Scientific Foundations for the Study of Computational Dynamics in AI Systems

Status: Draft 1
Date: 29 June 2026
Institution: Sirraya Labs
Identifier: SFS-0001
Supersedes: None
Version: 1.0-draft

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Table of Contents

1. Purpose and Scope
2. Ontology
3. Epistemology
4. Mathematical Framework
5. Principles
6. Scientific Questions and Hypotheses
7. Validation Framework
8. Research Debt Register
9. Notation
10. Governance

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1. Purpose and Scope

1.1 Motivation

The study of artificial intelligence systems has produced substantial understanding in several dimensions: architectural analysis of how system structure enables computation, behavioral evaluation of system capabilities, and optimization theory of how training produces those capabilities. What remains underdeveloped is a systematic treatment of inference itself as a computational process with its own structure, dynamics, and control affordances.

This specification introduces such a treatment. It defines a scientific framework for studying the internal computational dynamics of AI systems, not as a metaphor but as a formal commitment that generates testable hypotheses and supports principled intervention design.

1.2 Claims

This specification asserts that:

1. AI systems that process information sequentially admit a state-space abstraction useful for prediction, estimation, and intervention.
2. This abstraction can be formalized with sufficient precision to generate falsifiable hypotheses.
3. Empirical regularities in computational trajectories can be identified and characterized.
4. These regularities can inform the design of inference-time interventions.

1.3 Limitations

This specification does not assert that:

• Internal states correspond to human-interpretable concepts, beliefs, or intentions.
• The mathematical model is the true generative process of the system.
• Controllers designed under this framework guarantee any behavioral property.
• The framework applies to systems where internal states are inaccessible.
• The framework replaces architectural, behavioral, or optimization perspectives.

1.4 Falsifiability

This framework would require substantial revision or abandonment if any of the following were demonstrated:

• F-001: Internal state representations cannot predict future computation better than input-level baselines.
• F-002: No stable notion of computational state exists across inputs, tasks, or model instances.
• F-003: Candidate observations carry negligible mutual information with downstream behavior.
• F-004: Admissible interventions cannot systematically influence computational trajectories beyond the effect of random perturbations of equal magnitude.
• F-005: Computational trajectory geometry is entirely input-specific with no transferable structure across related inputs or tasks.

1.5 Scope

This specification applies to AI systems satisfying four criteria:

1. The system processes inputs through a sequence of internal computational steps.
2. Internal representations at each step are accessible for measurement.
3. The system's architecture is fixed during inference (no online learning).
4. Behavior is generated from internal representations through a known mechanism.

In scope: transformer language models, diffusion models, state-space models, recurrent architectures, and any future architecture meeting the above criteria.

Out of scope: closed-API systems without internal state access, systems that learn during inference, and systems whose internal computation is not structured as sequential steps.

1.6 Relationship to Prior Work

This specification synthesizes and formalizes ideas from the residual stream perspective on transformers (Elhage et al., 2021), the linear representation hypothesis (Park et al., 2024; Marks and Tegmark, 2024), neural ODE formulations (Chen et al., 2018), control-theoretic analyses of neural networks (Adu and Gharesifard, 2024), inference-time intervention methods (Turner et al., 2023; Li et al., 2023), empirical trajectory geometry studies (Mir, 2024), state estimation theory (Kalman, 1960; Anderson and Moore, 1979), and model predictive control (Mayne et al., 2000).

It differs from all of the above by providing a unified formal framework that distinguishes computational state from observations from behavior, defines intervention channels explicitly, specifies validation criteria for every assumption, is architecture-agnostic at its core, and includes an explicit epistemology.

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2. Ontology

Ontology specifies what entities exist within the framework. These are scientific objects, not mathematical ones. They exist independently of any particular representation.

2.1 Entity Taxonomy

The framework recognizes six categories of entities. Every object in the theory belongs to exactly one category.

ENT-001: Computational State

A computational state is an internal configuration of the system at a specific point during inference that is sufficient to determine the probability distribution over future internal configurations and outputs, conditional on future inputs.

