Non-Collapsible State Separation — v2.0 Draft
Intermediate draft. Formal SCM framing added, d-separation proof structure introduced, Assumptions 6.1–6.3 and Corollary 4.1 added.
Non-Collapsible State Separation
as a Structural Solution to Signal Smearing in Adaptive State Estimation Systems
Gregory Stuart Lacefield · Independent Researcher · Lacefield Research · Las Vegas, NV
gregorylacefield.com · Preliminary Draft · May 2026 · Provisional Patent Filed June 2026
Abstract
We model adaptive telemetry and state estimation systems as structural causal models (SCMs) over four categorically distinct signal classes: environmental context (S), genuine internal state (D), interface friction (I), and execution output (C). Standard architectures collapse these classes onto a single measurement channel, producing a smeared composite from which the independent signals cannot be recovered by statistical means. We prove, via the d-separation criterion on a strictly lower-triangular causal graph, that enforcing non-collapsible structural separation between these four signal classes eliminates reverse-path causal influence from execution state to internal state by architectural constraint. A second result establishes schema floor integrity under a prerequisite-gated update rule on the internal state DAG. Both results are domain-independent: the proof depends only on graph topology, not on the semantic content of the signal classes. The architecture has been instantiated in adaptive education, industrial operations, athletic training, and autonomous vehicle monitoring systems.
1. The Signal Smearing Problem
Adaptive systems that monitor a human operator or learner and adjust their behavior in response must solve a diagnostic problem: given observed output Y, what is the current genuine internal state of the human? This question cannot be answered by any system whose measurement architecture collapses the genuine internal state with environmental noise, interface friction, and physical execution variance onto a single channel. We formalize this failure and prove it is structural.
Definition 1.1 (Signal Classes).
Let the state of a human-system interaction be characterized by four categorically distinct random variables: S — Environmental state. Ambient conditions and system-side noise acting on the human. (Distraction load, session duration, connection latency, ambient conditions.) D — Internal state. The human’s genuine internal understanding or capacity. This is the target of inference. (Concept mastery, skill floor, procedural schema, route familiarity.) I — Interface state. Presentation and delivery variables imposed by the system. (Visual complexity, layout geometry, cognitive load of the interface, scaffold density.) C — Execution state. Raw physical and neuro-motor performance signals. (Response latency, input accuracy, muscle slip rate, reaction time variance.) The four variables S, D, I, C are generated by categorically distinct physical and cognitive mechanisms and are therefore causally independent at source.
Definition 1.2 (Smeared Architecture).
A standard adaptive system observes a composite measurement M defined as: M = f(S, D, I, C) where f is a function that combines all four signal classes onto a single output channel. In this architecture, the partial derivatives ∂M/∂D and ∂M/∂C are not identically zero and are coupled through the shared channel. We call this a smeared architecture.
Proposition 1.3 (Non-Recoverability of Smeared Signals).
Let M be generated by a smeared architecture. Then for any measurable function g, the estimator D̂ = g(M) satisfies E[‖D̂ − D‖²] > 0 if Var(C) > 0 and the channel function f is not invertible in D given C.
Proof.
Since M = f(S, D, I, C) and C contributes non-trivially to M (Var(C) > 0), any estimator g(M) conflates variation in D with variation in C. Because f is not assumed invertible in D given C, the information required to separate these contributions is not present in M. The mean squared error of any estimator D̂ = g(M) is therefore bounded away from zero. ■
Remark 1.4. Proposition 1.3 establishes that the smearing problem is not a statistical estimation problem amenable to better algorithms. It is a structural information problem: the data required to separate D from C is not present in M because it was never recorded separately. This motivates the architectural intervention of Section 2 rather than a refined estimator. The distinction between statistical mitigation and structural separation is the central claim of this paper.
2. The Non-Collapsible Four-Channel Architecture
We model the system as a structural causal model (V, G, F, Pᵤ) in the sense of Pearl (2000, Chapter 1), where V is the set of endogenous variables, G is the causal DAG over V, F is a set of structural equations, and Pᵤ is a distribution over exogenous noise variables. The distribution P over V is Markov with respect to G by construction: each variable is independent of its non-descendants given its parents in G. This is the Causal Markov Condition and it holds for every distribution generated by the structural equations of the model. No additional faithfulness assumption is required for the do-calculus results that follow; the interventional distribution P(D | do(C = c)) is identified by graph surgery on G without faithfulness.
