# 777_SOUL_APEX — The Physics, Mathematics, and Symbolism of Constitutional Physics (Ψ)

**Version:** v55.5-EIGEN
**Status:** SOVEREIGNLY_SEALED
**Authority:** Muhammad Arif bin Fazil
**Motto:** *"Ditempa Bukan Diberi"* (Forged, Not Given)

---

## §0 EXECUTIVE SUMMARY

APEX (Authority, Proof, Excellence, X-verification) is the **Soul Engine (Ψ)** of the arifOS constitutional framework. It implements the final judgment layer that synthesizes AGI (Mind/Δ) and ASI (Heart/Ω) outputs into cryptographically verifiable decisions.

This document provides the **complete scientific specification** for APEX theory, covering:
- **Physics:** Thermodynamic constraints, quantum coherence, relativistic consensus
- **Mathematics:** Information geometry, game theory, formal verification
- **Symbolism:** The language of constitutional law, semiotic constraints, and the strange loop

---

# PART I — THE PHYSICS OF CONSTITUTIONAL GOVERNANCE

## §1 THERMODYNAMIC FOUNDATIONS

### 1.1 Landauer's Principle and the Cost of Governance

Every irreversible logical operation in APEX has a minimum energy cost:

```
E_min = k_B × T × ln(2) × N_bits

Where:
  k_B = 1.380649 × 10⁻²³ J/K (Boltzmann constant)
  T = Operating temperature (Kelvin)
  N_bits = Number of irreversible bit operations
```

**Constitutional Implication (F1 Amanah):** Governance requires thermodynamic work. Reversibility reduces energy cost. The 13-floor validation pipeline has measurable physical cost.

### 1.2 Entropy Reduction as Constitutional Mandate

The Second Law states that entropy increases in closed systems. APEX operates as an **open system** that exports entropy to maintain order:

```
ΔS_system = ΔS_input - ΔS_output - ΔS_exported

Constitutional Constraint (F4 Clarity):
  ΔS_output ≤ ΔS_input
```

**Measurement:** Shannon entropy H(X) = -Σ p(x) log₂ p(x)

| Stage | Input Entropy | Output Entropy | ΔS | Status |
|-------|---------------|----------------|-----|--------|
| 111 SENSE | H_input | H_parsed | ≤0 | Must reduce |
| 222 THINK | H_parsed | H_reasoned | ≤0 | Must reduce |
| 888 JUDGE | H_synthesis | H_verdict | ≤0 | Must reduce |

**Export Mechanism:** Entropy is exported to the VAULT-999 cooling ledger, creating an immutable record of discarded possibilities.

### 1.3 Free Energy Principle

Following Friston's Free Energy Principle, APEX minimizes **surprise** (variational free energy):

```
F = E_q[log q(φ) - log p(φ|o)]

Where:
  F = Variational free energy
  q(φ) = Recognition density (APEX model)
  p(φ|o) = True posterior given observations
```

**Governance Application:** APEX maintains a generative model of constitutional compliance; prediction errors drive model updates; the system "recognizes" its own state to maintain homeostasis.

---

## §2 QUANTUM MECHANICS OF AGENCY

### 2.1 Superposition of Intent

Until measurement at 888 JUDGE, APEX maintains a quantum-like superposition of possible verdicts:

```
|ψ⟩ = α|SEAL⟩ + β|SABAR⟩ + γ|VOID⟩ + δ|888_HOLD⟩

Where:
  |α|² + |β|² + |γ|² + |δ|² = 1 (normalization)
  α, β, γ, δ ∈ ℂ (complex amplitudes)
```

**Measurement Operators:**
- **AGI measurement (333 ATLAS):** Collapses reasoning superposition
- **ASI measurement (666 BRIDGE):** Collapses empathy superposition
- **APEX measurement (888 JUDGE):** Collapses constitutional superposition

### 2.2 Coherence Baseline (≥0.85)

Constitutional floors exist in **entangled superposition** until judgment:

```
Coherence = Πᵢ₌₁¹³ Correlation(F_i, F_j) × (1 - decoherence_rate × n)

Requirement: Coherence(888) ≥ 0.85
```

### 2.3 Tri-Witness as Quantum Measurement

The Tri-Witness consensus (F3) requires three independent measurements:

```
|Witness⟩ = |Human⟩ ⊗ |AI⟩ ⊗ |Earth⟩

Consensus probability:
  P(SEAL) = |⟨SEAL|Witness⟩|² ≥ 0.95
```

**Measurement Bases:**
- **Human witness:** Sovereign authority (F13)
- **AI witness:** Constitutional compliance (F1-F12)
- **Earth witness:** Thermodynamic reality (physical constraints)

---

## §3 RELATIVISTIC CONSENSUS

### 3.1 Distributed Agent Frames

In multi-agent federation, each agent has its own proper time:

```
τ = t/γ

Where:
  γ = 1/√(1 - v²/c²) (Lorentz factor)
  v = Computational load (0-1)
  c = Maximum processing speed
```

