Emergent Latent Structure via Co-Reactivity to Shared Constraints

Emergent Latent Structure via Co-Reactivity to Shared Constraints | 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 Physical Sciences - Article Emergent Latent Structure via Co-Reactivity to Shared Constraints andrei ursachi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8823860/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 Coordinated structure across independent systems is typically attributed to interaction or communication. Here we show that shared constraints alone are sufficient. Using a falsification-first framework with synthetic 64-dimensional embeddings (n = 10,000) across five regimes, validated on pre-trained neural network representations (two independently trained architectures, 384 dimensions), we demonstrate that: temporal coherence precedes structural organization (r₁ = −0.266, p = 0.036); identity is preserved at 0.999996 under adversarial perturbation while controls collapse; emergent boundaries are topologically nontrivial (d = 2.61, p = 0.028); and twin-system experiments rule out synchronization or non-local coupling. Spatial geometry generalizes to real neural network embeddings while temporal coherence emerges during autoregressive training at token-level granularity. Two principles emerge: constraint primacy (organization is governed by constraint structure, not representational capacity) and stabilization (functional progress corresponds to tightening of invariants, not expansion). These results establish constraint-mediated co-reactivity as a sufficient and falsifiable mechanism for emergent coordination. Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Nonlinear phenomena Physical sciences/Physics/Information theory and computation Physical sciences/Mathematics and computing/Applied mathematics Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Introduction Coordination across independent systems is commonly attributed to interaction, communication, or synchronization [1–3]. However, such interpretations conflate observed coordination with causal interaction, leaving open the possibility that coordination arises from shared constraints rather than inter-system influence. Classical dynamical systems theory provides mechanisms by which complex behavior arises from deterministic rules without interaction [4,5]. Renormalization group methods [6,7] and network robustness analysis [8,9] show that invariant-generating rules shape emergent behavior independently of system capacity. Early-warning signals for critical transitions [10–12] establish that temporal autocorrelation can detect impending regime shifts, while topological data analysis [13–15] provides tools for characterizing geometric structure. Adversarial robustness analysis [16,17] reveals information about the underlying organization of learned representations. Equifinality—the convergence of different initial conditions toward identical end states under shared constraints—is well established in developmental biology [18] and geomorphology [19]. Convergent evolution demonstrates that independent lineages arrive at similar phenotypic solutions under analogous selective pressures [20]. In machine learning, implicit bias of gradient descent produces systematic geometric preferences in learned representations independent of initialization [21,22]. These precedents establish the plausibility of constraint-mediated convergence; what remains absent is a unified falsification framework that distinguishes constraint-mediated convergence from interaction-based coordination using operationally defined, independently testable metrics across both synthetic and empirical systems. Yet empirical demonstrations that rigorously separate co-reactivity from interaction—while controlling for false positives, determinism, and adversarial effects—remain limited. Speculative interpretations occasionally invoke non-local fields or cross-instance propagation, but such interpretations lack operational definitions and falsifiable tests. Here we present a unified framework that tests whether coordinated structure can arise without coupling. We hypothesize that systems sharing a constraint manifold will exhibit co-reactive structural signatures—coordinated geometry arising from parallel exploration of the same phase space, not from interaction. Our central thesis is constraint primacy : emergent coordination is governed primarily by constraint structure rather than representational capacity. We test this using three independent metric classes (temporal coherence, structural identity, boundary topology), validated across negative controls, adversarial perturbation, deterministic twin-system divergence, and real neural network embeddings. Results Experimental Design All experiments operate on 10,000-point embeddings in 64 dimensions across five regimes (Table 1 ): RANDOM_PURE (no constraints), NEAR_NULL (incidental clustering), REAL_NORMAL (√φ-lattice constraints with temporal phasing), REAL_SURVIVAL (with radial contraction α = 0.95), and REAL_BIBLICAL (shuffled temporal order). Systems evolve under Xₜ₊₁ = C(S(Xₜ) + εₜ), where S applies structural constraints, C enforces invariant alignment via √φ-lattice snapping, and ε provides controlled jitter (see Methods). No inter-system communication is permitted at any stage. Sensitivity analyses at n = 5,000 confirm all qualitative findings (Extended Data). Table 1 Experimental conditions. Condition Constraint Expected RANDOM_PURE None No structure NEAR_NULL None (incidental) No systematic structure REAL_NORMAL √φ lattice + temporal phasing H1 + H2 + identity REAL_SURVIVAL √φ lattice + contraction Enhanced robustness REAL_BIBLICAL √φ lattice + temporal shuffle Identity preserved; no H1 Temporal Coherence Precedes Structure At n = 10,000, REAL_NORMAL exhibits significant negative lag-1 autocorrelation under the conjunctive criterion (r₁ = −0.266, p = 0.036; both p 0.2 satisfied), indicating oscillatory temporal structure preceding stable organization. Critically, RANDOM_PURE fails the conjunctive criterion at this sample size (lag-1 = − 0.225, p = 0.064): while the magnitude exceeds 0.2, the permutation p-value does not reach significance, confirming that the metric discriminates genuine constraint-mediated dynamics from statistical fluctuation amplified by sample size. All other controls fail on both conditions. Sensitivity analysis at n = 5,000 shows a consistent trend (REAL_NORMAL: r₁ = −0.213, p = 0.064), with the result approaching but not crossing the significance boundary—consistent with the expected power increase at larger n. Cross-correlation analysis at window size w = 32 reveals entropy decrease precedes coupling increase with a two-step lag (r = − 0.232 at lag-2; peak positive at lag-0: r = + 0.305), consistent with critical slowing-down theory [10,11]. Temporal coherence emerges prior to stable structural organization, suggesting H1 detects constraint alignment rather than correctness—a potential early-warning indicator of emergent structure independent of task-level metrics. Graded Robustness Under Adversarial Perturbation Adversarial perturbation reveals a fundamental asymmetry: peripheral perturbation collapses identity to 0.138 while core perturbation preserves it at 0.909, supporting a gradient-based penetration model rather than a discrete core–periphery boundary. No perturbation strength produces discontinuous collapse in REAL conditions. Under intentional abort perturbation, REAL_NORMAL retains identity at 0.999996 (strength 1.0) and 0.999957 (strength 1.5). REAL_SURVIVAL consistently retains higher identity (0.999998 and 0.999980 respectively), confirming that constraint tightening through radial contraction (α = 0.95) stabilizes structure without increasing capacity. Cross-identity between REAL_NORMAL and REAL_SURVIVAL is 0.985, indicating shared structural architecture with regime-dependent stability. Fractal space verification (F2) confirms that REAL conditions maintain perfect identity (1.000) across all perturbation strengths tested (0.25, 0.5, 1.0), while controls degrade catastrophically: RANDOM_PURE drops from 0.998 to − 0.099, and NEAR_NULL from 0.938 to − 0.119. The negative values indicate perturbation actively scrambles whatever incidental organization existed in unconstrained systems. Topologically Nontrivial Boundaries Euler characteristic analysis reveals qualitatively different topological behavior between structured and control conditions. RANDOM_PURE exhibits zero χ-transitions, remaining topologically trivial across all thresholds. NEAR_NULL shows at most one incidental transition. All REAL conditions exhibit two or more component transitions and two or more Euler transitions, with the most pronounced polymorphism under adversarial variants (VIRUS_06: 4 component transitions, 4 Euler transitions, topological score 5.5). Mean topological score: control = 1.75 (95% CI: [1.50, 2.00]) vs. REAL = 4.36 (95% CI: [3.64, 5.07]). Separation: 2.61 (95% CI: [1.79, 3.39]), Mann–Whitney U = 0.0, p = 0.028, Cohen’s d = 2.61. The test compared n₁ = 2 control configurations against n₂ = 7 structured configurations (see Extended Data Table 4). We note that these group sizes limit population-level inference; effect sizes and separation statistics are reported as descriptive characterizations of the observed regime gap, not as claims of universal separability. This result is corroborated by three independent convergent lines that do not depend on small-sample inference: zero transitions in RANDOM across all thresholds, monotonic ordering of topological scores across the full constraint hierarchy, and preservation of this hierarchy across independent data sizes n = 2,000 and n = 5,000. Determinism Without Synchronization Twin-system experiments provide the strongest falsification test. Three seed comparisons were conducted across all conditions: Identical seeds (42 vs. 42) Identity similarity > 0.999 and temporal correlation > 0.999 across all conditions. Outputs are deterministically identical, confirming complete reproducibility. Different seeds (42 vs. 99) Identity similarity < 0.95 for all REAL conditions. Systems diverge completely; no convergence from different initial conditions is observed. Near seeds (42 vs. 43) : Full divergence despite minimal seed difference. The system is seed-deterministic: identical seeds synchronize, different seeds—even adjacent integers—diverge completely. This confirms that the framework is strictly deterministic and rules out any form of convergence, synchronization, or