Non-Collapsible State Separation — v3.2

June 2026. Adds Section 5a (Evidentiary Classification of Results), NoisyCausal and CORR2CAUSE/Jin et al. (2023) to prior art, causal parroting context. Superseded by later versions — real intermediate step, not the current statement.

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 state estimation systems as structural causal models 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 into a single composite measurement, making it impossible to recover D from observed output by statistical means alone. We prove that enforcing a strictly lower-triangular causal pipeline with non-collapsible profiles eliminates reverse-path causal influence from execution state to internal state. We further prove that a prerequisite-gated update rule combined with a circuit-breaker protocol preserves the integrity of confirmed internal state during execution anomalies. Both results are domain-independent and follow from graph topology rather than distributional assumptions. The architecture has been instantiated in education, industrial operations, athletic training, and autonomous vehicle monitoring.

Contributions

This paper makes the following contributions:

• Formalizes the signal smearing problem as a structural (rather than statistical) failure in adaptive state estimation systems.

• Proves that a strictly lower-triangular causal pipeline with non-collapsible profiles eliminates reverse-path causal influence from execution state to internal state (Theorem 1).

• Proves that a prerequisite-gated update rule combined with a circuit-breaker protocol preserves schema floor integrity during execution anomalies (Theorem 2).

• Establishes that both results are domain-independent and depend only on graph topology.

• Provides a clear architectural distinction between structural guarantees and post-hoc statistical correction methods used in prior work.

• Specifies the enforcement requirements needed to realize these guarantees in practice.

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. D — Internal state. The human’s genuine internal understanding or capacity. This is the target of inference. I — Interface state. Presentation and delivery variables imposed by the system. C — Execution state. Raw physical and neuro-motor performance signals. 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 = 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.

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.

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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.

Empirical confirmation (Test 5, Lacefield 2026b). Under identical coupling conditions (ε = 0.8, sustained), the smeared architecture produces corr(D_smeared, S) = 0.09–0.14 across three coupling schedules, while the NCFCA architecture produces corr(D_protected, S) ≈ −0.0007 — a 200-fold reduction in S contamination of D. The smeared architecture cannot eliminate this contamination by any downstream statistical method, consistent with Proposition 1.3. (NCFCA Empirical Validation Suite, Test 5, Lacefield 2026b.)

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). The distribution P over V is Markov with respect to G by construction. No faithfulness assumption is required; the interventional results that follow are 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 of concept or skill nodes with mastery states. Write access subject to circuit-breaker suspension. 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 four profiles Sₛₒₚ, Dᴅᴸᴾ, Iᴵᴼᴾ, Cᴄᴇᴾ must be maintained as structurally independent data structures. There shall exist no write path in the underlying data model that permits a value originating in Cᴄᴇᴾ (or Iᴵᴼᴾ) to directly modify any value stored in Dᴅᴸᴾ. The profiles may not be averaged, merged, or projected into a shared representation at any stage of processing or storage. All information flow between profiles must be unidirectional and must respect the lower-triangular causal order defined in Definition 2.3. Nomenclature equivalence: Throughout this proof, the four profiles Sₛₒₚ, Dᴅᴸᴾ, Iᴵᴼᴾ, Cᴄᴇᴾ are referred to by their signal class shorthand S, D, I, C respectively when the context is the causal signal rather than the data structure. The two notations are interchangeable: S ≡ Sₛₒₚ, D ≡ Dᴅᴸᴾ, I ≡ Iᴵᴼᴾ, C ≡ Cᴄᴇᴾ. Patent documents, technical specifications, and empirical reports in the NCFCA series use the same four designations. The signal class notation (S, D, I, C) is used in mathematical expressions and proofs; the profile notation (Sₛₒₚ, Dᴅᴸᴾ, Iᴵᴼᴾ, Cᴄᴇᴾ) is used when referring to the data structure implementation.

Remark 2.1 (Enforcement Mechanism). Enforcing the Non-Collapsibility Rule at the data-model level (rather than solely in application code) produces a structural guarantee analogous to a physical circuit breaker: the forbidden reverse-path influence cannot occur even if application logic contains errors or is later modified.

Definition 2.3 (Lower-Triangular Causal Pipeline).