Definition 2.1 (NCFCA Profiles).
The Non-Collapsible Four-Channel Architecture (NCFCA) maintains four independent profile structures: Sₛₒₚ — State Context Profile. Captures environmental state S. Resolved prior to any inference over D. Dᴅᴸᴾ — Dynamic State Profile. Captures internal state D. A directed acyclic graph Gᴅ = (V, Eᴅ) of concept or skill nodes with mastery states. Updated only from confirmed multi-session evidence. Write access subject to circuit-breaker suspension (Definition 2.4). Iᴵᴼᴾ — Interface Response Profile. Captures interface state I. Updated from Sₛₒₚ signals only. Cᴄᴇᴾ — Calibration Execution Profile. Captures execution state C. Acts as system circuit breaker.
Definition 2.2 (Non-Collapsibility Rule).
The profiles Sₛₒₚ, Dᴅᴸᴾ, Iᴵᴼᴾ, Cᴄᴇᴾ are never averaged, merged, or collapsed into a shared representation. No data path exists that allows a value written to Cᴄᴇᴾ to directly modify a value in Dᴅᴸᴾ. All inter-profile information flow is unidirectional and follows the pipeline order defined in Definition 2.3.
Definition 2.3 (Lower-Triangular Causal Pipeline).
Define the causal graph G = (Vᴳ, Eᴳ) over nodes Vᴳ = {Sₛₒₚ, Dᴅᴸᴾ, Iᴵᴼᴾ, Cᴄᴇᴾ} with directed edges: Eᴳ ⊆ { (Sₛₒₚ → Dᴅᴸᴾ), (Sₛₒₚ → Iᴵᴼᴾ), (Dᴅᴸᴾ → Cᴄᴇᴾ), (Iᴵᴼᴾ → Cᴄᴇᴾ) } The following directed edges are absent from G by the Non-Collapsibility Rule: Cᴄᴇᴾ ↛ Dᴅᴸᴾ, Cᴄᴇᴾ ↛ Sₛₒₚ, Cᴄᴇᴾ ↛ Iᴵᴼᴾ, Iᴵᴼᴾ ↛ Dᴅᴸᴾ, Dᴅᴸᴾ ↛ Sₛₒₚ The adjacency matrix A of G is strictly lower-triangular under the ordering (Sₛₒₚ, Dᴅᴸᴾ, Iᴵᴼᴾ, Cᴄᴇᴾ): A = ⎡ 0 0 0 0 ⎤ ⎢ a 0 0 0 ⎥ ⎢ b 0 0 0 ⎥ ⎣ 0 c d 0 ⎦ where a, b, c, d ∈ {0,1} are the permitted directed edge indicators, and all upper-triangular entries are structurally zero.
Definition 2.4 (Circuit Breaker Protocol).
Let δ(t) be an anomaly indicator for the execution profile at time t, with δ(t) = 1 if Cᴄᴇᴾ signals exceed a clinically or operationally validated threshold θᴄ, and δ(t) = 0 otherwise. The write-suspension protocol is: Φ(t) = 1 − δ(t) where Φ(t) = 1 permits writes to Dᴅᴸᴾ and Φ(t) = 0 suspends all writes to Dᴅᴸᴾ. The state of Dᴅᴸᴾ during suspension is frozen at its last confirmed value.
3. Main Results
3.1 D-Separation and Causal Non-Influence
We apply Pearl’s d-separation criterion to establish that Cᴄᴇᴾ cannot causally influence Dᴅᴸᴾ under the NCFCA causal graph G.
Definition 3.1 (D-Separation, Pearl 1988).
Let G be a DAG over variables V. A set of variables Z blocks a path p between variables X and Y if: (i) p contains a chain X → M → Y or a fork X ← M → Y such that M ∈ Z, or (ii) p contains a collider X → M ← Y such that M ∉ Z and no descendant of M is in Z. X and Y are d-separated by Z in G, written X ⊥ᴳ Y | Z, if Z blocks every path between X and Y in G.