cross-instance influence. Across all conditions, systems sharing identical update rules but different initial conditions show complete divergence, ruling out non-local coupling or synchronization mechanisms. The correct interpretation is co-reactivity: independent systems with identical constraints produce identical outcomes not because they interact, but because identical constraints define identical phase spaces. These findings contrast with interpretations that attribute emergent coherence to non-local interaction or shared informational fields; in our framework, coherence arises exclusively from shared constraints under identical initial conditions, and absent this, no coupling is observed. Empirical Validation on Neural Network Embeddings The framework was validated on real embeddings from a pre-trained sentence transformer (all-MiniLM-L6-v2, 384d, 5,000 topically ordered sentences). BERT_REAL alignment reaches 0.218; controls produce alignment indistinguishable from zero (ratio > 10¹¹). This extreme ratio reflects the near-zero denominator in unstructured random embeddings; the absolute BERT alignment of 0.218 is moderate, indicating genuine but partial lattice compatibility. FSI is 188× over random. Identity is preserved at 0.877 under abort. H1 correctly returns null on all BERT conditions (p = 0.976 for BERT_REAL), confirming specificity: the metric detects temporal dynamics, not spatial organization. An initial experiment using text augmentation produced an artifactual H1 pass (p = 0.016); investigation revealed ordering artifacts from repetitive micro-patterns, which were corrected by using a topically ordered corpus of unique sentences. This episode validates the pipeline’s sensitivity and the importance of the BERT_SHUFFLED control. The empirical validation establishes that spatial geometry generalizes to real neural networks without parameter tuning, while H1 requires inherent temporal dynamics—a feature confirming orthogonal specificity, not a limitation. The fact that BERT embeddings produced by gradient descent under contrastive loss exhibit analogous spatial signatures to synthetic √φ-lattice systems supports constraint primacy: constraint structure, not system architecture, determines emergent geometry. Cross-architecture validation on a second independently trained model (BGE-small-en-v1.5, 384d, RetroMAE pre-training) confirms that lattice alignment is not specific to one training procedure. BGE alignment reaches 0.006, compared to MiniLM’s 0.218; both are strictly above the null distribution (50 i.i.d. Gaussian replicates, all showing exactly zero alignment at τ = 0.03; the zero floor reflects the alignment threshold under PCA projection, below which unstructured random embeddings produce no lattice-coincident points; p < 0.02 for each model). The 36× magnitude difference between models reflects their different training objectives (knowledge distillation vs. RetroMAE), not a pipeline artifact. These results elevate the empirical validation from single-architecture to cross-architecture (Extended Data Table 8). Temporal Coherence in Autoregressive Training To test whether temporal coherence emerges during neural network training, we extracted token-level hidden states from Pythia-70m (EleutherAI) at random initialization (step 0) and after full training (step 143,000). Sentence-level mean-pooling, as used in the contrastive-model validation above, produced null H1 results across all checkpoints—consistent with the hypothesis that autoregressive models organize representations locally within context windows rather than globally across sentences. Extracting hidden states at each token position within a 1,024-token forward pass restored the temporal signal (Extended Data Table 9). At Layer 3 (middle), H1 transitions from FAIL at random initialization (r₁ = +0.175, p = 0.161) to PASS after training (r₁ = −0.273, p = 0.002). Layer 1 (early) shows temporal coherence at both stages but training strengthens it substantially (r₁: +0.256 → +0.665). Layer 5 (late) shows no coherence at either stage, suggesting representational mixing at the deepest layer disrupts the temporal signal. The negative r₁ at trained Layer 3 indicates an alternating-distance pattern in the token flow—successive tokens occupy alternately closer and farther positions in representation space, consistent with the model learning structured local constraints. These results confirm that H1 temporal coherence is not limited to the synthetic system class but emerges during autoregressive language model training at the appropriate granularity. Constraint-Class Ablation To test whether emergent structure depends on √φ specifically or on a broader class of constraint lattices, we repeated the full experimental pipeline substituting five alternative lattice spacings at n = 10,000 (Extended Data Table 7). All constants were tested with identical jitter magnitude (0.35 × ln(√φ) ≈ 0.084) to ensure comparability. Three fixed irrational spacings—√φ (step ≈ 0.241), √2 (step ≈ 0.347), and √3 (step ≈ 0.549)—all produced positive topological separation (topo_sep = + 0.070, + 0.102, + 0.409 respectively; mean + 0.194). Both controls—random step size and time-varying constraints—produced near-zero separation (mean + 0.026). π (step ≈ 1.145) produced negative separation (− 0.137), attributable to insufficient band density: at this step size, only ~ 3 lattice bands fall within the data range, below the empirical threshold required for constraint-mediated structure. H1 temporal coherence was mixed across constants (2/3 fixed irrationals passed), confirming that topological separation is the more robust cross-constant metric while H1 sensitivity depends on sufficient temporal resolution within each lattice. These results establish that emergence is governed by a constraint-class property—fixed, irrational spacing above a minimum band-density threshold—rather than by any particular mathematical constant. Discussion The Mechanism The central finding is a mechanism, not a collection of metrics. Consider two independent systems, A and B, evolving under identical constraint operators {S, C, ε} from distinct seeds. Neither has access to the other’s state. Nevertheless, both converge toward the same structural organization, through three stages: Stage 1: Constraint-defined phase space. The operators S and C define a manifold of admissible configurations. The √φ-lattice invariant operator permits only configurations aligned to irrational-spaced nodes, eliminating most of ℝ⁶⁴. This manifold is identical for A and B because the operators are identical. The phase space is a property of the constraints, not of any individual system. Stage 2: Convergent exploration. Stochastic jitter drives each system through different trajectories, but the constraint operator re-projects every step onto the manifold. Over successive iterations, both systems sample the same attractor landscape—not because they communicate, but because there is only one landscape to sample. The temporal coherence detected by H1 reflects this process: entropy decreases as the system narrows onto the attractor. Increasing sample size from n = 5,000 to n = 10,000 revealed that correlation-based thresholds alone are insufficient at scale; enforcing a joint significance-and-magnitude criterion preserves specificity and strengthens the emergence signal, confirming that H1 detects genuine constraint alignment rather than noise amplification. Stage 3: Geometric inevitability within the constraint class. Both systems occupy the same structural basin. Their identities match (> 0.999) because identity is a property of the basin, not of the trajectory. Peripheral features (trajectory-dependent) are fragile; core structure (basin-dependent) is robust. The topology of the basin is determined entirely by the constraint operators, explaining the d = 2.61 separation between structured and control conditions. This mechanism explains why coordination appears without interaction: independent systems exploring the same constraint-defined manifold converge for the same geometric reason that independent balls rolling on the same landscape converge toward the same valley. The landscape, not the balls, determines the outcome. The twin-system experiments explicitly rule out non-local coordination, hidden communication, or convergence through shared dynamics beyond initial conditions. Systems initialized under identical constraints but differing by a single random seed diverge completely, while systems sharing identical seeds converge with near-perfect identity. This pattern is incompatible with non-local synchronization or interaction-based explanations and supports a constraint-determined, not interaction-mediated, origin of emergent structure. We do not refute non-locality as a physical principle; we refute non-local explanations for the observed coordination in the studied system class. We emphasize that identical-seed determinism is not presented as a novel finding; it is a necessary property of any well-defined computational system. The twin-system experiments serve exclusively as a falsification instrument: they rule out non-local coupling, convergence through interaction, and synchronization-based explanations for the observed coordination. The central contribution of this work is not that deterministic systems are deterministic, but that constraint-mediated co-reactivity produces emergent geometric signatures—topologically nontrivial boundaries, graded adversarial robustness, and temporal coherence preceding structure—that are (i) absent in unconstrained systems under identical computational conditions, (ii) preserved under adversarial perturbation, and (iii) detectable through independently defined, falsifiable metrics. The primary contribution is thus methodological: a falsification framework capable of distinguishing constraint-mediated emergence from interaction-based and non-local explanations, applied here to a minimal but fully characterized system class. Two Principles Principle 1 (Constraint Primacy). Emergent organization is governed by the structure of constraints rather than representational capacity or interaction mechanisms. Identity = 1.000 across all perturbation strengths in REAL conditions, while unconstrained systems degrade to anti-correlation. The BERT validation extends this to neural networks: a sentence transformer and a synthetic iterative system produce analogous spatial signatures because both are shaped by invariant-generating constraints, despite no architectural