Timing Inaccessibility and the Projection Bound: Resolving Maxwell's Demon for Continuous Biological Substrates

BioSystems Timing Inaccessibility and the Projection Bound: Resolving Maxwell’s Demon for Continuous Biological Substrates --Manuscript Draft-- Manuscript Number: BIOSYS-D-25-00903R1 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 show that the thermodynamic advantage of biological, continuous substrates over digital simulators arises from timing inaccessibility: below the Landauer threshold, temporal order cannot be irreversibly registered without dissipating >= k_B T ln 2 per binary decision. Consequently, exponentially many micro-trajectories map to the same observable outcome (path degeneracy). Continuous high-dimensional substrates exploit this by integrating sub-Landauer couplings during evolution and paying only at projection (dimensional collapse to a low-dimensional output). We derive a Projection Bound for quasistatic projection at effective temperature T_eff: E_collapse >= k_B T_eff [ ln(N_eps,pre / N_eps,post) - KL(p_pre || U_pre) + KL(p_post || U_post) ]. Under typical-set conditions this reduces to E_collapse >= k_B T_eff ln(N_eps,pre / N_eps,post). Combined with a Temporal Registration Bound (recording the total order of M bins requires log2(M!) bits), we quantify the gap: enumerative digital tracking scales exponentially with dimension, whereas projection cost scales like ln G ~ D. For biologically plausible parameters, we estimate degeneracies of 10^42–10^94 (protein folding) and 10^50–10^100 (neural dynamics) as upper bounds under independence assumptions. The framework reconciles stochastic resonance (amplitude detection) with order inaccessibility, and clarifies why analog/neuromorphic systems gain efficiency by deferring projection. Keywords: Maxwell's demon, timing inaccessibility, path degeneracy, Landauer principle, dimensional collapse Response to Reviewers: 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 18, 2025 Dr. Abir Igamberdiev Editor-in-Chief BioSystems Dear Dr. Igamberdiev, I am pleased to submit the revised manuscript BIOSYS-D-25-00903, “Timing Inaccessibility and the Projection Bound: Resolving Maxwell’s Demon for Continuous Biological Substrates,” in response to the reviewers’ major revision decision. I am grateful for the constructive feedback, which has substantially strengthened the work. Summary of major revisions: Reviewer #1: •Section 3.2 now provides complete operational definition of Teff with explicit measurement protocol via fluctuation-dissipation relation •All entropy expressions use dimensionless covering numbers Nε; dimensional consistency verified throughout •Section 6.3 integrates Voronel (2018) and Bormashenko (2022) on biological spatial constraints •Title revised to directly state mechanism, contribution, and domain Reviewer #2: •Section 5 presents Kuramoto oscillators as genuinely continuous example (on TN, no discrete states) •Section 6.2 substantially expanded to engage quantum-classical transition, citing AnnbyAndersson et al. (PRR 2024) and Lebedev et al. (PRA 2016) •Novelty clarified throughout: timing inaccessibility mechanism, camera-engine duality, and path degeneracy quantification with biologically realistic parameters (1042–1094 for proteins, 1050–10100 for neural) Additional enhancements: Three comprehensive appendices added (parameter sensitivity, biological implementations across scales, data methods); testable predictions sharpened to attojoulescale heat dissipation at decision points (∼0.03–1 aJ). The revised framework resolves the biological efficiency puzzle: continuous high-dimensional substrates exploit timing inaccessibility to defer thermodynamic cost from exploration to projection, achieving ∼105–108×practical efficiency advantage over CMOS without violating the second law. Cover Letter A detailed point-by-point response addressing all reviewer comments is attached. The manuscript is original, not under review elsewhere, and prepared with AI assistance (Claude 4.5 Sonnet, GPT-5, Grok) under full author responsibility. Sincerely, Ian Todd Sydney Medical School University of Sydney 2 Ian Todd Sydney Medical School University of Sydney Sydney, NSW, Australia [email protected] October 18, 2025 Professor Abir Igamberdiev Editor-in-Chief BioSystems Elsevier Dear Professor Igamberdiev, I am pleased to submit the revised version of manuscript BIOSYS-D-25-00903, titled “Timing Inaccessibility and the Projection