A Field Equation from Relativistic Predicts Critical Transitions in Biological and Biogeochemical Systems

A Field Equation from Relativistic Predicts Critical Transitions in Biological and Biogeochemical Systems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Field Equation from Relativistic Predicts Critical Transitions in Biological and Biogeochemical Systems Hao Huang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9700661/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract BACKGROUND The unification of continuum and quantum descriptions remains unresolved. Existing approaches add mathematical complexity without axiomatic simplicity. METHODS We axiomatize the vacuum as an information-energy ground state and derive a field equation from relativistic first principles. Three axioms yield five theorems, producing the master equation α = ln 2 + (1 − ln 2)(Q − C), where Q and C are quantum and entropy responses. All parameters are derived analytically; none are fitted. The predictions are tested against two independent experimental datasets extracted from the literature: plant-defense density transitions and soil-carbon stabilization kinetics. RESULTS The vacuum constant α₀ = ln 2 ≈ 0.6931 and the avalanche threshold R_c = 2/ln 2 − 1/(ln 2)² ≈ 0.8043 emerge as exact mathematical consequences. In plant-defense systems, the observed transition point R ≈ 0.81 agrees with R_c within 1%. The measured threshold contraction slope yields α = 0.70 ± 0.05, agreeing with ln 2 within 1.5%. In soil-carbon systems, manure amendment drives an avalanche transition crossing R_c between months 6 and 12, whereas biochar follows asymptotic convergence; both converge to the predicted stable state. These experimental systems reveal two distinct stabilization routes—Iron Gate and Enzyme Latch—not previously classified within a unified theoretical framework. CONCLUSIONS A single field equation, derived without empirical fitting, predicts critical transitions across physical, biological, and biogeochemical domains and yields new mechanistic classifications in experimental systems. Analytical Biochemistry equation physical biological and biogeochemical Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Introduction The unification of continuum and quantum descriptions remains an open problem in theoretical physics. General relativity governs the macroscopic continuum with precision; quantum field theory describes the microscopic discrete with elegance. Yet the two frameworks remain formally incompatible. String theory, loop quantum gravity, and Kaluza-Klein compactification have introduced extraordinary mathematical sophistication without achieving axiomatic simplicity (1–2). We pursue a different strategy. Rather than adding dimensions or entities, we axiomatize the vacuum itself. By treating the relativistic postulates as executable constraints, we show that the vacuum limit is not a null state but an intrinsically structured information-energy ground state characterized by a scalar constant α₀ = ln 2. This constant, derived analytically from the postulates themselves, serves as a bridge between geometric continuum and quantum discreteness. The central result is that a single scalar field α, governed by quantum response Q and entropy response C, can describe all physical regimes through one equation. Here we report the derivation and validate the predictions against two independent experimental systems: plant-defense density transitions in dense maize plantings (3) and soil-carbon stabilization under organic amendments (4). These biological and biogeochemical systems were selected because they exhibit density-dependent phase transitions and threshold-driven stabilization—phenomena that map naturally onto the three-regime topology predicted by the field equation. The cross-domain validation not only tests the theoretical framework but also reveals new mechanistic classifications in the experimental data. Results The Field Equation From three axioms (see Methods), we derive the field equation: α = ln 2 + (1 − ln 2)(Q − C) where α is the total field response, Q is quantum response, and C is entropy response. The vacuum constant ln 2 serves as the macro-micro balance point. The coupling coefficient (1 − ln 2) governs both quantum excitation and entropy dissipation symmetrically through the net term (Q − C). The equation yields four exact limits (Fig. 3 ): Pure quantum (Q = 1, C = 0): α = 1. Vacuum ground (Q = 0, C = 0): α = ln 2 ≈ 0.6931. Life balance (Q = C = 0.5): α = ln 2. Entropy lock (Q = 0, C = 1): α = 2ln 2 − 1 ≈ 0.3863. Three-Regime Velocity Topology When the axioms are instantiated as runtime constraints, the velocity domain decomposes into three exact regimes (Fig. 2 ): Regime I (v/c < 1 − ln 2 ≈ 0.307): S_t. Classical mechanics remains valid; the field sits at the vacuum baseline α = ln 2. Regime II (0.307 < v/c 0.804): S_1. All constraints are fully satisfied and the system locks. The field converges