Status: Primitive (axiomatic; see Principle 1)

Properties:

• Internal to the system
• Not directly observable (see Section 3, Epistemology)
• May admit multiple equivalent representations (see Principle 3)
• Existence is axiomatic; nature is empirical

Candidate representations (not definitional):

• The residual stream vector in a transformer
• The hidden state in an RNN
• The latent vector in a state-space model
• The concatenation of residual stream and key-value cache
• A compressed representation learned by an autoencoder

The distinction between object and representation is fundamental. "Computational state" is the scientific object. "Residual stream" and "hidden vector" are mathematical representations of that object. The framework does not commit to any particular representation.

ENT-002: Computation

A computation is the evolution of computational state from an initial configuration to a final configuration, together with any outputs generated during or after this evolution.

Status: Primitive

Properties:

• Occurs over discrete or continuous time
• Transforms inputs and initial state into outputs and final state
• May be deterministic or stochastic
• Is the primary object of scientific study in this framework

ENT-003: Observation

An observation is any quantity derived from the computational state that is measurable during or after inference without modifying the state.

Status: Defined

Properties:

• Directly or indirectly measurable
• May be noisy, incomplete, or lossy
• Does not alter the computational state
• Is the only epistemic access to computational state (see Epistemology)

Examples: next-token probability distribution, entropy of the output distribution, norm of the residual stream, attention pattern statistics, outputs of trained probes, sparse autoencoder activations.

ENT-004: Intervention

An intervention is any externally applied modification to the computational state or its evolution during inference.

Status: Defined

Properties:

• Applied during inference
• Modifies state evolution
• May be state-dependent or state-independent
• Enters through specified channels
• Is the primary mechanism for influencing computation from outside

Examples: adding a vector to the residual stream, modifying attention weights, editing specific activations, applying a transformation to key or value vectors.

ENT-005: Behavior

Behavior is any externally visible output of the system.

Status: Defined

Properties:

• Directly observable
• Generated from computational state
• The primary object of evaluation
• May be evaluated against external criteria

Examples: generated token sequences, probabilities assigned to specific outputs, response latency, factual correctness as judged by external evaluation.

ENT-006: Environment

The environment comprises all external inputs to the system that are fixed during a given inference pass.

Status: Defined

Properties:

• Externally specified
• Not modified by the system during inference
• Includes prompts, retrieved context, system instructions, and tool outputs if pre-computed
• Does not include state-dependent interactions, which are part of computation

2.2 Entity Relationships

The relationships among entities are as follows:

Environment → [initializes] → Computational State
Computational State → [evolves via] → Computation
Computational State → [generates] → Behavior
Computational State → [produces] → Observation
Intervention → [modifies] → Computation
Observation → [informs] → Intervention (in closed-loop systems)

2.3 Exclusions

The following are explicitly excluded from the ontology:

• Concepts, beliefs, or intentions: Interpretative labels that may be applied to states but are not themselves entities.
• Truth: An evaluation criterion applied to behavior, not a property of computational state.
• Meaning: Semantic content is an interpretation, not an entity.
• Agents or selves: The framework studies computation, not agency.

This exclusion is methodological. It keeps the framework grounded in measurable quantities and prevents category errors in scientific claims.

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3. Epistemology

Epistemology specifies what can be known about the entities defined in Section 2, and how such knowledge can be acquired.

3.1 Knowledge Classification

Every claim within the framework falls into exactly one of the following categories.

K-001: Directly Observable

Quantities measurable from system outputs or exposed internals without inference or estimation.

Examples: generated tokens, logit vectors via unembedding, attention weights if the architecture exposes them, residual stream vectors if model internals are accessible, computation time.

Epistemic status: Measurement, not inference. Error comes from measurement precision, not model uncertainty.

K-002: Indirectly Observable

Quantities not directly measurable but estimable from directly observable quantities using a validated model.

Examples: semantic entropy estimated from logits, truthfulness of a representation estimated from probes, distance to a reference trajectory estimated from a geometric model, uncertainty of a state estimate.

Epistemic status: Estimation with quantifiable uncertainty. Claims must report confidence intervals or posterior distributions.

K-003: Latent

Quantities that are part of the theoretical framework but cannot be directly or indirectly observed with current methods.

Examples: the true minimal sufficient state if different from accessible representations, the exact disturbance at each step, the complete causal structure of state evolution.

Epistemic status: Theoretical construct. Claims about latent quantities are permissible in mathematical derivations but must be distinguished from empirical claims.

K-004: Mathematical

Truths derived from axioms and definitions by logical deduction.