Bound: Resolving Maxwell’s Demon for Continuous Biological Substrates.” I am grateful to you and the reviewers for the constructive feedback, which has substantially strengthened the manuscript. *Core Contribution This paper resolves a fundamental puzzle in biological thermodynamics: how do continuous biological systems achieve computational efficiency that appears to violate Landauer’s principle? We show that the answer lies in timing inaccessibility—below the Landauer threshold (Tln 2 ≈ 2.87×10−21 J at 300 K), temporal order cannot be irreversibly registered without dissipating energy per binary timing decision. This creates exponential path degeneracy: many micro-trajectories map to the same observable outcome. The framework introduces two key results: 1. Temporal Registration Bound (TRB): Irreversibly recording the order of Mtemporal bins requires ≥Tln 2 ·log2(M!) energy dissipation (Lemma 1, Section 2). 2. Projection Bound (PB): Quasistatic dimensional collapse from Nε,pre to Nε,post distinguishable states dissipates (Theorem 1, Section 3): Ecollapse ≥ln(Nε,pre/Nε,post)−hKL(ppre∥Upre)−KL(ppost∥Upost)i reducing to ln(Npre/Npost) under typical-set conditions, generalizing Landauer’s principle to continuous manifolds. These bounds explain why continuous high-dimensional substrates (proteins, neural populations) are thermodynamically superior to digital simulation for computing high-dimensional dynamics: they explore exponentially many paths in parallel via thermal noise, paying only at dimensional collapse to discrete outputs. The efficiency gap is ∼160–330×at theoretical Landauer limits, rising to ∼105–108×compared to contemporary CMOS. *Major Revisions Addressing Reviewer Concerns *Reviewer #1 Response to Reviewers 1. Effective temperature definition (Eq. 3): Section 3.2 now provides complete operational definition via fluctuation-dissipation relation with explicit measurement protocol. The text now states clearly: “Re[Γ(ω)] is the dissipative component of linear response, and Γ(ω) describes how the system responds to weak perturbations: ⟨x(ω)⟩= Γ(ω)F(ω).” A four-step measurement protocol is provided showing how to extract (ω0) from noise spectra and perturbation response as an observable quantity, not a fitted parameter. Additionally, we now note that in active biological systems, can exceed ambient temperature Tand may be frequency dependent (citing Seifert 2012). 2. Dimensional consistency (Eq. 5 and throughout): All entropy expressions now use dimensionless covering numbers Nε(counting ε-balls covering the manifold). Section 3.1 explicitly states: “For continuous manifolds, Nε∼(L/ε)Deff where L/ε is dimensionless.” The notation table (Table 1) clarifies: “L: System size (same units as ε;L/ε is dimensionless).” Theorem 1 includes the corollary with dimensionless ratios: “With power-law covering, Nε∼(L/ε)D(with L/ε dimensionless): Ecollapse ≥(Deff −D′) ln(L/ε).” All logarithms are now taken of dimensionless quantities throughout. 3. Prior biological applications: Section 6.3 (Spatial Scales and Information Capacity) now integrates Voronel (2018) and Bormashenko (2022), explicitly citing both papers and showing how spatial constraints combine with our temporal path degeneracy: Ecollapse ≥(Dspatial +Dtemporal) ln(L/ξ) The discussion notes: “Voronel (2018) and Bormashenko (2022) established that biological informational capacity is constrained by spatial scales. Biological systems exploit both spatial and temporal fine-graining.” 4. Title revision: Changed from “Maxwell’s Demon in a High-Dimensional Universe: Coherence Collapse and the Limits of Landauer Erasure” to “Timing Inaccessibility and the Projection Bound: Resolving Maxwell’s Demon for Continuous Biological Substrates” to directly state the mechanism (timing inaccessibility), the contribution (projection bound), and the domain (continuous biological substrates). This better reflects the paper’s content and biological focus. *Reviewer #2 1. Continuous system example: Section 5 now presents Kuramoto coupled oscillators as a genuinely continuous example operating on the N-torus TNwith continuous phase variables and no discrete states: dθi dt =ωi+K N N X j=1 sin(θj−θi)+ηi(t) We demonstrate dimensional collapse from Deff ≈N(incoherent regime) to Deff ≈2 (synchronized on the Ott-Antonsen manifold for Lorentzian frequency distributions), with explicit collapse cost: Ecollapse ≥(Deff −D′) ln(2π/∆θ) This is “the continuous analog of Landauer’s bound—fully geometric, no discrete bits.” The example includes citations to Ott & Antonsen (2008, 2009) for the low-dimensional manifold reduction and notes that recent computational work (Bian et al. 2025) encounters numerical failures precisely where temporal resolution demands exceed thermodynamic accessibility, providing computational evidence for our bounds. 