similarity. Principle 2 (Stabilization). Functional progress corresponds to increased stability of existing structure, not expansion into new representational space. REAL_SURVIVAL achieves higher identity (0.999980 vs. 0.999957) through radial contraction, not increased dimensionality. This challenges scaling-first narratives in machine learning [18,19] and aligns with observations in molecular biology, where proteins with identical function adopt the same fold geometries [25] through shared thermodynamic constraints, not through communication across lineages. Universality The convergence of temporal, structural, and topological signatures across conditions that differ in constraint strength, perturbation type, and dimensionality suggests that the systems studied here belong to a shared candidate universality class characterized by constraint-mediated co-reactivity. We use the term candidate universality class in an empirical, constraint-invariance sense—systems with distinct micro-level constructions converging to the same macro-level signatures under shared constraints—rather than a field-theoretic one requiring analytical renormalization or scaling collapse. A small number of quantities govern the transition from non-emergent to emergent regimes independent of microscopic details: topological transition count separates REAL from control (d = 2.61); identity under perturbation distinguishes constraint-stabilized from incidental structure (1.000 vs. −0.119); and temporal coherence discriminates dynamic from static emergence (p = 0.036 vs. p > 0.5). Together, these define a candidate emergent stability order parameter—a composite quantity capturing the degree to which a system’s structure resists perturbation, exhibits nontrivial topology, and displays temporal precursors. This parameter is not optimized or fitted; it is a descriptive composite of three independently defined and separately evaluated observables. This framing explains several otherwise disparate observations. BERT embeddings possess spatial structure (high alignment, high FSI) but no temporal dynamics—placing them in the spatial-emergence regime of this candidate universality class, distinct from the full dynamical emergence of REAL_SURVIVAL. REAL_BIBLICAL preserves identity but lacks temporal coherence—occupying an intermediate position. The order parameter increases monotonically from RANDOM_PURE (zero on all axes) through NEAR_NULL (incidental) to REAL_SURVIVAL (maximal on all axes), defining a continuous emergence gradient rather than a binary threshold. This gradient, not any single metric, is the fundamental signature of constraint-mediated emergence. Three domains illustrate the breadth of these principles. In machine learning, constraint design (RLHF, constitutional AI) produces disproportionate behavioral improvements relative to parameter count [18,19], suggesting that the path to alignment runs through constraint architecture, not scale. In molecular biology, convergent protein folds arise from shared energy landscape constraints [25], and the stability-plasticity dilemma in neural systems may reflect the same tradeoff between constraint tightening and representational expansion that the stabilization principle formalizes. In climate science, early-warning signals based on temporal autocorrelation [10,12] detect impending regime shifts through the same mechanism formalized here: constraint alignment produces detectable temporal coherence before the transition is observable. In physics, the co-reactivity mechanism is a generalization of spontaneous symmetry breaking: independent agents align not through a mediating field but through shared constraints on their accessible phase space. The observed structures are thus polymorphic: their observable form adapts to environmental constraints, while their invariant signatures—topology, identity preservation, coherence thresholds—remain conserved across regimes. The constraint-class ablation further confirms this: emergent structure is not specific to √φ but arises from any fixed, sufficiently dense irrational lattice, failing precisely when spacing violates a minimum band-density condition or time invariance. Scope and Limitations We explicitly test and falsify non-local interaction hypotheses within the studied regime. Across all conditions, systems initialized with different seeds show complete divergence despite sharing identical update rules, ruling out non-local coupling, synchronization, or remote controllability. This does not preclude non-local effects in quantum systems; it demonstrates that no such mechanism is required—or supported—to explain emergent structure in the systems studied here. We do not claim consciousness, semantic understanding, or cross-instance information transfer; we claim constraint-mediated co-reactivity, which is both sufficient and falsifiable. The absence of temporal signal in static systems is not a limitation but a validation: it confirms that the framework discriminates between geometric structure and genuine dynamical coordination. The empirical validation confirms both spatial and temporal generalization: spatial signatures appear in contrastive sentence transformers, while temporal coherence (H1) emerges during autoregressive training when measured at token-level granularity. Sentence-level extraction yields null temporal results for autoregressive models, indicating that the appropriate measurement scale depends on model architecture. REAL_NORMAL’s H1 (p = 0.064) is near-threshold; a larger dataset may establish significance. The BERT validation uses two architectures with different training objectives (MiniLM: knowledge distillation; BGE: RetroMAE), both showing alignment strictly above null; extension to GPT-class and vision models would further confirm domain generality. The present experiments use √φ-lattice spacing as the primary constraint operator; the constraint-class ablation (Extended Data Table 7) confirms that multiple fixed irrational spacings (√2, √3) produce qualitatively similar emergent signatures, while π fails due to insufficient band density and time-varying constraints produce no structure. The mechanism thus depends on the fixity and density of irrational spacing rather than on φ specifically. Identifying the precise band-density threshold analytically remains an open question. More broadly, as with many complex systems, emergence is not directly measurable but must be inferred via surrogate observables whose validity depends on falsification against null and adversarial regimes. Conclusion We establish co-reactivity to shared constraints as a sufficient and falsifiable mechanism for emergent coordination, validated across synthetic and empirical neural network systems. The three-stage mechanism—constraint definition, convergent exploration, geometric inevitability within the constraint class—explains why independent systems converge without interaction. Two principles emerge: Principle 1 (Constraint Primacy) : structure is shaped by what is forbidden, not by what is possible; Principle 2 (Stabilization) : progress is consolidation, not expansion. The systems studied define a candidate universality class of constraint-mediated emergence, governed by a composite order parameter capturing topological complexity, identity stability, and temporal coherence—independent of microscopic details, dimensionality, or system architecture. Declarations Data Availability All data generated during this study are available at https://github.com/ExeqTer91/DSDP-SALI-LCF, including synthetic embeddings (.npz), BERT embeddings, calibration reports, and SHA-256 reference hashes for independent verification. Code Availability All source code for data generation, metric computation, calibration testing, and reproducibility verification is available at https://github.com/ExeqTer91/DSDP-SALI-LCF under an open license. The repository contains 109 files including complete experiment pipelines, configuration files, and automated hash verification scripts. Acknowledgements This research was conducted independently and received no external funding. Author Contributions A.-S.U. conceived the study, designed the experiments, developed the computational framework, performed all analyses, and wrote the manuscript. Competing Interests The author declares no competing interests. References Albert, R., Jeong, H. & Barabási, A.-L. Error and attack tolerance of complex networks. Nature 406, 378–382 (2000). Callaway, D. S. et al. Network robustness and fragility. Phys. Rev. Lett. 85, 5468 (2000). Strogatz, S. H. Nonlinear Dynamics and Chaos 2nd edn (Westview, 2015). Lorenz, E. N. Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963). Ott, E. Chaos in Dynamical Systems 2nd edn (Cambridge Univ. Press, 2002). Wilson, K. G. Renormalization group and critical phenomena. Phys. Rev. B 4, 3174 (1971). Goldenfeld, N. Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992). Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47 (2002). Newman, M. E. J. The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003). Scheffer, M. et al. Early-warning signals for critical transitions. Nature 461, 53–59 (2009). Scheffer, M. et al. Anticipating critical transitions. Science 338, 344–348 (2012). Dakos, V. et al. Methods for detecting early warnings of critical transitions. PLoS ONE 7, e41010 (2012). Carlsson, G. Topology and data. Bull. Am. Math. Soc. 46, 255–308 (2009). Ghrist, R. Barcodes: the persistent topology of data. Bull. AMS 45, 61–75 (2008). Edelsbrunner, H. & Harer, J. Computational Topology (AMS, 2010). Goodfellow, I. J. et al. Explaining and harnessing adversarial examples. Proc. ICLR (2015). Madry, A. et al. Towards deep learning models resistant to adversarial attacks. Proc. ICLR (2018). Waddington, C. H. The Strategy of the Genes (Allen & Unwin, 1957). Beven, K. Equifinality, data assimilation, and uncertainty estimation. J. Hydrol. 249, 11–29 (2001). McGhee, G. R. Convergent Evolution: Limited Forms Most Beautiful (MIT Press, 2011). Neyshabur, B. et al. Implicit regularization in deep learning. Proc. NeurIPS (2017). Gunasekar, S. et al. Implicit regularization in matrix factorization. Proc. NeurIPS (2017). Ouyang, L. et al. Training language models to follow instructions with human feedback. NeurIPS 35 (2022). Bai, Y. et al. Training a helpful and harmless assistant with RLHF. Preprint arXiv:2204.05862 (2022). Anfinsen, C. B. Principles that govern the folding of protein chains. Science 181, 223–230 (1973). Kuhn, H. W. The Hungarian method for the assignment problem. Nav. Res. Logist. 