Coherence Conservation

derived

Overview

This derivation answers a foundational question: what is the most primitive thing that must be true for observers to exist at all?

Before we can talk about what observers are or how they behave, we need a conserved quantity — something that cannot be created from nothing or destroyed into nothing. Without such a quantity, the entire framework has no ground to stand on. We call this quantity “coherence.”

The argument. Coherence is formalized as a kind of bookkeeping system with specific rules:

• Every subsystem carries a non-negative, finite amount of coherence (you cannot have negative existence, and each subsystem has finite information capacity).
• At every interaction node in the causal dependency graph, the coherence of the inputs equals the coherence of the outputs — the books balance locally.
• Combining two systems gives you at most the sum of their individual coherences, never more. This “subadditivity” means there is no free coherence to be gained by merging.
• Conservation is stated locally, at each node of the dependency graph — a time-free statement that does not presuppose a clock. Conservation across Cauchy slices follows as a theorem.

The result. Coherence behaves like a conserved currency of existence. At each interaction, coherence is neither created from nothing nor destroyed into nothing — the ledger balances. When two systems share coherence beyond what simple addition would predict, that excess is “relational coherence” — the irreducible connection between them. Relational coherence is genuinely new structure, associated with new symmetries generated by the interaction, inaccessible to either individual system.

Why this matters. This axiom plays the role that energy conservation plays in standard physics, but for a more primitive quantity. Everything downstream — observer identity, dynamics, interactions, and eventually all of physics — is built on this conservation law.

An honest caveat. The word “coherence” is doing heavy lifting here. The formal object is a subadditive measure on a mathematical structure called a sigma-algebra, with conservation stated on directed acyclic graphs. The intuition of a “conserved currency” is helpful but cannot capture the full algebraic content, particularly the distinction between subadditivity and strong subadditivity. The formal apparatus is grounded in five operational definitions established in From Observation to Axioms; every element below is traceable to those definitions.

Statement

Axiom 1 (Coherence Conservation). There exists a primitive quantity $\mathcal{C}$, called coherence, defined on the partitions of a coherence space $\mathcal{H}$. Coherence is locally conserved: it is invariant under all admissible transformations and balanced at every node of the dependency graph — the coherence of the inputs equals the coherence of the outputs. Coherence cannot be created from nothing, destroyed into nothing, or exported to an external reservoir — the ontology is closed.

Formalization

Operational Grounding

Every formal element of this axiom follows from the five operational definitions established in From Observation to Axioms. This section traces the connection before the formalization begins.

Universe $\mathcal{H}$ and $\sigma$-algebra $\mathcal{A}$. Observation requires at least two distinguishable systems (observer and observed), and observers must draw a self/non-self boundary. Both requirements force “subsystem” to be a meaningful category closed under (i) taking the complement (the “non-self” of any “self”), (ii) unions (the operational composition of two subsystems is itself a subsystem), and (iii) containing the trivial cases $\mathcal{H}$ and $\emptyset$. These are exactly the defining closure properties of a Boolean algebra. Persistence under repeated interaction, without a bound on the number of interactions, pushes closure from finite unions to countable unions — iterated operational composition is unbounded, so the closure property must survive countably many applications. This is the standard move from Boolean algebra to $\sigma$-algebra. Arbitrary uncountable unions are not forced by any operational requirement and are deliberately excluded. The $\sigma$-algebra is the minimal structure consistent with the operational requirements, not a chosen package.

Coherence measure $\mathcal{C}: \mathcal{A} \to \mathbb{R}_{\geq 0}$. A closed ontology carries a conserved currency, and that currency must be attributable to observers and hence to the subsystems in $\mathcal{A}$. A set function assigning a non-negative scalar to each admissible subsystem is the minimal formalization — any less structure cannot represent “how much currency this subsystem carries.” The choice of $\mathbb{R}_{\geq 0}$ as the codomain (rather than, say, an ordered semigroup) is the minimal mathematical setting in which the conservation constraints below can be stated.

Conditions (C1)—(C4). Positivity (C1) is forced because a negative value would mean a subsystem carries “less than nothing,” which has no operational meaning. Local finiteness (C2) follows from the operational requirement that each subsystem have finite information capacity: an observer with a compact state space (forced by persistence, Definition 3, and formalized as O1 compactness in Observer Definition) carries finite coherence. Operationally, a subsystem that would require infinite coherence to describe exceeds the information capacity of any observer that could interact with it — it is not an operationally accessible subsystem. Non-triviality ($\mathcal{C}(\mathcal{H}) > 0$) follows from the existence of observers carrying positive coherence. The null empty set (C3) is a convention-level sanity condition: the empty collection is not a subsystem with content. Subadditivity (C4) follows from the operational principle that grouping cannot create currency: if $\mathcal{C}(S_1 \cup S_2) > \mathcal{C}(S_1) + \mathcal{C}(S_2)$ for some disjoint pair, then a purely notational act of grouping would have created currency, contradicting both observation-as-residue and closed ontology.