Examples: theorems about idealized systems, bounds derived from assumptions, representation equivalence results.

Epistemic status: Proof. Validity depends on axioms and definitions, not on empirical evidence.

K-005: Designed

Choices made by researchers not forced by evidence or logic.

Examples: using PCA rather than an autoencoder for manifold approximation, choosing a specific entropy threshold for intervention, selecting a particular control horizon.

Epistemic status: Engineering decision. Justified by pragmatics, not by truth claims.

3.2 Evidence Standards

Claim Type	Required Evidence	Standard
Measurement (K-001)	Instrument specification, precision analysis	Measurement error quantified
Estimation (K-002)	Validation on held-out data, confidence intervals	Statistical significance or Bayesian equivalent
Theorem (K-004)	Proof, explicit assumptions	Logical validity
Hypothesis test	Pre-registered protocol, effect size, replication	See Section 7
Engineering comparison (K-005)	Benchmark results, ablation, efficiency analysis	Fair comparison, reproducible

3.3 The Epistemic Gap

A central commitment of this framework is the explicit acknowledgment of the gap between computational state (ENT-001) and observations (ENT-003).

The Epistemic Gap Principle: No finite set of observations uniquely determines the computational state. All knowledge of computational state is mediated by observation models, which may be incomplete or incorrect.

Consequences:

1. State estimation is always inferential, never definitional.
2. Claims that a particular representation is the computational state are category errors.
3. The quality of state estimation must be evaluated empirically, not assumed.
4. Multiple observation models may be consistent with the same evidence.

3.4 Uncertainty Quantification

Every estimate of an indirectly observable quantity must be accompanied by an uncertainty quantification. Acceptable forms include confidence intervals, credible intervals, prediction intervals, bootstrap estimates of variability, and formal bounds derived from assumptions. Point estimates without uncertainty are permissible only for directly observable quantities (K-001).

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4. Mathematical Framework

This section introduces the mathematical formalism. All symbols defined here are to be used consistently across all papers in the research program.

4.1 State Space

The computational state at step $t$ is denoted $x_t$. The state space $\mathcal{X}$ is the set of all possible states.

Definition 4.1 (State Space): $\mathcal{X}$ is a measurable space. For transformer language models using the residual stream representation, $\mathcal{X} = \mathbb{R}^d$ with the Borel $\sigma$-algebra, where $d$ is the model dimension.

Status: Definition.

Note: $\mathbb{R}^d$ is a convenient representation. The true state space may be a submanifold of $\mathbb{R}^d$, a quotient space under some equivalence relation, or a space of lower intrinsic dimension. This is an empirical question (see Q-001).

4.2 Dynamics

Definition 4.2 (Open-Loop Dynamics): The evolution of the computational state in the absence of intervention is:

[ x_{t+1} = f_t(x_t) + w_t ]

where $f_t: \mathcal{X} \to \mathcal{X}$ is the nominal dynamics (the pretrained system, frozen during inference) and $w_t \in \mathcal{X}$ is the unmodeled disturbance.

Status: Definition.

Assumptions on $f_t$:

• A-DYN-001 (Determinism): $f_t$ is a deterministic function for a given model checkpoint. Status: Assumption, validated by rerunning the same input.
• A-DYN-002 (Lipschitz): $f_t$ is $K_t$-Lipschitz on the domain of observed states. Status: Hypothesis, validated under H-004.
• A-DYN-003 (Markov): The state evolution satisfies the first-order Markov property under the nominal dynamics. Status: Hypothesis, validated under H-002.

Note on $w_t$: The disturbance term is a modeling construct. It captures approximation error from the Markov assumption, effects of stochastic decoding when applicable, numerical precision effects, and any other deviation from nominal dynamics. No claim is made that $w_t$ is small, Gaussian, or independent across $t$. Its properties are empirical questions.

4.3 Observation Model

Definition 4.3 (Observation Model): An observation $y_t$ is related to the computational state by:

[ y_t = g(x_t) + v_t ]

where $g: \mathcal{X} \to \mathcal{Y}$ is the observation function, $\mathcal{Y}$ is the observation space, and $v_t$ is observation noise.

Status: Definition.

Candidate observations: See Section 6, Hypothesis H-005.

