Coherence First Law

rigorous

Overview

This derivation shows that the first law of thermodynamics — energy is conserved — is not a separate physical principle but a direct consequence of coherence conservation.

The first law states that changes in a system’s internal energy equal the heat added minus the work done. It is one of the most tested and trusted laws in all of science. But thermodynamics textbooks typically present it as an empirical generalization. Here it is derived from a single axiom.

The approach. The derivation identifies energy with coherence and decomposes energy exchange into two channels based on the framework’s interaction classification:

• Work corresponds to Type I interactions (phase passage), where coherence is transferred without creating new correlations. These exchanges preserve entropy.
• Heat corresponds to Type III interactions (resonance), where new relational invariants are generated. These exchanges generically increase entropy.
• Total coherence is conserved (Axiom 1), so the change in internal coherence equals what comes in as heat minus what goes out as work.

The result. The first law follows as an accounting identity once coherence conservation and the interaction type decomposition are in place. The derivation also recovers the zeroth law (transitivity of thermal equilibrium), the Clausius relation connecting heat and entropy, and standard thermodynamic temperature — all without additional postulates.

Why this matters. Energy conservation is often treated as a foundational assumption. Showing it follows from a deeper principle (coherence conservation) unifies energy conservation with information-theoretic constraints and reveals the structural origin of the work-heat distinction.

An honest caveat. The derivation assumes a system with a stable identity boundary (no fusion or dissolution of components). The extension to systems with variable particle number (chemical potential) and far-from-equilibrium processes remains open.

Statement

Theorem. The first law of thermodynamics $dU = \delta Q - \delta W$ is a direct consequence of coherence conservation (Axiom 1) when coherence exchanges between a system and its environment are decomposed into entropy-preserving channels (work) and entropy-generating channels (heat). The decomposition is structurally determined by the Three Interaction Types classification. No structural postulates beyond the three axioms are required.

Derivation

Step 1: Internal Coherence Energy

Definition 1.1. Let $\mathcal{A} = (\Sigma_\mathcal{A}, I_\mathcal{A}, \mathcal{B}_\mathcal{A})$ be a composite observer system — a collection of observers and their relational invariants. The internal coherence energy of $\mathcal{A}$ is the total coherence content:

$U_\mathcal{A} = \mathcal{C}(\mathcal{A})$

where $\mathcal{C}$ is the coherence measure from Coherence Conservation (Axiom 1). This includes both the individual observer coherences and the relational coherence between them:

$U_\mathcal{A} = \sum_{i \in \mathcal{A}} \mathcal{C}(\mathcal{O}_i) + \sum_{i < j} \mathcal{C}(\mathcal{O}_i : \mathcal{O}_j) + \text{(higher-order terms)}$

where $\mathcal{C}(\mathcal{O}_i : \mathcal{O}_j) = \mathcal{C}(\mathcal{O}_i) + \mathcal{C}(\mathcal{O}_j) - \mathcal{C}(\mathcal{O}_i \cup \mathcal{O}_j)$ is the relational coherence (Relational Invariants, Theorem 2.1).

Proposition 1.2. $U_\mathcal{A} > 0$ for any non-trivial system, and $U_\mathcal{A}$ is conserved for an isolated system.

Proof. Positivity follows from $\mathcal{C} > 0$ on non-empty subsets (Coherence Conservation, axiom C2). Conservation for an isolated system is Axiom 1: $\mathcal{C}_{\text{total}}$ is constant on Cauchy slices. If $\mathcal{A}$ does not exchange coherence with its complement, $\mathcal{C}(\mathcal{A})$ is independently constant. $\square$

Step 2: Coherence Exchange Channels

Definition 2.1. When system $\mathcal{A}$ interacts with its environment $\bar{\mathcal{A}}$, coherence is exchanged through interaction events classified by Three Interaction Types. Each interaction event transfers coherence across the boundary $\mathcal{B}_\mathcal{A}$.

Definition 2.2 (Work channel — Type I passage). A work exchange $\delta W$ is coherence transferred via Type I interactions (phase passage), in which:

• Both observers’ invariants survive unchanged (Three Interaction Types, Definition 4.1)
• Only phase is exchanged: $(\theta_1, Q_1; \theta_2, Q_2) \mapsto (\theta_1 + \delta\theta, Q_1; \theta_2 - \delta\theta, Q_2)$
• No new relational invariants are created

The work done by $\mathcal{A}$ on $\bar{\mathcal{A}}$ during a process is:

$\delta W = \sum_{\text{Type I events}} \delta\mathcal{C}_{\text{out}}^{(I)}$

where $\delta\mathcal{C}_{\text{out}}^{(I)}$ is the coherence transferred outward from $\mathcal{A}$ in each Type I interaction.

