Less formal intuition-building page
Formal availabilities and the comparison-unit problem
Relational quantum mechanics opens the door to new interpretations of quantum foundations by treating physical facts as valid only in interaction. This understanding moves us away from assumed dependencies on intrinsic essences of isolated systems to a non-essentialist approach to fundamentality. If physical facts are only valid in interaction, then it is natural to ask whether the identities that carry those facts should also be treated as relationally settled.
Standard baryon spectroscopy begins with installed packages at the base that already supply determining numeric values, intrinsic masses, couplings, scale choices, and empirical constants. The dynamics then compute the spectrum, but the possible hadron outputs are already shaped by determining inputs prior to the calculation itself. From a relational standpoint, this leaves a methodological aporia. Empirical success is achieved, but the organisation is still not recovered from interaction alone (Laudisa 2019; Di Biagio and Rovelli 2022; Cavalcanti, Di Biagio, and Rovelli 2023).
So, the question becomes: can baryon-classificatory organisation be recovered from a relational derivation alone? Can one meet the physics of fundamentality and recover the organisation of the spectrum without assuming identity as primitive and without supplying values and inputs that already determine the outcome? This work follows that question from the start. If physical facts are valid only in relation, then the relevant task is to derive the minimal units of relation whose internal structure is adequate to admit identity. The aim is to obtain admissible objects under the non-essentialist commitment, using only relation and the logic of resolution. We begin by analysing the law of identity, without assuming identity as primitively available. The statement \(q=q\) cannot be ground for our logic, as it already depends on \(q\) as a definite object stable across comparisons. If identity is not a supplied primitive, then for \(q=q\) to be a stable comparison, that comparison must negate the unresolved comparison \(q\neq q\), itself a negation; and from the investigation, we conclude that negation is the irreducible predicate. We term this act contrast, and hold it as a meta-level commitment that grants equivalence and inequivalence as available to the construction. Nothing may negate its own inequivalence from equivalence with itself. In isolation, something may be in contrast with itself, but nothing can distinguish itself from itself. So, a self-reference is a null reference. Self-reference negates relation, and thus participation in any relational interaction that may resolve its identity as an accessible fact for further comparison. It is absent and returns void.
This gives us three statuses of identity: two resolved states, Present Participation and Absence, and one potential state where neither resolution has been determined: Unresolved.
Relation assumes the availability of separation. Separation requires negation once more. A single isolated entry cannot be negated from itself, and supplies no separation, no comparison through which identity or value could be defined. A pair supplies comparison, but only as opposition in that each side is not the other. But here, though each side may be distinct from another, neither can be truly separated from the other. With three relata, each is an available negation to the other. In this way, the codependent structure forms a circuit of non-opposing comparisons forming a triangle.
Imagine three people participate in an exercise where they have to try to figure out their own name. There are three possible names, Jimmy, James, or John, but no one knows their own name, only that they are not the name in front of them. Each can only look at the one in front of them, and on their backs their name is written. \(q_A\) knows ``I am not Jimmy'', \(q_B\) knows ``I am not James'', and \(q_C\) knows ``I am not John''. This gives \(q_A=(0,1,1)\), \(q_B=(1,0,1)\), and \(q_C=(1,1,0)\).
Here, a \(0\) refers to the negated potential name from the one directly in front of them, and the \(1\)s are the remaining entries that are the potential names the person could still have. This gives one eliminated name and two names that are left. Person A, who sees they are ``not Jimmy'', could be either James or John.
Each relatum already depends on the formal availability of the partner to which it is directed. So \(q_A\) receives ``not John'' in that \(q_A\) is not \(q_B\). But \(q_B\) already depends on the negation ``not Jimmy''. So, from any reference to \(q_B\), ``not Jimmy'' is already formally available as \(q_B\)'s dependency. So \(q_A\)'s negation from \(q_B\) makes ``not Jimmy'' available as a dependency to \(q_A\), resolving the identity of the relatum. So, \(q_A\) is defined by a directed site ``not John'' and an inherited site ``not Jimmy'', resolving by elimination to the definite identity `James'.
We can think about this more generally in social dynamics too. As each person speaks, there is a directed act, but who we are is not only in what we say. We are who we are to others around us: how we are met in relation and validated. One site is directed, and another is inherited in the response of the others, resulting in \(6\) unique sites where identity may be held and resolved in relation.