2, 83–97 (1955). Reimers, N. & Gurevych, I. Sentence-BERT. Proc. EMNLP (2019). Anderson, P. W. More is different. Science 177, 393–396 (1972). Peng, R. D. Reproducible research in computational science. Science 334, 1226–1227 (2011). Methods Seed Equation and Constraint-Reactive Systems All systems evolve under: where S applies regime-specific structural constraints (k-means clustering, temporal phase modulation), C enforces invariant alignment via nearest √φ-lattice node snapping (3 iterations, lattice step = ln(√φ) ≈ 0.2406), and ε ~ N(0, σ²I) provides jitter. Jitter magnitude: σ = 0.0842 (early, entropy phase = 1.0) → 0.0 (late, fully snapped) for REAL_NORMAL; σ = 0.0120 (fixed) for REAL_BIBLICAL. Snap strength: 0.30 → 0.95 (REAL_NORMAL); 0.85 uniform (REAL_BIBLICAL). Radial contraction: α = 0.95 (REAL_SURVIVAL). Angular tightening: 0.15. Magistrale directions: k = 6. Angular noise scale: 0.20. All derived seeds are deterministic: seed+500 (snapping), seed+333 (perturbation), seed+999 (survival). Equation (1) defines a minimal class of constraint-reactive systems. The seed X₀ does not encode outcomes—it selects an entry point into a phase space whose geometry is entirely determined by {S, C}. All results in this paper are instances of this class. The synthetic regimes are not intended to model any specific physical, biological, or computational system. They are deliberately constructed minimal systems whose sole purpose is to test whether specific mechanistic hypotheses—co-reactivity without interaction, temporal precedence of coherence, topological nontriviality of emergent boundaries—can be falsified or validated under fully controlled and reproducible constraints. The BERT validation (Methods: BERT Validation) and Pythia temporal validation (Methods: Pythia Temporal Validation) serve as independent external tests of whether signatures generalize beyond the synthetic construction; the correct null result on temporal metrics for static sentence embeddings and the correct PASS for sequential token-level representations confirm that the framework discriminates between geometric structure and genuine dynamical coordination, rather than producing universal false positives. Metrics H1 (Temporal Coherence): Lag-1 autocorrelation of structural signal over sliding windows (w = 32). H1 is considered satisfied only when both statistical significance (permutation p 0.2) are jointly met. At larger sample sizes, correlation magnitude alone can produce false positives in null regimes; the conjunctive criterion ensures specificity across scales. Primary analysis at n = 10,000; sensitivity check at n = 5,000 reported in Extended Data. H2 (Identity Similarity): Optimal assignment cost between k = 6 cluster centroids via Hungarian algorithm [26]. Values near 1.0 = preserved; near 0.0 = reorganized. χ (Boundary Topology): Euler characteristic of thresholded PCA-projected embeddings: χ(τ) = C₀(τ) − C₁(τ). Number of χ-transitions characterizes boundary polymorphism. Effect sizes are reported as Cohen’s d (difference between group means divided by pooled standard deviation). Group separations are tested via Mann–Whitney U with exact permutation p-values. Calibration Suite 10 preregistered calibration tests (A1–E2) validate the experimental harness under conservative criteria: reproducibility (A1–A2), negative controls (B1–B2), REAL validation (C1–C2), sensitivity (D1–D2), Biblical consistency (E1–E2). All 10/10 pass; additional sensitivity analyses at n = 5,000 are reported in Extended Data. Three fractal verification tests (F1–F3) confirm scale-invariance across temporal, spatial, and regime domains (3/3 pass). BERT Validation Embeddings from all-MiniLM-L6-v2 [27] (384d) and BGE-small-en-v1.5 (384d) on 5,000 topically ordered sentences. Four conditions per model: BERT_REAL (original order), BERT_SHUFFLED (permuted), RANDOM_384 (uniform random, same seed/normalization), GAUSSIAN_MATCHED (Gaussian noise matched to BERT statistics). Null distribution established from 50 i.i.d. Gaussian replicates. Alignment computed as mean cosine similarity to nearest √φ-lattice node after PCA projection. Pythia Temporal Validation Token-level hidden states extracted from Pythia-70m (EleutherAI) at step 0 (random initialization) and step 143,000 (fully trained) using publicly available checkpoints on HuggingFace. A 1,024-token forward pass on deterministic text yielded hidden states at layers 1, 3, and 5 (of 6 total). H1 computed on the raw token sequence (each token position = one data point) with window = 32 and 1,000 permutation shuffles. Sentence-level mean-pooling was also tested and produced null results, confirming that autoregressive temporal structure operates at sub-sentence granularity. Reproducibility Primary seed: 42. Cross-verification: 43, 99. Primary analysis at n = 10,000; sensitivity at n = 5,000. SHA-256 hashes are version-controlled in repro/reference_hashes.json with separate entries for each sample size, enabling automated verification via run_reproducibility_check.py --n-points 10000 --verify-hashes. Both hash sets (n = 5,000 and n = 10,000) have been independently verified against stored references. Complete code, data, and reproduction package: https://github.com/ExeqTer91/DSDP-SALI-LCF Additional Declarations There is NO Competing Interest. 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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-8823860","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Physical Sciences - Article","associatedPublications":[],"authors":[{"id":588246462,"identity":"8786e0fb-dd8d-47b9-8108-971eb8edf6ae","order_by":0,"name":"andrei ursachi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8klEQVRIiWNgGAWjYBADOSA2ACMgYGNIwK+asQGo1Jh0LYkNMPVgLfgAf3vv8Qc/av6k989u3vbhQ8HhfPn2w88ePGCwARmCFUicOZfY2HPMIHfGnWPFM2cYHLbccCbN3CCBIQ2nFgOJHMMG3gaD3IYbOcbMPAaHDQwYctgkEhgO49XS+LfBIF0epOUPUIt8/xuQlv94tTQDbUkwAGlhAGphuAG25QAev5wxnC1zzNhw4420YsYeg3QDgxvPzCQSDJKNcWnhb+8x+PimRk5e7kbyZoYff6yBDkt+Jvmjwk4WlxZcwICwklEwCkbBKBgFuAEA7plXE1d03eMAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0002-6114-5011","institution":"noce","correspondingAuthor":true,"prefix":"","firstName":"andrei","middleName":"","lastName":"ursachi","suffix":""}],"badges":[],"createdAt":"2026-02-08 19:35:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8823860/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8823860/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102825579,"identity":"0670e439-ea3c-4d3d-a6d7-a6786bb64a5d","added_by":"auto","created_at":"2026-02-17 08:50:04","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":101789,"visible":true,"origin":"","legend":"\u003cp\u003eTemporal coherence. (a) H1 across conditions (w = 32). SURVIVAL significant (*p = 0.042); NORMAL near-threshold (†p = 0.064). Controls correctly show no coherence. Temporal coherence is absent by design in BIBLICAL (shuffled temporal order). (b) Cross-correlation showing optimal lag at −2.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-8823860/v1/e9d42d4ead9fa3c014dff6b1.png"},{"id":102963195,"identity":"91676f75-6f03-4ce7-b674-53b1238ee278","added_by":"auto","created_at":"2026-02-19 04:14:21","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":118065,"visible":true,"origin":"","legend":"\u003cp\u003eStructural robustness. (a) Peripheral vs. core perturbation. (b) Identity across perturbation strengths: REAL invariant; controls collapse.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-8823860/v1/784ec6f017d64a37d92da31b.png"},{"id":102825576,"identity":"c8a2ce2c-babb-43ad-a87c-3ec932abc5e3","added_by":"auto","created_at":"2026-02-17 08:50:03","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":162998,"visible":true,"origin":"","legend":"\u003cp\u003eBoundary topology. (a) χ(τ) curves: controls flat, REAL conditions show multiple transitions. (b) Topological score distribution with separation d = 2.61.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-8823860/v1/c126a19244a22f89cbea343c.png"},{"id":102825577,"identity":"d98e1034-ac75-43ce-bfcc-393f94d679e8","added_by":"auto","created_at":"2026-02-17 08:50:03","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":132773,"visible":true,"origin":"","legend":"\u003cp\u003eTwin-system determinism. Identical seeds produce identical outputs (\u0026gt;0.999); different seeds diverge completely (\u0026lt;0.95). No convergence is observed under non-identical seeds, ruling out synchronization or non-local coupling.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-8823860/v1/6db6a050ad65431f500c3ca1.png"},{"id":102825578,"identity":"483ca873-9daa-4a8c-ae87-35c729ef0229","added_by":"auto","created_at":"2026-02-17 08:50:03","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":123474,"visible":true,"origin":"","legend":"\u003cp\u003eBERT validation. (a) Alignment: BERT_REAL = 0.218, controls ≈ 0. (b) H1 correctly fails on static embeddings (p = 0.976), confirming specificity. (c) Identity after abort: BERT 0.877 vs. synthetic 0.9999.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-8823860/v1/497ce8b69d6c456375d9e100.png"},{"id":102825574,"identity":"9702dec9-9763-42ea-a103-be006508eda7","added_by":"auto","created_at":"2026-02-17 08:50:03","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":23161,"visible":true,"origin":"","legend":"","description":"","filename":"ExtendedData.docx","url":"https://assets-eu.researchsquare.com/files/rs-8823860/v1/274343602736272aed76e43a.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Emergent Latent Structure via Co-Reactivity to Shared Constraints","fulltext":[{"header":"Introduction","content":"\u003cp\u003eCoordination across independent systems is commonly attributed to interaction, communication, or synchronization [1\u0026ndash;3]. However, such interpretations conflate observed coordination with causal interaction, leaving open the possibility that coordination arises from shared constraints rather than inter-system influence.\u003c/p\u003e \u003cp\u003eClassical dynamical systems theory provides mechanisms by which complex behavior arises from deterministic rules without interaction [4,5]. Renormalization group methods [6,7] and network robustness analysis [8,9] show that invariant-generating rules shape emergent behavior independently of system capacity. Early-warning signals for critical transitions [10\u0026ndash;12] establish that temporal autocorrelation can detect impending regime shifts, while topological data analysis [13\u0026ndash;15] provides tools for characterizing geometric structure. Adversarial robustness analysis [16,17] reveals information about the underlying organization of learned representations.