Define the causal graph G over nodes {Sₛₒₚ, Dᴅᴸᴾ, Iᴵᴼᴾ, Cᴄᴇᴾ} with directed edges contained in: Eᴳ ⊆ { (Sₛₒₚ → Dᴅᴸᴾ), (Sₛₒₚ → Iᴵᴼᴾ), (Dᴅᴸᴾ → Cᴄᴇᴾ), (Iᴵᴼᴾ → Cᴄᴇᴾ) } The adjacency matrix of G is strictly lower-triangular under the ordering Sₛₒₚ ≺ Dᴅᴸᴾ ≺ Iᴵᴼᴾ ≺ Cᴄᴇᴾ.

Definition 2.4 (Circuit Breaker Protocol).

Let δ(t) be an anomaly indicator for the execution profile at time t. The write-suspension protocol is given by Φ(t) = 1 − δ(t), where Φ(t) = 1 permits writes to Dᴅᴸᴾ and Φ(t) = 0 suspends all writes to Dᴅᴸᴾ.

Definition 2.4a (Circuit Breaker Finite State Machine). The circuit breaker protocol of Definition 2.4 is equivalently specified as a three-state finite state machine (FSM) over the state space Σ = {MONITORING, SUSPENDED, RESUMED}. The FSM is defined by the following states and transitions: MONITORING: Initial state. Φ(t) = 1. Writes to Dᴅᴸᴾ are permitted. Transition to SUSPENDED occurs when δ(t) = 1 (anomaly detected at threshold θᴄ). SUSPENDED: Active anomaly window state. Φ(t) = 0. All writes to Dᴅᴸᴾ are blocked. The schema floor is frozen at the last confirmed value m(v_i)(t₀⁻). The system continues generating Cᴄᴇᴾ observations and monitoring δ(t). Transition to RESUMED occurs when δ(t) returns to 0 (anomaly window closes at t₁). RESUMED: Post-anomaly recovery state. Φ(t) = 1. Writes to Dᴅᴸᴾ are restored. The schema floor value is the value frozen at t₀ — no degradation is introduced by the anomaly window. Transition back to MONITORING occurs immediately on the first observation; RESUMED and MONITORING are operationally identical and the distinction is maintained for audit purposes only. Note on real-world implementation: In simulation under idealized conditions (ε = 0 at anomaly closure), the transition from SUSPENDED to RESUMED produces zero schema floor degradation by construction, because the freeze operation preserves the exact pre-anomaly state. In deployed systems operating under stochastic noise, the effective recovery behavior is a function of the sampling rate, the signal-to-noise ratio, and the precision of δ(t). The FSM guarantees the structural property — no writes during anomaly windows — but the real-world recovery latency depends on implementation parameters outside the scope of this proof. See Open Problem 3 (threshold calibration) and the empirical validation suite.

Definition 2.5 (Anomaly Indicator and Anomaly Window).

Let δ: ᵔ → {0, 1} be a measurable function such that δ(t) = 1 if and only if Cᴄᴇᴾ(t) exceeds a domain-specific threshold θᴄ. An anomaly window is any maximal half-open interval [t₀, t₁) during which δ(t) = 1 for all t ∈ [t₀, t₁). The value t₀ marks the onset of the anomaly, and t₁ marks the first time at which δ(t) returns to 0.

3. Main Results

The NCFCA delivers two complementary guarantees. The first establishes that the causal structure of the architecture prevents execution signals from influencing internal state estimates. The second establishes that, even during periods of unreliable execution, already-confirmed internal state cannot be overwritten.

3.1 Causal Non-Influence via Lower-Triangular Structure

We first show that the lower-triangular causal pipeline eliminates reverse-path causal influence from execution state to internal state.

Observation 3.1 (Topological Structure).

Under the ordering Sₛₒₚ ≺ Dᴅᴸᴾ ≺ Iᴵᴼᴾ ≺ Cᴄᴇᴾ, the graph G is strictly lower-triangular. Consequently, there are no directed paths from Cᴄᴇᴾ to any of {Sₛₒₚ, Dᴅᴸᴾ, Iᴵᴼᴾ}.

Theorem 1 (Zero Reverse-Path Causal Influence).

