AEGIS Mathematical Coherence Review¶

Version: 1.2.0 Date: 2026-01-31 Protocol: Rigor v1.0 + Claude-GPT Dialogical Collaboration Mode: ideation-critique GPT Reasoning: xhigh

Executive Summary¶

This document captures a multi-model validation of AEGIS's mathematical foundations to verify statistical methods, formula coherence, integration soundness, and identify potential gaps or failure modes.

Round 1: Claude Opus 4.5 Initial Analysis¶

1. Bayesian Risk/Profit Gates¶

1.1 Implementation Review¶

The conjugate normal-normal model is correctly implemented:

posterior_precision = 1/σ₀² + 1/σ_L²
posterior_variance = 1/posterior_precision
μ_post = posterior_variance × (μ₀/σ₀² + observed/σ_L²)
P(Δ ≥ threshold) = 1 - Φ((threshold - μ_post) / σ_post)

Mathematical Verification: - The precision-weighted update formula is the standard conjugate prior result - For Normal-Normal with known variance: this is textbook-correct - The CDF computation via 1 - Φ(z) correctly gives the upper tail probability

1.2 Concerns Identified¶

Concern B1: Model Appropriateness - Normal distribution assumes symmetric, unbounded risk deltas - Real-world risk deltas may be: - Bounded (can't lose more than 100%) - Skewed (tail risks are typically asymmetric) - Fat-tailed (extreme events more likely than normal predicts)

Recommendation: Consider log-normal or beta distribution for bounded deltas, or use robust estimators (e.g., Huber loss for outlier resistance).

Concern B2: Default Hyperparameters - μ₀=0 (no change expected): Reasonable neutral prior - σ₀=1.0: This implies 95% prior credible interval of [-1.96, 1.96] for normalized delta - σ_L=0.5: Observation noise ≈ 0.5 implies fairly high trust in observations

Issue: The ratio σ₀²/σ_L² = 1/0.25 = 4 means observations have 4× the precision of the prior. A single observation substantially shifts the posterior. This may be too reactive.

Recommendation: Document the implicit assumption that observations are high-signal. Consider σ_L=1.0 for noisier domains.

Concern B3: Boundary Behavior at P ≈ 0.95 - The gate passes if P(Δ ≥ 2) < 0.95 - At P = 0.9499, gate passes; at P = 0.9501, gate fails - This discrete threshold creates sensitivity to small changes

Recommendation: Consider graduated response (e.g., warning zone at 0.90-0.95) or hysteresis for state transitions.

2. PERT Three-Point Estimation¶

2.1 Implementation Review¶

E[X] = (a + 4m + b) / 6
Var[X] = ((b - a) / 6)²

Mathematical Verification: - The expected value formula is the modified PERT formula (standard) - The variance formula is the simplified PERT variance

2.2 Concerns Identified¶

Concern P1: Variance Approximation

The true PERT beta distribution has: - Shape parameters: α = 1 + 4×(m-a)/(b-a), β = 1 + 4×(b-m)/(b-a) - True variance: Var = (α×β) / ((α+β)²×(α+β+1)) × (b-a)²

The simplified formula ((b-a)/6)² is only accurate when m = (a+b)/2 (symmetric).

Quantitative Impact: - For symmetric (a=0, m=5, b=10): True Var ≈ 1.39, Simplified Var = 2.78 → 2× overestimate - For skewed (a=0, m=2, b=10): True Var ≈ 2.86, Simplified Var = 2.78 → Close - For highly skewed (a=0, m=1, b=10): True Var ≈ 3.47, Simplified Var = 2.78 → 20% underestimate

Recommendation: For governance where tail risks matter, use the true beta variance formula or document that the simplification is conservative for symmetric estimates but may underestimate for highly skewed ones.

Concern P2: Sensitivity to Mode (m)

The expected value E[X] = (a + 4m + b) / 6 weights m by 4×. - Small changes in m have large impact on expected value - No guidance on how to elicit m reliably

Recommendation: Add sensitivity analysis or document uncertainty in m as a separate variance component.

