Coherence Conservation

rigorous

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 amount of coherence (you cannot have negative existence).
• The total coherence across all of reality is fixed — it never increases or decreases.
• 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 holds across every “slice” of the causal dependency graph — a time-free statement that does not presuppose a clock.

The result. Coherence behaves like a conserved currency of existence. Observers can redistribute it through interactions, but the total ledger always balances. When two systems share coherence beyond what simple addition would predict, that excess is “relational coherence” — the irreducible connection between them.

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.

Statement

Axiom 1 (Coherence Conservation). There exists a primitive quantity $\mathcal{C}$, called coherence, defined on the partitions of a coherence space $\mathcal{H}$. The total coherence is globally conserved: it is invariant under all admissible transformations and constant across all slices of the dependency graph. Coherence cannot be created, destroyed, or exported — the ontology is closed.

Formalization

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 axioms (C1)–(C5) below

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

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

(C2) Normalization: $\mathcal{C}(\mathcal{H}) = C_0$ for a fixed constant $C_0 > 0$.

(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.

In the physical regime where $\mathcal{C}$ specializes to von Neumann entropy, $\delta$ equals the conditional mutual information $I(S_2; S_3 | S_1) \geq 0$, guaranteed by strong subadditivity. In the general framework, the sign of $\delta$ carries physical meaning: $\delta > 0$ when $S_2$ and $S_3$ redundantly encode their relationship with $S_1$; $\delta < 0$ when they synergistically encode it.

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) Across Cauchy slices. For every Cauchy slice $\Sigma$ of the dependency graph $\mathcal{G}$ (defined below):

$\mathcal{C}(\Sigma) \equiv \sum_{v \in \Sigma} \mathcal{C}(v) = C_0$

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) extends conservation to Cauchy slices that are not related by any global automorphism, analogous to energy conservation holding across all spacelike slices rather than just those related by Poincaré transformations. Neither part implies the other in general: (i) guarantees slice conservation only between automorphism-related slices, while (ii) asserts it universally.

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) that is postulated as part of the framework’s structure. 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 $\mathcal{G}$ is a structural postulate — it encodes the logical ordering of interactions. It is not derived from the coherence space alone, and it does not presuppose time. Time is derived later (in Time as Phase Ordering) as a monotonic parameterization of directed paths in $\mathcal{G}$. The existence of $\mathcal{G}$ is a co-axiom with the coherence space: together, $(\mathcal{H}, \mathcal{A}, \mathcal{C}, \mathcal{G})$ constitute the foundational structure.

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.

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 $\mathcal{C}(S) = |S|/n \cdot C_0$. Then:

• (C1): $|S| \geq 0$, so $\mathcal{C}(S) \geq 0$. ✓
• (C2): $\mathcal{C}(\mathcal{H}) = n/n \cdot C_0 = C_0$. ✓
• (C3): $\mathcal{C}(\emptyset) = 0/n \cdot C_0 = 0$. ✓
• (C4): For disjoint $S_1, S_2$: $\mathcal{C}(S_1 \cup S_2) = (|S_1| + |S_2|)/n \cdot C_0 = \mathcal{C}(S_1) + \mathcal{C}(S_2)$. 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 $\mathcal{C}(S) = f(|S|) \cdot C_0 / f(n)$ where $f(k) = \ln(1 + k)$. Then:

• (C1): $\ln(1 + |S|) \geq 0$ since $|S| \geq 0$. ✓
• (C2): $\mathcal{C}(\mathcal{H}) = f(n) \cdot C_0 / f(n) = C_0$. ✓
• (C3): $\mathcal{C}(\emptyset) = f(0) \cdot C_0/f(n) = \ln(1) \cdot C_0/f(n) = 0$. ✓
• (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 $S_1, S_2$: $\mathcal{C}(S_1 : S_2) = [f(a) + f(b) - f(a+b)] \cdot C_0/f(n) > 0$ since $\ln(1+a) + \ln(1+b) > \ln(1+a+b)$ 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. 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). $\square$

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: 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

Structural assumptions (co-axioms, not derived):

• The dependency graph $\mathcal{G}$ is a structural postulate (Definition 4.1, Remark). This is not a gap — it is an acknowledged foundational element alongside the coherence space.
• The vertex-to-subsystem identification $v \mapsto S_v$ connecting $\mathcal{G}$ to $(\mathcal{H}, \mathcal{A}, \mathcal{C})$ is postulated. 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 main structural postulate (the dependency graph) is acknowledged as a co-axiom. Conservation is stated as a single axiom with two parts — transformation isometry (i) and Cauchy slice invariance (ii) — with their logical independence noted in the remark following the axiom statement.

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 axiom (C6) is resolved by the operational identification developed in Coherence Operational. Theorem 4.1 of that derivation establishes a uniqueness result: under the axioms (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 von Neumann entropy $S(\rho) = -\text{tr}(\rho \ln \rho)$. 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. The question of whether non-monotone coherence measures are physically relevant is moot: the uniqueness theorem selects the von Neumann entropy uniquely, and von Neumann entropy is monotone.

Open Gaps

1. Uniqueness of $C_0$: Is the total coherence $C_0$ a free parameter or fixed by self-consistency? Without loss of generality, one may normalize $C_0 = 1$ for the abstract theory; the physical value of $C_0$ (if any) would be determined by the bootstrap fixed point.
2. 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 operational identification in Coherence Operational (Theorem 4.1) uniquely selects von Neumann entropy as the coherence functional. 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.