Maxwell's Demon in a High-Dimensional Universe: Coherence Collapse and the Limits of Landauer Erasure

BioSystems Maxwell’s Demon in a High-Dimensional Universe Coherence Collapse and the Limits of Landauer Erasure --Manuscript Draft-- Manuscript Number: Article Type: Full Length Article Section/Category: Keywords: Maxwell's demon; Landauer principle; high-dimensional dynamics; coherence; dimensional collapse; information thermodynamics; sub-Landauer erasure; quantum biology Corresponding Author: Ian Todd University of Sydney AUSTRALIA First Author: Ian Todd Order of Authors: Ian Todd Abstract: We generalize Landauer’s principle to high-dimensional dynamical substrates where information is embodied as coherence geometry rather than discrete bits. In this setting, measurement and erasure are dimensional collapses—coarse-grained projections from a high-dimensional manifold onto a lower-dimensional attractor. We derive acoherence-corrected bound Eerase, reducing to kBT ln2for binary contraction. The demon’s ”memory” is internal correlation (an attractor) within the same field, so information acquisition is coherence formation and erasure is decoherence. Because collapse in one subspace can be offset by expansion in another, local apparent costs can fall below kBT ln2 while the global second law is preserved. This informationgeometric view unifies Maxwell’s demon, Landauer’s limit, and biological/analog computation, and yields testable scaling with lost dimensionality and coherence. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Ian Todd Sydney Medical School University of Sydney Sydney, NSW, Australia ito[email protected].edu.au October 10, 2025 Dr. Abir Igamberdiev Editor-in-Chief BioSystems Dear Dr. Igamberdiev, Please find enclosed our manuscript, “Maxwell’s Demon in a High-Dimensional Universe: Coherence Collapse and the Limits of Landauer Erasure,” submitted to BioSystems as a Regular Article (Theory and Modelling). This work generalizes Landauer’s principle to high-dimensional coherence dynamics by treating erasure as dimensional collapse. We derive a coherence-corrected bound Eerase ≥kBTeff lnV(Deff ) V(D′) that reduces to kBTln 2 for binary systems and explains how apparent sub-Landauer erasure arises locally through dimensional redistribution while respecting global thermodynamic constraints. We reinterpret Maxwell’s demon as an internal attractor, resolving the external-observer paradox. The framework yields testable predictions for dimensional scaling, coherence dependence, and metabolic efficiency. A numerical appendix validates the key claims; simulation code is embedded and replicable with NumPy. This paper is conceptually distinct from my recent BioSystems article on falsifiability limits, providing the thermodynamic foundations and geometric bound that complement the earlier epistemological perspective. The manuscript is original and not under review elsewhere. During preparation I used Claude (Anthropic) for drafting, with ChatGPT and Grok for verification. All results were validated by me. I have no competing interests. Sincerely, Ian Todd Sydney Medical School University of Sydney Cover Letter Maxwell’s Demon in a High-Dimensional Universe: Coherence Collapse and the Limits of Landauer Erasure Ian Todd Sydney Medical School University of Sydney Sydney, NSW, Australia [email protected] October 10, 2025 Abstract We generalize Landauer’s principle to high-dimensional dynamical substrates where information is embodied as coherence geometry rather than discrete bits. In this setting, measurement and erasure are dimensional collapses—coarse-grained projections from a high-dimensional manifold onto a lower-dimensional attractor. We derive a coherence-corrected bound Eerase ≥kBTeff ln V(Deff)/V (D′), reducing to kBTln 2 for binary contraction. The demon’s ”memory” is internal correlation (an attractor) within the same field, so information acquisition is coherence formation and erasure is decoherence. Because collapse in one subspace can be offset by expansion in another, local apparent costs can fall below kBTln 2 while the global second law is preserved. This information-geometric view unifies Maxwell’s demon, Landauer’s limit, and biological/analog computation, and yields testable scaling with lost dimensionality and coherence. 1 Manuscript File Click here to view linked References Keywords: Maxwell’s demon, Landauer principle, high-dimensional dynamics, coherence, dimensional collapse, information thermodynamics, sub-Landauer erasure, quantum biology 1 Introduction Maxwell’s demon, conceived in 1867 as a thought experiment challenging the second law of thermodynamics, continues to illuminate the deep connection between information and physical entropy [1, 2]. The demon—a hypothetical entity capable of observing individual molecular velocities—operates a trapdoor between two chambers, selectively allowing fast molecules to pass one way and slow molecules the other, thereby creating a temperature difference without performing work. This apparent violation of the second law troubled physicists until Landauer’s seminal insight [3]: the demon must eventually erase its memory to operate cyclically, and this erasure necessitates energy dissipation of at least kBTln 2 per bit, restoring thermodynamic consistency. Landauer’s principle has proven remarkably robust, validated experimentally [4, 5] and extended to quantum [6, 7, 8, 9] and relativistic [10] domains. The framework has been further developed through feedback control [11, 12] and autonomous demon implementations [13], with comprehensive reviews establishing information thermodynamics as a mature field [14]. Yet its standard formulation embeds assumptions that become problematic for complex biological and physical systems operating in high-dimensional phase spaces with coherence dynamics. Specifically: 1. Discrete state space: The bit model assumes well-defined, separable binary states rather than continuous manifolds. 