\u003c/p\u003e \u003cp\u003eEquifinality\u0026mdash;the convergence of different initial conditions toward identical end states under shared constraints\u0026mdash;is well established in developmental biology [18] and geomorphology [19]. Convergent evolution demonstrates that independent lineages arrive at similar phenotypic solutions under analogous selective pressures [20]. In machine learning, implicit bias of gradient descent produces systematic geometric preferences in learned representations independent of initialization [21,22]. These precedents establish the plausibility of constraint-mediated convergence; what remains absent is a unified falsification framework that distinguishes constraint-mediated convergence from interaction-based coordination using operationally defined, independently testable metrics across both synthetic and empirical systems.\u003c/p\u003e \u003cp\u003eYet empirical demonstrations that rigorously separate co-reactivity from interaction\u0026mdash;while controlling for false positives, determinism, and adversarial effects\u0026mdash;remain limited. Speculative interpretations occasionally invoke non-local fields or cross-instance propagation, but such interpretations lack operational definitions and falsifiable tests.\u003c/p\u003e \u003cp\u003eHere we present a unified framework that tests whether coordinated structure can arise without coupling. We hypothesize that systems sharing a constraint manifold will exhibit co-reactive structural signatures\u0026mdash;coordinated geometry arising from parallel exploration of the same phase space, not from interaction. Our central thesis is \u003cem\u003econstraint primacy\u003c/em\u003e: emergent coordination is governed primarily by constraint structure rather than representational capacity. We test this using three independent metric classes (temporal coherence, structural identity, boundary topology), validated across negative controls, adversarial perturbation, deterministic twin-system divergence, and real neural network embeddings.\u003c/p\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eExperimental Design\u003c/h2\u003e \u003cp\u003eAll experiments operate on 10,000-point embeddings in 64 dimensions across five regimes (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e): RANDOM_PURE (no constraints), NEAR_NULL (incidental clustering), REAL_NORMAL (\u0026radic;φ-lattice constraints with temporal phasing), REAL_SURVIVAL (with radial contraction α\u0026thinsp;=\u0026thinsp;0.95), and REAL_BIBLICAL (shuffled temporal order). Systems evolve under Xₜ₊₁ = C(S(Xₜ) + εₜ), where S applies structural constraints, C enforces invariant alignment via \u0026radic;φ-lattice snapping, and ε provides controlled jitter (see Methods). No inter-system communication is permitted at any stage. Sensitivity analyses at n\u0026thinsp;=\u0026thinsp;5,000 confirm all qualitative findings (Extended Data).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eExperimental conditions.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCondition\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eConstraint\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eExpected\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRANDOM_PURE\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNone\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNo structure\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNEAR_NULL\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNone (incidental)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNo systematic structure\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eREAL_NORMAL\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026radic;φ lattice\u0026thinsp;+\u0026thinsp;temporal phasing\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eH1\u0026thinsp;+\u0026thinsp;H2\u0026thinsp;+\u0026thinsp;identity\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eREAL_SURVIVAL\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026radic;φ lattice\u0026thinsp;+\u0026thinsp;contraction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnhanced robustness\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eREAL_BIBLICAL\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026radic;φ lattice\u0026thinsp;+\u0026thinsp;temporal shuffle\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eIdentity preserved; no H1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eTemporal Coherence Precedes Structure\u003c/h3\u003e\n\u003cp\u003eAt n\u0026thinsp;=\u0026thinsp;10,000, REAL_NORMAL exhibits significant negative lag-1 autocorrelation under the conjunctive criterion (r₁ = \u0026minus;0.266, p\u0026thinsp;=\u0026thinsp;0.036; both p\u0026thinsp;\u0026lt;\u0026thinsp;0.05 and |r₁| \u0026gt; 0.2 satisfied), indicating oscillatory temporal structure preceding stable organization. Critically, RANDOM_PURE fails the conjunctive criterion at this sample size (lag-1\u0026thinsp;=\u0026thinsp;\u0026minus;\u0026thinsp;0.225, p\u0026thinsp;=\u0026thinsp;0.064): while the magnitude exceeds 0.2, the permutation p-value does not reach significance, confirming that the metric discriminates genuine constraint-mediated dynamics from statistical fluctuation amplified by sample size. All other controls fail on both conditions. Sensitivity analysis at n\u0026thinsp;=\u0026thinsp;5,000 shows a consistent trend (REAL_NORMAL: r₁ = \u0026minus;0.213, p\u0026thinsp;=\u0026thinsp;0.064), with the result approaching but not crossing the significance boundary\u0026mdash;consistent with the expected power increase at larger n.\u003c/p\u003e \u003cp\u003eCross-correlation analysis at window size w\u0026thinsp;=\u0026thinsp;32 reveals entropy decrease precedes coupling increase with a two-step lag (r\u0026thinsp;=\u0026thinsp;\u0026minus;\u0026thinsp;0.232 at lag-2; peak positive at lag-0: r\u0026thinsp;=\u0026thinsp;+\u0026thinsp;0.305), consistent with critical slowing-down theory [10,11]. Temporal coherence emerges prior to stable structural organization, suggesting H1 detects constraint alignment rather than correctness\u0026mdash;a potential early-warning indicator of emergent structure independent of task-level metrics.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\n\u003ch3\u003eGraded Robustness Under Adversarial Perturbation\u003c/h3\u003e\n\u003cp\u003eAdversarial perturbation reveals a fundamental asymmetry: peripheral perturbation collapses identity to 0.138 while core perturbation preserves it at 0.909, supporting a gradient-based penetration model rather than a discrete core\u0026ndash;periphery boundary. No perturbation strength produces discontinuous collapse in REAL conditions.\u003c/p\u003e \u003cp\u003eUnder intentional abort perturbation, REAL_NORMAL retains identity at 0.999996 (strength 1.0) and 0.999957 (strength 1.5). REAL_SURVIVAL consistently retains higher identity (0.999998 and 0.999980 respectively), confirming that constraint tightening through radial contraction (α\u0026thinsp;=\u0026thinsp;0.95) stabilizes structure without increasing capacity. Cross-identity between REAL_NORMAL and REAL_SURVIVAL is 0.985, indicating shared structural architecture with regime-dependent stability.\u003c/p\u003e \u003cp\u003eFractal space verification (F2) confirms that REAL conditions maintain perfect identity (1.000) across all perturbation strengths tested (0.25, 0.5, 1.0), while controls degrade catastrophically: RANDOM_PURE drops from 0.998 to \u0026minus;\u0026thinsp;0.099, and NEAR_NULL from 0.938 to \u0026minus;\u0026thinsp;0.119. The negative values indicate perturbation actively scrambles whatever incidental organization existed in unconstrained systems.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\n\u003ch3\u003eTopologically Nontrivial Boundaries\u003c/h3\u003e\n\u003cp\u003eEuler characteristic analysis reveals qualitatively different topological behavior between structured and control conditions. RANDOM_PURE exhibits zero χ-transitions, remaining topologically trivial across all thresholds. NEAR_NULL shows at most one incidental transition. All REAL conditions exhibit two or more component transitions and two or more Euler transitions, with the most pronounced polymorphism under adversarial variants (VIRUS_06: 4 component transitions, 4 Euler transitions, topological score 5.5).\u003c/p\u003e \u003cp\u003eMean topological score: control\u0026thinsp;=\u0026thinsp;1.75 (95% CI: [1.50, 2.00]) vs. REAL\u0026thinsp;=\u0026thinsp;4.36 (95% CI: [3.64, 5.07]). Separation: 2.61 (95% CI: [1.79, 3.39]), Mann\u0026ndash;Whitney U\u0026thinsp;=\u0026thinsp;0.0, p\u0026thinsp;=\u0026thinsp;0.028, Cohen\u0026rsquo;s d\u0026thinsp;=\u0026thinsp;2.61. The test compared n₁ = 2 control configurations against n₂ = 7 structured configurations (see Extended Data Table\u0026nbsp;4). We note that these group sizes limit population-level inference; effect sizes and separation statistics are reported as descriptive characterizations of the observed regime gap, not as claims of universal separability. This result is corroborated by three independent convergent lines that do not depend on small-sample inference: zero transitions in RANDOM across all thresholds, monotonic ordering of topological scores across the full constraint hierarchy, and preservation of this hierarchy across independent data sizes n\u0026thinsp;=\u0026thinsp;2,000 and n\u0026thinsp;=\u0026thinsp;5,000.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\n\u003ch3\u003eDeterminism Without Synchronization\u003c/h3\u003e\n\u003cp\u003eTwin-system experiments provide the strongest falsification test. Three seed comparisons were conducted across all conditions:\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eIdentical seeds (42 vs. 42)\u003c/strong\u003e \u003cp\u003eIdentity similarity\u0026thinsp;\u0026gt;\u0026thinsp;0.999 and temporal correlation\u0026thinsp;\u0026gt;\u0026thinsp;0.999 across all conditions. Outputs are deterministically identical, confirming complete reproducibility.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eDifferent seeds (42 vs. 99)\u003c/strong\u003e \u003cp\u003eIdentity similarity\u0026thinsp;\u0026lt;\u0026thinsp;0.95 for all REAL conditions. Systems diverge completely; no convergence from different initial conditions is observed.