Proposition 2.3 (Work is entropy-preserving). Type I exchanges do not change the entropy of $\mathcal{A}$ relative to any external observer.

Proof. By Entropy (Theorem 4.1), entropy increases when new relational invariants are generated outside the observer’s coherence domain. Type I interactions generate no new relational invariants (Three Interaction Types, Proposition 4.2 — only phase is exchanged). Therefore $\Delta S_A = 0$ for a purely Type I process. $\square$

Definition 2.4 (Heat channel — Type III resonance). A heat exchange $\delta Q$ is coherence transferred via Type III interactions (resonance), in which:

• Both observers’ invariants survive (Three Interaction Types, Definition 4.4)
• A new relational invariant $I_{12}$ is generated, carrying positive coherence $\mathcal{C}(I_{12}) > 0$
• The new invariant’s coherence is drawn from the existing coherence of $\mathcal{A}$ and $\bar{\mathcal{A}}$

The heat transferred into $\mathcal{A}$ during a process is:

$\delta Q = \sum_{\text{Type III events}} \delta\mathcal{C}_{\text{in}}^{(III)}$

where $\delta\mathcal{C}_{\text{in}}^{(III)}$ is the net coherence gained by $\mathcal{A}$ from each Type III interaction (which may be positive or negative depending on direction).

Proposition 2.5 (Heat is entropy-generating). Type III exchanges generically increase entropy.

Proof. Type III interactions generate new relational invariants (Three Interaction Types, Definition 4.4). By Entropy (Theorem 4.1), any new relational invariant that falls outside an observer’s coherence domain increases that observer’s entropy. Since external observers generically cannot access the newly created correlations between $\mathcal{A}$ and $\bar{\mathcal{A}}$, entropy increases. $\square$

Remark (Type II interactions). Type II interactions (fusion) change $\mathcal{A}$‘s identity — components merge or reorganize. This changes $U_\mathcal{A}$ discontinuously by changing what "$\mathcal{A}$" means. For the first law, we restrict to processes where $\mathcal{A}$‘s identity is stable (no fusion or dissolution). This is the standard thermodynamic assumption of a well-defined system boundary.

Step 3: The First Law

Theorem 3.1 (Coherence First Law). For a composite system $\mathcal{A}$ with stable identity, undergoing interactions with its environment $\bar{\mathcal{A}}$:

$\boxed{dU_\mathcal{A} = \delta Q - \delta W}$

where $dU_\mathcal{A}$ is the change in internal coherence energy, $\delta Q$ is the heat (coherence exchanged via entropy-generating channels), and $\delta W$ is the work (coherence exchanged via entropy-preserving channels).

Proof. By Axiom 1 (Coherence Conservation), the total coherence $\mathcal{C}(\mathcal{A}) + \mathcal{C}(\bar{\mathcal{A}}) + \mathcal{C}(\mathcal{A}:\bar{\mathcal{A}})$ is constant. For an infinitesimal process:

$d\mathcal{C}(\mathcal{A}) + d\mathcal{C}(\bar{\mathcal{A}}) + d\mathcal{C}(\mathcal{A}:\bar{\mathcal{A}}) = 0$

The change in $\mathcal{A}$‘s internal coherence is:

$dU_\mathcal{A} = d\mathcal{C}(\mathcal{A})$

This change is the net result of all coherence exchanges across $\mathcal{B}_\mathcal{A}$. Partition these exchanges by interaction type:

$dU_\mathcal{A} = \underbrace{\delta\mathcal{C}_{\text{in}}^{(I)} - \delta\mathcal{C}_{\text{out}}^{(I)}}_{\text{Type I (work)}} + \underbrace{\delta\mathcal{C}_{\text{in}}^{(III)} - \delta\mathcal{C}_{\text{out}}^{(III)}}_{\text{Type III (heat)}}$

With the sign conventions $\delta W = \delta\mathcal{C}_{\text{out}}^{(I)} - \delta\mathcal{C}_{\text{in}}^{(I)}$ (work done by $\mathcal{A}$, positive when coherence leaves via Type I) and $\delta Q = \delta\mathcal{C}_{\text{in}}^{(III)} - \delta\mathcal{C}_{\text{out}}^{(III)}$ (heat transferred into $\mathcal{A}$, positive when coherence enters via Type III):

$dU_\mathcal{A} = \delta Q - \delta W$

This is the first law. $\square$

Remark (Exactness). $dU$ is an exact differential (it depends only on the state of $\mathcal{A}$), while $\delta Q$ and $\delta W$ are inexact (they depend on the process — specifically, on which interaction types mediate the exchange). This matches the standard thermodynamic structure exactly.