2. Quantum demon literature and quantum-classical transition: Section 6.2 (Quantum Demons) is substantially expanded to engage with the quantum-classical transition, now citing: 2 •Annby-Andersson et al., Phys. Rev. Research 6, 043216 (2024) - “Maxwell’s demon across the quantum-to-classical transition” •Lebedev, Lesovik & Blatter, Phys. Rev. A 94, 052133 (2016) - entanglement and coherence in quantum state merging The discussion notes: “Recent quantum demons show coherence-entropy trade-offs with mathematical isomorphism S(ρ)↔Hε. Physical mechanisms differ—quantum superposition (mK, µs) vs. classical path degeneracy (300K, ms)—yet both yield isomorphic bounds. The dimensional geometry may unify quantum and classical regimes, differing in mechanism (coherent superposition vs thermal degeneracy) but not fundamental structure.” This addresses the quantum-classical transition as requested while acknowledging it as an active research area. 3. Novelty clarification: Enhanced throughout the manuscript: •Abstract: Now explicitly states the novel contribution upfront: “We show that the thermodynamic advantage of biological, continuous substrates over digital simulators arises from timing inaccessibility... Continuous high-dimensional substrates exploit this by integrating sub-Landauer couplings during evolution and paying only at projection.” •Section 1: Clearly distinguishes three possibilities for resolving the biological efficiency puzzle and establishes our thesis (option iii: different accounting via deferred projection). •Section 2.1: Introduces timing inaccessibility mechanism and explicitly contrasts detection capacity (AWGN) with recording/erasure cost (Landauer). •Section 2.2: Addresses stochastic resonance objection head-on, explaining why population integration detects amplitude but not order. •Section 6.1: Systematically contrasts our contribution with prior information-theoretic demons (Parrondo 1996, Vaikuntanathan 2009, Allahverdyan 2009), noting: “Our contribution: (i) classical biological substrate at 300K, (ii) metric-entropy covering numbers at finite resolution, (iii) timing inaccessibility as mechanism, (iv) path degeneracy from dimensional structure.” •Section 6.4 (Camera-Engine Duality): Presents novel reconceptualization of the demon as simultaneously sensing (camera) and steering (engine) environmental dynamics, with information existing as structural correlation rather than written bits. *Additional Enhancements Beyond Reviewer Requirements •Appendix A: Parameter sensitivity analysis with worked examples showing how log10 Ω varies from 1042–1094 (protein folding) and 1050–10100 (neural dynamics) across biologically plausible ranges for (Deff , κ, τc/∆t). Includes Table A.2 showing sensitivity across parameter space. •Appendix B: Cross-scale biological implementations from molecular (protein folding) to neural populations, with quantitative degeneracy estimates and experimental predictions. Presents neural population as camera-engine with detailed energetics. •Appendix C: Methods for estimating Deff and correlation factor κfrom experimental data using participation ratios and phase-shuffle surrogates, making the framework empirically 3 testable. •Section 2.3: Expanded explanation of high dimensionality enabling sub-Landauer sensing through population integration, with explicit calculation showing Etotal = 1000×10−3Tln 2 = Tln 2. •Section 2.4: New subsection explaining fundamental architectural difference preventing digital systems from exploiting sub-Landauer integration. •Section 6.5: Enhanced accuracy-efficiency tradeoff discussion showing what digital systems buy with energy cost (bit-exact computation, auditability) versus what continuous substrates sacrifice (path indeterminacy, thermal noise). •Section 6.7: Comprehensive limitations and caveats section addressing theoretical assumptions, task-dependence, and common objections with detailed replies. •Testable predictions: Significance section includes specific measurable quantities: (i) coherencetime dependence on accuracy, (ii) dimensional shifts with task, (iii) attojoule-scale heat dissipation at decision points (∼0.03–1 aJ). *Significance for BioSystems This framework addresses fundamental questions in biological computation and thermodynamics: 1. Resolves efficiency paradox: Explains how biological systems achieve computational efficiency without violating the second law through deferred projection and path degeneracy— biology doesn’t evade physics, it uses dimensional accounting. 