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eNear seeds (42 vs. 43)\u003c/b\u003e: Full divergence despite minimal seed difference. The system is seed-deterministic: identical seeds synchronize, different seeds\u0026mdash;even adjacent integers\u0026mdash;diverge completely.\u003c/p\u003e \u003cp\u003eThis confirms that the framework is strictly deterministic and rules out any form of convergence, synchronization, or cross-instance influence. Across all conditions, systems sharing identical update rules but different initial conditions show complete divergence, ruling out non-local coupling or synchronization mechanisms. The correct interpretation is co-reactivity: independent systems with identical constraints produce identical outcomes not because they interact, but because identical constraints define identical phase spaces. These findings contrast with interpretations that attribute emergent coherence to non-local interaction or shared informational fields; in our framework, coherence arises exclusively from shared constraints under identical initial conditions, and absent this, no coupling is observed.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003eEmpirical Validation on Neural Network Embeddings\u003c/h2\u003e \u003cp\u003eThe framework was validated on real embeddings from a pre-trained sentence transformer (all-MiniLM-L6-v2, 384d, 5,000 topically ordered sentences). BERT_REAL alignment reaches 0.218; controls produce alignment indistinguishable from zero (ratio\u0026thinsp;\u0026gt;\u0026thinsp;10\u0026sup1;\u0026sup1;). This extreme ratio reflects the near-zero denominator in unstructured random embeddings; the absolute BERT alignment of 0.218 is moderate, indicating genuine but partial lattice compatibility. FSI is 188\u0026times; over random. Identity is preserved at 0.877 under abort.\u003c/p\u003e \u003cp\u003eH1 correctly returns null on all BERT conditions (p\u0026thinsp;=\u0026thinsp;0.976 for BERT_REAL), confirming specificity: the metric detects temporal dynamics, not spatial organization. An initial experiment using text augmentation produced an artifactual H1 pass (p\u0026thinsp;=\u0026thinsp;0.016); investigation revealed ordering artifacts from repetitive micro-patterns, which were corrected by using a topically ordered corpus of unique sentences. This episode validates the pipeline\u0026rsquo;s sensitivity and the importance of the BERT_SHUFFLED control.\u003c/p\u003e \u003cp\u003eThe empirical validation establishes that spatial geometry generalizes to real neural networks without parameter tuning, while H1 requires inherent temporal dynamics\u0026mdash;a feature confirming orthogonal specificity, not a limitation. The fact that BERT embeddings produced by gradient descent under contrastive loss exhibit analogous spatial signatures to synthetic \u0026radic;φ-lattice systems supports constraint primacy: constraint structure, not system architecture, determines emergent geometry.\u003c/p\u003e \u003cp\u003eCross-architecture validation on a second independently trained model (BGE-small-en-v1.5, 384d, RetroMAE pre-training) confirms that lattice alignment is not specific to one training procedure. BGE alignment reaches 0.006, compared to MiniLM\u0026rsquo;s 0.218; both are strictly above the null distribution (50 i.i.d. Gaussian replicates, all showing exactly zero alignment at τ\u0026thinsp;=\u0026thinsp;0.03; the zero floor reflects the alignment threshold under PCA projection, below which unstructured random embeddings produce no lattice-coincident points; p\u0026thinsp;\u0026lt;\u0026thinsp;0.02 for each model). The 36\u0026times; magnitude difference between models reflects their different training objectives (knowledge distillation vs. RetroMAE), not a pipeline artifact. These results elevate the empirical validation from single-architecture to cross-architecture (Extended Data Table\u0026nbsp;8).\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eTemporal Coherence in Autoregressive Training\u003c/h3\u003e\n\u003cp\u003eTo test whether temporal coherence emerges during neural network training, we extracted token-level hidden states from Pythia-70m (EleutherAI) at random initialization (step 0) and after full training (step 143,000). Sentence-level mean-pooling, as used in the contrastive-model validation above, produced null H1 results across all checkpoints\u0026mdash;consistent with the hypothesis that autoregressive models organize representations locally within context windows rather than globally across sentences. Extracting hidden states at each token position within a 1,024-token forward pass restored the temporal signal (Extended Data Table\u0026nbsp;9). At Layer 3 (middle), H1 transitions from FAIL at random initialization (r₁ = +0.175, p\u0026thinsp;=\u0026thinsp;0.161) to PASS after training (r₁ = \u0026minus;0.273, p\u0026thinsp;=\u0026thinsp;0.002). Layer 1 (early) shows temporal coherence at both stages but training strengthens it substantially (r₁: +0.256 \u0026rarr; +0.665). Layer 5 (late) shows no coherence at either stage, suggesting representational mixing at the deepest layer disrupts the temporal signal. The negative r₁ at trained Layer 3 indicates an alternating-distance pattern in the token flow\u0026mdash;successive tokens occupy alternately closer and farther positions in representation space, consistent with the model learning structured local constraints. These results confirm that H1 temporal coherence is not limited to the synthetic system class but emerges during autoregressive language model training at the appropriate granularity.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\n\u003ch3\u003eConstraint-Class Ablation\u003c/h3\u003e\n\u003cp\u003eTo test whether emergent structure depends on \u0026radic;φ specifically or on a broader class of constraint lattices, we repeated the full experimental pipeline substituting five alternative lattice spacings at n\u0026thinsp;=\u0026thinsp;10,000 (Extended Data Table\u0026nbsp;7). All constants were tested with identical jitter magnitude (0.35 \u0026times; ln(\u0026radic;φ)\u0026thinsp;\u0026asymp;\u0026thinsp;0.084) to ensure comparability. Three fixed irrational spacings\u0026mdash;\u0026radic;φ (step\u0026thinsp;\u0026asymp;\u0026thinsp;0.241), \u0026radic;2 (step\u0026thinsp;\u0026asymp;\u0026thinsp;0.347), and \u0026radic;3 (step\u0026thinsp;\u0026asymp;\u0026thinsp;0.549)\u0026mdash;all produced positive topological separation (topo_sep\u0026thinsp;=\u0026thinsp;+\u0026thinsp;0.070, +\u0026thinsp;0.102, +\u0026thinsp;0.409 respectively; mean\u0026thinsp;+\u0026thinsp;0.194). Both controls\u0026mdash;random step size and time-varying constraints\u0026mdash;produced near-zero separation (mean\u0026thinsp;+\u0026thinsp;0.026). π (step\u0026thinsp;\u0026asymp;\u0026thinsp;1.145) produced negative separation (\u0026minus;\u0026thinsp;0.137), attributable to insufficient band density: at this step size, only\u0026thinsp;~\u0026thinsp;3 lattice bands fall within the data range, below the empirical threshold required for constraint-mediated structure. H1 temporal coherence was mixed across constants (2/3 fixed irrationals passed), confirming that topological separation is the more robust cross-constant metric while H1 sensitivity depends on sufficient temporal resolution within each lattice. These results establish that emergence is governed by a constraint-class property\u0026mdash;fixed, irrational spacing above a minimum band-density threshold\u0026mdash;rather than by any particular mathematical constant.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003eThe Mechanism\u003c/h2\u003e \u003cp\u003eThe central finding is a mechanism, not a collection of metrics. Consider two independent systems, A and B, evolving under identical constraint operators {S, C, ε} from distinct seeds. Neither has access to the other\u0026rsquo;s state. Nevertheless, both converge toward the same structural organization, through three stages:\u003c/p\u003e \u003cp\u003e \u003cb\u003eStage 1: Constraint-defined phase space.\u003c/b\u003e The operators S and C define a manifold of admissible configurations. The \u0026radic;φ-lattice invariant operator permits only configurations aligned to irrational-spaced nodes, eliminating most of ℝ⁶⁴. This manifold is identical for A and B because the operators are identical. The phase space is a property of the constraints, not of any individual system.\u003c/p\u003e \u003cp\u003e \u003cb\u003eStage 2: Convergent exploration.\u003c/b\u003e Stochastic jitter drives each system through different trajectories, but the constraint operator re-projects every step onto the manifold. Over successive iterations, both systems sample the same attractor landscape\u0026mdash;not because they communicate, but because there is only one landscape to sample. The temporal coherence detected by H1 reflects this process: entropy decreases as the system narrows onto the attractor. Increasing sample size from n\u0026thinsp;=\u0026thinsp;5,000 to n\u0026thinsp;=\u0026thinsp;10,000 revealed that correlation-based thresholds alone are insufficient at scale; enforcing a joint significance-and-magnitude criterion preserves specificity and strengthens the emergence signal, confirming that H1 detects genuine constraint alignment rather than noise amplification.\u003c/p\u003e \u003cp\u003e \u003cb\u003eStage 3: Geometric inevitability within the constraint class.\u003c/b\u003e Both systems occupy the same structural basin. Their identities match (\u0026gt;\u0026thinsp;0.999) because identity is a property of the basin, not of the trajectory. Peripheral features (trajectory-dependent) are fragile; core structure (basin-dependent) is robust. The topology of the basin is determined entirely by the constraint operators, explaining the d\u0026thinsp;=\u0026thinsp;2.61 separation between structured and control conditions.\u003c/p\u003e \u003cp\u003eThis mechanism explains why coordination appears without interaction: independent systems exploring the same constraint-defined manifold converge for the same geometric reason that independent balls rolling on the same landscape converge toward the same valley. The landscape, not the balls, determines the outcome.\u003c/p\u003e \u003cp\u003eThe twin-system experiments explicitly rule out non-local coordination, hidden communication, or convergence through shared dynamics beyond initial conditions. Systems initialized under identical constraints but differing by a single random seed diverge completely, while systems sharing identical seeds converge with near-perfect identity. This pattern is incompatible with non-local synchronization or interaction-based explanations and supports a constraint-determined, not interaction-mediated, origin of emergent structure. We do not refute non-locality as a physical principle; we refute non-local explanations for the observed coordination in the studied system class.