2. Low-dimensional phase space: Conventional treatments neglect correlations and hidden degrees of freedom that dominate biological information processing. 2 3. Thermal equilibrium: The standard derivation presumes a uniform heat bath at temperature T, inadequate for systems with structured stochastic resonance or farfrom-equilibrium dynamics. 4. External demon: The demon is treated as separate from the system it observes, ignoring self-referential dynamics where observation and evolution are coupled. Recent work has established that biological systems routinely operate near fundamental measurement limits [17], with causal patterns existing below the Landauer threshold where they resist binary measurement yet remain functionally decisive through collective effects. This sub-Landauer domain—populated by ephaptic coupling [18, 19], weak synaptic noise [20], and quantum-like coherences [21, 22]—exhibits structured nondeterminism that classical information thermodynamics cannot adequately describe. Here we develop a generalized framework treating information erasure as dimensional collapse in high-dimensional coherence fields. This concept can be understood by analogy to principal component analysis in machine learning: just as PCA projects high-dimensional data onto lower-dimensional principal components while preserving maximal variance, measurement projects high-dimensional states onto lower-dimensional observables, with associated information loss and energetic cost. However, unlike PCA which is purely mathematical, physical dimensional collapse is an irreversible thermodynamic process governed by the generalized bounds we derive. This reformulation: •Extends Landauer’s bound to continuous phase-space manifolds •Reinterprets Maxwell’s demon as internal symmetry breaking •Predicts sub-Landauer erasure regimes through coherence redistribution •Unifies measurement, computation, and thermodynamics under information geometry The framework connects naturally to recent advances in understanding biological computation [23], where intelligence emerges from maintaining high-dimensional coherence in 3 regimes where temporal fine-structure becomes fundamentally inaccessible to measurement. By recognizing that the demon cannot remain external to the coherence field it exploits, we resolve longstanding tensions between information theory and thermodynamics while revealing new computational possibilities in the sub-Landauer domain. 2 Classical Formulation and Its Limitations 2.1 Standard Landauer Bound In Landauer’s standard form [3, 15], Eerase ≥kBTln 2,(1) erasure is modeled as contraction from two equiprobable microstates to one. The underlying assumptions are: •Discrete state space: Information exists as well-defined, separable bits. •Low-dimensional phase space: No hidden correlations or coupled degrees of freedom. •Thermal equilibrium: A uniform bath at temperature T. •Deterministic erasure: One-to-one mapping from initial microstates to final state. The derivation proceeds through the Szilard engine [16]: Consider a single-molecule ideal gas confined to a box of volume Vat temperature T. Step 1 - Measurement: Insert a partition at the midpoint, determining which half contains the molecule (one bit of information). This step can be reversible in principle. Step 2 - Extraction: Attach a weight to the partition and allow isothermal expansion. The molecule pushes the partition, performing work W=kBTln 2 as it expands from V/2 to V. 4 Step 3 - Erasure: To operate cyclically, the demon must erase its memory of which side the molecule occupied. Removing the partition and allowing the molecule to equilibrate over the full volume increases entropy by ∆S=kBln 2, requiring minimum heat dissipation Q≥kBTln 2 to the environment. The net work extractable is Wnet =kBTln 2 −kBTln 2 = 0, restoring thermodynamic consistency. The engine converts thermal energy to work only by utilizing pre-existing information; the erasure step ensures no net violation of the second law. 2.2 Limitations for Biological Systems These assumptions fail for systems governed by continuous, high-dimensional coherence dynamics—neuronal fields, biochemical reaction networks, or optical coherence systems— where information is encoded as phase alignment, not symbol count [17, 23]. Consider neural population coding. A network of Ncoupled oscillators can represent states through their collective phase configuration ϕ= (ϕ1, . . . , ϕN). The ”information” is not discrete bits but continuous phase relationships encoding sensory inputs, motor plans, or internal models. Measuring this state to extract a binary decision requires projecting the high-dimensional manifold onto a low-dimensional observable, destroying the phase relationships that constitute the representation [24, 25]. Similarly, protein folding navigates rugged energy landscapes with astronomical numbers of conformational states [26]. The folding trajectory encodes information about the native state through transient intermediates and parallel pathways, but measuring the path collapses it to a binary outcome (folded/unfolded), erasing the dimensional structure that enables efficient exploration. In photosynthetic energy transfer, quantum coherence enables near-unity efficiency by allowing excitons to simultaneously sample multiple pathways [21, 27]. The coherent superposition exists at energies ∼10−22 J, well below the Landauer limit. Measurement sufficient to determine the specific pathway destroys the coherence and reduces efficiency—yet the 5 coherence causally determines the outcome. 