Study notes: The Given and the Found

A fixed point is a state that f_\theta leaves unchanged, so requiring one admits only answers that are self-consistent under the learned dynamics. In an aligned landscape, self-consistency correlates with correctness. The requirement is therefore a regularizer toward correctness, structurally like a simplicity bias toward generalization. Neither contains the answer. Each tilts the search toward it under favorable conditions and toward a confident mistake under unfavorable ones. A self-consistent state that is wrong is the analogue of the simplest hypothesis that is false.

The correctness condition is what makes an answer right: Sudoku’s constraints, a reward, an energy. It is a fact about the task, prior to any network. The landscape is the learned dynamical system, whose attractors are supposed to coincide with answers that satisfy the condition. Training produces it, and it is an imperfect surrogate. The search is the trajectory actually followed at inference, restarts included, paid for by iterations and parallel runs. The iteration acts on the landscape and the search; the condition comes from somewhere else. The interesting questions sit at the seams: where the condition comes from, how faithfully the landscape approximates it, and when traversing the landscape finds answers to the condition and not to its surrogate.

• Energy-based models. Someone wrote the energy down, and inference descends it. Convergence is a theorem.
• The Lattice Deduction Transformer. The architecture guarantees the form of soundness, since the dynamics live on a lattice and can only remove candidates. Training supplies the strength of the deduction.
• MuZero. The condition is given, because the environment supplies reward. The search runs on a learned, value-equivalent model that is faithful where MuZero already plays and unreliable off-policy. The policy prior keeps the search where the surrogate holds.
• Pure learned dynamics (Equilibrium Reasoners, HRM, TRM, Attractor Models, GRAM). There is no energy, no contraction and no guarantee. Training carries the whole burden of making attractors coincide with solutions.

Down the spectrum, correctness migrates from the architecture into a trained surrogate, and its guarantee weakens from “theorem” to “holds where we trained it”.

9\times9\times9=729. The concretization \gamma(a), intersected with the valid solutions, leaves the permutations (1,2,3) and (2,1,3). Abstracting back gives (\{1,2\},\{1,2\},\{3\}): candidates 1 and 2 leave the third cell because they survive in no valid solution. Deduction descends the lattice and only removes candidates. The trained operator approximates \mathit{ded}_p. A step may fail to derive a true fact, and it never derives a false one, so the model returns a correct answer or abstains. In the essay’s words, “Correctness is architectural; competence is trained.”

If the residual were a good per-instance signal of correctness, the lowest-residual run would win. That it loses means the residual says nothing about which answers are right. A falling residual can mean convergence to a spurious attractor, a stable state that decodes to the wrong answer. The dangerous regime is deployable accuracy paired with a confidence signal that cannot be read.

• 1. No reachable attractor decodes to a correct answer: misalignment, and no amount of compute helps.
• 2. Correct and spurious attractors coexist: the failure is basin selection.
• 3. The correct attractor exists and its basin is too narrow: reachability.
• 4. The well-aligned regime, where residual decay is coupled to error reduction. Only here is trusting convergence valid.

The essay generalizes: wherever the objective at inference is a learned surrogate, the signal of having “figured it out” is a trained artifact, calibrated only as far as its training data.

Depth is the number of iterations in one trajectory, and it entrenches whatever basin the trajectory has entered. Breadth is the number of independent restarts from different initializations, and it covers more basins. The inference budget is their product, but they do different work. Depth fails in the spurious-attractor regime: a trajectory settled on a wrong answer sinks deeper, and the only escape is to start elsewhere. It also fails under genuine multiplicity, where several attractors are all correct. Settling on one is then information loss, not error. Both are premature convergence, and the maxim is do not over-settle: keep the distribution alive until the task demands commitment. The brakes are randomized initialization with path noise, which also shape the landscape so that convergence can be trusted; GRAM’s stochastic transitions; and LDT’s decide-temperature with parallel chains.

As learning, the weights change to absorb computation that would otherwise happen at inference, and the ceiling attainable at fixed data rises. As efficiency, the ceiling stays and compute is reallocated: a halting head stops easy instances early, or more training buys fewer deployment steps. The discriminator is ceiling versus cost-to-ceiling. Equilibrium internalization is observed in Attractor Models. Trained only on next-token loss, the backbone’s initial guess drifts toward the fixed point, until the solver can be removed at inference with little degradation. The loop self-distills into the feedforward initialization. The same absorbed computation can be cashed as a higher ceiling at a fixed budget or as the same ceiling at lower deployment cost. Which one we observe depends on whether we hold accuracy or compute fixed.

