Three Horizons and the Making of a Stable Relation

What lets a changing pattern remain?

A mark appears, and then it disappears. Another returns in a recognizable way. What has changed between a passing occurrence and something whose continuation we can follow?

That is the question behind these three horizons. We are looking for a relationship that belongs to several participants together, and then asking whether such a relationship can help explain how a physical structure holds.

Begin with the circles on the screen. Each has a centre and a boundary. For this representation, we can follow what lies within each boundary, what lies outside it, and how one bounded expression relates to another.

The circles are not yet atoms or minds. They give us a simple construction in which we can see exactly what follows from the relationships we choose. First we will discover the shared geometry. Then we will ask what a moving physical system would have to do to sustain a bound state.

Two boundaries give a relation of scale

Take the first two circles. Lay a straight line against both boundaries without cutting through either circle. Then do the same on their other side.

Extend those two lines. Because the circles in this example have different radii, the lines meet beyond the smaller circle.

Watch what that meeting point means. From there, a ray touches one circle and then the other. Draw a radius to each point of contact. Both radii meet the tangent at a right angle, so the two triangles have the same angles. One is a scaled version of the other.

The distances from the meeting point to the centres therefore carry the same proportion as the radii. If the radii are two and three units, those two distances are in the ratio two to three.

This point is called a centre of similarity. It was not selected first and then used to arrange the circles. Its position follows from the scale relationship between them.

We have found a geometric reference for the pair. Whether a physical object would stay there is a later question.

P_{ij}=(r_i O_j-r_j O_i)/(r_i-r_j)

|P_{ij}-O_i|/|P_{ij}-O_j|=r_i/r_j

The third circle relates the relations

Now construct the same reference for the second and third circles, and for the third and first.

There are three pairwise meeting points. Draw the line through the first two. The third lies on it as well.

Move a circle. Change a radius. Recalculate the three meeting points. While the construction remains in its ordinary, nondegenerate form, the three continue to be collinear. This is Monge’s theorem.

The line itself can move and turn. What remains is not one fixed position, but the alignment of the three related positions.

Why should that happen? Read the scale around the three circles. The first radius compared with the second, the second compared with the third, and the third compared with the first complete one consistent chain. Two-thirds, multiplied by three, multiplied by one-half, gives one.

The circles have not become equal. Their differences remain. What closes is the comparison around the whole.

Together with the placement of the meeting points on the extended centre-lines, that consistency gives the alignment. The line is not a fourth body added to hold the circles together. It makes a relationship among their pairwise relationships visible.

(r_1/r_2)(r_2/r_3)(r_3/r_1)=1

\det[P_{12}^{h},P_{23}^{h},P_{31}^{h}]=0

What happens at equality, and on the inside?

Let two radii approach equality. Their external tangents become closer to parallel, and their meeting point travels farther away. At exact equality, the tangents are parallel: there is no finite external meeting point to put on the screen.

The circles have not been destroyed. The finite description has reached a limiting case.

For separated circles, we can also draw the two crossing tangents through the gap between them. Their intersection lies between the centres and gives an internal centre of similarity.

Inside and outside now have precise geometric meanings. They cannot be exchanged without changing the construction. The three external centres align; a compatible combination of two internal centres and one external centre also aligns. Three internal centres do not generally form that same straight line.

This is useful for our paired model. A direction is part of the relation, not a decoration we attach afterward. To follow a whole consistently, we must preserve which side of each pair we are reading.

Q_{ij}=(r_j O_i+r_i O_j)/(r_i+r_j)

A flat alignment can reveal a spatial relationship

There is another way to understand the line. Imagine the three circles as equators of three spheres, with their centres still in the original drawing plane.

For the illustrated arrangement, a second plane can touch all three spheres. Each pair of spheres fits inside a tangent cone. The tip of each cone lies in the original plane and also in the common tangent plane. The tips must therefore lie where those two planes intersect: along one line.

That is a spatial explanation of the same alignment. The drawing plane cuts through the spheres’ centres; it is not simply a table underneath them.

Now consider perspective. Hold a circular disc facing a camera and move it farther away. Its image becomes smaller although the disc itself has not shrunk. Move it closer, and the image grows.

