This discussion concerns the evolution of the relational framework itself rather than the geometric ⊗ operator developed elsewhere.

Toward a calculus of relationship composition

One of the more instructive moments in mathematical work is discovering that the thing you've been studying isn't the object, it's a shadow the object casts.

For several months I've been developing a relational framework for control and complex systems. I believed its central object was a relation, written a ⧖ b, expressing compatibility between two evolving system descriptions: not that they agree, but that each remains within what the other can accommodate as both evolve.

Then a reviewer asked the question that reorganized everything: what is the object? Not the notation, not the intuition, the mathematical object.

The uncomfortable answer was that ⧖ wasn't one. It was a predicate. It answered yes or no and carried no internal structure: nothing to evolve, nothing to compose, nothing to project a control law from. I had been treating an observable as if it were the thing observed.

So here is the reformulation, offered for criticism.

The candidate object

Fix a state space X, a control system ẋ = f(x, u) with admissible controls U, a horizon T, and an envelope map R assigning to each description its admissible region. Define the relationship object

ρ(a, b) = (R_a, R_b, C_ab),

where C_ab is the compatibility structure: the set of joint controlled trajectories on [0, T] along which each system remains within the other's envelope throughout;

C_ab = { (x_a, x_b, u_a, u_b) : dynamics and control constraints hold, and x_a(t) ∈ R(x_b(t)), x_b(t) ∈ R(x_a(t)) for all t }.

This is a controlled joint-viability construction, so existence questions land in known territory (Aubin). The original relation survives only as a derived observable: a ⧖ b ⟺ C_ab ≠ ∅.

Why compatibility is not intersection. Two constructions motivate separating the relationship from instantaneous geometry:

1. Overlap without compatibility. Place two substantially overlapping regions around a saddle point of the flow. The intersection is nonempty now, yet every joint trajectory through it exits at least one envelope in finite time: C_ab = ∅.
2. Compatibility without overlap. Two disjoint regions on a cylinder under controlled rotation — a rendezvous configuration. Instantaneous intersection is empty, yet admissible controls carry both descriptions into sustained mutual admissibility: C_ab ≠ ∅.

If these hold as generally as I believe they do, compatibility is a property of the triple (regions, regions, flow) and cannot factor through the regions alone.

The open structural questions. These are finite, and each has a definite answer:

• Composition: given ρ(a,b) and ρ(b,c) composed through a shared middle trajectory, how does the composite differ from ρ(a,c) directly defined? Is the defect δ(a,b,c) a computable, nontrivial object? (Note ⧖ is deliberately non-transitive; the defect is meant to measure that failure, not quotient it away.)
• Grading: what scalar observables on C_ab (margin to constraint boundary, measure, sustainable duration) support a persistence calculus under the time-evolution semigroup?
• Generation: can a projection Π from relationship trajectories to control laws be constructed, so that controllers are derived from relationship dynamics rather than postulated?
• Reduction, the question I most want answered. Behaviors in Willems' sense compose by intersection, which is associative and idempotent with zero defect. Approximate bisimulation (Girard–Pappas) relates trajectories within tolerance but is built toward equivalence. Does (ρ, composition, δ, evolution) admit a faithful embedding into behavioral interconnection, viability theory, approximate bisimulation, or relation algebra? If yes, I want the reduction stated precisely — that equivalence would itself be worth having. If no, I want to know exactly where the factorization fails.

The genuinely speculative extension

The version above holds the envelope map R fixed. The framework I'm ultimately after does not, the envelopes evolve in response to the history of the relationship itself,

R_a(t) = G(R_a(0), ρ|[0,t]),

so the relationship reshapes the admissible regions while the regions reshape the relationship. This coupled construction is a fixed-point problem, and its well-posedness is unproven, I suspect monotone G with a Knaster–Tarski argument is the tractable case, but I don't yet have the theorem. It is also, I believe, the part that most clearly cannot be a pre-given behavior, since the constraint set is a function of the unfolding relation rather than data fixed in advance. I hold this part the most loosely and consider it the most likely to be either the real contribution or the real mistake.

An invitation

The purpose of publishing these notes is to expose the proposed object to mathematical scrutiny before investing further in the framework. At this stage, I'm less interested in defending the construction than in discovering whether it survives contact with existing mathematics.

The questions I would most like answered are:

• Does the proposed relationship object reduce to an existing formalism?
• If so, where is that reduction most naturally expressed?
• If not, where does the reduction first fail?
• If the object is still ill-typed, what is the minimal additional structure required to make it mathematically well-defined?

Perhaps, static ρ is known mathematics, well-dressed; the defect calculus is open and plausibly new; the participation loop is either new or reducible only vandalously (a reduction is vandalous if it embeds the objects but fails to preserve the observables that motivated their construction). Maybe.

Whether that assessment ultimately proves correct is exactly the question. If the static construction reduces cleanly, that's a result. If the defect calculus survives reduction, that's a result. And if the participation loop resists reduction except by embedding it into a larger formalism that discards the very observables motivating its introduction, I'd like to understand that precisely rather than rhetorically.

I'm throwing this into the ring because I suspect the answer already exists, or because it doesn't. Either way, I'd rather find out now than after another year of building on the wrong object.

For readers without a background in control theory or topology, the section below explains the intuition rather than the formal mathematics. The definitions and equations in the main essay remain the authoritative version.

When the Wrong Mathematical Object Refuses to Stay Put

Suppose you want to write down mathematical rules for a relationship, not a human one, but a relationship between moving systems: two automated vehicles holding formation, two drones coordinating a handoff.

The mistake I'd been making was treating that relationship as a yes-or-no question: are these systems compatible right now? A yes-or-no answer is a shadow. It has no internal structure, nothing that changes over time, nothing you could steer with. I realized I had been treating an observable as if it were the thing observed. The relation answered a question, but it wasn't the mathematical object itself

So the reformulation gives the relationship itself a body. The object ρ bundles three things. Each system's safe zone, and, the important part, the compatibility structure, the full set of future paths along which both systems can keep operating inside each other's safe zones, given what the physics and their controls actually allow.

Why isn't compatibility just "the safe zones overlap"?

Two pictures:

Overlap without compatibility. Two boats sit side by side where a current forks, water converging from behind, splitting violently ahead. Their positions overlap perfectly at this instant, but every path forward drags them to opposite sides of the fork faster than their engines can fight it. Overlapping now; compatible never.

Compatibility without overlap. Two dancers stand on opposite sides of a spinning carousel. No overlap at all right now, but the spin is predictable, and with modest timing they meet in ten seconds. Disjoint now; deeply compatible.

So compatibility is a fact about regions and motion together, never about a snapshot.

The purpose of introducing ρ is not to invent new notation. It is to ask whether relationships themselves can become mathematical objects that evolve, compose, and eventually generate control decisions. Whether that object already exists under another name, or whether it genuinely adds something new, is exactly the question I'm hoping others will help answer.

© 2026 Symfield PBC, Nicole Flynn. All rights reserved.

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