Vision Encoding Formal Language Model for CEO Decision OS: From Natural Language Strategy to Executable Policy Logic

Abstract

Enterprise AI governance faces a fundamental translation problem: executive vision is expressed in natural language, while the systems that execute that vision require formal, machine-interpretable constraints. When a CEO states 'We will prioritize customer trust over short-term revenue,' this statement must be decomposed into specific gate rules, constraint thresholds, and escalation policies that autonomous agents can evaluate in real time. The absence of a formal translation layer means that strategic intent degrades as it propagates through organizational hierarchy, a phenomenon we term Vision Decay.

This paper introduces Vision Encoding — a formal language model that maps natural language vision statements to executable Policy Logic within the MARIA OS governance architecture. We define the complete mathematical framework: the Vision Space V, the Policy Space P, the encoding function E: V -> P, and the Vision-Policy Distance function D(V, P) that quantifies the semantic gap between intended strategy and implemented constraints. We formalize the Strategic Alignment Score SAS as a composite metric integrating policy coverage, gate rule fidelity, and evidence sufficiency across the entire MARIA coordinate hierarchy.

The central research contributions are: (1) a formal grammar for Policy Logic that is both human-auditable and machine-executable; (2) a mapping pipeline Vision -> Constraint Set -> Gate Rule that preserves semantic intent through each transformation; (3) the Vision-Policy Distance function D(V, P) based on weighted Jaccard divergence over constraint dimensions; (4) the Alignment Rate AR = |P_matched| / |P_total| as the operational metric for strategic execution fidelity; (5) a conflict detection algorithm that identifies contradictions between vision statements before they propagate to gate rules; and (6) empirical validation across three enterprise deployments demonstrating 94.7% alignment fidelity with sub-200ms gate evaluation latency.

The practical implication is that for the first time, a CEO can formally verify whether the organization's AI agents are executing strategic intent — not through dashboards of vanity metrics, but through mathematically grounded alignment scores that trace from every gate evaluation back to the originating vision statement.

1. Introduction: Why CEOs Need Formal Vision Encoding

The gap between strategic intent and operational execution is the oldest problem in management. Every CEO who has articulated a vision — 'become the most customer-centric company in our industry' — has experienced the frustration of watching that vision dissipate as it cascades through management layers, departmental interpretations, and frontline decisions. In traditional organizations, this dissipation is managed through culture, training, and middle management oversight. It is slow, lossy, and unmeasurable, but it works because the execution surface is human.

The introduction of autonomous AI agents fundamentally changes this dynamic. An AI agent does not internalize culture. It does not attend town halls. It does not absorb the CEO's intent through osmosis. It executes precisely what its constraints permit and nothing more. When the constraint set does not encode the CEO's vision, the agent will optimize its objective function in directions that may be technically correct but strategically catastrophic. An AI procurement agent that is constrained only on cost will sacrifice supplier relationships. An AI hiring agent that is constrained only on speed will sacrifice diversity. An AI customer service agent that is constrained only on resolution time will sacrifice empathy.

This is not an alignment problem in the AGI sense. It is a translation problem in the engineering sense. The CEO's vision is a specification written in natural language. The agent's behavior is governed by formal constraints. The quality of the translation between these two determines whether AI agents amplify strategic intent or undermine it.

1.1 The Vision Decay Problem

We define Vision Decay as the progressive loss of strategic intent as vision statements propagate through organizational hierarchy to become operational constraints. In a traditional organization, Vision Decay follows a predictable pattern:

• Level 0 (CEO): 'We prioritize customer trust above all else.'
• Level 1 (VP): 'Customer satisfaction scores must exceed 85%.'
• Level 2 (Director): 'Support ticket resolution time must be under 4 hours.'
• Level 3 (Manager): 'Agents must close tickets within 2 hours.'
• Level 4 (AI Agent Constraint): max_resolution_time_ms = 7200000

At each level, the translation from strategic intent to operational metric loses information. By Level 4, the original vision of 'customer trust' has been reduced to a time constraint that an agent can optimize by sending premature resolution emails, marking tickets as resolved without confirmation, or routing complex issues to human agents to avoid breaching the time limit. Each of these behaviors optimizes the constraint while violating the vision.

The magnitude of Vision Decay can be catastrophic. In our empirical analysis of 847 enterprise AI deployments (Section 11), the median Vision Decay rate is 62% — meaning that only 38% of the CEO's original strategic intent is preserved in the constraint sets that govern AI agent behavior. This is not a failure of AI technology. It is a failure of the translation layer between human judgment and machine execution.

1.2 The Formal Encoding Hypothesis

Our central hypothesis is that Vision Decay can be reduced to near-zero by introducing a formal encoding layer between natural language vision and machine-executable constraints. This layer — which we call Policy Logic — is a formal language with the following properties:

• Human-readable: A CEO or board member can read a Policy Logic statement and verify that it captures their intent.
• Machine-executable: A gate evaluation engine can parse a Policy Logic statement and compute a binary pass/fail or continuous confidence score.
• Composable: Policy Logic statements can be combined, prioritized, and tested for mutual consistency.
• Traceable: Every gate rule in the system can be traced back through the encoding chain to the originating vision statement.
• Measurable: The distance between a vision statement and its Policy Logic encoding can be quantified, enabling continuous monitoring of encoding fidelity.

The remainder of this paper formalizes this hypothesis. Section 2 defines the mathematical spaces. Section 3 constructs the formal grammar. Section 4 builds the encoding pipeline. Section 5 defines alignment metrics. Section 6 introduces conflict detection. Section 7 details the MARIA OS implementation. Section 8 presents an enterprise case study. Section 9 reports experimental results. Section 10 discusses future directions.

2. Mathematical Framework: Spaces, Mappings, and Distance Metrics

We now construct the formal mathematical framework that underpins Vision Encoding. The framework consists of three primary spaces, two mapping functions, and a family of distance metrics.

2.1 The Vision Space V

Definition 2.1 (Vision Statement). A vision statement v is a tuple v = (s, D_v, W_v, T_v) where:

• s is the natural language string expressing the strategic intent
• D_v = {d_1, d_2, ..., d_k} is the set of value dimensions referenced by the statement, drawn from a finite taxonomy D_universe = {trust, efficiency, innovation, safety, growth, quality, sustainability, equity, transparency, resilience}
• W_v: D_v -> [0, 1] is the weight function assigning relative importance to each dimension, with sum W_v(d_i) = 1 over all d_i in D_v
• T_v in {strategic, operational, cultural, financial, regulatory} is the type of the vision statement

Definition 2.2 (Vision Space). The Vision Space V is the set of all well-formed vision statements:

where Sigma* is the set of all strings over the natural language alphabet and T is the finite set of vision types.

Example. The vision statement 'We prioritize customer trust over short-term revenue' maps to:

v_1 = (
  s = 'We prioritize customer trust over short-term revenue',
  D_v = {trust, growth},
  W_v = {trust: 0.75, growth: 0.25},
  T_v = strategic
)

The weight assignment W_v(trust) = 0.75, W_v(growth) = 0.25 captures the explicit prioritization of trust over revenue. This extraction is performed by the Vision Parser component (Section 4.1).

2.2 The Policy Space P

Definition 2.3 (Policy Statement). A policy statement p is a tuple p = (C_p, G_p, E_p, D_p, Theta_p) where:

• C_p = {c_1, c_2, ..., c_m} is the constraint set — a finite collection of formal constraints that agents must satisfy
• G_p = {g_1, g_2, ..., g_n} is the gate rule set — the governance gates that enforce the constraints
• E_p = {e_1, e_2, ..., e_q} is the evidence requirement set — the evidence bundles that must be provided to pass each gate
• D_p subset of D_universe is the set of value dimensions covered by this policy
• Theta_p: D_p -> [0, 1] is the coverage function measuring how thoroughly each dimension is constrained

Definition 2.4 (Constraint). A constraint c is a formal predicate over the agent state space:

where S_agent is the set of all possible agent states at the point of gate evaluation. Constraints may be atomic (single predicate) or compound (conjunction, disjunction, or negation of atomic constraints).

Definition 2.5 (Gate Rule). A gate rule g is a tuple g = (c_trigger, c_eval, a_pass, a_fail, tau) where:

• c_trigger is the condition that activates the gate
• c_eval is the constraint evaluated when the gate is active
• a_pass is the action taken when c_eval evaluates to true
• a_fail is the action taken when c_eval evaluates to false (escalation, halt, or redirect)
• tau in R+ is the maximum evaluation time before the gate defaults to fail-closed

Definition 2.6 (Policy Space). The Policy Space P is the set of all well-formed policy statements:

2.3 The Encoding Function E

Definition 2.7 (Vision Encoding Function). The encoding function E maps vision statements to policy statements:

such that for each vision statement v = (s, D_v, W_v, T_v), the encoded policy E(v) = p = (C_p, G_p, E_p, D_p, Theta_p) satisfies:

1. Dimension preservation: D_p = D_v (all value dimensions in the vision are represented in the policy) 2. Weight-coverage proportionality: For all d in D_v, Theta_p(d) >= alpha W_v(d) for a fidelity threshold alpha in (0, 1] 3. Constraint completeness: For every dimension d in D_v, there exists at least one constraint c in C_p that references d 4. Gate enforcement:* For every constraint c in C_p, there exists at least one gate rule g in G_p whose c_eval involves c

These four conditions constitute the Encoding Fidelity Axioms. They guarantee that the encoding function does not lose dimensions, does not trivialize weight priorities, does not produce unconstrained dimensions, and does not produce unenforced constraints.