Assumptions on $g$:

• A-OBS-001 (Computability): $g$ can be evaluated during inference without additional forward passes through the full model. Status: Engineering constraint.
• A-OBS-002 (Differentiability): $g$ is differentiable almost everywhere with respect to $x_t$, enabling gradient-based control. Status: Assumption, varies by observable.

4.4 Intervention Model

Definition 4.4 (Controlled Dynamics): The evolution of the computational state under intervention is:

[ x_{t+1} = f_t(x_t, B_t u_t) + w_t ]

where $u_t \in \mathcal{U}$ is the control input, $B_t$ is the intervention channel operator specifying how control enters the computation, and $\mathcal{U}$ is the control space.

Status: Definition.

Special case (Additive Residual Intervention): For interventions that add directly to the residual stream: $x_{t+1} = f_t(x_t) + B_t u_t + w_t$.

Assumptions on $B_t$:

• A-INT-001 (Locality): Interventions at layer $t$ affect only the state from layer $t$ onward. Status: By construction.
• A-INT-002 (Channel Structure): $B_t$ may be rank-deficient, reflecting that interventions cannot modify the state in arbitrary directions. Status: Hypothesis, validated under H-006.

4.5 Control Policy

Definition 4.5 (Control Policy): A control policy $\pi$ maps available information to control inputs:

[ u_t = \pi(\hat{x}t, x_t^{\text{ref}}, \mathcal{H}{t-1}) ]

where $\hat{x}t$ is an estimate of the current state, $x_t^{\text{ref}}$ is a reference state from a calibration set of successful executions, and $\mathcal{H}{t-1}$ is the history of previous states, observations, and interventions.

Status: Definition.

Policy Taxonomy:

Class	Name	Form	Examples
P-1	Open-Loop Static	$u_t = \alpha v \cdot \mathbf{1}[t = t^*]$	Activation Addition, CAA
P-2	Open-Loop Scheduled	$u_t = \alpha_t v$	Identified gap in literature
P-3	Proportional Feedback	$u_t = K_p e_t$	ITI, Feedback Controller
P-4	Predictive Control	$u_t = \arg\min \sum_k \ell(\hat{x}{t+k}, u{t+k})$	SemMPC
P-5	Learned Feedback	$u_t = \pi_\theta(\hat{x}_t, x_t^{\text{ref}})$	Future work

4.6 Cost Functional

Definition 4.6 (Cost Functional): The quality of a controlled trajectory is measured by:

[ J(\tau, {u_t}) = \sum_{t=0}^{L} \ell(x_t, u_t) + \Phi(x_L) ]

where $\ell: \mathcal{X} \times \mathcal{U} \to \mathbb{R}{\geq 0}$ is the stage cost, $\Phi: \mathcal{X} \to \mathbb{R}{\geq 0}$ is the terminal cost, and $L$ is the number of computational steps.

Status: Definition.

Typical stage cost: $\ell(x_t, u_t) = D_s(x_t, x_t^{\text{ref}}) + \mu |u_t|^2$, where $D_s$ is a semantic divergence measure and $\mu > 0$ is the control cost weight.

4.7 Reference Trajectories

Definition 4.7 (Reference Trajectory): For a given input and evaluation criterion $\mathcal{C}$, a reference trajectory is:

[ \tau^{\text{ref}} = (x_0^{\text{ref}}, x_1^{\text{ref}}, \ldots, x_L^{\text{ref}}) ]

where each $x_t^{\text{ref}}$ is the state at step $t$ from an execution that satisfies $\mathcal{C}$.

Status: Definition.

Note: $\mathcal{C}$ is defined externally. The framework is agnostic to the specific criterion. "Reference" denotes membership in a calibration set of successful executions, not a claim about optimality or uniqueness.

4.8 Semantic Divergence

Definition 4.8 (Semantic Divergence): $D_s: \mathcal{X} \times \mathcal{X} \to \mathbb{R}_{\geq 0}$ is a non-negative function satisfying $D_s(x, x) = 0$, $D_s(x, x') \geq 0$, and differentiability in both arguments almost everywhere.

Status: Definition.

Candidate measures: Euclidean distance, cosine distance, Mahalanobis distance, projection distance. The choice of $D_s$ is an engineering decision (K-005), validated by its utility in control and prediction.

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5. Principles

Principles are organizing commitments that are neither axioms nor hypotheses. They express the conceptual foundation of the framework.