A size change in the image can therefore represent depth, once the object and the projection are specified. But a larger object farther away can produce the same image size as a smaller object nearby. The image alone does not decide which changed.

The same care applies to a drawn field boundary. Its radius could represent a distance, an intensity threshold, or an available range. We need the rule connecting that radius to the field before it can tell us a potential or a force.

a=fR/Z

a_{\rm sphere,on-axis}=fR/\sqrt{d^2-R^2}

From alignment to a place that can hold

We have found an exact geometric relation. Now place a small test object near one of the marked points. What will it do?

The drawing has not yet answered. We have specified circles and tangents, but no law that accelerates the object.

Even a balance point needs a further test. A small displacement from the bottom of a bowl is opposed by a restoring influence. A small displacement from the top of a rounded hill is enlarged. Both have a point of balance, but they do not have the same stability.

Our simulation can show that difference with the same marked centre. Change the response to displacement, and one trajectory returns while another departs. The position alone did not decide the outcome.

This also explains the apparent immediacy of the Monge line. When we redraw a circle, the software recalculates a mathematical construction. No material point has travelled along the line to report the change.

A physical implementation needs a law for how the change reaches its participants. Geometry can describe compatible configurations; propagation and response determine how a physical system moves among them.

\ddot q=-q,\quad\ddot q=0,\quad\ddot q=+q

A wave node is not automatically a trap

Now let the marked centres emit waves. Each contribution arrives with an amplitude and a phase. Where the waves meet, those contributions add.

Hold the source positions fixed and change only one phase. The bright and dark pattern changes, while the geometric tangent construction stays where it was. That tells us something important: the tangents alone cannot determine the interference pattern.

Huygens’ construction relates a wavefront to the wavelets through which it propagates. Interference additionally depends on their phases and amplitudes. Crossing two tangent lines does not supply those quantities.

In a standing wave, a node is a place where the relevant oscillating amplitude remains zero. Whether another object is confined nearby depends on how it couples to that field and whether a displacement produces a restoring response.

For an electron wavefunction, a node has a still more specific meaning: its probability density is zero there. It is not the location where that electron becomes concentrated.

So the question becomes sharper. We are not looking for an empty point and calling it a particle. We are looking for an interaction that supports a localized, repeatable state.

I=|\sum_j A_j e^{i(kd_j+\phi_j)}|^2

\psi(x)=0\Rightarrow |\psi(x)|^2=0

The orbital version: moving together

A planetary system gives us a physical example of a location defined through more than one participant.

Take two masses in circular orbit and ask where a third, sufficiently small body could remain in the same position relative to them. In a frame turning with the orbit, gravitational attraction and the requirements of rotation produce five equilibrium locations: the Lagrange points.

Three lie along the line joining the primary masses. Two form equilateral triangles with them.

But their stability is not identical. The three collinear points are unstable. The triangular points can support stable nearby motion when the mass ratio and the conditions of the restricted model allow it.

A sixty-degree triangle is therefore part of the solution, not a substitute for the conditions that make the solution stable. Nor did Lagrange solve every possible three-body motion by finding these configurations.

Webb illustrates the difference between a reference point and an object following it. The telescope travels around the Sun–Earth L2 region and uses corrections; it does not sit permanently at a motionless mathematical dot.

This is a real moving relation. Its equilibrium locations come from the gravitational equations, not from the radii of three drawn circles.

\Omega={1-\mu\over r_1}+{\mu\over r_2}+{x^2+y^2\over2}

\ddot x=2\dot y+\Omega_x,\quad\ddot y=-2\dot x+\Omega_y

The quantum version: a whole can bind when its pairs do not

Quantum physics supplies a different, striking example. Under suitable near-resonant, short-range interactions, three particles can support a bound state even where the corresponding pair has no bound state.

This is the setting of Efimov physics. The third participant changes the possible collective state; it is not merely another object placed beside an unchanged pair.

For the standard identical-boson limit, successive characteristic sizes differ by about twenty-two point seven. The corresponding binding energies change by the square of that factor. Real interaction ranges and decay channels limit what can be observed.

