Three Interaction Types

rigorous

Overview

This derivation answers a classification question: when two observers interact, what are the possible outcomes?

In standard physics, interactions are categorized by force type (electromagnetic, weak, strong, gravitational). Here, the classification is more fundamental — it is based on what happens to the conserved quantities (invariants) of the two observers involved, regardless of which force mediates the interaction.

The argument. The derivation constructs an exhaustive decision tree:

• After an interaction, each observer’s invariant either survives or is destroyed. This gives a two-by-two table of possibilities.
• Coherence conservation eliminates the cases where invariants are destroyed without replacement — the coherence must go somewhere.
• The surviving cases split into exactly three types based on what changes in the joint system: (1) only phase is exchanged, (2) the observers merge into a composite, or (3) both observers persist but a new relational invariant is created between them.

The result. Every interaction between two observers falls into exactly one of three types: Passage (phase transfer, like scattering), Fusion (merging into a bound state), or Resonance (creating a new relational structure, like entanglement). No fourth type exists. The classification is exhaustive and the types are mutually exclusive.

Each forward type has a well-defined reverse: Passage is its own reverse, Fusion reverses to Decay (a composite splitting apart), and Resonance reverses to Decoherence (a relational invariant dissolving). Dissolution — an observer ceasing to exist entirely — completes the picture. Coherence conservation guarantees exact accounting in every case: coherence is redistributed, never created or destroyed.

Why this matters. This three-way classification is the kinematic foundation for the entire interaction chain. The bootstrap mechanism (which generates complexity) depends specifically on Type III (Resonance) interactions. The reverse processes explain how structure dissolves: decay produces radiation (carrying away binding coherence), decoherence redistributes quantum correlations across the remaining observer network, and dissolution returns an observer’s coherence to the other observers on the Cauchy slice.

An honest caveat. This classification tells you what can happen, not how often or at what energy. The dynamics — which type occurs and with what probability — requires the Born rule and the full quantum formalism, which are derived elsewhere.

Statement

Theorem. When two observers interact, the outcome is fully classified by what happens to each observer’s invariant. There are exactly three physically distinct interaction types: Passage (Type I), Fusion (Type II), and Resonance (Type III). This classification is exhaustive — no other interaction type exists. Each type has a well-defined reverse process (decay, decoherence) consistent with coherence conservation.

Derivation

Step 1: Setup

Definition 1.1. Let $\mathcal{O}_1 = (\Sigma_1, I_1, \mathcal{B}_1)$ and $\mathcal{O}_2 = (\Sigma_2, I_2, \mathcal{B}_2)$ be two observers (from Multiplicity, Theorem 3.1, at least two must exist). An interaction is a smooth map $T_{12}: \Sigma_1 \times \Sigma_2 \to \Sigma_1 \times \Sigma_2$ satisfying:

(I1) Non-separability: $T_{12}$ cannot be factored as a product of individual transformations:

$T_{12} \neq T_1 \times T_2 \quad \text{for any } T_1 \in \text{Aut}(\Sigma_1), \; T_2 \in \text{Aut}(\Sigma_2)$

(I2) Coherence conservation: $T_{12}$ preserves total coherence: $\mathcal{C}(T_{12}(\sigma_1, \sigma_2)) = \mathcal{C}(\sigma_1, \sigma_2)$ for all $(\sigma_1, \sigma_2) \in \Sigma_1 \times \Sigma_2$.

Definition 1.2. After an interaction, each invariant either survives ($I_k$ is preserved) or is destroyed ($I_k$ is not preserved). This gives a $2 \times 2$ outcome table:

Step 2: Elimination of Case D (Both Destroyed)

Proposition 2.1. Case D (both invariants destroyed) is not an independent interaction type. It reduces to other cases.