Principle 1: State Evolution

Every computation evolves through a sequence of internal state transitions. The state at each step contains the information necessary to determine future evolution, conditional on future inputs.

Status: Principle.

Justification: This is the defining commitment of the state-space perspective. Without it, the framework reduces to behavioral analysis. It is not empirically testable in itself; it is the lens through which empirical questions are formulated.

Principle 2: Behavioral Emergence

Observable behavior emerges from computational state. The relationship between state and behavior may be complex, stochastic, and mediated by architectural mechanisms, but behavior is always a function of state, not an independent entity.

Status: Principle.

Justification: This principle grounds behavioral claims in computational ones. Understanding behavior requires understanding the states that generate it.

Principle 3: Representation Independence

Computational state admits multiple equivalent mathematical representations. If two representations are related by a bijection that preserves the dynamics and observation structure, they are equivalent for all scientific and engineering purposes within the framework.

Status: Principle.

Consequence: The framework does not commit to the residual stream, the KV cache, or any particular representation as the state. The choice of representation is a pragmatic decision evaluated by its utility.

Principle 4: Measurement Limitation

No finite set of observations perfectly reveals computational state. All knowledge of state is inferential, mediated by observation models with finite precision and possible systematic error.

Status: Principle.

Justification: This principle distinguishes the framework from approaches that treat hidden states as directly accessible. It mandates uncertainty quantification and motivates state estimation as a central problem.

Principle 5: Intervention Locality

Interventions influence computation only through specified channels at specified points. An intervention at step $t$ cannot affect states at steps prior to $t$, and its effect on subsequent states is mediated by the dynamics.

Status: Principle.

Justification: This principle defines the causal structure of intervention. It is what makes control design possible: interventions have predictable, local effects that propagate forward through the dynamics.

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6. Scientific Questions and Hypotheses

This section organizes the research program into a structured set of questions, each associated with testable hypotheses and validation criteria.

6.1 Question Hierarchy

Q-001: What is the computational state?

• H-001: Residual stream sufficiency
• H-001b: Dimensionality of state (future)

Q-002: What is the structure of dynamics?

• H-002: Markov approximation
• H-003: Local linearizability
• H-004: Lipschitz continuity

Q-003: What can be observed?

• H-005: Observation informativeness
• H-006: State estimability

Q-004: What is the geometry of successful computation?

• H-007: Low-dimensional concentration
• H-008: Cross-task transfer

Q-005: Can computation be controlled?

• H-009: Local controllability
• H-010: Intervention efficacy
• H-011: Channel comparison

Q-006: Can trajectories be stabilized?

• H-012: Idealized stability
• H-013: Robust stability

Q-007: Can estimation and control be combined?

• H-014: Closed-loop benefit
• H-015: Computational feasibility

6.2 Detailed Hypotheses

H-001: Residual Stream Sufficiency

Statement: The residual stream after block $t$ is a sufficient state representation for predicting future tokens, in the sense that a model using $x_t$ as input predicts $x_{t+1}$ better than a model using only the input token embedding.

Scientific Question: Q-001
Status: Hypothesis
Knowledge Class: K-002 (indirectly observable, via prediction accuracy)
Validation Criterion: A predictor $\hat{x}{t+1} = h(x_t)$ achieves lower mean squared error than a baseline predictor $\hat{x}{t+1} = h_0(e)$, where $e$ is the token embedding, on held-out trajectories. Effect size: MSE reduction exceeds 20 percent. Significance: $p < 0.001$.
Protocol Reference: TSI-0001, Protocol P-001

H-002: Markov Approximation

Statement: The state evolution satisfies a first-order Markov approximation in the sense that adding history does not substantially improve prediction of the next state.

Scientific Question: Q-002
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: A first-order model predicting $x_{t+1}$ from $x_t$ achieves MSE within 5 percent of a model using the full history $(x_0, \ldots, x_t)$.
Protocol Reference: TSI-0001, Protocol P-002

H-003: Local Linearizability

Statement: The transformer block $f_t$ is well-approximated by its first-order Taylor expansion in a neighborhood of reference trajectories.

Scientific Question: Q-002
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: The empirical linearization error is bounded by twice the theoretical bound on 95 percent of states within distance $\delta$ of reference trajectories, where the theoretical bound is derived from the estimated Lipschitz constant.
Protocol Reference: TSI-0001, Protocol P-004

H-004: Lipschitz Constant Estimation

Statement: The transformer block $f_t$ admits a Lipschitz constant $K_t$ that can be estimated from trajectory data and is finite on the domain of observed states.