2. Quantitative predictions: Provides testable scaling laws for energy dissipation, dimensional reduction, and coherence dependence across molecular to neural scales. Example: ordering M= 10 temporal bins requires 6.3×10−20 J at 300 K. 3. Unifying framework: Connects Landauer’s principle, Maxwell’s demon, stochastic resonance, and biological information processing through geometric thermodynamics. Shows how timing inaccessibility creates path degeneracy as physical mechanism, not epistemic limitation. 4. Methodological contribution: Camera-engine duality and timing inaccessibility provide new conceptual tools for understanding biological organization, computational irreducibility, and the physical basis of anticipatory systems (Igamberdiev 2014). 5. Cross-scale applicability: Framework spans protein folding (1042–1094 path degeneracy), neural dynamics (1050–10100), and cellular coordination, with consistent thermodynamic principles. The timing inaccessibility mechanism has broad implications for understanding life’s computational architecture: biological systems are optimized not for bit-exact computation but for robust pattern recognition through high-dimensional coherence maintenance and deferred projection. The framework respects fundamental thermodynamic limits while explaining apparent violations through rigorous dimensional accounting. *Technical Improvements •All mathematical notation unified using consistent macros (, ) •Notation table (Table 1) enhanced with units and distinctions between T(bath temperature 4 for TRB) and (operational, for PB) •Typical-set approximation linked to Cover & Thomas definition •Dimensionless ratios explicitly noted throughout (L/ε, covering numbers) •Cross-references added linking abstract claims to appendix derivations *Manuscript Status All figures, tables, and appendices are complete and integrated. The manuscript includes comprehensive citations (46 references), methods for experimental validation, and clear statements of assumptions and limitations. Generative AI use (Claude 4.5 Sonnet for drafting, GPT-5 and Grok for feedback) is declared transparently per journal policy, with full author responsibility for scientific content. I believe the revised manuscript makes a significant contribution to understanding thermodynamic foundations of biological computation and addresses all reviewer concerns comprehensively and rigorously. The timing inaccessibility framework provides testable explanations for biological efficiency while opening new research directions in information thermodynamics, natural computation, and the physics of living systems. Thank you for the opportunity to revise this work. I look forward to your decision. Sincerely, Ian Todd 5 Timing Inaccessibility and the Projection Bound: Resolving Maxwell’s Demon for Continuous Biological Substrates Ian Todd Sydney Medical School University of Sydney Sydney, NSW, Australia [email protected] October 18, 2025 Abstract We show that the thermodynamic advantage of biological, continuous substrates over digital simulators arises from timing inaccessibility: below the Landauer threshold, temporal order cannot be irreversibly registered without dissipating ≥kBTln 2per binary decision. Consequently, exponentially many micro-trajectories map to the same observable outcome (path degeneracy). Continuous high-dimensional substrates exploit this by integrating sub-Landauer couplings during evolution and paying only at projection (dimensional collapse to a low-dimensional output). We derive a Projection Bound for quasistatic projection at effective temperature Teff: Ecollapse ≥kBTeff ln Nε,pre Nε,post −KL(ppre ∥Upre) + KLppost ∥Upost(0.1) 1 Manuscript File Click here to view linked References For partial order distinguishing equivalence classes of sizes mj: Etime min ≥kBTln 2 ·log2 M! Qjmj!!