3. Utility Function (Rubric v2.1)¶

3.1 Implementation Review¶

U = (ΔP_H + γ·ΔV_L) + κ·ΔR - (φ_S·ΔC_S + φ_D·ΔC_D) - ΔOPEX
LCB = U - z_α × √Var(U)
Var(U) = Var(P) + γ²·Var(V) + κ²·Var(R)

Mathematical Verification: - The LCB formula assumes U is approximately normal (CLT-based justification) - Variance propagation uses: Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X,Y)

3.2 Concerns Identified¶

Concern U1: Independence Assumption

The variance calculation:

variance = Var(P) + γ²·Var(V) + κ²·Var(R)

Implicitly assumes Cov(P,V) = Cov(P,R) = Cov(V,R) = 0

This is likely violated: - Profit and Value are often positively correlated - Profit and Risk may be negatively correlated (high-risk = high-reward) - Value and Risk may have domain-specific relationships

Impact: If components are positively correlated, true variance is underestimated, leading to: - LCB being too optimistic - More proposals passing that shouldn't

Recommendation: 1. At minimum, add a correlation adjustment factor: Var(U) = Var(P) + γ²Var(V) + κ²Var(R) + ρ_adj × σ_P × σ_V × γ 2. Or document the independence assumption explicitly and monitor empirically

Concern U2: Kappa (κ) Only for Risk Reduction

kappa_effective = self.kappa if delta_R < 0 else 0.0

This means: - Risk reduction (ΔR < 0) adds κ×|ΔR| to utility - Risk increase (ΔR > 0) has zero penalty beyond gate failure

Logical Inconsistency: If risk increases, shouldn't there be a negative utility contribution? The current design relies entirely on the risk gate to block high-risk proposals, but doesn't penalize moderate risk increases in the utility calculation.

Recommendation: Consider: kappa_effective = self.kappa * delta_R (always apply, negative for increase, positive for decrease).

Concern U3: Complexity Cost Exclusion from Variance

Complexity costs (φ_S·ΔC_S + φ_D·ΔC_D) are treated as deterministic constants in variance.

Issue: Complexity estimates also have uncertainty. If complexity is misestimated by 20%, the utility swing could be significant (φ_D = 2000 $/point).

Recommendation: If complexity estimates have standard errors, include them: Var(Complexity) = φ_S²·Var(C_S) + φ_D²·Var(C_D).

4. KL Divergence Drift Detection¶

4.1 Implementation Review¶

def _compute_kl_divergence(self, p, q):
    kl = 0.0
    for p_i, q_i in zip(p, q):
        p_i = max(p_i, epsilon)  # epsilon = 1e-10
        q_i = max(q_i, epsilon)
        if p_i > 0:
            kl += p_i * math.log(p_i / q_i)
    return kl

Mathematical Verification: - The formula D_KL(P||Q) = Σ P(x) log(P(x)/Q(x)) is correct - Epsilon smoothing prevents log(0)

4.2 Concerns Identified¶

Concern K1: Asymmetry of KL Divergence

D_KL(P||Q) ≠ D_KL(Q||P)

In drift detection: - P = current distribution (what we're monitoring) - Q = baseline (historical reference)

Issue: KL(P||Q) measures how much P diverges from Q, but: - If Q has zeros where P doesn't → KL undefined (epsilon masks this) - The direction matters: KL is sensitive to mode-seeking vs mode-covering

Recommendation: Consider Jensen-Shannon divergence: JS = (KL(P||M) + KL(Q||M))/2 where M = (P+Q)/2. This is symmetric and bounded [0, log(2)].

Concern K2: Histogram Binning Sensitivity

num_bins: int = 50

Issues: - 50 bins may be too many for small samples (sparse bins → noisy KL) - 50 bins may be too few for high-resolution differences - Bin edges are determined by data range, not fixed percentiles

Recommendation: - Use Freedman-Diaconis rule for adaptive binning: bins = (max-min) / (2×IQR×n^(-1/3)) - Or use kernel density estimation for smoother distributions

Concern K3: Non-Stationarity

Issue: KL divergence assumes both P and Q are stationary. If the baseline drifts gradually (regime change), the detection may: - Fire false positives during transition - Miss slow drifts that don't exceed threshold

Recommendation: Add trend-adjusted baselines or exponentially-weighted moving baselines.

5. Six-Gate Integration¶

5.1 Implementation Review¶

Gates evaluated in sequence: 1. Risk Gate (Bayesian posterior) 2. Profit Gate (Bayesian posterior) 3. Novelty Gate (Logistic) 4. Complexity Floor (Hard constraint) 5. Quality Gate (Threshold + subscores) 6. Utility Gate (LCB threshold)

Override eligibility: not all_passed and gates[COMPLEXITY].passed

5.2 Concerns Identified¶

Concern G1: Gate Ordering

Question: Does evaluation order matter?

Analysis: In current implementation, all gates are evaluated regardless of earlier failures. Order doesn't affect pass/fail outcome.

However: If short-circuit evaluation were added (for performance), order would matter. Current design is robust.