Scientific Question: Q-002
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: Estimated $K_t$ from trajectory pairs satisfies $|f_t(x) - f_t(x')| \leq K_t |x - x'|$ for 99 percent of held-out pairs, and $K_t$ is stable across different estimation methods.
Protocol Reference: TSI-0001, Protocol P-003

H-005: Observation Informativeness

Statement: At least one candidate observation has estimated mutual information with downstream task success exceeding 0.1 bits.

Scientific Question: Q-003
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: Estimated mutual information exceeds 0.1 bits for at least one observable at some layer $t$, estimated via mutual information neural estimation or equivalent, with bootstrap confidence interval excluding zero.
Protocol Reference: TSI-0001, Protocol P-005

Candidate Observations Table:

ID	Observable	Definition
OBS-001	Next-token entropy	$-\sum_v p_v \log p_v$
OBS-002	Logit margin	$\max_v z_v - \text{second}\max_v z_v$
OBS-003	Residual norm	$|x_t|_2$
OBS-004	Residual change	$|x_t - x_{t-1}|_2$
OBS-005	Cosine to reference	$\cos(x_t, x_t^{\text{ref}})$
OBS-006	Subspace distance	$|x_t - \Pi_{\mathcal{M}}(x_t)|_2$
OBS-007	Attention entropy	Entropy of attention distribution
OBS-008	Probe output	$\sigma(w^\top x_t + b)$

H-006: State Estimability

Statement: The computational state $x_t$ can be estimated from a window of observations with error significantly lower than a naive baseline.

Scientific Question: Q-003
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: A state estimator $\hat{x}t = h(y{t-k}, \ldots, y_t)$ achieves lower error than using $y_t$ directly by at least 20 percent on average.
Protocol Reference: TSI-0001, Protocol P-006

H-007: Low-Dimensional Concentration

Statement: Reference trajectories concentrate near a subspace of dimension $r < d/2$, where $d$ is the ambient dimension, with the top-$r$ principal components explaining more than 80 percent of variance.

Scientific Question: Q-004
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: $r/d < 0.5$ with explained variance ratio exceeding 0.8, replicated across at least three model families and three tasks.
Protocol Reference: TSI-0001, Protocol P-007

H-008: Cross-Task Transfer

Statement: The low-dimensional subspace learned from reference trajectories on one task partially transfers to related tasks.

Scientific Question: Q-004
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: Distance to the transferred subspace has point-biserial correlation with task success exceeding 0.3 on the new task.
Protocol Reference: TSI-0001, Protocol P-008

H-009: Local Controllability

Statement: The linearized system $(A_t, B_t)$ where $A_t = I + J_{f_t}(x_t^{\text{ref}})$ is controllable in a neighborhood of the reference trajectory; that is, the controllability Gramian is full rank.

Scientific Question: Q-005
Status: Hypothesis
Knowledge Class: K-002 (for empirical Gramian), K-004 (for rank condition)
Validation Criterion: The condition number of the controllability Gramian is less than $10^6$ at more than 90 percent of layers for at least one intervention channel.
Protocol Reference: TSI-0001, Protocol P-009

H-010: Intervention Efficacy

Statement: State-dependent interventions reduce distance to a reference trajectory more than norm-matched random perturbations.

Scientific Question: Q-005
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: A state-dependent intervention reduces divergence by more than 20 percent relative to a norm-matched random perturbation, averaged over layers and prompts.
Protocol Reference: TSI-0001, Protocol P-010

H-011: Channel Comparison

Statement: Different intervention channels differ in their efficacy for trajectory steering, and the most effective channel depends on the layer.

Scientific Question: Q-005
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: At least one channel outperforms the mean across channels by more than 10 percent in divergence reduction at some layer.
Protocol Reference: TSI-0001, Protocol P-011

H-012: Idealized Stability

Statement: For the idealized linearized system with no disturbances, there exists a control policy that drives the state toward the reference trajectory at a geometric rate with contraction factor less than one.

