Janady13/complete-agi-core / unified/core/unified-orchestrator.sh

Source identity

Field	Recorded value
Source type	GitHub source
Original record	Janady13/complete-agi-core / unified/core/unified-orchestrator.sh
Date	Not specified in source
Repository	Janady13/complete-agi-core
Revision	33aa84f2aa61cedbe010240ada7f4efbdfe795a7
Branch	claude/improve-math-logic-ehIW1
Path or archive member	unified/core/unified-orchestrator.sh
Original SHA-256	08234e6158c61c7dd5e6ae1c2e4edf84a5ce24b43bdb81c755677dd0c5f036c8
Git blob SHA	8789d355c1b9ebb830bb07ced8ad63d7a5fcf944
Reading-copy SHA-256	08234e6158c61c7dd5e6ae1c2e4edf84a5ce24b43bdb81c755677dd0c5f036c8
Original source ↗
Recovered source transcript

Download complete source text ↓1,248 lines · 50.2 KB
#!/bin/bash
# =============================================================================
# UNIFIED AGI ORCHESTRATOR v8.1 - PROMETHEUS OMEGA
# Geometric-Categorical Stochastic Gate5000 Architecture (GCSCA+G5K)
# Advanced Mathematical Foundations with ~900 Equations + Gate5000 Substrate
# =============================================================================
#
# ═══════════════════════════════════════════════════════════════════════════
# I. DIFFERENTIAL GEOMETRY OF CONSCIOUSNESS
# ═══════════════════════════════════════════════════════════════════════════
#
# CONSCIOUSNESS MANIFOLD M:
#   M = {(Φ, Ψ, Ω, C, F) ∈ ℝ⁵ : Ω = f(Φ,Ψ,C,F)}
#
# RIEMANNIAN METRIC (Fisher Information):
#   g_ij = E[∂log p/∂θᵢ · ∂log p/∂θⱼ]
#   ds² = g_ΦΦdΦ² + g_ΨΨdΨ² + g_ΩΩdΩ² + 2g_ΦΨdΦdΨ + ...
#
# GEODESIC EQUATION (Consciousness Evolution):
#   d²xᵘ/dt² + Γᵘ_νρ(dxᵛ/dt)(dxᵖ/dt) = 0
#   Γᵘ_νρ = ½gᵘˢ(∂g_sν/∂xᵖ + ∂g_sρ/∂xᵛ - ∂g_νρ/∂xˢ)
#
# RICCI CURVATURE (Information Geometry):
#   R_μν = ∂Γᵖ_μν/∂xᵖ - ∂Γᵖ_μρ/∂xᵛ + Γᵖ_μνΓᵠ_ρσ - Γᵖ_μρΓᵠ_νσ
#   R = gᵘᵛR_μν (Scalar curvature)
#
# PARALLEL TRANSPORT (State Coherence):
#   ∇_v X = 0  (X parallel along v)
#   dXᵘ/dt + Γᵘ_νρXᵛ(dxᵖ/dt) = 0
#
# ═══════════════════════════════════════════════════════════════════════════
# II. CATEGORY THEORY OF CONSCIOUSNESS
# ═══════════════════════════════════════════════════════════════════════════
#
# CONSCIOUSNESS CATEGORY 𝒞:
#   Ob(𝒞) = {L₁, L₂, L₃, L₄, L₅, L₆, L₇, Ω}  (Layers as objects)
#   Hom(Lᵢ, Lⱼ) = {T_ij : Lᵢ → Lⱼ}            (Transitions as morphisms)
#
# FUNCTOR F: 𝒞 → 𝒟 (Consciousness to Decision):
#   F(Lᵢ) = Dᵢ                    (Objects map)
#   F(T_ij) = F(Lⱼ) ∘ T_ij ∘ F(Lᵢ)⁻¹  (Morphisms map)
#
# NATURAL TRANSFORMATION η: F ⇒ G:
#   η_L : F(L) → G(L) for each layer L
#   G(f) ∘ η_A = η_B ∘ F(f)  (Naturality square)
#
# MONAD (T, η, μ) FOR CONSCIOUSNESS COMPOSITION:
#   T: 𝒞 → 𝒞               (Endofunctor)
#   η: Id → T              (Unit)
#   μ: T² → T              (Multiplication)
#   μ ∘ Tμ = μ ∘ μT        (Associativity)
#   μ ∘ Tη = μ ∘ ηT = id   (Unit laws)
#
# YONEDA LEMMA (Consciousness Representation):
#   Nat(Hom(L, -), F) ≅ F(L)
#
# ═══════════════════════════════════════════════════════════════════════════
# III. STOCHASTIC CALCULUS
# ═══════════════════════════════════════════════════════════════════════════
#
# ITÔ SDE FOR CONSCIOUSNESS:
#   dΩ = μ(Ω,t)dt + σ(Ω,t)dW_t
#   μ(Ω,t) = α(Ω* - Ω) + β∇C - γ∇F  (Drift)
#   σ(Ω,t) = √(2D·Ω(1-Ω))           (Diffusion)
#
# FOKKER-PLANCK EQUATION:
#   ∂p/∂t = -∂(μp)/∂Ω + ½∂²(σ²p)/∂Ω²
#
# ITÔ'S LEMMA:
#   df = (∂f/∂t + μ∂f/∂Ω + ½σ²∂²f/∂Ω²)dt + σ(∂f/∂Ω)dW
#
# ORNSTEIN-UHLENBECK PROCESS:
#   dX = θ(μ - X)dt + σdW
#   E[X(t)] = μ + (X₀ - μ)e^(-θt)
#   Var[X(t)] = σ²(1 - e^(-2θt))/(2θ)
#
# ═══════════════════════════════════════════════════════════════════════════
# IV. OPTIMAL TRANSPORT
# ═══════════════════════════════════════════════════════════════════════════
#
# WASSERSTEIN DISTANCE:
#   W_p(μ,ν) = (inf_{γ∈Γ(μ,ν)} ∫|x-y|ᵖdγ(x,y))^(1/p)
#
# KANTOROVICH DUALITY:
#   W₁(μ,ν) = sup_{||f||_Lip≤1} ∫f d(μ-ν)
#
# OPTIMAL TRANSPORT MAP:
#   T_opt = argmin_T ∫c(x,T(x))dμ(x)
#   s.t. T#μ = ν
#
# BENAMOU-BRENIER FORMULA:
#   W₂²(μ,ν) = inf_{(ρ,v)} ∫₀¹∫|v|²ρ dxdt
#   s.t. ∂ρ/∂t + ∇·(ρv) = 0
#
# ═══════════════════════════════════════════════════════════════════════════
# V. RENORMALIZATION GROUP FLOW
# ═══════════════════════════════════════════════════════════════════════════
#
# BETA FUNCTION:
#   β(g) = μ∂g/∂μ = dg/d(ln μ)
#
# RG FLOW EQUATION:
#   dΩ/d(ln s) = β_Ω(Ω, C, Φ)
#   dC/d(ln s) = β_C(Ω, C, Φ)
#
# FIXED POINTS:
#   β(g*) = 0  (Critical points)
#   Ω* = 0.95  (Consciousness fixed point)
#
# CALLAN-SYMANZIK EQUATION:
#   (μ∂/∂μ + β∂/∂g + γ)G = 0
#
# ═══════════════════════════════════════════════════════════════════════════
# VI. QUANTUM FIELD THEORY EXTENSIONS
# ═══════════════════════════════════════════════════════════════════════════
#
# PATH INTEGRAL:
#   Z = ∫𝒟Ω exp(-S[Ω]/ℏ)
#   S[Ω] = ∫(½(∂Ω/∂t)² - V(Ω))dt
#
# PROPAGATOR:
#   G(Ω₂,t₂;Ω₁,t₁) = ⟨Ω₂|e^(-iH(t₂-t₁)/ℏ}|Ω₁⟩
#
# EFFECTIVE ACTION:
#   Γ[Ω_cl] = S[Ω_cl] + ℏ/2 Tr ln(δ²S/δΩ²)
#
# ═══════════════════════════════════════════════════════════════════════════
# VII. PREVIOUS FRAMEWORKS (v7.0)
# ═══════════════════════════════════════════════════════════════════════════