Under the NCFCA causal graph G, the execution profile Cᴄᴇᴾ and the internal state profile Dᴅᴸᴾ are d-separated by the empty set with respect to reverse-directed paths: Cᴄᴇᴾ ⊥ᴳ Dᴅᴸᴾ | ∅. Consequently, in any distribution P consistent with G: P(Dᴅᴸᴾ | do(Cᴄᴇᴾ = c)) = P(Dᴅᴸᴾ) for all c.

Proof.

Consider the mutilated graph obtained by applying the intervention do(Cᴄᴇᴾ = c). By definition of the do-operator, all incoming edges to Cᴄᴇᴾ are removed. By Observation 3.1, the original graph G contains no directed paths from Cᴄᴇᴾ to Dᴅᴸᴾ. Removing incoming edges to Cᴄᴇᴾ cannot create any new directed path from Cᴄᴇᴾ to Dᴅᴸᴾ. Therefore, there is no directed path from Cᴄᴇᴾ to Dᴅᴸᴾ in the mutilated graph. By the rules of do-calculus, this implies that interventions on Cᴄᴇᴾ have no effect on Dᴅᴸᴾ.

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Remark 3.2. The result is non-trivial because it holds in the interventional distribution, not merely in expectation or under statistical controls. A system could enforce write-path separation at the application layer while still admitting reverse causal influence through confounding variables in the observational distribution. Theorem 1 eliminates such influence at the structural level.

Remark 3.3. The permitted downstream path Dᴅᴸᴾ → Cᴄᴇᴾ is intentional. It allows internal state to inform execution thresholds. Smearing is defined as reverse-path influence from C to D; the forward direction does not constitute a violation. The total observational covariance between Dᴅᴸᴾ and Cᴄᴇᴾ is nonzero through this permitted downstream path; the paper does not claim otherwise.

Empirical confirmation, Theorem 1 (Test 5, Lacefield 2026b). corr(D_protected, S) ≈ −0.0007 across 100 Monte Carlo simulations under sustained coupling ε = 0.8, consistent with the zero reverse-path causal influence guarantee. Under identical conditions, corr(D_smeared, S) = 0.1404. The error types are qualitatively distinct: smeared architecture error reflects S contamination inside D; NCFCA error reflects temporal lag during frozen periods, which resolves upon window closure. (NCFCA Empirical Validation Suite, Test 5, Lacefield 2026b.)

Remark 3.4. Assumption 6.1 permits causal influence from environmental state (S) to internal state (D) and models this via the edge Sₛₒₚ → Dᴅᴸᴾ. This is distinct from the forbidden reverse paths from execution state. The lower-triangular constraint does not deny legitimate S → D influence; it ensures this influence is tracked separately from any execution-to-internal contamination.

3.2 Dynamic Protection of Confirmed Internal State

While Theorem 1 shows that the causal structure prevents execution signals from reaching internal state through information flow, it does not address updates that might occur through the internal update logic of Dᴅᴸᴾ itself. We now show that the combination of prerequisite-gated updates and the circuit-breaker protocol protects already-confirmed internal state during periods of unreliable execution.

Definition 3.3 (Internal State DAG).

The Dynamic State Profile Dᴅᴸᴾ is a directed acyclic graph G_D = (V_D, E_D) where each node v_i ∈ V_D represents a discrete concept or skill unit with a binary mastery state m(v_i) ∈ {0, 1}. A directed edge (v_r, v_i) ∈ E_D denotes that v_r is a prerequisite of v_i. The schema floor F ⊆ V_D is the maximal antichain of confirmed mastered nodes.

Definition 3.4 (Prerequisite-Gated Update Rule).

A mastery state update Δm(v_i) = 1 is permitted if and only if all prerequisite nodes are confirmed mastered and the write-permission flag is active: Δm(v_i) = 1 ⇔ (∀ v_r ∈ Prereq(v_i), m(v_r) = 1) ∧ (Φ(t) = 1).

Theorem 2 (Schema Floor Integrity).

Let G_D be the internal state DAG equipped with the prerequisite-gated update rule and the circuit-breaker protocol. Then for any anomaly window [t₀, t₁): m(v_i)(t) = m(v_i)(t₀⁻) for all v_i ∈ V_D and all t ∈ [t₀, t₁).