Scientific Question: Q-006
Status: Hypothesis
Knowledge Class: K-004 (mathematical)
Validation Criterion: A theorem proving this statement under explicitly stated assumptions including H-003 and H-009.
Protocol Reference: Not applicable (theoretical result)

H-013: Robust Stability

Statement: Under bounded disturbances, the closed-loop system maintains the state within a bounded distance of the reference trajectory.

Scientific Question: Q-006
Status: Hypothesis
Knowledge Class: K-004 (mathematical)
Validation Criterion: A theorem or formal bound.
Protocol Reference: Not applicable (theoretical result)

H-014: Closed-Loop Benefit

Statement: Closed-loop control, combining state estimation and feedback, maintains closer proximity to reference trajectories than the best open-loop baseline on held-out prompts.

Scientific Question: Q-007
Status: Hypothesis
Knowledge Class: K-002
Validation Criterion: Average divergence over the trajectory is reduced by more than 10 percent compared to the best open-loop baseline, with $p < 0.01$.
Protocol Reference: TSI-0001, Protocol P-012

H-015: Computational Feasibility

Statement: Closed-loop control can be implemented with less than 50 percent additional inference latency compared to the uncontrolled system on standard hardware.

Scientific Question: Q-007
Status: Hypothesis
Knowledge Class: K-001 (directly measurable)
Validation Criterion: Wall-clock latency overhead below 50 percent on an NVIDIA A100 or equivalent GPU, batch size 1.
Protocol Reference: TSI-0001, Protocol P-013

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7. Validation Framework

7.1 Hypothesis Lifecycle

Every hypothesis transitions through defined states:

Proposed → Under Test → Supported / Rejected / Unresolved

• Proposed: Hypothesis is formulated with a validation criterion.
• Under Test: A pre-registered experiment is in progress.
• Supported: The experiment met the validation criterion and was replicated at least once.
• Rejected: The experiment failed to meet the criterion and the failure was replicated.
• Unresolved: No experiment has been conducted, or results are inconsistent across replications.

7.2 Experiment Pre-Registration

Before conducting an experiment to test a hypothesis, the protocol must be specified in TSI-0001, sample sizes and statistical tests must be pre-specified, and the pre-registration must be timestamped and immutable. Exploratory analysis is permitted and encouraged but cannot serve as the primary evidence for transitioning a hypothesis from Proposed to Supported.

7.3 Replication Standard

A result is considered replicated if a second experiment by a different researcher, or the same researcher on a different dataset split, produces the same conclusion using the same protocol but different data, with the effect in the same direction and statistically significant. Single-experiment results are noted as preliminary until replicated.

7.4 Reporting Standards

All experimental papers must report effect sizes, confidence intervals for all estimated quantities, sample sizes and exclusion criteria, pre-registration reference if applicable, specification version, and computational resources used.

7.5 Negative Results

Negative results are considered scientific contributions equal in value to positive results. They must be reported with the same rigor, published or archived, and used to refine or reject hypotheses in the specification.

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8. Research Debt Register

The following questions are unresolved and tracked as research debt.

ID	Question	Status	Owner	Target
RD-001	Does the residual stream satisfy the Markov property?	Open	Paper 1	Q1 2027
RD-002	What is the minimal sufficient state representation?	Open	Paper 2	Q2 2027
RD-003	Which observations are informationally sufficient for state estimation?	Open	Paper 2	Q2 2027
RD-004	Is the system locally controllable through residual additions?	Open	Paper 3	Q3 2027
RD-005	Can observability be proven theoretically?	Open	Paper 2	Q3 2027
RD-006	What is the geometry of reference trajectories across tasks?	Open	Paper 1	Q1 2027
RD-007	Does the Lipschitz constant admit a tight computable bound?	Open	Paper 1	Q1 2027
RD-008	How does the framework extend to mixture-of-experts architectures?	Open	Paper 4+	TBD
RD-009	How does the framework extend to state-space model architectures?	Open	Paper 4+	TBD
RD-010	Can the framework inform training objectives?	Open	Paper 5+	TBD
RD-011	How do different divergence measures compare?	Open	Paper 1	Q1 2027
RD-012	Is the reference subspace consistent across model scales?	Open	Paper 3	Q3 2027
RD-013	Can controllers transfer across model families?	Open	Paper 4	Q4 2027
RD-014	What is the optimal control horizon for MPC?	Open	Paper 4	Q4 2027

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9. Notation

All papers in the research program must use the following notation. Deviations must be explicitly justified.