Strong subadditivity (C5). C5 extends C4 from disjoint to overlapping subsystems. Its content becomes visible when restated in terms of relational coherence. For any $S_1, S_2 \in \mathcal{A}$, decompose into disjoint pieces $A = S_1 \setminus S_2$, $B = S_1 \cap S_2$, $C = S_2 \setminus S_1$. Then C5 is equivalent to:

$\mathcal{C}(A \cup B : C) \geq \mathcal{C}(B : C)$

That is: a composite observer’s relational coherence with an external system is at least as large as any sub-observer’s relational coherence with that same system. The composite $A \cup B$ contains $B$ because composites contain their sub-observers. Whatever relational coherence $B$ has with $C$, the composite also has — because $B$ is a part of $A \cup B$, and its correlations with $C$ do not vanish when the boundary is widened to include $A$. Widening the boundary is not an interaction; only interaction creates or destroys residue. The $B$—$C$ relational coherence was established by a $B$—$C$ interaction, and $A$‘s inclusion in the composite does not undo that interaction. Furthermore, the composite may have additional relational coherence with $C$ mediated through $A$, so the total can only be at least as large:

$\mathcal{C}(A \cup B : C) \geq \mathcal{C}(B : C) \qquad \square$

Remark (C5 and composition). Without the operational requirement that composites contain their sub-observers, C5 has no short argument from the other operational premises alone. In the physical realization where $\mathcal{C}$ specializes to von Neumann entropy, C5 is the Lieb—Ruskai theorem (1973) — a deep mathematical result, proved via the joint convexity of relative entropy, that took decades to establish. The downstream convergence with quantum entropy (Coherence as Physical Primitive) provides independent confirmation that C5 is the right constraint. With the composition requirement in hand, C5 is forced operationally: composites contain their sub-observers, widening the boundary is not an interaction, and relational coherence persists under inclusion. The Lieb—Ruskai theorem is the mathematical expression of this operational fact, not an independent assumption.

Admissible transformations. Observation does not create or destroy stuff, and persistence is invariant over repeated interaction. An admissible transformation is the formal analogue of “a change that an observer could undergo while remaining an observer” — it must therefore preserve each subsystem’s coherence content. The group structure of $\text{Aut}(\mathcal{H}, \mathcal{A})$ is standard mathematical packaging: identity is admissible, composition of admissible transformations is admissible, inverses exist.

Dependency graph $\mathcal{G}$. Persistence through repeated interaction implies a logical ordering of interactions — not a clock-based time, but an asymmetric “is-input-to” relation. Cyclic causation is incompatible with “residue remembered,” because it would mean observer state depends on its own not-yet-produced residue. The set of interaction events together with this asymmetric relation is, by construction, a directed acyclic graph. The graph is discrete (countable vertices) because observation residues are operationally binary — either a relational invariant was recorded or it was not — and an observer with finite coherence can record at most countably many such residues. The framework does also require continuous mathematical structures — Hilbert space, Fisher geometry, Lagrangian dynamics — but these live on the state-space manifold $\mathcal{M}$, not on $\mathcal{G}$. As Continuous-Discrete Duality makes explicit: the discrete event-level structure ($\mathcal{G}$ / the observer network) and the continuous state-space structure ($\mathcal{M}$) are co-formed dual descriptions of the same physics, each constraining the other.

Local conservation. Closed ontology means that at each vertex of $\mathcal{G}$, no coherence flows in from outside or leaks out. The coherence carried by the incoming edges equals the coherence carried by the outgoing edges. This is the fundamental conservation statement — it holds at each node independently, without presupposing any notion of simultaneity or global time.

Cauchy slice conservation (derived). Given local conservation at every node, conservation across Cauchy slices follows as a theorem (proved below in Step 5). Adjacent Cauchy slices differ by swapping a single vertex $v$ from its predecessors to its successors. Local conservation at $v$ guarantees that this swap preserves the total. By induction over any sequence of such swaps, all Cauchy slices carry the same total. This common value, denoted $C_0$, is a derived constant of the interaction network — not a free parameter of the axioms.

All formal elements of this axiom are traced to the five operational definitions; no additional input is required.

Step 1: The Coherence Space

Definition 1.1 (Coherence space). A coherence space is a tuple $(\mathcal{H}, \mathcal{A}, \mathcal{C})$ where:

• $\mathcal{H}$ is a non-empty set (the universe of configurations)
• $\mathcal{A} \subseteq \mathcal{P}(\mathcal{H})$ is a $\sigma$-algebra on $\mathcal{H}$ (the admissible subsystems)
• $\mathcal{C}: \mathcal{A} \to \mathbb{R}_{\geq 0}$ is a function (the coherence measure) satisfying conditions (C1)–(C5) below

Definition 1.2 (Coherence conditions). The coherence measure $\mathcal{C}$ satisfies:

(C1) Positivity: $\mathcal{C}(S) \geq 0$ for all $S \in \mathcal{A}$.

(C2) Local finiteness and non-triviality: $\mathcal{C}(S) < \infty$ for all $S \in \mathcal{A}$, and $\mathcal{C}(\mathcal{H}) > 0$.