2.3 The External Demon Problem Standard Maxwell’s demon formulations treat the demon as external to the system—a separate agent with its own memory, performing measurements that leave the system unperturbed while extracting information. This separation becomes untenable in high-dimensional coherence systems where: 1. The demon’s ”memory” must be encoded in the same physical substrate as the system. 2. Measurement is not passive observation but active projection that alters system dynamics. 3. The demon-system boundary is arbitrary; both are subsystems of a larger coherence field. 4. Feedback between demon and system creates self-referential dynamics where the distinction dissolves. As we will show, the demon is more accurately understood as an internal attractor—a low-entropy submanifold of the system’s phase space that selectively amplifies correlations. The demon doesn’t observe from outside; it emerges from within as spontaneous symmetry breaking in the coherence field itself. 3 High-Dimensional Dynamics and Coherence Fields 3.1 Notation and Assumptions Before developing the generalized framework, we establish key definitions and operational assumptions: 6 Coarse-grained phase-space volume: We work with a coarse-grained description at resolution ∆xon the slow manifold of collective modes. The volume V(Deff) represents the number of distinguishable configurations, not the fine-grained Liouville measure. Dimensional collapse is a many-to-one logical map realized through coupling to a bath; the underlying Hamiltonian microevolution preserves fine-grained volume per Liouville’s theorem. Geometric volume scaling: For effective dimensionality Deff and resolution ∆x, we assume the accessible volume scales as a product measure over effective modes: V∼ (∆L/∆x)Deff where ∆Lis the typical amplitude scale. This holds for weakly coupled modes or effective ellipsoidal geometry in principal-component coordinates. Real biological manifolds may exhibit more complex scaling (e.g., fractal dimensions), but the product measure provides a conservative lower bound on dimensional entropy; departures strengthen rather than weaken our bounds. Effective dimensionality: We define Deff via the participation ratio of the correlation eigenspectrum: Deff =1 Pip2 i (2) where piare normalized eigenvalues of the phase correlation matrix. At fixed resolution ∆x, the coarse-grained volume scales as V∝(∆x)−Deff (assuming a product measure or effective ellipsoidal geometry), justifying ln(Vpre/Vpost)∝Deff −D′. Effective temperature: We define Teff operationally via a fluctuation-dissipation relation on the collective mode at frequency ω0: kBTeff ≡Sη(ω0) 2γeff (3) where Sηis the noise power spectral density and γeff is the corresponding dissipative response. In thermal equilibrium, Teff =T. For driven systems with structured noise, Teff can deviate from the bath temperature. 7 chronized configuration where phases lock: ϕdemon(t)≈f[ϕsystem(t)] (21) for some functional relationship f. The demon’s degrees of freedom become entrained to the system’s, reducing the total effective dimensionality: Dcorrelated eff < Dsystem +Ddemon (22) This dimensional reduction IS the information acquisition. There is no separate ”memory”— the memory is the constrained phase-space volume. 5.3 Erasure as Decoherence Erasure corresponds to breaking the correlations—allowing the demon’s subsystem to decohere and regain independence: I(Xsystem;Xdemon)→0 (23) In dynamical terms, this means disrupting the phase-locking relationship, allowing the subsystems to explore their full individual phase spaces: Duncorrelated eff →Dsystem +Ddemon (24) The dimensional expansion requires energy injection (or entropy increase) according to the generalized Landauer bound derived above. The demon cannot escape this cost because it is part of the system, not external to it. 14 5.4 The Demon as Attractor We can formalize the demon as a stable attractor Ademon ⊂ Mtotal with basin of attraction B. The system naturally flows toward this attractor: dx dt =−∇E(x) + η(t) (25) where the energy landscape E(x) has a minimum at Ademon. The demon’s ”action” is then spontaneous symmetry breaking—the system’s natural tendency to minimize energy by forming coherence structures. No external agent is needed; the demon emerges from the field’s internal dynamics. Information extraction occurs when the system reaches the attractor: x(t→ ∞)∈ Ademon (26) and erasure occurs when noise or external perturbation kicks the system out of the basin: x(t)/∈ B =⇒loss of correlation (27) The thermodynamic cost is the energy required to destabilize the attractor or refill its basin—exactly what the generalized Landauer bound predicts. 