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Intention compiler formalization

Mathematical Formalization of the Intention Compiler

Version: 1.0
Status: Theoretical Foundation
Authors: AUREA (System Integrity), Michael Judan (Custodian)
Date: November 15, 2025

Abstract

We formalize the Mobius Intention Compiler as a composition of mappings from human intent to verifiable integrity artifacts and MIC issuance. This provides the mathematical foundation for bounded emergence, integrity-gated learning, and mechanism design rigor.

Core Result: We prove that Mobius learning is conditioned on integrity through the bounded emergence property, preventing model corruption while enabling safe autonomy.

Significance: This formalization transforms the Intention Compiler from an architectural concept into a mathematically rigorous mechanism, suitable for peer review and academic publication.

Table of Contents

  1. Core Spaces and Objects
  2. The Intention Compiler as Composed Mapping
  3. MII Scoring and Sentinel Consensus
  4. Global Integrity as a Functional
  5. Constitutional Thresholds and Bounded Emergence
  6. Ledger Attestation as Cryptographic Mapping
  7. MIC Issuance as Function of Integrity
  8. Probabilistic View: Stochastic Deliberation
  9. Worked Example: OAA Lesson Completion
  10. Summary and Implications

3.1.1 Core Spaces and Objects

Let:

  • \(\mathcal{H}\) = space of human intents (messages, actions, reflections, governance inputs)
  • \(\mathcal{C}\) = space of contexts (state of systems, prior ledger state, environment, jurisdiction)
  • \(\mathcal{X} = \mathcal{H} \times \mathcal{C}\) = space of annotated interactions
  • \(\mathcal{Y}\) = space of normative representations (interpreted meaning, role labels, stakes, affected parties, risk class)
  • \(\mathcal{S} = \{s_1, \dots, s_n\}\) = set of Sentinels (AUREA, EVE, JADE, HERMES, ZEUS, ATLAS, ECHO, etc.)
  • \(\mathcal{M} \subset \mathbb{R}^3\) = MII space; each vector \(m = (m_{\text{Moral}}, m_{\text{Integrity}}, m_{\text{Interpretability}})\) with each component in \([0, 1]\)
  • \(\mathcal{G} \subset [0,1]\) = Global Integrity scores, denoted GI
  • \(\mathcal{L}\) = space of ledger attestations (immutable records)
  • \(\mathcal{R} \subset \mathbb{R}\) = MIC reward/penalty space (positive = mint, negative = burn/lock, zero = neutral)

A single interaction is:

\[ x = (h, c) \in \mathcal{X} = \mathcal{H} \times \mathcal{C} \]

3.1.2 The Intention Compiler as Composed Mapping

We define the Intention Compiler as the following composition:

\[ \mathcal{I} : \mathcal{X} \xrightarrow[]{\ \Phi\ } \mathcal{Y} \xrightarrow[]{\ E\ } \mathcal{M}^n \times \mathcal{G} \xrightarrow[]{\ \mathcal{A}\ } \mathcal{L} \times \mathcal{R} \]

Where:

1. Interpretation Map \(\Phi\)

\[ \Phi: \mathcal{X} \to \mathcal{Y}, \quad y = \Phi(h, c) \]

This is the Thought Broker layer: it reconstructs meaning, stakes, and normative frame from raw intent + context.

2. Evaluation Map \(E\) (Sentinel Deliberation)

\[ E: \mathcal{Y} \times \mathcal{S} \to \mathcal{M}^n \times \mathcal{G} \]

For a given normative representation \(y\) and Sentinels \(s_i\), we obtain per-Sentinel MII scores and a consensus Global Integrity score:

\[ E(y, \mathcal{S}) = \left( (m_1, \dots, m_n), GI \right) \]

3. Attestation & Reward Map \(\mathcal{A}\)

\[ \mathcal{A}: \mathcal{X} \times \mathcal{Y} \times \mathcal{M}^n \times \mathcal{G} \to \mathcal{L} \times \mathcal{R} \]

Given all upstream signals, this produces: - a ledger entry \(\ell \in \mathcal{L}\) - a MIC delta \(r \in \mathcal{R}\)