(2.3) At sub-Landauer energies where Emeas ≪kBTln 2, the number of binary timing decisions that can be irreversibly registered approaches zero. Proof sketch. There are M !possible total orders of M bins. Uniquely specifying one order requires H = log2 ( M !) bits (by Stirling: log2 ( M !) ≈Mlog2M−Mlog2e ). Irreversibly writing these bits dissipates at least kBTln 2 per bit by Landauer’s principle [1]. Clarification: A reversible comparator that leaves no persistent record can, in principle, avoid dissipation. The energetic bound applies when the order is stabilized as memory (logically irreversible). Concrete example: At T = 300 K, ordering M = 10 bins minimally dissipates log2 (10!) · kBTln 2 ≈21.8×2.87 ×10−21 J≈6.3×10−20 J. 2.2 Stochastic Resonance: Why It Detects Amplitude But Not Order Natural objection: If stochastic resonance enables detection of sub-Landauer signals through population integration [5, 6], doesn’t this eliminate path degeneracy? Answer: No. Stochastic resonance provides: •Amplitude/presence detection (SNR ∼√Nscaling) •Population-level integration across Ncoupled units •Knowledge that coordination occurred It cannot provide: •Timing of individual events (which fluctuation occurred when) 8 •Temporal ordering (did event A precede B?) •Which specific coordination pattern was realized For M = 10 temporal bins, distinguishing all orderings requires log2 (10!) ≈ 21 . 8bits, costing ≥ 6 . 3 × 10 −20 J. If individual signal energies are E∼ 10 −23 J (sub-Landauer), measurement energy exceeds signal energy by 6000 × . Population integration can detect that 10 events occurred, but not when or in what order. 2.3 High Dimensionality: The Key to Sub-Landauer Sensing The puzzle: If individual events are sub-Landauer, how does the system sense anything? The solution: The physical substrate is formally infinite-dimensional with power-law mode distribution. At any finite energy/temporal resolution, Deff is finite and grows with precision. At energy threshold Ethresh , a finite number of modes Deff ( Ethresh )carry energy ≥Ethresh. With Deff accessible modes, a signal of energy Esignal ≪kBTln 2per component can still be detected if integrated: Etotal ≈Neff Esignal, Neff ∝Deff ×ln τc ∆t(2.4) Even if Esignal ≪kBTln 2 per channel, Etotal ≳kBTln 2 achieves detectability. Example: Neural population with Deff ∼ 1000 accessible modes can detect signals where each mode receives only Esignal ∼ 10 −3kBTln 2because population integration yields effective energy Etotal ∼1000 ×10−3kBTln 2 = kBTln 2, meeting the detection threshold. Ephaptic coupling: Extracellular fields ∼ 0 . 1–1mV/mm over neuronal membrane ( Cm∼ 1 µ F/cm 2 ) yield work ∼ 10 −23 –10 −22 J per neuron ( ≈ 3 × 10 −3 –3 × 10 −2 times kBTln 2) (see [37] for field strengths and scaling). 9 2.4 Why Digital Systems Cannot Do This Fundamental architectural difference: Digital systems: •Binary bit requires ≥kBTln 2 to be irreversibly registered •Must receive supra-Landauer signals per input bit • Cannot integrate sub-Landauer signals across channels without first amplifying each channel above threshold (costs energy) •Must discretize each input above Landauer threshold before processing Continuous high-D systems: •Can couple weakly (sub-Landauer per link) to many sources •Integrate via population dynamics •Collapse only at output •Pay for dimensional reduction, not per-interaction bit registration The degeneracy is not a bug—it’s what enables weak coupling to be computationally useful. Path degeneracy and sensing capability are coupled: exponentially many microstates allow sensitivity to distributed sub-Landauer perturbations. 3 The Projection Bound 3.1 What Are We Paying For? When forcing a high-dimensional system to produce a low-dimensional output, you must dissipate heat. The cost has three components: 10 1. Geometric squashing ( ln Nε,pre/Nε,post ): How many configurations are you destroying? For continuous manifolds, Nε∼(L/ε)Deff where L/ε is dimensionless. 2. Pre-collapse non-uniformity ( −KL ( ppre∥Upre )): Were you using all states uniformly? If the system had a heavy-tailed distribution, you were effectively more constrained than geometry suggests. This lowers the bound (you’ve already "paid" some entropy reduction). 3. Post-collapse specificity (+ KL ( ppost∥Upost )): How peaked is your final state? This increases the bound—you’re creating more order. Typical-set approximation: For high-dimensional systems with many weakly-interacting modes, equipartition and maximum entropy drive toward near-uniform occupancy. When both preand post-collapse distributions are approximately uniform, KL terms vanish and you get the clean geometric bound. 