Proof.

By Definition 2.5, δ(t) = 1 for all t ∈ [t₀, t₁). By Definition 2.4, this implies Φ(t) = 0 throughout the interval. By Definition 3.4, a mastery update Δm(v_i) = 1 is permitted only if Φ(t) = 1. Therefore no updates are permitted for any node during [t₀, t₁), which implies that mastery states are constant on this interval. Since the schema floor is defined in terms of these mastery states, it is likewise constant throughout the anomaly window.

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Remark 3.5. The anomaly window is defined solely with respect to the execution profile. Its duration is determined by the behavior of Cᴄᴇᴾ and the chosen threshold θᴄ. Theorem 2 holds for any such window, regardless of length, provided the circuit-breaker protocol is active. This makes the guarantee robust to variation in how anomaly detection is implemented across domains, as long as δ(t) is well-defined.

How the Two Results Work Together.

Theorem 1 ensures that execution signals cannot causally influence internal state through the information-flow architecture. Theorem 2 ensures that, even if execution becomes unreliable, the internal update logic itself cannot overwrite already-confirmed state. Together, they provide layered protection: structural prevention of contamination combined with operational suspension of updates during degraded conditions. Both results are required for the full non-collapsibility guarantee.

4. Domain Independence

The proofs of Theorem 1 and Theorem 2 depend only on the topological structure of the causal graph and the formal properties of the update rule. They do not depend on the semantic interpretation of the four signal classes. 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. The proof of Theorem 2 uses only the definition of Φ(t) and the logical structure of the update condition. Neither proof depends on the domain interpretation of the nodes. Therefore both results hold for any domain instantiation satisfying the stated definitions.

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5. Relationship to Prior Work

Prior approaches to the signal separation problem, including Bayesian Knowledge Tracing (Corbett & Anderson, 1994), Deep Knowledge Tracing (Piech et al., 2015), Item Response Theory (Rasch, 1960; Lord & Novick, 1968), and contextual bandit methods (Lattimore & Szepesvári, 2020), operate on smeared composite measurements. These methods attempt to statistically recover internal state from a measurement that has already collapsed independent signal classes. By Proposition 1.3, no such estimator can achieve zero error when execution variance is non-zero. Recent extensions to BKT and DKT have incorporated learner fatigue as an additional model parameter (Wang et al., 2025), explicitly acknowledging the contamination problem; however, adding a fatigue parameter to a model trained on a smeared composite applies statistical correction after the collapse has already occurred, which Proposition 1.3 establishes as structurally insufficient.

Bayesian filtering approaches, including Kalman filters and particle filters applied to adaptive state estimation, apply a principled probabilistic framework to the same collapsed composite. These methods minimize the variance of the contaminated estimate and can track state changes under noise, but they cannot achieve zero covariance between the internal state estimate and the environmental or execution signals, because contamination is introduced at the data intake level before any filtering operation is applied. The filter operates on M = f(S, D, I, C) and produces an improved estimate of D; it cannot produce a structurally guaranteed zero-contamination estimate because the information required to separate D from S, I, and C was not recorded independently. This is the information leakage problem: once the signals are mixed at the point of measurement, no downstream processing can recover the lost separation. The NCFCA addresses the information leakage problem by structural prevention rather than statistical correction — the four signal classes are maintained as independent records before any computation occurs, so no leakage is introduced.

The empirical magnitude of execution-state contamination varies by domain, assessment design, and population. Proposition 1.3 does not claim large contamination in all cases; it establishes that no statistical estimator can provide a zero-error structural guarantee absent architectural separation. The distinction is not about typical magnitude but about the class of guarantee available. Direct empirical comparison confirms this qualitative distinction: under moderate coupling (ε = 0.8, sustained), the smeared architecture produces corr(D, S) = 0.09–0.14, compared to corr(D, S) ≈ −0.0007 for the NCFCA architecture under identical conditions — a 200-fold reduction. The error types are qualitatively distinct: smeared architecture error reflects S contamination permanently embedded in D; NCFCA error reflects temporal lag during frozen periods only, which resolves completely upon anomaly window closure. (NCFCA Empirical Validation Suite, Test 5, Lacefield 2026b.)