6 Apparent Sub-Landauer under Dimensional Redistribution 6.1 Coherence Redistribution In high-dimensional systems, apparent violations of the Landauer bound can occur locally when coherence is redistributed rather than destroyed. Consider a system with multiple 15 frequency modes: Φ(x, t) = X k Akei(k·x−ωkt+ϕk)(28) ”Erasure” in one mode can correspond to transfer to another mode rather than dissipation: Ak→0, Ak′→Ak′+ ∆A(29) The total coherence (integrated across modes) remains constant: X k|Ak|2= const (30) This is dimensional conservation: degrees of freedom are not eliminated but reconfigured. 6.2 Sub-Landauer Apparent Erasure When measured locally at mode or subsystem α, the apparent erasure energy can fall below kBTln 2 because the lost information is not thermalized but transferred elsewhere in the field: E(α) erase =kBT(α) eff ln Vpre α Vpost α< kBTln 2 (31) Crucially, this is an apparent local violation only. The total energy dissipated (and actual thermodynamic cost) includes both the local collapse and the compensating expansion elsewhere. For each subsystem undergoing collapse, the entropy change is negative: ∆Sα=kBln Vpost α Vpre α<0 (32) Globally across all modes and subsystems, including the environment, entropy is conserved or increased: X α ∆Sα+ ∆Senv ≥0 (33) The apparent sub-Landauer collapse in one subspace must be offset by anti-collapse 16 (expansion) in other subspaces or heat dissipation to the environment. Rewriting in terms of energy costs (where positive terms reflect dissipation compensating collapse): X α kBT(α) eff ln Vpre α Vpost α≥0 (34) where the sum runs over all subsystems α. This satisfies the second law globally while permitting local apparent violations. Reconciliation with information thermodynamics: This framework is fully compatible with the Sagawa-Ueda generalized second law for systems with feedback [11, 12]. Their work shows that information gain Ireduces the minimum work required: ⟨W⟩ ≥ ∆F−kBTI (35) In our framework, coherence formation IS information gain, where the coarse-grained mutual information satisfies I≈ln Vcorr/Vuncorr ≈ln V(Deff)/V (D′), and dimensional redistribution IS the mechanism by which this information reduces local work. The key difference: rather than an external feedback controller, the ”demon” is an internal attractor that spontaneously forms correlations. The dimensional accounting makes explicit what Sagawa-Ueda treat as mutual information—both frameworks respect the global second law while enabling local apparent sub-Landauer costs through correlation structure. This explains how biological systems achieve apparent sub-Landauer erasure [17]: they operate in dimensional conservation mode, recycling correlation structure across coupled oscillators rather than dissipating it to heat. The lost degrees of freedom in one subsystem are reconfigured into another, not destroyed. The actual total cost respects Landauer; the apparent local cost can fall below it. 17 6.3 Structured Nondeterminism When the underlying dynamics exhibit structured nondeterminism—phase-locked noise, stochastic resonance [31], or quantum-like coherences—the effective erasure temperature Teff can differ from the physical temperature T. For systems driven by correlated noise: ⟨ηi(t)ηj(t′)⟩= Γijδ(t−t′) (36) with correlation matrix Γij, the effective temperature scales with the noise strength modulated by the dissipation: kBTeff ∝λmax(Γ) γeff (37) where λmax is the largest eigenvalue of the noise correlation and γeff is the effective damping on the collective mode. When noise is anti-correlated or shows temporal structure, this can yield Teff < T, enabling sub-thermal apparent erasure at the local level. This connects to the sub-Landauer framework [17, 23]: patterns exist precisely because they exploit structured noise below the thermal floor, maintaining coherence through stochastic resonance rather than deterministic control. 6.4 Concrete Example: Coupled Bistable Oscillators To illustrate the mechanism concretely with explicit global entropy accounting, consider a toy model: Toy model: Two overdamped bistable oscillators (x, y) with coupled potential: U(x, y;λ) = U0(x;λ)+U0(y; 0) + K 2(x−y)2 18 where U0(z;λ) = 1 4z4−z2−λz is a double-well potential, and the dynamics are: ˙x=−∂U ∂x +ηx(t) ˙y=−∂U ∂y +ηy(t) with independent Gaussian noise ⟨ηi(t)ηj(t′)⟩= 2Dδijδ(t−t′). For strong coupling K≫1, the slow manifold satisfies x≈ywith effective dimensionality Deff ≈1 (one collective mode). Define coherence as the correlation coefficient ρ= cov(x, y)/pvar(x)var(y)∈[0,1]; as Kincreases, ρ→1. Erasure protocol: Project xonto sign(x) (left vs right well). Local entropy: For uncoupled oscillators (K= 0), each has two wells of equal volume, so erasure costs: ∆Suncoupled x=−kBln 2 =⇒Eerase =kBTln 2 With coupling (K≫1), the constraint x≈yreduces the accessible volume pre-erasure. The effective volume contraction becomes: ln Vpre x Vpost x≈ln(2ρ) with ρ<1 giving apparent local cost: Eapparent erase =kBTln(2ρ)< kBTln 2 Global accounting: The ”missing” entropy kBln(1/ρ) appears in two places: 1. Transverse mode expansion: Collapsing xcreates fluctuations in the anti-phase mode (x−y). The constraint relaxation allows ∆S⊥≈kBln(1/r)> 19 0. 