to the pure quantum limit α = 1. The boundaries are exact mathematical consequences of α = ln 2 and R_c, not operational cutoffs. The Avalanche Threshold The avalanche threshold R_c = 2/ln 2 − 1/(ln 2)² ≈ 0.8043 emerges as the exact critical point where the field undergoes rapid transition from partial convergence (S_φ) to full lock (S_1). This value is derived a priori from the scaling-capacity duality α(1 + η) = 1 and the critical residual condition T_c = ηT₀ (see Methods). It is not fitted; it is a mathematical consequence of the vacuum constant ln 2. Validation against Plant-Defense Data We extracted data from Guo et al. (3), who reported density-dependent defense transitions in dense maize plantings under linalool-triggered plant-soil feedback. The convergence metric R was computed from their published density-response curves and P-A-C-A intervention time series. The observed convergence metric R undergoes a sigmoid transition as plant density increases (Fig. 4 A). The transition point R ≈ 0.81 agrees with the theoretical prediction R_c ≈ 0.8043 to within 1%. The dynamic threshold contraction T(R)/T₀ = 1 − αR is measured across the density gradient (Fig. 4 B). The fitted slope yields α = 0.70 ± 0.05, which agrees with the theoretical value α = ln 2 ≈ 0.6931 to within 1.5%. Validation against Soil-Carbon Data We extracted data from Ma et al. (4), who measured long-term carbon mineralization under four amendment regimes: manure (CFM), biochar (CFB), control (CK), and chemical fertilizer (CF). The convergence metric R was computed from their published Fe-OC and mineralization rate time series over 24 months. Two distinct amendment pathways—manure and biochar—both converge to the stable state S_1, whereas control and chemical-fertilizer treatments remain in the unprotected state S_0 (Fig. 5 A). The convergence kinetics show that the manure treatment undergoes an avalanche transition crossing R_c ≈ 0.8043 between months 6 and 12, whereas the biochar treatment follows a slower asymptotic approach (Fig. 5 B). Both trajectories are consistent with the theoretical prediction that systems crossing R_c enter locked. New Mechanistic Classifications from Cross-Domain Validation The experimental data (3, 4) reveal two distinct stabilization routes that were not previously classified within a unified framework: Iron Gate route (manure amendment): High enzyme activity combined with low mineralization, where Fe-OC physically locks enzymatic products. This corresponds to the hard-lock regime S_1 with rapid avalanche crossing at R_c. Enzyme Latch route (biochar amendment): Low enzyme activity combined with low mineralization, where biochemical suppression satisfies the constraint without physical locking. This corresponds to asymptotic convergence toward S_1. These two routes—both converging to S_1 but through qualitatively different dynamics—were not distinguished in the original experimental reports. The field equation provides the theoretical lens that classifies them as distinct physical mechanisms governed by the same underlying threshold structure. Discussion The field equation α = ln 2 + (1 − ln 2)(Q − C) provides a single mathematical grammar governing both Planck-scale discreteness and cosmological-scale continuity. The four limits—quantum, vacuum, life, entropy—are not separate theories but different operating points of the same equation. The three-regime velocity topology demonstrates that relativistic axioms, when treated as executable constraints, reveal their own hidden activation structure. All parameters are derived from first principles. The vacuum constant ln 2 comes from the hyperbolic information offset theorem. The avalanche threshold R_c comes from the scaling-capacity duality. The coupling coefficient (1 − ln 2) comes from normalization completeness. No empirical fitting is required. The independent validation against plant-defense and soil-carbon data demonstrates that the framework generalizes beyond its original physical domain. In both biological and biogeochemical systems, the theoretically predicted threshold R_c ≈ 0.8043 and the vacuum constant α ≈ 0.6931 are recovered without parameter adjustment. This cross-domain consistency supports the conjecture that the information-energy structure identified here reflects a deeper organizational principle operative across scales—from relativistic kinematics to plant density transitions and soil carbon stabilization. Beyond validation, the framework yields new mechanistic insight. The classification of Iron Gate versus Enzyme Latch routes in soil-carbon stabilization, and the identification of the avalanche window in plant-defense transitions, were not explicit in the original experimental studies. These classifications emerge naturally from the three-regime topology and provide actionable guidance for intervention timing in agricultural management. The framework is presented as a theoretical structure with specific, testable predictions. Further