2\times21=42 block applications, so inference compute is matched, and the tied model has about 21 times fewer parameters. At fixed compute and far fewer parameters, iterative structure raises the ceiling. The gain comes from the structure and not from raw depth, and it appears where data is scarce, which is the signature of help to learning. The headline comparison does not hold test-time effort constant. It shows test-time depth extrapolation: an operator trained at 16 iterations generalizes to over a thousand, which a feedforward network cannot do at all. That is a capability claim, not a measurement of the learning effect.

Against a fixed correct label, a confidently correct prediction has (prediction − target) near zero and contributes almost nothing. The confidence sits on the target side: a confident intermediate computation manufactures a peaked target. In Attractor Models the target is the loop’s converged state. In MuZero it is the search’s visit counts. In LDT it is the \alpha-aggregate of still-consistent solutions. The peaked target gives (-0.45,+0.45) and the diffuse one (-0.05,+0.05), a push nine times weaker. The confidence of the manufactured target sets the amplitude. Whether the target is grounded in truth sets the sign.

Grounded targets include a supervised label, LDT’s \alpha-aggregate computed from real solutions, and an actual game outcome. Against one, a confident wrong convergence meets a large error and is pushed back down: the gradient flattens the wrong basin. With an ungrounded target, such as a policy head regressed onto its own confidently mistaken search, the same strong push writes the error into the weights and biases the next search the same way.

Grounding does not prevent the loss of alternatives. When each step’s target is built from the previous iterate, correctly signed pushes compound, and a strategy slightly below the best disappears. Nor does it prevent spurious attractors. They come from underdetermination: output-level supervision constrains the decoded answer only along the trajectories training visits, and the rest of the latent landscape is free to hold stable states the loss never sees. That is why the remedies are geometric. Randomized initialization, path noise and breadth manage coverage of the landscape, not the sign of the push.

The single-step limit is the feedforward network, which solves a small fraction of the puzzles the iterated operator solves. Hypothesis (1), amplification: compressing toward fewer, bigger steps is self-reinforcing, because a more decisive operator manufactures a more peaked target. Hypothesis (2), learnability: at a fixed parameter count, an operator that must make a bigger change per step is harder to fit. The claim is about fitting, not about a fixed operator computing a big step badly. Both forces act on the operator during training with opposite signs, so they settle at an optimal granularity, finer where the target map is harder. Early in training only a small step is learnable; as competence accrues the step coarsens and the loop compresses. The prescription is to amortize no faster than learnability permits. Deep supervision at every iteration is the field paying to keep each step learnable, and LDT’s \alpha-operator is the most principled case: it hands over the most precise sound refinement of the current state as the target.

MuZero’s policy prior is a confidence gate. It lets amplification act only where the learned model is reliable, which confines the search and the targets it manufactures to the region the surrogate fits. LDT’s candidate-elimination head uses an asymmetric cross-entropy that weights false eliminations eight times more heavily than false retentions, because a wrongly eliminated candidate cannot be recovered. That is deliberate de-amplification in the unsound direction. The reduction is 0.78/0.028\approx28-fold. Training bought completeness and speed. Soundness never moved, because the architecture installed it.

Everything in the earlier sections amortizes search over a correctness condition that is given. None of these systems amortizes the discovery of the condition. The line is structural, not a limit on compute or parameters. Some problems hand over the condition: the rules, the energy, the reward, the task metric, the Galois connection. Others require it to be inferred per instance before the machinery can run. ARC is of the second kind, since each task carries its own rules. A naive port of LDT plateaus and gets no benefit from test-time search, because the conflict head becomes unreliable and the search cannot tell good branches from bad. The authors imagine an abstract domain “over the programs that produce solutions, rather than over the solution states themselves”. The essay reads that as the move from deduction to abduction and induction. The lattice, the contraction and the energy each encode a fixed condition, and that is what the iteration exploits. Without a fixed condition the search has no structure to descend.

The prediction: scaling these architectures keeps winning on Sudoku-, maze- and SAT-shaped problems and keeps hitting a wall on ARC-shaped rule induction, at essentially any scale. The dividing line is givenness, not difficulty and not size. It would be wrong if a system in this family cracked open-ended rule induction by scaling depth, breadth and amortized training alone, with no new mechanism for representing and searching over candidate conditions.

• A limit cycle as the right terminal behavior for inputs with many solutions is “our reading, not theirs”. No architecture in the cluster makes a non-converging recurrent set its terminal competence.
• That the binding constraint on step size is learnability and not forward expressivity is isolated by none of the papers. Deep supervision confounds learnability with optimization stability, and the separating experiment is described, not run.
• Whether iterating a locally competent operator or compressing it carries the headline quality is unsettled. The two experiments that would tell have not been run cleanly: “We think it; we have not proven it.”
• Settling certifies nothing about the world. A system can settle with every internal signal of completion into a spurious attractor.