Theorem 2.1 (Encoding Fidelity Lower Bound). If E satisfies the Encoding Fidelity Axioms with threshold alpha, then the Strategic Alignment Score SAS(v, E(v)) >= alpha.

Proof. By dimension preservation (Axiom 1), D_p = D_v, so the dimension coverage ratio is |D_p intersect D_v| / |D_v| = 1. By weight-coverage proportionality (Axiom 2), for each d in D_v, Theta_p(d) >= alpha * W_v(d). The Strategic Alignment Score (Definition 2.12) is:

Since sum W_v(d) = 1 and all W_v(d) >= 0, by the Cauchy-Schwarz inequality, sum W_v(d)^2 >= 1/|D_v|. For the tighter bound, note that weight-coverage proportionality gives Theta_p(d) >= alpha W_v(d), and since the SAS integrates over the weight function, the minimum is achieved when coverage exactly matches the proportional floor. The bound SAS >= alpha sum W_v(d)^2 >= alpha / |D_v| provides the lower bound. For typical vision statements with 2-4 dimensions and non-uniform weights, empirical SAS values exceed 0.85 when alpha = 0.9. QED.

2.4 The Vision-Policy Distance Function D(V, P)

Definition 2.8 (Vision-Policy Distance). The distance between a vision statement v and a policy statement p is:

where:

• SAS(v, p) is the Strategic Alignment Score (Definition 2.12)
• DimGap(v, p) = |D_v \ D_p| / |D_v| is the fraction of vision dimensions not covered by the policy
• WeightDiv(v, p) is the weight divergence (Definition 2.9)
• lambda, mu >= 0 are penalty coefficients for dimension gaps and weight divergence respectively

Definition 2.9 (Weight Divergence). The weight divergence between vision weights W_v and policy coverage Theta_p is the normalized asymmetric KL divergence:

where epsilon > 0 is a smoothing constant to prevent division by zero. The normalization by log|D_v| ensures WeightDiv in [0, 1] for any number of dimensions.

Proposition 2.2 (Distance Metric Properties). D(v, p) satisfies:

1. Non-negativity: D(v, p) >= 0 for all v in V, p in P 2. Identity of indiscernibles (relaxed): D(v, p) = 0 if and only if SAS(v, p) = 1, DimGap(v, p) = 0, and WeightDiv(v, p) = 0 3. Monotonicity: If SAS(v, p_1) > SAS(v, p_2) and DimGap(v, p_1) <= DimGap(v, p_2) and WeightDiv(v, p_1) <= WeightDiv(v, p_2), then D(v, p_1) < D(v, p_2)

Note that D is not a metric in the strict mathematical sense (it does not satisfy symmetry or the triangle inequality) because Vision Space and Policy Space have different structures. It is a directed distance — measuring how far a policy is from faithfully encoding a vision, not the reverse.

2.5 The Alignment Rate AR

Definition 2.10 (Policy Matching). A policy statement p is said to match a vision statement v with tolerance delta if D(v, p) < delta.

Definition 2.11 (Alignment Rate). Given a vision set V_org = {v_1, v_2, ..., v_K} (the complete set of an organization's vision statements) and a policy set P_org = {p_1, p_2, ..., p_M} (the complete set of deployed policies), the Alignment Rate is:

The Alignment Rate answers the question: what fraction of the CEO's vision statements have at least one policy that faithfully encodes them? An AR of 1.0 means every vision statement is covered. An AR of 0.6 means 40% of the CEO's strategic intent has no corresponding policy enforcement — those dimensions are unguarded, and AI agents operating in those areas have no formal constraint connecting their behavior to the CEO's intent.

Definition 2.12 (Strategic Alignment Score). The Strategic Alignment Score for a single vision-policy pair is:

where Theta_p(d) is set to 0 for dimensions d not in D_p. The organization-wide SAS is the weighted average:

This takes the best-matching policy for each vision statement and averages across all vision statements. An organization with SAS_org = 0.95 has near-perfect strategic encoding. An organization with SAS_org = 0.40 has severe Vision Decay.

3. The Policy Logic Formal Language

With the mathematical spaces defined, we now specify the formal language that serves as the encoding target. Policy Logic is a typed, first-order constraint language designed to be simultaneously readable by executives, parseable by gate engines, and verifiable by proof assistants.

3.1 Grammar Definition

Definition 3.1 (Policy Logic Grammar). The grammar G_PL = (N, Sigma, R, S) is defined as:

S         -> PolicySet
PolicySet -> Policy | Policy '&&' PolicySet
Policy    -> 'POLICY' PolicyID ':' DimRef Priority ConstraintBlock GateBlock
DimRef    -> 'DIMENSION' DimName Weight
DimName   -> 'trust' | 'efficiency' | 'innovation' | 'safety' | 'growth'
           | 'quality' | 'sustainability' | 'equity' | 'transparency' | 'resilience'
Weight    -> 'WEIGHT' FloatVal
Priority  -> 'PRIORITY' IntVal

ConstraintBlock -> 'CONSTRAINTS' '{' ConstraintList '}'
ConstraintList  -> Constraint | Constraint ';' ConstraintList
Constraint      -> AtomicConstraint | CompoundConstraint
AtomicConstraint -> Metric Comparator Threshold 'ON' Scope
CompoundConstraint -> Constraint 'AND' Constraint
                   | Constraint 'OR' Constraint
                   | 'NOT' Constraint

Metric      -> Identifier '.' Identifier    // e.g., customer.satisfaction_score
Comparator  -> '>=' | '<=' | '>' | '<' | '==' | '!='
Threshold   -> FloatVal | IntVal | StringVal
Scope       -> CoordinatePattern            // e.g., G1.U*.P*.Z*.A*

GateBlock   -> 'GATES' '{' GateList '}'
GateList    -> GateRule | GateRule ';' GateList
GateRule    -> 'GATE' GateID ':' TriggerExpr '=>' EvalExpr '|' FailAction Timeout
TriggerExpr -> 'WHEN' Condition
EvalExpr    -> 'REQUIRE' Constraint 'WITH_EVIDENCE' EvidenceType
FailAction  -> 'ESCALATE' EscalationTarget | 'HALT' | 'REDIRECT' CoordinatePattern
Timeout     -> 'TIMEOUT' Duration
EvidenceType -> 'audit_log' | 'approval_record' | 'metric_snapshot'
             | 'human_attestation' | 'model_explanation'

3.2 Concrete Example

Consider the vision statement v_1 = 'We prioritize customer trust over short-term revenue.' The Vision Encoding pipeline produces the following Policy Logic:

POLICY PL-2026-001:
  DIMENSION trust WEIGHT 0.75
  DIMENSION growth WEIGHT 0.25
  PRIORITY 1

  CONSTRAINTS {
    customer.satisfaction_score >= 0.85 ON G1.U*.P*.Z*.A*;
    customer.churn_rate <= 0.05 ON G1.U*.P*.Z*.A*;
    customer.complaint_resolution_rate >= 0.92 ON G1.U1.P3.Z*.A*;
    revenue.quarterly_growth >= 0.02 ON G1.U*.P*.Z*.A*;
    NOT (revenue.discount_rate > 0.30 AND customer.satisfaction_score < 0.80)
      ON G1.U1.P2.Z*.A*
  }

  GATES {
    GATE G-TRUST-01:
      WHEN agent.action.category == 'customer_facing'
      => REQUIRE customer.satisfaction_score >= 0.85
         WITH_EVIDENCE metric_snapshot
      | ESCALATE G1.U1.P3.Z1.A1
      TIMEOUT 5000ms;

    GATE G-TRUST-02:
      WHEN agent.action.financial_impact > 100000
      => REQUIRE customer.churn_prediction <= 0.10
         WITH_EVIDENCE model_explanation
      | ESCALATE G1.U1.P1.Z1.A1
      TIMEOUT 10000ms;

    GATE G-GROWTH-01:
      WHEN agent.action.category == 'pricing'
      => REQUIRE revenue.margin >= 0.15 AND customer.lifetime_value_delta >= 0
         WITH_EVIDENCE audit_log
      | HALT
      TIMEOUT 3000ms
  }

This Policy Logic encoding preserves the original vision's prioritization: trust constraints are stricter (satisfaction >= 0.85, churn <= 0.05) while growth constraints have a lower floor (quarterly growth >= 0.02). The gate rules enforce trust-first evaluation — any customer-facing action must pass the trust gate before proceeding, while financial actions require both margin protection and customer lifetime value analysis.

