We present FoilForm, a two-stage neural surrogate for predicting aerodynamic lift and drag coefficients (Cl, Cd) directly from airfoil contour geometry across angle-of-attack sweeps. The first stage is an ultra-compact autoregressive transformer (4,470 parameters) with a novel pairwise outer-product attention mechanism that captures second-order geometric feature interactions. The second stage is a lightweight convolutional correction network (34,290 parameters) that refines predictions using the full-resolution contour. Trained on 1,768 airfoils from a compiled dataset of 2,946 profiles at Re = 105, the full pipeline achieves validation MAE of 0.040 on Cl and 0.0081 on Cd across 1,178 held-out airfoils, while running at 0.3 ms per airfoil in batched CPU inference. Ablation studies demonstrate that pairwise interactions, minimal dropout (p = 0.05), and 4-layer depth are critical design choices. We benchmark against NeuralFoil (xxxlarge) and show 9× lower Cl error and 19× lower L/D error on matched evaluation, while using 8.6× fewer parameters.

Manipulation benchmarks are often interpreted as tests of perception and language-conditioned control, but a score can also reflect regularities in the evaluation states. We introduce the placement radius R, the radius of the smallest disc containing an object’s shipped positions. The condition R ≤δ is exactly the geometric condition for one task-indexed constant to lie within distance δ of every shipped position; behavioral equivalence additionally requires an embodiment-specific robustness assumption. Applied to LIBERO, this audit reveals a shipped-state defect in LIBERO-Object. Its tasks declare 5×5 cm target-sampling regions, yet the shipped states span only 0.0–1.9 cm, and six of ten tasks place the target at one point to floating-point precision. In a paired intervention on one task, sampling the declared region reduces the lookup-goal condition from 25/30 to 10/30 successes (p = 0.0007). A reset-informed condition supplied the true target position and succeeded on all 30 repaired states. Thus the same motion controller can solve the repaired states when supplied the correct target. A sparse displacement ladder shows a dose response but does not identify a uniform tolerance δ: outcomes change even at the smallest nonzero level, and the experiment varies direction and initial state together. Accordingly, placement radius is a pre-training diagnostic of geometric shortcut opportunity, not by itself a certificate of policy behavior. The causal intervention is limited to one task, and we do not establish that a trained policy exploits the defect.

People choose a neural network's architecture before training. Architecture search automates that choice, but it ranks whole designs by their final loss, and loss cannot tell a part that captures structure from one that fits noise. We present TONN, which builds sequence models from small circuits, such as delays, running sums, products and attention. Inside one training run, TONN keeps a circuit only if switching it off raises the error, in bits, on data the network has not trained on. We call that rise the circuit's worth. On pure noise, growing a network by this rule added nothing, while five other growth rules added 18 to 24 units. On induction, where the next token is whatever followed the last copy of the current token, TONN matched a transformer (0.018 against 0.017 bits per token, 3 seeds). It did so by denoising, meaning it started from a large random network and kept replacing its least useful circuits. Even when no single circuit could do the lookup alone, it composed an induction head, the copying circuit found in trained transformers, from delays, products and attention. From one random start, it kept short-range circuits for proteins and long-range lookups for movie reviews, so the architecture followed the data. At larger scale, how worth is measured decides whether the search helps. On Long Range Arena Text, which classifies movie reviews read as 4,096 raw bytes, a cheap one-pass estimate of worth lost to the random starts (69.7 against 75.5 percent, 3 seeds). Exact worth reached 78.9 percent against 74.9 for its start, in the one seed we ran; a gentler schedule reached 73.5. That network ran as fast as a transformer of similar size on one GPU. Our transformers, trained on the same budget, reached only 58 to 61 percent, against 79.1 published. On proteins the search added 6 points over its random start but trailed designed networks by 2 to 4 points. On web text, a same-size transformer is 0.13 bits per byte ahead.

Fast lift and drag estimates along airfoil polars matter for design sweeps, but high-fidelity tools are slow at scale. We show that a causal sequence model with only 6394 trainable param- eters can match or beat two public NeuralFoil checkpoints (xxsmall and xxxlarge) on the same withheld BigFoil-derived test rows when each method sees identical preprocessing and the decoder is fed ground-truth past coefficients along the polar (teacher forcing). On the first 4096 rows of that split, pooled mean Cl MAE is 0.051 54 and Cd MAE is 0.002 99, below both baselines at much smaller width. FoilCORE walks the polar in angle-of-attack order. Geometry and (α, Re, M, Cl, Cd) tokens embed at width d=8. A masked causal trunk, path nonlinearities, gated outer products, and a two-output head predict the next step, so the curve is an ordered sweep rather than unrelated (α, Re) samples. Greedy decoding with K=0 raises validation MAE versus teacher forcing; Section 7.4 describes a NeuralFoil xxxlarge warmstart for open-loop use. Scope and caveats are in Section 9.