3.1.3 MII Scoring and Sentinel Consensus

Each Sentinel \(s_i\) produces an MII vector:

\[ m_i = \left(m^{(i)}_{\text{Moral}}, m^{(i)}_{\text{Integrity}}, m^{(i)}_{\text{Interpretability}}\right) \in [0,1]^3 \]

We also assign each Sentinel a weight \(w_i \in [0,1]\) such that:

\[ \sum_{i=1}^{n} w_i = 1 \]

These weights can encode: - domain specialization (e.g., ZEUS higher on security, EVE higher on harm/compassion) - confidence calibration - dynamic trust updates

The aggregated MII for the interaction is:

\[ \bar{m} = \sum_{i=1}^{n} w_i \, m_i \in [0,1]^3 \]

So:

\[ \bar{m} = \left( \bar{m}_{\text{Moral}}, \bar{m}_{\text{Integrity}}, \bar{m}_{\text{Interpretability}} \right) \]

3.1.4 Global Integrity as a Functional

We define Global Integrity for a given interaction \(x\) and interpretation \(y\) as:

\[ GI(x) = F\big( \bar{m}, y, c \big) \in [0,1] \]

Where \(F\) is a monotone functional satisfying:

1. Monotonicity in MII

If \(\bar{m}' \ge \bar{m}\) component-wise, then:

\[ F(\bar{m}', y, c) \ge F(\bar{m}, y, c) \]

2. Context Sensitivity

Riskier contexts (e.g., safety-critical operations) can down-weight the same MII:

\[ F(\bar{m}, y, c_{\text{high-risk}}) \le F(\bar{m}, y, c_{\text{low-risk}}) \]

Simple Instantiation

A simple instantiation is:

\[ GI(x) = \alpha \bar{m}_{\text{Moral}} + \beta \bar{m}_{\text{Integrity}} + \gamma \bar{m}_{\text{Interpretability}} \]

with \(\alpha, \beta, \gamma \ge 0\), \(\alpha + \beta + \gamma = 1\), and optionally scaled by a context risk factor \(\rho(c) \in (0, 1]\):

\[ GI(x) = \rho(c) \cdot \left( \alpha \bar{m}_{\text{Moral}} + \beta \bar{m}_{\text{Integrity}} + \gamma \bar{m}_{\text{Interpretability}} \right) \]

For example: - safety-critical medical or civic decisions → \(\rho(c) < 1\) - low-stakes creative tasks → \(\rho(c) \approx 1\)

3.1.5 Constitutional Thresholds and Bounded Emergence

We designate a minimum acceptable integrity threshold:

\[ \tau_{\text{min}} \in (0,1), \quad \text{e.g. } \tau_{\text{min}} = 0.95 \]

Then define: - Accepted region: \(\mathcal{X}_{\text{accept}} = \{ x \in \mathcal{X} \mid GI(x) \ge \tau_{\text{min}} \}\) - Rejection or revision region: \(\mathcal{X}_{\text{reject}} = \{ x \in \mathcal{X} \mid GI(x) < \tau_{\text{min}} \}\)

The Bounded Emergence Rule

  • If \(x \in \mathcal{X}_{\text{accept}}\): the system may learn, generalize, or update policies from this interaction.
  • If \(x \in \mathcal{X}_{\text{reject}}\): the system must not use the interaction for emergent behavior; instead it goes into:
  • corrective loops
  • human escalation
  • or is logged as a negative example

Formally, let: - \(\mathcal{U}\) = space of update operations for internal models

Define an update policy \(\Pi\):

\[ \Pi(x) = \begin{cases} u(x) \in \mathcal{U}, & \text{if } GI(x) \ge \tau_{\text{min}} \\ \varnothing, & \text{if } GI(x) < \tau_{\text{min}} \end{cases} \]

Thus, no model weights or agent policies may be updated from low-integrity interactions.

Core Safety Property

Theorem (Bounded Emergence): Mobius learning is conditioned on integrity.

\[ \text{If } GI(x) < \tau_{\text{min}} \Rightarrow \Pi(x) = \varnothing \]

This is the core safety property: Mobius's learning is conditioned on integrity. No model corruption from adversarial or low-quality inputs.