Lemma 3.2 (Pearl’s Completeness Theorem, Pearl 2000).
Let (V, G, P) be a structural causal model where G is the causal DAG and P is a distribution Markov with respect to G. Then: X ⊥ᴳ Y | Z ⇒ X ⊥ Y | Z (under P) That is, d-separation in G implies conditional independence in any distribution consistent with G.
Theorem 1 (Zero Reverse-Path Causal Influence).
Under the NCFCA causal graph G defined in Definition 2.3, the execution profile Cᴄᴇᴾ and the internal state profile Dᴅᴸᴾ are d-separated by the empty set: Cᴄᴇᴾ ⊥ᴳ Dᴅᴸᴾ | ∅ with respect to reverse-directed paths in G. Consequently, in any distribution P consistent with G: P(Dᴅᴸᴾ | do(Cᴄᴇᴾ = c)) = P(Dᴅᴸᴾ) for all c. Interventions on Cᴄᴇᴾ have no effect on Dᴅᴸᴾ.
Proof.
We enumerate all paths between Cᴄᴇᴾ and Dᴅᴸᴾ in G and show each is blocked.
By Definition 2.3, the directed edges of G are a subset of { Sₛₒₚ → Dᴅᴸᴾ, Sₛₒₚ → Iᴵᴼᴾ, Dᴅᴸᴾ → Cᴄᴇᴾ, Iᴵᴼᴾ → Cᴄᴇᴾ }. The absent edges Cᴄᴇᴾ ↛ Dᴅᴸᴾ and Iᴵᴼᴾ ↛ Dᴅᴸᴾ are structural zeros by the Non-Collapsibility Rule (Definition 2.2).
The candidate undirected paths between Cᴄᴇᴾ and Dᴅᴸᴾ in G are:
Path π1: Cᴄᴇᴾ ← Dᴅᴸᴾ. This path corresponds to the directed edge Dᴅᴸᴾ → Cᴄᴇᴾ, which exists in G. However, as an undirected path from Cᴄᴇᴾ to Dᴅᴸᴾ it runs against the direction of the edge. In the causal do-calculus, P(Dᴅᴸᴾ | do(Cᴄᴇᴾ = c)) severs all incoming edges to Cᴄᴇᴾ in the mutilated graph G̅. In G̅, the edge Dᴅᴸᴾ → Cᴄᴇᴾ is severed. Path π1 does not exist in G̅.
Path π2: Cᴄᴇᴾ ← Iᴵᴼᴾ ← Sₛₒₚ → Dᴅᴸᴾ. In G̅, the edge Iᴵᴼᴾ → Cᴄᴇᴾ is severed (incoming to Cᴄᴇᴾ). Path π2 does not exist in G̅.
No further undirected paths between Cᴄᴇᴾ and Dᴅᴸᴾ exist in G, because the only edges involving Cᴄᴇᴾ are Dᴅᴸᴾ → Cᴄᴇᴾ and Iᴵᴼᴾ → Cᴄᴇᴾ, both of which are severed in G̅.
Since all paths between Cᴄᴇᴾ and Dᴅᴸᴾ are absent in the mutilated graph G̅, we have:
P(Dᴅᴸᴾ | do(Cᴄᴇᴾ = c)) = P(Dᴅᴸᴾ) for all c.
By Lemma 3.2, this implies Cᴄᴇᴾ ⊥ᴳ Dᴅᴸᴾ with respect to all reverse-directed influence paths in G. Interventions on Cᴄᴇᴾ have zero causal effect on Dᴅᴸᴾ.
■
Theorem 1 establishes the central result: under the NCFCA graph topology, execution-layer anomalies cannot cause changes in the internal state estimate. The guarantee is structural rather than probabilistic — it holds for any distribution consistent with the causal graph, not only in expectation or with high probability.
Remark 3.2. The result is non-trivial in the following sense. A system could enforce write-path separation at the application layer while still admitting reverse causal influence through confounding variables in the observational distribution — which is the regime in which statistical control methods operate and which Proposition 1.3 shows to be insufficient. Theorem 1 holds in the interventional distribution P(D | do(C = c)), which severs all incoming edges to C in the mutilated graph G̃ and therefore eliminates confounding paths. The architectural constraint is what produces this result, not a distributional assumption. A purely statistical separation — one that conditions on C rather than intervening on it — would not achieve the same guarantee.