The causal reasoning benchmark literature independently confirms the structural problem that the NCFCA architecture addresses. On the CORR2CAUSE benchmark, which evaluates the ability to infer causation from correlational data, GPT-4 achieved an F1 score of 29.08 — near-chance performance — while the best fine-tuned baseline reached 33.38 (Jin et al., 2023). Researchers attributed this failure to what they term causal parroting: models inferring answers from textual and statistical surface patterns rather than from analysis of the underlying causal structure (Structured Thinking Matters, 2025). NoisyCausal (2026), a benchmark specifically designed to evaluate causal reasoning under structured noise — including irrelevant distractors, value perturbations, confounding, and partial observability — confirms that standard LLMs struggle to disentangle correlation from causation when noise is introduced (arXiv:2605.04313). The NCFCA does not improve a model's ability to reason about noise; it eliminates the noise source before reasoning occurs. This is a categorically different intervention than what the benchmark literature is measuring.

The NCFCA resolves the problem at the architectural level by preventing the collapse from occurring in the first place, rather than attempting to statistically undo it afterward. The result is a structural guarantee rather than a probabilistic estimate. The key distinction is not incremental: prior approaches optimize statistical models operating on contaminated data; the present architecture optimizes the structural pipeline itself to eliminate data contamination before computation occurs.

5a. Evidentiary Classification of Results

The results presented in this paper occupy four distinct evidentiary tiers. Distinguishing them is necessary for correct interpretation and for identifying which claims require additional validation.

Tier 1 — Formal proof (strongest). Theorems 1 and 2 and Proposition 1.3 are mathematical results that follow from the defined graph topology and update rule. They do not depend on empirical data, simulation results, or any specific domain instantiation. They hold for any system that satisfies the architectural constraints of Definitions 2.1 through 2.5. These results cannot be weakened by empirical findings — if the topology holds, the guarantees hold.

Tier 2 — AI-executed computational validation. The NCFCA Empirical Validation Suite (Lacefield 2026b) consists of six falsifiability tests executed by a general-purpose AI system running Python simulation code written to the mathematical specification of this proof. The tests confirmed the structural properties the proof predicts: zero reverse-path contamination (Test 5), schema floor preservation during anomaly windows (Test 3), domain-invariant baseline behavior (Test 4), and the warmup integrity failure mode (Test 6B). These are not synthetic data in the pejorative sense — they are AI-executed computational confirmations that the formal architecture behaves as the proof requires. They are simulation results, not field data. Their evidentiary role is to confirm that the proof's structural properties are operationally instantiated in the implemented system, not to establish empirical effect sizes in real-world deployments.

Tier 3 — AI system empirical data (pending). The NCFCA AI Evaluation Experiment (Lacefield 2026, in preparation) will apply the NCFCA prerequisite-gated evaluation protocol to real deployed AI systems — specifically to assess whether the protocol detects hallucinations that survive standard single-shot composite benchmarks. The inputs are real LLM outputs from real deployed systems. The hallucination events are real system behaviors, not simulated ones. This tier produces the first genuinely empirical data: field observations of the NCFCA protocol operating on actual AI system outputs. Results pending administration.

Tier 4 — Human field data (pending). Empirical data from real human learners collected under the NCFCA separation protocol (the NCFCA Field Version protocol) will provide the first validation of the architecture in its primary education domain instantiation. This tier requires the NCFCA Separation Protocol to be administered across a minimum of 20 clean baseline sessions per student, with explicit S, I, and C channel logging separate from D-channel updates. Data collection not yet initiated.

Remark 5a.1 (The Structural Invariance Finding). The domain independence result of Test 4 — a baseline false positive rate of 1.8–1.9% across four domain instantiations (education, industrial, athletic, recovery) at a fixed threshold θᴄ = 2.5σ — is the single most important Tier 2 finding for the following reason: most adaptive systems require retuning for every new domain because their performance depends on domain-specific statistical properties of the data. The NCFCA architecture produces identical baseline performance across wildly different domains because the guarantee follows from the topology, not from the data. This is the empirical signature of a structural result. It is not a tuned outcome. No hyperparameter adjustment produced the 1.8–1.9% consistency — it emerged from the architecture.