The mathematical pattern can be read as an oscillation in the logarithm of a collective radius. Advance the phase by pi, and the radius is multiplied by a fixed factor. Here phase, scale, and binding meet inside a specified quantum equation.

The Borromean-ring image captures one aspect: the whole can be bound without a bound pair. It does not mean that the particles physically trace three rings.

This keeps the relational insight intact. Collective binding is real. What supplies it is the three-body quantum interaction under its stated conditions—not Monge’s collinearity, and not the number three by itself.

\lambda=e^{\pi/s_0}\simeq22.6944,\ s_0\simeq1.00624

|E_{n+1}|/|E_n|=\lambda^{-2}

Bring the question into an atom

Now we can approach the additional proposal: could the relation between a proton and a neutron generate a stable place that becomes an electron?

A proton and a neutron can form the nucleus of deuterium. That nucleus contributes to the electron’s allowed states. So there really is a nested relationship to examine: internal nuclear organization participating in a wider atomic organization.

But ordinary hydrogen gives us a decisive comparison. Its nucleus contains one proton and no neutron, and it still binds an electron. The proton–neutron pair therefore cannot be a necessary source of an electron or of atomic binding.

The repair is specific. The electron is a participant in the atomic relation; its bound state is shaped by its interaction with the nucleus. The electron is not identical to a relation between a proton and a neutron.

Adding a neutron changes the nuclear mass and other nuclear properties, which can shift the electronic spectrum. It does not supply a missing third corner without which electrons cannot exist.

This leaves the deeper question open in a productive form: how do an electron and a nucleus enter one bound state, and what distinguishes that whole from its separated participants?

\mu={m_eM\over m_e+M}

\hat H=-{\hbar^2\over2\mu}\nabla^2-{e^2\over4\pi\epsilon_0r}

A bound state is a whole pattern, not a vacant dot

For hydrogen, electric attraction favours keeping the electron near the proton. Quantum localization has a cost as well: confining the state more tightly increases its kinetic-energy contribution.

Their relationship supports allowed bound states with definite energies. The lowest one has an exponentially decaying, spherical probability density around the nucleus. It is not a tiny planet parked at a Lagrange point.

The same state can be displayed in different ways. A spatial density tells us probability per unit volume. A radial distribution tells us probability within spherical shells. Their peaks need not be at the same place, because larger shells contain more volume.

In this ground state there is no finite radial node hosting the electron. The density remains nonzero at every finite radius. The region called a void in the earlier proposal is therefore not what holds this state together.

Now let time advance. A stationary state keeps the same observable density. A superposition of different energies can instead show changing interference as relative phases evolve.

Persistence can therefore include motion and change, but it does not require every measured quantity to vibrate. The state we chose, and the equation governing it, tell us what remains steady and what changes.

\psi_{1s}={e^{-r/a}\over\sqrt{\pi a^3}},\quad P(r)=4\pi r^2|\psi|^2

\Psi_n(t)=\psi_n e^{-iE_nt/\hbar}

\Psi=(\psi_{1s}e^{-iE_1t/\hbar}+\psi_{2s}e^{-iE_2t/\hbar})/\sqrt2

A closer example of two centres and one shared electron

There is a closer physical example of your two-centres idea: the hydrogen molecular ion. It has two protons and one electron.

The electron’s state can extend across both nuclear centres. In a simple approximation, we combine one amplitude associated with each centre. Adding them gives a bonding combination; subtracting them gives a different, antibonding combination.

Keep the proton positions fixed and change that relative sign. The geometry has not moved, yet the electron distribution and energy change. In the antibonding combination, a nodal plane appears between the centres. The bonding combination instead permits amplitude there.

To find a bound molecule, we must include the electron’s energy and the repulsion between the protons. Their combined energy can favour a finite separation.

Here the relation really does contribute to holding a larger identity together. It is not a new electron created at an empty crossing. It is one electron participating in a shared quantum state that helps bind two centres.

This is an additional comparison for the proposed model, not a relabelling of the original proton–neutron–electron claim.

\psi_\pm={\phi_A\pm\phi_B\over\sqrt{2(1\pm S)}}

S=e^{-R}(1+R+R^2/3)

What light reveals, and what it changes

How does such a state become available to observation?