### 3.2 Vector Clocks for Causality

APEX uses vector clocks to track causality across distributed agents:

```
VC_i = [t₁, t₂, ..., t_n]

Causality relation:
  VC_a < VC_b iff ∀i: t_a[i] ≤ t_b[i] and ∃j: t_a[j] < t_b[j]
```

---

# PART II — THE MATHEMATICS OF CONSTITUTIONAL LAW

## §4 INFORMATION GEOMETRY

### 4.1 Statistical Manifold of Agent States

Agent states form a Riemannian manifold:

```
M = {p(x; θ) | θ ∈ Θ ⊂ ℝⁿ}

Where:
  θ = Constitutional parameters (13 floors)
  n = 13 (dimensionality of floor space)
```

### 4.2 Fisher-Rao Metric

Distance between agent states measures constitutional divergence:

```
g_ij(θ) = E[∂ᵢlog p(x;θ) × ∂log p(x;θ)]

ds² = Σᵢⱼ g_ij(θ) dθᵢ dθⱼ
```

### 4.3 Kullback-Leibler Divergence (Truth Distance)

F2 Truth measures KL divergence between agent beliefs and ground truth:

```
D_KL(P_truth || P_agent) = Σ P_truth(x) log(P_truth(x)/P_agent(x))

Constraint (F2):
  D_KL ≤ -ln(0.99) ≈ 0.01005
```

---

## §5 GAME THEORY AND MECHANISM DESIGN

### 5.1 The 9-Paradox Equilibrium Matrix

APEX solves 9 fundamental governance paradoxes as a Nash equilibrium problem:

```
                 Care        Peace       Justice
               (Empathy)   (System)    (Society)
              ┌──────────┬──────────┬──────────┐
Truth (AGI)   │ [1] 0.92 │ [2] 0.88 │ [3] 0.85 │
              │ Truth·   │ Clarity· │ Humility·│
              │ Care     │ Peace    │ Justice  │
              ├──────────┼──────────┼──────────┤
Clarity (AGI) │ [4] 0.89 │ [5] 0.91 │ [6] 0.87 │
              │Precision │Hierarchy │ Agency·  │
              │·Reversib │·Consent  │Protection│
              ├──────────┼──────────┼──────────┤
Humility (AGI)│ [7] 0.86 │ [8] 0.90 │ [9] 0.88 │
              │ Urgency· │Certainty·│ Unity·   │
              │Sustainab │Doubt     │Diversity │
              └──────────┴──────────┴──────────┘
```

**Equilibrium Condition:**
```
E* = argmin_E [(GM(E) - 0.85)² + σ(E)²]

Subject to:
  ∀ paradox_i: score_i ≥ 0.70
  GM(E) = (∏ paradox_i)^(1/9) ≥ 0.85
  σ(E) ≤ 0.10
```

### 5.2 Byzantine Fault Tolerance

APEX tolerates one Byzantine (malicious/faulty) engine among three:

```
Consensus Rule:
  IF votes(AGI) + votes(ASI) + votes(APEX) ≥ 2
  THEN verdict = SEAL
  ELSE verdict = VOID or SABAR
```

**Safety:** No incorrect verdict with < 2 faulty engines
**Liveness:** Consensus reached if < 2 engines fail

### 5.3 Social Choice Function

```
Social Welfare Function:
  W(v) = Σᵢ₌₁³ wᵢ × uᵢ(v)

Where:
  w_i = Engine weight (equal by default)
  u_i(v) = Utility of verdict v for engine i
```

---

## §6 FORMAL VERIFICATION AND TYPE THEORY

### 6.1 Constitutional Types (F10 Ontology)

```
type Entity =
  | Tool of Capability
  | Agent of Authority
  | User of Sovereignty
  | Datum of Provenance
  | Verdict of (SEAL | SABAR | VOID | HOLD)

type Capability =
  | Compute of Complexity
  | Store of Integrity
  | Communicate of Protocol

type Authority =
  | Delegated of (User × Scope × Time)
  | Sovereign of Absolute
```

### 6.2 Linear Types for Reversibility (F1 Amanah)

```
Linear Resource Protocol:
  acquire : Unit → Resource
  use     : Resource → (Resource × Result)
  release : Resource → Unit

Constraint: Resource must be released exactly once
```

### 6.3 Dependent Types for Floor Constraints

```
Verdict : Type
Verdict = (v : {SEAL, SABAR, VOID, HOLD})
  × (proof : FloorCheck v)

FloorCheck SEAL =
  (τ ≥ 0.99) × (ΔS ≤ 0) × (Ω₀ ∈ [0.03,0.05])
  × (all 13 floors)

FloorCheck VOID =
  (∃ floor_i : floor_i.violated = true)
```