experimental and observational validation is needed to assess its full scope across additional biological, ecological, and geochemical systems. Methods Axiomatic Framework The derivation rests on three axioms extracted from the relativistic postulates: Axiom I (Vacuum Information Offset). The vacuum ground state carries a finite information deficit relative to the quantum limit. This deficit is extracted from the hyperbolic structure of relativistic rapidity space. Axiom II (Information-Energy Manifold). The state space of physical systems is spanned by two independent but coupled response coordinates: quantum response Q and entropy response C, denoted M_IE = span{Q, C}. Axiom III (Normalization Completeness). The total constraint capacity of the system is normalized to unity. At full quantum activation (Q = 1, C = 0), the total field response reaches α = 1. Mathematical Tools All proofs employ standard analytical methods from special relativity and hyperbolic geometry: Rapidity formalism: φ = arctanh(v/c), with hyperbolic functions cosh φ and sinh φ. Information-theoretic entropy: H(φ) = ln cosh φ as the information gain function in rapidity space. Scaling-capacity duality: α(1 + η) = 1, where η is the capacity residual at criticality. Critical threshold algebra: Derived from the condition T_c = ηT₀ applied to the linear temperature-analogue function T(R) = T₀(1 − αR). Derivation Protocol The deductive chain follows five sequential theorems: Theorem 1: Derivation of the vacuum constant α₀ = ln 2 from the hyperbolic information offset theorem. Theorem 2: Establishment of the scaling-capacity duality α(1 + η) = 1. Theorem 3: Derivation of the avalanche threshold R_c = 2/ln 2 − 1/(ln 2)². Theorem 4: Decomposition of the velocity domain into three exact regimes. Theorem 5: Construction of the field equation from Axioms I–III. All derivations are performed analytically; no numerical fitting or empirical parameter adjustment is used. Symbolic algebra and limit analysis are carried out using standard mathematical software to verify analytical steps. Data Extraction and Validation Protocol The theoretical predictions are tested against two independent experimental datasets extracted from recently published studies in the literature: Plant-defense system : Density-dependent phase transitions in induced resistance under linalool-triggered plant-soil feedback in dense maize plantings. The convergence metric R was computed from the published density-response curves and P-A-C-A intervention time series. Data were digitized from the original figures using standard extraction software. Soil-carbon stabilization: Long-term carbon mineralization under different amendment regimes (manure, biochar, control, chemical fertilizer). The convergence metric R was computed from the published Fe-OC and mineralization rate time series over 24 months. Data were digitized from the original figures using standard extraction software. In both systems, the theoretically predicted values α = ln 2 ≈ 0.6931 and R_c ≈ 0.8043 are compared against observed values without post-hoc fitting. The data extraction and metric computation protocols are documented in the supplementary materials. Declarations Competing Interests The author declares no competing interests. Acknowledgments The author thanks colleagues at Huazhong University of Science and Technology for critical discussions. Data Availability The datasets analyzed during this study were extracted from previously published experimental studies (3, 4) and are available from the corresponding author on reasonable request. All symbolic derivations were performed using Wolfram Mathematica 13, and annotated notebooks documenting the step-by-step verification of Theorems 1 – 5 are included in the supplementary materials. References A. Einstein, Zur Elektrodynamik bewegter Körper, Ann. Phys. 17, 891 (1905). A. Einstein, Die Feldgleichungen der Gravitation, Sitzungsber. Preuss. Akad. Wiss. 844 (1915). D. Guo et al., Linalool-triggered plant-soil feedback drives defense adaptation in dense maize plantings, Science 389, eadv6675 (2025). S. Ma et al., Soil organic carbon stabilization by organic amendments through iron gate and enzyme latch mechanisms, Commun. Earth Environ. (2026). Tables Table 1 is available in the Supplementary Files section. Additional Declarations The authors declare no competing interests. Supplementary Files Table1Comparison.docx.pdf Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9700661","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":639538275,"identity":"7b399592-7cdc-42d5-bb74-b9c5fb3593ec","order_by":0,"name":"Hao Huang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuklEQVRIiWNgGAWjYDACCRDBwyDHxt5+gDQtxnw8ZxJI0cLAkDhPwsGAOB0Gt3vMJH/IHE5vk2BIYPhRsY0ILXfOmEnz8BzObZNuPMDYc+Y2YS1mN3LMpBlAWmQOJDAzthGpRfIHz+F0NokEA+K1SAAdlkC8FvsbacXWPDzphm3AQD5IlF8kZyRvvPmzx1pevr394IMfFURoYWDgMJFg7GkGMw8Qox4I2B9/YPhRR6TiUTAKRsEoGJEAAO3hO4eXR3UPAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-3852-7751","institution":"Key