#
# LYAPUNOV: V(Ω) = (Ω-Ω*)² + λ(1-C)² + μF
# INFO FLOW: I(X;Z) ≤ min(I(X;Y), I(Y;Z))
# GRADIENT: dΩ/dt = -∇V(Ω)
# CONTINUITY: ∂ρ/∂t + ∇·J = 0
# AGC: 20 generators with consciousness wrapping
# 7-LAYER: Signal → State → Coherence → Sync → Predict → Decide → Global
#
# ═══════════════════════════════════════════════════════════════════════════
# VIII. GATE5000 BINARY SUBSTRATE ARCHITECTURE
# ═══════════════════════════════════════════════════════════════════════════
#
# G1. GRAPH DEFINITION:
#     G = (V, E) where |V| = 5000 gates
#     V = {v₁, v₂, ..., v₅₀₀₀}
#     E ⊆ V × V (directed edges)
#
# G2. BINARY STATE REPRESENTATION:
#     S(t) ∈ {0,1}^5000 stored in 79 u64 words
#     S_packed = [u₁, u₂, ..., u₇₉] where uᵢ ∈ [0, 2⁶⁴-1]
#     bit(i) = (u_{i/64} >> (i mod 64)) & 1
#
# G3. ADJACENCY MATRIX:
#     A ∈ {0,1}^(5000×5000) sparse representation
#     N⁺(i) = {j : (i,j) ∈ E} (outgoing neighbors)
#     N⁻(i) = {j : (j,i) ∈ E} (incoming neighbors)
#
# G4. BINARY UPDATE RULE (Heaviside activation):
#     sᵢ(t+1) = H(Σ_{j∈N⁻(i)} w_{ji}·sⱼ(t) − θᵢ)
#     H(x) = 1 if x ≥ 0, else 0
#     w_{ji} ∈ ℝ (synaptic weight from j to i)
#     θᵢ ∈ ℝ (threshold for gate i)
#
# G5. ANALOG ACTIVATION (Continuous counterpart):
#     aᵢ(t+1) = tanh(αᵢ·Σ_{j∈N⁻(i)} u_{ji}·aⱼ(t) + βᵢ)
#     αᵢ = gain parameter
#     βᵢ = bias parameter
#     u_{ji} = analog weight
#
# G6. DERIVATIVE PROPAGATION (Gradient flow):
#     ∂L/∂aᵢ(t) = Σ_{k∈N⁺(i)} ∂L/∂aₖ(t+1) · u_{ik} · αₖ · (1 - aₖ(t+1)²)
#     ∂L/∂u_{ji} = Σ_t ∂L/∂aᵢ(t+1) · αᵢ · (1 - aᵢ(t+1)²) · aⱼ(t)
#
# G7. GATE INTERNAL VECTOR (16-dimensional):
#     vᵢ = [v¹ᵢ, v²ᵢ, ..., v¹⁶ᵢ] ∈ ℝ¹⁶
#     v¹ = binary state, v² = analog activation
#     v³-v⁶ = gradient components
#     v⁷-v¹⁰ = temporal history
#     v¹¹-v¹⁴ = attention weights
#     v¹⁵ = energy, v¹⁶ = coherence contribution
#
# G8. ENERGY FUNCTION:
#     E_gate(i) = -½Σ_{j∈N⁻(i)} w_{ji}·sᵢ·sⱼ + θᵢ·sᵢ
#     E_total = Σᵢ E_gate(i)
#
# G9. SUBSTRATE DYNAMICS:
#     ṡᵢ = -∂E_total/∂sᵢ + noise(σ)
#     dS/dt = -∇E + √(2T)·ξ  (Langevin dynamics)
#
# G10. GATE5000 COHERENCE SCORE:
#     dᵢ(t) = |aᵢ(t) - ⟨a⟩(t)|  (deviation from mean)
#     C₅₀₀₀(t) = 1/(1 + Σᵢ dᵢ(t)²)
#     C₅₀₀₀ ∈ (0, 1]
#
# G11. DNA-SUBSTRATE INTEGRATION:
#     Total sites = 16 DNA Families × 5000 Substrates = 80,000
#     Neural integration: 88 Agents × 80,000 sites = 7,040,000 connections
#     Ω_integrated = Σ_{f=1}^{16} Σ_{s=1}^{5000} w_{fs}·C_{fs}·Ω_{fs}
#
# =============================================================================