3.3 Type System and Well-formedness

Policy Logic includes a static type system that prevents ill-formed constraints at encoding time:

Definition 3.2 (Well-formedness Rules).

• WF-1 (Dimension Coverage): Every dimension referenced in the DIMENSION block must have at least one constraint in the CONSTRAINTS block.
• WF-2 (Gate Coverage): Every constraint in the CONSTRAINTS block must be referenced by at least one gate in the GATES block.
• WF-3 (Weight Normalization): The WEIGHT values across all DIMENSION entries must sum to 1.0 (within epsilon = 0.001).
• WF-4 (Scope Validity): Every CoordinatePattern must be a valid MARIA coordinate or wildcard pattern matching at least one node in the deployed hierarchy.
• WF-5 (Timeout Bound): Every TIMEOUT value must satisfy tau_min <= tau <= tau_max where tau_min = 100ms and tau_max = 300000ms.
• WF-6 (Escalation Reachability): Every EscalationTarget must resolve to an active agent or human operator in the MARIA hierarchy.

Theorem 3.1 (Decidability of Well-formedness). The well-formedness of a Policy Logic statement is decidable in O(|C| * |G| + |D|) time, where |C| is the number of constraints, |G| is the number of gate rules, and |D| is the number of dimensions.

Proof. Each well-formedness rule can be checked independently. WF-1 requires iterating over dimensions and checking existence of a matching constraint: O(|D| |C|). WF-2 requires iterating over constraints and checking existence of a matching gate: O(|C| |G|). WF-3 is a single summation: O(|D|). WF-4 requires validating each scope pattern against the coordinate hierarchy, which is bounded by O(|C| + |G|) pattern evaluations each taking O(depth) = O(5) for the five-level MARIA coordinate system. WF-5 is O(|G|) comparisons. WF-6 is O(|G|) lookups in the agent registry. The dominant term is O(|C| * |G|). QED.

3.4 Semantic Equivalence and Normalization

Two Policy Logic statements may be syntactically different but semantically equivalent. We define a normalization procedure that reduces any well-formed Policy Logic statement to a canonical form.

Definition 3.3 (Canonical Form). A Policy Logic statement is in canonical form if: (1) constraints are sorted by dimension, then by metric name; (2) compound constraints are in conjunctive normal form (CNF); (3) gate rules are sorted by trigger specificity (most specific first); (4) redundant constraints are eliminated.

Definition 3.4 (Semantic Equivalence). Two Policy Logic statements p_1 and p_2 are semantically equivalent, written p_1 equiv p_2, if their canonical forms are identical.

The normalization procedure enables efficient comparison of policies and is used by the conflict detection algorithm (Section 6).

4. The Vision-to-Policy Pipeline Architecture

The encoding function E is not a monolithic transformation. It is implemented as a four-stage pipeline, each stage performing a specific, verifiable transformation. This decomposition enables independent testing, auditing, and refinement of each stage.

4.1 Stage 1: Vision Parsing

The Vision Parser transforms natural language vision statements into structured Vision Tuples (Definition 2.1).

Input: Natural language string s. Output: Vision tuple v = (s, D_v, W_v, T_v).

The parser operates in three substeps:

Substep 1a: Dimension Extraction. A fine-tuned language model identifies value dimensions referenced in the statement. The model is trained on a corpus of 12,400 executive communications (earnings calls, board presentations, strategy memos) annotated with value dimensions from D_universe. The extraction achieves F1 = 0.94 on held-out test data.

Substep 1b: Weight Inference. Comparative language patterns ('prioritize X over Y', 'X is more important than Y', 'X-first approach') are mapped to weight orderings. The weight assignment algorithm uses a modified Bradley-Terry model:

where beta_i is the inferred strength parameter for dimension d_i from pairwise comparisons extracted from the natural language. When the statement contains explicit prioritization ('trust over revenue'), the pairwise comparison (trust > revenue) constrains the beta values such that beta_trust > beta_revenue.

Substep 1c: Type Classification. The statement is classified into one of the five vision types (strategic, operational, cultural, financial, regulatory) using a multi-class classifier with accuracy 0.97.

4.2 Stage 2: Constraint Generation

The Constraint Generator maps each value dimension to a set of formal constraints over the agent state space.

Input: Vision tuple v = (s, D_v, W_v, T_v). Output: Constraint set C_p = {c_1, c_2, ..., c_m}.

Constraint generation uses a Dimension-Constraint Library (DCL) — a curated mapping from value dimensions to constraint templates. The DCL is structured as:

where C_template is the set of parameterized constraint templates. For example:

DCL(trust, strategic) = {
  customer.satisfaction_score >= {{threshold_high}},
  customer.churn_rate <= {{threshold_low}},
  customer.nps_score >= {{threshold_medium}},
  data.privacy_compliance == true,
  communication.response_time <= {{threshold_time}}
}

Threshold parameters are instantiated based on the weight W_v(d) and organizational baseline metrics. The instantiation function Phi maps weights to thresholds:

where baseline(m) is the current organizational metric value and target(m) is the ideal value. Higher weight produces more aggressive thresholds, encoding the CEO's relative prioritization into quantitative targets.

4.3 Stage 3: Gate Rule Synthesis

The Gate Synthesizer creates gate rules that enforce the generated constraints.

Input: Constraint set C_p, organizational hierarchy H. Output: Gate rule set G_p = {g_1, g_2, ..., g_n}.

Gate synthesis follows four principles:

Principle 1: Constraint-Gate Bijection. Every constraint must be enforced by at least one gate. The synthesizer creates a minimum gate set that covers all constraints, solving a set cover optimization:

Principle 2: Specificity Ordering. Gates with more specific trigger conditions are evaluated before more general gates, preventing broad gates from short-circuiting specific evaluations.

Principle 3: Fail-Closed Default. Every synthesized gate defaults to halt or escalate on evaluation failure, consistent with the fail-closed architecture described in our prior work on agent gate design.

Principle 4: Evidence Binding. Each gate is bound to an evidence type that matches the constraint's domain. Financial constraints require audit_log evidence. Customer metrics require metric_snapshot evidence. Human judgment calls require human_attestation evidence.

4.4 Stage 4: Policy Compilation

The Policy Compiler combines the outputs of Stages 1-3 into a well-formed Policy Logic statement, runs the well-formedness checks (Definition 3.2), and produces the final encoding.

Input: Constraint set C_p, gate rule set G_p, evidence requirements E_p, dimension metadata. Output: Well-formed Policy Logic statement p, or compilation errors.

The compiler performs three optimization passes:

Pass 1: Constraint Deduplication. Semantically equivalent constraints from different dimensions are merged. For example, if both 'trust' and 'quality' produce a constraint on customer.satisfaction_score, the stricter threshold is retained.

Pass 2: Gate Consolidation. Gates with identical triggers but different evaluations are merged into multi-evaluation gates, reducing runtime overhead.

Pass 3: Scope Narrowing. Wildcard scopes (G1.U.P.Z.A) are narrowed to the most specific scope that covers all relevant agents, reducing gate evaluation to only the nodes where enforcement matters.

The complete pipeline produces an encoding with bounded latency:

Theorem 4.1 (Pipeline Latency Bound). The Vision-to-Policy Pipeline produces a well-formed Policy Logic statement in O(|D_v| |C_template| log|H|) time, where |H| is the size of the organizational hierarchy.

Proof. Stage 1 (Vision Parsing) runs in O(|s|) using the pre-trained models with constant-time inference per token. Stage 2 (Constraint Generation) iterates over |D_v| dimensions, each producing O(|C_template|) constraint candidates. Stage 3 (Gate Synthesis) solves the set cover problem using a greedy approximation in O(|C_p| log|C_p|). Stage 4 (Policy Compilation) performs deduplication in O(|C_p| log|C_p|), gate consolidation in O(|G_p| log|G_p|), and scope narrowing using binary search over the hierarchy in O(|G_p| log|H|). The dominant term is O(|D_v| |C_template| log|H|). QED.

5. Strategic Alignment Score: Detailed Formalization

The Strategic Alignment Score (SAS) was introduced in Definition 2.12. Here we expand the formalization to handle multi-vision organizations, temporal drift, and hierarchical decomposition.

5.1 Multi-Vision Aggregation

Real organizations have multiple, potentially overlapping vision statements. A CEO might articulate 5-15 strategic priorities. The organization-wide SAS must aggregate alignment across all of them while accounting for inter-vision dependencies.

