Prior Research Report: Positioning GhostDrift Theory in Operations Research and Its Theoretical Innovations

1. Executive Summary

This report defines the positioning of GhostDrift Theory within the field of Operations Research (OR). GhostDrift is not an incremental improvement of existing optimization algorithms; rather, it is a structural expansion of OR that provides a "Pre-decision Constraint"—a necessary condition for the social and mathematical validity of a decision. While traditional OR focuses on "reducing or absorbing" uncertainty, GhostDrift focuses on the "fixation of unknowability." By doing so, it mathematically prevents "Responsibility Evaporation" (the dilution of accountability through post-hoc rationalization).

2. Current State and the "Mathematical Gap" in OR

2.1 The Implicit Assumption in Existing Research

Mainstream OR formalizes uncertainty via (1) probability distributions, (2) uncertainty sets, or (3) scenarios/adaptation. However, these models fail to consider the "expansion of post-hoc explanations." After an incident, the freedom to re-rationalize the causal links, criteria, and responsibility boundaries can be expanded indefinitely. This leaves two critical failure modes outside the model:

1. Arbitrariness of Premises: The freedom to redefine "why this set/distribution was chosen" after the fact.

2. Over-supply of Explanations: The dissipation of responsibility by adding new parameters or model interpretations post-incident.

2.2 Isomorphic Mapping and Gaps with OR Concepts

GhostDrift extends the "freedom constraints" familiar to OR (used to ensure feasibility) into the channels of explanation and accountability.

• (1) Non-anticipativity / Information Filtration:

  • Existing: Decision $x_t$ cannot depend on future information $\mathcal{F}_t$.

  • GhostDrift: Post-hoc rationalization is a violation of "explanation non-anticipativity." GhostDrift restricts the generation of explanation $e$ to the catalog $E_{pre}$ fixed at the decision time $\mathcal{F}_{decision}$.

• (2) Policy Space Constraints:

  • Existing: Decisions must be chosen from an allowed policy class $\Pi$ ($\pi \in \Pi$).

  • GhostDrift: Justifications must be chosen from an allowed explanation catalog $E_{pre}$ ($e \in E_{pre}$).

• (3) Commitment / Time Consistency:

  • Existing: Rules ensuring that future re-optimization does not destroy the current plan.

  • GhostDrift: Demands "Time Consistency of Explanation Procedures." It prohibits the redistribution of responsibility through post-hoc redefinition.

3. Mathematical Definition: Fixation as a Specification

3.1 Minimal Formalism

• Decision Variable: $x \in X$

• Event (Scenario): $\omega \in \Omega$

• Observation/Evidence (Audit Log): $s \in S$

• Explanation (Justification Procedure): $e \in E$

• Verifier: $V_e : X \times S \to \{0,1\}$

3.1.1 Closure of the Explanation Catalog $E_{pre}$

$E_{pre} := \{ e_1, \dots, e_M \}$ is a finite set of justification IDs fixed before the decision. Each $e_i$ is a tuple $(id_i, spec_i, V_i)$, where $V_i$ is a hard-coded verification logic.

3.1.2 Freeze and Acceptance Conditions

Before the decision, a commitment value $h$ is created and published:

$$h := \text{Commit}(E_{pre}, U, \text{Schema}_S)$$

The acceptance condition is defined as:

$$\text{Accept}(h, x, s, e) := [\text{Verify}(h, s)=1] \land [e \in E_{pre}(h)] \land [V_e(x, s)=1]$$

3.2 Theorem: Post-hoc Impossibility (Procedural Infeasibility)

【Attacker Model: Post-hoc Rationalization】 An operation $o : (x, s) \mapsto (x, s', e')$ represents an attempt to evaporate responsibility (RE) via:

• (RE1) Explanation Expansion: $e' \notin E_{pre}(h)$

• (RE2) Field Deviation: Deviating from fixed fields like $\text{metric\_id}$ or $\text{uncertainty\_spec\_id}$.

• (RE3) Recomposition: Re-mapping causal or responsibility boundaries post-incident.