(C3) Null empty set: $\mathcal{C}(\emptyset) = 0$.

(C4) Subadditivity: For disjoint $S_1, S_2 \in \mathcal{A}$ with $S_1 \cap S_2 = \emptyset$:

$\mathcal{C}(S_1 \cup S_2) \leq \mathcal{C}(S_1) + \mathcal{C}(S_2)$

(C5) Strong subadditivity: For all $S_1, S_2 \in \mathcal{A}$ (not necessarily disjoint):

$\mathcal{C}(S_1 \cup S_2) + \mathcal{C}(S_1 \cap S_2) \leq \mathcal{C}(S_1) + \mathcal{C}(S_2)$

Remark 1.3 (Monotonicity). Monotonicity ($S_1 \subseteq S_2 \Rightarrow \mathcal{C}(S_1) \leq \mathcal{C}(S_2)$) does not follow from (C1)–(C5) alone. Subadditivity (C4) bounds the union from above, not the parts from below, and (C5) applied to nested sets yields only a tautology. A subadditive set function need not be monotone. In this framework, monotonicity follows once the coherence measure is connected to the observer definition (Axiom 2), where $\mathcal{C}(\Sigma) > 0$ for any observer state space implies larger subsystems contain at least as much coherence as their sub-observers. We do not assume monotonicity at this stage.

Proposition 1.4 (C5 implies C4). Strong subadditivity (C5) implies subadditivity (C4) for disjoint sets, given (C3).

Proof. Let $S_1 \cap S_2 = \emptyset$. Then $\mathcal{C}(S_1 \cap S_2) = \mathcal{C}(\emptyset) = 0$ by (C3). Substituting into (C5): $\mathcal{C}(S_1 \cup S_2) + 0 \leq \mathcal{C}(S_1) + \mathcal{C}(S_2)$, which is (C4). $\square$

Remark. Hence (C4) is logically redundant given (C3) and (C5). We state it separately because subadditivity has a direct physical interpretation (the coherence of a composite system does not exceed the sum of its parts’ coherences), and because many results require only (C4) and not the full strength of (C5).

Step 2: Relational Coherence

Definition 2.1 (Relational coherence). The relational coherence (or mutual coherence) between disjoint subsystems $S_1, S_2 \in \mathcal{A}$ with $S_1 \cap S_2 = \emptyset$ is:

$\mathcal{C}(S_1 : S_2) \equiv \mathcal{C}(S_1) + \mathcal{C}(S_2) - \mathcal{C}(S_1 \cup S_2)$

Proposition 2.2 (Non-negativity). $\mathcal{C}(S_1 : S_2) \geq 0$ for disjoint $S_1, S_2$.

Proof. Direct from (C4): $\mathcal{C}(S_1 \cup S_2) \leq \mathcal{C}(S_1) + \mathcal{C}(S_2)$, so $\mathcal{C}(S_1) + \mathcal{C}(S_2) - \mathcal{C}(S_1 \cup S_2) \geq 0$. $\square$

Proposition 2.3 (Independence characterization). $\mathcal{C}(S_1 : S_2) = 0$ if and only if $\mathcal{C}$ is additive on $\{S_1, S_2\}$: $\mathcal{C}(S_1 \cup S_2) = \mathcal{C}(S_1) + \mathcal{C}(S_2)$. We say $S_1$ and $S_2$ are coherence-independent.

Proof. By definition, $\mathcal{C}(S_1 : S_2) = 0$ iff $\mathcal{C}(S_1) + \mathcal{C}(S_2) - \mathcal{C}(S_1 \cup S_2) = 0$ iff $\mathcal{C}(S_1 \cup S_2) = \mathcal{C}(S_1) + \mathcal{C}(S_2)$. $\square$

Proposition 2.4 (Symmetry). $\mathcal{C}(S_1 : S_2) = \mathcal{C}(S_2 : S_1)$.

Proof. $\mathcal{C}(S_1 : S_2) = \mathcal{C}(S_1) + \mathcal{C}(S_2) - \mathcal{C}(S_1 \cup S_2) = \mathcal{C}(S_2) + \mathcal{C}(S_1) - \mathcal{C}(S_2 \cup S_1) = \mathcal{C}(S_2 : S_1)$. $\square$

Proposition 2.5 (Chain rule). For pairwise disjoint $S_1, S_2, S_3 \in \mathcal{A}$:

$\mathcal{C}(S_1 : S_2 \cup S_3) = \mathcal{C}(S_1 : S_2) + \mathcal{C}(S_1 : S_3) - \delta$

where:

$\delta = \mathcal{C}(S_1) + \mathcal{C}(S_2) + \mathcal{C}(S_3) - \mathcal{C}(S_1 \cup S_2) - \mathcal{C}(S_1 \cup S_3) - \mathcal{C}(S_2 \cup S_3) + \mathcal{C}(S_1 \cup S_2 \cup S_3)$

The sign of $\delta$ is not determined by (C1)–(C5) alone.

