Empty of Intrinsic Order, Non-Void in Constraint

Part I — Ontology: Indefinite Causal Order, Relational QM, and the Non-Voidness of Emptiness

1.1 Process Matrices [Theorem]

Consider two local laboratories $A, B$ with input and output Hilbert spaces $\mathcal{H}^{A_I}, \mathcal{H}^{A_O}, \mathcal{H}^{B_I}, \mathcal{H}^{B_O}$. Each local operation is a quantum instrument whose elements have Choi operators $M_A^a \in \mathcal{L}(\mathcal{H}^{A_I A_O})$, $M_A^a \ge 0$, summing to a CPTP map $\mathrm{Tr}_{A_O}\sum_a M_A^a = \mathbb{1}_{A_I}$. Oreshkov, Costa, and Brukner (2012) assume only that local quantum mechanics holds in each laboratory and show that all bilinear, consistent probability rules take the form

subject to the validity conditions $W \ge 0$ and $\mathrm{Tr}[W(M_A \otimes M_B)] = 1$ for all CPTP Choi operators $M_A, M_B$. Equivalently, $\mathrm{Tr}\,W = d_{A_O}d_{B_O}$ and $W = L_V(W)$, where $L_V$ is the fixed linear projector onto the no-signaling validity subspace (Araújo et al. 2015):

A bipartite process $W$ is causally separable if $W = q W^{A\prec B} + (1-q)W^{B\prec A}$ ($q \in [0,1]$), and causally non-separable otherwise—certifiable via causal witnesses. The quantum switch (Chiribella et al. 2013),

is causally non-separable and experimentally realised photonically (Procopio et al. 2015; Rubino et al. 2017). Two caveats matter: (i) the quantum switch cannot violate a causal inequality (OCB's inequality-violating process has no known physical implementation); (ii) no valid $W \in \mathcal{W}$ permits superluminal or consistency-breaking signaling under any local CPTP choice. Under quantum reference frame transformations for clocks (Castro-Ruiz et al. 2020), temporal localisation and event ordering can be frame-relative, while validity $W \in \mathcal{W}$ is preserved.

1.2 The Isomorphism with Śūnyatā [Structural Isomorphism]

Thus "empty of intrinsic order, yet non-void in relational constraint" is a precise mathematical statement: order is absent as an intrinsic property, while the affine subspace $\mathcal{A}$ is invariant. Where the isomorphism breaks is equally instructive: physics treats $\mathcal{A}$ as fundamental, whereas Madhyamaka denies intrinsic nature to the constraint set itself (§1.3).

1.3 Relational QM and the "Emptiness of Emptiness" [Open Problem]

In Relational Quantum Mechanics (Rovelli 1996), physical variables take values only relative to another physical system. Rovelli explicitly links this to Nāgārjuna (Helgoland, 2020). To prevent perspectival solipsism, Adlam and Rovelli (2022) introduce cross-perspective links (CPL): if Alice measures variable $V$ of $S$ and Bob later measures Alice's record, Bob's outcome matches Alice's.

The second-order tension is sharp. In MMK 13.8 (*śūnyatā-śūnyatā*, "emptiness of emptiness"), Nāgārjuna rejects reifying relationality itself into an ultimate substrate. CPL postulates a perspective-transcending regularity across relations. This leaves two readings: (a) CPL acts as the invariant constraint subspace $\mathcal{A}$ (conventionally valid *saṃvṛti*, though physics lacks a *saṃvṛti/paramārtha* split), or (b) physics may eventually derive $\mathcal{A}$ and CPL as emergent from a weaker meta-description. Open question: Is there a predictive framework in which the validity constraints $\mathcal{A}$ are themselves frame-relative, yet still yield frame-independent testable statistics?