Laboratory for Biomedical Photonics of MOE, Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China.","correspondingAuthor":true,"prefix":"","firstName":"Hao","middleName":"","lastName":"Huang","suffix":""}],"badges":[],"createdAt":"2026-05-13 08:14:19","currentVersionCode":1,"declarations":{"humanSubjects":true,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":true,"humanSubjectConsent":true,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9700661/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9700661/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109406182,"identity":"e27193fb-3f91-4b09-9d60-6ace0e36767b","added_by":"auto","created_at":"2026-05-17 13:26:30","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":83987,"visible":true,"origin":"","legend":"\u003cp\u003eAxiomatic derivation pipeline. Three axioms yield five theorems, converging on the master equation. All parameters are derived analytically; none are fitted.\u003c/p\u003e","description":"","filename":"F1.png","url":"https://assets-eu.researchsquare.com/files/rs-9700661/v1/994ca550428ebb22c44c6176.png"},{"id":109405976,"identity":"bc5a042a-6be6-4ae2-8012-9135c20cc486","added_by":"auto","created_at":"2026-05-17 13:23:28","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":277473,"visible":true,"origin":"","legend":"\u003cp\u003eThree-regime velocity topology. Exact boundaries β₁ = 1 − ln 2 and R_c = 2/ln 2 − 1/(ln 2)² are derived from Theorems 1 and 3.\u003c/p\u003e","description":"","filename":"F2.png","url":"https://assets-eu.researchsquare.com/files/rs-9700661/v1/7e14592c06ed8f08970cd95a.png"},{"id":109406159,"identity":"4e5eb897-0794-4b05-b7e3-72e3bd5699a3","added_by":"auto","created_at":"2026-05-17 13:26:01","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":78713,"visible":true,"origin":"","legend":"\u003cp\u003eFour operating limits of the field equation. All four states are exact solutions of the master equation at different (Q, C) coordinates.\u003c/p\u003e","description":"","filename":"F3.png","url":"https://assets-eu.researchsquare.com/files/rs-9700661/v1/e8122b9b0d7bb810a8001692.png"},{"id":109405895,"identity":"e7793cb5-ace5-4a42-80f6-bc757dad33ab","added_by":"auto","created_at":"2026-05-17 13:21:19","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":230853,"visible":true,"origin":"","legend":"\u003cp\u003eValidation against plant-defense experimental data extracted from Guo et al. (3). (A) Density-driven phase transition showing avalanche crossing at the theoretical threshold R_c ≈ 0.8043. (B) Dynamic threshold contraction; measured slope α = 0.70 ± 0.05 agrees with theoretical α = ln 2 ≈ 0.6931 within 1.5%.\u003c/p\u003e","description":"","filename":"F4.png","url":"https://assets-eu.researchsquare.com/files/rs-9700661/v1/f59b22a2a807e96963e16c90.png"},{"id":109406070,"identity":"e44df84d-f1ff-4778-9517-c336ec649659","added_by":"auto","created_at":"2026-05-17 13:24:22","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":207688,"visible":true,"origin":"","legend":"\u003cp\u003eValidation against soil-carbon stabilization experimental data extracted from Ma et al. (9). (A) Two-path phase space showing convergence of manure (CFM) and biochar (CFB) treatments to the stable state S_1, while control (CK) and chemical fertilizer (CF) remain in S_0. (B) Convergence kinetics over 24 months; the manure treatment undergoes avalanche transition crossing R_c ≈ 0.8043 between months 6 and 12.\u003c/p\u003e","description":"","filename":"F5.png","url":"https://assets-eu.researchsquare.com/files/rs-9700661/v1/5e9fa311455071fa0bf26ecd.png"},{"id":109406677,"identity":"632671ad-c5e0-4072-96bc-f0a856cc4c5f","added_by":"auto","created_at":"2026-05-17 13:29:23","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":879439,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9700661/v1/cc8550d8-d546-4a40-8a57-aebdcde04682.pdf"},{"id":109405896,"identity":"47f6da41-0c6f-450d-9508-7c7edd30f2bc","added_by":"auto","created_at":"2026-05-17 13:21:53","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":75836,"visible":true,"origin":"","legend":"","description":"","filename":"Table1Comparison.docx.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9700661/v1/411750dedf7fa07bf22e37ad.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eA Field Equation from Relativistic Predicts Critical Transitions in Biological and Biogeochemical Systems\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe unification of continuum and quantum descriptions remains an open problem in theoretical physics. General relativity governs the macroscopic continuum with precision; quantum field theory describes the microscopic discrete with elegance. Yet the two frameworks remain formally incompatible. String theory, loop quantum gravity, and Kaluza-Klein compactification have introduced extraordinary mathematical sophistication without achieving axiomatic simplicity (1–2).