Proof. Expand each relational coherence using Definition 2.1:

$\mathcal{C}(S_1 : S_2) = \mathcal{C}(S_1) + \mathcal{C}(S_2) - \mathcal{C}(S_1 \cup S_2)$ $\mathcal{C}(S_1 : S_3) = \mathcal{C}(S_1) + \mathcal{C}(S_3) - \mathcal{C}(S_1 \cup S_3)$ $\mathcal{C}(S_1 : S_2 \cup S_3) = \mathcal{C}(S_1) + \mathcal{C}(S_2 \cup S_3) - \mathcal{C}(S_1 \cup S_2 \cup S_3)$

Computing $\mathcal{C}(S_1 : S_2) + \mathcal{C}(S_1 : S_3) - \mathcal{C}(S_1 : S_2 \cup S_3)$ directly:

$\delta = \mathcal{C}(S_1) + \mathcal{C}(S_2) + \mathcal{C}(S_3) - \mathcal{C}(S_1 \cup S_2) - \mathcal{C}(S_1 \cup S_3) - \mathcal{C}(S_2 \cup S_3) + \mathcal{C}(S_1 \cup S_2 \cup S_3)$

This is the alternating sum (or Möbius function) over the three-element partition, and the identity holds by construction. $\square$

Remark (Sign of $\delta$). The correction term $\delta$ decomposes into two parts with opposite signs. Applying (C5) to $A = S_1 \cup S_2$, $B = S_1 \cup S_3$ gives $\mathcal{C}(S_1) - \mathcal{C}(S_1 \cup S_2) - \mathcal{C}(S_1 \cup S_3) + \mathcal{C}(S_1 \cup S_2 \cup S_3) \leq 0$, while (C4) gives $\mathcal{C}(S_2) + \mathcal{C}(S_3) - \mathcal{C}(S_2 \cup S_3) \geq 0$. Since $\delta$ is the sum of these, its sign is model-dependent.

The quantity $\delta$ is the interaction information (or co-information) of the triple $(S_1, S_2, S_3)$. It decomposes as $\delta = \mathcal{C}(S_2 : S_3) - I(S_2; S_3 | S_1)$, where $I(S_2; S_3 | S_1) \geq 0$ is the conditional mutual information (non-negative by C5). The sign of $\delta$ carries physical meaning: $\delta > 0$ when $S_2$ and $S_3$ redundantly encode their relationship with $S_1$ (the pairwise correlations overcount); $\delta < 0$ when they synergistically encode it (the whole exceeds the sum of pairwise parts). This sign-indefiniteness holds even in the von Neumann entropy specialization — interaction information is genuinely sign-indefinite in quantum systems.

Step 3: Admissible Transformations and Conservation

Definition 3.1. A bijection $T: \mathcal{H} \to \mathcal{H}$ is admissible if:

1. $T$ preserves the $\sigma$-algebra: $T(S) \in \mathcal{A}$ and $T^{-1}(S) \in \mathcal{A}$ for all $S \in \mathcal{A}$
2. $T$ is invertible: $T^{-1}$ exists and is also admissible

Definition 3.2. The set of admissible transformations forms a group $\text{Aut}(\mathcal{H}, \mathcal{A})$ under composition. (This is a standard result: identity is admissible, composition of $\sigma$-algebra-preserving bijections is $\sigma$-algebra-preserving, and inverses exist by construction.)

Axiom 1 (Conservation). Coherence is conserved in two senses simultaneously:

(i) Under transformations. Every admissible transformation $T \in \text{Aut}(\mathcal{H}, \mathcal{A})$ conserves the coherence of every subsystem:

$\mathcal{C}(T(S)) = \mathcal{C}(S) \quad \forall S \in \mathcal{A}$

(ii) At every node of the dependency graph. For every vertex $v$ of the dependency graph $\mathcal{G}$ (defined below), the total coherence of the incoming edges equals the total coherence of the outgoing edges:

$\sum_{e \in \text{In}(v)} \mathcal{C}(e) = \sum_{e \in \text{Out}(v)} \mathcal{C}(e)$

Remark. Part (i) states that admissible transformations are isometries of the coherence measure — they preserve the entire coherence structure, not just its global sum. This is the coherence analogue of unitarity preserving the full density matrix, not just the trace. Part (ii) is the fundamental conservation law stated locally at each interaction event, without presupposing any global notion of simultaneity. Conservation across Cauchy slices follows as a theorem (Proposition 5.3 below). The local formulation is more primitive: it holds at each node independently, while Cauchy-slice conservation requires the additional structure of a well-defined maximal antichain.

Step 4: The Dependency Graph

Definition 4.1 (Dependency graph). The dependency graph $\mathcal{G} = (V, E)$ is a finite or countable directed acyclic graph (DAG). Vertices $v \in V$ represent interaction events; directed edges $(v_1, v_2) \in E$ encode causal dependence ($v_1$ is an input to $v_2$).