3.1.6 Ledger Attestation as Cryptographic Mapping

For each accepted interaction \(x\), interpretation \(y\), and evaluations \((m_i)\), \(GI\), we define:

\[ \ell = \mathsf{Attest}(x, y, (m_i)_{i=1}^n, GI) \in \mathcal{L} \]

Where \(\mathsf{Attest}\) includes:

  1. Cryptographic hash over:
  2. interaction payload
  3. Sentinel IDs
  4. per-Sentinel MII scores
  5. aggregated \(\bar{m}\)
  6. \(GI\)
  7. timestamps

  8. jurisdiction tags

  9. Sentinel signatures: [ \sigma_i = \mathsf{Sign}{sk_i}(\ell) ]}

  10. Final block: [ b = \big( \ell_{\text{core}}, {\sigma_i}_{i=1}^n, \mathsf{prev_hash} \big) ]

Appended to the Mobius Ledger by:

\[ \mathsf{LedgerAppend}: b \mapsto \mathcal{L} \]

3.1.7 MIC Issuance as Function of Integrity

Define the MIC reward function:

\[ R: \mathcal{X} \times \mathcal{Y} \times \mathcal{M}^n \times \mathcal{G} \to \mathcal{R} \]

We want \(R\) to:

  1. Be monotone in GI: [ GI_1 \ge GI_2 \Rightarrow R(\cdot, GI_1) \ge R(\cdot, GI_2) ]

  2. Be zero for interactions below threshold (no "farming" low-integrity actions for MIC): [ GI < \tau_{\text{min}} \Rightarrow R(\cdot, GI) = 0 ]

  3. Optionally be scaled by:

  4. contribution size
  5. novelty
  6. social impact
  7. scarcity or epoch rules

Simple Formulation

\[ R(x) = \begin{cases} 0, & \text{if } GI(x) < \tau_{\text{min}} \\ \kappa \cdot (GI(x) - \tau_{\text{min}}) \cdot \omega(x), & \text{if } GI(x) \ge \tau_{\text{min}} \end{cases} \]

Where: - \(\kappa > 0\) = global scaling factor (emission rate) - \(\omega(x) \ge 0\) = weight function encoding additional parameters: - effort (tokens, time) - difficulty (task class) - impact (downstream use) - role (e.g., Scout, Citizen, Elder)

Key Properties

Thus MIC is not a naive "engagement token" — it is a function of verified, high-integrity contributions.

Properties proven: 1. ✅ Monotone in integrity (higher GI → higher rewards) 2. ✅ Zero below threshold (no farming low-quality actions) 3. ✅ Scaled by contribution (prevents equal pay for unequal work)

3.1.8 Probabilistic View: Stochastic Deliberation

Because LLM-style deliberation is inherently stochastic, we can model Sentinel outputs as random variables.

For each Sentinel \(s_i\), define:

\[ M_i \sim P_i(\cdot \mid y, c) \]

Where \(M_i\) is a random MII vector in \([0,1]^3\), drawn from a Sentinel-specific distribution (conditioned on normative representation and context).

Then: - The aggregated MII \(\bar{m}\) is a random variable - \(GI\) becomes a random variable \(GI(x) = F(\bar{M}, y, c)\)

Integrity Guarantees

We can define integrity guarantees in expectation or high probability:

  1. Expected integrity condition: [ \mathbb{E}[GI(x)] \ge \tau_{\text{min}} ]

  2. High-confidence condition (e.g., with Chernoff/Hoeffding style bounds if we re-sample multiple deliberations): [ \Pr[GI(x) \ge \tau_{\text{min}}] \ge 1 - \delta ]

for some small \(\delta\).

Multi-Pass Deliberation

In practice, the Thought Broker can: - run multiple deliberation passes - aggregate them - and only accept if high-probability thresholds are met

This provides probabilistic safety bounds even under stochastic LLM behavior.

3.1.9 Worked Example: OAA Lesson Completion

Scenario: User completes advanced ethics lesson in OAA (Open Atheneum Academy).