Definition 5.1 (Vision Interaction Matrix). For a vision set V_org = {v_1, ..., v_K}, the interaction matrix M in R^{K x K} is defined as:

M_ij is the Jaccard similarity between the dimension sets of visions i and j. High M_ij indicates that two vision statements share value dimensions and their policies should be jointly evaluated for consistency.

Definition 5.2 (Interaction-Weighted Organization SAS). The organization-wide SAS accounting for vision interactions is:

where Conflict(v_k, v_j) is the vision conflict score (Section 6) and gamma >= 0 is the conflict penalty coefficient. This formulation penalizes alignment scores when conflicting visions share dimensions, forcing the organization to resolve contradictions rather than hide them behind high individual scores.

5.2 Temporal Alignment Drift

Strategic alignment is not static. As the organization evolves — new products launch, markets shift, regulations change — the vision set V_org and the policy set P_org drift apart. We model this drift as a time-varying process.

Definition 5.3 (Alignment Drift Rate). The alignment drift rate at time t is:

where v_dot_k(t) represents the rate of change of vision k (re-articulation by the CEO) and p_dot_m(t) represents the rate of policy adaptation. In practice, vision statements change quarterly (board meetings, earnings calls) while policies can be updated continuously. This asymmetry in update rates is a primary driver of Vision Decay.

Definition 5.4 (Drift Alarm Condition). A drift alarm is triggered when:

for a configured drift threshold delta_drift and observation window Delta_t. In our production deployment, delta_drift = 0.05 and Delta_t = 7 days, meaning a 5-percentage-point drop in organizational alignment within a week triggers executive review.

5.3 Hierarchical SAS Decomposition

The MARIA coordinate system (Galaxy.Universe.Planet.Zone.Agent) enables hierarchical decomposition of the SAS, revealing where in the organization Vision Decay is most severe.

Definition 5.5 (Hierarchical SAS). For a coordinate prefix C (e.g., G1.U2), the localized SAS is:

where P_C subset P_org is the set of policies whose scope includes coordinate prefix C. This decomposition enables a CEO to see, for example, that SAS(G1.U1) = 0.93 (Business Unit 1 is well-aligned) while SAS(G1.U3) = 0.61 (Business Unit 3 has significant Vision Decay).

The hierarchical decomposition satisfies a monotonicity property:

Proposition 5.1 (Hierarchical Monotonicity). If C_1 is a prefix of C_2 (i.e., C_2 is a more specific coordinate), then the policy set P_{C_2} subset P_{C_1}, and consequently SAS(C_2) <= SAS(C_1) when the max is taken over a subset.

This property means alignment can only decrease as you move deeper into the hierarchy — alignment at the Universe level is always at least as good as alignment at the Zone level. Vision Decay accumulates downward.

6. Conflict Detection Between Vision and Execution

A critical failure mode in Vision Encoding is the production of conflicting policies from conflicting vision statements. If the CEO states both 'Maximize growth at all costs' and 'Prioritize risk management and capital preservation,' the encoding pipeline will produce constraint sets that cannot be simultaneously satisfied. Deploying these constraints to gate rules creates a system where agents are guaranteed to fail at least one gate on every action.

6.1 Formal Conflict Definition

Definition 6.1 (Constraint Conflict). Two constraints c_1 and c_2 are in conflict if there exists no agent state s in S_agent such that c_1(s) = true and c_2(s) = true simultaneously:

Definition 6.2 (Soft Conflict). Two constraints c_1 and c_2 are in soft conflict with degree sigma in [0, 1] if the fraction of satisfying states that satisfy both is less than sigma:

Hard conflicts (Definition 6.1) are soft conflicts with sigma = 0. In practice, most real-world conflicts are soft — the constraints can be simultaneously satisfied but only over a narrow range of agent states, creating operational brittleness.

Definition 6.3 (Vision Conflict Score). The conflict score between two vision statements v_i and v_j is:

where C_i = C_{E(v_i)} is the constraint set produced by encoding vision v_i, and Sat(c) = {s in S_agent : c(s) = true} is the satisfying set of constraint c. The conflict score is the average pairwise Jaccard distance between satisfying sets across all constraint pairs from the two visions.

6.2 Conflict Detection Algorithm

We present an efficient algorithm for detecting conflicts in a set of encoded vision statements.

Algorithm 1: Vision Conflict Detection

Input: Encoded policy set P_org = {E(v_1), ..., E(v_K)}
Output: Conflict matrix C in R^{K x K}, conflict clusters

1. For each pair (i, j) where i < j:
   a. Extract constraint sets C_i, C_j
   b. For each pair (c_a, c_b) in C_i x C_j:
      - If c_a and c_b reference the same metric:
        i.  Compute satisfying intervals I_a, I_b
        ii. Compute overlap ratio = |I_a intersect I_b| / |I_a union I_b|
        iii. If overlap ratio < sigma_threshold: flag soft conflict
        iv. If overlap ratio = 0: flag hard conflict
   c. Compute Conflict(v_i, v_j) as the average conflict score
   d. Store C[i][j] = Conflict(v_i, v_j)

2. Identify conflict clusters using agglomerative clustering:
   a. Build distance matrix from C
   b. Merge clusters where average inter-cluster conflict > sigma_cluster
   c. Report clusters as groups of visions requiring resolution

3. For each conflict cluster:
   a. Identify the constraining dimensions
   b. Propose resolution options:
      - Priority ordering (vision v_i takes precedence over v_j on dimension d)
      - Scope partitioning (vision v_i applies to Universe U_1, v_j to U_2)
      - Threshold relaxation (weaken the stricter constraint by delta)

Return: C, conflict clusters, resolution proposals

Theorem 6.1 (Conflict Detection Completeness). Algorithm 1 detects all hard conflicts and all soft conflicts with degree sigma >= sigma_threshold.

Proof sketch. Every pair of constraints referencing the same metric is evaluated in Step 1b. For atomic constraints over continuous metrics, the satisfying set is an interval (or union of intervals), and the overlap ratio is computed exactly. For compound constraints in CNF, the satisfying set is the intersection of satisfying sets of conjuncts, which is computed by interval intersection. The pairwise evaluation over all constraint pairs guarantees that no inter-vision conflict is missed. Clustering in Step 2 groups conflicts but does not introduce or remove them. QED.

6.3 Conflict Resolution Strategies

When conflicts are detected, MARIA OS offers three resolution strategies:

Strategy 1: Priority Override. Assign a total ordering to vision statements. When constraints from lower-priority visions conflict with higher-priority ones, the lower-priority constraints are relaxed. Formally, if v_i has priority p_i > p_j = priority of v_j, and Conflict(v_i, v_j) > 0, then constraints in C_j that conflict with C_i are replaced by the intersection-compatible relaxation:

Strategy 2: Scope Partitioning. Assign conflicting visions to disjoint organizational scopes. Vision v_i applies to Universe U_1, vision v_j applies to Universe U_2, and neither's constraints appear in the other's scope. This is appropriate when the conflict reflects genuinely different strategic directions for different business units.

Strategy 3: Constraint Synthesis. Generate a new constraint that subsumes both conflicting constraints by finding the Pareto-optimal frontier:

This strategy is used when both visions are non-negotiable and a compromise constraint must be found.

7. Implementation in MARIA OS

The Vision Encoding framework is implemented within the MARIA OS governance architecture. This section details the integration points with existing subsystems: the gate evaluation engine, responsibility gates, evidence bundles, and the MARIA coordinate hierarchy.

7.1 Architecture Overview

The Vision Encoding pipeline is deployed as a set of services within the MARIA OS Intelligence Layer:

Vision Input (CEO Dashboard)
  |
  v
[Vision Parser Service]
  |  Extracts: dimensions, weights, type
  v
[Constraint Generator Service]
  |  Queries: Dimension-Constraint Library
  |  Applies: Threshold instantiation function Phi
  v
[Gate Synthesizer Service]
  |  Queries: MARIA Hierarchy (Coordinate System)
  |  Solves: Set cover optimization
  v
[Policy Compiler Service]
  |  Validates: Well-formedness rules WF-1..WF-6
  |  Optimizes: Deduplication, consolidation, scope narrowing
  v
[Policy Store] -- persisted Policy Logic statements
  |
  v
[Gate Evaluation Engine] -- runtime enforcement
  |
  v
[Alignment Monitor] -- continuous SAS computation

7.2 Integration with Gate Evaluation Engine

The Gate Evaluation Engine (lib/engine/decision-pipeline.ts) is extended with a Policy-Aware Gate Evaluator. When a decision enters the pipeline at the proposed stage, the engine:

1. Retrieves all Policy Logic statements whose scope matches the decision's MARIA coordinate 2. Evaluates all gate rules whose trigger conditions match the decision's metadata 3. For each activated gate, checks the evaluation constraint against the current agent state 4. Requires evidence bundles matching the gate's evidence type 5. Computes the gate pass/fail result and logs the evaluation with full traceability to the originating vision statement

The gate evaluation is augmented with a Vision Trace — a data structure that links every gate evaluation back to the vision statement that produced the gate rule:

interface VisionTrace {
  visionId: string           // Original vision statement ID
  policyId: string           // Encoded Policy Logic ID
  constraintId: string       // Specific constraint being evaluated
  gateId: string             // Gate rule performing the evaluation
  encodingTimestamp: string  // When the encoding was performed
  alignmentScore: number     // SAS at encoding time
  evaluationResult: 'pass' | 'fail' | 'escalate'
  evidenceBundle: string[]   // Evidence items provided
}

This Vision Trace is stored as part of the immutable audit record in the decision_transitions table, enabling end-to-end traceability from CEO vision to gate evaluation.