3.2 Effective Temperature: Operational Definition For systems out of equilibrium, define Teff operationally via the fluctuation-dissipation relation [7]: Sη(ω) = 2kBTeff(ω) Re[Γ(ω)] (3.1) where Sη ( ω )is noise power spectral density, Re [Γ( ω )] is the dissipative component of linear response, and Γ( ω )describes how the system responds to weak perturbations: ⟨x ( ω ) ⟩ = Γ(ω)F(ω). Measurement protocol: 1. Measure noise spectrum Sη(ω) 2. Apply weak sinusoidal perturbation at ω0 (e.g., inject 50–200 pA membrane-equivalent drive in cortical slice) 3. Measure phase-resolved response to extract Re[Γ(ω0)] 4. Ratio gives Teff(ω0)=Sη(ω0)/(2kBRe[Γ(ω0)])—an observable, not a fitted parameter 11 In equilibrium: Teff = T . Out of equilibrium: Teff ( ω )is operational, evaluated at collapse mode ω0 . Note that in active biological systems driven by ATP hydrolysis or other energy sources, Teff can exceed ambient temperature T [ 7 ], reflecting enhanced effective noise from non-thermal fluctuations. 3.3 Main Theorem Theorem 3.1 (Projection Bound for Continuous Systems).Let {Ci}Nε i=1 be an ε -coarse graining of the accessible manifold and ppre, ppost be the pre/post distributions over cells. For a quasistatic projection that reduces support from Nε,pre to Nε,post , the mean dissipated heat at effective temperature Teff (evaluated at the dominant collapse mode ω0) satisfies: Ecollapse ≥kBTeff ln Nε,pre Nε,post −KL(ppre ∥Upre) + KLppost ∥Upost(3.2) where Upre/post are uniform on their supports. (This reproduces (1.2) under the typical-set approximation.) Under typical-set approximation (equiprobable cells): Ecollapse ≥kBTeff lnNε(Deff)/Nε(D′)(3.3) With power-law covering, Nε∼(L/ε)Dfor fixed L/ε: Ecollapse ≥kBTeff(Deff −D′) ln(L/ε)(3.4) Proof outline. Step 1: Define coarse-grained entropy as Sε = kB ( ln Nε−KL ( p∥U )) where U is uniform over the Nεcells. 12 Step 2: The projection reduces support and alters p: ∆Ssys =Spost ε−Spre ε(3.5) =kB[ln Nε,post −ln Nε,pre −KL(ppost∥Upost) + KL(ppre∥Upre)] (3.6) Step 3: By the second law: ∆Senv ≥ −∆Ssys. Step 4: Integrating heat at Teff gives Ecollapse = Teff ∆ Senv ≥ −Teff ∆ Ssys , yielding equation (3.2). Classical limit: Binary system with Nε=2→1recovers kBTln 2. Finite-time correction: Finite-time protocols add thermodynamic-length term L2/ (4 τ ) [ 30 ]. For neural decisions at τ∼ ms timescales with typical L ∼ 10, excess dissipation ∼10−20 J is small compared to quasistatic bound (∼10−19–10−18 J for Deff ∼50–200). 4 Path Degeneracy: Quantifying the Inaccessible 4.1 Definition and Scaling At coarse-graining ε over window with temporal resolution ∆ t ,Ω(∆ t )is the number of resolution-distinguishable micro-trajectories consistent with the same macro-outcome. For continuous manifold M, the metric ε-entropy [8]: Hε= ln Nε(M), Nε∼L εDeff (4.1) with L/ε dimensionless. As ε decreases, Nε grows exponentially. Systems with large Deff exhibit exponentially more distinguishable configurations. For oscillator system with coherence time τc and temporal resolution ∆ t , the time13 bandwidth product τc/∆tbounds independent temporal modes [9], yielding: Ω(∆t)∼exp Deff(∆t) κln τc ∆t(4.2) where κ≥1accounts for correlations. 4.2 Neural-Scale Example Consider neural dynamics with: •Coherence time: τc∼100 ms •Coarse temporal resolution: ∆tcoarse = 20 ms •Fine temporal resolution: ∆tfine = 0.1ms •Dimensionality increase: ∆Deff ≈50 (consistent with [10]) •Correlation redundancy: κ∼2(estimated from phase-shuffle surrogates) Path degeneracy ratio: Ratio ∼exp ∆Deff κln τc/∆tfine τc/∆tcoarse  = exp 50 2×ln(200)≈1058 (4.3) Interpretation: One observable spike train at coarse resolution (20 ms bins) corresponds to ∼ 10 58 distinguishable micro-trajectories at fine resolution (0.1 ms bins). This falls within the 10 50 –10 100 range cited in the abstract (see Appendix A for full parameter sensitivity showing how variations in Deff,κ, and temporal resolution produce this range). 