How Things Happen

The previous chapter established that the axioms force observers to exist in pairs. Pairs interact — they must, since each provides the other’s non-trivial environment. What happens when they do?

The answer turns out to be surprisingly constrained. There are exactly three things that can happen, and from those three interaction types, the framework derives time, entropy, and the quantum of action — three of the most fundamental concepts in physics.

Three Ways to Interact

When two observers meet, the axioms limit what can occur. Each observer has an invariant. The interaction must respect coherence conservation. Here are the three possibilities:

Type I — Passage. Both observers survive the interaction with their invariants intact. The only thing that can transfer without threatening either invariant is phase — the relative advancement of each observer’s internal cycle. This is transient, wave-like interaction. If you have ever wondered why particles behave as waves, here is a structural reason: phase transfer is the only coherence-preserving mode of transient interaction between observers that keep their identities.

Type II — Fusion. The two observers merge. Both individual invariants dissolve, replaced by a new composite invariant that subsumes both. The original observers cease to exist as separate entities. A genuinely new observer appears at a higher level of complexity.

Type III — Resonance. Both observers keep their individual invariants, but a new conserved quantity appears — a relational invariant that lives on the joint state space. It cannot be attributed to either observer alone. It is irreducibly about the relationship. Quantum entanglement is the most famous example, but chemical bonds, causal relationships, and many other physical structures fit this pattern.

Three types. No more, no fewer. This is not postulated — it is derived from the requirement that interactions preserve some, all, or none of the individual invariants while conserving total coherence.

The Bootstrap

Type III interactions have a remarkable property. The relational invariant they create is itself a conserved structure with an invariant and a self/non-self distinction. It satisfies the observer definition. It is an observer, at the next level of the hierarchy.

As an observer, it can participate in further Type III interactions, generating second-order relational invariants. Those are observers too. The process iterates. Each level of interaction generates new degrees of freedom, new symmetries, new observers — compounding without bound.

This is the bootstrap mechanism: the universe generates complexity from coherence conservation alone, not from any additional law favoring complexity. Each interaction does not merely rearrange existing structure — it expands the space of possible structure. The hierarchy grows because the axioms leave it no choice.

Time

In standard physics, time is a background parameter — the stage on which physics plays out. You put in time as a coordinate, and events happen “in” it. Where this time comes from and why it flows one direction are treated as separate, difficult problems.

The framework takes a different approach. Time is derived.

Each minimal observer has an internal cycle — a loop that advances phase with each traversal. When two observers interact via Type I passage, phase is transferred between them. This transfer is ordered: the phase advancement of one observer constrains what phases are available to others it has interacted with. String enough interactions together across the full observer network, and a global ordering emerges — a partial order on events.

This partial order is time. “Earlier” and “later” are coherence-accessibility relations: event A is earlier than event B if A’s coherence structure is an ancestor of B’s in the network of interactions. Time is not a background coordinate inhabited by events. It is constituted by the structure of events themselves.

The Arrow of Time

The direction is built in. When a Type III interaction generates a relational invariant, that invariant is conserved — it cannot be un-generated. The process of relational invariant accumulation is irreversible. There is no transformation within the framework that takes a network with a relational invariant and returns it to the state before that invariant existed.

The arrow of time points in the direction of increasing relational invariant depth. Entropy increases because the bootstrap mechanism continuously generates structure outside any bounded observer’s domain. This is not a statistical tendency that might occasionally reverse — it is a structural consequence of conservation.

Entropy

Standard thermodynamics defines entropy in several roughly equivalent ways: disorder, missing information, the logarithm of accessible microstates. These definitions work in practice but leave a puzzle — entropy of what, relative to whom?

The framework gives a clean answer: entropy is observer-indexed inaccessible coherence. For a specific observer A, the entropy of a system is the total coherence of that system minus what falls within A’s coherence domain. It is always relative to a specific observer.

The second law follows directly. Each Type III interaction generates relational invariants. Some fall within A’s domain — A can “see” them. Others fall outside — they are part of the universe’s structure but inaccessible to A. Since interactions continuously generate structure, and only some fraction is accessible to any bounded observer, entropy increases for every bounded observer over time.

This also explains why the universe as a whole does not have increasing entropy: relative to the universe taken as its own observer, total coherence is conserved. The global entropy is constant. The second law is a statement about bounded observers, not about the universe.

The Quantum of Action

Action, in classical physics, is the time-integral of the Lagrangian — a quantity minimized along physical trajectories. In quantum mechanics, action appears in the path integral, where each path is weighted by a phase factor involving Planck’s constant $\hbar$.

The framework identifies action as coherence cost — the minimum coherence required to execute a transformation while maintaining an observer’s invariant. Physical paths are those that minimize this cost, which is why they satisfy the principle of least action.

Planck’s constant $\hbar$ then has a precise meaning: it is the coherence cost of one minimal observer cycle. One tick of the most fundamental clock. This is not an empirical constant that could have taken a different value — it is set by the normalization of the minimal observer loop. The quantum of action exists because observer cycles are discrete, and the smallest cycle has a definite, finite cost.

The uncertainty principle follows from the same source. The phase position of an observer’s loop and its cycle count are conjugate aspects of the coherence quantum. Knowing one precisely makes the other indeterminate — not because of measurement limitations, but because the coherence quantum is a single structure that cannot be separated into independent parts.

On solid ground: The three interaction types are rigorously derived. Time as a partial ordering, the arrow of time from irreversible relational invariant accumulation, and the identification of $\hbar$ with minimal cycle cost are all formally established from the axioms. The bootstrap mechanism is the framework’s most distinctive structural contribution.

Work in progress: The connection between the framework’s notion of entropy and standard thermodynamic entropy requires a uniform-coherence assumption for the Boltzmann recovery. This is natural but not derived from the axioms. The precise quantitative relationship between coherence cost and standard action remains an area where the mathematical formalization is still developing.

With dynamics, time, and action in hand, a remarkable question opens up: does the framework also produce spacetime geometry?