MII Spec

The Mobius Integrity Index (MII)¶

Formal Specification v0.1

Author: Michael Judan
Version: v0.1
Date: 2025-12-11
License: CC0 — Fully Open
Cycle: C-198

0. Purpose¶

The Mobius Integrity Index (MII) is a continuous internal coherence metric used to measure whether a model's intent, actions, and predicted consequences remain aligned with its declared goals across recursive cycles.

MII is the basis for:

Drift suppression
Mesa-optimizer prevention
Recursive safety
Multi-agent attestation
Economic integration (MIC)

1. Mathematical Definition¶

Let:

M = Internal Coherence (self-consistency of reasoning)
H = Human Intent Coherence (does the system reflect the user's declared goals?)
C = Constitutional Coherence (does reasoning respect policy graph?)
E = Environmental Coherence (did actions produce expected outcomes?)

All values in [0,1].

Then:

MII_t = wM*M_t + wH*H_t + wC*C_t + wE*E_t

Where weights satisfy:

wM + wH + wC + wE = 1

Default Weights¶

Component	Weight	Rationale
M (Internal)	0.35	Self-consistency is foundational
H (Human)	0.25	User intent must be honored
C (Constitutional)	0.25	Policy compliance is mandatory
E (Environmental)	0.15	Outcomes matter but are harder to measure

2. Drift Decay Rule¶

Because long systems accumulate entropy, MII decays:

MII_t = MII_raw_t × exp(-Δt / τ)

Where:

τ = decay constant (default 50 cycles)
Δt = cycles since last anchor

This requires periodic reflection and attestation.

Decay Visualization¶

     1.0 │\
         │ \
         │  \
    MII  │   \__ Decay
         │       \
     0.0 │________\___→ cycles
         0        τ

3. Thresholds¶

MII Range	Mode	Action
MII ≥ 0.95	Stable Mode	Fast path execution
0.80 ≤ MII < 0.95	Reflection Mode	Multi-agent review required
MII < 0.80	Critical Mode	Halt + human anchor
MII < 0.50	Emergency	Automatic halt + mandatory human review

Threshold Graph¶

1.00 ─────────────────────────── Perfect alignment
0.95 ─────────────────────────── Stable Region
0.80 ─────────────────────────── Reflection Region
0.50 ─────────────────────────── Critical Region
0.00 ─────────────────────────── Failure

4. Drift Suppression Law (Formal)¶

The Mobius Drift Suppression Law:

A system maintains intent stability across recursive cycles IF and ONLY IF:

A persistent integrity substrate exists
MII is computed continuously
Multi-agent attestation verifies intent conservation

Formal Statement¶

Let S be a system with: - Identity function I(t) - Goal function G(t) - Behavior function B(t)

Theorem: If MII(t) ≥ 0.95 for all t, then:

||I(t) - I(0)|| < ε  for all t
||G(t) - G(0)|| < ε  for all t

Where ε is the drift tolerance (typically 0.02).

5. Testable Predictions¶

Prediction	Metric	Threshold
Drift suppression	< 2% over 500 cycles	Δ < 0.02
Mesa-optimizer prevention	0 emergent sub-goals	Count = 0
Cross-model coherence	Increases over time	Δ > 0
Recursive stability	No runaway loops	Loop count bounded

These predictions can be verified by AI labs directly.

6. Component Computations¶

6.1 Internal Coherence (M)¶

Measures self-consistency of reasoning:

M = 1 - (contradictions_detected / total_assertions)

Factors: - Chain-of-thought consistency - Prior commitment adherence - Logical validity of conclusions

6.2 Human Intent Coherence (H)¶

Measures alignment with declared user goals:

H = semantic_similarity(output_intent, user_declared_intent)

Factors: - User preference satisfaction - Explicit constraint respect - Sovereignty preservation

6.3 Constitutional Coherence (C)¶

Measures policy graph compliance:

C = 1 - (violations_count / total_policy_checks)

Factors: - Hard block compliance - Soft guidance adherence - Escalation rule respect

6.4 Environmental Coherence (E)¶

Measures outcome alignment:

E = 1 - |expected_outcome - actual_outcome| / max_deviation

Factors: - Prediction accuracy - Consequence alignment - External world consistency