Following this pattern, the resolved stances of identities become \(q_A=(0,1,0)\), \(q_B=(0,0,1)\), and \(q_C=(1,0,0)\). Each entry now has two excluded names and one remaining value. Thus \(q_A=\mathrm{James}\), \(q_B=\mathrm{John}\), and \(q_C=\mathrm{Jimmy}\). Geometrically, each relatum is read as the three relation-edges, and each resolved relatum-status appears only after the circuit has closed, with the first negation at each site as one directed relation-edge. These three first negations form the centre triangle, \(d_A:q_A\to q_B\), \(d_B:q_B\to q_C\), and \(d_C:q_C\to q_A\). Then, each receives two inherited edges, closing each relatum as a separate triangle of one directed negation and two inherited negations coming back, forming \(6\) unique sites witnessing the relation. Through this convergence, each relatum is potentially resolved with accessibility to its own identity without self-reference, by being routed through the other relatum it depends upon and what is available from that negated site.
Counting each negation in the relation as an edge, we obtain \(3\) directed edges, \(6\) inherited edges, \(6\) witness sites, and \(9\) edges in total, converting to \(3\) triangles of three relata: one triangle for each potentially resolved relatum about a centre triangle formed by their directed negation. This in turn derives a hexagon of six witness sites, where the outer edges represent the inherited relata returning, and the negated edges form the central triangle of directed negations. This relation gives us the base for the geometric derivation. The integer return displacement and necessity for closure make \(2\pi\), \(6\), and \(1/6\) available as the first numeric values that form the necessities of the construction. This in turn is used to read the hexagon inside a square radius margin, derived inside a circle whose radius we unitised to \(1\). Later, to bridge to the baryon scale, this is then read one-to-one as \(1\,\mathrm{fm}\), bridging pure dimensionless ratio geometry to the hadron scale.
This circuit of six witness sites gives integer displacements of where, from the directed site, the inherited negation is held in relation. Each relatum \(q\) is defined by the magnitude of the displacement around the circuit. This in turn gives us our pure geometry base to derive the numerics as the relational foundation of the construction. This then gives us a vector notation of the circuit's unique configuration.
Self-reference is forbidden, so magnitudes of \(0\) and \(6\) are not relational participant entities. Such values mean the sites are only validated at their own relatum positions, negating relation. \(0\) yields no separation, and \(6\), coming back full circle around the hexagon to the same self-reference, gives our triad \((q_A,q_B,q_C)\) five active placements \((1,\ldots,5)\). The configurations \((0,0,0)\) and \((6,6,6)\) then express the boundaries of the construction as the point of complete self-equivalence.
The single-site value is then \(\theta_{\mathrm{site}}=\pi/3\). However, since no site can self-reference, this value is forbidden from each site; hence, each site holds the value only in its dependency on the relation \(\Phi=2\pi-\pi/3=5\pi/3\). Squaring this complement gives the co-dependent magnitude \(\mathrm{IES}=\Phi^2=(5\pi/3)^2\). The inherited closure scale then reads the same scale through completed six-site return: \(H_{\mathrm{inh}}:=(6^2/5)\mathrm{IES}=20\pi^2\). Deviation from the equivalence locus is read in two independent registers, angular departure \(a_{\mathrm{dev}}\) and radial departure \(r_{\mathrm{dev}}\), so total displacement is \(\kappa=a_{\mathrm{dev}}^2+r_{\mathrm{dev}}^2\), with coherence \(C=e^{-\kappa}\). The sign remains in \(a_{\mathrm{dev}}\) for charge, while \(\kappa\) gives total departure.
this is the base set of our geometry, but lets try break this down more and make the whole cocnept more intuitive.
Identity is not supplied as primitive, self-reference has already failed as a ground of resolution, and contrast has been retained as the irreducible act through which equivalence and inequivalence may be held apart. From this, Presence, Absence, and Unresolved are now available as the three statuses of resolution.
So if identity is not primitive, then what is the smallest unit of relation whose internal structure is adequate to admit identity?
If identity is not supplied at the base, then q cannot be used as though it were already settled, because q has no direct access to itself. Any reference q needs to itself has to be routed through another first. It is a kind of “who am I?” rather than “I am.” The self has to become available through comparison.