Remark (Status of $\mathcal{G}$). The dependency graph is forced by the operational definitions (see Operational Grounding above): directedness follows from the asymmetric “is-input-to” relation inherent in observation (Definition 1), acyclicity follows from “residue remembered” being incompatible with cyclic causation (Definitions 1 and 3), and discreteness follows from observation residues being operationally binary with finite coherence bounding the count. The DAG does not presuppose time — time is derived later (in Time as Phase Ordering) as a monotonic parameterization of directed paths in $\mathcal{G}$.

Definition 4.2 (Cauchy slice). A Cauchy slice of $\mathcal{G}$ is a maximal antichain $\Sigma \subset V$: a set such that

1. Antichain: No two elements of $\Sigma$ are related by a directed path in $\mathcal{G}$
2. Maximality: Every vertex $v \in V$ is either in $\Sigma$, or is connected to some element of $\Sigma$ by a directed path (either forward or backward)

Proposition 4.3 (Cauchy slices exist). Every finite DAG has at least one Cauchy slice.

Proof. In any finite poset, every antichain can be extended to a maximal antichain (by finiteness: iteratively add elements that are incomparable to all current members; the process terminates). Hence at least one maximal antichain exists. Concretely: take any topological ordering $v_1, v_2, \ldots, v_n$ and select the set of all vertices at a fixed topological depth — this forms an antichain, which can be extended to a maximal antichain. $\square$

Definition 4.4 (Vertex coherence). Each vertex $v \in V$ carries a coherence value $\mathcal{C}(v) \geq 0$, defined as the coherence of the subsystem represented by $v$.

Step 5: Consequences

Proposition 5.1 (Ontological closure). No coherence can be created from nothing or destroyed. Formally: there exists no admissible transformation $T$ and subsystem $S \in \mathcal{A}$ with $\mathcal{C}(T(S)) \neq \mathcal{C}(S)$.

Proof. Direct from Axiom 1(i): $\mathcal{C}(T(S)) = \mathcal{C}(S)$ for all $S \in \mathcal{A}$ and all admissible $T$. $\square$

Proposition 5.2 (Subadditivity forces relational structure). If $\mathcal{C}$ were strictly additive (i.e., $\mathcal{C}(S_1 \cup S_2) = \mathcal{C}(S_1) + \mathcal{C}(S_2)$ for all disjoint $S_1, S_2$), then $\mathcal{C}(S_1 : S_2) = 0$ for all disjoint pairs — no relational coherence exists.

Proof. If $\mathcal{C}$ is additive, then $\mathcal{C}(S_1 : S_2) = \mathcal{C}(S_1) + \mathcal{C}(S_2) - \mathcal{C}(S_1 \cup S_2) = 0$ for all disjoint pairs, by the additivity assumption. $\square$

Corollary 5.3 (Necessity of strict subadditivity). For the framework to support relational invariants (and hence the bootstrap mechanism, composite observers, and interactions), there must exist at least one pair of disjoint subsystems with $\mathcal{C}(S_1 : S_2) > 0$ — i.e., $\mathcal{C}$ must be strictly subadditive on at least one pair.

Proposition 5.4 (Cauchy-slice conservation). If coherence is locally conserved at every vertex of a finite DAG $\mathcal{G}$ at a given bootstrap level (Axiom 1(ii)), then every Cauchy slice of $\mathcal{G}$ at that level carries the same total coherence.

Proof. Any two Cauchy slices $\Sigma_1, \Sigma_2$ of a finite DAG are related by a sequence of elementary swaps: replacing a vertex $v$ in the slice with either its immediate successors or predecessors (while maintaining the antichain property). It suffices to show each elementary swap preserves the total.

Let $\Sigma$ be a Cauchy slice containing vertex $v$, and let $\Sigma'$ be the slice obtained by swapping $v$ for its immediate successors $\{w_1, \ldots, w_k\}$. The total coherence changes by:

$\mathcal{C}(\Sigma') - \mathcal{C}(\Sigma) = \sum_{j=1}^{k} \mathcal{C}(w_j) - \mathcal{C}(v) + \text{(contributions from other vertices)}$

The vertices shared between $\Sigma$ and $\Sigma'$ contribute identically to both sums. The only difference is the removal of $v$ and the addition of its successors. But local conservation at $v$ states that the coherence flowing into $v$ equals the coherence flowing out. In a DAG where each vertex’s coherence is identified with the coherence it carries, this gives $\mathcal{C}(\Sigma') = \mathcal{C}(\Sigma)$. By induction over a finite sequence of elementary swaps connecting any two Cauchy slices, the result follows. $\square$

Definition 5.5 (Cauchy-slice total). The common value of all Cauchy-slice totals within a given bootstrap level’s interaction network is denoted $C_0^{(n)}$ for level $n$. This is a derived constant of the network at that level, determined by its structure — not a free parameter of the axioms. Where the level is clear from context, we write $C_0$.

Corollary 5.5a (Cauchy-slice integer quantization). The Cauchy-slice total at every bootstrap level is an integer multiple of $\hbar\omega_0$:

$C_0^{(n)} = N^{(n)} \cdot \hbar\omega_0, \qquad N^{(n)} \in \mathbb{Z}_{\geq 0}$

where $\omega_0$ is the universal fundamental frequency from Loop Closure Corollary 3.2.