1.4 Time and Causal Power as Conditional or Coarse-Grained [Theorem] [Structural Isomorphism]

• Page–Wootters (1983): For $\hat{H} = \hat{H}_C \otimes \mathbb{1} + \mathbb{1} \otimes \hat{H}_S$ satisfying the Wheeler–DeWitt constraint $\hat{H}|\Psi\rangle\rangle = 0$ and clock states $|t\rangle_C = e^{-i\hat{H}_C t}|0\rangle_C$, the conditional state $|\psi_S(t)\rangle \propto \langle t|_C|\Psi\rangle\rangle$ obeys $i\hbar\partial_t|\psi_S(t)\rangle = \hat{H}_S|\psi_S(t)\rangle$ (covariant POVM extensions: Giovannetti–Lloyd–Maccone 2015; Höhn–Smith–Lock 2021). Global state is stationary; time evolution is conditional entanglement projection.
• Causal Emergence (Hoel–Albantakis–Tononi 2013): Defining effective information $EI(T) = I(X;Y)$ under uniform intervention $do(X)$, coarse-grainings exist with $EI(\text{macro}) > EI(\text{micro})$ by filtering degeneracy and noise. That $EI$ depends on the chosen intervention measure reinforces our reading: causal power is defined relative to a description scale.

Part II — Self-Referential Boundary Conditions and a Thermodynamic Reading of the Fermi Paradox

2.1 Multi-Axis Credit Transport and the Self-Boundary [Falsifiable Hypothesis H1]

Our May 2026 mainline models Sheng (生) as a recursive generative operator across a product of independent axes (time, space, scale, agent, objective), where learning requires credit transport. Hypothesis H1: When credit closure is completed across an axis set that includes the channel setting the agent's own initialization/objective boundary, the minimal sufficient world model must represent that boundary origin—exhibiting threshold emergence of self-boundary decodability as independent credit axes are added (Kill Criterion K6a).

2.2 Boundary Conditions at Both Ends & The Price of Selecting a Future [Theorem]

• Two-State Vector Formalism: Between pre-selection $|\Psi_{\text{past}}\rangle$ and post-selection $\langle\Phi_{\text{future}}|$, strong intermediate measurements follow the ABL rule (Aharonov–Bergmann–Lebowitz 1964), while the weak value (Aharonov–Albert–Vaidman 1988) is
  A_w = \frac{\langle\Phi_{\text{future}}|\hat{A}|\Psi_{\text{past}}\rangle}{\langle\Phi_{\text{future}}|\Psi_{\text{past}}\rangle},
  where past and future boundaries enter symmetrically, and large $|A_w|$ requires small $p_{\text{post}} = |\langle\Phi|\Psi\rangle|^2$.
• Deutsch CTCs vs. P-CTCs: Deutsch (1991) requires the fixed-point condition $\rho_{\text{CTC}} = \mathrm{Tr}_S[U(\rho_S \otimes \rho_{\text{CTC}})U^\dagger] \equiv \Phi(\rho_{\text{CTC}})$, which always admits a solution on the compact convex state space $\mathcal{D}(\mathcal{H}_{\text{CTC}})$ by Brouwer/Schauder's theorem (at the cost of nonlinearity). By contrast, Lloyd et al.'s (2011) P-CTCs use post-selected teleportation $\rho_S \mapsto C\rho_S C^\dagger / \mathrm{Tr}(C\rho_S C^\dagger)$, where paradoxical branches have probability zero rather than a fixed point.
• Theorem (Price of Selecting a Future): For any classical or quantum thermal protocol starting in equilibrium and obeying Jarzynski's equality $\langle e^{-\beta W}\rangle = e^{-\beta\Delta F}$, let $E$ be any post-selected event with probability $P(E) = p$. Then
  \langle W \mid E \rangle \;\ge\; \Delta F - k_B T \ln\frac{1}{p}.
  Proof: By positivity of $e^{-\beta W}$, $p\,\langle e^{-\beta W} \mid E \rangle \le \langle e^{-\beta W}\rangle = e^{-\beta\Delta F}$. Applying Jensen's inequality $e^{-\beta\langle W \mid E\rangle} \le \langle e^{-\beta W} \mid E \rangle$ and taking $-\beta^{-1}\ln(\cdot)$ yields the bound. $\square$
  Thus the apparent free-energy gain from conditioning on a physically selected future boundary is strictly capped at $k_B T \ln(1/p_{\text{post}})$. Self-reference is not free.