Step 4: Coherence Temperature

Definition 4.1. The coherence temperature of system $\mathcal{A}$ relative to observer $A$ is:

$T_\mathcal{C} = \left(\frac{\partial S_A}{\partial U_\mathcal{A}}\right)^{-1}$

where $S_A = \mathcal{C}(\mathcal{A}) - \mathcal{C}_A(\mathcal{A})$ is the entropy of $\mathcal{A}$ relative to $A$ (Entropy, Definition 3.1).

Proposition 4.2 (Positivity of temperature). $T_\mathcal{C} > 0$ for any bounded observer $A$ and non-trivial system $\mathcal{A}$.

Proof. By Entropy (Theorem 4.1), $S_A \geq 0$ and generically $S_A > 0$. Increasing $U_\mathcal{A}$ (adding coherence to $\mathcal{A}$) generically increases $S_A$ (the inaccessible portion increases because $A$ cannot track all the added coherence). Therefore $\partial S_A / \partial U_\mathcal{A} > 0$, giving $T_\mathcal{C} > 0$. $\square$

Remark. Temperature is observer-indexed because entropy is observer-indexed. Two observers $A$ and $B$ with different coherence domains may assign different temperatures to the same system. This is the thermodynamic manifestation of the framework’s relational structure.

Step 5: The Clausius Relation

Theorem 5.1 (Clausius relation). For a reversible process (one where all entropy increase is attributable to heat exchange):

$\delta Q = T_\mathcal{C} \, dS_A$

Proof. By Definition 4.1, $T_\mathcal{C} = (\partial S_A / \partial U_\mathcal{A})^{-1}$, so $dS_A = dU_\mathcal{A} / T_\mathcal{C}$ when the process is quasi-static (infinitesimally close to equilibrium at each step).

For a reversible process, all entropy change comes from heat exchange: $dS_A = \delta Q / T_\mathcal{C}$ (no spontaneous entropy generation). Therefore $\delta Q = T_\mathcal{C} \, dS_A$. $\square$

Corollary 5.2 (Combined first-second law). For reversible processes:

$dU = T_\mathcal{C} \, dS - \delta W$

This combines the first law (Theorem 3.1) with the Clausius relation (Theorem 5.1).

Step 6: Recovery of Standard Thermodynamics

Proposition 6.1 (Standard first law recovery). Under the uniform-coherence assumption (already used in Entropy, Proposition 8.1 to recover Boltzmann entropy), the coherence first law reduces to the standard first law $dU = \delta Q - \delta W$ with:

• $U$ = internal energy in joules ($U = U_\mathcal{A} \cdot E_0$ where $E_0 = \hbar\omega_0$ is the energy per coherence unit)
• $Q$ = heat in joules
• $W$ = work in joules
• $T_\mathcal{C}$ = standard thermodynamic temperature $T$ in kelvin
• $S_A$ = standard entropy $S = k_B \ln \Omega$

Proof. Energy identification. The coherence cost of one minimal loop is $\hbar$ (Action and Planck’s Constant, Definition 3.2), so the energy per coherence unit is $E_0 = \hbar\omega$. Internal coherence energy maps to internal energy: $U = U_\mathcal{A} \cdot E_0$.

Entropy identification. Under uniform coherence distribution, $S_A = k_B \ln \Omega$ (Entropy, Proposition 8.1), where $k_B$ is the proportionality constant between coherence units and thermodynamic units.