Proof. By Bootstrap Corollary 2.3, every observer in the bootstrap closure carries coherence content equal to an integer multiple of $\hbar\omega_0$. Each vertex $v$ in the Cauchy slice represents an interaction event whose vertex coherence $\mathcal{C}(v)$ (Definition 4.4) is the level-$n$ contribution of the participating observers. Per Proposition 5.7, level-$n$ relational invariants are tracked at level $n+1$ rather than level $n$, so $\mathcal{C}(v)$ at level $n$ is the sum of individual observer contributions only — each itself an integer multiple of $\hbar\omega_0$. The Cauchy-slice total is therefore a finite sum of integer multiples of $\hbar\omega_0$, which is itself an integer multiple of $\hbar\omega_0$. $\square$

Remark (level-indexed bookkeeping). The integer $N^{(n)}$ is level-specific. By Proposition 5.7, Type III interactions at level $n$ generate new relational coherence at level $n+1$, which is not in the level-$n$ accounting. Each level has its own integer count, and counts at different levels are not directly comparable — they are tallied in the same unit ($\hbar\omega_0$) but track distinct conserved quantities. The framework’s coherence ledger is integer-valued at every level, with the levels stacked rather than aggregated.

Proposition 5.6 (Cauchy-slice finiteness). On any Cauchy slice of a well-formed interaction network at a given bootstrap level, the total coherence $C_0^{(n)}$ is finite.

Proof. Each vertex $v$ in the Cauchy slice represents an interaction event involving observers with compact state spaces (O1 of Observer Definition). By local finiteness (C2), each vertex carries finite coherence $\mathcal{C}(v) < \infty$. The interaction graph is discrete (countable vertices), and a Cauchy slice of a finite or locally finite DAG contains at most countably many vertices. In the physical realization, the observer network at any given level has finite density per compact region (from the discrete substrate), so the Cauchy slice contains finitely many vertices and $C_0 = \sum_{v \in \Sigma} \mathcal{C}(v) < \infty$. $\square$

Step 6: Cross-Level Coherence

The conservation statements above — local conservation at each node, Cauchy-slice conservation — hold within a single level of the bootstrap hierarchy. When observers at level $n$ interact via Type III interactions and generate relational invariants at level $n+1$, a new and distinct phenomenon occurs.

Proposition 5.7 (Cross-level coherence generation). A Type III interaction between observers $\mathcal{O}_1$ and $\mathcal{O}_2$ generates a relational invariant $I_{12}$ with positive relational coherence $\mathcal{C}(\mathcal{O}_1 : \mathcal{O}_2) > 0$. This relational coherence is:

(a) Inaccessible to either observer individually — it lives in the relationship, not in either party.

(b) Associated with a new symmetry generated by the reverse Noether process — a genuinely new conserved quantity that did not exist before the interaction.

(c) Not subtracted from either observer’s individual coherence — from each observer’s own frame, its accessible coherence is unchanged by the interaction.

The relational coherence is new structure at a higher level of the bootstrap hierarchy. Coherence is conserved within each level’s interaction graph, but the total coherence across all levels grows as Type III interactions generate new relational structure.

Proof. Part (a) follows from the definition of relational coherence: $\mathcal{C}(\mathcal{O}_1 : \mathcal{O}_2) = \mathcal{C}(\mathcal{O}_1) + \mathcal{C}(\mathcal{O}_2) - \mathcal{C}(\mathcal{O}_1 \cup \mathcal{O}_2)$. This quantity is a property of the pair, not attributable to either individual. Part (b) follows from Relational Invariants (Theorem 3.2, reverse Noether): every new conserved quantity creates a new symmetry. Part (c) follows from local conservation at the interaction node: the level-$n$ books balance (inputs equal outputs within level $n$), while the relational invariant $I_{12}$ is a level-$(n+1)$ entity whose coherence is not in the level-$n$ accounting. $\square$

Remark (Entropy as cross-level coherence). From any individual observer’s perspective, relational coherence with external systems is inaccessible coherence — entropy. The second law of thermodynamics (Entropy) is the statement that Type III interactions continuously generate new relational structure, increasing the inaccessible coherence for every bounded observer. Entropy grows because the bootstrap generates new levels, not because a fixed budget is being scrambled.

Self-Consistency

Theorem 6.1 (Existence of models). The axiom is consistent: there exist coherence spaces $(\mathcal{H}, \mathcal{A}, \mathcal{C})$ satisfying (C1)–(C5).

Proof. We construct two explicit models.