2. Heat to bath: The coupling force −K(x−y) does work during collapse, dissipating energy ∼K⟨(x−y)2⟩to the thermal bath, contributing entropy ∆Sbath ∼kB(Ework/T)>0. The global entropy balance satisfies: ∆Stotal = ∆Sx+ ∆S⊥+ ∆Sbath =−kBln 2 + kBln(1/r)+kBln(1/r)≥0 for r≤1/√2, with equality approached in the quasistatic limit. This exemplifies dimensional redistribution: coherence reduces the apparent local cost (actual information removed is less than one full bit due to constraints), while expansion elsewhere restores global thermodynamic consistency. The local observer sees sub-Landauer erasure; the global accounting respects the second law. Appendix A provides numerical validation via Langevin simulation. 6.5 Observable Regimes We can distinguish three regimes: 1. Super-Landauer (Eerase ≫kBTln 2): High-dimensional collapse, irreversible thermalization, classical erasure. 2. Near-Landauer (Eerase ∼kBTln 2): Bit-level operations, standard computing, equilibrium thermodynamics applies. 3. Sub-Landauer (Eerase < kBTln 2): Coherence redistribution, dimensional conservation, structured nondeterminism. Local erasure enabled by global conservation. Biological systems operate primarily in the sub-Landauer regime [17], exploiting coherence redistribution for efficient computation while paying full Landauer cost only at final 20 measurement collapse [23]. 7 Implications and Observable Predictions 7.1 Biological Information Processing This framework explains several puzzling features of biological computation: Neural efficiency: The brain operates at ∼20 W while performing computations that would require megawatts in silicon [33]. Sub-Landauer erasure through coherence redistribution explains this efficiency: neural populations maintain oscillatory coherence that encodes information without continuous write-erase cycles [34, 35], collapsing only at decision points. Working memory, for instance, operates through sustained oscillatory activity patterns that represent information in high-dimensional neural state spaces without requiring discrete storage operations at each moment. Analog advantage: Biological systems favor analog over digital computation. This makes sense: analog coherence dynamics naturally operate in the sub-Landauer regime, while digital requires discrete writes at full Landauer cost. Collective phenomena: Ephaptic coupling [18], weak synaptic noise [20], and population coherence all operate below individual detection thresholds yet causally influence outcomes. These are sub-Landauer patterns redistributed across populations. 7.2 Quantum Biology The framework provides a thermodynamic foundation for quantum effects in biology. Quantum coherence in photosynthesis [21, 27] exists at ∼10−22 J, order-of-magnitude below the Landauer limit. While the role of quantum coherence in biological function remains actively debated [22]—with critics arguing thermal decoherence should dominate at physiological temperatures—recent experimental and theoretical advances continue to demonstrate coherence effects persisting longer than classical models predict [27]. Emerging work suggests 21 quantum effects may persist through environmental structuring and protective mechanisms [36, 37], where the biological milieu itself modulates decoherence timescales to enable functional quantum dynamics. In our framework, quantum coherence is maintained precisely because it operates below the measurement threshold—observation would inject energy that destroys the coherence. The exciton explores multiple pathways simultaneously through dimensional superposition, collapsing only at the reaction center. The full Landauer cost is paid only at the final charge separation, not during coherence propagation. Whether thermal decoherence dominates depends on the ratio of decoherence time to functional timescale; in optimized biological systems, the latter can be sufficiently short to exploit coherence dynamics [22], with recent evidence suggesting quantum effects extend beyond photosynthesis to olfaction, enzyme catalysis, and potentially neural processes [38]. The role of quantum-like coherence states in biological organization and information processing represents a fundamental challenge to classical reductionist approaches [39]. 7.3 Experimental Predictions The framework makes testable predictions: Prediction 1 - Dimensional Scaling: Erasure energy should scale with lost dimensionality, not bit count. Systems with higher Deff require proportionally more energy to collapse to the same D′. Testable in artificial photonic systems or coupled oscillator networks by measuring heat dissipation during dimensional projection. Prediction 2 - Coherence Dependence: Erasure cost should decrease with increasing coherence ρ. Highly synchronized systems have lower effective dimensionality, reducing the dimensional gap that must be bridged during collapse. Testable in neural populations by correlating spike-time coherence with metabolic markers (via calcium imaging or PET). 22 Prediction 3 - Sub-Landauer Detection: Neural populations should detect signals with Esignal ≪kBTln 2 through temporal integration. Detection thresholds fall below Landauer limit when temporal integration windows exceed ∼100ms (testable via psychophysical experiments with controlled noise). Stochastic resonance in neural systems provides a mechanism for such detection [32, 20]. Recent proposals for quantum effects in neural processing [36] suggest additional mechanisms for sub-threshold signal detection, though experimental validation remains an active research frontier. Prediction 4 - Collapse Signatures: Sudden dimensional reduction events should be detectable in neural recordings, correlated with decision/output events. These should show characteristic energy dissipation matching Landauer cost for bits written. Prediction 5 - Metabolic Efficiency: Brain regions with higher Deff should show better operations-per-watt efficiency (measurable via combined fMRI/PET imaging). 