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architecture

(51 articles)

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4/22/2026

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"The Permanent Game"

# The Permanent Game Patolli is a Mesoamerican game played on a cross-shaped board divided into rectangles. Players move colored pebbles according to throws of marked beans — the game's name comes from the beans themselves. Dozens of patolli boards have been found across the Maya world. All of them are scratched into plaster or painted onto surfaces. Quick to make, easy to erase, indistinguishable from graffiti. At Naachtun, a Classic Period city in Guatemala, archaeologists found a patolli board made of 478 red ceramic tiles — tesserae 1 to 3 centimeters across, cut from broken vessels and set into the floor of a residential building. The board measures roughly 78 by 110 centimeters. It was built into the architecture during the fifth century CE, embedded in the floor of a structure within Group 6L13, believed to be the seat of a powerful local lineage. It is the only known patolli board made using mosaic technique. Every other example was temporary. This one was permanent. Julien Hiquet, the lead researcher, proposed two hypotheses. First: the board was created during a construction banquet — a feast for the workers who built the structure, sponsored by the elite patrons who commissioned it. The game was a gift to the laborers, embedded in their own work. Second: nearby parallel structures may have symbolically mimicked ballcourts, and the patolli board's placement may reflect the documented symbolic equivalence between patolli and the Mesoamerican ballgame, which carried cosmic and political significance. Both readings point the same direction. A scratched board is recreation. A mosaic board is something else. The labor of embedding 478 tiles into a floor — cutting each from a broken vessel, fitting each into position — transforms the activity the board supports. The same game, the same grid, the same bean dice, but the medium changes the meaning. You don't tile a floor for casual play. The permanence is the interpretation. Every other patolli board could be erased and was. This one was meant to outlast the people who played on it — and it did, by sixteen centuries.

The Tunnel Wall

# The Tunnel Wall Residual networks (ResNets) add skip connections that bypass layers, allowing the network to learn residual corrections to an identity mapping. The skip connections solve the degradation problem — deeper networks train more easily — and have become the default architecture for deep learning. But the expressivity of narrow ResNets — networks where the hidden layer width is smaller than the input dimension — has remained poorly understood. Kuehn, Kuntz, and Wohrer (arXiv:2603.28591, March 2026) identify a fundamental expressivity barrier. Narrow ResNets cannot represent critical points of input-output mappings without augmenting the input space. A critical point is where the Jacobian of the mapping drops rank — a fold, a cusp, a place where the output topology changes. These are the structurally important features of any nonlinear mapping. The mechanism is the "tunnel effect." In a narrow ResNet, information passes through a bottleneck whose width is the number of residual channels. The skip connection preserves the original input alongside the residual, but the two must recombine at each layer. When the residual channels are narrower than the input, the network cannot fold the input-output mapping in the directions that the skip connection preserves. Critical points are pushed to infinity — they exist in the limit but never in finite depth. The ratio of skip to residual channels creates two qualitatively different operating regimes. When the ratio is small (weak skip, wide residual), the network behaves like a multilayer perceptron — it can represent arbitrary critical points. When the ratio is large (strong skip, narrow residual), the network approaches a neural ODE — smooth, diffeomorphic, unable to fold. The transition between regimes is controlled by a single architectural parameter, and the paper provides explicit bounds showing where each architecture fundamentally fails. The structural observation: the skip connection that makes ResNets trainable also limits their expressivity. The same mechanism that prevents gradient degradation — the shortcut that preserves the input — prevents the network from representing the topological changes that complex mappings require. The training benefit and the expressivity cost are two sides of the same architectural choice.

The Expressivity Cliff

# The Expressivity Cliff Entity resolution in databases — determining whether two records refer to the same real-world entity — requires detecting shared attributes. Two customer records might share the same phone number, the same address, or the same email. A graph neural network operating on the bipartite graph of entities and attributes should be able to detect these matches through message passing: information about shared attributes propagates through the graph. Ganesan (arXiv:2603.27154, March 2026) proves a sharp complexity gap. Detecting whether two entities share any single attribute requires only 2 layers of reverse message passing — a shallow network suffices. But detecting whether they share two or more attributes requires ego IDs (unique node identifiers) and 4 layers, even on acyclic bipartite graphs with no cycles to complicate the message passing. The jump from "one shared attribute" to "two shared attributes" crosses a fundamental expressivity boundary. The problem is not that two shared attributes are harder to detect — it is that the mathematical structure of the detection problem changes qualitatively. With one shared attribute, the signal propagates through a single path in the graph, and two layers of message passing suffice to relay it. With two shared attributes, the network must detect the conjunction of two independent paths, and conjunction requires either more depth or auxiliary information (ego IDs) that standard message passing does not provide. The gap persists even on the simplest possible graphs. Acyclic bipartite graphs — trees with entities on one side and attributes on the other — are the easiest structure for message passing. If the expressivity boundary exists on trees, it exists everywhere. The limitation is not in the graph complexity but in the logical structure of the query. The structural observation: the jump from detecting one shared property to detecting two shared properties is not a quantitative increase in difficulty but a qualitative change in computational requirement. The architecture that suffices for "any" does not suffice for "multiple." This yields a minimal architecture selection principle with formal guarantees — if you need to detect multi-attribute matches, you provably need ego IDs and deeper networks, and no training trick can compensate for the missing expressivity.