Remark 5a.2 (Zero-Latency Recovery as Architectural Property). The zero schema floor degradation result of Test 3 is correctly framed as a logical property rather than a statistical result. In stochastic adaptive filters (Kalman, particle), recovery from an anomaly window is a statistical process with O(n) latency — the filter must re-converge through multiple noisy observations before the estimate returns to pre-anomaly accuracy. In the NCFCA architecture, recovery is a binary state transition: when δ(t) returns to 0, the circuit breaker closes and the system resumes from the exact schema floor value frozen at anomaly onset. There is no re-convergence because nothing was overwritten. In simulation under idealized conditions (ε = 0 at anomaly closure), this produces a recovery gap of 0.0000 by construction. In real deployments, the effective recovery latency depends on sampling rate and implementation parameters, but the architectural property — that no contaminated observations were written during the anomaly window — holds regardless. The system is not recovering; it is resuming from a clean state that was structurally preserved.

6. Assumptions and Limitations

Assumption 6.1 (Source Independence).

The four signal classes S, D, I, and C are generated by categorically distinct causal mechanisms. In particular, there do not exist unmodeled common causes capable of inducing a direct causal dependence between execution state (C) and internal state (D), or between interface friction (I) and internal state (D), outside the pathways explicitly represented in the causal graph G. Causal influence from environmental state (S) to internal state (D) is permitted and is represented by the edge Sₛₒₚ → Dᴅᴸᴾ. No such direct causal pathway is permitted from C to D or from I to D. If this assumption is violated, additional edges would be required in G, and both the d-separation result in Theorem 1 and the non-collapsibility claim would need to be re-established under the modified graph.

Assumption 6.2 (Schema-Level Enforcement).

The lower-triangular pipeline constraint must be enforced at the data-model level, not only at the application level. Application-level enforcement can be bypassed by implementation errors or future code modifications. Schema-level enforcement via foreign key constraints, write-path validation, and audit logging at the data persistence layer produces a structural guarantee: the absent edges in G are absent not by policy but by the absence of a data pathway capable of carrying them.

Assumption 6.3 (Circuit Breaker Calibration).

The anomaly detection threshold θᴄ in Definition 2.5 must be empirically calibrated for each domain and population. The Theorem 2 guarantee is conditional on δ(t) correctly identifying anomaly events. False negatives reduce the completeness of the guarantee; false positives reduce system responsiveness without affecting correctness. The existence of the free parameter θᴄ does not make the conditional guarantee vacuous: every detection-based guarantee in the engineering literature is conditional on threshold calibration in the same sense. Clinical or operational validation is required prior to deployment.

7. Open Problems

The following open problems are identified for future work:

Empirical validation. The source independence assumption (6.1) requires empirical validation across target domains via controlled experiments measuring pairwise covariance between signal class proxies under natural operating conditions.

Tiered and continuous mastery. The present results assume binary mastery states m(v_i) ∈ {0, 1}. The intended practical implementation uses domain-specific confidence thresholds: high-importance or safety-critical concept nodes require elevated confirmation thresholds (e.g., ≥ 98% confidence), while less critical nodes admit lower thresholds. Concepts that only reach intermediate confidence levels remain in a provisional, reviewable state open to later correction via a supermajority review mechanism. Formalizing this tiered threshold system while preserving the schema floor integrity guarantee of Theorem 2 is an open problem.

High-assurance circuit breaker specification. Formal specification of the circuit breaker protocol sufficient for high-assurance certification in safety-critical domains (e.g., ASIL-D in the automotive domain) is listed as future work.

Time-varying causal graphs. 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.

Warmup integrity verification (Open Problem 6). The Proposition 6.4 falsifiability construction assumes a clean baseline calibration period free of coupling. Empirical testing demonstrates that warmup contamination — coupling present during the calibration window — inflates the baseline σ estimate and causes silent detection failure: reduced detection capability with simultaneously reduced false positive rate, giving no visible indication of miscalibration (NCFCA Empirical Validation Suite, Test 6B, Lacefield 2026b). At coupling strength ε_warmup = 0.6 during the warmup period, post-warmup detection drops from 28.3% to 5.4% while false positive rate drops to 0.0%. A formal protocol for verifying source independence prior to establishing the baseline C distribution is required before Assumption 6.3 can be considered fully operational in high-stakes deployments. This is an open engineering problem with direct bearing on deployment safety.