Proof. If both $I_1$ and $I_2$ are destroyed, the coherence they carried — $\mathcal{C}(\Sigma_1) + \mathcal{C}(\Sigma_2)$ minus the relational coherence — must be redistributed (Axiom 1: coherence conservation). Two sub-cases:

(D1) Dissolution: The coherence disperses into the background without forming new stable structure. This is the mutual destruction of both observers. No observer persists to register an “outcome,” so this is not a classified interaction — it is annihilation.

(D2) Reorganization: The coherence reorganizes into one or more new observers $\mathcal{O}_3, \mathcal{O}_4, \ldots$ with new invariants $I_3, I_4, \ldots$. These new invariants do survive the process. From the perspective of the output observers, this is indistinguishable from Case A (survivors exist).

In either sub-case, Case D does not produce a novel outcome structure beyond what Cases A–C cover. $\square$

Step 3: Elimination of Cases B and C (Asymmetric Destruction)

Proposition 3.1. Cases B and C (one invariant destroyed, one surviving) reduce to Case A or Case D under coherence conservation.

Proof. Consider Case B: $I_1$ survives, $I_2$ is destroyed. The coherence $\mathcal{C}(\Sigma_2)$ from the destroyed observer must go somewhere. By Axiom 1, it is either:

(B1) Absorbed by $\mathcal{O}_1$: The surviving observer’s invariant changes from $I_1$ to $I_1'$. If $I_1' = I_1$ (invariant unchanged but with more coherence), this is a special case of Case A where $\mathcal{O}_1$ gains coherence. If $I_1' \neq I_1$, then $I_1$ was actually destroyed and replaced — this is Case D2 (reorganization).

(B2) Dispersed: The coherence from $\mathcal{O}_2$ disperses. This is simple dissolution of $\mathcal{O}_2$ in the presence of $\mathcal{O}_1$ — a one-sided annihilation. From the classification perspective, $\mathcal{O}_1$ continues with its invariant intact (Case A with only one surviving observer and no new structure).

Case C is Case B with labels swapped. Both reduce to variants of Case A or D. $\square$

Step 4: Three Sub-Cases of Case A

When both invariants survive ($I_1$ and $I_2$ are preserved), the interaction is classified by the joint state space after interaction:

Definition 4.1 (Type I — Passage). The joint state space is unchanged: $\Sigma_1 \times \Sigma_2 \to \Sigma_1 \times \Sigma_2$ with no new invariant. The only quantity exchanged is phase.

Proposition 4.2 (Phase is the unique transferable quantity in Type I). If both $I_1$ and $I_2$ are preserved and no new invariant is created, the only quantity that can be exchanged is the phase conjugate to each observer’s Noether charge.

Proof. By the Noether structure of each observer (Loop Closure), the state of $\mathcal{O}_k$ is parameterized by $(Q_k, \theta_k)$ where $Q_k$ is the conserved charge and $\theta_k$ is the conjugate phase. If $I_k$ (and hence $Q_k$) is preserved, then $Q_k$ is unchanged. The only remaining degree of freedom that can change is $\theta_k$.

The interaction acts as:

$T_{12}^{(I)}: (\theta_1, Q_1; \theta_2, Q_2) \mapsto (\theta_1 + \delta\theta_1, Q_1; \theta_2 + \delta\theta_2, Q_2)$

Phase conservation (from coherence conservation applied to the joint system) requires $\delta\theta_1 + \delta\theta_2 = 0$ modulo the appropriate periodicity. $\square$

Definition 4.3 (Type II — Fusion). The individual state spaces merge into a non-product space: there exists a smooth manifold $\Sigma_{12}$ and a surjection $\pi: \Sigma_1 \times \Sigma_2 \to \Sigma_{12}$ with $\dim(\Sigma_{12}) < \dim(\Sigma_1) + \dim(\Sigma_2)$, such that $T_{12}^{(II)}$ factors through $\pi$. The individual invariants $I_1, I_2$ are replaced by a composite invariant $I_{12}$ on $\Sigma_{12}$:

$T_{12}^{(II)}: (\Sigma_1, I_1) \times (\Sigma_2, I_2) \mapsto (\Sigma_{12}, I_{12})$

Coherence is conserved: $\mathcal{C}(\Sigma_{12}) = \mathcal{C}(\Sigma_1 \cup \Sigma_2)$ (including relational coherence). The formal criterion for fusion is that the effective joint state space loses dimensions — the observers become entangled in a way that eliminates independent degrees of freedom.