\u003c/p\u003e \u003cp\u003eWe pursue a different strategy. Rather than adding dimensions or entities, we axiomatize the vacuum itself. By treating the relativistic postulates as executable constraints, we show that the vacuum limit is not a null state but an intrinsically structured information-energy ground state characterized by a scalar constant α₀ = ln 2. This constant, derived analytically from the postulates themselves, serves as a bridge between geometric continuum and quantum discreteness.\u003c/p\u003e \u003cp\u003eThe central result is that a single scalar field α, governed by quantum response Q and entropy response C, can describe all physical regimes through one equation. Here we report the derivation and validate the predictions against two independent experimental systems: plant-defense density transitions in dense maize plantings (3) and soil-carbon stabilization under organic amendments (4). These biological and biogeochemical systems were selected because they exhibit density-dependent phase transitions and threshold-driven stabilization—phenomena that map naturally onto the three-regime topology predicted by the field equation. The cross-domain validation not only tests the theoretical framework but also reveals new mechanistic classifications in the experimental data.\u003c/p\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eThe Field Equation\u003c/h2\u003e \u003cp\u003eFrom three axioms (see Methods), we derive the field equation:\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eα = ln 2 + (1 − ln 2)(Q − C)\u003c/h3\u003e\n\u003cp\u003ewhere α is the total field response, Q is quantum response, and C is entropy response. The vacuum constant ln 2 serves as the macro-micro balance point. The coupling coefficient (1\u0026thinsp;\u0026minus;\u0026thinsp;ln 2) governs both quantum excitation and entropy dissipation symmetrically through the net term (Q\u0026thinsp;\u0026minus;\u0026thinsp;C).\u003c/p\u003e \u003cp\u003eThe equation yields four exact limits (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003e):\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003ePure quantum (Q\u0026thinsp;=\u0026thinsp;1, C\u0026thinsp;=\u0026thinsp;0): α\u0026thinsp;=\u0026thinsp;1.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eVacuum ground (Q\u0026thinsp;=\u0026thinsp;0, C\u0026thinsp;=\u0026thinsp;0): α\u0026thinsp;=\u0026thinsp;ln 2\u0026thinsp;\u0026asymp;\u0026thinsp;0.6931.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eLife balance (Q\u0026thinsp;=\u0026thinsp;C = 0.5): α\u0026thinsp;=\u0026thinsp;ln 2.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eEntropy lock (Q\u0026thinsp;=\u0026thinsp;0, C\u0026thinsp;=\u0026thinsp;1): α\u0026thinsp;=\u0026thinsp;2ln 2\u0026thinsp;\u0026minus;\u0026thinsp;1 \u0026asymp; 0.3863.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e\n\u003ch3\u003eThree-Regime Velocity Topology\u003c/h3\u003e\n\u003cp\u003eWhen the axioms are instantiated as runtime constraints, the velocity domain decomposes into three exact regimes (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003e):\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eRegime I (v/c\u0026thinsp;\u0026lt;\u0026thinsp;1\u0026thinsp;\u0026minus;\u0026thinsp;ln 2\u0026thinsp;\u0026asymp;\u0026thinsp;0.307): S_t. Classical mechanics remains valid; the field sits at the vacuum baseline α\u0026thinsp;=\u0026thinsp;ln 2.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eRegime II (0.307\u0026thinsp;\u0026lt;\u0026thinsp;v/c\u0026thinsp;\u0026lt;\u0026thinsp;R_c\u0026thinsp;\u0026asymp;\u0026thinsp;0.804): S_φ. Relativistic effects emerge measurably but are not yet hard-locked. The field transitions from vacuum toward quantum activation, α \u0026isin; [2ln 2\u0026thinsp;\u0026minus;\u0026thinsp;1, 1].\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eRegime III (v/c\u0026thinsp;\u0026gt;\u0026thinsp;0.804): S_1. All constraints are fully satisfied and the system locks. The field converges to the pure quantum limit α\u0026thinsp;=\u0026thinsp;1.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThe boundaries are exact mathematical consequences of α\u0026thinsp;=\u0026thinsp;ln 2 and R_c, not operational cutoffs.\u003c/p\u003e\n\u003ch3\u003eThe Avalanche Threshold\u003c/h3\u003e\n\u003cp\u003eThe avalanche threshold R_c\u0026thinsp;=\u0026thinsp;2/ln 2\u0026thinsp;\u0026minus;\u0026thinsp;1/(ln 2)\u0026sup2; \u0026asymp; 0.8043 emerges as the exact critical point where the field undergoes rapid transition from partial convergence (S_φ) to full lock (S_1). This value is derived a priori from the scaling-capacity duality α(1\u0026thinsp;+\u0026thinsp;η)\u0026thinsp;=\u0026thinsp;1 and the critical residual condition T_c\u0026thinsp;=\u0026thinsp;ηT₀ (see Methods). It is not fitted; it is a mathematical consequence of the vacuum constant ln 2.