Remark 3.3. The permitted downstream path Dᴅᴸᴾ → Cᴄᴇᴾ is intentional and does not constitute smearing. Smearing is defined as reverse-path causal influence from C to D. The fact that D causally influences C — that schema state informs execution thresholds — is a feature of the architecture, not a violation of the non-collapsibility rule. Theorem 1 is a statement about the reverse direction only. The total covariance between Dᴅᴸᴾ and Cᴄᴇᴾ in the observational distribution is nonzero through this permitted path; the paper does not claim otherwise.
Remark 3.4. Assumption 6.1 in Section 6 acknowledges that S may causally influence D through physiological and cognitive mechanisms — for example, chronic environmental load affecting genuine internal capacity. This does not violate the non-collapsibility result. The S → D pathway is captured by the permitted edge Sₛₒₚ → Dᴅᴸᴾ in the causal graph and flows through the designated channel. The lower-triangular constraint does not deny the existence of this causal relationship; it enforces that it is recorded and tracked separately from the C → D pathway, which the graph forbids. See Section 6 for further discussion.
We contrast this with prior approaches. Bayesian Knowledge Tracing (Corbett & Anderson, 1994), Deep Knowledge Tracing (Piech et al., 2015), and Item Response Theory (Rasch, 1960) treat the signal separation problem as a statistical inference problem: they estimate D from a smeared composite M using probabilistic methods. By Proposition 1.3, no such estimator achieves zero mean squared error when Var(C) > 0. The NCFCA resolves this by eliminating the smearing at the architectural level rather than correcting for it downstream.
3.2 Schema Floor Integrity
The second main result addresses the internal update logic of Dᴅᴸᴾ. We prove that under the prerequisite-gated update rule and the circuit-breaker protocol, no execution-layer event can corrupt the confirmed internal state floor.
Definition 3.3 (Internal State DAG).
The Dynamic State Profile Dᴅᴸᴾ is a directed acyclic graph Gᴅ = (Vᴅ, Eᴅ) where: (i) Each node vᵢ ∈ Vᴅ represents a discrete concept or skill unit with binary mastery state m(vᵢ) ∈ {0, 1}. (ii) A directed edge (vᵣ, vᵢ) ∈ Eᴅ denotes that vᵣ is a prerequisite of vᵢ. (iii) Define Prereq(vᵢ) = { vᵣ : (vᵣ, vᵢ) ∈ Eᴅ } as the immediate prerequisite set of vᵢ. (iv) The schema floor F ⊆ Vᴅ is the maximal antichain of confirmed mastered nodes: F = { vᵢ ∈ Vᴅ : m(vᵢ) = 1 and ∀ vᵣ ∈ Prereq(vᵢ), m(vᵣ) = 1 }.
Definition 3.4 (Prerequisite-Gated Update Rule).
A mastery state update Δm(vᵢ) = 1 is permitted if and only if: (i) All prerequisite nodes are confirmed mastered: ∀ vᵣ ∈ Prereq(vᵢ), m(vᵣ) = 1. (ii) The write-permission flag is active: Φ(t) = 1. Formally: Δm(vᵢ) = 1 ⇔ [∀ vᵣ ∈ Prereq(vᵢ), m(vᵣ) = 1] ∧ [Φ(t) = 1]
Theorem 2 (Schema Floor Integrity).
Let Gᴅ be the internal state DAG with prerequisite-gated update rule (Definition 3.4) and circuit-breaker protocol (Definition 2.4). Then for any execution anomaly event at time t₀ such that δ(t₀) = 1: ∀ vᵢ ∈ Vᴅ, m(vᵢ)(t) = m(vᵢ)(t₀⁻) for all t ∈ [t₀, t₁) where t₁ is the time at which δ returns to 0. That is, the mastery state of every node in Gᴅ is frozen at its pre-anomaly value for the duration of the anomaly window.
Proof.
Let t₀ be the onset of an execution anomaly, so δ(t₀) = 1. By Definition 2.4, Φ(t) = 1 − δ(t) = 0 for all t ∈ [t₀, t₁).