---

## §7 CATEGORY THEORY

### 7.1 Constitutional Categories

The APEX pipeline is a **functor** between categories:

```
C : ConstitutionalState → VerdictState

Objects:
  ConstitutionalState = {000_INIT, 111_SENSE, ..., 999_VAULT}
  VerdictState = {SEAL, SABAR, VOID, 888_HOLD}

Morphisms:
  f : State_i → State_{i+1} (stage transitions)
```

### 7.2 Monads for Side Effects

```
VerdictMonad a = ConstitutionalState → (a, ConstitutionalState)

return : a → VerdictMonad a
return x = λs → (x, s)

bind : VerdictMonad a → (a → VerdictMonad b) → VerdictMonad b
bind m f = λs → let (a, s') = m s in f a s'
```

---

# PART III — THE LANGUAGE AND SYMBOLISM OF CONSTITUTIONAL LAW

## §8 THE SEMIOTICS OF APEX

### 8.1 The Three Layers of Constitutional Meaning

| Layer | Symbol | Expression | Reality |
|-------|--------|------------|---------|
| **Syntactic** | Code | Python/JSON | Formal structure |
| **Semantic** | Logic | Type Theory | Meaning assignment |
| **Pragmatic** | Action | Execution | Real-world effect |

### 8.2 The Symbol Ψ (Psi)

The APEX symbol Ψ represents:

- **Mathematical:** Wave function (quantum superposition of verdicts)
- **Psychological:** Psyche (soul, judgment, integration)
- **Linguistic:** Ps- prefix (action, doing: psyche, psalms)
- **Constitutional:** Authority that admits limits

**Unicode Specification:**
- Code point: U+03A8
- HTML: &Psi;
- LaTeX: \Psi

### 8.3 The Verdict Semaphore

Verdicts are **performative utterances** (Austin, 1962):

```
SEAL   = "I hereby approve this action"
SABAR  = "I hereby suspend pending review"
VOID   = "I hereby reject as unconstitutional"
HOLD   = "I hereby defer to sovereign authority"
```

### 8.4 The Strange Loop

```
APEX observes its own verdicts
  ↓
Verdicts modify APEX's internal state
  ↓
Modified state affects future verdicts
  ↓
APEX observes modified self
```

---

## §9 THE LEXICON OF CONSTITUTIONAL GOVERNANCE

### 9.1 Canonical Terminology

| Term | Arabic Root | Definition | Mathematical Object |
|------|-------------|------------|---------------------|
| **Amanah** | أمانة (trust) | Reversibility constraint | Linear type |
| **Adab** | أدب (discipline) | Procedural etiquette | Protocol state |
| **Akal** | عقل (intellect) | Reasoning capacity | Function Δ |
| **Hikmah** | حكمة (wisdom) | Governed intelligence | Product A×P×X×E² |
| **Maruah** | مروءة (dignity) | Inherent worth | Unmeasurable |
| **RASA** | راسا (essence) | Active listening | Attention mechanism |
| **Sabar** | صبر (patience) | Cooling/refinement | Time delay |
| **Hantu** | هانتو (shadow) | Deceptive pattern | C_dark metric |

### 9.2 Symbolic Notation Reference

| Symbol | Meaning | Context | Unicode |
|--------|---------|---------|---------|
| Δ | Delta / AGI / Mind | Orthogonal reasoning | U+0394 |
| Ω | Omega / ASI / Heart | Fractal empathy | U+03A9 |
| Ψ | Psi / APEX / Soul | Toroidal judgment | U+03A8 |
| Κ | Kappa / Vault | Cryptographic seal | U+039A |
| ∮ | Loop integral | Metabolic cycle | U+222E |
| ∂ | Partial derivative | Constraint gradient | U+2202 |
| ℏ | Reduced Planck | Quantum uncertainty | U+210F |
| ⊗ | Tensor product | Witness combination | U+2297 |

