Reasoning shortcuts are rule solutions that reach correct predictions through unintended concepts. A recent framework of Takemura, Inoue, and Nishino analyzes them through an automorphism group of value relabelings, asking when rules pin concepts down. Its key definition, one value permutation shared across all positions, does not apply as stated to any of its four heterogeneous benchmarks, and the most direct embedding, padding, produces confident false pathology: 90.91% of solution pairs unexplained on CLE4EVR, versus 0% under every well-defined rung of the componentwise hierarchy we introduce; the padded verdict rotates under configuration-file ordering. Across eleven rule families under fifteen pre-specified predictions (thirteen confirmed), unexplained-pair rates span 0% to 99.9999% and track provable structure: six theorems give sufficient conditions for transitivity and its failure. For circuit-given rules, symmetry-inertness of a coordinate is coNP-complete; automorphism existence is coNP-hard under randomized reductions, lies in $\Sigma_2^p$, is not $\Sigma_2^p$-complete in the Boolean case unless PH collapses, and is coNP-complete on monotone circuits. Boolean transitivity is classified exactly: automorphisms explain everything iff the solution set is an affine coset. Weakly supervised models place all 94 observed shortcuts at the one level the theory flags, none at the 48 it certifies transitive, and none at twelve typed-ambiguous levels. Relocating the absorbing element moves every shortcut with it; a confusion null attributes the location to geometry while the observed rate exceeds it by half again. Trained end to end on CLE4EVR's rule and heterogeneous domains through a synthetic prototype front end, models produce 20,223 label-preserving errors with zero different-orbit exceptions, as transitivity predicts, where the padded instrument would misreport 78-88% of them.
A central premise of latent world models is that predicting the future forces a representation to internalize the physics of its environment. Which physical quantities does a trained latent actually contain, and what decides this? We answer with controlled interventions in POKEWORLD, an interactive environment whose visually identical objects hide mass, drag, and contact stiffness. A certificate-gated protocol first certifies each parameter as recoverable from raw observations, then measures whether it enters the latent, so a null result can be attributed to the objective rather than to the environment. The resulting identifiability map has two organizing mechanisms and one frontier. Inputs limit what can be known, while prediction targets decide what is retained. Stiffness enters the latent only when touch is forecast ($R^2=0.50$, compared with $-0.02$ when the same signal is merely fused into the input), and under single-step prediction a vision-only latent discards even perfectly visible object state. Drag marks the frontier. It carries a recoverability certificate of 0.89 yet plateaus near 0.13 under every deterministic prediction objective we test, while a supervised head on the same trunk reaches 0.45. Parameters whose readout is slow and ratio-type under the sensed coordinates fall outside what these objectives acquire. On RH20T, an input-target factorial across scaling curves reproduces both mechanisms across two robots and 4,258 episodes. Every arm missing information or prediction pressure stays flat over a fivefold data range, and only the full multimodal objective forecasts force beyond a persistence baseline, with held-out gains that grow with scale. Objective structure determines which physical parameters a latent acquires, and additional data improves only the parameters it already acquires.
Empirical work on algorithmic collusion asks one question of the data: are prices supracompetitive? We show this can be answered"no"by a conspiracy that is nonetheless profitable. Consider bidding agents that couple only through the joint distribution of their unexplained bid components, leaving every agent's own bid law exactly at the competitive law. Any test whose input is a single agent's price or bid history then has power exactly equal to its false-positive rate, for every coupling strength up to comonotonicity. The published detection methodology is therefore blind to this conduct by construction rather than underpowered, and no sample size repairs it. Three empirical results follow. First, the mechanism appears in real language-model agents: twenty models from nineteen independent developers, three deployment prompts each, show residual correlation of $+0.053$ between two deployments of one model against $+0.0001$ across models, with a 95% interval clustered by developer of $[0.030, 0.078]$, under an auditor that sees every order feature and is fitted out of sample. Second, the coupling falls monotonically as sampling temperature rises ($p=0.002$), turning a deployment parameter into a candidate mitigation. Third, on 24 days of Ethereum block-building auction data covering 77,684 bids from 39 bidders, the honest population of bidder pairs is itself so dependent that a screen held at a 5% false-positive rate must sit above a floor of $+0.50$ to $+0.81$, which is 20 to 32 times the family-wise sampling threshold and does not fall as the audit window grows. Since lawful multi-identity operation and conspiracy are behaviourally indistinguishable here, the tractable regulatory target is not detection but counting: resolving 40 bidding identities into 23 operators raises the Herfindahl index by 247.5%, and adding behavioural clusters from public bid streams reaches 324.5%.