Concern G2: Novelty Gate Threshold Analysis

exponent = -k * (N - N0)  # k=10, N0=0.7
gate_value = 1 / (1 + exp(exponent))
passed = gate_value >= 0.8

Solving for passing threshold: - G(N) ≥ 0.8 requires: 1/(1+e^(-10(N-0.7))) ≥ 0.8 - e^(-10(N-0.7)) ≤ 0.25 - -10(N-0.7) ≤ -1.386 - N ≥ 0.7 + 0.139 = 0.839

The novelty gate passes for N ≥ 0.839 (approximately)

Concern: The logistic is very steep (k=10). Between N=0.8 and N=0.85: - N=0.80: G=0.731 (FAIL) - N=0.84: G=0.802 (PASS) - N=0.85: G=0.818 (PASS)

Very small changes in novelty score cross the threshold.

Recommendation: Consider lower k (e.g., k=5) for smoother transition, or document that novelty scoring must be precise.

Concern G3: Override Logic Soundness

override_eligible = not all_passed and gates[COMPLEXITY].passed

Interpretation: Override is possible if: - Not all gates passed (obviously, else no need for override) - Complexity floor passed (complexity is non-overridable)

This is logically sound. The complexity gate serves as a hard barrier that human override cannot bypass.

Edge Case: What if ONLY complexity fails? - all_passed = False, gates[COMPLEXITY].passed = False - override_eligible = False ✓ Correct

What if complexity passes but utility and risk fail? - all_passed = False, gates[COMPLEXITY].passed = True - override_eligible = True ✓ Correct

Verified: The override logic is sound.

Concern G4: Confidence Aggregation

min_confidence = min(confidences) if confidences else 1.0

Issue: Minimum confidence across gates loses information. If 5 gates have 0.99 confidence and 1 has 0.5, the min is 0.5.

Alternative: Consider weighted geometric mean or use the most relevant confidence for the failing gate(s).

6. Numerical Stability¶

6.1 Implementation Review¶

• exp() clamped to [-700, 700]: Prevents overflow (e^709 ≈ 10^308)
• epsilon = 1e-10 for log smoothing: Prevents log(0)
• epsilon = 0.01 for baseline normalization: Prevents division by zero

6.2 Concerns Identified¶

Concern N1: Epsilon Inconsistency

Different epsilon values are used: - 1e-10 for KL divergence (very small) - 0.01 for baseline normalization (much larger)

Issue: 1e-10 could cause numerical instability with log: - log(1e-10) = -23 - log(1e-10 / 1e-10) = 0 ✓ - log(0.01 / 1e-10) = log(10^8) = 18.4

This is within safe ranges, but the inconsistency could be confusing.

Recommendation: Unify epsilon handling with documented rationale for each context.

Concern N2: CDF Tail Accuracy

def _standard_normal_cdf(self, z: float) -> float:
    return 0.5 * (1.0 + math.erf(z / math.sqrt(2.0)))

Analysis: math.erf is accurate to 15-16 significant digits (double precision).

For extreme tails (|z| > 8), the CDF approaches 0 or 1 with limited precision: - Φ(8) ≈ 1 - 6.22e-16 (at machine epsilon) - Φ(-8) ≈ 6.22e-16

For governance thresholds at P=0.95 (z ≈ 1.645), accuracy is excellent. No concern.

7. Integration Point: pcw_decide()¶

7.1 Concerns Identified¶

Concern I1: Default Utility Result

if utility_result is None:
    utility_result = UtilityResult(
        raw=0.0,
        variance=0.01,
        lcb=0.1,
        components=default_components,
        decision_path="investment",
    )

Issue: Default LCB=0.1 means missing utility input → gate passes (if threshold=0).

Recommendation: Make utility_result required, or set lcb=-∞ for missing to force failure.

Concern I2: Confidence Aggregation Semantics

Using min(confidences) conflates different types of confidence: - Bayesian posterior probability (risk/profit gates) - Logistic gate output (novelty) - Binary pass/fail (complexity) - LCB margin (utility)

These are not directly comparable.

Recommendation: Report confidence per gate type rather than aggregating across semantically different measures.

