We present the Awen Grid Digital Collider, a numerical instrument that ev olv es two coupled ensembles("ledgers") of states on unit hy perspheres: a Real Ledger of up to 10â· unit quaternions on SÂł, and an ImaginaryLedger of equally many unit octonions on Sâ·, distributed across two consumer GPUs. The Real Ledger ev olv esby an observ er-weighted map built from the general SO(4) sandwich rotation Ï â q_bÂ·Ï Â·q_aâ»Âč; the ImaginaryLedger ev olv es by unit-octonion Cay leyâDickson rotation; "collisions" between the sectors are measured by theoctonion associator [x ,y ,z] = (x y )z â x (y z), an observ able that is prov ably zero on the quaternionic subalgebraand therefore self-calibrating. All structural theorems the instrument relies on â the SU(2) representation of theHamilton product, associativ ity of â, alternativ ity and non-associativ ity of đ, and the composition-algebra law|x y | = |x ||y | underly ing ex act norm conserv ation â are machine-v erified to 10â»Âčâ”â10â»Âčâ¶.Three empirical results follow. First, the observ er-weighted map possesses a global ring attractor on SÂł: from auniform random beam, thousands of independently ev olv ing states v isibly self-organize into a single ring within~27 ticks, after which ev ery measured observ able phase-locks. Second, the locked observ ables are inv ariantacross a 200Ă range of beam sizes (5Ă10âŽâ10â· nodes), ten random seeds, two arithmetic precisions(float32/float64), two backends (CPU/dual-GPU), and two independent operators: collision rift 1 .1 7 90 ±0.001 1 , lion ratio 1 1 .554 ± 0.006, mass index 0.99627 (across-seed range 5Ă10â»â”). Third, a Lev el II sweep ofthe rotation:fold mix ing weight maps the attractor landscape and finds a genuine interior resonance at w â 0.865(lion 39.4), while demonstrating that prev iously published constants of the framework (a claimed fold-amplituderesonance at 0.48, and a legacy "Lion constant" of 0.5352) are respectiv ely not reproduced under a preregistered 10â·-node blind sweep and unreachable any where on the measured slice â and whose documentaryorigins we identify from the primary sources (an algebraic identity and a bookkeeping snapshot, respectiv ely ;Section 6.5). A 40-item falsifiability audit of the RHC corpus (18 v erified · 8 false · 5 contradictions · 6 ex ternalmismatches quantified · 1 not reproduced · 1 open · 1 untestable) and a measured correction of the framework'scompression claims (delta pre-transform: â30 to â37 % on correlated telemetry ; â0% bey ond entropy on anydata) are included. The instrument computes geometry on simulated states; it does not act on phy sical matter,and no claim to the contrary is made.
We report, in a finite designed reactionâdiffusion medium on a 64Ă64 lattice, a distributed physical memory that lives in the configuration of a flow-constructed boundary rather than in any object, label, or instruction store. A directed flow writes a local orientation field into boundary material; after the originating flow is removed and the medium relaxes, an identical weak, direction-free scalar input recovers opposite motions depending only on the boundary's history. The effect is causally necessary: resetting the boundary, shuffling its local orientation, or disabling its plasticity abolishes recovery, and a fixed-protocol permutation test (statistic and one-sided alternative fixed in advance) gives per-seed p †5Ă10â»Âł across eight independent seeds with large effect sizes. In an autonomous extension, boundary memory continuously gates permeability, acquired resource drives motion, and motion pays for repair of the boundary that stores the memory, closing a self-maintaining loop that survives repeated damage. Results rest on gates and thresholds fixed before the confirmation runs, disjoint development/confirmation seed families, and a conserved resource ledger in the autonomous stage (B1). We do not claim natural occurrence, open-ended evolution, spontaneous birth of the carrier, or new physical laws; the contribution is the causal-necessity protocol and its transportable operational definition of configuration memory. This study is one component of a broader, open-ended research program exploring the possibility space of non-genomic organization â memory, self-maintenance, and selection-like dynamics that need not rely on a genome. The present preprint establishes only the distributed-memory component and makes no claim about evolution, inheritance, life, or natural occurrence.