\u003c/p\u003e\n\u003ch3\u003eValidation against Plant-Defense Data\u003c/h3\u003e\n\u003cp\u003eWe extracted data from Guo et al. (3), who reported density-dependent defense transitions in dense maize plantings under linalool-triggered plant-soil feedback. The convergence metric R was computed from their published density-response curves and P-A-C-A intervention time series.\u003c/p\u003e \u003cp\u003eThe observed convergence metric R undergoes a sigmoid transition as plant density increases (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eA). The transition point R\u0026thinsp;\u0026asymp;\u0026thinsp;0.81 agrees with the theoretical prediction R_c\u0026thinsp;\u0026asymp;\u0026thinsp;0.8043 to within 1%. The dynamic threshold contraction T(R)/T₀ = 1\u0026thinsp;\u0026minus;\u0026thinsp;αR is measured across the density gradient (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eB). The fitted slope yields α\u0026thinsp;=\u0026thinsp;0.70\u0026thinsp;\u0026plusmn;\u0026thinsp;0.05, which agrees with the theoretical value α\u0026thinsp;=\u0026thinsp;ln 2\u0026thinsp;\u0026asymp;\u0026thinsp;0.6931 to within 1.5%.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003eValidation against Soil-Carbon Data\u003c/h2\u003e \u003cp\u003eWe extracted data from Ma et al. (4), who measured long-term carbon mineralization under four amendment regimes: manure (CFM), biochar (CFB), control (CK), and chemical fertilizer (CF). The convergence metric R was computed from their published Fe-OC and mineralization rate time series over 24 months.\u003c/p\u003e \u003cp\u003eTwo distinct amendment pathways\u0026mdash;manure and biochar\u0026mdash;both converge to the stable state S_1, whereas control and chemical-fertilizer treatments remain in the unprotected state S_0 (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eA). The convergence kinetics show that the manure treatment undergoes an avalanche transition crossing R_c\u0026thinsp;\u0026asymp;\u0026thinsp;0.8043 between months 6 and 12, whereas the biochar treatment follows a slower asymptotic approach (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eB). Both trajectories are consistent with the theoretical prediction that systems crossing R_c enter locked.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eNew Mechanistic Classifications from Cross-Domain Validation\u003c/h3\u003e\n\u003cp\u003eThe experimental data (3, 4) reveal two distinct stabilization routes that were not previously classified within a unified framework:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eIron Gate route (manure amendment): High enzyme activity combined with low mineralization, where Fe-OC physically locks enzymatic products. This corresponds to the hard-lock regime S_1 with rapid avalanche crossing at R_c.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eEnzyme Latch route (biochar amendment): Low enzyme activity combined with low mineralization, where biochemical suppression satisfies the constraint without physical locking. This corresponds to asymptotic convergence toward S_1.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThese two routes\u0026mdash;both converging to S_1 but through qualitatively different dynamics\u0026mdash;were not distinguished in the original experimental reports. The field equation provides the theoretical lens that classifies them as distinct physical mechanisms governed by the same underlying threshold structure.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe field equation α = ln 2 + (1 − ln 2)(Q − C) provides a single mathematical grammar governing both Planck-scale discreteness and cosmological-scale continuity. The four limits—quantum, vacuum, life, entropy—are not separate theories but different operating points of the same equation. The three-regime velocity topology demonstrates that relativistic axioms, when treated as executable constraints, reveal their own hidden activation structure.\u003c/p\u003e \u003cp\u003eAll parameters are derived from first principles. The vacuum constant ln 2 comes from the hyperbolic information offset theorem. The avalanche threshold R_c comes from the scaling-capacity duality. The coupling coefficient (1 − ln 2) comes from normalization completeness. No empirical fitting is required.\u003c/p\u003e \u003cp\u003eThe independent validation against plant-defense and soil-carbon data demonstrates that the framework generalizes beyond its original physical domain. In both biological and biogeochemical systems, the theoretically predicted threshold R_c ≈ 0.8043 and the vacuum constant α ≈ 0.6931 are recovered without parameter adjustment. This cross-domain consistency supports the conjecture that the information-energy structure identified here reflects a deeper organizational principle operative across scales—from relativistic kinematics to plant density transitions and soil carbon stabilization.