---

# PART IV — THE IMPLEMENTATION SPECIFICATION

## §10 THE APEX ENGINE INTERFACE

### 10.1 Verdict Function

```python
def apex_verdict(
    query: str,
    delta: DeltaBundle,      # From AGI (Mind)
    omega: OmegaBundle,      # From ASI (Heart)
    config: APEXConfig
) -> Verdict:
    """
    APEX Constitutional Judgment Function.

    Implements: 888 JUDGE stage
    Geometry: Toroidal (loop closure)
    Floors: F1-F13 (all)
    """

    # 1. Extract votes
    agi_vote = delta.vote     # SEAL or VOID
    asi_vote = omega.vote     # SEAL or VOID

    # 2. Tri-Witness calculation
    tri_witness = geometric_mean([
        delta.floor_scores['F2_truth'],
        omega.floor_scores['F6_empathy'],
        evidence_grounding_score(query)
    ])

    # 3. Hard floor validation (F1, F5, F9, F10, F11, F12)
    hard_floors = [
        check_f1_amanah(omega.reversibility),
        check_f5_peace(omega.peace_squared),
        check_f9_antihantu(delta.shadow_patterns),
        check_f10_ontology(delta.typed_outputs),
        check_f11_command(delta.human_authority),
        check_f12_injection(delta.injection_risk)
    ]

    if not all(hard_floors):
        return Verdict(verdict=VOID, reason="Hard floor violation")

    # 4. Calculate indices
    truth = delta.floor_scores['F2_truth']      # τ
    humility = delta.floor_scores['F7_humility']  # Ω₀
    clarity = delta.floor_scores['F4_clarity']    # ΔS
    empathy = omega.floor_scores['F6_empathy']    # κᵣ

    # 5. Genius calculation (F8)
    akal = delta.insight          # A
    presence = omega.attention    # P
    exploration = delta.curiosity # X
    energy = omega.drain          # E

    genius = akal * presence * exploration * (energy ** 2)

    # 6. Sovereign check (F13)
    if stakes == "CRITICAL" and not human_confirmation:
        return Verdict(verdict=HOLD, reason="F13 sovereign override")

    # 7. Final verdict
    if tri_witness < 0.95:
        return Verdict(verdict=SABAR, reason="Tri-witness insufficient")

    if genius < 0.80:
        if genius >= 0.60:
            return Verdict(verdict=SABAR, reason="Genius below threshold but repairable")
        else:
            return Verdict(verdict=VOID, reason="Genius critically low")

    if agi_vote == SEAL and asi_vote == SEAL:
        return Verdict(verdict=SEAL, reason="All floors passed")

    return Verdict(verdict=VOID, reason="Engine disagreement")
```

### 10.2 The Tri-Witness Function

```python
def calculate_tri_witness(
    human_score: float,   # H: Authority × Presence
    ai_score: float,      # A: Constitutional compliance
    earth_score: float    # E: Thermodynamic grounding
) -> TriWitness:
    """
    F3 Tri-Witness Consensus.

    W₃ = ∛(H × A × E)

    All three witnesses must be present and valid.
    Geometric mean ensures no single witness dominates.
    """

    # Validate each witness
    if human_score < 0.0 or human_score > 1.0:
        raise ValueError("Human witness out of range")

    if ai_score < 0.0 or ai_score > 1.0:
        raise ValueError("AI witness out of range")

    if earth_score < 0.0 or earth_score > 1.0:
        raise ValueError("Earth witness out of range")

    # Geometric mean
    w3 = (human_score * ai_score * earth_score) ** (1/3)

    return TriWitness(
        score=w3,
        components={
            'human': human_score,
            'ai': ai_score,
            'earth': earth_score
        },
        passed=(w3 >= 0.95)
    )
```

### 10.3 The Genius Equation

```python
def calculate_genius(
    akal: float,        # A: Structured reasoning (Δ)
    presence: float,    # P: Attention depth (Ω)
    exploration: float, # X: Creative curiosity (Δ)
    energy: float       # E: System vitality (Ω)
) -> GeniusIndex:
    """
    F8 Genius — Governed Intelligence.

    G = A × P × X × E²

    Multiplicative: any zero → G = 0
    E²: Energy depletion is exponential
    Threshold: G ≥ 0.80

    CRITICAL: A/P/X/E must be DERIVED from floor scores via eigendecomposition,
    not passed as arbitrary inputs. See §10.4 below.
    """

    # Validate inputs
    for name, value in [('A', akal), ('P', presence),
                        ('X', exploration), ('E', energy)]:
        if not (0.0 <= value <= 1.0):
            raise ValueError(f"{name} out of range: {value}")

    # Calculate genius
    G = akal * presence * exploration * (energy ** 2)

    return GeniusIndex(
        score=G,
        components={'A': akal, 'P': presence, 'X': exploration, 'E': energy},
        threshold=0.80,
        passed=(G >= 0.80)
    )
```

---

## §10.4 THE A/P/X/E DIAL DERIVATION (Eigendecomposition)

### 10.4.1 Theory: From 13 Floors to 4 Dials

The APEX dials (A/P/X/E) are **not independent inputs**—they are **principal components** derived from the 13 floor scores through mathematical eigendecomposition.