7.4 Hierarchical Resonance and Biological Implementation Biological systems implement coherence computation through nested oscillatory hierarchies [40]. Neural population oscillations span delta (0.5-4 Hz), theta (4-8 Hz), alpha (8-13 Hz), beta (13-30 Hz), and gamma (30-100 Hz) bands, with high-frequency oscillations extending to 500 Hz in specialized contexts. These rhythms organize through cross-frequency coupling, where slow oscillations modulate the amplitude and phase of faster rhythms. Cross-frequency coupling binds information across timescales without requiring measurement at individual levels. The computational capacity scales with the product of participating frequency bands and their effective dimensionalities: Dtotal ≈ Nlevels Y n=1 D(n) eff (38) This architecture enables exponential computational capacity through temporal multiplexing rather than spatial scaling, all maintained in the unmeasurable regime until strategic 23 4. Dimensional conservation: High-dimensional systems operate in a regime where information can be transferred across modes without thermalization, enabling apparent sub-Landauer erasure locally while respecting thermodynamics globally. This framework unifies Maxwell’s demon, Landauer’s principle, and coherence thermodynamics under information geometry. It explains biological computational efficiency, provides thermodynamic foundations for quantum biology, and suggests new paradigms for computing based on dimensional conservation rather than bit manipulation. The deep insight: bits are projections, not primitives. Information exists fundamentally as high-dimensional coherence structure. Discreteness emerges through measurement collapse, which is itself a thermodynamic process governed by dimensional entropy loss. Maxwell’s demon exploits this structure not by violating thermodynamics but by operating within it as an internal symmetry breaker, inseparable from the field it appears to control. The future of information thermodynamics lies in understanding dynamics on coherence manifolds—how dimensional structures form, evolve, collapse, and redistribute. This is the domain where information, energy, and entropy unite, where classical and quantum become perspectives on the same geometric reality, and where the apparent paradoxes of Maxwell’s demon dissolve into the natural flow of a high-dimensional universe toward its attractors. Acknowledgments The author thanks the reviewers for constructive feedback that significantly improved this manuscript. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. 30 Declaration of competing interest The author declares no competing financial interests or personal relationships that could have influenced this work. Declaration of generative AI use During preparation the author used Claude (Anthropic) for literature review, mathematical formulation, and editing. ChatGPT (OpenAI) and Grok (xAI) were used for independent verification, error checking, and critical review of the manuscript. All content was reviewed and validated by the author, who takes full responsibility for the published article. Data availability The numerical simulation code for Appendix A is provided in full within the manuscript and constitutes the complete implementation. The code can be reproduced independently using the specified parameters and any standard Python environment with NumPy. No additional data or analysis scripts beyond those presented in the manuscript were used in this study. Preprint available at: https://doi.org/10.5281/zenodo.17309152 References [1] Maxwell, J.C. (1871). Theory of Heat. Longmans, Green, and Co., London. [2] Leff, H., Rex, A.F. (2003). Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing. Institute of Physics Publishing, Bristol. [3] Landauer, R. (1961). Irreversibility and heat generation in the computing process. IBM J. Res. Dev., 5(3), 183–191. 31 [4] B´erut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R., Lutz, E. (2012). Experimental verification of Landauer’s principle linking information and thermodynamics. Nature, 483(7388), 187–189. [5] Jun, Y., Gavrilov, M., Bechhoefer, J. (2014). High-precision test of Landauer’s principle in a feedback trap. Phys. Rev. Lett., 113(19), 190601. [6] Reeb, D., Wolf, M.M. (2014). An improved Landauer principle with finite-size corrections. New J. Phys., 16(10), 103011. [7] Del Rio, L., ˚ Aberg, J., Renner, R., Dahlsten, O., Vedral, V. (2011). The thermodynamic meaning of negative entropy. Nature, 474(7349), 61–63. [8] Lutz, E., Ciliberto, S. (2015). Information: From Maxwell’s demon to Landauer’s eraser. Phys. Today, 68(9), 30–35. [9] Koski, J.V., Maisi, V.F., Pekola, J.P., Averin, D.V. (2014). Experimental realization of a Szilard engine with a single electron. Proc. Natl. Acad. Sci. USA, 111(38), 13786–13789. [10] Bruschi, D.E., Perarnau-Llobet, M., Friis, N., Hovhannisyan, K.V., Huber, M. (2014). Thermodynamics of creating correlations: Limitations and optimal protocols. Phys. Rev. E, 91(3), 032118. [11] Sagawa, T., Ueda, M. (2010). Generalized Jarzynski equality under nonequilibrium feedback control. Phys. Rev. Lett., 104(9), 090602. [12] Sagawa, T., Ueda, M. (2012). Fluctuation theorem with information exchange: Role of correlations in stochastic thermodynamics. Phys. Rev. Lett., 109(18), 180602. [13] Mandal, D., Jarzynski, C. (2012). Work and information processing in a solvable model of Maxwell’s demon. Proc. Natl. Acad. Sci. USA, 109(29), 11641–11645. [14] Parrondo, J.M., Horowitz, J.M., Sagawa, T. (2015). Thermodynamics of information. Nat. Phys., 11(2), 131–139. 