Feasibility Gap Theory

1. Motivation

Problem: Binary pass/fail flags do not distinguish between: - Minor rounding errors (δ = $1) - Material misstatements (δ = $1B)

Solution: Compute a continuous severity score based on the distance a filing must travel to re-enter the conservation manifold.

2. Mathematical Definition

Feasibility Set

$$\mathcal{F} = \{ \mathbf{y} \in \mathbb{R}^n : \mathbf{A} \mathbf{y} = \mathbf{0} \}$$

where: - $\mathbf{A} \in \mathbb{Z}^{m \times n}$ is the constraint matrix (currently the simplified $m = 15$ system) - $\mathbf{y} \in \mathbb{R}^n$ stacks the XBRL tag values for a filing

Distance to Feasibility

$$\delta^*(\mathbf{y}) = \min_{\Delta \mathbf{y}} \|\Delta \mathbf{y}\|_1 \quad \text{s.t.} \quad \mathbf{A}(\mathbf{y} + \Delta \mathbf{y}) = \mathbf{0}$$

Interpretation: Minimum L1-norm adjustment required to restore feasibility.

3. LP Formulation

Using variable splitting with non-negative vectors $\mathbf{u}, \mathbf{v} \geq \mathbf{0}$ such that $\Delta \mathbf{y} = \mathbf{u} - \mathbf{v}$:

$$\begin{aligned} \min_{\mathbf{u}, \mathbf{v}} \quad & \mathbf{1}^\top \mathbf{u} + \mathbf{1}^\top \mathbf{v} \\ \text{s.t.} \quad & \mathbf{A}(\mathbf{y} + \mathbf{u} - \mathbf{v}) = \mathbf{0} \\ & \mathbf{u}, \mathbf{v} \geq \mathbf{0} \end{aligned}$$

This yields a convex program solved efficiently with interior-point methods (ECOS) or first-order methods (SCS) via CVXPY.

4. Normalization

The raw gap $\delta^*$ carries units of dollars and cannot be compared across firms of different scale.

Scaled gap:

$$\delta^*_{\text{scaled}} = \frac{\delta^*(\mathbf{y})}{\|\mathbf{y}\|_1}$$

Interpretation: Fraction of total tag magnitude requiring adjustment.

5. Materiality Thresholds

Severity	δ*_scaled Range	Interpretation
Pass	< 0.001 (0.1%)	Rounding / precision issues
Low	0.001 – 0.01 (0.1% – 1%)	Minor, likely immaterial
Medium	0.01 – 0.05 (1% – 5%)	Investigate
High	0.05 – 0.10 (5% – 10%)	Possible control deficiency
Critical	> 0.10 (>10%)	Material misstatement risk

Calibration Source: SEC Staff Accounting Bulletin No. 99 (SAB 99) — materiality guidelines.

6. Theoretical Properties

Theorem 1 — Existence and Boundedness

If $\operatorname{rank}(\mathbf{A}) = m < n$, the feasibility set is a non-empty affine subspace and the L1 minimization is bounded.

Sketch: The constraint system is consistent (per TASK_02 rank analysis). The feasible region for $\Delta \mathbf{y}$ is affine and the L1 norm is coercive over this space.

Theorem 2 — Sensitivity

For any coordinate $i$ with optimal solution $\Delta \mathbf{y}^*$:

$$\frac{\partial \delta^*}{\partial y_i} \in [-1, 1]$$

Meaning small adjustments to individual tags cause bounded changes to the feasibility gap, ensuring stability against minor restatements.

7. Computational Complexity

Solver: CVXPY + ECOS (default) or SCS (fallback).
Variables: $n$ (equal to number of tags retained in the simplified universe, currently $n = 10$).
Complexity: $O(n^{2.5})$ for dense interior-point methods; empirically 30–50ms per filing.
Throughput: 5,533 filings process in under 10 minutes on a modern workstation.

8. Comparison to Alternative Metrics

Metric	Pros	Cons
L1 feasibility gap (ours)	Sparse adjustments; interpretable; aligns with journal entry adjustments	Optimal $\Delta \mathbf{y}$ may be non-unique
L2 feasibility gap	Unique minimizer; smooth	Distributes adjustments across many tags; less audit-aligned
L∞ feasibility gap	Captures worst-case single adjustment	Ignores aggregate misstatement magnitude
Norm of residual ‖Ay‖	Cheap to compute	Does not quantify adjustments needed for compliance

9. Limitations

Intent: LP quantifies magnitude, not fraud intent or control root cause.
Tag granularity: Mixes GAAP violations with XBRL tagging errors; requires analyst review.
Context: 1% of $10M$ and 1% of $100B$ yield same scaled score; auditors still interpret absolute dollars.

Mitigation: Combine with gold-labeled review (TASK_07), temporal drift analysis (TASK_05), and peer benchmarks by industry.

Farrell, M. J. (1957). The Measurement of Productive Efficiency. Journal of the Royal Statistical Society.
Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research.
Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.

Novelty: Applies feasibility distance to accounting identity enforcement with IFRS/GAAP-aware decomposition (simplified 15-constraint model).

References

Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
SEC Staff Accounting Bulletin 99 (1999). Materiality.
Farrell, M. J. (1957). “The Measurement of Productive Efficiency”. Journal of the Royal Statistical Society.