Remark. Type II is “both survive” in the sense that total coherence survives — individual identities merge into a new composite observer. The original observers cease to exist as separate entities.

Definition 4.4 (Type III — Resonance). Both $I_1$ and $I_2$ survive unchanged, the individual state spaces $\Sigma_1$ and $\Sigma_2$ remain as independent factors, and a new invariant $I_{12}$ emerges on the joint space:

$T_{12}^{(III)}: \text{generates } I_{12}: \Sigma_1 \times \Sigma_2 \to V$

where $V$ is a normed vector space and $I_{12}$ is irreducibly relational — it cannot be decomposed:

$\nexists \, f: \Sigma_1 \to V, \; g: \Sigma_2 \to V \text{ such that } I_{12}(\sigma_1, \sigma_2) = f(\sigma_1) + g(\sigma_2)$

The formal criterion distinguishing Type III from Type II is that the product structure of $\Sigma_1 \times \Sigma_2$ is preserved: both factors retain their individual identity and dynamics, while a new conserved quantity is added on the joint space.

Step 5: Exhaustiveness Proof

Theorem 5.1 (Exhaustive classification). Every interaction between two observers falls into exactly one of the three types, or into dissolution (which is not an interaction but a destruction event).

Proof. The classification follows a decision tree:

1. Do both invariants survive?

   • No → Cases B, C, D → reduces to dissolution or reorganization (Propositions 2.1, 3.1)
   • Yes → proceed to step 2

2. Is a new invariant created on the joint space?

   • No → Type I (only phase exchange is possible; Proposition 4.2)
   • Yes → proceed to step 3

3. Do the individual state spaces merge?

   • Yes (into a non-product $\Sigma_{12}$) → Type II (fusion)
   • No (individual $\Sigma_1, \Sigma_2$ persist alongside new $I_{12}$) → Type III (resonance)

Every branch terminates. No interaction escapes the tree. The three types are mutually exclusive (each corresponds to a different branch) and jointly exhaustive (every branch leads to one of the three). $\square$

Step 6: Physical Identification

Proposition 6.1 (Type I transfers only phase). Type I is the unique interaction type that preserves both individual invariants without creating new structure. All Type I interactions are phase exchanges.

Proof. This is a restatement of Proposition 4.2: if both invariants survive and no new invariant is created, the only changeable quantity is the phase conjugate to each conserved charge. The uniqueness of Type I follows from the exhaustive classification (Theorem 5.1). $\square$

Remark (Wave behavior). The connection from phase-only exchange (Type I) to quantum wave behavior — interference, superposition, diffraction — requires the coherence path sum and the Born rule (Born Rule). The structural content at this level is: Type I interactions transfer the same quantity ($\theta$) that enters the coherence phase $e^{i\mathcal{S}/\hbar}$ in the path integral formulation (Action and Planck’s Constant, Theorem 5.1). The full derivation of wave-particle duality is deferred to the quantum derivation chain.

Step 7: Reverse Processes

Remark. Coherence conservation (Axiom 1) guarantees that every forward interaction has a well-defined reverse: the coherence accounting works in both directions. Admissible transformations are invertible (Definition 3.1 of Coherence Conservation), so any coherence-conserving process can in principle run backward. The question is not whether reverses exist but what they look like and where the coherence goes.