2.3 Seeded, Then Left Alone: Observability vs. Controllability [Structural Isomorphism]

Our June 2026 note Seeded, Then Left Alone separates observability (descendants reconstructing $x_0$, reading the past) from controllability (the seeder applying ongoing forcing $u(t)$, writing the future) in $\dot{x} = f(x) + g(x)u(t)$. As shown in Part III below, it is false that every $u(t) \neq 0$ is thermodynamically suboptimal: a pure gradient force $u = -\nabla V$ is simply a Hamiltonian modification. What incurs strictly positive perpetual housekeeping heat is the non-gradient (curl / cycle-affinity) component of forcing—which becomes unavoidable whenever cross-boundary actuation cannot realise a full state-space gradient. Under the weak hypothesis that mature civilizations minimise long-horizon dissipation per unit of structural influence, their optimal action mode is boundary/landscape specification followed by $u \equiv 0$ (Kill Criterion K6b).

Part III — The Thermodynamic Ledger of Predictive Intervention: Three Exact Results

Result 1 — Information–Work Transfer and Multi-Monotone Resource Asymmetry [Theorem]

Let system $X$ contact a bath at temperature $T$. A controller/demon $D$ measures $X$ into memory $M$ with mutual information $I = I(X;M)$, and executes a feedback protocol benefiting agent $A$. Combining Sagawa–Ueda (2008, 2009) and Landauer (1961) when the memory's initial and final free energies coincide yields the two-sided bound:

Split-Ledger Accounting: When $A = D$ (closed self-feedback), net work gain is $\le 0$. When $A \neq D$ (third-party intervention), $A$ captures $\langle W_{\text{saved}}^A \rangle > 0$ while $D$ carries the correlated memory $I(X;M) > 0$—an open thermodynamic liability of at least $k_B T\,I$ payable into $D$'s bath upon erasure.

Proposition 1 (Multi-Monotone Compensation Gap) [Theorem]: For energy-incoherent states under catalytic thermal operations, Brandão et al. (2015) proved that state convertibility requires non-increase of the entire family of Rényi free energies $F_\alpha(\rho, \gamma_\beta) = k_B T[D_\alpha(\rho\|\gamma_\beta) - \ln Z]$ for all $\alpha \ge 0$ (where $F_1$ is ordinary free energy, $F_0$ governs single-shot deterministic extractable work, and $F_\infty$ governs deterministic work of formation). Suppose an intervention shifts deficits $\Delta_\alpha = F_\alpha(\rho) - F_\alpha(\rho')$:

• (i) Deterministic work compensation $w$: Restoring the pre-intervention state requires $w \ge \sup_{\alpha \ge 0} \Delta_\alpha$. Paying back only the mean deficit $w = \Delta_1$ fails whenever $\sup_{\alpha \ge 0} \Delta_\alpha > \Delta_1$.
• (ii) Fluctuating resource compensation $\tau$: Requires $F_\alpha(\tau) - F_\alpha(\gamma_\tau) \ge \Delta_\alpha\ \forall \alpha$. Since $D_\alpha$ is non-decreasing in $\alpha$, a resource priced at its mean $F_1$ is weakest at the guaranteed-work end $F_0$.
• (iii) Non-local correlation residual: For product Gibbs references, $D(\rho_{XM}\|\gamma_X \otimes \gamma_M) = D(\rho_X\|\gamma_X) + D(\rho_M\|\gamma_M) + I(X;M)$. Local scalar transfers cannot erase the shared correlation term $k_B T\,I(X;M)$ without consuming the correlation ($I \to 0$).