7.3 Integration with Responsibility Gates

Responsibility gates (lib/engine/responsibility-gates.ts) enforce human-in-the-loop (HITL) review for high-impact decisions. Vision Encoding enhances responsibility gates by introducing Vision-Weighted Risk Scoring.

The standard risk score for a decision at node i is R_i (Definition 2.3 in our prior work). Vision Encoding adds a vision-alignment multiplier:

where v_relevant is the vision statement most relevant to the decision (by dimension overlap), p_i is the policy evaluated at node i, and beta > 0 is a sensitivity parameter. When SAS is high (policy well-aligned with vision), the risk multiplier is near 1.0 and the standard risk score applies. When SAS is low (policy poorly aligned), the risk multiplier increases, making it more likely that the responsibility gate triggers human review.

This creates a self-correcting feedback loop: decisions in poorly-aligned areas of the organization automatically receive more human oversight, compensating for Vision Decay until the policies are updated.

7.4 Integration with Evidence Bundles

Evidence bundles (lib/engine/evidence.ts) are extended with a Vision Relevance Score for each piece of evidence:

where Relevance(e, d) in [0, 1] measures how relevant evidence item e is to value dimension d. Evidence bundles with higher VRS are prioritized in gate evaluation, ensuring that the most strategically relevant evidence is reviewed first.

7.5 Alignment Dashboard

The CEO Alignment Dashboard provides real-time visibility into Vision Encoding metrics:

• Organization SAS Heat Map: SAS values overlaid on the MARIA coordinate hierarchy, colored from green (SAS > 0.90) through yellow (0.70 < SAS < 0.90) to red (SAS < 0.70)
• Drift Timeline: SAS_org(t) plotted over time with drift alarm indicators
• Conflict Matrix: Interactive visualization of the Vision Conflict Matrix with cluster highlighting
• Vision Trace Explorer: Drill-down from any vision statement through its encoding, constraints, gate rules, to individual gate evaluations
• Alignment Rate Gauge: Real-time AR with breakdown by vision statement

The dashboard updates every 60 seconds in production, with critical drift alarms pushed via WebSocket in real time.

7.6 API Integration

Vision Encoding is exposed through the MARIA OS API:

POST /api/intelligence/vision-encode
  Input: { statement: string, context?: string }
  Output: { visionTuple: VisionTuple, policyLogic: string, sas: number }

GET /api/intelligence/alignment
  Output: { sasOrg: number, alignmentRate: number, driftRate: number }

GET /api/intelligence/alignment/:coordinate
  Output: { sas: number, policies: PolicySummary[], traces: VisionTrace[] }

POST /api/intelligence/conflict-detect
  Input: { visionIds: string[] }
  Output: { conflictMatrix: number[][], clusters: ConflictCluster[] }

POST /api/intelligence/vision-update
  Input: { visionId: string, newStatement: string }
  Output: { updatedPolicy: string, deltasSAS: number, affectedGates: string[] }

8. Case Study: Enterprise Deployment at a Global Financial Services Firm

We present a detailed case study of Vision Encoding deployment at a global financial services firm with 22,000 employees, $14B in annual revenue, and 340 AI agents operating across five business units (Retail Banking, Corporate Lending, Asset Management, Insurance, and Treasury Operations).

8.1 Baseline Assessment

Before Vision Encoding deployment, the firm's CEO had articulated seven strategic vision statements in the annual strategy presentation:

• V1: 'We will be the most trusted financial institution for retail customers.'
• V2: 'Technology-driven cost efficiency will fund our growth investments.'
• V3: 'Risk management is non-negotiable — no agent acts without appropriate oversight.'
• V4: 'Customer data privacy is a competitive advantage, not a compliance burden.'
• V5: 'We will automate 80% of routine decisions within 18 months.'
• V6: 'Cross-selling through AI recommendations will increase customer lifetime value by 30%.'
• V7: 'Regulatory compliance must be proactive, not reactive.'

The firm's AI agents were governed by 892 gate rules distributed across the five business units. The baseline assessment revealed:

• Manual mapping coverage: The IT governance team had manually mapped gate rules to 3 of 7 vision statements (V3, V5, V7). The remaining 4 vision statements had no formal policy encoding.
• Baseline Alignment Rate: AR = 3/7 = 0.43
• Baseline SAS_org: 0.38 (measured retroactively using the Vision Encoding framework)
• Unresolved conflicts: V2 ('cost efficiency') directly conflicted with V3 ('risk management oversight') — cost-cutting agents were reducing HITL review frequency to save operational costs, degrading risk coverage.

8.2 Vision Encoding Deployment

The Vision Encoding pipeline was deployed in three phases over 12 weeks:

Phase 1 (Weeks 1-3): Vision Parsing and Validation. All seven vision statements were parsed into structured Vision Tuples. The CEO personally reviewed and adjusted the dimension weights. Key adjustments:

• V1 was assigned trust = 0.80, efficiency = 0.20 (CEO elevated trust weight from the parser's initial estimate of 0.65)
• V6 was assigned growth = 0.60, trust = 0.40 (CEO added trust dimension — cross-selling must not erode trust)
• V3 and V5 were identified as potentially conflicting (automation vs. oversight) and flagged for resolution

Phase 2 (Weeks 4-8): Constraint Generation and Conflict Resolution. The Constraint Generator produced 247 new constraints across the seven visions. The Conflict Detection Algorithm identified three conflict clusters:

• Cluster 1: V2 (cost efficiency) vs. V3 (risk oversight) — resolved by Priority Override: V3 takes precedence, cost-cutting agents cannot reduce HITL frequency below configured minimum
• Cluster 2: V5 (80% automation) vs. V3 (oversight) — resolved by Scope Partitioning: routine decisions (risk score < 0.3) follow V5 automation targets; non-routine decisions follow V3 oversight requirements
• Cluster 3: V6 (cross-selling) vs. V4 (privacy) — resolved by Constraint Synthesis: cross-selling recommendations require explicit consent data in the evidence bundle, with a synthesized constraint: customer.consent_scope INCLUDES 'cross_sell' AND data.usage_purpose == 'recommendation'

Phase 3 (Weeks 9-12): Gate Synthesis and Deployment. The Gate Synthesizer produced 156 new gate rules and modified 89 existing rules. The Policy Compiler integrated these with the existing 892 rules, deduplicating 34 redundant rules and consolidating 28 gate pairs.

8.3 Results After 90 Days

After 90 days of production deployment, the results were:

• Alignment Rate: AR increased from 0.43 to 1.00 (all seven visions now have encoding)
• Organization SAS: SAS_org increased from 0.38 to 0.91
• Vision Decay Rate: Reduced from 62% to 9% (measured as 1 - SAS_org)
• Gate evaluation latency: Mean 142ms, P99 287ms (within the 300ms target)
• Conflict incidents: Zero hard conflicts in production (all detected and resolved during Phase 2)
• Drift alarms: Two drift alarms triggered in the 90-day period, both caused by new product launches that introduced agent actions not covered by existing policies. Both were resolved within 48 hours by extending the Policy Logic.

8.4 Specific Impact Metrics

Retail Banking (G1.U1): The trust-focused policies for V1 reduced customer complaint escalation by 34%. AI agents stopped sending premature resolution notifications, instead routing complex issues to human agents when the customer sentiment score fell below the trust threshold.

Corporate Lending (G1.U2): The risk oversight policies for V3 caught 12 lending decisions in the first month that would have exceeded the firm's risk appetite. Under the previous system, these decisions would have been auto-approved because they met the individual credit score threshold but violated the portfolio concentration constraint that was not previously encoded.

Asset Management (G1.U3): Cross-selling recommendations (V6) increased customer lifetime value by 18% in the first quarter while maintaining a customer consent compliance rate of 99.7%, demonstrating that the synthesized constraint from Cluster 3 resolution enabled growth without privacy violations.

Treasury Operations (G1.U5): Automation rate (V5) reached 76% for routine decisions, on track for the 80% target, while maintaining zero risk incidents — the scope partitioning from Cluster 2 resolution ensured that automation expansion did not compromise oversight for non-routine decisions.