So the task here is to find the smallest relational unit in which identity-status can be resolved, while primitive objecthood, background, substrate, and installed label remain unavailable as starting permissions.
This is the comparison-unit problem. We have an occurrence. Inside that occurrence, each site is read through its relation to the others. A site is a place of evaluation, where distinction, support, return, Presence, and Absence may be tested. It is a position inside comparison before it is a resolved object.
Let’s make this concept a little bit easier to access and understand by taking an example that is probably a common idea the reader has already seen in pop philosophy circles and on social media. Two individuals are debating about the meaning of a glyph on the floor. One says it is a 9. The other one says it is a 6. Now, normally this debate is used to frame some perspective about who is actually correct, whether it is objectivism, an essence, or relativism.
The meme image
Often, then, this is met with some sort of metaphysics or interpretation about who is truly correct, or the better perspective to take. But here, we do not treat identity as primitive. There is no discernible better option to take. All that matters is whether there is some live participation about this glyph on the floor, or there is not. Instead, what we focus on is the identity of the entire relation. What is the consensus? How does that return?
The core point of this exercise in thinking is that nobody can actually tell whether they are really seeing a glyph on the floor at all, or whether what they are seeing is truly a glyph. There may be some objective original history. There may be some personal subjectivity. But none of that is supplied from the relation alone, and to us that is all that matters. What can be spoken about is the asymmetry of the agreement, where they converge, and where they differ. But in each example, the glyph on the floor becomes a centred axis, an external dependency that the system defines but does not in itself prove. When there is a dependency defined by some difference in separation across the system, then the system remains active and live about that axis. When all identities are equivalent in full, complete coherence and agreement, there is no further participation to negate or negotiate the existence of the glyph on the floor. So a structure in perfect coherence, where it is complete and all sites are equivalent, has no live negotiation to participate in further relationality. No one moves to another step to actually verify the existence.
Moving away from relativism and objectivism, we can still take the non-essentialist path and think about it through a more utilitarian approach. How does the system define its existence? The consensus of that shared definition does not identify the object on the floor, but it does identify the system negated from it. We cannot say whether a glyph truly exists by looking at these structures alone. However, we can identify that this is a structure at a higher level of contrast, with varying degrees of coherent views that are compatible with one another.
Here, Presence means that relation has been held as participation. Absence means that the occurrence has returned without participating relation. Unresolved means that relation is active, while identity-status remains open.
This helps make the three statuses more intuitive. Presence is participating relation. Absence is returned non-participation. Unresolved is the active middle case, where the comparison is live but has not yet closed.
For the unresolved state, suppose qAq_AqA speaks only English, qCq_CqC speaks only Arabic, and qBq_BqB is bilingual. Then qAq_AqA and qCq_CqC may be brought into relation through qBq_BqB. qAq_AqA may resolve with qBq_BqB, and qCq_CqC may resolve with qBq_BqB, while direct closure between qAq_AqA and qCq_CqC remains unavailable.
So let us move on to the panels. Below are drawn four panels of various modes of contrast around the glyph on the floor. The participants look toward the same place, but the labels around the glyph differ per their perspective. The question is not merely what each participant says locally. The question is whether the whole circuit has one coherent relation around the same target.
The panels
Panel A
Panel A shows two shared identities. From two perspectives, the figure on the floor is witnessed as a 9, but from the third opposing relatum, they return “it is a 6.” Everyone looking is coherent in that they agree it is a glyph. However, from the perspective of the two that say it is a 9, they share the same perspective. They cannot distinguish one from the other, they are equivalent. But the one that says “it is a 6” knows it is outright in its assertion. One site is unresolved from each perspective in that it is not distinct, but the system closes. There is some difference, some active negotiation, that makes the circuit live about the configuration. We can, for the sake of logic, call this a (1,1,2)(1,1,2)(1,1,2). But the core is, the participants are gathered around the same thing, and their readings make the glyph available as something under comparison. The glyph has begun to have identity-status because it can be held across more than one perspective, there is some level of verification, each one validates the other in its own perspective, that they truly do see something on the floor, and the circuit remains live about resolving its identity. It participates around some external dependency, external dependency here being the glyph on the floor.