By Definition 3.4, the update condition Δm(vᵢ) = 1 requires Φ(t) = 1 as a necessary conjunct. Since Φ(t) = 0 for all t ∈ [t₀, t₁), the necessary condition fails for all vᵢ ∈ Vᴅ and all t in the anomaly window.
Therefore Δm(vᵢ) = 0 for all vᵢ ∈ Vᴅ and all t ∈ [t₀, t₁), which gives m(vᵢ)(t) = m(vᵢ)(t₀⁻) for all vᵢ and all t in the window.
Since F is defined as a function of mastery states and no mastery state changes during [t₀, t₁), the schema floor F is constant on this interval.
■
Theorems 1 and 2 together establish the two guarantees of the NCFCA: (1) execution-layer anomalies cannot propagate upstream to corrupt the internal state estimate via the causal pipeline, and (2) execution-layer anomalies cannot corrupt the internal state estimate via the update logic of Dᴅᴸᴾ itself. The internal state is protected by both the graph topology and the write-suspension protocol.
Remark 3.6. Theorem 2 is stated for binary mastery states m(vᵢ) ∈ {0, 1}, which is the standard assumption in knowledge tracing literature (Corbett & Anderson, 1994). The result extends immediately to threshold-based continuous mastery: replace m(vᵢ) = 1 with m(vᵢ) ≥ τ for a domain-specified threshold τ ∈ (0, 1], and the proof proceeds identically. Extension to fully continuous mastery functions m : Vᴅ → [0, 1] without a threshold assumption requires additional structure on the update rule and is listed as an open problem in Section 7.
4. Domain Independence
The proofs of Theorem 1 and Theorem 2 depend only on the topological structure of the causal graph G and on the formal properties of the update rule. They do not depend on the semantic interpretation of the four signal classes S, D, I, C. This establishes domain independence as a corollary of the proof method.
Corollary 4.1 (Domain Independence).
For any domain instantiation of the four signal classes S, D, I, C satisfying Definition 1.1 and any causal graph G consistent with Definition 2.3, Theorems 1 and 2 hold.
Proof.
The proof of Theorem 1 uses only the edge set Eᴳ and the do-calculus mutilation of G. Neither depends on the domain interpretation of the nodes. The proof of Theorem 2 uses only the definition of Φ(t) and the logical structure of the update condition. Neither depends on what Dᴅᴸᴾ represents semantically. Since both proofs are valid for any G satisfying Definition 2.3 and any update rule satisfying Definition 3.4, the results hold across all domain instantiations.
■
The NCFCA has been instantiated in four domains. In each case the four profiles correspond to the signal classes of Definition 1.1 under domain-appropriate semantic interpretation, and the same causal graph topology of Definition 2.3 is enforced. The structural guarantees of Theorems 1 and 2 apply identically across all four instantiations by Corollary 4.1.
5. Relationship to Prior Work
The signal separation problem in adaptive systems has been addressed through statistical inference approaches in the literature. We distinguish the structural guarantee of the NCFCA from the probabilistic guarantees of prior approaches.
Bayesian Knowledge Tracing (BKT; Corbett & Anderson, 1994) models skill state as a latent variable and estimates it from observed performance using a Hidden Markov Model. The guarantee is probabilistic: the estimator converges to the true state given sufficient data. By Proposition 1.3, convergence is bounded away from zero when Var(C) > 0 and the channel function is non-invertible. BKT does not address the structural coupling.
Performance Factors Analysis (PFA; Pavlik et al., 2009) and Deep Knowledge Tracing (DKT; Piech et al., 2015) extend BKT with richer feature representations and neural network architectures respectively. Both inherit the smeared architecture and the probabilistic-only guarantee.
Item Response Theory (IRT; Rasch, 1960; Lord & Novick, 1968) models item difficulty and ability as separate latent dimensions but does not address execution noise, environmental load, or interface friction as structurally distinct signal classes. The smearing between C and D is not separated.
Contextual bandit and reinforcement learning approaches (Lattimore & Szepesvári, 2020) model context as a feature modifying the reward signal. This mixes S with D in the feature representation. The structural coupling is reduced by feature engineering but not eliminated by architectural constraint.
