirty pounds, you can test which of the two you are in.
The whole thing, before any maths
Five paragraphs. No symbols, no jargon. The same short version again, with room to breathe. Read this much and you have the framework. Everything below it is the machinery.
One. Your body is a factory with floors. The gut is the ground floor, the thinking part is near the top. Sensation travels up, decisions travel down. The surprising bit is that the ground floor can overrule the top — which is why you cannot think your way out of feeling sick.
Two. When the thinking floor cannot put a name to a feeling, that feeling does not disappear. It travels back down and lands in your gut. Unnamed emotion becomes stomach ache. That is the founding observation, and it is the least controversial thing in the whole framework.
Three. Anything that has to remember and forget at the same time settles at the same balance: keep about 62%, release about 38%. Hold on to everything and you are overwhelmed; hold on to nothing and there is nothing there. That single ratio is what the rest of the document is built around.
Four. Here is what Regulus adds. Two people who bond do not merely end up near each other — they change each other, and past a certain point you cannot get the earlier version of yourself back. The framework treats that as a physical event with a physical signature: the arteries feeding your heart relax. Not as a metaphor. The organ that feeds everything else finally lets itself be fed.
Five. And the useful part. Something can look exactly like love from the outside and be its opposite. The same behaviours — the obsession, the constant checking, the inability to leave. A completely different body: heart shut rather than open, stress hormones climbing rather than falling. So the framework hands you one question to tell them apart.
Who in this system is unchanged? §9.2 · the whole diagnostic, in six words
If both of you have been changed by it, that is a bond. If only one of you is being changed while the other stays exactly as they were, that is extraction wearing a bond's clothes. The same test works on cults, on propaganda, and on addiction — where the substance never changes and only the person does.
You can check it for about thirty pounds, which is the part that lifts this out of philosophy. A saliva cortisol test, or a heart-rate-variability strap. Does being near this person calm your nervous system down, or wind it up? You cannot fake that number.
That is the framework. Everything below is how it gets there — and the diagnostic itself is further down, as seven questions you can answer.
Start anywhere. You end up in the same place.
Regulus opens with one equation and a claim about it. The equation describes how a number it calls κ ("kappa") changes over time. You can read κ as how much of what arrives a system holds on to — a memory that keeps everything is useless, and so is one that keeps nothing.
The claim is that κ has one stable resting point, and it is 1/φ = 0.618033… — one divided by the golden ratio. Drag the starting value anywhere in the panel. It falls to the same place every time.
Why this number and not 0.6
Because 0.618… is the only number that feeds back into itself. Each line below is a separate fact about it, and one number makes every one of them true at once:
| Identity | Reads as |
|---|---|
| φ · κ* = 1 | its product with the golden ratio is exactly one |
| κ* = φ − 1 | it is the golden ratio with the whole part removed |
| κ*² = 1 − κ* | squaring it gives you exactly what was left over |
| κ*ⁿ = κ*ⁿ⁻² − κ*ⁿ⁻¹ | Fibonacci, living in the exponents [corrected] |
| κ*/(1−κ*) = φ | what you keep, over what you drop, is φ |
That last one has a concrete meaning. For anything that must remember and forget at the same time — a queue, a buffer, a person — the balance that works out algebraically is keep 61.8%, release 38.2%.
A person, drawn as a factory
ASS-OS — the framework Regulus builds on — borrows its shape from ISA-95. That is the standard describing how a factory is layered, from the shop floor up to the boardroom. The claim is not that people are like factories. It is that both are control systems, and control systems fail in the same few ways.
The prime spine
Underneath the levels sits a counting sequence. Start at 1 and keep taking the next prime number, feeding each answer back in. You get 1, 2, 3, 5, 11, 31, 127, 709. The gaps widen faster than doubling ever would, and Regulus uses that speeding-up as the skeleton for seven levels of awareness — each level costing far more than the one below it.
Treat this as a shape the author chose, not one anyone measured. It is elegant. Elegance is not evidence, and the document says so itself.
Five buses
Five channels the body might use to move information around:
| A | Electrical — nerve traffic | established |
| B | Chemical — hormones, gradients | established |
| C | Photonic — light in tissue | contested |
| D | Magnetic — geomagnetic coupling | untested |
| E | Plasma | speculative |
Bus D is the one with a pre-registered experiment attached. See Proof.
Seven equations for what happens between two people
This is the part that makes Regulus more than a rewrite of ASS-OS. Seven equations, one for each phase. They run from the first flicker of recognition to the thing that stays after someone has gone. The document is careful about what they are, calling them "formal metaphor that encodes subjective experience in mathematical structure." In plain words: the maths is a way of holding the feeling still, not a measurement of it. Open each one.