Summary of Findings (Post-Dialogue Consensus)¶

Critical Issues (Require Immediate Attention)¶

Significant Issues (Should Address)¶

Minor Issues (Nice to Have)¶

Verified Sound (Both Models Agree)¶

• ✓ Bayesian posterior formula (mathematically correct for latent parameter)
• ✓ Override logic (logically sound)
• ✓ Numerical stability (adequate for governance thresholds)
• ✓ Gate ordering (independent, no short-circuit)
• ✓ CDF tail accuracy (excellent for P=0.95)
• ✓ PERT expected value formula (correct)

Round 2: GPT 5.2 Pro Critique¶

Status: COMPLETE ✅ Duration: 734 seconds (12.2 minutes) Reasoning Level: xhigh Timestamp: 2026-01-31T08:48:34

Corrections to Claude's Analysis¶

GPT 5.2 Pro identified several errors in the initial Claude analysis:

1. PERT Variance Numerical Errors (CRITICAL)¶

Claude's claim that the simplified PERT variance can be off by "2×" is incorrect.

GPT's Correction: For Beta-PERT with λ=4 (α = 1+4(m-a)/(b-a), β = 1+4(b-m)/(b-a)): - α + β = 6 always - True variance: αβ/252 × (b-a)² - Simplified variance: (b-a)²/36 - Ratio: αβ/7

Since αβ ∈ [5, 9], the ratio is [0.714, 1.286], meaning: - Worst case: overestimates by 40% (when m=a or m=b) - Or underestimates by 22% (when m is central) - NOT 2× as Claude claimed

Corrected Examples (a=0, b=10): | Mode (m) | α | β | True Var | Simplified Var | Error | |----------|---|---|----------|----------------|-------| | 5 (symmetric) | 3 | 3 | 3.571 | 2.778 | -22% (under) | | 2 | 1.8 | 4.2 | 3.000 | 2.778 | -8% (under) | | 1 | 1.4 | 4.6 | 2.556 | 2.778 | +9% (over) |

2. Covariance Formula Incomplete¶

Claude's suggested correlation adjustment was mathematically incomplete. Missing: - Factor of 2 in covariance terms - All covariance pairs

Correct Formula:

Var(U) = Var(P) + γ²Var(V) + κ²Var(R)
       + 2γ·Cov(P,V) + 2κ·Cov(P,R) + 2γκ·Cov(V,R)

3. Distribution Recommendations Underspecified¶

Claude's suggestion to use "log-normal or beta for bounded deltas" is problematic: - Log-normal has positive support only - Vanilla beta is [0,1] - Deltas can be negative

Better Alternatives: - Transformed Beta on [L, U] - Logit-normal / logistic transform - Truncated / skew-normal - Student-t likelihood for fat tails (recommended)

GPT's Agreement/Disagreement Matrix¶

Additional Issues Claude Missed (GPT)¶

GPT identified 9 additional mathematical issues:

A. Posterior vs Posterior Predictive (CRITICAL)¶

Claude verified the posterior for the latent parameter θ, but decisions should use the posterior predictive for future realized values.

Current (potentially wrong):

P(θ ≥ t | y) = 1 - Φ((t - μ_post) / σ_post)

Correct for realized outcomes:

P(Y_new ≥ t | y) = 1 - Φ((t - μ_post) / √(σ_post² + σ_L²))

The current implementation may be overconfident about realized outcomes.

B. Sample Size / Aggregation¶

If "observed" is an average of n observations, likelihood variance should be σ_L²/n. Many systems accidentally treat an average as a single observation.

C. Unknown σ_L (Fat Tails)¶

Taking σ_L as fixed is the main fragility. A Normal-Inverse-Gamma prior yields Student-t posterior predictive, which is more realistic for risk governance.

D. LCB Normal Approximation Weak¶

U is sum of ~3 uncertain components. CLT justification is thin. If any component is skewed/heavy-tailed: - "LCB = mean - zσ" is not the α-quantile - Better: Monte Carlo empirical quantile, or distribution-free bound (Cantelli)

E. Variance Propagation Ignores Nonlinearity¶

The κ toggle (only when ΔR < 0) makes the mapping nonlinear. If ΔR is uncertain near 0, treating κ as toggled by point estimate sign is mathematically inconsistent.

Recommendation: Incorporate P(ΔR < 0) or use smooth penalty.

F. KL Bin Alignment Essential¶

KL requires P and Q over the same discrete support. If histograms use different bin edges, zip(p, q) produces nonsense. This is a common implementation bug.

G. KL Threshold Calibration¶

Empirical KL between finite histograms is noisy. Without calibration (bootstrap under "no drift"), "KL > τ" has unknown false alarm rate.

H. Multiple Testing / Selection Bias¶

With many gates, overall false rejection/acceptance behavior is not obvious. Should reason about joint error if confidences are probabilistic.

I. Units/Scale Coherence¶

Utility combines profit/value/risk with weights (γ, κ, φ_S, φ_D). Mathematically valid only if on commensurate scales (all dollars or all normalized). Needs dimensional analysis.