【Theorem: Infeasibility of Post-hoc Explanation Expansion】 Even if an attacker modifies logs or explanations (operation $o$), as long as $\text{Verify}$ and $\text{Binding}$ are maintained, the Acceptance is preserved, and any modification satisfying RE is incompatible with the acceptance conditions.

【Proof Sketch】 The validity of $\text{Accept}$ relies on the axioms (C1–C5). Any $e'$ or $s'$ produced by the attacker will be rejected by $\text{Verify}$'s referential closure check if it references IDs outside $E_{pre}$, and the binding of fields to $spec_i$ blocks the substitution of semantics. Thus, responsibility evaporation is procedurally infeasible. ■

4. Operational Design and Threat Model

4.1 Assumptions for Implementation

The security of this protocol relies on:

• Cryptographic Assumptions: Collision resistance of hash functions and unforgeability of digital signatures.

• Log Integrity: Use of WORM (Write-Once-Read-Many) ledgers or transparency logs to ensure immutability.

4.2 Axioms of Commit/Verify

Under these assumptions, $\text{Verify}(h, s)=1$ guarantees the following:

1. (C1) Binding: $s$ satisfies $\text{Schema}_S(h)$ and fixed fields match $\text{Bind}(h)$.

2. (C2) Referential Closure: Any $id$ referenced by $s$ is in $E_{pre}(h)$.

3. (C3) Authenticity: Approver IDs and signatures align with the pre-agreed policy $U(h)$.

4. (C4) Data Binding: Hashes for training data and model specs match $\text{Bind}(h)$.

5. (C5) Metric Binding: All metric and threshold IDs match $\text{Bind}(h)$.

5. Comparison: GhostDrift vs. MRM

The fundamental difference between traditional Model Risk Management (MRM) and GhostDrift lies in the "Zeroing of Degrees of Freedom," not just the intensity of management.

Comparison	MRM (SR 11-7 / PRA SS1/23)	GhostDrift
Approach	"Management and Control" of explanations	"Dimensionality Collapse" of explanation freedom
Expansion Freedom	Metrics and models are tracked, but post-hoc re-rationalization remains possible.	Explanation paths are closed via finite ID reference, excluding "additional explanations."
Recomposition	Narrative reconstruction is possible based on context.	Explanation space is closed; only identity transformations are allowed.
Mathematical Property	Post-hoc audit trail	Procedural Infeasibility (Attack Infeasibility)

GhostDrift is not a framework for "better governance." It is a constraint that closes the explanation space to a finite set of IDs before the decision, making responsibility evaporation procedurally unacceptable.

6. Selected References

• Ben-Tal, A., et al. (2009). Robust Optimization. Princeton University Press.

• Bertsimas, D., & Sim, M. (2004). The Price of Robustness. Operations Research, 52(1), 35-53.

• Bertsimas, D., et al. (2011). Theory and applications of robust optimization. SIAM Review, 53(3), 464-501.

• Delage, E., & Ye, Y. (2010). Distributionally robust optimization under moment uncertainty. Operations Research, 58(3), 595-612.

• Goh, J., & Sim, M. (2010). DRO and its tractable approximations. Operations Research, 58(4), 902-917.

• Artzner, P., et al. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228.

• Rockafellar, R. T., & Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk, 2(3), 21-42.

• Shapiro, A. (2012). Time consistency of dynamic risk measures. Operations Research Letters, 40(6), 436-439.

• Marchau, V. A. W. J., et al. (2019). Decision Making under Deep Uncertainty. Springer.

• De Bock, K. W., et al. (2024). Explainable AI for Operational Research (XAIOR). European Journal of Operational Research, 317(2), 571-588.

• Elmachtoub, A. N., & Grigas, P. (2022). Smart “Predict, then Optimize”. Management Science, 68(1), 9-26.

• Federal Reserve System (2011). SR 11-7: Guidance on Model Risk Management.

• Bank of England / PRA (2023). SS1/23: Model Risk Management Principles for Banks.