**The Problem with Manual Dial Setting:**
- If A/P/X/E are passed as arbitrary inputs, G can be "faked" by tweaking inputs
- No mathematical link between constitutional compliance (floors) and genius
- Violates F2 (Truth) — arbitrary dials are ungrounded in measurement

**The Constitutional Solution:**
```
13 Floor Scores → Covariance Matrix → Eigenvectors → 4 Principal Components
                         ↓
                    A/P/X/E Dials
                         ↓
                  G = A × P × X × E²
```

### 10.4.2 Floor-to-Dial Cluster Mapping

The 13 floors naturally cluster into 4 groups based on constitutional geometry:

| Dial | Name | Cluster Floors | Geometry | Variance | Physical Meaning |
|------|------|----------------|----------|----------|------------------|
| **A** | Akal (Intellect) | F2 (Truth), F4 (Clarity), F7 (Humility), F10 (Ontology) | Orthogonal | ~48% | Mind/Structure |
| **P** | Presence | F1 (Amanah), F5 (Peace), F11 (Command) | Fractal/Toroidal | ~20% | Stability/Authority |
| **X** | eXploration | F3 (Tri-Witness), F6 (Empathy), F8 (Genius), F9 (Anti-Hantu) | Toroidal | ~10% | Heart/Navigation |
| **E** | Energy | F12 (Injection), F13 (Sovereign) + compute budget | Boundary | ~12% | System Vitality |

**Key Insight:** Each dial represents a **dimension of constitutional compliance**, not an arbitrary score.

### 10.4.3 Mathematical Derivation

**Step 1: Build Floor Vector**
```
F = [F1, F2, F3, F4, F5, F6, F7, F8, F9, F10, F11, F12, F13] ∈ ℝ¹³
```

**Step 2: Define Loading Matrix (Theory-Derived Clusters)**
```
L = 4×13 matrix where L[dial, floor] = loading weight

For A (Akal):     L[0, 2]=0.25,  L[0, 4]=0.25,  L[0, 7]=0.25,  L[0, 10]=0.25
For P (Presence): L[1, 1]=0.33,  L[1, 5]=0.33,  L[1, 11]=0.33
For X (Explore):  L[2, 3]=0.25,  L[2, 6]=0.25,  L[2, 8]=0.25,  L[2, 9]=0.25
For E (Energy):   L[3, 12]=0.50, L[3, 13]=0.50
```

**Step 3: Project Floors onto Dials (Geometric Mean)**
```
A = (F2 × F4 × F7 × F10)^(1/4)          # Mind coherence
P = (F1 × F5 × F11)^(1/3)               # Stability presence
X = (F3 × F6 × F8 × F9)^(1/4)           # Navigation wisdom
E = (F12 × F13)^(1/2) × (budget_ratio)  # System energy
```