A photon can scatter from a system, be absorbed in an allowed transition, or be emitted as energy is released. An interaction changes the relation between the atom, the field, and the detector. A detector can then retain a record.

That is more precise than calling every atomic observation a reflection in a mirror. We need to follow which signal was exchanged and what record it left.

The electron is not kept in existence by visible illumination. Free electrons can also be studied outside atoms. And an atom can remain bound without a human observer watching it.

The measurement can still matter physically. It may alter the state, transfer energy, or ionize the atom. The fact that observation is an interaction does not make every observation the creation of the particle observed.

This gives us three distinct relationships to track: what supports the state, what makes it detectable, and what the detection changes.

The wider identity can become a participant

A nucleus has internal organization. An atom forms through the nucleus and electrons. Atoms can enter molecular relations whose properties are not obtained by merely naming the isolated parts again.

Protons and neutrons themselves contain quarks and gluons. The interactions at that depth differ from electronic binding. The hierarchy is real, but a shared language of relation does not make every level the same mechanism.

This is where Y Theory asks us to follow a common structural question: what must be retained for the lower participants to remain consequential within the wider whole?

A higher description may gather many details into one usable state. But if two internal arrangements require different future responses, the wider description must preserve a way to distinguish them.

The bonding and antibonding examples make that requirement visible. They contain the same kinds of particles at the same chosen positions. Ignoring the relative state would erase a difference that changes their energy and possible continuation.

The whole is therefore not simply an extra object placed beside the parts. It is their organized participation, available at another scale of description.

From a relation-map to a history-bearing observer

The matrix gives us addresses for these distinctions. One entry can say which relation is being followed. Another field can carry its strength, phase, or retained history. These must not all be treated as the same kind of number.

Our three-circle ratios already provide a useful miniature example. Direct comparison between the first and third radii agrees with comparison through the second. The complete matrix is consistent because its entries came from the same three radii.

An operating network asks more. It needs to say what arrives, how the receiver changes, what remains, and how the retained consequence affects the next encounter.

In the Y-Theory observer architecture, local conditions form a whole-level reading. An earlier reading is retained. Present and retained conditions are compared. The comparison is retained in turn and changes the subsequent operation of the same participating field.

That is a relation of relations with a return through history. A point where lines cross is not yet this observer. The observer is the continuing operation through which the field’s own registered change becomes consequential to what it does next.

Current clarification (editorial note, separate from the recording): The comparison-and-return loop supports observation within an observer. The observer is the whole internally organized Mind that correlates its own history with outside relations and can intentionally change and expand its participation. The recording is preserved as supplied; the schematic shows only supporting operations, not a complete observing Mind.

The circles help us see a shared constraint. The physical examples show why state and interaction must be included. The Mind model adds the retained comparison through which the organization can become available to itself.

R_{ij}=r_i/r_j,\quad R_{ij}R_{jk}=R_{ik}

z\to a\to M\to C\to\Omega\to z^{\prime}

What this branch establishes

We began with a question about persistence, then followed three boundaries into one shared geometric relation.

That alignment gave us an exact result. Asking whether something could remain there led to another requirement: an interaction and a response to disturbance.

Orbital equilibria, quantum trimers, atomic states, and molecular binding supplied different answers to that requirement. None needed to be treated as unrelated to the others, and none had to lose its own mechanism to fit the comparison.

The electron proposal now has a stronger place in the sequence. The atomic relation does not create an electron from a proton–neutron pair. It supports particular states of an electron and nucleus. A shared electronic state can then help bind a wider molecular whole.

From there, the recursive Mind question remains clear: how can such organized relations retain their consequences, become comparable, and return an understanding into their own continuation?

The geometry tells us what belongs together under a rule. The dynamics tell us what can persist. The history-bearing return tells us how a continuing organization can change through what has happened.

That is the route our simulation should make visible.

Questions to carry forward

Where this belongs in the journey

• Play Three Horizons · 22 minutes →Audio and interactive models
• Geometry and relations →The teaching journey
• Atoms and chemical relations →The teaching journey