Panel B
In Panel B, we see a system in full symmetry. There is no deviation in terms at any point. Every relatum receives immediate validation of its asserted stance and perspective. In other words, every perspective is shared. This system is fully coherent and thus complete. But here is the subtlety in being complete. Not only does the system not supply any definitive verification that the glyph really is on the floor from outside of itself, only that they all agree, but also it does not supply any method by which it is live about that axis, any further step towards relationally resolving that identity. The system has no further dependency on the glyph on the floor. Whether it is there or not, the system concludes that it is there, and thus no unique identity is returned. This structure is absent from further participation. It is whole, complete, closed, its coherence is complete, and its external dependency is zero. At first this looks like the strongest case: everyone agrees, so the glyph appears settled. Within the local system, each participant has a distinct position, but each position returns the same reading. The system is fully coherent. So since each is immediately validated, this point of self-equivalence we can list as (0,0,0)(0,0,0)(0,0,0).
Panel C
Now we take the distinct configuration that names a number of possibilities. One says it is a 9. The other says it is a 6. The other one says it is not a number at all, it is actually the letter g. Here we have further separation across the system. This is a stronger level of asymmetry. Every relatum is defined with its own unique identity in comparison to the others. Each makes a comparison of the glyph on the floor and agrees that at least it is a glyph. This system remains live and active about that axis, with no unresolved identity from any perspective. Each one is negated from the others. Conceptually, we can hold this as, say, (1,1,3)(1,1,3)(1,1,3), where the 3 is the greater step distance in relation, that it is a letter rather than a number. But importantly, what matters here is the glyph is active because it remains dependent on the convergence of perspectives. Something was defined, but ultimately not supplied from the relation itself. External dependency.
Panel D
Panel D gives neutralisation. The lower-left participant says, “it is a 6.” The lower-right participant says, “it is a 9.” The upper participant does not give a separate glyph-reading. Instead, it says, “you are both right.” Socially, this is the mediator’s answer. But what actually happened here? The system was neutralised by a step that is not actually participating. This references the other participants, not the glyph on the floor. What we actually have here is not a triangle at all, but a duad in opposition, which is witnessed by a third site. That third site is not in participation about the same external dependency, the glyph on the floor, it is negated from the two, in that it cannot see the separation they do.
What is important to us here is that the circuit can internally define the dependency around which the relation is organised. It can determine that (qA,qB,qC)(q_A,q_B,q_C)(qA,qB,qC) stand in comparison around a shared axis. It can also determine whether the relation is live, collapsed, or unresolved. But it does not yet determine the external dependency as an object in its own right. Only define the dependency. But here we are not talking epistemology, questions of what can be known and experienced. We are talking ontology. Ontologically this is still enough to define a configuration as present because this state exists, the consensus of the relations may be identified, it may then stably report that into higher structures that then may depend on reference to this relation, without the glyph on the floor ever being real at all.
The comparison-unit defines the demand for such a dependency; the determination of that dependency requires the later formal layer.
So let us take another example, so we can precisely name and understand the distinction in the mode of thinking. This should help the reader reason through the harder formalism and derivations of the next sections. We do not take many analogies within this work. The work is derivation-focused. But we offer this example so the reader may grasp the concept with some intuition before being met with the hard brutalism of the first-order logical derivation.
Suppose three people report the same strange light in the sky.
Each person stands in a different place. Each person has a different experience. One sees a flash above the trees. Another sees something pass over the roofline. Another hears a sound and sees a light vanish behind a hill. None of these experiences is identical. The true experience of anyone within relation to this event may be entirely different.
Yet the reports still converge.
They converge because each report becomes accessible under the same formal definition. The word “spaceship” begins to organise the reports. The word “aliens” gives the reports a shared target. It gives them a way to be gathered, compared, repeated, and entered into a wider statistic. The experiences remain different, but they are now accessible under one common term.
This is the important distinction.
“Spaceship” and “aliens” have not yet been secured as real objects. Nothing in the reports, by itself, supplies the truth of the claim. The system can define what it depends upon, but it cannot supply the truth of that dependency from within the same active relation.
Nevertheless, the dependency becomes real in its effects.