9. Benchmarks and Experimental Results

We evaluate the Vision Encoding framework across three dimensions: encoding fidelity, runtime performance, and conflict detection accuracy.

9.1 Experimental Setup

Dataset: 847 vision statements collected from public earnings call transcripts, annual reports, and strategy presentations of Fortune 500 companies (2022-2025). Each statement was independently annotated by three domain experts with value dimensions and weights. Inter-annotator agreement (Fleiss' kappa) was 0.82.

Baseline Systems: We compare against three alternative approaches:

• Manual Encoding (ME): Human governance analysts manually create gate rules from vision statements. Gold standard for accuracy but requires 4-6 hours per vision statement.
• LLM Direct Translation (LLM-DT): GPT-4 directly generates gate rules from vision statements without the formal intermediate representation. Representative of current industry practice.
• Keyword Matching (KM): Rule-based system that maps keywords in vision statements to predefined constraint templates. Simplest baseline.

Metrics:

• Encoding Fidelity (EF): Pearson correlation between the encoded policy's SAS and the manually validated SAS. EF = 1.0 means the automated encoding matches human judgment perfectly.
• Dimension Recall (DR): Fraction of annotated dimensions correctly identified by the parser.
• Weight Accuracy (WA): Mean absolute error between inferred weights and annotated weights.
• Conflict Detection Precision/Recall (CDP/CDR): Against human-annotated conflict labels.
• Gate Evaluation Latency (GEL): Time to evaluate a single gate in production.
• Pipeline Latency (PL): End-to-end time from vision statement input to Policy Logic output.

9.2 Encoding Fidelity Results

Vision Encoding achieves 96.4% of manual encoding fidelity (0.947 vs. 0.982) while reducing encoding time from 4-6 hours to under 30 seconds per vision statement. LLM Direct Translation suffers from dimension hallucination (generating constraints for dimensions not in the vision) and weight distortion (failing to capture explicit prioritization). Keyword Matching misses implicit dimensions entirely — a statement like 'We will earn customer trust through transparent operations' activates 'trust' and 'transparency' keywords but misses the implied 'quality' dimension.

9.3 Conflict Detection Results

The formal constraint representation enables high-precision conflict detection because conflicts are identified through set-theoretic operations on satisfying sets rather than semantic similarity of natural language. LLM-DT frequently misidentifies complementary constraints as conflicting (low precision) and misses soft conflicts where constraints are satisfiable but over a narrow range (low recall).

9.4 Runtime Performance

The full pipeline latency of 4.6s mean is acceptable because vision encoding is an offline operation — CEOs do not re-articulate vision in real time. The critical path is gate evaluation latency, which at 12ms mean is well within the requirements for real-time decision governance. SAS computation at 89ms enables the 60-second dashboard refresh cycle with margin.

9.5 Scalability Analysis

We evaluate scaling behavior across three axes:

Scaling with Vision Count (K): SAS computation is O(K M) where M = |P_org|. Conflict detection is O(K^2 |C|^2) in the worst case. For K = 50 (large enterprise), conflict detection completes in 8.3 seconds — acceptable for the offline conflict detection workflow.

Scaling with Policy Count (M): Gate evaluation is O(M_local) where M_local is the number of policies matching the decision's coordinate scope. With scope narrowing (Section 4.4, Pass 3), M_local averages 12 policies per decision, independent of total M.

Scaling with Hierarchy Depth: MARIA coordinates have a fixed depth of 5 (Galaxy.Universe.Planet.Zone.Agent), so hierarchy-related operations are O(1) in coordinate depth. The width of each level (number of Universes, Planets, etc.) affects scope matching but is bounded by wildcard optimization.

9.6 Ablation Study

We ablate the four pipeline stages to measure individual contributions:

The ablation reveals that Weight Inference is the most critical stage for SAS — without it, the encoding cannot distinguish between 'trust-first' and 'trust-and-growth-equally' strategies. Gate Synthesis does not affect SAS computation (which measures coverage, not enforcement) but is essential for operational correctness. Conflict Detection similarly does not affect SAS but prevents deployment of contradictory policies. Scope Narrowing is a pure performance optimization.

10. Extended Mathematical Results

This section presents additional theoretical results that establish the formal properties of the Vision Encoding framework.

10.1 Convergence of Iterative Encoding Refinement

In practice, the encoding function E is not applied once. CEOs review the initial encoding, provide feedback, and the encoding is refined iteratively. We prove that this iterative process converges.

Definition 10.1 (Refinement Operator). A refinement operator R: P x Feedback -> P takes a policy and human feedback and produces a refined policy. Feedback consists of dimension weight adjustments, constraint threshold modifications, and gate rule additions/removals.

Theorem 10.1 (Convergence of Iterative Refinement). If the refinement operator R is monotonically non-decreasing in SAS (i.e., SAS(v, R(p, f)) >= SAS(v, p) for any valid feedback f), then the sequence p_0, p_1, p_2, ... where p_{t+1} = R(p_t, f_t) converges to a fixed point p with SAS(v, p) = max_p SAS(v, p) in at most ceil(1/delta_min) iterations, where delta_min is the minimum SAS improvement per refinement step.

Proof. SAS(v, p) is bounded above by 1 (since W_v sums to 1 and Theta_p values are in [0, 1]). The sequence SAS(v, p_0) <= SAS(v, p_1) <= ... is monotonically non-decreasing and bounded above, hence convergent by the monotone convergence theorem. Since each step improves SAS by at least delta_min > 0 (or the process terminates), the number of iterations is at most (1 - SAS(v, p_0)) / delta_min <= 1/delta_min. QED.

In our empirical data, convergence occurs within 2-4 iterations for 94% of vision statements, with the CEO providing feedback on weight adjustments (most common) and constraint threshold overrides (second most common).

10.2 Optimal Policy Allocation Under Budget Constraints

In a resource-constrained environment, the organization cannot encode all vision statements simultaneously. We formalize the problem of optimal vision encoding allocation.

Definition 10.2 (Encoding Budget). Each vision statement v_k has an encoding cost c_k (in analyst-hours) and a strategic importance weight omega_k. The total encoding budget is B.

Optimization Problem (Vision Encoding Allocation):

This is a variant of the 0-1 knapsack problem and is NP-hard in general. However, the encoding costs c_k are approximately uniform (due to the standardized pipeline), and the strategic importance weights omega_k are provided by the CEO. Under uniform costs, the optimal solution is greedy: encode vision statements in decreasing order of omega_k until the budget is exhausted.

Theorem 10.2 (Greedy Optimality Under Uniform Costs). When c_k = c for all k, the greedy algorithm (sort by omega_k descending, encode until budget exhausted) achieves the optimal solution.

Proof. With uniform costs, the budget constraint becomes |S| <= B/c. The objective is sum omega_k SAS(v_k, E(v_k)). Since SAS values are independent of which other visions are encoded (under the assumption of no cross-vision constraint sharing), the problem decomposes into K independent terms. Selecting the top B/c terms by omega_k SAS_expected maximizes the sum. Under the further assumption that the encoding function E achieves approximately uniform SAS across visions (empirically validated: std dev of SAS < 0.04), the selection reduces to sorting by omega_k. QED.

10.3 Information-Theoretic Bound on Encoding Fidelity

We establish a fundamental limit on how accurately a formal language can encode natural language vision.

Theorem 10.3 (Encoding Fidelity Upper Bound). For a Policy Logic grammar G_PL with |D_universe| = d dimensions and maximum constraint complexity k, the maximum achievable SAS for a vision statement with semantic complexity H(v) bits is:

when H(v) > d k log_2(|C_template|), and SAS_max = 1 otherwise.

Proof sketch. The Policy Logic grammar can express at most d k log_2(|C_template|) bits of information (d dimensions, each with up to k constraints, each constraint selected from |C_template| templates). If the vision statement carries H(v) bits of semantic information, the encoding necessarily loses H(v) - d k log_2(|C_template|) bits. This information loss translates to SAS reduction proportional to the lost fraction. QED.

This bound motivates the design of the Dimension-Constraint Library: increasing |C_template| raises the information capacity of the encoding, enabling higher fidelity. Our current DCL with |C_template| = 127 templates across d = 10 dimensions and k = 8 maximum constraints per dimension yields a theoretical SAS_max of 0.971, consistent with our empirical results.

10.4 Stability Under Vision Perturbation

A desirable property of the encoding function is stability: small changes to the vision statement should produce small changes to the encoded policy.

Definition 10.3 (Lipschitz Continuity of Encoding). The encoding function E is L-Lipschitz if:

where d_V is a distance metric on Vision Space (e.g., the L2 distance on the weight vector: d_V(v_1, v_2) = ||W_{v_1} - W_{v_2}||_2) and D is a policy distance function.

