Posture Dispersion (Z)

The framework

Most legal AI asks whether a machine can decide a case. Posture Dispersion asks a narrower question: how open is this question? The instrument, called Ariadne in the source paper, never rules. It measures whether a contested procedural disposition stays stable when the legal materials are held fixed and only the judge's posture changes, and it returns a contestability artifact for human review.

The method has nine components.

1. A fixed, verified substrate. Every reasoner receives identical legal materials, verified before any model scores the case: which authorities are binding, persuasive, limited or overruled; which propositions each supports; who bears which burden; which standard controls. If one run cites a live case, another invents one and a third forgets a controlling rule, the resulting spread is input failure, not openness.
2. A validity gate. Briefs resting on dead, invented or misattributed authority are halted or quarantined before any number is computed. Validity is reported per proposition, and no language model writes a citation or a validity verdict.
3. A declared panel of posture-conditioned reasoners. A posture is a stable decision filter over a shared list of moves: what a judge admits, how it ranks conflicting moves, and at what generality it pitches the controlling principle. At the trial level the paper conjectures four dispositional axes (see A-axes). Posture is not ideology and not a party preference. The panel is declared, versioned and held fixed.
4. Procedural gates on a constrained scale. Each posture scores its disposition at each gate (pleading screen, threshold exits, discovery, class certification, expert admissibility, summary judgment, trial, remedy) on an integer scale from 1 (let the claim through) to 10 (screen it out), with a one-line rationale. A posture may instead record decline-to-reach, because forcing scores at unreached gates manufactures dispersion.
5. Replicates and a noise floor. Each posture runs more than once. The pooled within-posture variance is the instrument's own noise floor.
6. Noise-corrected dispersion. The variance of posture means overstates the posture effect by the noise divided by the replicate count, so that amount is subtracted.
7. Normalization to [0, 1]. The corrected variance is divided by the largest variance any quantity on a width-9 scale can have (Popoviciu's bound), so Z can be computed for a single new motion with no reference corpus.
8. Abstention. If the between-posture signal does not clear the noise at a declared level, the instrument issues no Z.
9. Aggregation and output contract. Gate values combine by a weighted average whose weights can encode appellate durability, yielding an all-gate Z, a durable Z over deferentially reviewed gates, and a durable kill point. Output always includes the interval, noise floor, abstention status, panel declaration and rationale ledger. A bare score is a prohibited output.

In the plain-text notation the paper uses:

Model (gate g, posture p, replicate r):
  S(p, r, g) = mu(g) + alpha(p, g) + eps(p, r, g)

m postures, R replicates each:
  V_between(g) = sample variance of the m posture means   (divisor m - 1)
  V_within(g)  = pooled within-posture variance           (divisor m(R - 1))

Noise-corrected posture variance:
  sigma2_posture_hat(g) = V_between(g) - V_within(g) / R

Normalized openness (scores 1 to 10, so W = 9 and W^2 / 4 = 20.25):
  Z(g) = min(1, max(0, sigma2_posture_hat(g)) / 20.25)

Abstention test (report Z only if F exceeds the upper critical value):
  F(g) = R * V_between(g) / V_within(g),  df = (m - 1, m(R - 1))

Case level, weights w(g) >= 0:
  Z(c) = sum of w(g) * Z(g) / sum of w(g)

The reading rule. Low Z: the merits overdetermine the outcome, and the law decides. High Z: the judge's posture is the tie-breaker. Noise-level spread: the instrument abstains.

The formal properties are proved as five theorems: the correction is unbiased; Z is bounded by a tight normalizer; population Z is zero exactly when posture is predictively irrelevant at the gate; the aggregate is bounded and monotone; and abstention controls the false-openness rate. These are properties of the ruler, not facts about judges.

How to apply it

The paper walks one logged run end to end: an engineered New York Commercial Division motion to dismiss a fraudulent inducement claim, with invented parties and real authorities, built on four knife edges (duplication, present fact versus future promise, reliance by a sophisticated party, pleading particularity). It is a mechanics demonstration, not evidence about real dockets.

Five postures scored the pleading gate over byte-identical materials:

Step 1, between-posture variance. The scores 2, 2, 4, 7, 8 have mean 4.6 and unbiased sample variance 7.8.

Step 2, noise floor. Two control profiles ran three times each: a Median Judge scored 4, 4, 4 and a Before-Coffee Judge scored 5, 5, 4. The pooled floor is 0.167.

Step 3, correct honestly. The run deviated from protocol: panel postures ran once each, so their scores carry the full noise, not its R-th part. The paper therefore subtracts the whole floor: 7.8 - 0.167 = 7.633.

Step 4, normalize. 7.633 / 20.25 gives Z of approximately 0.38, slightly above the roughly 0.33 a uniform spread across the whole scale would produce.

Step 5, qualify. With five posture means, the interval is wide, roughly 0.14 to 1. The formal abstention test could not be run, because an unreplicated panel has no within-posture degrees of freedom. The honest report is "open gate, extent uncertain, protocol deviation disclosed."

For contrast, the paper's illustrative (not measured) numbers show abstention working: with 8 postures, 4 replicates, V_between 0.30 and V_within 0.50, Z is about 0.009 and F is 2.4, so the instrument stays silent. Raise V_between to 9.0 and Z is about 0.44 with F of 72: a confident open reading.

Known limitations and critiques

Persona collapse. The strongest objection is empirical: frontier models may behave as one formalist judge under different labels, so every panel converges and Z reads low for the wrong reason. Grounding can mask this, since identical personas over identical law also produce a clean low Z. The validation plan adds a clone discriminator: on cases experts rate as hard, the panel must still disperse above the noise floor.

Separability versus fidelity. Models that respect the substrate may collapse to formalism, while models that vary may introduce legal error. Only runs that demand both at once can test this.

Panel-relativity. The normalizer is absolute but the numerator is not. A panel of near-clones reads low; a panel stretched to the lawful corners reads high. A Z without its versioned panel declaration is rhetoric, not measurement.

What dispersion conflates. Z does not separate Hartian penumbra, discretion the law deliberately delegates, and ordinary disagreement among competent postures. Only the rationale ledger lets a reviewer see which is in play.

Small designs are weak. At five postures and three replicates, the power to detect a posture effect equal to the noise is only about one half, and reporting only on rejection inflates values near the threshold.

Substrate and selection. If the substrate is wrong, Z measures the openness of the wrong law. Litigated motions overrepresent uncertainty, because predictable disputes settle.

Not yet validated. The architecture is implemented, but whether the axes are separable, the substrate faithful at scale, and Z aligned with expert and docket judgments are six stated conjectures awaiting staged tests. The authors are affiliated with Legawrite.AI and disclose that interest; the axis names also vary between versions of the paper.