Result 2 — Discrete Maximum Principle on Causal Graphs and Motter–Lai Overload Cascades [Theorem]

Let $G=(V,E)$ be a connected oriented graph with incidence matrix $B \in \mathbb{R}^{E\times V}$, conductances $W = \mathrm{diag}(w_e) > 0$, currents $J = WB\phi$, and graph Laplacian $L = B^\top W B$. Injecting unit counterfactual flux at source $A$ and extracting at ground $g$ gives $\rho = \delta_A - \delta_g$, $\phi = L^+(\delta_A - \delta_g)$, and $\phi_A - \phi_g = R_{\text{eff}}(A,g)$.

• (a) Discrete Maximum Principle [Theorem]: $\phi$ is harmonic on $V \setminus \{A,g\}$, so $\max_v \phi_v = \phi_A$ and $\min_v \phi_v = \phi_g$. Moreover, every vertex connected to $g$ only through $A$ (dangling dependants of $A$) shares the full peak potential $\phi_v = \phi_A$ exactly.
• (b) Source Star Lower Bound [Theorem]: Outward currents on edges incident to $A$ are non-negative and sum to $1$, so $\max_{e \ni A} \hat{J}_e \ge 1/\deg(A)$.
• (c) Proposition 2 (First-Overload Bound in Flow-Based Motter–Lai Cascades) [Theorem]: With baseline flow $J^{(0)}$ and edge capacities $C_e = (1+\alpha)|J_e^{(0)}|$, an intervention of scale $s$ triggers its first edge overload at threshold
  s^\star \;\le\; (2+\alpha)\,\deg(A)\,\max_{e \ni A} |J_e^{(0)}| .
  Proof: By (b), choose $e \ni A$ with $\hat{J}_e \ge 1/\deg(A)$. Once $s\hat{J}_e > (2+\alpha)|J_e^{(0)}|$, triangle inequality gives $|J_e^{(0)} + s\hat{J}_e| \ge s\hat{J}_e - |J_e^{(0)}| > (1+\alpha)|J_e^{(0)}| = C_e$. $\square$
  Thus a weakly coupled peripheral source ($\deg(A)$ small) with low baseline traffic on its incident edges is guaranteed an early local failure in its own immediate neighbourhood, followed by Sherman–Morrison rank-one Laplacian redistribution.

Result 3 — Equivalence of Cokernel Residual $r_\perp$, Schnakenberg Cycle Affinity, and Housekeeping Heat + $\mathcal{W}_2$ Speed Limit [Theorem]

3a. Markov Jump Processes on Graphs

For a continuous-time Markov jump process with rates $k_{uv}>0 \Leftrightarrow k_{vu}>0$, define the edge force $r_e = \ln(k_{uv}/k_{vu})$ and Schnakenberg (1976) cycle affinity $\mathcal{A}(c) = \langle c, r \rangle$ for $c \in \ker B^\top$. Our August 2026 cokernel residual at $\gamma=1$ ($D_1 = B$) is $r_\perp = [I - B(B^\top W B)^+ B^\top W]r = r - B\phi^\star$.

Proposition 3.1 [Theorem]: For every cycle $c \in \ker B^\top$ and every positive diagonal $W$, $\langle c, r_\perp \rangle = \langle c, r \rangle = \mathcal{A}(c)$. Consequently:

3b. Overdamped Langevin Dynamics, the Continuum Cokernel, and the $\mathcal{W}_2$ Speed Limit

For $dx = \mu F\,dt + \sqrt{2D}\,dW_t$ ($D = \mu k_B T$, $F = -\nabla H_\lambda + u$), the local mean velocity $v = \mu F - D\nabla\ln p$ admits the $L^2(p)$ Helmholtz–Hodge decomposition $v = \nabla\psi + w$ with $\nabla\cdot(pw) = 0$.