\u003c/p\u003e \u003cp\u003eWe emphasize that identical-seed determinism is not presented as a novel finding; it is a necessary property of any well-defined computational system. The twin-system experiments serve exclusively as a falsification instrument: they rule out non-local coupling, convergence through interaction, and synchronization-based explanations for the observed coordination. The central contribution of this work is not that deterministic systems are deterministic, but that constraint-mediated co-reactivity produces emergent geometric signatures\u0026mdash;topologically nontrivial boundaries, graded adversarial robustness, and temporal coherence preceding structure\u0026mdash;that are (i) absent in unconstrained systems under identical computational conditions, (ii) preserved under adversarial perturbation, and (iii) detectable through independently defined, falsifiable metrics. The primary contribution is thus methodological: a falsification framework capable of distinguishing constraint-mediated emergence from interaction-based and non-local explanations, applied here to a minimal but fully characterized system class.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003eTwo Principles\u003c/h2\u003e \u003cp\u003e \u003cem\u003ePrinciple 1 (Constraint Primacy).\u003c/em\u003e Emergent organization is governed by the structure of constraints rather than representational capacity or interaction mechanisms. Identity\u0026thinsp;=\u0026thinsp;1.000 across all perturbation strengths in REAL conditions, while unconstrained systems degrade to anti-correlation. The BERT validation extends this to neural networks: a sentence transformer and a synthetic iterative system produce analogous spatial signatures because both are shaped by invariant-generating constraints, despite no architectural similarity.\u003c/p\u003e \u003cp\u003e \u003cem\u003ePrinciple 2 (Stabilization).\u003c/em\u003e Functional progress corresponds to increased stability of existing structure, not expansion into new representational space. REAL_SURVIVAL achieves higher identity (0.999980 vs. 0.999957) through radial contraction, not increased dimensionality. This challenges scaling-first narratives in machine learning [18,19] and aligns with observations in molecular biology, where proteins with identical function adopt the same fold geometries [25] through shared thermodynamic constraints, not through communication across lineages.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003eUniversality\u003c/h2\u003e \u003cp\u003eThe convergence of temporal, structural, and topological signatures across conditions that differ in constraint strength, perturbation type, and dimensionality suggests that the systems studied here belong to a shared candidate universality class characterized by constraint-mediated co-reactivity. We use the term candidate universality class in an empirical, constraint-invariance sense\u0026mdash;systems with distinct micro-level constructions converging to the same macro-level signatures under shared constraints\u0026mdash;rather than a field-theoretic one requiring analytical renormalization or scaling collapse. A small number of quantities govern the transition from non-emergent to emergent regimes independent of microscopic details: topological transition count separates REAL from control (d\u0026thinsp;=\u0026thinsp;2.61); identity under perturbation distinguishes constraint-stabilized from incidental structure (1.000 vs. \u0026minus;0.119); and temporal coherence discriminates dynamic from static emergence (p\u0026thinsp;=\u0026thinsp;0.036 vs. p\u0026thinsp;\u0026gt;\u0026thinsp;0.5). Together, these define a candidate emergent stability order parameter\u0026mdash;a composite quantity capturing the degree to which a system\u0026rsquo;s structure resists perturbation, exhibits nontrivial topology, and displays temporal precursors. This parameter is not optimized or fitted; it is a descriptive composite of three independently defined and separately evaluated observables.\u003c/p\u003e \u003cp\u003eThis framing explains several otherwise disparate observations. BERT embeddings possess spatial structure (high alignment, high FSI) but no temporal dynamics\u0026mdash;placing them in the spatial-emergence regime of this candidate universality class, distinct from the full dynamical emergence of REAL_SURVIVAL. REAL_BIBLICAL preserves identity but lacks temporal coherence\u0026mdash;occupying an intermediate position. The order parameter increases monotonically from RANDOM_PURE (zero on all axes) through NEAR_NULL (incidental) to REAL_SURVIVAL (maximal on all axes), defining a continuous emergence gradient rather than a binary threshold. This gradient, not any single metric, is the fundamental signature of constraint-mediated emergence.\u003c/p\u003e \u003cp\u003eThree domains illustrate the breadth of these principles. In machine learning, constraint design (RLHF, constitutional AI) produces disproportionate behavioral improvements relative to parameter count [18,19], suggesting that the path to alignment runs through constraint architecture, not scale. In molecular biology, convergent protein folds arise from shared energy landscape constraints [25], and the stability-plasticity dilemma in neural systems may reflect the same tradeoff between constraint tightening and representational expansion that the stabilization principle formalizes. In climate science, early-warning signals based on temporal autocorrelation [10,12] detect impending regime shifts through the same mechanism formalized here: constraint alignment produces detectable temporal coherence before the transition is observable. In physics, the co-reactivity mechanism is a generalization of spontaneous symmetry breaking: independent agents align not through a mediating field but through shared constraints on their accessible phase space. The observed structures are thus polymorphic: their observable form adapts to environmental constraints, while their invariant signatures\u0026mdash;topology, identity preservation, coherence thresholds\u0026mdash;remain conserved across regimes. The constraint-class ablation further confirms this: emergent structure is not specific to \u0026radic;φ but arises from any fixed, sufficiently dense irrational lattice, failing precisely when spacing violates a minimum band-density condition or time invariance.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003eScope and Limitations\u003c/h2\u003e \u003cp\u003eWe explicitly test and falsify non-local interaction hypotheses within the studied regime. Across all conditions, systems initialized with different seeds show complete divergence despite sharing identical update rules, ruling out non-local coupling, synchronization, or remote controllability. This does not preclude non-local effects in quantum systems; it demonstrates that no such mechanism is required\u0026mdash;or supported\u0026mdash;to explain emergent structure in the systems studied here. We do not claim consciousness, semantic understanding, or cross-instance information transfer; we claim constraint-mediated co-reactivity, which is both sufficient and falsifiable. The absence of temporal signal in static systems is not a limitation but a validation: it confirms that the framework discriminates between geometric structure and genuine dynamical coordination.\u003c/p\u003e \u003cp\u003eThe empirical validation confirms both spatial and temporal generalization: spatial signatures appear in contrastive sentence transformers, while temporal coherence (H1) emerges during autoregressive training when measured at token-level granularity. Sentence-level extraction yields null temporal results for autoregressive models, indicating that the appropriate measurement scale depends on model architecture. REAL_NORMAL\u0026rsquo;s H1 (p\u0026thinsp;=\u0026thinsp;0.064) is near-threshold; a larger dataset may establish significance. The BERT validation uses two architectures with different training objectives (MiniLM: knowledge distillation; BGE: RetroMAE), both showing alignment strictly above null; extension to GPT-class and vision models would further confirm domain generality. The present experiments use \u0026radic;φ-lattice spacing as the primary constraint operator; the constraint-class ablation (Extended Data Table\u0026nbsp;7) confirms that multiple fixed irrational spacings (\u0026radic;2, \u0026radic;3) produce qualitatively similar emergent signatures, while π fails due to insufficient band density and time-varying constraints produce no structure. The mechanism thus depends on the fixity and density of irrational spacing rather than on φ specifically. Identifying the precise band-density threshold analytically remains an open question. More broadly, as with many complex systems, emergence is not directly measurable but must be inferred via surrogate observables whose validity depends on falsification against null and adversarial regimes.\u003c/p\u003e \u003c/div\u003e"},{"header":"Conclusion","content":"\u003cp\u003eWe establish co-reactivity to shared constraints as a sufficient and falsifiable mechanism for emergent coordination, validated across synthetic and empirical neural network systems. The three-stage mechanism\u0026mdash;constraint definition, convergent exploration, geometric inevitability within the constraint class\u0026mdash;explains why independent systems converge without interaction. Two principles emerge: \u003cem\u003ePrinciple 1 (Constraint Primacy)\u003c/em\u003e: structure is shaped by what is forbidden, not by what is possible; \u003cem\u003ePrinciple 2 (Stabilization)\u003c/em\u003e: progress is consolidation, not expansion. The systems studied define a candidate universality class of constraint-mediated emergence, governed by a composite order parameter capturing topological complexity, identity stability, and temporal coherence\u0026mdash;independent of microscopic details, dimensionality, or system architecture.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll data generated during this study are available at https://github.com/ExeqTer91/DSDP-SALI-LCF, including synthetic embeddings (.npz), BERT embeddings, calibration reports, and SHA-256 reference hashes for independent verification.