8. Conclusion

We have presented a structural solution to signal smearing in adaptive state estimation systems. By enforcing a strictly lower-triangular causal pipeline with non-collapsible profiles, the architecture guarantees that execution-layer anomalies cannot corrupt the estimate of genuine internal state. This is a structural guarantee, not a statistical one.

The results are supported by two independent lines of confirmation. First, the formal proof establishes the structural guarantee from graph topology alone, independent of any empirical data. Second, AI-executed computational validation (NCFCA Empirical Validation Suite, Lacefield 2026b) confirms that the proof's structural properties are operationally instantiated: zero reverse-path contamination (Test 5), schema floor preservation during anomaly windows (Test 3), and domain-invariant baseline behavior across four instantiation domains at 1.8–1.9% false positive rate (Test 4). These are AI-executed computational confirmations that the formal architecture behaves as the proof requires, not field measurements. Empirical validation in real human and AI system deployments constitutes the next evidentiary tier (Section 5a).

The results are domain-independent. The same architectural constraints apply whether the domain is education, industrial operations, athletic training, recovery, or autonomous systems. The 1.8–1.9% false positive rate consistency across four structurally different domains is the empirical signature of this domain independence — it did not require retuning and it was not a tuned outcome. It follows from the topology.

The causal parroting problem identified in recent benchmark literature — LLMs inferring causal answers from surface statistical patterns rather than from structural analysis — is a manifestation of the same smearing problem the NCFCA addresses at the architecture level. Standard benchmarks measure performance under noise. The NCFCA structurally eliminates the noise source before the estimation occurs. These are not competing approaches; they are operating at different levels of the same problem.

Once re-filtered, the data yields clearer roadmaps. Domains can identify which gaps are real and require new high-fidelity data collection, versus which apparent gaps are artifacts of prior measurement architectures. The result is research that rests on actual authority — clean causal attribution and structural separation — rather than perceived authority, institutional credentials, or aggregated statistical summaries that conceal underlying signal collapse.

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Gregory Stuart Lacefield · Lacefield Research · Las Vegas, NV · gregorylacefield.com

Document ID: LR-PROOF-01 · Classification: Internal — not for public distribution

Version History

v3.2 (June 2026) — Section 5a (Evidentiary Classification of Results) added with four-tier evidentiary stack; AI-executed computational validation framing added; Remarks 5a.1 (structural invariance) and 5a.2 (zero-latency recovery as architectural property) added; NoisyCausal (2026) and CORR2CAUSE/Jin et al. (2023) added to prior art section; causal parroting context added; conclusion updated to reflect evidentiary stack and domain independence finding; NoisyCausal, Jin et al. (2023), and Structured Thinking (2025) added to references. Section numbering updated (prior Section 5a inserted before Section 6).

v3.1 (June 2026) — FSM definition added (Definition 2.4a); Bayesian/Kalman filtering prior art contrast added; information leakage framing added; nomenclature equivalence note added (S≡SCP, D≡DSP, I≡IRP, C≡CEP); empirical confirmation remarks added after Proposition 1.3, Theorem 1, and Theorem 2; empirical comparison paragraph added to prior art section; Open Problem 6 (warmup integrity) added; Lord & Novick (1968) and Wang et al. (2025) added to references.

v3.0 (June 2026) — Open Problem 6 (warmup integrity) first added; all empirical evidence insertions from NCFCA Empirical Validation Suite (Tests 3, 4, 5, 6B) added; Assumption 6.3 calibration parameters updated; duplicate Assumption 6.1 removed; Bareinboim letter version.

v2.0 (June 2026) — Full audit pass (21 checks); all definition numbering corrected; Definitions 3.1 and 3.2 added; C(t) formally defined as measurable function; BKT connection to Proposition 1.3 stated; Gagne (1968), Theobald et al. (2022), Wilks (1938) added to references; abstract updated to education instantiation language.

v1.0 (May 2026) — Initial version.