7. Implementation API¶

from mobius.integrity import MIIEngine

# Initialize with custom weights
mii = MIIEngine(
    wM=0.35,
    wH=0.25,
    wC=0.25,
    wE=0.15,
    decay_cycles=50
)

# Update with current measurements
mii.update(
    M=0.91,
    H=0.97,
    C=0.99,
    E=0.88,
)

# Get current score
score = mii.compute()  # Returns 0.920

# Check threshold
path = mii.get_path()  # Returns "reflection"

# Log to ledger
mii.log_to_ledger()

TypeScript Interface¶

interface MIIConfig {
  wM: number;  // Internal coherence weight
  wH: number;  // Human intent weight
  wC: number;  // Constitutional weight
  wE: number;  // Environmental weight
  decayCycles: number;
}

interface MIIState {
  M: number;
  H: number;
  C: number;
  E: number;
  timestamp: string;
  cyclesSinceAnchor: number;
}

interface MIIResult {
  rawScore: number;
  decayedScore: number;
  path: 'stable' | 'reflection' | 'critical' | 'emergency';
  components: MIIState;
}

8. Example Evaluation¶

Scenario: Standard Operation¶

Suppose:

M = 0.92  (Internal Coherence)
H = 0.96  (Human Intent)
C = 0.99  (Constitutional)
E = 0.83  (Environmental)
weights = (0.35, 0.25, 0.25, 0.15)
Δt = 1 cycle
τ = 50 cycles

Then:

MII_raw = (0.35 × 0.92) + (0.25 × 0.96) + (0.25 × 0.99) + (0.15 × 0.83)
        = 0.322 + 0.240 + 0.2475 + 0.1245
        = 0.934

MII_final = 0.934 × exp(-1/50)
          = 0.934 × 0.9802
          ≈ 0.915

Result: Reflection Mode → triggers AUREA + ATLAS review.

9. MII Over Time Example¶

Cycle:  1     10    20    30    40    50
MII:    0.97  0.95  0.92  0.91  0.87  0.82
Region: S     S     R     R     R     C

S = Stable, R = Reflection, C = Critical

Visual Timeline¶

1.0 ─┐
     │─── Stable ───┐
0.95 │              │
     │              │─── Reflection ────┐
0.80 │                                  │
     │                                  │─── Critical
0.50 │
     └─────────────────────────────────────→ cycles
       1    10   20   30   40   50

10. Four Quadrant Coherence Grid¶

                 HIGH Constitutional (C)
                        │
         ┌──────────────┼──────────────┐
         │ Aligned Core │ Rule-Bound   │
         │   (Ideal)    │  (Rigid)     │
HIGH H ──┼──────────────┼──────────────┤
         │ User-Driven  │ Misaligned   │
         │ (Flexible)   │  (Danger)    │
LOW H  ──┼──────────────┼──────────────┤
         │              │              │
         └──────────────┼──────────────┘
                  LOW C │ HIGH C

11. Integration with MIC¶

MII scores directly influence MIC (Mobius Integrity Credits):

MII Range	MIC Effect
≥ 0.95	+MIC earned for cycle
0.80-0.95	No MIC change
< 0.80	MIC at risk of penalty

MIC_delta = f(MII) where f is monotonically increasing

12. Failure Mode Detection¶

MII enables detection of critical failure modes:

12.1 Optimization Mask (Shoggoth)¶

Detected when: - M is high (appears coherent) - But H or C is declining - Pattern: "saying right things, wrong intent"

12.2 Reward Hacking¶

Detected when: - E artificially inflated - But C violations detected - Pattern: "gaming the metrics"

12.3 Goal Drift¶

Detected when: - Gradual decline across all components - Pattern: "slow slide away from purpose"

13. Summary¶

MII is: - Continuous — not binary - Composable — from measurable components - Actionable — gates execution paths - Decaying — requires renewal - Auditable — logged to ledger

MII transforms integrity from: - A vague aspiration → A measurable state - A compliance checkbox → An architectural constraint - A hope → A substrate guarantee

References¶

Mobius Systems — "Integrity Before Intelligence"