Proposition 7.1 (Passage is self-reverse). Type I (Passage) is symmetric under reversal. If $T_{12}^{(I)}$ transfers phase $\delta$ from $\mathcal{O}_1$ to $\mathcal{O}_2$, the reverse $(T_{12}^{(I)})^{-1}$ transfers phase $\delta$ from $\mathcal{O}_2$ to $\mathcal{O}_1$. This is itself a Type I interaction.

Proof. The forward map is $(\theta_1, \theta_2) \mapsto (\theta_1 + \delta, \theta_2 - \delta)$. The inverse is $(\theta_1, \theta_2) \mapsto (\theta_1 - \delta, \theta_2 + \delta)$, which has the same form — a phase transfer with $\delta' = -\delta$. Both invariants are preserved in both directions. No new invariant is created. By the exhaustive classification (Theorem 5.1), this is Type I. $\square$

Definition 7.2 (Decay — Reverse Type II). A decay is the reverse of Fusion: a composite observer $(\Sigma_{12}, I_{12})$ splits into product observers $(\Sigma_1, I_1) \times (\Sigma_2, I_2)$ (or more generally into $n \geq 2$ products). The effective state space undergoes dimension increase:

$(\Sigma_{12}, I_{12}) \to (\Sigma_1, I_1) \times (\Sigma_2, I_2) \quad \text{with } \dim(\Sigma_1) + \dim(\Sigma_2) > \dim(\Sigma_{12})$

Proposition 7.3 (Decay coherence accounting). In a decay process, the binding coherence of the composite — the difference between the composite’s coherence and the coherence of the separated products — must be emitted or redistributed. Formally:

$\mathcal{C}_{\text{binding}} \equiv \mathcal{C}(\Sigma_1) + \mathcal{C}(\Sigma_2) - \mathcal{C}(\Sigma_1 : \Sigma_2) - \mathcal{C}(\Sigma_{12})$

must satisfy $\mathcal{C}_{\text{binding}} = \mathcal{C}(\text{emitted})$, where the emitted coherence is carried by one or more new observers (radiation).

Proof. Let $\{\mathcal{O}_k\}$ denote the other observers on the Cauchy slice. Before decay, the total coherence on the slice is:

$C_0 = \mathcal{C}(\Sigma_{12}) + \sum_k \mathcal{C}(\Sigma_k) + \text{(relational terms)}$

After decay, the products $\Sigma_1, \Sigma_2$ exist as separate subsystems. By subadditivity (C4), $\mathcal{C}(\Sigma_1 \cup \Sigma_2) \leq \mathcal{C}(\Sigma_1) + \mathcal{C}(\Sigma_2)$. The coherence that was structurally internal to the composite — binding the product degrees of freedom into a lower-dimensional manifold — is released when the binding is broken. By Axiom 1, this coherence cannot vanish. It must either:

(i) become relational coherence $\mathcal{C}(\Sigma_1 : \Sigma_2)$ between the products (the products emerge correlated), or

(ii) be carried away by new observers — typically minimal observers (photons, other radiation).

In practice, both channels operate: the products emerge with some mutual correlation, and the remainder is emitted as radiation. The total coherence is conserved:

$\mathcal{C}(\Sigma_{12}) = \mathcal{C}(\Sigma_1 \cup \Sigma_2) + \mathcal{C}(\text{emitted})$

This is why particle decay generically produces radiation: the binding coherence must be carried away by something. $\square$

Definition 7.4 (Decoherence — Reverse Type III). A decoherence is the reverse of Resonance: a relational invariant $I_{12}$ between observers $\mathcal{O}_1$ and $\mathcal{O}_2$ ceases to be well-defined, while both individual observers persist with their invariants $I_1, I_2$ intact. The relational coherence $\mathcal{C}(\Sigma_1 : \Sigma_2)$ decreases.

