The Internet taught us that the value of a network depends on \emph{how} its nodes connect: broadcast stars scale as $V\!\propto\!N$ (Sarnoff), fully-connected meshes as $N^2$ (Metcalfe), and group-forming networks as $2^{N}$ (Reed). We ask the analogous question for networks of AI agents. We model the net value of connection as a function of coordination-group size, derive from it the properties an optimal collaboration protocol must have, and introduce ANet Patu-1 -- a self-organizing consensus protocol in which the network continuously re-forms its own coalitions, adaptively riding the upper envelope of all three regimes at $O(1)$ parallel consensus rounds. To measure value without opinion-grading, we score an emergent protocol by formally specifying it and deriving its complexity, the way distributed algorithms are analyzed. Two results follow. (i)~Emergence -- a crowd of the \emph{cheapest} model, when heterogeneous, starts weak but its collective value compounds with $N$ and \emph{overtakes} a crowd of a far \emph{stronger} model that is homogeneous: a crossover that marks a scaling law for collaboration rather than for scale. (ii)~Reflexivity -- a heterogeneous network, given only its own problem and no design hints, converges on ANet Patu-1 itself, reconstructing the high-dimensional law that governs its own connective value.
Disordered metamaterials feature microstructures with inherent randomness and irregularity, enabling them to achieve broader property coverage and superior performance unavailable in their regular counterparts. Despite their promise, designing disordered microstructures is substantially harder than designing regular ones. Their design remains trapped between manual parameterizations with limited expressiveness, and generative AI that is data-hungry and struggles to generalize. To address these limitations, we propose a generative design framework based on Neural Cellular Automata that dynamically grows complex microstructures through learned local interaction rules, inspired by the self-organizing processes in natural materials. This framework requires only a single training template, yet accommodates diverse disordered microstructures and adapts to irregular domains and arbitrary discretizations. By manipulating the learned local rules, we can steer the growth process to generate microstructures unseen during training, providing control over orientation, anisotropy, and directional thickness without retraining. As a dynamic, local growth process, it naturally produces spatially varying microstructures that transition smoothly to enable location-specific mechanical properties. We demonstrate this in a multiscale mechanical cloaking design, where microstructures vary across the space to meet an optimized heterogeneous property distribution. Our design enables excellent cloaking performance without complicated post-processing and incompatible assembly common in existing methods. This data-efficient, generalizable approach opens access to previously intractable disordered materials for biomedical implants and soft robotics.
We ask a structural question: given unreliable elementary problem-solvers, what organizations of them solve hard problems reliably, and what are the limits? We develop a $decomposition~algebra$: elementary solvers are morphisms in a stochastic category, and four combinators (sequential composition, parallel ensembling, verification gating, and recursive reduction) generate the space of compound solvers. We equip this algebra with two homomorphisms, a $reliability$ valuation into the ordered monoid $([0,1],\le)$ and a $cost$ valuation into a commutative semiring, and we derive the composition laws that govern how reliability flows through structure. Our central results are (i) a $verification~odds~law$ (the result that names this report), showing that a verification gate multiplies the odds of correctness by the verifier's likelihood ratio $Λ$, so that $k$ conditionally independent gates yield geometric amplification; (ii) a $reliability~amplification~theorem$, giving target reliability $1-δ$ at $O(\log 1/δ)$ verification depth whenever $Λ>1$; and (iii) a $threshold~dichotomy$: above the critical parameters reliability can be driven arbitrarily close to one at logarithmic cost, while at or below them no amplification is possible. We then show that $self-organization$ is the least fixed point of a monotone improvement operator on the complete lattice of strategies, and that this fixed point equalizes marginal log-odds gain per unit cost. Finally, we prove matching limits: an information ceiling bounds per-gate amplification by a divergence quantity; shared error causes create a strictly positive voting floor, so diversity is $necessary$ for unbounded amplification. Reliability, in short, is neither free nor magical: it is bought with independent information, arranged by composition, and bounded by the verifier.