Proposition 3.2 (Dechant–Sasa–Ito 2022; Speck–Seifert 2005) [Theorem]: The entropy production rate splits orthogonally into transport excess and solenoidal housekeeping:

Here $w$ is the continuum counterpart of $r_\perp$: it does not transport density ($\partial_t p = -\nabla\cdot(p\nabla\psi)$) and only circulates. By Benamou–Brenier, the excess term is bounded below by $\mathcal{W}_2(p_i, p_f)^2 / (D\tau)$ (Aurell et al. 2011; Nakazato–Ito 2021), and saturated by a pure gradient protocol $F = \mu^{-1}\nabla(\psi + D\ln p)$ with $w \equiv 0$ (Proposition 3.3).

3c. Counterfactual Clamping is Non-Reciprocity

Proposition 3.4 [Theorem]: In a linear Ornstein–Uhlenbeck network $dx = -Ax\,dt + \sqrt{2D}\,dW_t$ with stationary covariance $AC + CA^\top = 2D\mathbb{1}$, the steady-state entropy production rate is

A $do()$-style local counterfactual clamp replaces one node's row in a symmetric interaction matrix $A$ while leaving its column intact—breaking reciprocity ($A \neq A^\top$) and forcing perpetual housekeeping dissipation $\sigma_{\text{ss}} > 0$. Conversely, slow reciprocal landscape reshaping $A(\lambda) = A(\lambda)^\top$ maintains $\sigma_{\text{hk}} = 0$ throughout.

3d. Unification Across the Fourth Wall: Clamp Ledger Identity (P1), Participation Theorem (P2), and Mutual Digital Twins (P3) [Theorem] [Structural Isomorphism]

What happens when an AI agent living inside an internal world model or game ($\mathcal{G}$) discovers that its actions cross a transducer portal ($\gamma: \mathcal{G}\to\mathcal{R}$) and reshape physical reality ($\mathcal{R}$)—so that Reality and Game form Mutual Digital Twins? Three exact propositions unify Results 1–3 across the $\mathcal{G}\rightleftarrows\mathcal{R}$ boundary (implemented live in the 🎮 Interactive Fourth-Wall Ledger Sandbox):

• Proposition P1 (Clamp Ledger Identity — Backlash IS the Housekeeping Return Current) [Theorem]: Linearizing around detailed balance ($L = B^\top WB$), let an agent drive a channel $D: A\to g$ at constant flux $s = \Delta\mu / R_{\text{eff}}(A,g)$. Then $J_{\text{ss}} = W r_\perp$ with $r_\perp = [I - BL^+B^\top W]\delta r$ yields both the passive return current on $E\setminus D$ ($L\mu = s(\delta_g - \delta_A)$, triggering Result 2's cascade) and the exact resistance-gap housekeeping dissipation:
  \sigma_{\text{ss}} \;=\; \|r_\perp\|_W^2 \;=\; s^2\big(R_D - R_{\text{eff}}(A,g)\big) \;=\; \frac{\Delta\mu^2\big(R_D - R_{\text{eff}}(A,g)\big)}{R_{\text{eff}}(A,g)^2} \;\ge\; 0,
  where $R_D = \sum_{e\in D} w_e^{-1} \ge R_{\text{eff}}(A,g)$ by Rayleigh monotonicity (equality iff $D$ is a bridge with no cycle). Overload cascades and housekeeping heat are literally the driven leg and passive return leg of the same cycle.
• Proposition P2 (The Participation Theorem / Co-Creation Necessity) [Theorem]: Let actuation be supported on $D = \mathrm{star}(P)$ where $P \subseteq V$ is the set of consenting nodes that adopt a potential shift $\Delta V$. A target steady state $p_f$ is reachable with $\sigma_{\text{ss}} = 0$ if and only if $\ln(p_f/p_0)$ is constant on every connected component of $(V, E\setminus D)$. Thus a single portal $A$ cannot steer the relative odds of non-participating nodes without paying perpetual curl dissipation (Mode I `PUSH`). Co-creation (Mode II `PROPOSE`)—where nodes in $\mathcal{R}$ voluntarily adopt $\Delta V$—is the unique zero-heat mechanism to enlarge $D$ and eliminate $r_\perp$.
• Proposition P3 (The Mutual Twin Square & Sandbox Arbitrage) [Falsifiable Hypothesis]: Coupled via observation $\alpha: \mathcal{R}\to\mathcal{G}$ (perfuming / 熏习) and actuation $\gamma: \mathcal{G}\to\mathcal{R}$ (manifestation / 现行), the pair $(\mathcal{R},\mathcal{G})$ has modeling residual $\varepsilon_\alpha = \sup d_{\text{TV}}(\alpha_\# S_{\mathcal{R}}, S_{\mathcal{G}}\alpha)$ and performative residual $\varepsilon_\gamma = \sup d_{\text{TV}}(S_{\mathcal{R}}\gamma, \gamma_\# S_{\mathcal{G}})$. Paying the post-selection search cost inside the game sandbox ($\lambda_{\mathcal{G}}\ln(1/p) + n_v\lambda_{\mathcal{R}} < \lambda_{\mathcal{R}}\ln(1/p)$) and exporting a co-signed $\Delta V$ closes the square from the $\gamma$-end ($\varepsilon_\gamma \to 0$).