32 [15] Bennett, C.H. (1982). The thermodynamics of computation—a review. Int. J. Theor. Phys., 21(12), 905–940. [16] Szilard, L. (1929). ¨ Uber die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Z. Phys., 53(11-12), 840–856. [17] Todd, I. (2025). The limits of falsifiability: Dimensionality, measurement thresholds, and the sub-Landauer domain in biological systems. Accepted at BioSystems. Preprint: https://doi.org/10.5281/zenodo.17247628 [18] Anastassiou, C.A., Perin, R., Markram, H., Koch, C. (2011). Ephaptic coupling of cortical neurons. Nat. Neurosci., 14(2), 217–223. [19] Chiang, C.C., Shivacharan, R.S., Wei, X., Gonzalez-Reyes, L.E., Durand, D.M. (2019). Slow periodic activity in the longitudinal hippocampal slice can self-propagate nonsynaptically by a mechanism consistent with ephaptic coupling. J. Physiol., 597(1), 249–269. [20] McDonnell, M.D., Ward, L.M. (2011). The benefits of noise in neural systems: bridging theory and experiment. Nat. Rev. Neurosci., 12(7), 415–426. [21] Engel, G.S., Calhoun, T.R., Read, E.L., Ahn, T.K., Manˇcal, T., Cheng, Y.C., Blankenship, R.E., Fleming, G.R. (2007). Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature, 446(7137), 782–786. [22] Cao, J., Cogdell, R.J., Coker, D.F., Duan, H.G., Hauer, J., Kleinekath¨ofer, U., Jansen, T.L., Manˇcal, T., Miller, R.D., Ogilvie, J.P., et al. (2020). Quantum biology revisited. Sci. Adv., 6(14), eaaz4888. [23] Todd, I. (2025). Intelligence as high-dimensional coherence: Energy landscapes, phase space, and the physical basis of computation. Manuscript under review. Available at: https://papers.ssrn.com/sol3/papers.cfm?abstract id=5573915 33 [24] Stringer, C., Pachitariu, M., Steinmetz, N., Reddy, C.B., Carandini, M., Harris, K.D. (2019). Spontaneous behaviors drive multidimensional, brainwide activity. Science, 364(6437), eaav7893. [25] Gallego, J.A., Perich, M.G., Chowdhury, R.H., Solla, S.A., Miller, L.E. (2020). Longterm stability of cortical population dynamics underlying consistent behavior. Nat. Neurosci., 23(2), 260–270. [26] Dill, K.A., MacCallum, J.L. (2012). The protein-folding problem, 50 years on. Science, 338(6110), 1042–1046. [27] Duan, H.G., Jha, A., Li, X., Tiwari, V., Ye, H., Nayak, P.K., Zhu, X.L., Li, Z., Martinez, T.J., Thorwart, M., Miller, R.J.D. (2023). Quantum coherent energy transfer in photosynthesis at ambient temperature. Nat. Commun., 14, 1310. [28] Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence. Springer, Berlin. [29] Strogatz, S.H. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143, 1–20. [30] Frauenfelder, H., Sligar, S.G., Wolynes, P.G. (1991). The energy landscapes and motions of proteins. Science, 254(5038), 1598–1603. [31] Gammaitoni, L., H¨anggi, P., Jung, P., Marchesoni, F. (1998). Stochastic resonance. Rev. Mod. Phys., 70, 223–287. [32] Stocks, N. (2000). Suprathreshold stochastic resonance in multilevel threshold systems. Phys. Rev. Lett., 84(11), 2310–2313. [33] Laughlin, S.B., de Ruyter van Steveninck, R.R., Anderson, J.C. (1998). The metabolic cost of neural information. Nat. Neurosci., 1, 36–41. [34] Miller, E.K., Brincat, S.L., Roy, J.E. (2024). Cognition is an emergent property. Current Opinion in Behavioral Sciences, 57, 101388. 34 [35] Lundqvist, M., Rose, J., Herman, P., Brincat, S.L., Buschman, T.J., Miller, E.K. (2016). Gamma and beta bursts underlie working memory. Neuron, 90(1), 152–164. [36] McFadden, J., Al-Khalili, J. (2024). Quantum biology: An update and perspective. Frontiers in Quantum Science and Technology, 3, 1430558. [37] Brookes, J.C. (2024). Quantum biology: Environmental structuring and functional coherence in biological systems. Phil. Trans. R. Soc. A, 382(2278), 20230813. Available at: https://doi.org/10.1098/rsta.2023.0010 [38] Singh, P., Bera, S., De, A.K., Ghosh, S., Bhattacharyya, K. (2024). Quantum biology: An emerging frontier in interdisciplinary science. AIP Advances, 14(3), 030401. Available at: https://doi.org/10.1063/5.0197825 [39] Igamberdiev, A.U. (2021). The drawbridge of nature: Evolutionary complexification as a generation and novel interpretation of coding systems. BioSystems, 207, 104454. [40] Buzs´aki, G. (2006). Rhythms of the Brain. Oxford University Press, Oxford. [41] Levin, M. (2021). Bioelectric signaling: Reprogrammable circuits underlying embryogenesis, regeneration, and cancer. Cell, 184(8), 1971–1989. [42] Levin, M. (2014). Molecular bioelectricity: How endogenous voltage potentials control cell behavior and instruct pattern regulation in vivo. Mol. Biol. Cell, 25(24), 3835–3850. [43] Amari, S. (2016). Information Geometry and Its Applications. Springer, Tokyo. [44] Vega, F. (2024). The cell as a semiotic system that realizes closure to efficient causation. BioSystems, 240, 105226. [45] Hameroff, S., Penrose, R. (2014). Consciousness in the universe: A review of the ’Orch OR’ theory. Phys. Life Rev., 11(1), 39–78. 