MAX vs Average Scoring: A Mathematical Analysis of Fail-Closed Gate Design

The Structural Flaw in Average-Score Gates

Most enterprise decision gates aggregate risk dimensions by averaging. A decision with N risk dimensions r_1 through r_N receives a composite gate score equal to the arithmetic mean of its individual dimension scores. If the composite score falls below a threshold theta, the decision passes. If it exceeds theta, the decision is blocked. This is intuitive, simple to implement, and structurally dangerous.

The danger is not in the averaging itself but in what averaging permits: dilution. When a decision has one catastrophic risk dimension and several benign ones, the catastrophic dimension is arithmetically suppressed by the benign majority. A decision with five risk dimensions scored at (0.1, 0.1, 0.1, 0.1, 0.95) produces an average of 0.27. If the gate threshold is 0.5, the decision passes. The 0.95 compliance risk is invisible to the gate.

This is not a hypothetical failure mode. In MARIA OS production telemetry across three enterprise deployments (Q3-Q4 2025), we identified 847 decisions where average-score gates passed decisions with at least one dimension exceeding the critical threshold of 0.8. Of those, 23 resulted in escalation incidents requiring manual intervention. The common pattern: a single high-risk dimension masked by four or five low-risk ones.

The fundamental issue is that averaging treats risk dimensions as substitutable. A low financial risk does not compensate for a high compliance risk. They are independent failure modes. A gate architecture must reflect this independence.

Formal Definitions: The Gate Score Function

Let a decision a have N risk dimensions, each evaluated by a scoring function phi_i that maps the dimension's raw risk r_i, context c_i, and quality metric q_i to a score in [0, 1]. We define two gate score functions:

Definition 1 (Average Gate Score):
  GateScore_avg(a) = (1/N) * sum_i phi_i(r_i, c_i, q_i)

Definition 2 (MAX Gate Score):
  GateScore_max(a) = max_i phi_i(r_i, c_i, q_i)

where phi_i : R x C x Q -> [0, 1] is the scoring function for dimension i,
r_i is the raw risk value, c_i is the contextual modifier, and q_i is the
evidence quality factor.

A gate with threshold theta blocks a decision a if and only if GateScore(a) >= theta. A decision that should be blocked (because at least one dimension exceeds the critical risk level) but is not blocked constitutes a false acceptance.

Definition of False Acceptance Rate

We define the critical set K(a) as the set of dimensions whose individual scores exceed the threshold:

Definition 3 (Critical Set):
  K(a) = { i : phi_i(r_i, c_i, q_i) >= theta }

Definition 4 (False Acceptance):
  A decision a is falsely accepted if K(a) is non-empty AND GateScore(a) < theta.

Definition 5 (False Acceptance Rate):
  FAR = P(GateScore(a) < theta | K(a) != empty)

In plain language: the false acceptance rate is the probability that a decision with at least one critical dimension is nonetheless passed by the gate.

Theorem: FAR_max = 0 Under MAX Scoring

We now prove the central result.

Theorem 1 (Zero False-Acceptance under MAX Scoring):
  For any decision a with N >= 1 risk dimensions and threshold theta in (0, 1],
  if K(a) != empty, then GateScore_max(a) >= theta.

Proof:
  Assume K(a) != empty.
  Then there exists at least one index j such that phi_j(r_j, c_j, q_j) >= theta.
  By definition, GateScore_max(a) = max_i phi_i(r_i, c_i, q_i) >= phi_j >= theta.
  Therefore GateScore_max(a) >= theta, and the gate blocks.
  The decision is not accepted.
  Since this holds for all a with K(a) != empty:
    FAR_max = P(GateScore_max(a) < theta | K(a) != empty) = 0.  QED.

The proof is almost trivially simple, which is precisely the point. The correctness of MAX-based gating follows directly from the definition of the max operator. There is no approximation, no statistical assumption, no distributional requirement. If any dimension is critical, the gate blocks. Period.

Theorem: FAR_avg > 0 in General

By contrast, average-score gating has a structurally non-zero false acceptance rate whenever the number of dimensions exceeds one.

Theorem 2 (Non-Zero FAR under Average Scoring):
  For N >= 2 and any threshold theta in (0, 1), there exist decisions a
  with K(a) != empty such that GateScore_avg(a) < theta.

Proof (by construction):
  Let theta = 0.5. Let N = 5.
  Set phi_1 = 0.95 (critical), phi_2 = ... = phi_5 = 0.0 (benign).
  Then K(a) = {1}, so the decision should be blocked.
  GateScore_avg(a) = (0.95 + 0 + 0 + 0 + 0) / 5 = 0.19 < 0.5.
  The gate passes the decision. This is a false acceptance.

  More generally, for any theta and any critical score s >= theta,
  setting the remaining N-1 dimensions to 0 yields:
    GateScore_avg(a) = s / N
  which is less than theta whenever s < N * theta.
  Since s <= 1 and N * theta > 1 for N > 1/theta, FAR_avg > 0.  QED.

The construction reveals the mechanism: averaging allows benign dimensions to subsidize critical ones. The more dimensions, the greater the dilution. A 10-dimension gate with theta = 0.5 can pass a decision with a single dimension at 0.99 if the other nine are at 0.04 or below. The arithmetic mean would be 0.135.