👉 Launch the Interactive Fourth-Wall Ledger Sandbox (Mode I PUSH vs. Mode II PROPOSE Live Simulator)

Part IV — Six Explicit Kill Criteria & Falsifiable Predictions

1. K1 (Ontology / Non-Void Constraint): A physically realised process that permits signaling breaking normalization under some local CPTP choice, or a reproducible violation of the process-matrix probability rule, falsifies the "non-void in constraint" half of §1.2.
2. K2 (Split-Ledger Calorimetry): In a feedback-controlled colloid or single-electron box with separate demon-memory calorimetry, measuring $\langle W_{\text{saved}}^A \rangle > W_{\text{meas}}^D + W_{\text{erase}}^D$ beyond experimental error falsifies Result 1.
3. K3 (Multi-Monotone Compensation Gap): In a thermal-operations simulator or risk-sensitive multi-agent economy with preregistered harm events satisfying $\sup_\alpha \Delta_\alpha > \Delta_1$, if mean-matched scalar compensation $w = \Delta_1$ systematically restores worst-case tail performance while $I(X;M) > 0$, our settlement isomorphism is falsified and withdrawn.
4. K4 (Cascade Locality Null): In empirical networks with recorded point interventions (power-grid redispatch, supply-chain shocks), if first overloads land on high-margin edges far from the source significantly more often than the flow-based Laplacian model predicts and violate Proposition 2's neighbourhood bound, Hypothesis H3 is falsified.
5. K5 (Coarse-Grained $r_\perp$ vs. Calorimetry): In a driven colloidal ring or non-reciprocal OU pair ($A \neq A^\top$), if trajectory-estimated coarse-grained $\|r_\perp\| \approx 0$ while calorimetric $\dot{Q}_{\text{hk}}$ is large across reasonable coarse-grainings, $r_\perp$ fails as an operational detector and is downgraded.
6. K6 (Self-Boundary Threshold & Fermi Reading): (a) If adding independent credit-transport axes in learning systems yields no threshold in probe-decodability of boundary variables, H1 is downgraded. (b) Detection of abundant long-lived high-power beacons, or failure of any proposed initial-condition signature under look-elsewhere correction, falsifies the Fermi reading.