We present \textbf{ORCHID} (\textit{Orchestrated Reduction Consensus for Hash-based Integrity in Distributed Ledgers}), a novel bio-inspired consensus protocol that maps the neuroscientific \emph{binding problem} -- how the brain integrates distributed neural oscillations into a unified conscious percept -- onto the distributed systems \emph{consensus problem}, how blockchain nodes agree on a single ledger state under Byzantine faults. Grounded in the Penrose--Hameroff Orchestrated Objective Reduction (Orch~OR) hypothesis and the Kuramoto synchronisation model, ORCHID equips each node with a quantum-noisy phase oscillator; consensus is triggered when the network's order parameter $r(t)$ crosses a \emph{binding threshold} $Ξ_b$, mirroring the gamma-band binding event in conscious perception. ORCHID is further strengthened by a coherence-weighted Quantum Secret Sharing (QSS) layer, extending the survey framework of Weinberg to a concrete consensus application. Simulation results on Watts--Strogatz small-world networks ($n=10$--$150$) demonstrate: (i)~the Kuramoto order parameter reaches $r_{\max}=0.988$ under coupling $K=3.0$, well above the theoretical critical coupling $K_c \approx 1.41$; (ii)~a sharp QSS fidelity phase transition at coherence $c^*\approx 0.82$, confirming Theorem~2; (iii)100\% consensus rate at all tested Byzantine fractions (0\%--40\%), with median convergence under 4~s for $n=30$; and (iv)~ORCHID achieves $O(n{\cdot}k)$ message complexity, outperforming PBFT's $O(n^2)$ at $n\geq150$. These results establish ORCHID as a scalable, biologically plausible, and quantum-augmented consensus mechanism for post-quantum distributed ledgers.
We present a production-ready framework for verifiable self-organization in networks of 768 coupled quantum oscillators (the âCrystal Brainâ), integrating sixfine-tuned components: adaptive SPSA optimization with automatic plateau escape, multi-resolution Louvain community detection, non-deterministic proof seeds,normalized causal efficacy metrics, dynamic Merkle root hashing, and a semanticproof-tagging API. We map the Ising regime classification of Bhalla et al. (2026)onto a Kuramoto-type phase dynamics and demonstrate convergence to a coherentCAPTURE regime with 84.7% capture fraction. A companion octonionic atlas of 50nuclides (OctoSpec v0.4) reveals a moderate anti-correlation (r = â0.54, p < 0.001)between the Octonionic Anomaly Index and nuclear binding energy. All coherencemilestones are certified by zero-knowledge proofs generated at 80-bit security viathe ZEE200 backend and registered immutably on the ARKHE OCTRA chain.
The Genesis Ledger is a computational framework for scalable coordination in complex systems. It demonstrates that stable large-scale organisation cannot be achieved through fixed control alone, but requires three coupled mechanisms: continuous adaptive control, memory of past disturbances, and cross-layer verification of system state. Using a lattice-based model, we show that coordination exhibits a non-linear dependence on coupling strength, with both under-constrained (disordered) and over-constrained (rigid) regimes leading to failure. Adaptive modulation of coupling enables systems to navigate this trade-off, maintaining coherence under dynamic conditions. We further introduce a memory variable that allows the system to reduce recovery time under repeated perturbations, effectively adapting its baseline response to environmental complexity. Finally, we demonstrate that single-layer feedback systems are vulnerable to deceptive or misleading signals, while cross-layer verificationâcomparing independent representations of system stateâenables robust filtering of false inputs. Together, these results define a general principle for scalable coordination: Stable systems must continuously regulate constraint, adapt based on experience, and verify signals through consistency across independent representations. This repository provides a reproducible simulation framework, figure generation pipeline, and structured package suitable for further research, validation, and application in distributed systems.