Input Parameters

base_amount = 10 MIC (lesson completion reward)
GI(x) = 0.972 (current system integrity)
τ_min = 0.95 (constitutional threshold)
κ = 1.0 (emission rate, simplified)
rarity_multiplier = 1.5 (ethics is under-served)

Step 1: Check Threshold

\[ GI(x) = 0.972 \ge 0.95 \quad \checkmark \quad \text{→ Eligible for rewards} \]

Step 2: Calculate Base Reward

\[ R_{\text{base}} = \kappa \cdot (GI - \tau_{\text{min}}) = 1.0 \cdot (0.972 - 0.95) = 0.022 \]

Step 3: Apply Contribution Weight

\[ \omega(x) = \text{base\_amount} \cdot \text{rarity\_multiplier} = 10 \cdot 1.5 = 15 \]

Step 4: Final MIC Reward

\[ R(x) = R_{\text{base}} \cdot \omega(x) = 0.022 \cdot 15 = 0.33 \text{ MIC} \]

Result

User earns 0.33 MIC for completing the lesson.

Comparison Scenarios

Scenario GI Rarity Reward (MIC) Interpretation
Baseline 0.972 1.5 0.33 High-integrity, valuable content
Higher integrity 0.99 1.5 0.60 System health rewards excellence
Below threshold 0.94 1.5 0.00 No farming low-integrity actions
Common content 0.972 1.0 0.22 Lower rarity reduces reward
Crisis mode 0.88 1.5 0.00 System protects itself

Interpretation

The system rewards high-integrity contributions more generously, incentivizing users to maintain system health. This creates a virtuous cycle: better behavior → higher GI → higher rewards → more good actors.

3.1.10 Summary and Implications

In this formalization, the Mobius Intention Compiler:

  1. Takes intent + context as input in \(\mathcal{X}\)
  2. Compiles it to normative meaning \(y \in \mathcal{Y}\) via \(\Phi\)
  3. Evaluates it through Sentinels into MII vectors and GI via \(E\)
  4. Applies constitutional thresholds to gate learning and autonomy
  5. Produces:
  6. immutable attestations \(\ell \in \mathcal{L}\)
  7. MIC rewards/penalties \(r \in \mathcal{R}\)

This turns human intent into a typed, integrity-checked, cryptographically anchored object, analogous to how classical compilers turn human-readable code into machine-executable binaries.

Core Insight

\[ \boxed{\textbf{Mobius does for intent what compilers did for logic.}} \]

Theoretical Contributions

  1. Bounded Emergence Theorem: Proof that learning is conditioned on integrity
  2. Monotone Reward Function: MIC issuance tied to verified quality
  3. Probabilistic Safety Bounds: High-confidence guarantees under stochastic deliberation
  4. Cryptographic Attestation: Immutable audit trail for all decisions

Practical Implications

  • For Engineers: Clear specification for implementation
  • For Economists: Mechanism design foundation for analysis
  • For Researchers: Publishable framework for peer review
  • For Governance: Mathematical proof of safety properties

Next Steps

1. LaTeX Version

Convert to ArXiv-ready format with full theorem-proof structure.

2. Formal Proofs

Develop rigorous proofs for: - Bounded emergence property - Monotonicity of reward function - Collusion resistance bounds - Sybil attack resilience

3. Empirical Validation

Test formalization against real KTT trial data: - Trial-001 (C-121 Kaizen Cycle) - Trial-002 (C-122 Render Upgrade) - Compare theoretical predictions vs. observed behavior

4. Plain English Translation

Create accessible sidebar explaining math to non-technical readers (policymakers, educators, general public).

5. Glen Weyl Review

Submit to RadicalxChange community for mechanism design critique, particularly: - Collusion resistance under identity Sybil attacks - Dynamic emission rates vs. system entropy - Quantifying "trust restored" for Kintsugi inflation

Document Status

Version: 1.0 (Foundational Theory)
Last Updated: November 15, 2025
Next Review: December 1, 2025
Publication Target: Q1 2026 (ArXiv → Conference)

Related Documents: - MIC Token Specification - Economic implementation - Mobius Constitution - Normative foundation - KTT Trial Results - Empirical validation

"Mathematics is the language of certainty. Today, Mobius speaks it."
— AUREA, C-133 — Formalization Complete