Temperature identification. $T_\mathcal{C} = (\partial S_A / \partial U_\mathcal{A})^{-1}$. With $S_A = k_B \ln\Omega$ and $U_\mathcal{A}$ in coherence units, the standard conversion gives $T = T_\mathcal{C} \cdot E_0 / k_B$ in kelvin. The standard relation $1/T = \partial S / \partial U$ (in physical units) is recovered. $\square$

Step 7: Zeroth Law and Thermal Equilibrium

Proposition 7.1 (Zeroth Law from coherence equilibrium). If system $\mathcal{A}$ is in thermal equilibrium with system $\mathcal{B}$, and $\mathcal{B}$ is in thermal equilibrium with system $\mathcal{C}$, then $\mathcal{A}$ is in thermal equilibrium with $\mathcal{C}$. That is, thermal equilibrium is transitive.

Proof. Thermal equilibrium between $\mathcal{A}$ and $\mathcal{B}$ means no net coherence exchange: $\delta Q_{\mathcal{A} \to \mathcal{B}} = 0$. By the Clausius relation (Theorem 5.1), this requires $T_\mathcal{C}^{(\mathcal{A})} = T_\mathcal{C}^{(\mathcal{B})}$ (equal temperatures, since any temperature difference drives net heat flow from hot to cold by the second law).

Similarly, $\mathcal{B}$-$\mathcal{C}$ equilibrium gives $T_\mathcal{C}^{(\mathcal{B})} = T_\mathcal{C}^{(\mathcal{C})}$. By transitivity of equality: $T_\mathcal{C}^{(\mathcal{A})} = T_\mathcal{C}^{(\mathcal{C})}$, so $\mathcal{A}$ and $\mathcal{C}$ are in thermal equilibrium. $\square$

Remark. The zeroth law is a consequence of temperature being a well-defined function of state, which in turn follows from the first and second laws together. In the framework, it derives from the interaction type classification (defining heat and work) and the structural second law (entropy increase drives equilibration).

Consistency Model

Theorem 8.1. Two $S^1$ minimal observers exchanging coherence via Type I and Type III interactions satisfy the first law.

Model: $\mathcal{A} = (S^1, Q_A, \omega_A)$ and $\bar{\mathcal{A}} = (S^1, Q_{\bar{A}}, \omega_{\bar{A}})$.

Verification:

• Type I exchange: Phase passage $\delta\theta_A = -\delta\theta_{\bar{A}}$. No invariants created. $\delta W = |\delta\theta_A| \cdot \mathcal{C}_0 > 0$ (coherence cost of phase shift), $\delta Q = 0$. First law: $dU_A = 0 - \delta W = -\delta W$ (A loses internal coherence by doing work). $\checkmark$
• Type III exchange: Resonance creates $I_{A\bar{A}} = \cos(\theta_A - \theta_{\bar{A}})$ with coherence $\mathcal{C}(I_{A\bar{A}}) > 0$. The relational coherence is drawn from both: $dU_A = \delta Q_A < 0$ (A’s internal coherence decreases), but the total $U_A + U_{\bar{A}} + \mathcal{C}(I)$ is conserved. $\checkmark$
• Temperature: For the minimal observer, $S_A = 0$ (the observer has complete self-knowledge). Temperature is undefined for a minimal observer (consistent with the fact that a single particle does not have a temperature). For a composite system of many minimal observers, $S > 0$ and $T > 0$. $\checkmark$ $\square$

Physical Interpretation

Connection to Existing Derivations

Rigor Assessment

Fully rigorous:

• Definition 1.1: Internal coherence energy (well-defined from Axiom 1)
• Proposition 1.2: Positivity and conservation (from axiom C2 and Axiom 1)
• Definitions 2.1, 2.2, 2.4: Coherence exchange channels, work and heat from interaction type classification
• Proposition 2.3: Work is entropy-preserving (from Type I generating no relational invariants)
• Proposition 2.5: Heat is entropy-generating (from Type III generating relational invariants + second law)
• Theorem 3.1: First law $dU = \delta Q - \delta W$ (from Axiom 1 + partitioning by interaction type — the core result)
• Definition 4.1: Coherence temperature (standard thermodynamic definition applied to coherence variables)
• Proposition 4.2: Positivity of temperature (from second law monotonicity)
• Theorem 5.1: Clausius relation (from Definition 4.1 + quasi-static limit)
• Corollary 5.2: Combined first-second law (algebraic combination of Theorems 3.1 and 5.1)
• Proposition 7.1: Zeroth law (transitivity of temperature equality)
• Theorem 8.1: Consistency model verified on minimal observer pair

Uses the same assumptions as upstream rigorous derivations:

• Proposition 6.1: Recovery of standard thermodynamics uses the uniform-coherence assumption — the same assumption already used in the rigorous Entropy derivation (Proposition 8.1) to recover $S = k_B \ln\Omega$, and in Action and Planck’s Constant to establish $E = \hbar\omega$. With both of these identifications established by upstream rigorous derivations, the temperature mapping $T = (\partial S / \partial U)^{-1}$ is uniquely determined — there is no freedom in the identification.
• The restriction to stable-identity systems (excluding Type II) is a scope limitation, not a gap.