14 5 Example: Kuramoto Oscillators Consider Ncoupled phase oscillators: dθi dt =ωi+K N N X j=1 sin(θj−θi)+ηi(t)(5.1) where ⟨ηi(t)ηj(t′)⟩= 2Dδijδ(t−t′). State space is the N-torus TN. Order parameter: reiΨ=1 NPN j=1 eiθj At angular resolution ∆θ, one macroscopic configuration (r, Ψ) corresponds to: N∆θ∼2π ∆θDeff (5.2) distinguishable microstates. Effective dimensionality: • Synchronized regime (strong coupling, r≈ 1): Deff ≈ 2(Ott-Antonsen manifold for Lorentzian g(ω)under analyticity assumptions [35, 36]) •Incoherent regime (weak coupling, r≈0): Deff ≈N •Intermediate coupling: 2< Deff < N When the system collapses from exploring the full Deff -dimensional manifold to a measurement outcome: Ecollapse ≥kBTeff(Deff −D′) ln(2π/∆θ)(5.3) This is the continuous analog of Landauer’s bound—fully geometric, no discrete bits. Recent computational approaches encounter numerical failures ("covariance explosion") precisely where temporal resolution demands exceed thermodynamic accessibility [ 11 ]—computational evidence for the bounds derived here. 15 6 Relation to Prior Work and Implications 6.1 Information-Theoretic Demons Prior continuous demons [ 12 , 13 , 14 ] quantify costs via mutual information. Our contribution: (i) classical biological substrate at 300K, (ii) metric-entropy covering numbers at finite resolution, (iii) timing inaccessibility as mechanism, (iv) path degeneracy from dimensional structure. Bennett [ 27 ] established that logical reversibility bounds energy only if computation is quasi-static and low-noise. Still et al. [ 28 ] showed predictive information confers thermodynamic advantage, but creating/storing it has minimal work cost. Kolchinsky & Wolpert [ 29 ] demonstrated that minimal work depends on changes in non-equilibrium distributions—aligning with our Projection Bound via ln Nεand KL terms. 6.2 Quantum Demons Recent quantum demons [ 15 , 16 ] show coherence-entropy trade-offs with mathematical isomorphism S ( ρ ) ↔Hε . Physical mechanisms differ—quantum superposition (mK, µ s) vs. classical path degeneracy (300K, ms)—yet both yield isomorphic bounds. The dimensional geometry may unify quantum and classical regimes, differing in mechanism (coherent superposition vs thermal degeneracy) but not fundamental structure. 6.3 Spatial Scales and Information Capacity Voronel [ 24 ] and Bormashenko [ 25 ] established that biological informational capacity is constrained by spatial scales. Biological systems exploit both spatial and temporal finegraining. Total capacity: Ecollapse ≥kBTeff(Dspatial +Dtemporal) ln(L/ξ)(6.1) 16 6.4 Camera-Engine Duality: A New Demon Mechanism These prior frameworks quantify costs but don’t fully capture how biological substrates defer projection. This motivates the camera-engine duality below, which reframes the demon mechanism physically. The classical demon is straightforward: a computer with memory that it periodically erases, paying kBTln 2 per bit. The demon is the memory. The biological demon is more subtle. 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A Parameter Sensitivity Analysis for Path Degeneracy This appendix provides worked examples showing how parameter choices lead to the headline degeneracy ranges cited in the main text. A.1 Protein Folding Example Parameters: •Micro-level states: Nmicro ∼1048–10100 (rotamer libraries to continuous torsions) •Meso-level states: Nmeso ∼106(folding intermediates) •Temporal coherence: τc∼10−6–1s (folding time) •Fine resolution: ∆tfine ∼10−12 s (bond vibration) •Coarse resolution: ∆tcoarse ∼10−9s (conformational transition) •Effective dimensionality: Deff ∼20–50 (backbone degrees of freedom) •Correlation factor: κ∼5–10 (folding funnel constraint) 26 Calculation: log10 Ω = Deff κlog10 τc ∆tfine + log10 Nmicro Nmeso (A.1) ≈20–50 5–10 ×log10(106–1012) + log10(1042–1094)(A.2) ≈42–94 (A.3) A.2 Neural Population Example Parameters: •Population size: N∼1000 neurons •Coherence time: τc∼100 ms •Fine resolution: ∆tfine ∼0.1ms (spike timing precision) •Coarse resolution: ∆tcoarse ∼20 ms (behavioral bins) •Effective dimensionality: Deff ∼50–200 (from large-scale recordings) • Correlation factor: κ∼ 2–5(estimated as Dshuffle eff /Ddata eff under phase-shuffle surrogates) Calculation: log10 Ω = ∆Deff κlog10 τc/∆tfine τc/∆tcoarse (A.4) =50–200 2–5×log10(200) (A.5) ≈23–92 (A.6) Note: To avoid double counting, we do not add a separate "voltage microstate" multiplier; the scaling in equation (1.3) already counts resolution-distinguishable micro-trajectories at the chosen (ε, ∆t). 27 A.3 Sensitivity Table Table 2: log10 Ωacross parameter ranges for neural example (∆Deff, κ)τc/∆t= 50 τc/∆t= 200 τc/∆t= 500 (30, 2) 25 34 40 (50, 2) 42 58 68 (80, 2) 68 92 108 (50, 3) 28 38 45 (50, 5) 17 23 27 Key observations: •Degeneracy remains exponentially large (Ω∼1017–10108) across plausible ranges •Correlations (higher κ) reduce log10 Ωproportionally • Time-bandwidth ratio has logarithmic effect, so moderate changes in τc/ ∆ t produce manageable shifts • Upper bounds assume independence; real biological systems have additional constraints B Appendix B: Biological Implementations Across Scales The timing inaccessibility framework applies from molecular to organismal scales wherever continuous dynamics operate below the Landauer threshold. B.1 Molecular Scale: Protein Folding Protein folding exemplifies path degeneracy at the molecular level. A typical protein explores ∼ 10 48 –10 100 conformational microstates en route to its native fold, yet folding time is only ∼10−6–1s. Energetics: Thermal energy kBT≈ 4 . 