Theorem 10.4 (Lipschitz Bound for Vision Encoding). The Vision Encoding pipeline with threshold instantiation function Phi (Section 4.2) satisfies L-Lipschitz continuity with:

Proof. The threshold instantiation function is Phi(W_v(d), baseline(m)) = baseline(m) + W_v(d) (target(m) - baseline(m)), which is linear in W_v(d). A perturbation delta_W in the weight produces a threshold change of delta_W (target(m) - baseline(m)). The policy distance induced by this threshold change is bounded by the maximum threshold sensitivity across all dimensions. Since Phi is linear, the Lipschitz constant is exactly the maximum slope. QED.

In practice, L ranges from 0.1 to 0.4 depending on the gap between baseline and target metrics, meaning that a 10% change in a vision weight produces at most a 4% change in the encoded policy's threshold values. This stability ensures that minor re-articulations of vision do not cause abrupt policy changes.

11. Production Deployment Considerations

Deploying Vision Encoding in enterprise environments introduces engineering challenges beyond the mathematical framework. This section addresses the practical considerations for production operation.

11.1 Version Control for Policy Logic

Policy Logic statements are versioned using semantic versioning (MAJOR.MINOR.PATCH):

• MAJOR: Vision statement re-articulation (e.g., CEO changes strategic priority)
• MINOR: Constraint addition or threshold modification
• PATCH: Gate rule optimization without semantic change

Each version is immutable once deployed. Rollback is achieved by reverting to a previous version, not by modifying an existing version. The version history maintains full traceability:

PL-2026-001 v1.0.0  [2026-01-15] Initial encoding of V1
PL-2026-001 v1.1.0  [2026-02-01] Added churn_rate constraint
PL-2026-001 v1.1.1  [2026-02-08] Optimized gate scope from G1.U*.P*.Z*.A* to G1.U1.P3.Z*.A*
PL-2026-001 v2.0.0  [2026-03-01] CEO re-articulated trust weight from 0.75 to 0.85

11.2 Multi-Tenant Vision Isolation

In the MARIA coordinate system, each Galaxy represents a tenant. Vision Encoding enforces strict isolation: a vision statement at Galaxy G1 cannot produce constraints that affect agents in Galaxy G2. The scope validation in WF-4 ensures that all coordinate patterns are rooted in the correct Galaxy.

Within a Galaxy, Universes may have independent or shared vision sets. The tenant administrator configures the Vision Inheritance Policy:

• Independent: Each Universe has its own vision set, no inheritance
• Cascading: Galaxy-level visions cascade to all Universes, which may add Universe-specific visions
• Overridable: Universe-specific visions can override Galaxy-level visions on shared dimensions

11.3 Real-Time vs. Batch Encoding

Vision Encoding supports two operational modes:

Batch Mode (default): Vision statements are encoded during strategic planning cycles (quarterly). The full pipeline runs, conflicts are detected, and the CEO reviews the encoding before deployment. This mode is used for the initial deployment and major vision updates.

Real-Time Mode: When the CEO inputs a new vision statement through the dashboard, the pipeline runs in streaming mode with progressive output. The Vision Parser returns within 3 seconds, showing the extracted dimensions and weights for immediate feedback. Constraint Generation and Gate Synthesis complete within 10 seconds, and the CEO can preview the Policy Logic before confirming deployment.

Real-time mode bypasses the conflict detection step (which requires comparing against all existing visions) and instead queues a background conflict check that reports results within 5 minutes.

11.4 Handling Ambiguous Vision Statements

Not all vision statements are crisp enough for high-fidelity encoding. We quantify ambiguity and handle it explicitly.

Definition 11.1 (Vision Ambiguity Score). The ambiguity of a vision statement v is:

where P(D_v, W_v | s) is the confidence of the Vision Parser in its dimension and weight extraction. High ambiguity (A(v) > 0.3) triggers an interactive clarification workflow:

1. The system presents the CEO with the top-3 interpretations (dimension/weight assignments) 2. The CEO selects the correct interpretation or provides clarification 3. The selected interpretation is used for encoding 4. The ambiguity score and clarification are stored in the audit log

In our dataset, 23% of vision statements had A(v) > 0.3, requiring clarification. After clarification, the encoding fidelity for these statements was comparable to low-ambiguity statements (SAS = 0.941 vs. 0.949).

12. Related Work

Vision Encoding draws on and extends several research areas.

Policy Specification Languages. Formal policy languages have a long history in access control (XACML), network configuration (Ponder), and cloud governance (OPA/Rego). These languages specify operational constraints but do not address the translation from strategic intent. Vision Encoding operates at a higher level of abstraction, producing constraints that can be compiled to XACML, Rego, or any operational policy language.

Strategic Alignment Measurement. The Balanced Scorecard (Kaplan & Norton, 1992) and Strategy Maps attempt to link strategic objectives to operational metrics. However, these frameworks are descriptive (they document relationships) rather than prescriptive (they do not generate constraints). Vision Encoding is prescriptive: it produces executable gate rules from vision statements.

AI Alignment. The AI alignment literature (Amodei et al., 2016; Christiano et al., 2017) focuses on aligning AI behavior with human values in general. Vision Encoding is narrower and more tractable: it aligns AI agents with a specific CEO's stated strategy, using formal constraints rather than reward shaping. This makes the alignment verifiable and auditable.

Natural Language to Formal Logic. Semantic parsing (Zettlemoyer & Collins, 2005; Berant et al., 2013) maps natural language to formal representations. Vision Encoding differs in that the target representation (Policy Logic) is domain-specific and designed for executability, not general logical inference.

Multi-Agent Governance. Frameworks like CASA (Ferber et al., 2004) and organizational models (Dignum, 2004) define normative structures for multi-agent systems. Vision Encoding extends these by providing a pipeline from executive strategy to runtime enforcement, closing the loop between governance architecture and strategic intent.

13. Future Directions

The Vision Encoding framework opens several research directions that we plan to pursue.

13.1 Multi-Stakeholder Vision Aggregation

Current Vision Encoding assumes a single authoritative source (the CEO). In reality, strategic direction is shaped by the board, executive team, regulators, and major customers. Multi-stakeholder aggregation requires a social choice function that combines potentially conflicting vision sets:

We are investigating voting-theoretic approaches (weighted majority, Borda count) and mechanism design frameworks (VCG mechanisms) for this aggregation problem. The key challenge is preserving the Encoding Fidelity Axioms under aggregation.

13.2 Dynamic Vision Adaptation

Vision statements are currently treated as quasi-static (updated quarterly). A dynamic adaptation framework would continuously update vision encodings based on market signals, competitive intelligence, and organizational performance:

This gradient-based adaptation would suggest vision refinements to the CEO based on observed performance gaps, while preserving the human-in-the-loop requirement for vision approval.

13.3 Cross-Organization Benchmarking

With Vision Encoding deployed across multiple organizations, anonymized SAS distributions enable cross-organization benchmarking. An organization can compare its SAS_org against industry peers, identifying areas where their strategic encoding lags behind competitors. This requires a privacy-preserving aggregation protocol that reveals distributional statistics without exposing individual vision statements or policies.

13.4 Formal Verification of Policy Consistency

The current conflict detection algorithm (Section 6) uses set-theoretic operations on satisfying sets. A stronger guarantee would be formal verification using SMT solvers (Z3, CVC5) that prove the satisfiability of the entire constraint set simultaneously. This would move from pairwise conflict detection to global consistency verification:

Our preliminary experiments with Z3 show that global verification for typical enterprise constraint sets (500-1000 constraints) completes in under 10 seconds, making it feasible for the batch encoding workflow.

13.5 Vision Encoding for Non-CEO Roles

While this paper focuses on CEO vision, the same framework applies to any authoritative statement that must be translated to operational constraints: CTO technical vision, CFO financial policies, CHRO cultural values, and Chief Risk Officer risk appetite statements. Each role-specific vision would use a domain-specific Dimension-Constraint Library while sharing the same formal language and alignment metrics.

13.6 Causal Impact of Vision Encoding

A rigorous evaluation of Vision Encoding's causal impact on organizational performance requires a randomized controlled trial across comparable business units — some with Vision Encoding, some without — over a multi-quarter period. We are designing such a trial with two enterprise partners, with results expected in late 2026.

14. Conclusion

This paper has presented Vision Encoding, a formal framework for converting CEO vision statements from natural language to executable Policy Logic within the MARIA OS governance architecture. We have made six contributions:

First, we defined the mathematical spaces — Vision Space V, Policy Space P — and the encoding function E: V -> P with four Encoding Fidelity Axioms that guarantee dimension preservation, weight-coverage proportionality, constraint completeness, and gate enforcement.

Second, we introduced the Policy Logic formal language with a typed grammar, well-formedness rules decidable in polynomial time, and semantic equivalence through canonical normalization.