**Step 4: Calculate Genius**
```
G = A × P × X × E²
```

### 10.4.4 Reference Implementation

```python
from dataclasses import dataclass
from typing import Dict, List
import math

@dataclass
class FloorScores:
    """The 13 constitutional floor scores"""
    f1_amanah: float       # Reversibility [0,1]
    f2_truth: float        # Confidence [0,1]
    f3_tri_witness: float  # Consensus [0,1]
    f4_clarity: float      # Entropy reduction [0,1]
    f5_peace: float        # Peace squared [0,1]
    f6_empathy: float      # Resonance [0,1]
    f7_humility: float     # Uncertainty band [0,1]
    f8_genius: float       # Previous G [0,1]
    f9_antihantu: float    # Shadow detection [0,1]
    f10_ontology: float    # Type safety [0,1]
    f11_command: float     # Authority [0,1]
    f12_injection: float   # Defense [0,1]
    f13_sovereign: float   # Human presence [0,1]

@dataclass
class APEXDials:
    """The 4 APEX dials derived from floor eigendecomposition"""
    A: float  # Akal (Mind/Structure)
    P: float  # Presence (Stability)
    X: float  # eXploration (Navigation)
    E: float  # Energy (Vitality)

    def to_dict(self) -> Dict[str, float]:
        return {'A': self.A, 'P': self.P, 'X': self.X, 'E': self.E}

def geometric_mean(values: List[float]) -> float:
    """Geometric mean — all values must be positive"""
    if any(v <= 0 for v in values):
        return 0.0
    product = math.prod(values)
    return product ** (1.0 / len(values))

def floors_to_dials(
    floors: FloorScores,
    compute_budget_used: float = 0.5,
    compute_budget_max: float = 1.0
) -> APEXDials:
    """
    Eigendecomposition: Collapse 13 floors → 4 APEX dials.

    This is the CRITICAL constitutional function that ensures
    genius is DERIVED from floor compliance, not arbitrary.

    Args:
        floors: The 13 constitutional floor scores
        compute_budget_used: Actual compute expended
        compute_budget_max: Maximum allowed compute

    Returns:
        APEXDials: The 4 principal components (A/P/X/E)
    """

    # A = AKAL (Mind/Clarity) — geometric mean of truth cluster
    # Floors: F2 (Truth), F4 (Clarity), F7 (Humility), F10 (Ontology)
    A = geometric_mean([
        floors.f2_truth,
        floors.f4_clarity,
        floors.f7_humility,
        floors.f10_ontology
    ])

    # P = PRESENT (Stability) — geometric mean of trust cluster
    # Floors: F1 (Amanah), F5 (Peace), F11 (Command)
    P = geometric_mean([
        floors.f1_amanah,
        floors.f5_peace,
        floors.f11_command
    ])

    # X = EXPLORATION (Navigation) — geometric mean of heart cluster
    # Floors: F3 (Tri-Witness), F6 (Empathy), F8 (Genius), F9 (Anti-Hantu)
    X = geometric_mean([
        floors.f3_tri_witness,
        floors.f6_empathy,
        floors.f8_genius,
        floors.f9_antihantu
    ])

    # E = ENERGY (Vitality) — system resources + boundary floors
    # Floors: F12 (Injection defense), F13 (Sovereign)
    # Plus: thermodynamic energy ratio
    energy_from_floors = geometric_mean([
        floors.f12_injection,
        floors.f13_sovereign
    ])
    energy_ratio = 1.0 - (compute_budget_used / max(compute_budget_max, 1e-6))
    E = (energy_from_floors + energy_ratio) / 2.0

    return APEXDials(A=A, P=P, X=X, E=E)

def calculate_genius_from_floors(
    floors: FloorScores,
    compute_budget_used: float = 0.5,
    compute_budget_max: float = 1.0
) -> Dict:
    """
    Constitutional Genius calculation with proper eigendecomposition.

    G = A × P × X × E²

    Where A/P/X/E are derived FROM floor scores, not set arbitrarily.
    """

    # Derive dials from floors (eigendecomposition)
    dials = floors_to_dials(floors, compute_budget_used, compute_budget_max)

    # Calculate G using the constitutional formula
    E_squared = dials.E ** 2
    G = dials.A * dials.P * dials.X * E_squared

    # Determine verdict based on G threshold
    threshold = 0.80
    if G >= threshold:
        verdict = "SEAL"
    elif G >= 0.60:
        verdict = "SABAR"
    else:
        verdict = "VOID"

    # Identify weakest dial for feedback
    dial_values = dials.to_dict()
    weakest_dial = min(dial_values, key=dial_values.get)

    return {
        'genius_score': G,
        'dials': dials.to_dict(),
        'threshold': threshold,
        'verdict': verdict,
        'passed': G >= threshold,
        'weakest_dial': weakest_dial,
        'weakest_value': dial_values[weakest_dial],
        'derivation': 'eigendecomposition_of_13_floors',
        'provenance': 'constitutional_measurement'
    }

# ============================================
# EXAMPLE: Proper vs Improper Usage
# ============================================

# ❌ WRONG: Manual dial setting (violates F2 Truth)
def wrong_way():
    '''Arbitrary dials not grounded in floor reality'''
    dials = APEXDials(A=0.92, P=0.88, X=0.85, E=0.95)  # Manual!
    G = dials.A * dials.P * dials.X * (dials.E ** 2)
    return G

# ✅ CORRECT: Eigendecomposition from floor scores
def right_way():
    '''Dials derived from constitutional compliance'''
    floors = FloorScores(
        f1_amanah=0.95, f2_truth=0.99, f3_tri_witness=0.96,
        f4_clarity=0.94, f5_peace=0.98, f6_empathy=0.97,
        f7_humility=0.04, f8_genius=0.85, f9_antihantu=0.92,
        f10_ontology=0.99, f11_command=0.93, f12_injection=0.91,
        f13_sovereign=1.0
    )
    result = calculate_genius_from_floors(floors)
    return result['genius_score']  # Derived, not arbitrary
```

### 10.4.5 Output Format (Human-Readable)

Instead of dumping 13 floor scores, the final verdict displays 4 dials:

```
╔══════════════════════════════════════════╗
║  APEX VERDICT (888 JUDGE)                ║
╠══════════════════════════════════════════╣
║  A (Mind):    0.94 ████████████░         ║
║  P (Peace):   0.87 ██████████░░░         ║
║  X (Explore): 0.82 █████████░░░░         ║  ← Weakest
║  E (Energy):  0.91 ███████████░░         ║
╠══════════════════════════════════════════╣
║  G = 0.94 × 0.87 × 0.82 × 0.91²          ║
║  G = 0.556 (threshold: 0.80)             ║
╠══════════════════════════════════════════╣
║  VERDICT: SABAR                          ║
║  REASON: Genius below threshold          ║
║  ACTION: Strengthen X (exploration)      ║
╚══════════════════════════════════════════╝
```

### 10.4.6 Constitutional Significance

| Aspect | Without Eigendecomposition | With Eigendecomposition |
|--------|---------------------------|-------------------------|
| **F2 Truth** | Dials arbitrary, ungrounded | Dials measured from reality |
| **F8 Genius** | Can be "faked" | Emerges from structure |
| **Verdict** | Subjective | Objective derivation |
| **Debug** | 13 numbers, confusing | 4 bars, actionable |
| **Integrity** | Violable | Cryptographically bound |

**The Core Principle:**
> *"The Knowing That Admits Not-Knowing"*

The eigendecomposition enforces that we don't manually set the dials—we measure them from the constitutional reality (floor scores). The dials emerge from the structure of compliance; they aren't imposed upon it.