Symbol	Meaning	Section
$x_t$	Computational state at step $t$	4.1
$\mathcal{X}$	State space	4.1
$d$	Ambient dimension of $\mathcal{X}$ (for $\mathbb{R}^d$ representations)	4.1
$L$	Number of computational steps	4.2
$\tau$	State trajectory $(x_0, \ldots, x_L)$	4.1
$f_t$	Nominal dynamics at step $t$	4.2
$w_t$	Unmodeled disturbance at step $t$	4.2
$K_t$	Lipschitz constant of $f_t$	4.2
$y_t$	Observation at step $t$	4.3
$g$	Observation function	4.3
$\mathcal{Y}$	Observation space	4.3
$v_t$	Observation noise at step $t$	4.3
$u_t$	Control input at step $t$	4.4
$\mathcal{U}$	Control space	4.4
$B_t$	Intervention channel operator at step $t$	4.4
$\pi$	Control policy	4.5
$\hat{x}_t$	State estimate at step $t$	4.5
$x_t^{\text{ref}}$	Reference state at step $t$	4.7
$\tau^{\text{ref}}$	Reference trajectory	4.7
$D_s$	Semantic divergence	4.8
$J$	Cost functional	4.6
$\ell$	Stage cost	4.6
$\Phi$	Terminal cost	4.6
$\mu$	Control cost weight	4.6
$A_t$	Jacobian of $f_t$ at reference: $I + J_{f_t}(x_t^{\text{ref}})$	H-009
$W_c$	Controllability Gramian	H-009
$N_h$	Prediction horizon	4.5
$\mathcal{M}$	Reference subspace	H-007
$\Pi_{\mathcal{M}}$	Projection onto reference subspace	H-007
$\gamma$	Contraction rate	H-012
$\delta$	Disturbance bound	H-013

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10. Governance

10.1 Ownership

SFS-0001 is owned and maintained by Sirraya Labs. It is an internal scientific document that may be released publicly at the discretion of the research director.

10.2 Amendment Process

Any researcher may propose an amendment to SFS-0001. Amendments must specify the sections to be modified, justify the modification with reference to new evidence, new hypotheses, corrections, or clarifications, be reviewed by at least one other researcher, and be versioned according to the scheme below.

10.3 Versioning

• Major versions (v1.0, v2.0): Changes to axioms, principles, or core definitions. Requires director approval.
• Minor versions (v1.1, v1.2): New hypotheses, revised validation criteria, notation additions, research debt updates. Requires peer review.
• Patch versions (v1.0.1): Typographical corrections, clarifications, formatting. No review required.

10.4 Paper Compliance

A paper is compliant with this specification if it references the SFS version used, states which scientific questions it addresses, specifies which hypotheses it tests or relies upon, uses the notation defined in Section 9, reports results against the validation criteria in Section 6, and classifies each claim according to the knowledge taxonomy in Section 3.

10.5 Living Document

SFS-0001 is a living document. It is expected to evolve as experiments validate or reject hypotheses, as new architectures emerge, and as theoretical understanding deepens. The version history will serve as a record of the research program's intellectual development.

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References

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• Anderson, B.D.O. and Moore, J.B. Optimal Filtering. Prentice-Hall, 1979.
• Chen, R.T.Q., Rubanova, Y., Bettencourt, J., and Duvenaud, D. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 2018.
• Elhage, N. et al. A mathematical framework for transformer circuits. Transformer Circuits Thread, 2021.
• Kalman, R.E. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 1960.
• Li, K. et al. Inference-time intervention: Eliciting truthful answers from a language model. Advances in Neural Information Processing Systems, 2023.
• Marks, S. and Tegmark, M. The geometry of truth: Emergent linear structure in large language model representations. International Conference on Machine Learning, 2024.
• Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O.M. Constrained model predictive control: Stability and optimality. Automatica, 2000.
• Mir, A.H. The geometry of truth: Layerwise semantic dynamics for LLMs. Sirraya Labs Technical Report, 2024.
• Park, K., Choe, Y.J., and Veitch, V. The linear representation hypothesis and the geometry of large language models. arXiv:2311.03658, 2024.
• Turner, A.M. et al. Steering language models with activation engineering. arXiv:2308.10248, 2023.

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End of SFS-0001 Draft 1