1 × 10 −21 J at 300 K; individual hydrogen bonds ∼ 2–10 kBT ; hydrophobic contacts ∼ 1–5 kBT . While individual bond energies are supra28 Landauer, the energetic cost of irreversibly registering the timing and order of the full conformational trajectory is sub-Landauer per timing decision. Path degeneracy: For a 100-residue protein with ∼ 3conformations per residue, microstate space ∼ 3 100 ≈ 10 48 to 10 100 , but coarse-grained intermediate states ∼ 10 3 –10 6 . Path degeneracy G∼ 10 42 –10 94 (upper bounds; folding funnels reduce effective degeneracy via correlation factor κ∼5–10). Collapse: Thermodynamic cost appears at native state stabilization. Estimated ∆ Gfold ∼ 5–15 kcal/mol ≈ (3 . 5 × 10 −20– 1 . 0 × 10 −19 )J, consistent with collapsing Deff ∼ 20–50 effective dimensions. B.2 Neural Population as Camera-Engine Consider a neural population making a perceptual decision. The substrate is formally infinitedimensional (continuous electromagnetic fields, cross-frequency phase coupling, continuous membrane voltage), but at typical recording resolutions ( 1ms, 100 µ V sensitivity), only Deff ∼100 is thermodynamically accessible. Camera phase (0–100 ms): Environmental photon arrivals at ∼ 10 6 rods/cones. Each photon at 500–600 nm has E∼ (3 . 3 – 4 . 0) × 10 −19 J ( ∼ 80 – 96 kBT ), but timing is sub-Landauer. Accessible modes Deff ∼ 100 integrate timing patterns. Internal voltage landscape becomes isomorphic to environmental light pattern through weak coupling. No bits written—structural correlation emerges. Engine phase (concurrent): Attention modulates sensory gain via weak top-down connections. Each feedback synapse ∼ 10 −22 J ≪kBTln 2, but Deff ∼ 100 feedback channels coordinately bias processing. Environment steered toward task-relevant features using internal map built during camera phase. Collapse (at decision, ∼ 150 ms): Motor output: "left" vs "right" (binary). Dimensional collapse: accessible slice Deff ∼ 100 →D′ = 1. Dissipation: ∼ 100 kBTln 2 ≈ 3 × 10 −19 J. Digital would pay this during every timestep; biological pays once at decision. 29 B.3 Cross-Scale Pattern Table 3: Path degeneracy across biological scales System Event energy Deff Estimated log10 Ω Protein folding 1–10kBT20–50 42–94 Ca2+ waves 5–20kBT30–100 25–40 Gene regulation 5–15kBT10–30 15–20 Neural (ephaptic) ∼10−3kBTln 2 50–200 50–100 Common mechanism: Thermal noise explores vast micro-trajectory spaces; weak fields bias ensembles; thermodynamic cost concentrates at collapse to discrete outputs. C Appendix C: Estimating Deff and κfrom Data Data and preprocessing. Acquire X∈RT×M (time × channels). Band-limit to the frequency band of interest with zero-phase FIR; extract analytic signal zm ( t )via Hilbert transform for phase-based measures. Feature matrix. Either (i) use covariance C = 1 T−1 ( X−¯ X ) ⊤ ( X−¯ X ), or (ii) build a phase-coherence matrix W with entries Wij = 1 TPtei(ϕi(t)−ϕj(t)) (inter-site phase coherence, ISPC [10]). Normalize Wto unit trace to avoid scale artifacts. Eigen-spectrum and participation ratio. Let {λk} be eigenvalues of C (or W ), sorted descending. Define Deff =(Pkλk)2 Pkλ2 k . Report Deff ( t )in a sliding window (e.g., 200 ms with 10 ms hop) to track task-locked changes. Correlation factor κ .Construct S surrogate datasets by phase-shuffling each channel: zm ( t ) 7→ eiθmzm ( t )with random θm , preserving power spectra but destroying cross-channel 30 timing. Compute D(sur) eff on each surrogate; define κ≡Ddata eff mediansD(sur) eff (κ≥1). Use κin Ω-scaling as in equation (1.3). Robustness notes. (1) Z-score channels before C to prevent amplitude-dominated modes. (2) Repeat analyses across bands (theta/alpha/beta), aligning with task epochs. (3) Control for sample size by fixing window length and applying Ledoit–Wolf shrinkage on C if M is large relative to T. 31 LaTeX Source File Click here to access/download LaTeX Source File maxwell timing inaccessibility.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