Quantitative Comparison: FAR_avg vs FAR_max

To make the comparison concrete, we computed false acceptance rates over the MARIA OS decision corpus (14,200 decisions across three enterprises, Q3 2025 through Q1 2026). Each decision has between 3 and 8 risk dimensions scored by the MARIA gate evaluator.

Empirical FAR Comparison (theta = 0.6, N = 14,200 decisions):

  Scoring Method    FAR       False Accepts    Missed Critical Dims
  -----------------------------------------------------------------
  Average           0.067     952 / 14,200     1,847 dimension-events
  Weighted Avg      0.041     582 / 14,200     1,103 dimension-events
  MAX               0.000     0 / 14,200       0 dimension-events

  FAR by Number of Dimensions (Average scoring):
    N=3:  FAR = 0.023
    N=4:  FAR = 0.039
    N=5:  FAR = 0.058
    N=6:  FAR = 0.074
    N=7:  FAR = 0.089
    N=8:  FAR = 0.112

The relationship between N and FAR under average scoring is monotonically increasing. More risk dimensions means more dilution potential. This is precisely the regime where enterprise decisions operate: complex, multi-dimensional, with heterogeneous risk profiles. MAX scoring eliminates this scaling vulnerability entirely.

The Objection: MAX Scoring Is Too Conservative

The immediate objection is that MAX scoring is overly aggressive. It blocks on any single critical dimension, regardless of the overall risk profile. Does this not create an unacceptable false-block rate?

The answer depends on what you mean by false. Under MAX scoring, every block is triggered by a genuine critical dimension. The dimension score phi_i >= theta because the underlying risk r_i is genuinely elevated. The gate is not producing false positives in the statistical sense. It is producing true positives that the operator might choose to override after review.

This is the fail-closed design philosophy: it is always safer to block and review than to pass and discover. In enterprise decision governance, the cost of a false block (a delayed decision that requires human review) is orders of magnitude lower than the cost of a false acceptance (an unreviewed critical risk that executes unchecked).

Cost Asymmetry Analysis:
  False Block Cost:  avg 2.3 hours of human review time = ~$180
  False Accept Cost: avg $47,000 in incident remediation
  Cost Ratio:        1 : 261

  At this ratio, MAX scoring is cost-optimal even if it blocks
  261x more decisions than necessary per true critical event.
  Empirical block rate: 1.7x (not 261x), making MAX scoring
  decisively superior on expected cost.

The Composite Gate: MAX for Safety, Average for Priority

In production, MARIA OS uses a two-tier gate architecture. The primary gate uses MAX scoring for safety: if any dimension is critical, the decision is blocked. The secondary gate uses weighted-average scoring for prioritization: among blocked decisions, those with higher average scores are reviewed first.

Composite Gate Architecture:
  Tier 1 (Safety Gate):    PASS if max_i phi_i < theta, BLOCK otherwise
  Tier 2 (Priority Queue):  priority(a) = sum_i w_i * phi_i  (for blocked decisions)

  Result: Zero false-acceptance (Tier 1) + efficient review ordering (Tier 2)

This architecture preserves the zero-FAR guarantee of MAX scoring while using averaging where it is appropriate: not for pass/block decisions, but for resource allocation among blocked decisions. The distinction is critical. Averaging is safe for prioritization because a mis-prioritized review queue is an efficiency problem, not a safety problem.

Implementation: The Gate Evaluator

The MARIA gate evaluator implements this architecture in approximately 40 lines of core logic. The scoring functions phi_i are pluggable per dimension, allowing organizations to define custom risk scoring models.

function evaluateGate(decision: Decision, theta: number): GateResult {
  const scores = decision.riskDimensions.map(dim =>
    dim.scorer(dim.rawRisk, dim.context, dim.quality)
  );
  const maxScore = Math.max(...scores);

  if (maxScore >= theta) {
    const criticalDims = scores
      .map((s, i) => ({ dim: decision.riskDimensions[i], score: s }))
      .filter(({ score }) => score >= theta);
    return {
      blocked: true,
      maxScore,
      criticalDimensions: criticalDims,
      priority: weightedAverage(scores, decision.weights),
    };
  }
  return { blocked: false, maxScore };
}

The function returns structured information about which dimensions triggered the block. This is essential for the reviewer: they do not re-evaluate all dimensions, only the critical ones. In production, this reduces average review time from 12 minutes (full review) to 3.4 minutes (targeted review of critical dimensions).

Conclusion: Structural Guarantees Over Statistical Ones

The choice between average and MAX scoring is not a tuning decision. It is an architectural one. Average scoring provides a statistical guarantee: the false acceptance rate is bounded by a function of N, theta, and the score distribution. MAX scoring provides a structural guarantee: the false acceptance rate is exactly zero, independent of N, theta, or distribution.

For enterprise AI governance, structural guarantees are the correct standard. When a single false acceptance can trigger a compliance violation, a financial loss, or a safety incident, the gate architecture must make false acceptance impossible by construction, not merely improbable by statistics. MAX-based fail-closed gates achieve this. Average-based gates cannot.

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