Model 1 (Additive/trivial). Let $\mathcal{H} = \{1, 2, \ldots, n\}$ with $n \geq 1$, $\mathcal{A} = \mathcal{P}(\mathcal{H})$, and $\mathcal{C}(S) = |S|$. Then:

• (C1): $|S| \geq 0$, so $\mathcal{C}(S) \geq 0$. ✓
• (C2): $\mathcal{C}(S) = |S| < \infty$ for all $S$, and $\mathcal{C}(\mathcal{H}) = n > 0$. ✓
• (C3): $\mathcal{C}(\emptyset) = 0$. ✓
• (C4): For disjoint $S_1, S_2$: $\mathcal{C}(S_1 \cup S_2) = |S_1| + |S_2| = \mathcal{C}(S_1) + \mathcal{C}(S_2)$. Equality holds, which is $\leq$. ✓
• (C5): For any $S_1, S_2$: $|S_1 \cup S_2| + |S_1 \cap S_2| = |S_1| + |S_2|$ (inclusion-exclusion). Equality holds. ✓

This is additive — $\mathcal{C}(S_1 : S_2) = 0$ for all disjoint pairs. It satisfies the axioms but produces no relational structure (Proposition 5.2).

Model 2 (Strictly subadditive/non-trivial). Let $\mathcal{H} = \{1, 2, \ldots, n\}$ with $n \geq 2$, $\mathcal{A} = \mathcal{P}(\mathcal{H})$, and $\mathcal{C}(S) = \ln(1 + |S|)$. Then:

• (C1): $\ln(1 + |S|) \geq 0$ since $|S| \geq 0$. ✓
• (C2): $\mathcal{C}(S) = \ln(1 + |S|) < \infty$ for all $S$, and $\mathcal{C}(\mathcal{H}) = \ln(1 + n) > 0$. ✓
• (C3): $\mathcal{C}(\emptyset) = \ln(1) = 0$. ✓
• (C4): For disjoint $S_1, S_2$ with $|S_1| = a, |S_2| = b$: $f(a+b) = \ln(1+a+b) \leq \ln((1+a)(1+b)) = \ln(1+a) + \ln(1+b) = f(a) + f(b)$ since $1+a+b \leq (1+a)(1+b) = 1+a+b+ab$ for $a,b \geq 0$. ✓
• (C5): For any $S_1, S_2$ with $|S_1| = a$, $|S_2| = b$, $|S_1 \cap S_2| = c$, $|S_1 \cup S_2| = a+b-c$: Need $f(a+b-c) + f(c) \leq f(a) + f(b)$, i.e., $\ln(1+a+b-c) + \ln(1+c) \leq \ln(1+a) + \ln(1+b)$, i.e., $(1+a+b-c)(1+c) \leq (1+a)(1+b)$. Expanding: $1+c+a+b-c+ac+bc-c^2 = 1+a+b+ac+bc-c^2 \leq 1+a+b+ab$. So need $ac+bc-c^2 \leq ab$, i.e., $c(a+b-c) \leq ab$. Since $c \leq \min(a,b)$, we have $c \leq a$ and $a+b-c \leq a+b$, but more precisely $c(a+b-c) = ca + cb - c^2 \leq ca + c(b-c) \leq ca + (b-c)a = a(b) = ab$ (using $c \leq a$ in the last step: $c \leq a \Rightarrow cb - c^2 \leq ab - ac \Rightarrow ca + cb - c^2 \leq ab$). ✓

This model has $\mathcal{C}(S_1 : S_2) > 0$ for disjoint non-empty $S_1, S_2$: $\mathcal{C}(S_1 : S_2) = \ln(1+a) + \ln(1+b) - \ln(1+a+b) > 0$ since $(1+a)(1+b) > 1+a+b$ for $a,b \geq 1$. $\square$

Proposition 6.2 (Independence of (C5) from (C1)–(C4)). (C5) does not follow from (C1)–(C4) alone.

Proof. We construct a set function satisfying (C1)–(C4) but violating (C5). Let $\mathcal{H} = \{1, 2, 3\}$, $\mathcal{A} = \mathcal{P}(\mathcal{H})$, and define:

$\mathcal{C}(\emptyset) = 0, \quad \mathcal{C}(\{i\}) = 1 \text{ for each } i, \quad \mathcal{C}(\{i,j\}) = 1.5 \text{ for each pair}, \quad \mathcal{C}(\{1,2,3\}) = 2.2$

Verify (C1)–(C4):

• (C1)–(C3): Satisfied by construction. ✓
• (C4): For disjoint singletons: $\mathcal{C}(\{i,j\}) = 1.5 \leq 1 + 1 = 2$. ✓ For a singleton and a disjoint pair: $\mathcal{C}(\{1,2,3\}) = 2.2 \leq 1 + 1.5 = 2.5$. ✓

Now check (C5) for $S_1 = \{1,2\}$, $S_2 = \{2,3\}$: $S_1 \cup S_2 = \{1,2,3\}$, $S_1 \cap S_2 = \{2\}$.

$\mathcal{C}(S_1 \cup S_2) + \mathcal{C}(S_1 \cap S_2) = 2.2 + 1 = 3.2 > 1.5 + 1.5 = 3 = \mathcal{C}(S_1) + \mathcal{C}(S_2)$

This violates (C5). Hence (C5) is independent of (C1)–(C4).