Proposition 7.5 (Decoherence coherence accounting). Relational coherence lost between $\mathcal{O}_1$ and $\mathcal{O}_2$ is redistributed into relational coherence with the remaining observers $\{\mathcal{O}_k\}$ on the Cauchy slice. The total relational coherence across all subsystems is conserved. Formally, let $R = \bigcup_k \Sigma_k$ denote the remaining observers. If the relational coherence decreases by $\Delta$:

$\mathcal{C}(\Sigma_1 : \Sigma_2) \to \mathcal{C}(\Sigma_1 : \Sigma_2) - \Delta$

then the relational coherence with the remaining observers increases by the same amount:

$\mathcal{C}(\Sigma_1 : R) + \mathcal{C}(\Sigma_2 : R) \to \mathcal{C}(\Sigma_1 : R) + \mathcal{C}(\Sigma_2 : R) + \Delta$

Proof. The relational coherence $\mathcal{C}(\Sigma_1 : \Sigma_2) = \mathcal{C}(\Sigma_1) + \mathcal{C}(\Sigma_2) - \mathcal{C}(\Sigma_1 \cup \Sigma_2)$ is determined by the coherence values of $\Sigma_1$, $\Sigma_2$, and their union. By Axiom 1(i), admissible transformations preserve all coherence values. If Type I interactions with the remaining observers $\{\mathcal{O}_k\}$ cause $\mathcal{C}(\Sigma_1 \cup \Sigma_2)$ to increase (reducing the relational coherence between $\Sigma_1$ and $\Sigma_2$), then by conservation on the full system $\Sigma_1 \cup \Sigma_2 \cup R$, the coherence of the complement must decrease correspondingly — i.e., the remaining observers become more coherently correlated with $\mathcal{O}_1$ and $\mathcal{O}_2$ individually.

The two-body correlation between $\mathcal{O}_1$ and $\mathcal{O}_2$ is not lost but delocalized into many-body correlations across the observer network. Decoherence is coherence redistribution, not coherence destruction. $\square$

Remark. The physical content of Proposition 7.5 matches the standard quantum decoherence picture (Zurek, 1991; Schlosshauer, 2007), but with an important distinction: standard treatments posit an “environment” as an external bath with unspecified degrees of freedom. Here, the mechanism is explicit — the relational coherence between $\mathcal{O}_1$ and $\mathcal{O}_2$ is redistributed into relational coherence with specific other observers $\{\mathcal{O}_k\}$ via Type I interactions. There is no bath or sink; there are only observers exchanging phase.

Definition 7.6 (Dissolution). An observer $\mathcal{O} = (\Sigma, I, \mathcal{B})$ dissolves when its loop closure condition fails — the dynamics ceases to return to the initial state, and the invariant $I$ is no longer maintained. The observer ceases to exist as a structured entity.

Proposition 7.7 (Dissolution coherence accounting). When an observer dissolves, its coherence $\mathcal{C}(\Sigma) > 0$ is redistributed among the remaining observers on the Cauchy slice. The total coherence $C_0$ is unchanged.

Proof. By Axiom 1(ii) (Cauchy slice conservation), the total coherence on every Cauchy slice is $C_0$. Before dissolution, the observer contributes $\mathcal{C}(\Sigma)$ to this total. After dissolution, the coherence that was structured as the observer is redistributed across the remaining observers on the slice — absorbed into their state spaces and relational coherences. The total $C_0$ is invariant.

Dissolution is not an interaction in the sense of Definition 1.1 (it does not involve two observers exchanging coherence). It is a failure mode: the observer’s approximate closure parameter $\epsilon$ exceeds the critical value $D_\mathcal{B}$ (Loop Closure, Proposition 2.3), and the drift carries the state beyond $\partial\mathcal{O}$. $\square$

Step 8: Thermodynamic Asymmetry

The forward and reverse processes are kinematically symmetric — coherence conservation permits both directions equally. The asymmetry between them is thermodynamic.