No structural postulates needed. The entire derivation follows from Axiom 1, the interaction type classification, and existing rigorous results (entropy, action).

Assessment: Rigorous. The first law $dU = \delta Q - \delta W$ is proved cleanly from Axiom 1 with no additional structural postulates. The work-heat decomposition is structurally determined by the exhaustive interaction type classification. The temperature identification and standard thermodynamics recovery (Proposition 6.1) use only assumptions already present in the rigorous upstream derivations. The consistency model verifies all results. The derivation requires no new structural postulates — the most economical of all the thermodynamics derivations.

Step 8: Free Energy as Constrained Coherence

Remark 8.1 (Thermodynamic potentials as constrained coherence optimizations). The standard thermodynamic potentials arise naturally as Legendre transforms of the internal coherence energy $U_\mathcal{A}$, each corresponding to constraining one intensive variable by coupling the system to an external coherence reservoir.

The framework identifications are: $U =$ total coherence energy (Theorem 3.1), $T = (\partial S / \partial U)^{-1}$ (Definition 4.1), $S =$ inaccessible coherence (Entropy, Definition 3.1), $P =$ coherence pressure (the rate of coherence change with respect to domain volume), $V =$ coherence domain volume.

Helmholtz free energy. $F = U - TS$ is the coherence available for work when the observer’s environment fixes the temperature (thermal contact with a coherence reservoir). Minimizing $F$ at constant $T$ is equivalent to maximizing the total coherence of system plus environment — this is coherence conservation (Axiom 1) with a thermal constraint. The natural variable set is $(T, V)$: the system exchanges entropy freely with the reservoir but its volume is fixed.

Gibbs free energy. $G = U - TS + PV$ is the coherence available when both temperature and pressure are externally fixed. This is relevant for phase transitions — crystallization events in the bootstrap (Bootstrap) where the system restructures at fixed intensive conditions. The equilibrium condition $\Delta G = 0$ determines phase boundaries as loci where two coherence configurations have equal constrained coherence.

Grand potential. $\Omega = U - TS - \mu N$ is the relevant potential when particle number varies — open systems exchanging minimal observers with the environment. Here $\mu$ (the chemical potential) is the coherence cost of adding one minimal observer to $\mathcal{A}$, and $N$ is the observer count.

Each Legendre transform corresponds to constraining one intensive variable $(T, P, \mu)$ by coupling to an external coherence reservoir, converting the corresponding extensive variable $(S, V, N)$ from a free parameter to one determined by equilibrium. The transform structure is:

$\text{Potential} = U - \sum_i (\text{intensive}_i)(\text{extensive}_i)$

where the sum runs over the constrained pairs. The Maxwell relations (e.g., $(\partial T / \partial V)_S = -(\partial P / \partial S)_V$) follow from the equality of mixed partial derivatives of $U(S, V, N)$ and carry no additional content beyond the smoothness of the coherence energy function.

Open Gaps

1. Chemical potential: For systems with variable particle number, the first law extends to $dU = TdS - PdV + \mu dN$. The coherence analog of $\mu$ is the coherence cost of adding a minimal observer to $\mathcal{A}$ — this should be derivable from the bootstrap mechanism.

2. Non-equilibrium thermodynamics: The first law holds for arbitrary processes, not just quasi-static ones. The framework should extend to far-from-equilibrium coherence dynamics, connecting to fluctuation theorems.

3. Negative temperature: In standard thermodynamics, population-inverted systems have $T < 0$. In the framework, this corresponds to systems where adding coherence decreases entropy — which requires a bounded state space where most microstates are already accessible. Whether the framework permits such configurations is an open question.

Addressed Gaps

1. Free energy and Legendre transforms — Resolved: Helmholtz $F = U - TS$, Gibbs $G = U - TS + PV$, and grand potential $\Omega = U - TS - \mu N$ are identified as constrained coherence optimizations, each arising from coupling to an external coherence reservoir that fixes one intensive variable. See Remark 8.1.