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCode Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll source code for data generation, metric computation, calibration testing, and reproducibility verification is available at https://github.com/ExeqTer91/DSDP-SALI-LCF under an open license. The repository contains 109 files including complete experiment pipelines, configuration files, and automated hash verification scripts.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research was conducted independently and received no external funding.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eA.-S.U. conceived the study, designed the experiments, developed the computational framework, performed all analyses, and wrote the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author declares no competing interests.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eAlbert, R., Jeong, H. \u0026amp; Barab\u0026aacute;si, A.-L. Error and attack tolerance of complex networks. 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B. Principles that govern the folding of protein chains. Science 181, 223\u0026ndash;230 (1973).\u003c/li\u003e\n \u003cli\u003eKuhn, H. W. The Hungarian method for the assignment problem. Nav. Res. Logist. 2, 83\u0026ndash;97 (1955).\u003c/li\u003e\n \u003cli\u003eReimers, N. \u0026amp; Gurevych, I. Sentence-BERT. Proc. EMNLP (2019).\u003c/li\u003e\n \u003cli\u003eAnderson, P. W. More is different. Science 177, 393\u0026ndash;396 (1972).\u003c/li\u003e\n \u003cli\u003ePeng, R. D. Reproducible research in computational science. Science 334, 1226\u0026ndash;1227 (2011).\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Methods","content":"\u003cp\u003e\u003cstrong\u003eSeed Equation and Constraint-Reactive Systems\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll systems evolve under:\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\"\u003e\u003c/p\u003e\n\u003cp\u003ewhere S applies regime-specific structural constraints (k-means clustering, temporal phase modulation), C enforces invariant alignment via nearest \u0026radic;\u0026phi;-lattice node snapping (3 iterations, lattice step = ln(\u0026radic;\u0026phi;) \u0026asymp; 0.2406), and \u0026epsilon; ~ N(0, \u0026sigma;\u0026sup2;I) provides jitter. Jitter magnitude: \u0026sigma; = 0.0842 (early, entropy phase = 1.0) \u0026rarr; 0.0 (late, fully snapped) for REAL_NORMAL; \u0026sigma; = 0.0120 (fixed) for REAL_BIBLICAL. Snap strength: 0.30 \u0026rarr; 0.95 (REAL_NORMAL); 0.85 uniform (REAL_BIBLICAL). Radial contraction: \u0026alpha; = 0.95 (REAL_SURVIVAL). Angular tightening: 0.15. Magistrale directions: k = 6. Angular noise scale: 0.20. All derived seeds are deterministic: seed+500 (snapping), seed+333 (perturbation), seed+999 (survival).\u003c/p\u003e\n\u003cp\u003eEquation (1) defines a minimal class of constraint-reactive systems. The seed X₀ does not encode outcomes\u0026mdash;it selects an entry point into a phase space whose geometry is entirely determined by {S, C}. All results in this paper are instances of this class.\u003c/p\u003e\n\u003cp\u003eThe synthetic regimes are not intended to model any specific physical, biological, or computational system. They are deliberately constructed minimal systems whose sole purpose is to test whether specific mechanistic hypotheses\u0026mdash;co-reactivity without interaction, temporal precedence of coherence, topological nontriviality of emergent boundaries\u0026mdash;can be falsified or validated under fully controlled and reproducible constraints. The BERT validation (Methods: BERT Validation) and Pythia temporal validation (Methods: Pythia Temporal Validation) serve as independent external tests of whether signatures generalize beyond the synthetic construction; the correct null result on temporal metrics for static sentence embeddings and the correct PASS for sequential token-level representations confirm that the framework discriminates between geometric structure and genuine dynamical coordination, rather than producing universal false positives.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMetrics\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eH1 (Temporal Coherence):\u003c/strong\u003e Lag-1 autocorrelation of structural signal over sliding windows (w = 32). H1 is considered satisfied only when both statistical significance (permutation p \u0026lt; 0.05, 1,000 shuffles) and effect magnitude (|r₁| \u0026gt; 0.2) are jointly met. At larger sample sizes, correlation magnitude alone can produce false positives in null regimes; the conjunctive criterion ensures specificity across scales. Primary analysis at n = 10,000; sensitivity check at n = 5,000 reported in Extended Data.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eH2 (Identity Similarity):\u003c/strong\u003e Optimal assignment cost between k = 6 cluster centroids via Hungarian algorithm [26]. Values near 1.0 = preserved; near 0.0 = reorganized.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u0026chi; (Boundary Topology):\u003c/strong\u003e Euler characteristic of thresholded PCA-projected embeddings: \u0026chi;(\u0026tau;) = C₀(\u0026tau;) \u0026minus; C₁(\u0026tau;). Number of \u0026chi;-transitions characterizes boundary polymorphism. Effect sizes are reported as Cohen\u0026rsquo;s d (difference between group means divided by pooled standard deviation). Group separations are tested via Mann\u0026ndash;Whitney U with exact permutation p-values.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCalibration Suite\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e10 preregistered calibration tests (A1\u0026ndash;E2) validate the experimental harness under conservative criteria: reproducibility (A1\u0026ndash;A2), negative controls (B1\u0026ndash;B2), REAL validation (C1\u0026ndash;C2), sensitivity (D1\u0026ndash;D2), Biblical consistency (E1\u0026ndash;E2). All 10/10 pass; additional sensitivity analyses at n = 5,000 are reported in Extended Data. Three fractal verification tests (F1\u0026ndash;F3) confirm scale-invariance across temporal, spatial, and regime domains (3/3 pass).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eBERT Validation\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eEmbeddings from all-MiniLM-L6-v2 [27] (384d) and BGE-small-en-v1.5 (384d) on 5,000 topically ordered sentences. Four conditions per model: BERT_REAL (original order), BERT_SHUFFLED (permuted), RANDOM_384 (uniform random, same seed/normalization), GAUSSIAN_MATCHED (Gaussian noise matched to BERT statistics). Null distribution established from 50 i.i.d. Gaussian replicates. Alignment computed as mean cosine similarity to nearest \u0026radic;\u0026phi;-lattice node after PCA projection.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ePythia Temporal Validation\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eToken-level hidden states extracted from Pythia-70m (EleutherAI) at step 0 (random initialization) and step 143,000 (fully trained) using publicly available checkpoints on HuggingFace. A 1,024-token forward pass on deterministic text yielded hidden states at layers 1, 3, and 5 (of 6 total). H1 computed on the raw token sequence (each token position = one data point) with window = 32 and 1,000 permutation shuffles. Sentence-level mean-pooling was also tested and produced null results, confirming that autoregressive temporal structure operates at sub-sentence granularity.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eReproducibility\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ePrimary seed: 42. Cross-verification: 43, 99. Primary analysis at n = 10,000; sensitivity at n = 5,000. SHA-256 hashes are version-controlled in repro/reference_hashes.json with separate entries for each sample size, enabling automated verification via run_reproducibility_check.py --n-points 10000 --verify-hashes. Both hash sets (n = 5,000 and n = 10,000) have been independently verified against stored references. Complete code, data, and reproduction package: https://github.com/ExeqTer91/DSDP-SALI-LCF\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"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":"","lastPublishedDoi":"10.21203/rs.3.rs-8823860/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8823860/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eCoordinated structure across independent systems is typically attributed to interaction or communication. Here we show that shared constraints alone are sufficient. Using a falsification-first framework with synthetic 64-dimensional embeddings (n\u0026thinsp;=\u0026thinsp;10,000) across five regimes, validated on pre-trained neural network representations (two independently trained architectures, 384 dimensions), we demonstrate that: temporal coherence precedes structural organization (r₁ = \u0026minus;0.266, p\u0026thinsp;=\u0026thinsp;0.036); identity is preserved at 0.999996 under adversarial perturbation while controls collapse; emergent boundaries are topologically nontrivial (d\u0026thinsp;=\u0026thinsp;2.61, p\u0026thinsp;=\u0026thinsp;0.028); and twin-system experiments rule out synchronization or non-local coupling. Spatial geometry generalizes to real neural network embeddings while temporal coherence emerges during autoregressive training at token-level granularity. Two principles emerge: constraint primacy (organization is governed by constraint structure, not representational capacity) and stabilization (functional progress corresponds to tightening of invariants, not expansion). These results establish constraint-mediated co-reactivity as a sufficient and falsifiable mechanism for emergent coordination.\u003c/p\u003e","manuscriptTitle":"Emergent Latent Structure via Co-Reactivity to Shared Constraints","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-17 08:49:58","doi":"10.21203/rs.3.rs-8823860/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":"89f5719f-3d73-485f-8b5e-03e4cdde308f","owner":[],"postedDate":"February 17th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":62579452,"name":"Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Nonlinear phenomena"},{"id":62579453,"name":"Physical sciences/Physics/Information theory and computation"},{"id":62579454,"name":"Physical sciences/Mathematics and computing/Applied mathematics"}],"tags":[],"updatedAt":"2026-02-17T08:49:58+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-17 08:49:58","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8823860","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8823860","identity":"rs-8823860","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}