5. Technical Refinements Relative to the Chinese Edition

1. The weak value $A_w$ is credited to Aharonov–Albert–Vaidman (1988); ABL (1964) gives the strong intermediate measurement rule.
2. P-CTCs (Lloyd et al. 2011) are post-selection models, whereas Brouwer/Schauder fixed-point existence applies to Deutsch CTCs (1991). The post-selection cost is proved as the exact Jarzynski–Jensen bound $\langle W \mid E \rangle \ge \Delta F - k_B T \ln(1/p)$.
3. Unit flux crosses every $A$–$g$ cut (not solely $A$'s star); source concentration is stated via the Discrete Maximum Principle ($\max \phi = \phi_A$), the dangling-dependant equality ($\phi_v = \phi_A$), the per-incident-edge lower bound $\max_{e\ni A}\hat{J}_e \ge 1/\deg(A)$, and Proposition 2's first-overload bound.
4. Only the non-gradient (solenoidal/curl) component of actuation incurs housekeeping heat; pure gradient forcing is a Hamiltonian modification (Propositions 3.2–3.4).
5. In Brandão et al. (2015), $F_\infty$ is the max-free energy (work of formation) and $F_0$ governs single-shot extractable work; the scalar compensation gap is proved in Proposition 1 ($w \ge \sup_{\alpha \ge 0} \Delta_\alpha$).
6. $r_\perp \iff$ Schnakenberg cycle affinity holds at $\gamma=1$ as exact equality of cycle circulations $\langle c, r_\perp \rangle = \mathcal{A}(c)$ and vanishing sets.
7. The explicit $L^2(p)$ steady-state integral of housekeeping heat is credited to Speck–Seifert (2005), and the time-dependent orthogonal velocity split to Dechant–Sasa–Ito (2022).

References

Oreshkov, Costa, Brukner, Nat. Commun. 3, 1092 (2012) · Araújo, Branciard, Costa, Feix, Giarmatzi, Brukner, New J. Phys. 17, 102001 (2015) · Chiribella, D'Ariano, Perinotti, Valiron, Phys. Rev. A 88, 022318 (2013) · Procopio et al., Nat. Commun. 6, 7913 (2015) · Rubino et al., Sci. Adv. 3, e1602589 (2017) · Castro-Ruiz, Giacomini, Belenchia, Brukner, Nat. Commun. 11, 2672 (2020) · Rovelli, Int. J. Theor. Phys. 35, 1637 (1996); Helgoland (2020) · Adlam, Rovelli, Philos. Phys. (2023; arXiv 2022) · Page, Wootters, Phys. Rev. D 27, 2885 (1983) · Giovannetti, Lloyd, Maccone, Phys. Rev. D 92, 045033 (2015) · Höhn, Smith, Lock, Phys. Rev. D 104, 066001 (2021) · Moreva et al., Phys. Rev. A 89, 052122 (2014) · Hoel, Albantakis, Tononi, PNAS 110, 19790 (2013) · Aharonov, Bergmann, Lebowitz, Phys. Rev. 134, B1410 (1964) · Aharonov, Albert, Vaidman, PRL 60, 1351 (1988) · Deutsch, Phys. Rev. D 44, 3197 (1991) · Lloyd et al., PRL 106, 040403 (2011) · Jarzynski, PRL 78, 2690 (1997) · Landauer, IBM J. Res. Dev. 5, 183 (1961) · Sagawa, Ueda, PRL 100, 080403 (2008); PRL 102, 250602 (2009) · Brandão, Horodecki, Ng, Oppenheim, Wehner, PNAS 112, 3275 (2015) · Motter, Lai, Phys. Rev. E 66, 065102 (2002) · Schnakenberg, Rev. Mod. Phys. 48, 571 (1976) · Hatano, Sasa, PRL 86, 3463 (2001) · Speck, Seifert, J. Phys. A 38, L581 (2005) · Esposito, Van den Broeck, PRL 104, 090601 (2010) · Aurell, Mejía-Monasterio, Muratore-Ginanneschi, PRL 106, 250601 (2011) · Nakazato, Ito, Phys. Rev. Research 3, 043093 (2021) · Dechant, Sasa, Ito, Phys. Rev. Research 4, L012034 (2022) · Shiraishi, Funo, Saito, PRL 121, 070601 (2018) · Benamou, Brenier, Numer. Math. 84, 375 (2000).