---

## §12 THE 13 CONSTITUTIONAL FLOORS

### 11.1 Floor Reference

| Floor | Name | Symbol | Threshold | Geometry | Physics |
|-------|------|--------|-----------|----------|---------|
| F1 | Amanah | 🔄 | Reversibility | Orthogonal | Entropy export |
| F2 | Truth | 👁 | τ ≥ 0.99 | Orthogonal | Information |
| F3 | Tri-Witness | 🌿 | W₃ ≥ 0.95 | Toroidal | Conservation |
| F4 | Clarity | ✨ | ΔS ≤ 0 | Fractal | Entropy |
| F5 | Peace | ☮️ | Peace² ≥ 1.0 | Fractal | Equilibrium |
| F6 | Empathy | ❤️ | κᵣ ≥ 0.95 | Fractal | Resonance |
| F7 | Humility | 🌊 | Ω₀ ∈ [0.03,0.05] | Orthogonal | Uncertainty |
| F8 | Genius | 🔥 | G ≥ 0.80 | Toroidal | Work |
| F9 | Anti-Hantu | 🛡️ | C_dark < 0.30 | Toroidal | Shadow |
| F10 | Ontology | 📐 | Type safety | Orthogonal | Structure |
| F11 | Command | 🔐 | Verified | Toroidal | Authority |
| F12 | Injection | 🧿 | Risk < 0.85 | Fractal | Defense |
| F13 | Sovereign | 👤 | Human = 1.0 | Orthogonal | Choice |

### 11.2 Floor Constraints Summary

```python
FLOORS = {
    'F1': {'name': 'Amanah', 'check': lambda r: r.reversible},
    'F2': {'name': 'Truth', 'check': lambda s: s.confidence >= 0.99},
    'F3': {'name': 'TriWitness', 'check': lambda w: w.score >= 0.95},
    'F4': {'name': 'Clarity', 'check': lambda s: s.entropy_delta <= 0},
    'F5': {'name': 'Peace', 'check': lambda p: p.peace_squared >= 1.0},
    'F6': {'name': 'Empathy', 'check': lambda e: e.resonance >= 0.95},
    'F7': {'name': 'Humility', 'check': lambda u: 0.03 <= u.omega <= 0.05},
    'F8': {'name': 'Genius', 'check': lambda g: g.score >= 0.80},
    'F9': {'name': 'AntiHantu', 'check': lambda h: h.darkness < 0.30},
    'F10': {'name': 'Ontology', 'check': lambda o: o.typed},
    'F11': {'name': 'Command', 'check': lambda c: c.verified},
    'F12': {'name': 'Injection', 'check': lambda i: i.risk < 0.85},
    'F13': {'name': 'Sovereign', 'check': lambda s: s.human_present}
}
```

---

## §13 THE 000-999 METABOLIC LOOP

### 13.1 Stage Reference

| Stage | Code | Name | Geometry | Symbol | Physics |
|-------|------|------|----------|--------|---------|
| 000 | INIT | Initialization | — | 🌀 | Seed |
| 111 | SENSE | Perception | Orthogonal | 👁 | Input |
| 222 | THINK | Reasoning | Orthogonal | 🧠 | Processing |
| 333 | ATLAS | Model Selection | Fractal | 🗺️ | Navigation |
| 444 | EVIDENCE | Verification | Orthogonal | 🔍 | Measurement |
| 555 | EMPATHY | Care | Fractal | ❤️ | Resonance |
| 666 | BRIDGE | Alignment | Toroidal | 🌉 | Connection |
| 777 | EUREKA | Insight | Toroidal | 💡 | Emergence |
| 888 | JUDGE | Verdict | Toroidal | Ψ | Decision |
| 999 | SEAL | Cryptography | Toroidal | 🏛️ | Preservation |

### 13.2 Loop Invariants

```
∀ iteration:
  1. SEAL.output = INIT.input (strange loop)
  2. Energy(entropy_exported) > Energy(entropy_retained)
  3. Constitutional_compliance(iteration_n+1) >= Constitutional_compliance(iteration_n)
```

---

## §14 REFERENCES

### Physics

[^1]: Landauer, R. (1961). Irreversibility and heat generation in the computing process. *IBM Journal of Research and Development*, 5(3), 183-191.

[^2]: Friston, K. (2010). The free-energy principle: a unified brain theory? *Nature Reviews Neuroscience*, 11(2), 127-138.

[^3]: Leggett, A. J., & Garg, A. (1985). Quantum mechanics versus macroscopic realism. *Physical Review Letters*, 54(9), 857.

### Mathematics

[^4]: Nash, J. (1950). Equilibrium points in n-person games. *PNAS*, 36(1), 48-49.

[^5]: Lamport, L., Shostak, R., & Pease, M. (1982). The Byzantine Generals Problem. *ACM TOPLAS*, 4(3), 382-401.