GPT's Overall Assessment¶

Local coherence: Many individual formulas (Normal-Normal posterior update, logistic mapping, LCB form) are standard in isolation.

Global coherence is weaker because the system mixes: - Bayesian tail probabilities (clear probabilistic semantics) - Heuristic scoring functions (logistic novelty "confidence") - Deterministic hard constraints - Normal-approximate LCB using partial variance propagation

without a single consistent probabilistic or decision-theoretic interpretation tying them together.

Most Serious Mathematical Risks (Priority Order): 1. Overconfidence from posterior vs posterior predictive 2. Uncalibrated uncertainty propagation in utility 3. PERT variance confusion (unclear distribution definition) 4. KL drift implementation hazards (bin alignment, calibration)

Bottom Line: AEGIS is a reasonable heuristic multi-gate framework, but not yet mathematically "tight" as a probabilistic governance system.

Round 3: Claude Revision¶

Status: COMPLETE ✅ Timestamp: 2026-01-31

Acknowledged Corrections¶

I acknowledge GPT 5.2 Pro's corrections to my initial analysis:

1. PERT Variance: My numerical examples were incorrect. The true error range is ±22-40%, not 2×. I accept this correction.

2. Covariance Formula: I omitted the factor of 2 and additional covariance pairs. The complete formula GPT provided is correct.

3. Distribution Recommendations: My suggestion of "log-normal or beta" was underspecified for signed deltas. Student-t likelihood is a better recommendation for robustness.

Revised Summary of Findings¶

Based on the Claude-GPT dialogue, here is the consolidated assessment:

Critical Issues (Require Immediate Attention)¶

Significant Issues (Should Address)¶

Minor Issues (Nice to Have)¶

Revised Verification Status¶

Consensus Points (Claude + GPT Agreement)¶

Both models agree on:

1. Independence assumption is problematic - variance calculation ignores covariance
2. Default utility should fail closed - fail-open on missing is dangerous
3. KL divergence should be symmetric - Jensen-Shannon is better
4. Confidence aggregation is semantically flawed - mixing incomparable measures
5. Novelty gate is steep - small changes cross threshold
6. Override logic is sound - complexity non-override is consistent
7. Numerical stability is adequate - for governance thresholds

Disagreement Points (Resolved)¶

Appendix A: Mathematical Derivations¶

A.1 Bayesian Posterior for Normal-Normal¶

Given: - Prior: θ ~ N(μ₀, σ₀²) - Likelihood: x | θ ~ N(θ, σ_L²)

Posterior: θ | x ~ N(μ_post, σ_post²)

Where: - σ_post² = 1/(1/σ₀² + 1/σ_L²) - μ_post = σ_post² × (μ₀/σ₀² + x/σ_L²)

A.2 PERT Beta Distribution¶

The PERT beta distribution on [a, b] with mode m has: - α = 1 + 4(m-a)/(b-a) - β = 1 + 4(b-m)/(b-a)

Mean: μ = (a + 4m + b)/6 Variance: σ² = (b-a)² × αβ / ((α+β)²(α+β+1))

A.3 LCB Derivation¶

For U = Σ w_i X_i where X_i are independent: - E[U] = Σ w_i E[X_i] - Var(U) = Σ w_i² Var(X_i) (if independent)

LCB at confidence 1-α: - LCB = E[U] - z_α × √Var(U)

Where z_α = Φ⁻¹(1-α). For α=0.05, z_α ≈ 1.645.

Appendix B: Test Evidence¶

Change Log¶

Appendix C: Dialogue Metadata¶

Model Contributions¶

Actionable Outcomes¶

1. ✅ Priority 1: Fix posterior vs posterior predictive in Bayesian gates (NEW-A) - IMPLEMENTED v3.11.0
2. Added compute_posterior_predictive() method
3. Added use_predictive parameter to existing methods
4. See: ADR-006-posterior-predictive.md
5. ✅ Priority 2: Implement full covariance matrix in utility variance (U1+) - IMPLEMENTED v3.11.0
6. Added optional cov_pv, cov_pr, cov_vr parameters
7. Factor of 2 included per variance formula
8. ✅ Priority 3: Document PERT variance limitations or implement true beta (P1) - DOCUMENTED v3.11.0*
9. Enhanced docstring on ThreePointEstimate.variance
10. Provides true Beta-PERT variance formula for reference
11. ✅ Priority 4: Change default utility to fail-closed (I1) - IMPLEMENTED v3.11.0
12. Default lcb=float('-inf') ensures missing utility fails gate
13. Consider: ADR for statistical model improvements