Investigating the timeâfrequency-based phase synchronization between nonstationary time series of the cryptocurrenciesâ prices can be a suitable tool to reveal their complicated interactions at different frequencies. In this work, the phase synchronization between 25 cryptocurrencies with the highest capitalization from December 1, 2021 to June 1, 2022 is calculated using the phase-locking value method based on wavelet transform. Then, utilizing the graph theory, the cryptocurrency networks are constructed, and their topological features like path length (PL), clustering coefficient (CC) and node strength are evaluated in various frequencies. This research indicates a strong phase synchronization between the investigated cryptocurrencies, especially in low frequencies. Also, the networksâ high average CC and short PL compared to their equivalent regular and random networks display the small-worldness of the networks. We observe from the obtained results that Bitcoin, Ethereum and Binance currencies, among the most popular cryptocurrencies, have the highest average node strengths at different frequencies. Also, TRON shows the lowest CC and node strength among all currencies, representing its limited phase interaction with other currencies.
Sebastian MĂŒller, Isabel Amigo, Alexandre Reiffers-Masson, Santiago Ruano-RincĂłn
In directed acyclic graph (DAG)-based distributed ledgers, unreferenced blocks (tips) form the backlog of a distributed queueing system. Each new block creates one tip and attempts to remove up to $k$ existing tips by referencing them. With heterogeneous propagation delays, these service decisions are made from delayed local information, so nodes may disagree on the backlog and some reference attempts are wasted. We study a continuous-time Poisson model with bounded heterogeneous delays and uniform tip selection. We prove that the embedded tip-configuration chain is irreducible, aperiodic, and positive Harris recurrent, and hence admits a unique stationary regime. The observer and local tip-pool sizes have stationary exponential moments, converge to their stationary limits, and satisfy almost-sure ergodic averages. We also derive a Little-type identity relating the stationary mean observer tip count to the mean time until a typical block is first referenced. Simulations are included as qualitative illustrations of the effects of delay variability and issuance heterogeneity.
Open access
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math.PR
cs.DC
Mathematical and Theoretical Epidemiology and Ecology Models
In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a flat blow-up profile, already proved by Bricmont and Kupainen together with Herrero and VelĂĄzquez. Though our approach relies on the well-celebrated method, based on the reduction of the problem to a finite-dimensional one, then the use of a topological 'shooting method' to solve the latter, the novelty of our approach lays in the use of a modulation technique to control the projection of the zero eigenmode arising in the problem. Up to our knowledge, this is the first time where modulation is used with this kind of profiles. We do hope that this simplifies the argument.
Open access
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Stability and Controllability of Differential Equations
Interactions between stock and cryptocurrency markets have experienced shifts and changes in their dynamics. In this paper, we study the connection between S&P500 and Bitcoin in higher-order moments, specifically up to the fourth conditional moment, utilizing the time-scale perspective of the wavelet coherence analysis. Using data from 19 August 2011 to 14 January 2022, the results show that the co-movement between Bitcoin and S&P500 is moment-dependent and varies across time and frequency. There is very weak or even non-existent connection between the two markets before 2018. Starting 2018, but mostly 2019 onwards, the interconnections emerge. The co-movements between the volatility of Bitcoin and S&P500 intensified around the COVID-19 outbreak, especially at mid-term scales. For skewness and kurtosis, the co-movement is stronger and more significant at mid- and long-term scales. A partial-wavelet coherence analysis underlines the intermediating role of economic policy uncertainty (EPU) in provoking the Bitcoin-S&P500 nexus. These results reflect the co-movement between US stock and Bitcoin markets beyond the second moment of return distribution and across time scales, suggesting the relevance and importance of considering fat tails and return asymmetry when jointly considering US equity-Bitcoin trading or investments and the policy formulation for the sake of US market stability.
For users of the Ethereum network, the gas price is a crucial parameter that determines how swiftly the decentralized consensus protocol confirms a transaction. This paper studies the statistics of the Ethereum gas price. We start with some conceptual discussion of the gas price notion in view of the actual transaction-selection strategies used by Ethereum miners. Subsequently, we provide the descriptive statistics of what we call the threshold gas price. Finally, we identify and estimate a seasonal ARIMA (SARIMA) model for predicting the hourly median of the threshold gas price.