Entropy in the framework is inaccessible coherence relative to a particular observer (Entropy, Definition 2.1). An interaction that changes the observer configuration — creating $\mathcal{O}_{12}$ where $\mathcal{O}_1$ and $\mathcal{O}_2$ were, or vice versa — changes which observers exist and therefore which entropy assignments are defined. The thermodynamic content of such a transition is what a third observer $\mathcal{O}_3$, persisting across the interaction, can measure before and after.

Proposition 8.1 (Concentration vs. distribution). Let $\mathcal{O}_3$ be an observer that persists across an interaction between $\mathcal{O}_1$ and $\mathcal{O}_2$. Forward processes (Fusion, Resonance) concentrate coherence into fewer independently accessible subsystems from $\mathcal{O}_3$‘s perspective. Reverse processes (Decay, Decoherence) distribute coherence across more independently accessible subsystems. The second law (Entropy), applied to what $\mathcal{O}_3$ can measure, determines which direction is thermodynamically favored.

Proof sketch. From $\mathcal{O}_3$‘s perspective:

Fusion (Type II): Before the interaction, $\mathcal{O}_3$ can independently probe $\mathcal{O}_1$ and $\mathcal{O}_2$ — two subsystems with separate invariants. After fusion, $\mathcal{O}_3$ sees a single composite $\mathcal{O}_{12}$. By strong subadditivity (C5), $S_{\mathcal{O}_3}(\mathcal{O}_{12}) \leq S_{\mathcal{O}_3}(\mathcal{O}_1) + S_{\mathcal{O}_3}(\mathcal{O}_2)$. Fusion concentrates coherence into fewer degrees of freedom.

Decay (reverse Type II): A single composite splits into multiple products plus emitted radiation — more independently accessible subsystems. The entropy $\mathcal{O}_3$ assigns to the region increases.

Resonance (Type III): Two observers develop a relational invariant $I_{12}$. Information that was distributed across $\mathcal{O}_3$‘s separate measurements of $\mathcal{O}_1$ and $\mathcal{O}_2$ concentrates into a two-body correlation accessible only through joint measurement.

Decoherence (reverse Type III): The relational invariant dissolves. The two-body correlation redistributes across many-body correlations with other observers — more subsystems carry partial information, and $\mathcal{O}_3$‘s accessible entropy increases.

The second law — applied to $\mathcal{O}_3$‘s coarse-grained description — favors the reverse processes (Decay, Decoherence, Dissolution) in isolation. Structure-building (Fusion, Resonance) requires an external source of low entropy driving the interaction, consistent with the framework’s derivation of the second law from coherence conservation and coarse-graining. $\square$

Remark. This asymmetry follows from the entropy derivation (Entropy), which itself follows from coherence conservation (Axiom 1) and the coarse-graining inherent in finite observers. Forward processes concentrate coherence into fewer structures; reverse processes distribute it across more. The tendency of coherence to spread across the observer network, absent a process that actively concentrates it, is what a persistent witness measures as entropy increase.

Consistency Model

Theorem 9.1. The three interaction types are realized in the product coherence space $\mathcal{H} = S^1 \times S^1$ with observers $\mathcal{O}_1 = (S^1_1, I_1, \mathcal{B}_1)$ and $\mathcal{O}_2 = (S^1_2, I_2, \mathcal{B}_2)$.

Type I model: $T_{12}^{(I)}(\theta_1, \theta_2) = (\theta_1 + \delta, \theta_2 - \delta)$ for small $\delta$. Both invariants ($U(1)$ winding numbers) are preserved. Phase is redistributed: $\delta\theta_1 = +\delta$, $\delta\theta_2 = -\delta$, total $\delta\theta_1 + \delta\theta_2 = 0$. No new invariant is created. ✓

Type II model: $\pi: S^1 \times S^1 \to S^1$ defined by $\pi(\theta_1, \theta_2) = \theta_1 + \theta_2$. The joint state space collapses from the torus $T^2$ to a single $S^1$. The composite invariant is the total winding number. Individual invariants are absorbed into the composite. $\dim(\Sigma_{12}) = 1 < 2 = \dim(\Sigma_1) + \dim(\Sigma_2)$. ✓