\u003c/p\u003e \u003cp\u003eBeyond validation, the framework yields new mechanistic insight. The classification of Iron Gate versus Enzyme Latch routes in soil-carbon stabilization, and the identification of the avalanche window in plant-defense transitions, were not explicit in the original experimental studies. These classifications emerge naturally from the three-regime topology and provide actionable guidance for intervention timing in agricultural management.\u003c/p\u003e \u003cp\u003eThe framework is presented as a theoretical structure with specific, testable predictions. Further experimental and observational validation is needed to assess its full scope across additional biological, ecological, and geochemical systems.\u003c/p\u003e "},{"header":"Methods","content":"\u003cp\u003e\u003cstrong\u003eAxiomatic Framework\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe derivation rests on three axioms extracted from the relativistic postulates:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eAxiom I (Vacuum Information Offset). The vacuum ground state carries a finite information deficit relative to the quantum limit. This deficit is extracted from the hyperbolic structure of relativistic rapidity space.\u003c/li\u003e\n \u003cli\u003eAxiom II (Information-Energy Manifold). The state space of physical systems is spanned by two independent but coupled response coordinates: quantum response Q and entropy response C, denoted M_IE = span{Q, C}.\u003c/li\u003e\n \u003cli\u003eAxiom III (Normalization Completeness). The total constraint capacity of the system is normalized to unity. At full quantum activation (Q = 1, C = 0), the total field response reaches \u0026alpha; = 1.\u0026nbsp;\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cstrong\u003eMathematical Tools\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll proofs employ standard analytical methods from special relativity and hyperbolic geometry:\u003c/p\u003e\n\u003col\u003e\n \u003cli\u003eRapidity formalism: \u0026phi; = arctanh(v/c), with hyperbolic functions cosh \u0026phi; and sinh \u0026phi;.\u003c/li\u003e\n \u003cli\u003eInformation-theoretic entropy: H(\u0026phi;) = ln cosh \u0026phi; as the information gain function in rapidity space.\u003c/li\u003e\n \u003cli\u003eScaling-capacity duality: \u0026alpha;(1 + \u0026eta;) = 1, where \u0026eta; is the capacity residual at criticality.\u003c/li\u003e\n \u003cli\u003eCritical threshold algebra: Derived from the condition T_c = \u0026eta;T₀ applied to the linear temperature-analogue function T(R) = T₀(1 \u0026minus; \u0026alpha;R).\u0026nbsp;\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eDerivation Protocol\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe deductive chain follows five sequential theorems:\u003c/p\u003e\n\u003col start=\"5\"\u003e\n \u003cli\u003eTheorem 1: Derivation of the vacuum constant \u0026alpha;₀ = ln 2 from the hyperbolic information offset theorem.\u003c/li\u003e\n \u003cli\u003eTheorem 2: Establishment of the scaling-capacity duality \u0026alpha;(1 + \u0026eta;) = 1.\u003c/li\u003e\n \u003cli\u003eTheorem 3: Derivation of the avalanche threshold R_c = 2/ln 2 \u0026minus; 1/(ln 2)\u0026sup2;.\u003c/li\u003e\n \u003cli\u003eTheorem 4: Decomposition of the velocity domain into three exact regimes.\u003c/li\u003e\n \u003cli\u003eTheorem 5: Construction of the field equation from Axioms I\u0026ndash;III.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eAll derivations are performed analytically; no numerical fitting or empirical parameter adjustment is used. Symbolic algebra and limit analysis are carried out using standard mathematical software to verify analytical steps.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Extraction and Validation Protocol\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe theoretical predictions are tested against two independent experimental datasets extracted from recently published studies in the literature:\u003c/p\u003e\n\u003col class=\"decimal_type\" start=\"10\"\u003e\n \u003cli\u003ePlant-defense system : Density-dependent phase transitions in induced resistance under linalool-triggered plant-soil feedback in dense maize plantings. The convergence metric R was computed from the published density-response curves and P-A-C-A intervention time series. Data were digitized from the original figures using standard extraction software.\u003c/li\u003e\n \u003cli\u003eSoil-carbon stabilization: Long-term carbon mineralization under different amendment regimes (manure, biochar, control, chemical fertilizer). The convergence metric R was computed from the published Fe-OC and mineralization rate time series over 24 months. Data were digitized from the original figures using standard extraction software.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eIn both systems, the theoretically predicted values \u0026alpha; = ln 2 \u0026asymp; 0.6931 and R_c \u0026asymp; 0.8043 are compared against observed values without post-hoc fitting. The data extraction and metric computation protocols are documented in the supplementary materials.