Attractor Calculus: A Formal Operator Language for Steering Cognition Across Substrates

In the paper's words

Prompt engineering, neural stimulation, and cognitive intervention all face the same underlying problem: steering an information-processing system into a desired dynamical state, with no principled language for saying what "steering" means or what it should cost. Practitioners in each domain currently work system-by-system, with no way to state whether a control primitive discovered in one substrate transfers to another. This paper formalizes that gap.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The annihilation occurs when the cubic $x^{3}-x-u=0$ has a double root, i.e., when both the equilibrium condition and its derivative vanish: $$ x^{3}-x=u,\qquad 3x^{2}-1=0. $$ The relevant double root on the left branch is $x=-1/\sqrt{3}$, giving the critical tilt $$ u_{c}=\Bigl(-\frac{1}{\sqrt{3}}\Bigr)^{3}-\Bigl(-\frac{1}{\sqrt{3}}\Bigr)=-\frac{1}{3\sqrt{3}}+\frac{1}{\sqrt{3}}=\frac{2}{3\sqrt{3}}\approx0.384900. $$ The cost rate of the minimal sustained input is $$ u_{c}^{2}=\frac{4}{27}\approx0.148148\ \text{per unit time}, $$ and the total cost of a tilt held for time $T$ is $E_{\mathrm{tilt}}(T)=\frac{4}{27}T$.

   from 4.3 SHIFT_BOUNDARY: minimal constant-tilt cost

2. Under noise intensity $D=0.05$ and a dwell-time reliability requirement of failure probability $\leq10^{-3}$ over $T_{d}=10^{6}$ time units, DEEPEN_BASIN must add $\delta d\approx0.786138$ of barrier depth, raising impulse cost by $2\delta d\approx1.572277$.

   from Abstract

3. Closed-form minimal-energy derivations for basin transition in a canonical bistable potential: impulse cost $E_{\mathrm{imp}}=2\Delta V$, tilt cost rate $u_{c}^{2}=4/27$, break-even horizon $T^{*}=27/8$, and their generalization to the family $V(x)=x^{4}/4-ax^{2}/2$ giving $T^{*}(a)=27/(8a)$ (Section 4).

   from 1. Introduction

4. LEXR shows that a formal language can be fitted to a trained recurrent system after the fact; our calculus asks the inverse question, namely which formal operators can be fitted before the fact as control commands. [11] quantifies what formal structure buys when used as an inductive bias during control: imposing formal-language constraints on Markov decision processes, combined with PPO and Lagrangian enforcement in Safety Gym, reduced the accumulated cost rate often by between almost one half and an order of magnitude, at the price of decreased performance in the earliest training steps while the constraint embedding was learned.

   from 2. Background and Related Work

5. Our operator set is designed with the same reuse goal: each operator's cost theorem should be proved once and instantiated per substrate. [9] demonstrates the maturity such an approach can reach: Wasm SpecTec generates a formal specification, prose pseudocode, and mechanized definitions from a single source of truth written in a domain-specific language, and the extracted Wasm semantics passes all 23,778 applicable tests in the official Wasm test suite.

   from 2. Background and Related Work

6. This criterion is deliberately analogous to the inheritance criterion of [10]: language-convexity there, cost-invariance here, both asking when a property survives a change of carrier.

   from 3.5 Universality criterion

How it is done

1. We define an Attractor Calculus: a small operator set (CREATE_ATTRACTOR, DEEPEN_BASIN, SHIFT_BOUNDARY, PERTURB, HOLD) over attractor-based dynamical systems, together with a cost functional and a composition algebra.

   from Abstract

2. The contributions are: A definition of the Attractor Calculus: a state-space semantics, five operators (CREATE_ATTRACTOR, DEEPEN_BASIN, SHIFT_BOUNDARY, PERTURB, HOLD), a quadratic cost functional, and a composition algebra with an additivity theorem (Section 3).

   from 1. Introduction

3. Formal language theory has repeatedly been used to describe the behavior of learned neural systems, and each use exposes both the power and the limits of the approach. [3] surveys the connection from Chomsky's hierarchy to modern NLP and records a caution that directly constrains our claims: constructions in which recurrent networks encode large computations exploit arbitrary numerical precision, such encodings are highly sensitive to slight perturbations of network parameters, and they correspond to steep valleys in parameter space that gradient-based optimization is unlikely to find.

   from 2. Background and Related Work

4. We define an operator's cost function $c_{\mathrm{op}}$ to be portable across a class of substrates if $c_{\mathrm{op}}$ is a function only of invariant quantities of the attractor landscape, such as the barrier height $\Delta V$, the noise intensity $D$, and basin volumes, and not of substrate-specific quantities such as the local curvature profile of the barrier or the parameterization of the control channel.

   from 3.5 Universality criterion

What would falsify it

1. If it fails, the failure itself is informative: it would show that steering levers are irreducibly substrate-specific and that cross-substrate transfer claims in prompting and neurostimulation should be treated with suspicion.

   from 1. Introduction

2. It builds the formal layer that makes the conjecture testable, and it derives the first closed-form results that the formal layer predicts.

   from 1. Introduction

3. An engineering pathway: the smallest experiment that would de-risk or falsify the calculus on a real trained network, the readiness level of the current work, and the binding constraints (Section 7).

   from 1. Introduction

Where the claim stops

1. Every result in this paper is a theorem about a one-dimensional model potential plus labeled assumptions; no experiment on a trained network, an LLM, or a neural circuit is reported.

   from 6. Discussion

2. The results rest on the following assumptions, each with a stated sensitivity: Canonical potential.

   from 6. Discussion

3. Section 4.2 assumes an underdamped steering coordinate in which an impulse delivers momentum $p$ with cost $p^{2}$.

   from 6. Discussion

Where to read

1. Abstract218 words · 1 min
2. 1. Introduction484 words · 2 min
3. 2. Background and Related Work1,043 words · 5 min
4. 3. Methods518 words · 2 min
5. 4. Analysis1,095 words · 5 min
6. 5. Results246 words · 1 min
7. 6. Discussion719 words · 3 min
8. 7. Practical Implications and Engineering Pathways833 words · 4 min
9. 8. Conclusion202 words · 1 min
10. Provenance and AI Involvement225 words · 1 min