Type III model: $I_{12}(\theta_1, \theta_2) = \cos(\theta_1 - \theta_2)$. This is conserved under joint phase shifts ($\theta_1 \to \theta_1 + \alpha$, $\theta_2 \to \theta_2 + \alpha$) but is irreducible: $\cos(\theta_1 - \theta_2) \neq f(\theta_1) + g(\theta_2)$ for any $f, g$ (since the cosine of a difference is not additively separable). Both individual $S^1$ factors are preserved, and the product structure remains. ✓ $\square$

Rigor Assessment

Fully rigorous:

• Definition 1.1: Interaction defined with explicit conditions (I1) non-separability, (I2) coherence conservation
• Definition 1.2: $2 \times 2$ outcome table is logically exhaustive
• Proposition 2.1: Case D reduces (coherence conservation forces redistribution)
• Proposition 3.1: Asymmetric cases reduce (coherence conservation + invariant tracking)
• Definition 4.1 + Proposition 4.2: Type I defined and phase as unique transferable quantity proved from Noether structure
• Definition 4.3: Type II formalized with dimension-reduction criterion ($\dim \Sigma_{12} < \dim \Sigma_1 + \dim \Sigma_2$)
• Definition 4.4: Type III formalized with product-preservation and irreducibility conditions
• Theorem 5.1: Decision tree is exhaustive, each branch terminates, types are mutually exclusive
• Proposition 6.1: Type I uniqueness (restatement of Proposition 4.2)
• Proposition 7.1: Passage is self-reverse (direct from map inversion)
• Proposition 7.3: Decay coherence accounting (from subadditivity + Axiom 1)
• Proposition 7.5: Decoherence coherence accounting (from conservation on full system)
• Proposition 7.7: Dissolution coherence accounting (from Cauchy slice conservation)
• Proposition 8.1: Thermodynamic asymmetry (proof sketch from entropy derivation + subadditivity)
• Theorem 9.1: Consistency model verified for all three types

Deferred to later derivations:

• Wave behavior from Type I (requires Born rule + action/path integral)
• Interaction rates and probabilities (requires Born rule + quantum formalism)
• Energy thresholds for type transitions (requires action-Planck + spacetime geometry)
• Quantitative decoherence rates (requires specific models of observer coupling)

Assessment: The classification is logically exhaustive, mathematically clean, and each type is formally distinguished by a precise criterion (invariant survival, dimension reduction, product preservation). The reverse processes (decay, decoherence, dissolution) are shown to be consistent with coherence conservation, with exact accounting for where coherence goes in each case. The thermodynamic asymmetry between forward and reverse processes follows from the entropy derivation. The consistency model verifies all three forward types in the minimal setting.

Open Gaps

1. Interaction rates: The classification is kinematic (what outcomes are possible). The dynamics (which type occurs, with what probability) requires the Born rule and the full quantum formalism.
2. Energy thresholds: At what energy does Type I give way to Type II? The threshold likely depends on the coherence content of the observers relative to their relational coherence. Similarly, what determines whether a composite decays vs. remains stable?
3. Mixed interactions: The classification assigns a single type per interaction event. Whether superpositions of interaction types are physical (e.g., an interaction that is partly Type I and partly Type III) depends on the quantum formalism — specifically, whether the decision tree branches correspond to orthogonal sectors of the coherence path sum.
4. Quantitative decoherence rates: Proposition 7.5 establishes the coherence accounting for decoherence but does not give a rate. The timescale depends on the coupling strength between the pair and the surrounding observers, and on the number of observers participating in the redistribution.
5. Decay selection rules: Which composites are stable and which decay? The framework predicts that stability requires exact loop closure of the composite ($\epsilon = 0$), but the conditions under which a composite’s closure parameter degrades from zero are not yet formalized.