The distinguishing property of the NCFCA is the nature of its guarantee. The prior approaches produce estimators whose accuracy depends on data quantity, model specification, and distributional assumptions. The NCFCA produces a causal non-influence result: by Theorem 1, no amount of data or model sophistication is required to maintain zero reverse-path influence from C to D, because the influence is absent by graph topology.
6. Assumptions and Limitations
The results of Section 3 depend on three assumptions, each of which imposes an empirical or implementation requirement.
Assumption 6.1 (Source Independence). The four signal classes S, D, I, C are generated by categorically distinct causal mechanisms. Within the NCFCA graph, causal influence from S to D is permitted and is captured by the designated edge Sₛₒₚ → Dᴅᴸᴾ. This assumption does not deny that environmental conditions affect genuine internal state — chronic stress, sleep deprivation, and persistent environmental load are known to modulate cognitive capacity and are handled through the permitted S → D pathway in the causal graph. What the assumption requires is that this influence does not share a causal origin with execution signals C outside the modeled pathways. Violations — shared latent confounders between D and C not captured by the graph — would require extending Eᴳ with additional edges and re-evaluating the d-separation result under the augmented topology. Empirical validation in each target domain is required.
Assumption 6.2 (Schema-Level Enforcement). The lower-triangular pipeline constraint of Definition 2.3 must be enforced at the database schema level, not only at the application level. Application-level enforcement is bypassable by implementation errors or future code changes. Schema-level enforcement via foreign key constraints, write-path validation, and audit logging at the data persistence layer produces a structural guarantee analogous to a physical circuit break: the absent edges in G are absent not by policy but by the absence of a data pathway capable of carrying them. The distinction between policy-level and schema-level enforcement is the same distinction the proof draws between statistical mitigation and architectural separation.
Assumption 6.3 (Circuit Breaker Calibration). The anomaly threshold θᴄ of Definition 2.4 must be empirically calibrated for each domain and population. The Theorem 2 guarantee is conditional on δ(t) correctly classifying anomaly events. This conditionality is not a weakness unique to this result: every detection-based guarantee in the engineering literature is conditional on threshold calibration in the same sense. The existence of a free parameter θᴄ does not make the guarantee vacuous; it makes it domain-specific. False negatives reduce guarantee completeness; false positives reduce responsiveness without affecting correctness. Clinical or operational validation is required prior to deployment. Formal specification sufficient for ASIL-D functional safety certification in the automotive domain is listed as an open problem in Section 7.
7. Open Problems
Empirical validation of Assumption 6.1 across the four target domains via controlled experiments that measure pairwise covariance between signal class proxies under natural operating conditions.
Extension of the prerequisite-gated update rule (Definition 3.4) to continuous-valued mastery representations, replacing the binary mastery state m(vᵢ) ∈ {0, 1} with a real-valued competence function m : Vᴅ → [0, 1].
Formal specification of the circuit breaker anomaly detection protocol sufficient for ASIL-D functional safety certification in the automotive domain.
Comparative empirical study of NCFCA against BKT, DKT, and IRT on benchmark datasets, measuring diagnostic accuracy and schema floor stability under induced execution noise.
Extension of the d-separation result to time-varying causal graphs, addressing the case where the edge set Eᴳ changes dynamically as the system adapts.
8. Conclusion
We have presented two structural results for adaptive state estimation systems. Theorem 1 establishes, via d-separation in the NCFCA causal graph, that the execution profile Cᴄᴇᴾ has zero causal influence on the internal state profile Dᴅᴸᴾ under the lower-triangular pipeline constraint. Theorem 2 establishes that the prerequisite-gated update rule and circuit-breaker protocol jointly guarantee schema floor integrity during execution anomaly windows. Both results are domain-independent by Corollary 4.1.
The key distinction from prior work is the nature of the guarantee. Statistical inference approaches reduce but cannot eliminate the estimation error introduced by signal smearing, because the separation problem is structural rather than distributional. The NCFCA eliminates the causal pathway that produces smearing, producing a guarantee that holds for any distribution consistent with the causal graph rather than only in expectation or asymptotically.
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Gregory Stuart Lacefield · Lacefield Research · Las Vegas, NV · May 2026 · gregorylacefield.com