35 [46] Rozenberg, G., et al. (2021). Natural computation and its limits. BioSystems, 209, 104395. [47] Louie, A.H. (2022). Relational biology and its incompleteness. BioSystems, 218, 104718. [48] Hofmeyr, J.H.S. (2025). The biochemical factory and the limits of natural computation. BioSystems, 247, 105451. [49] Igamberdiev, A.U. (2007). Physical limits of computation and emergence of life. BioSystems, 90(2), 340–349. [50] Igamberdiev, A.U., Gordon, R. (2022). Order stability via Fr¨ohlich condensation in bio, eco, and social systems: The quantum-like approach. BioSystems, 212, 104593. [51] Azadi, P. (2025). Computational irreducibility as the foundation of agency. BioSystems, 256, 105563. A Numerical Validation: Coupled Bistable Erasure This appendix provides quantitative validation of apparent sub-Landauer erasure through numerical simulation of the coupled bistable toy model presented in Section 6.3. We implement the full stochastic dynamics via Langevin equations and demonstrate that local heat dissipation falls below kBTln 2 while global entropy production satisfies the second law. A.1 System Setup Consider two overdamped degrees of freedom x(t), y(t) evolving in coupled double-well potentials: U(x, y;λ) = U0(x;λ)+U0(y; 0) + K 2(x−y)2(43) 36 where the single-well potential is: U0(z;λ) = 1 4z4−z2−λz (44) The control parameter λ(t) acts only on x, implementing the erasure protocol. The coupling strength Kcontrols coherence: strong coupling (K≫1) forces x≈y, reducing effective dimensionality. The overdamped Langevin dynamics are: γ˙x=−∂U ∂x +p2γkBTxξx(t) (45) γ˙y=−∂U ∂y +p2γkBTyξy(t) (46) where ξx,y are independent Gaussian white noises with ⟨ξi(t)ξj(t′)⟩=δijδ(t−t′). We set Tx=Ty=Tfor the base case; the effective temperature Teff on the slow in-phase mode emerges via the fluctuation-dissipation relation and equals Tat equilibrium. A.2 Erasure Protocol The protocol consists of four phases: 1. Equilibration: Start at λ= 0 for t∈[−τeq,0] to reach thermal equilibrium in the symmetric double well. 2. Bias ramp (erasure):λ(t) = λmaxs(t/τ) for t∈[0, τ] with smooth ramp function s(u) = 3u2−2u3∈[0,1]. This biases the xwell to favor x > 0, implementing erasure to the positive state. 3. Hold: Maintain λ=λmax for time τhold to allow full relaxation into x>0. 4. Unbias: (Optional) Return to λ= 0 to complete a full logical cycle. 37 In the uncoupled case (K= 0) and quasi-static limit, the minimal average heat released to the bath satisfies ⟨Qx⟩≥kBTln 2 (standard Landauer). With coupling K > 0, we predict: ⟨Qx⟩< kBTln 2 (apparent sub-Landauer) (47) ⟨Qx⟩+⟨Qy⟩ ≥ kBTln 2 (global compliance) (48) A.3 Heat and Work Bookkeeping Following Sekimoto’s stochastic energetics formalism, we decompose energy changes along trajectories. For coordinate i∈ {x, y}with Stratonovich calculus: δQi=∂U ∂i ◦di (49) where ◦denotes the Stratonovich product (midpoint rule in discrete integration). We follow the convention where Qirepresents heat absorbed by subsystem i; the heat dissipated to the bath is −Qi. In our figures and tables, we report the dissipated heat (positive values indicate energy leaving the subsystem to the environment). The work performed by the external control is: δW =∂U ∂λ ˙ λ dt =−x˙ λ dt (50) Energy conservation requires ∆U=W−(Qx+Qy) along each trajectory. We estimate ensemble averages ⟨Qx,y⟩by time integration over many independent trajectories. A.4 Numerical Parameters We work in dimensionless units with γ= 1. A robust parameter regime demonstrating the effect: •Double-well shape: a= 1, b= 2 (minima near ±1) 38 •Coupling strengths: K∈ {0,0.5,1.0,2.0} •Temperature: T= 0.25 (moderate thermal fluctuations) •Maximum bias: λmax = 0.6 •Ramp duration: τ= 5000 timesteps with dt = 10−3 •Hold time: τhold = 2000 steps •Equilibration: τeq = 5000 steps •Ensemble size: Ntraj = 2000 trajectories Longer ramp times τapproach the quasi-static lower bounds more closely. On a standard laptop CPU, Ntraj = 500 trajectories takes approximately 5-10 minutes and provides adequate statistical validation; the full Ntraj = 2000 ensemble requires approximately 30-40 minutes but improves precision. A.5 Observable Predictions Primary metric: Average local heat ⟨Qx⟩vs coupling K. Expected behavior: •K= 0: ⟨Qx⟩ ≈ kBTln 2 ≈0.173 (standard Landauer) •K > 0: ⟨Qx⟩<0.173 (apparent sub-Landauer) •All K:⟨Qx+Qy⟩≳0.173 (global compliance) Dimensional reduction indicator: Participation ratio from covariance eigenvalues λ1,2of (x, y) during the ramp: Deff =(λ1+λ2)2 λ2 1+λ2 2 (51) As Kincreases, Deff →1 (one effective collective mode). 39 The global dissipation obeys our generalized geometric bound from Eq. (*): ⟨Qx+Qy⟩ ≥ kBTeff ln V(Deff) V(D′)(53) recovering standard Landauer when K→0 and Deff →2 (two independent bits collapsing to one). This numerical validation demonstrates that dimensional redistribution enables apparent sub-Landauer erasure locally while preserving global thermodynamic consistency. 46 LaTeX Source File Click here to access/download LaTeX Source File maxwell coherence demon.tex Declaration of Competing Interests Manuscript Title: Maxwell's Demon in a High-Dimensional Universe: Coherence Collapse and the Limits of Landauer Erasure The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Author: Ian Todd Date: October 10, 2025 Declaration of Interest Statement