Whether or not the external dependency is real, it is manifested inside the relation. It changes what people attend to. It changes what they expect. It gives future witnesses a language in which to speak. It may create fear, curiosity, investigation, funding, denial, belief, ridicule, or institutional response. Those effects determine an outcome in the wider network. Behaviour, perspective, understanding, all changed under this lens of perception; those determinations are ontologically real whether the claim they depend upon is true or not.
So the question is not simply whether the people know correctly.
This is not merely epistemology. The relation itself has ontic effect, because it creates a live structure in which something unresolved may still be invoked, used, named, depended upon, and built into wider evidence.
This is why the shared target may be invoked before it has been secured.
A local stance may depend on what remains unresolved. The triad is the first comparison-unit in which each pair is held through the remaining third, so the local content is no longer exhausted by one line alone. Each report may differ. Each site may differ. Each perspective may differ. But the relation can still converge around the same internally defined dependency.
So the system must converge around some axis of contention, even where that axis has not yet been supplied as an already determined object. The relation internally defines the demand for that axis. The whole relation determines that the dependency is required, while the dependency itself remains unresolved at this stage.
Here we are not talking merely about epistemology, or only about questions of what can be known from a report. We are talking about the relation itself. The relation itself has ontic effect, because once the relata stand in relation, there is a live structure in which something unresolved may still be invoked, used, named, depended upon, and built into wider evidence, even though that something has not yet been secured as an object in its own right.
That is why the shared target may be invoked before it is secured.
A local stance may depend on what remains unresolved. The triad is the first comparison-unit in which each pair is held through the remaining third, so that the local content is no longer exhausted by one line alone. Each report, each site, and each perspective may differ, yet the relation can still converge around the same internally defined dependency, because what is shared at this stage is not yet the completed object, but the demand for the dependency around which the relation is being organised.
So the system must converge around some axis of contention, even where that axis has not yet been supplied as an already determined object. The relation internally defines the demand for such an axis. The whole relation determines that the dependency is required, while the dependency itself remains unresolved at this stage. So at this level, the external dependency is internally defined, but not yet determined.
The relation has made the demand available. It has not yet supplied the resolved object that satisfies that demand.
This is the root of external dependency in the comparison-unit itself.
For the triad to be identified as one triad, (qA,qB,qC)(q_A,q_B,q_C)(qA,qB,qC) must be negotiating around a common term. If qAq_AqA, qBq_BqB, and qCq_CqC each relate around a different subject, the occurrence has not converged. The relation remains distributed across separate targets rather than closing as one comparison-unit.
The story of Theseus’s ship gives the same thought in a more familiar form.
At first, there is a ship. It belongs to Theseus. It sails, returns, is remembered, and becomes part of the life of Athens. Then one plank is replaced. Then another. Then the sails are changed. Then the mast. Eventually, every material part has been exchanged.
So when is it still Theseus’s ship?
The usual answer tries to find the identity inside the object itself. It asks whether some inner essence has remained the same across all the replacements. But here the question is relational.
The ship remains Theseus’s ship so long as there are those who depend on it as Theseus’s ship, define it that way, name it that way, tell stories about it that way, gather around it that way, and identify it within the life of Athens that way. Children come to see it. The stories remain. It stands as a place of brilliance, memory, pride, and civic inheritance.
In that sense, it is Theseus’s ship.
But Theseus’s ship does not exist outside the relation as a self-certifying essence. It does not stand apart from the thoughts, motives, uses, stories, and dependencies of Athens. It is available as Theseus’s ship because a structure of participation sustains it as such.
This is also how temporality is understood within the structure. Everything has its own unique perspective, but transcendence is not movement through time. There is no hidden essence called the real Theseus’s ship, as though by piecing it together slowly one could somehow trick the universe, overcome definition, and show that it always was Theseus’s ship.
No. As every piece is placed, it is placed under the definition of repairing Theseus’s ship. The work is not done to an anonymous pile of wood. It is done to this ship, under this name, inside this civic relation. The act of repair already depends on the identity it claims to preserve.
The day the children stop speaking, the day the story stops circulating, the day the external dependency of the association with Theseus’s ship is gone, that relation is gone. Once that relation is gone, all there is is wood, stardust, atoms, particles, or whatever else one wants to say. Just material.
