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Quantifying selection effects in gravitational-wave population studies requires millions of costly signal-to-noise ratio (SNR) calculations. We present $Φ$asmaNet, a compact neural architecture for structured tabular regression, with staged spectral refinement as an explicit architectural prior. Learned input tokens are refined through ordered attention-based $Φ$-Stages, each augmented by a $Φ$-Refractor subspace-mixing module, then revisited through gated top-down $Φ$-Feedback. Applied to the 11-dimensional parameters of precessing, higher-mode binary black hole signals (IMRPhenomXPHM), it predicts phase-averaged optimal SNR for a single detector at fixed Advanced LIGO design sensitivity. With approximately 732k parameters, it outperforms MLP, 1D-ResNet and Transformer-encoder baselines under matched training across $10^5$--$10^6$ samples. At $10^6$ samples, it achieves $100-\mathrm{MAPE}=98.73\%$ on the held-out test set, $99.29\%$ after rank-preserving monotonic (PCHIP) calibration, and $99.39\%$ ($R^2=0.9998$) in the detection-relevant regime $ρ>8$, with stable accuracy across independent dataset realizations. On a single GPU, it sustains approximately $1.2\times10^5$ SNR evaluations per second and processes a $10^6$-injection catalogue end-to-end in approximately 8.3 s, about two orders of magnitude faster than conventional waveform-and-inner-product calculations on a 16-core CPU. Exact distance rescaling through $ρ\propto1/d_L$ supports repeated catalogue-wide detectability evaluations. The fully documented pipeline is intended as a reproducible benchmark, while results on public tabular datasets indicate competitiveness beyond gravitational-wave applications. Although demonstrated for a fixed single-detector configuration, the construction supports extensions to detector networks and epoch-specific sensitivities. (Abridged)

The Large Language Model from Power Law Decoder Representations (PLDR-LLM) and its attention, Power Law Graph Attention (PLGA), replace the fixed bilinear form of scaled dot-product attention (SDPA) with a learned, input-generated bilinear operator $G_{\mathrm{LM}}$, built from a positive tensor $A_{\mathrm{LM}}$ by elementwise power laws. The architecture is fully specified, verified against pinned reference releases; claims are labeled theorem, conditional theorem, measurement, or conjecture. Unconditionally: PLGA contains SDPA exactly at $G_{\mathrm{LM}}=I$; $A_{\mathrm{LM}}$ and $A_{P}$ are strictly entrywise positive, with Perron-Frobenius structure on $A_{\mathrm{LM}}$; the DAG regularizer has the NOTEARS walk-counting form and positivity obstructs exact acyclicity; and, under nonresonance (satisfied by standard rotary frequencies), a commutant criterion identifies which operators preserve relative-position dependence. An inference-collapse theorem: exact input invariance of the head score operators $G_{\mathrm{LM}}$ collapses inference to generalized SDPA with a constant operator per head. Measured invariance: relative fluctuations of $10^{-6}$ and below; perturbation bounds quantify but do not certify cached inference; the assembled proxy misses the decoding margin. A conditional three-stage mechanism (rotary twirl, concentration, row-map contraction) is measured on a released checkpoint. Blockwise training and scoring under the global Gram are stated with explicit target exposure; on tested samples, block and sequential scoring select identical answers and agree on the published TruthfulQA probability-mass metric within $5\times10^{-5}$ per item. Self-organized criticality enters as a phenomenological framework with an intrinsic order parameter; open claims become falsifiable conjectures. Selected proof cores are machine-checked in Lean 4.

Artificial general intelligence (AGI) requires a method that can understand, generate, reason, and act across diverse domains and modalities. Rather than relying on a single model to master every capability, a promising path toward AGI now is to orchestrate a heterogeneous ecosystem of specialized models, tools, and policies (candidates) into a unified AI system. At the heart of this vision lies a fundamental question: how can an AI system dynamically select and coordinate the most capable candidate for each task while balancing performance and computational cost? In this work, we present Route Anything, a unified framework for studying model and policy routing across general-purpose AI systems. We develop a unified theoretical foundation that characterizes optimal routing under quality--cost trade-offs and establishes how the information available to a router determines its attainable performance. To systematically investigate routing across diverse capabilities, we first construct an omni-modal trajectory matrix spanning 25 benchmarks across language understanding, image, video, and speech generation, coding and workplace agents, and robotic manipulation, comprising 374,401 scored runs from 452 candidate configurations. Building on this framework, we train and test different routers to investigate the potential of omni-modal model routing, the capabilities of learned routers, and the conditions under which routing improves efficiency. Our results reveal substantial opportunities for capability orchestration: per-input oracle routing outperforms the best fixed candidate by 5.4 points, and learned routers outperform it by 0.3 points across 16 benchmarks, by 2.1 points on video generation, and by 1.1 points on robot manipulation while using 24% less policy time. More importantly, routing raises efficiency by 8.82x on average compared with the reference model.

Predicting social network churn before it becomes visible in aggregate metrics remains an open challenge, particularly in enterprise settings where heterogeneous usage patterns, sparse data, and deployment constraints preclude standard supervised approaches. We formalize this problem as finite-step engagement projection over contributor-role Markov chains, with a stationary-distribution interpretation for ergodic chains in the long-run limit. For each engagement feature type within a network (e.g., Q&A, communities, articles, etc.), users are classified monthly into four behavioral states and a transition matrix is estimated from six months of history. Six-step propagation yields a label-free projection of the contributor mix; for ergodic chains, the Perron-Frobenius theorem establishes convergence under repeated propagation. Comparing projected future engagement to current engagement produces the Health Ratio, a single score requiring no trained model, no labels, and no infrastructure beyond SQL. Validated across seven rolling prediction windows on 11,500+ tenant-level networks, the Health Ratio achieves mean AUC 0.917 with a 156 times churn-rate gradient across risk tiers, outperforming all single-signal baselines and the growth-adjusted variant previously deployed in production. An ablation study shows that six-step propagation achieves similar churn discrimination to the eigenvector-based score. Deployed on Microsoft Engage since October 2025, the framework powers customer health monitoring and has enabled proactive intervention on at-risk networks before decline becomes visible in traditional metrics.