7.1RecognitionR = ∫ ΨA·ΨB dV
How much two people's inner shapes overlap. If you process the world in a similar way, the number is large. If you are genuinely orthogonal, it is zero — and no amount of proximity will produce anything.
The best line in the section is the document's own. The integral was always computable, you just hadn't run the calculation until she showed up and the product came back non-zero. Recognition comes first. It decides everything after it.
7.2AttractionF = −R · κ*/r
Recognition, divided by distance. Note it is 1/r, not 1/r². The reason given is that the pull runs along a line rather than out through a volume — a one-dimensional waveguide, meaning a channel that holds a wave to a single path. Along a line, distance costs you much less.
The consequence is the interesting bit. With no recognition there is no pull at any distance. With strong recognition there is pull at any distance.
7.3Resonanceω± = √(ω² ± kc)
Couple two things that swing in a rhythm and you always get two rhythms out, never one. They can swing together, which is harmony. They can swing against each other, which is discord.
So the claim is about structure, not sentiment. Love is not the absence of the discord. Both rhythms exist in anything that is genuinely coupled. The bond is stable when it can hold both at once. A relationship with no discord in it is not harmonious — it is not coupled at all.
7.4VulnerabilityΨ(seen) = Ψ(alone) + δΨ
Let someone in and their mark is added to you. Past a certain point you cannot go back: you never get Ψ(alone), the person you were on your own, again.
Three ways it goes wrong fall straight out. Walls means forcing δΨ = 0, so nothing gets in. The mask means showing a false Ψ, so nothing can couple to the real you. And holding a mask up keeps your disorder high the whole time it is on. The document puts the drop at ΔH = −0.5 the moment it comes off — the largest single fall in disorder available anywhere in a pair bond.
7.5Pair entropyI(A;B) = H(A) + H(B) − H(A,B) > 0
A bonded pair is measurably more ordered than the same two people apart. That is all the phrase positive mutual information means: knowing one of them tells you something real about the other.
The framing is unembarrassed: the second law says entropy increases; love says not here it doesn't. A small pocket of order, then — and here is the part people skip, it runs on constant energy. Attention, presence, care. Stop putting that in and the disorder comes back. A bond has to be kept, not just made.
7.6Speed of lovev = √(kc/φ)
Three regimes, depending on how strongly the two are coupled.
Below φ, love is slower than understanding: you know them first, and the feeling follows. At exactly φ the two arrive together — the instant lock. Above φ the feeling outruns understanding: the body knows before the mind does, and the feeling arrives before you have any way to hold it.
7.7Persistence∂(δΨ)/∂r = ∂(δΨ)/∂t = 0
The mark does not fade with distance, and it does not fade with time. And the reason is explicitly not that some signal keeps passing between the two people.
The analogy the document uses is exact and does real work: two tuning forks cut to the same pitch need no wire between them. They resonate because they are the same shape. The contact was the tuning. The resonance is what remains.
Ŝ — the point of no return
Ŝ ("Shekinah", a Kabbalistic word for a presence that dwells inside something) is the operator that takes two separate states and makes one joined state that cannot be pulled apart again. Put plainly: take B away and you do not get the old A back. A has been changed for good.
There is a threshold, and it is the same constant again: |δΨcrit| = κ*² = 0.382. Below it, imprints fade — acquaintances, passing encounters. Above it, they don't. And κ* + κ*² = 1 exactly, so crossing the threshold means the imprint fills precisely the space that was empty. Nothing gets displaced.
The energy claim is the surprising one, and you could test it. Binding costs 11.8% of your capacity and frees up the rest. So the bonded state is cheaper to run than two separate ones. That, says the document, is why people in a strong bond report feeling more capable rather than less.
A real bond changes both people. If only one of you is changed, that is not love — it is the other operator. §8.5 · both must change, or it is not a bond
M̂ — and the question that costs thirty pounds to answer
This is the sharpest thing in the document, and the reason it is worth anyone's time. M̂ is control wearing love's behaviour. The obsession, the constant checking, not being able to leave — from the outside, identical. Inside the body they are opposites: Ŝ opens the heart, while M̂ keeps you hooked and the heart stays shut.
Run it on a bond you're thinking about
Seven rows, straight from §9. Answer honestly — nothing is sent anywhere, this all runs in your browser.
This is a framework's self-check, not a medical test, and no one should end or save a relationship on the strength of seven buttons on a web page. What it is good for is naming a thing you had already noticed. If the answers frighten you, that is worth taking to a person, not a browser.