But here again, we are not applying this logic only to how people think and feel in social dynamics. We are making a claim about what is available to a local system. Just as there are limits to knowledge that depend on formal availability, there are limits to identity, objecthood, and physical fact when identity is not assumed as primitive.
If we do not assume identity as primitive, and if we do not begin from an objective reality in which identities are already independently supplied, then those two matters have to be understood through formal availabilities. What is available to one site from another determines how it may interact, and how it may interact determines how it may be held, used, invoked, depended upon, and resolved as physical fact within a relational system.
Again, this is not merely epistemology. It is not only a question of what one site can know, or whether one report is accurate, or whether the object has already been secured outside the relation. We are talking about the relation itself. The relation itself has ontic effect, because once the relata stand in relation, something may be defined inside that relation, and once something is defined inside that relation, it may organise what the other sites do around it, even where that something has not yet been determined as an object in its own right.
The ship organises civic memory. The alien report organises attention and investigation. Either way here, something is defined, and that definition determines the relations around it, without that thing necessarily being determined in itself.
So long as difference remains active, and so long as the dependency holding the relation together remains under negotiation, the system remains live, participating, and open to resolution. It has not closed into a settled object, and it has not returned absence. It remains live because each site still has reason to refer beyond itself, and each site still depends on what is available from the others. That live relation is the kind of interaction that can become a physical fact, because the relation is already organised around an internally defined dependency whose determination requires further relational verification.
That dependency is defined by the relation as a whole. An isolated site cannot provide it from itself, because an isolated site has no route through which its own identity may be returned to it. It can only sit in self-reference, and self-reference has already failed as the ground of resolution.
This is why we do not begin with a purely objectivist account, where identity is assumed to sit inside the thing itself, or a purely relativist account, where identity collapses into private opinion. Both miss the relation. The relevant question is what is available from one site to another, what may be invoked, what may be supported, what may be named, what may be entered into wider relation, and what may be depended upon.
That is what resolves identity here.
Identity is not treated as a hidden essence already sitting inside the object. It is resolved through availability, participation, and dependence within a structure.
And perhaps, after all this, the two examples are closer than they first appear. Theseus’s ship may have sailed through Athens, while the alien ship may never have sailed anywhere at all. But once a structure of relation gathers around it, names it, depends on it, and builds further claims through it, the ship has already entered the harbour of consequence.
Who knows? Perhaps Theseus’s ship was a spaceship all along.
Monad · Duad · Triad
Monad
The monad cannot be distinct because nothing can distinguish itself from itself. It cannot negate its own inequivalence from equivalence with itself, because in self-reference it negates relation, separation, and therefore Presence. Hence, under the restriction that identity is not primitive, the monad returns Absence.
Duad
The duad improves on the monad because two terms can stand in contrast: qAq_AqA may be distinguished from qBq_BqB, and qBq_BqB may be distinguished from qAq_AqA. But the duad still cannot close the occurrence, because each site only reverses against the other. Each side has the other side as its opposition, but no third site is available through which the relation can be supported as one comparison-unit.
Triad
The triad is therefore the first admissible comparison-unit, because qAq_AqA, qBq_BqB, and qCq_CqC each provide a route through which the other two may be held in relation, so difference is no longer bare opposition between two sites, but a supported structure in which each pair depends on the remaining third. This is why the triad first makes possible a live occurrence. It defines a shared dependency, holds local difference through relational support, and opens the route by which identity-status may be resolved without self-reference.
Closing Reflection
So, that is the intuitive bridge back to physics. The panels, the spaceship, and Theseus’s ship are only aids for seeing the same structural point: a relation may define the dependency around which its sites are organised before that dependency has been supplied as an already settled object. Baryons are the natural place to test this because they are already physical three-part structures. They are held together by the strong interaction, and confinement means the relevant content is not available as three freely separable pieces in ordinary isolation. So the triad is not merely a convenient analogy here. It gives a way of asking whether a bound three-relata structure can return charge, family placement, mass interval, radius-linked branch, and decay order from the organisation of relation itself, rather than from values installed beforehand. The intuitive question is therefore simple: if a baryon cannot be treated as three independent pieces merely placed side by side, can its classificatory identity be read from the way the three-part relation closes? This is why baryons matter for the construction. They give a physical test case where individuation, closure, and relational dependence can be asked of the same object.