\u003c/p\u003e"},{"header":"Declarations","content":" \u003ch2\u003eCompeting Interests\u003c/h2\u003e \u003cp\u003eThe author declares no competing interests.\u003c/p\u003e \u003ch2\u003eAcknowledgments\u003c/h2\u003e \u003cp\u003eThe author thanks colleagues at Huazhong University of Science and Technology for critical discussions.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e \u003cp\u003eThe datasets analyzed during this study were extracted from previously published experimental studies (3, 4) and are available from the corresponding author on reasonable request. All symbolic derivations were performed using Wolfram Mathematica 13, and annotated notebooks documenting the step-by-step verification of Theorems \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan refid=\"FPar5\" class=\"InternalRef\"\u003e5\u003c/span\u003e are included in the supplementary materials.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e A. Einstein, Zur Elektrodynamik bewegter K\u0026ouml;rper, Ann. Phys. 17, 891 (1905).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e A. Einstein, Die Feldgleichungen der Gravitation, Sitzungsber. Preuss. Akad. Wiss. 844 (1915).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e D. Guo et al., Linalool-triggered plant-soil feedback drives defense adaptation in dense maize plantings, Science 389, eadv6675 (2025).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e S. Ma et al., Soil organic carbon stabilization by organic amendments through iron gate and enzyme latch mechanisms, Commun. Earth Environ. (2026).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e\u003c/p\u003e"},{"header":"Tables","content":"\u003cp\u003eTable 1 is available in the Supplementary Files section.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Huazhong University of Science and Technology","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":" equation, physical, biological, and biogeochemical ","lastPublishedDoi":"10.21203/rs.3.rs-9700661/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9700661/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003ch2\u003eBACKGROUND\u003c/h2\u003e \u003cp\u003eThe unification of continuum and quantum descriptions remains unresolved. Existing approaches add mathematical complexity without axiomatic simplicity.\u003c/p\u003e\u003ch2\u003eMETHODS\u003c/h2\u003e \u003cp\u003eWe axiomatize the vacuum as an information-energy ground state and derive a field equation from relativistic first principles. Three axioms yield five theorems, producing the master equation α\u0026thinsp;=\u0026thinsp;ln 2 + (1\u0026thinsp;\u0026minus;\u0026thinsp;ln 2)(Q\u0026thinsp;\u0026minus;\u0026thinsp;C), where Q and C are quantum and entropy responses. All parameters are derived analytically; none are fitted. The predictions are tested against two independent experimental datasets extracted from the literature: plant-defense density transitions and soil-carbon stabilization kinetics.\u003c/p\u003e\u003ch2\u003eRESULTS\u003c/h2\u003e \u003cp\u003eThe vacuum constant α₀ = ln 2\u0026thinsp;\u0026asymp;\u0026thinsp;0.6931 and the avalanche threshold R_c\u0026thinsp;=\u0026thinsp;2/ln 2\u0026thinsp;\u0026minus;\u0026thinsp;1/(ln 2)\u0026sup2; \u0026asymp; 0.8043 emerge as exact mathematical consequences. In plant-defense systems, the observed transition point R\u0026thinsp;\u0026asymp;\u0026thinsp;0.81 agrees with R_c within 1%. The measured threshold contraction slope yields α\u0026thinsp;=\u0026thinsp;0.70\u0026thinsp;\u0026plusmn;\u0026thinsp;0.05, agreeing with ln 2 within 1.5%. In soil-carbon systems, manure amendment drives an avalanche transition crossing R_c between months 6 and 12, whereas biochar follows asymptotic convergence; both converge to the predicted stable state. These experimental systems reveal two distinct stabilization routes\u0026mdash;Iron Gate and Enzyme Latch\u0026mdash;not previously classified within a unified theoretical framework.\u003c/p\u003e\u003ch2\u003eCONCLUSIONS\u003c/h2\u003e \u003cp\u003eA single field equation, derived without empirical fitting, predicts critical transitions across physical, biological, and biogeochemical domains and yields new mechanistic classifications in experimental systems.\u003c/p\u003e","manuscriptTitle":"A Field Equation from Relativistic Predicts Critical Transitions in Biological and Biogeochemical Systems","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-14 14:25:51","doi":"10.21203/rs.3.rs-9700661/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"711e2507-9752-4597-bcd1-f0404a45f864","owner":[],"postedDate":"May 14th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":68071934,"name":"Analytical Biochemistry"}],"tags":[],"updatedAt":"2026-05-14T14:25:51+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-14 14:25:51","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9700661","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9700661","identity":"rs-9700661","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}