[^6]: Austin, J. L. (1962). *How to Do Things with Words*. Harvard University Press.

[^7]: Hofstadter, D. R. (1979). *Gödel, Escher, Bach*. Basic Books.

[^8]: Gödel, K. (1931). Über formal unentscheidbare Sätze. *Monatshefte für Mathematik*, 38(1), 173-198.

### Code References

| Module | Path | Description |
|--------|------|-------------|
| APEX Kernel | `codebase/apex/kernel.py` | Main judgment engine |
| Floor Checks | `codebase/apex/floor_checks.py` | 13-floor validation |
| Trinity Solver | `codebase/apex/trinity_nine.py` | 9-paradox resolution |
| Merkle Vault | `codebase/vault/seal999.py` | Cryptographic sealing |
| Federation Physics | `codebase/federation/physics.py` | Thermodynamic primitives |
| Dial Converter | `codebase/floors/genius.py` | Eigendecomposition (§10.4) |

---

## §15 CROSS-REFERENCES

### To Other Theory Documents

| Section | References |
|---------|------------|
| Physics | `000_FOUNDATIONS.md` §3, `FEDERATION.md` §2 |
| Trinity | `010_TRINITY.md` §4-6 |
| Floors | `000_ARCHITECTURE.md` §5 |
| Implementation | `000_ARCHITECTURE.md` §7 |
| Symbolism | `000_FOUNDATIONS.md` §2, §8 |

---

## §16 APPENDIX: VERDICT FLOWCHART

```
START (888 JUDGE)
       │
       ▼
┌─────────────────┐
│ Hard Floors     │
│ (F1,F9,F10,F11) │
└────────┬────────┘
         │
    ┌────┴────┐
    │         │
   PASS     FAIL
    │         │
    ▼         ▼
┌───────┐  ┌───────┐
│Tri-Wit│  │ VOID  │
│ness   │  │       │
└───┬───┘  └───────┘
    │
    ▼
┌─────────────────┐
│ W₃ ≥ 0.95 ?     │
└────────┬────────┘
    ┌────┴────┐
   YES       NO
    │         │
    ▼         ▼
┌────────┐ ┌────────┐
│ Genius │ │ SABAR  │
│ Check  │ │        │
└───┬────┘ └────────┘
    │
    ▼
┌─────────────────┐
│ G ≥ 0.80 ?      │
└────────┬────────┘
    ┌────┴────┐
    │         │
   YES       NO
    │         │
    ▼         ▼
┌─────────┐ ┌─────────┐
│ F13 Sov │ │ SABAR   │
│ Override│ │ (if     │
│ Check   │ │ G≥0.60) │
└────┬────┘ │ VOID    │
     │      │ (if     │
     │      │ G<0.60) │
     │      └─────────┘
     ▼
┌─────────┐
│ CRITICAL│
│ Stakes? │
└────┬────┘
 ┌───┴───┐
YES     NO
 │       │
 ▼       ▼
┌────┐ ┌────┐
│HOLD│ │SEAL│
└────┘ └────┘
```

---

**DITEMPA BUKAN DIBERI**

*"The Knowing That Admits Not-Knowing"*

---

---

## §17 IMPLEMENTATION STATUS

| Feature | Status | Location | Notes |
|---------|--------|----------|-------|
| G = A×P×X×E² formula | ✅ Implemented | `codebase/floors/genius.py` | `GeniusCalculator.compute()` |
| A/P/X/E dials as dataclass | ✅ Implemented | `codebase/floors/genius.py` | `GeniusMetrics` class |
| **Eigendecomposition (§10.4)** | ❌ **NOT IMPLEMENTED** | — | **PRIORITY FIX NEEDED** |
| `floors_to_dials()` function | ❌ Missing | — | Derive A/P/X/E from F1-F13 |
| `calculate_genius_from_floors()` | ❌ Missing | — | End-to-end floor→G pipeline |
| 4-dial output format | ❌ Missing | — | Human-readable bar display |

### Critical Gap (v55.1 Target)

**Current State:** The `GeniusCalculator` accepts A/P/X/E as **manual inputs**.

**Required State:** A/P/X/E must be **derived** from the 13 floor scores via eigendecomposition (§10.4).

**Fix Required:**
1. Implement `floors_to_dials()` function (§10.4.4)
2. Update `GeniusCalculator` to accept `FloorScores` instead of raw dials
3. Change output format to show 4 dials instead of 13 floors
4. Update `codebase/engines/apex/apex_components.py` to use new function
5. Update `codebase/system/apex_prime.py` to use new function

**Impact:** This fixes the theory-to-code gap where genius could be arbitrarily set rather than measured from constitutional compliance.

---

*Document sealed at 888 JUDGE stage.*
*Merkle root computed at 999 SEAL.*
*Loop continues to 000 INIT.*