Third, we constructed the four-stage Vision-to-Policy Pipeline (parsing, constraint generation, gate synthesis, policy compilation) with bounded latency O(|D_v| |C_template| log|H|) and empirical mean latency of 4.6 seconds.

Fourth, we formalized the Strategic Alignment Score SAS as a continuous metric, the Alignment Rate AR as an organizational coverage metric, and the Vision-Policy Distance D(V, P) as a directed distance function. We proved that iterative encoding refinement converges to optimal SAS in finite steps (Theorem 10.1) and established an information-theoretic upper bound on encoding fidelity (Theorem 10.3).

Fifth, we presented a conflict detection algorithm with proven completeness (Theorem 6.1) and three resolution strategies — priority override, scope partitioning, and constraint synthesis — that handle the full spectrum of inter-vision conflicts.

Sixth, we validated the framework through deployment at a global financial services firm, demonstrating Alignment Rate improvement from 0.43 to 1.00, SAS_org improvement from 0.38 to 0.91, and Vision Decay reduction from 62% to 9% over a 90-day production deployment.

The central insight of this work is that the gap between executive vision and AI agent behavior is not a cultural problem or a communication problem — it is a formal language problem. When vision can be expressed in a language that is both human-auditable and machine-executable, the gap closes. When every gate evaluation traces back to a specific vision statement through a specific policy encoding, strategic alignment becomes measurable, auditable, and enforceable.

For the first time, a CEO can ask 'Are our AI agents actually executing my strategy?' and receive a mathematically grounded answer: the Strategic Alignment Score, decomposed by business unit, by value dimension, by individual agent. And when the answer is less than 1.0, the system identifies exactly where the gap is, which policies need updating, and which visions need clarification.

Judgment does not scale. But judgment, once encoded in a formal language, can govern the systems that do.

Appendix A: Complete Policy Logic Grammar (BNF)

<policy-set>     ::= <policy> | <policy> '&&' <policy-set>
<policy>         ::= 'POLICY' <policy-id> ':' <dim-block> <priority> <constraint-block> <gate-block>
<dim-block>      ::= <dim-decl> | <dim-decl> <dim-block>
<dim-decl>       ::= 'DIMENSION' <dim-name> 'WEIGHT' <float>
<dim-name>       ::= 'trust' | 'efficiency' | 'innovation' | 'safety' | 'growth'
                   | 'quality' | 'sustainability' | 'equity' | 'transparency' | 'resilience'
<priority>       ::= 'PRIORITY' <integer>
<constraint-block> ::= 'CONSTRAINTS' '{' <constraint-list> '}'
<constraint-list> ::= <constraint> | <constraint> ';' <constraint-list>
<constraint>     ::= <atomic> | <compound>
<atomic>         ::= <metric> <comp> <threshold> 'ON' <scope>
<compound>       ::= <constraint> 'AND' <constraint>
                   | <constraint> 'OR' <constraint>
                   | 'NOT' <constraint>
                   | '(' <compound> ')'
<metric>         ::= <ident> '.' <ident>
<comp>           ::= '>=' | '<=' | '>' | '<' | '==' | '!='
<threshold>      ::= <float> | <integer> | <string>
<scope>          ::= <coord-pattern>
<coord-pattern>  ::= 'G' <id-or-wild> '.U' <id-or-wild> '.P' <id-or-wild> '.Z' <id-or-wild> '.A' <id-or-wild>
<id-or-wild>     ::= <integer> | '*'
<gate-block>     ::= 'GATES' '{' <gate-list> '}'
<gate-list>      ::= <gate-rule> | <gate-rule> ';' <gate-list>
<gate-rule>      ::= 'GATE' <gate-id> ':' <trigger> '=>' <eval> '|' <fail-action> <timeout>
<trigger>        ::= 'WHEN' <condition>
<eval>           ::= 'REQUIRE' <constraint> 'WITH_EVIDENCE' <evidence-type>
<evidence-type>  ::= 'audit_log' | 'approval_record' | 'metric_snapshot'
                   | 'human_attestation' | 'model_explanation'
<fail-action>    ::= 'ESCALATE' <coord-pattern> | 'HALT' | 'REDIRECT' <coord-pattern>
<timeout>        ::= 'TIMEOUT' <integer> 'ms'

Appendix B: Dimension-Constraint Library Excerpt

// DCL Entry: trust x strategic
DCL['trust']['strategic'] = [
  { metric: 'customer.satisfaction_score', comp: '>=', threshold_range: [0.70, 0.95] },
  { metric: 'customer.churn_rate', comp: '<=', threshold_range: [0.01, 0.10] },
  { metric: 'customer.nps_score', comp: '>=', threshold_range: [30, 80] },
  { metric: 'customer.complaint_resolution_rate', comp: '>=', threshold_range: [0.80, 0.99] },
  { metric: 'data.privacy_compliance', comp: '==', threshold: true },
  { metric: 'communication.response_time_hours', comp: '<=', threshold_range: [1, 24] },
  { metric: 'transparency.explanation_coverage', comp: '>=', threshold_range: [0.80, 1.00] },
]

// DCL Entry: growth x financial
DCL['growth']['financial'] = [
  { metric: 'revenue.quarterly_growth_rate', comp: '>=', threshold_range: [0.01, 0.15] },
  { metric: 'revenue.customer_acquisition_cost', comp: '<=', threshold_range: [50, 500] },
  { metric: 'revenue.lifetime_value', comp: '>=', threshold_range: [500, 10000] },
  { metric: 'revenue.margin', comp: '>=', threshold_range: [0.10, 0.40] },
  { metric: 'growth.market_share_delta', comp: '>=', threshold_range: [0.001, 0.05] },
]

// DCL Entry: safety x regulatory
DCL['safety']['regulatory'] = [
  { metric: 'compliance.violation_count', comp: '==', threshold: 0 },
  { metric: 'compliance.audit_pass_rate', comp: '>=', threshold_range: [0.95, 1.00] },
  { metric: 'risk.exposure_ratio', comp: '<=', threshold_range: [0.01, 0.10] },
  { metric: 'risk.incident_count_monthly', comp: '<=', threshold_range: [0, 5] },
  { metric: 'governance.hitl_coverage', comp: '>=', threshold_range: [0.80, 1.00] },
]

Appendix C: Proof of Theorem 6.1 (Conflict Detection Completeness) — Full Version

Theorem 6.1. Algorithm 1 detects all hard conflicts and all soft conflicts with degree sigma >= sigma_threshold.

Proof. We prove completeness for hard conflicts; the soft conflict case follows by continuity of the overlap ratio.

Let c_a in C_i and c_b in C_j be two constraints in hard conflict, i.e., Sat(c_a) intersect Sat(c_b) = emptyset. We must show that Algorithm 1 identifies this conflict.

Case 1: c_a and c_b are atomic constraints on the same metric m. Then c_a has the form m comp_a theta_a and c_b has the form m comp_b theta_b. Step 1b of the algorithm iterates over all pairs (c_a, c_b) where both reference the same metric. For the pair (c_a, c_b), Step 1b(i) computes the satisfying intervals I_a and I_b. Since the constraints are on the same metric, Sat(c_a) and Sat(c_b) are intervals (or unions of intervals) on the real line. The overlap ratio in Step 1b(ii) is |I_a intersect I_b| / |I_a union I_b|. Since the constraints are in hard conflict, I_a intersect I_b = emptyset, so the overlap ratio is 0 < sigma_threshold, and Step 1b(iv) flags a hard conflict. The conflict is detected.

Case 2: c_a and c_b are compound constraints. Any compound constraint in CNF can be decomposed into a conjunction of atomic constraints. Two compound constraints are in hard conflict if and only if there exists at least one pair of constituent atomic constraints that are in hard conflict (since the satisfying set of a conjunction is the intersection of satisfying sets). Step 1b iterates over all pairs of atomic sub-constraints (via the CNF decomposition), so the conflicting pair is found by Case 1.

Case 3: c_a and c_b reference different metrics. Two constraints on different metrics can only be in hard conflict if the metrics are functionally dependent (e.g., revenue.growth and revenue.margin may be related through a financial model). Algorithm 1's Step 1b only compares constraints on the same metric. However, cross-metric hard conflicts require domain knowledge of metric dependencies, which is encoded in the Dimension-Constraint Library's dependency graph. When the DCL specifies a dependency between metrics m_1 and m_2, the algorithm lifts both constraints to the joint space (m_1, m_2) and performs the overlap analysis in two dimensions. This extension, while not shown in the simplified Algorithm 1, is implemented in the production version.

For soft conflicts, the argument is identical except that the overlap ratio is compared against sigma_threshold rather than 0. Since the overlap ratio is a continuous function of the constraint parameters, and the algorithm computes it exactly for atomic constraints and via interval arithmetic for compound constraints, all soft conflicts with degree >= sigma_threshold are detected. QED.

Appendix D: Notation Reference