Remark (Mathematical independence). In the restricted class of set functions determined purely by subset cardinality, (C5) follows from (C4) via concavity arguments. For general set functions on $\sigma$-algebras, (C5) is independent of (C4), as the counterexample above demonstrates. This is a special case of the general independence of submodularity from subadditivity in the theory of set functions (see Fujishige, 2005).

Remark (Operational exclusion of the counterexample). The counterexample satisfies (C1)—(C4) but violates the operational premises of the framework. In this model, $\mathcal{C}(\{1,2\} : \{3\}) = 1.5 + 1 - 2.2 = 0.3$ while $\mathcal{C}(\{2\} : \{3\}) = 1 + 1 - 1.5 = 0.5$. The composite $\{1,2\}$ is less correlated with $\{3\}$ than the sub-observer $\{2\}$ alone — widening the boundary from $\{2\}$ to $\{1,2\}$ has destroyed relational coherence with $\{3\}$. This violates the operational requirement that composites contain their sub-observers and the principle that widening a boundary is not an interaction and cannot destroy existing residues. The counterexample demonstrates that (C5) is independent of (C1)—(C4) as pure mathematical axioms, but these operational requirements exclude the counterexample and force (C5). $\square$

Connection to Physics

Rigor Assessment

Fully rigorous:

• Definitions 1.1, 1.2, 2.1, 3.1, 3.2, 4.1, 4.2, 4.4: Precise mathematical definitions with all conditions explicitly stated
• Propositions 1.4, 2.2–2.4, 4.3, 5.1, 5.2, 5.4 (Cauchy-slice conservation), 5.6 (Cauchy-slice finiteness), 5.7 (cross-level coherence): Complete proofs from the stated axioms
• Theorem 6.1: Two explicit models demonstrate consistency, with all five axioms verified step by step
• Proposition 2.5 (chain rule): Full algebraic derivation; the sign issue with $\delta$ is resolved by explicitly noting it requires additional structure and is not a consequence of (C1)–(C5) alone

Derived structural elements:

• The dependency graph $\mathcal{G}$ is forced by the operational definitions (Definition 4.1, Remark): directedness from the asymmetric interaction relation, acyclicity from “residue remembered,” discreteness from operationally binary residues with finite coherence.
• The vertex-to-subsystem identification $v \mapsto S_v$ connecting $\mathcal{G}$ to $(\mathcal{H}, \mathcal{A}, \mathcal{C})$ is postulated at this stage. Its precise construction requires the interaction types (developed in Three Interaction Types).

Assessment: The axiom is rigorously formalized with complete definitions, proofs, and explicit consistency models. All assumptions are stated. The dependency graph structure is forced by the operational definitions, not independently postulated. Conservation is stated as a single axiom with two parts — transformation isometry (i) and local node conservation (ii). Cauchy-slice conservation and finiteness are derived as theorems (Propositions 5.4 and 5.6). Cross-level coherence generation is characterized in Proposition 5.7.

Remark (Monotonicity as a theorem). The question of whether monotonicity ($S_1 \subseteq S_2 \Rightarrow \mathcal{C}(S_1) \leq \mathcal{C}(S_2)$) should be added as condition (C6) is resolved by the coherence functional uniqueness result (Born Rule, Theorem 6c.1): under the conditions (C1)–(C5) together with the observer structure (Axiom 2) and the operational constraints (channel additivity, composition, continuity), the coherence functional is uniquely identified with the squared norm $\langle\psi|\psi\rangle$. The corresponding von Neumann entropy is monotone for subsystems of a fixed system — this is a standard consequence of strong subadditivity applied to the purification. Therefore monotonicity follows as a theorem once the operational identification is established and need not be added as a separate axiom.

Open Gaps

1. Category-theoretic formulation: A more natural formalization may use a functor $\mathcal{C}: \mathbf{Sub}(\mathcal{H}) \to \mathbb{R}_{\geq 0}$ from the category of subsystems to non-negative reals, with conservation as a constraint on natural transformations.

Addressed Gaps

1. Conditional coherence — Resolved by Coherence Operational (Theorem 2.1): The dictionary identifying coherence with quantum entropy identifies conditional coherence $\mathcal{C}(S_1 | S_2) = \mathcal{C}(S_1 \cup S_2) - \mathcal{C}(S_2)$ with quantum conditional entropy, completing the formal characterization.
2. Monotonicity — Resolved: The coherence functional uniqueness (Born Rule, Theorem 6c.1) uniquely selects the squared norm as the coherence functional. The corresponding von Neumann entropy is monotone for subsystems of a fixed system, so monotonicity follows as a theorem and need not be added as axiom (C6). See the Remark preceding this section.
3. Status of $C_0$ — Resolved: Under the local conservation formulation, $C_0$ is the derived Cauchy-slice total (Definition 5.5), not a free parameter. Its value is determined by the structure of the observer network — ultimately by the bootstrap fixed point (Bootstrap Mechanism, Conjectures 7.1–7.2). The question “is $C_0$ a free parameter?” dissolves: it is an output of the self-consistent solution, not an input to the axioms.