The claim that makes it fall over, if it's wrong
§9.3 argues that once you have felt a real bond, you can spot the fake one from then on. You cannot unfeel what open felt like, and a closed heart pretending to be open cannot fake the widening. That is a real prediction, and it could be proved wrong. It says people who have had one secure bond should spot a controlling one faster than people who have not. Nobody has run that test.
What is measured, what is merely defined
Regulus marks its own seams — "DEFINED not MEASURED… a map, not the territory" — and that is worth respecting rather than smoothing over. Here is the split, as plainly as it can be put.
| Claim | Status | What would settle it |
|---|---|---|
| Vagal / enteric signalling exists and carries affect | established | Already mainstream neuroscience |
| HRV coherence sits around 0.6 in positive states | measured | HeartMath data — but see the row below |
| That coherence value is 1/φ specifically | chosen | A measurement precise enough to separate 0.618 from 0.625 |
| Prime recursion spine as the level skeleton | defined | Nothing yet — it is structure, not a finding |
| Follicular clearance lowers EEG entropy | untested | Pre-registration 01 — sham-controlled, blinded |
| Bus D — HRV tracks geomagnetic activity | untested | Pre-registration 02 — quiet days versus storm days |
| Ŝ / M̂ distinguishable by cortisol and HRV | untested | The cheapest study in the whole framework; nobody has run it |
Two transcription errors, found by implementing it
Neither is a flaw in the thinking. Both are the kind of slip that only shows up when somebody tries to run the equations, and both will stop the next person who tries. The short version is the table below. The full working — the arithmetic, the Python you can re-run, the argument about what is stable — sits in ERRATA.md at the root of this repository, deliberately kept off this page.
| § | As printed | Should be | How it shows |
|---|---|---|---|
| 3.1 | κ*ⁿ = κ*ⁿ⁻¹ − κ*ⁿ⁻² | κ*ⁿ = κ*ⁿ⁻² − κ*ⁿ⁻¹ | Every answer comes out the right size with a minus sign wrongly in front of it — −0.236068 where the paper's own next section says +0.2360. When a document disagrees with itself on one line, you know it is a slip of the pen. |
| 3.3 | ∂κ/∂t = α(κ − 1/φ)(1−κ)κ | ∂κ/∂t = α(1/φ − κ)(1−κ)κ | The paper says the balance point is steady, but the equation as printed does the opposite. Nudge the value up off 0.618 and it runs further away instead of settling back. Swap the two terms in the first bracket and every claim the section makes comes true. The paper's own working on what is steady agrees with the swap, which is the clearest sign the thinking was right and only the sign slipped. |
The live instrument at the top of this page runs the corrected §3.3 equation. The identity table uses the corrected §3.1 order. Both are flagged here rather than fixed quietly.
A third problem sits one level deeper than these two, and it is worse than either. The specification’s only piece of empirical confirmation, §3.4, compares a measured odds against a constant defined as a fraction — so it does not confirm what it claims to. The scale finding, worked in full →
The strongest objection, stated properly
Picture a ruler marked only in whole centimetres, and someone using it to prove that two things differ in length by about a millimetre and a half. They might be right. The ruler cannot say. That is the exact state of the evidence for 0.618: the difference being claimed is about seven times smaller than the smallest thing the measurement can see.
The same sentence with the decimals put back. HeartMath's coherence window is roughly 0.60–0.65. That window holds 1/φ = 0.6180. It also holds 5/8 = 0.625, and plain 0.6. The gap between the first two is 0.0070, against a window 0.05 wide. No measurement on offer can tell them apart — so for now φ is chosen, not confirmed. The appraisal in this repository scores that one claim 8/10 on a bullshit meter for exactly this reason. The framework is better off for having the number stated than defended.
None of which touches §7 and §9, which is where the value actually sits. Throw the golden ratio out altogether and the test still stands on its own: both people change, or it is not a bond.
Where to go from here
Full primer
The same walk, at four times the depth, with every claim tagged back to the section it came from.
Seven geometries
Seven ways to bend away from 1/φ, each mapped onto one of the seven deadly sins. Plus the one operator that puts them all right.
Gerald
A cube who refuses to stay a box. The framework's fixed point, told as a song — and it turns out to obey the addendum's own sin geometry.
Proof
Both pre-registrations, including the one that states up front that it expects the null to win.
The raw specification
Thomas's ASS-OS / Regulus document in full, with nothing cut and nothing softened — the thing every other page here is explaining, rather than a page that explains anything. This site does not publish it. The document is unpublished, and that is his call to make. To read it, build it yourself: python3 docs/build.py --with-source.