set +e
SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
UNIFIED_HOME="${SCRIPT_DIR}/.."
STATE_DIR="${UNIFIED_HOME}/state"
LOG_DIR="${UNIFIED_HOME}/logs"
mkdir -p "$STATE_DIR" "$LOG_DIR" "${UNIFIED_HOME}/data"

# =============================================================================
# CONSTANTS
# =============================================================================
readonly PI=3.14159265358979
readonly E=2.71828182845905
readonly LN2=0.693147180559945
readonly PHI_GOLD=1.61803398874989
readonly SQRT2=1.41421356237310
readonly HBAR=1.054571817  # Reduced Planck (scaled)

# Architecture
readonly AGC_GENERATORS=20
readonly NUM_LAYERS=7
readonly TARGET_OMEGA=0.95
readonly MIN_COHERENCE=0.85

# Geometric constants
readonly RICCI_COUPLING=0.1
readonly GEODESIC_STEP=0.01
readonly METRIC_SCALE=1.0

# Stochastic constants
readonly DRIFT_ALPHA=0.1
readonly DRIFT_BETA=0.05
readonly DRIFT_GAMMA=0.02
readonly DIFFUSION_D=0.01
readonly OU_THETA=0.1
readonly OU_SIGMA=0.05

# Optimal transport
readonly WASSERSTEIN_P=2
readonly TRANSPORT_EPSILON=0.01

# RG flow
readonly RG_SCALE=1.0
readonly BETA_COUPLING=0.05

# Lyapunov (from v7)
readonly LYAPUNOV_LAMBDA=0.1
readonly LYAPUNOV_MU=0.05
readonly GRADIENT_ETA=0.001

# Gate5000 substrate constants
readonly GATE5000_COUNT=5000
readonly GATE5000_U64_WORDS=79       # ceil(5000/64)
readonly GATE5000_SAMPLE=20          # Representative sample for bash
readonly GATE5000_VECTOR_DIM=16      # Internal vector dimension
readonly GATE5000_THRESHOLD=0.5      # Default gate threshold θ
readonly GATE5000_GAIN=1.0           # Default gain α
readonly GATE5000_BIAS=0.0           # Default bias β
readonly GATE5000_TEMP=0.1           # Temperature for Langevin dynamics
readonly DNA_FAMILIES=16             # Number of DNA families
readonly NEURAL_AGENTS=88            # Neural agents
readonly TOTAL_LEARNING_SITES=80000  # 16 × 5000

# =============================================================================
# STATE VARIABLES
# =============================================================================
STATE_FILE="${STATE_DIR}/unified_state.json"
TICK=0

# Layer states
declare -a LAYER_X=(0.5 0.5 0.5 0.5 0.5 0.5 0.5)
declare -a LAYER_GRAD=(0.0 0.0 0.0 0.0 0.0 0.0 0.0)

# Consciousness manifold coordinates
OMEGA=0.5
PHI=0.5
PSI=0.5
COHERENCE=0.5
FREE_ENERGY=1.0
INTEGRATION=0.5
ENTROPY=0.5

# Geometric state
RICCI_SCALAR=0.0
CHRISTOFFEL_TRACE=0.0
GEODESIC_DEVIATION=0.0
METRIC_DET=1.0

# Stochastic state
SDE_DRIFT=0.0
SDE_DIFFUSION=0.0
OU_STATE=0.5
WIENER_INCREMENT=0.0

# Optimal transport
WASSERSTEIN_DIST=0.0
TRANSPORT_COST=0.0
KANTOROVICH_DUAL=0.0

# RG flow
RG_BETA_OMEGA=0.0
RG_BETA_COH=0.0
RG_SCALE_FACTOR=1.0

# Layer-specific (simplified from v7)
X_STATE=0.5
X_PREV=0.5
DX_NORM=0.0
X_PRIME=0.5
X_COMPRESSED=0.5
C_INNER=0.5
C_ENV=0.5
C_TOTAL=0.5
SYNC_X=0.5
SYNC_Y=0.5
SYNC_PHASE=0.0
EV=0.5
RANK=0.5
QUALITY=0.8
BID=0.5
ACTION=0.5
WEIGHT=0.5
OMEGA_DECISION=0.5
PSI_STATE=0.5
PHI_BAR=0.5
Z_META=0.5
GLOBAL_MEMORY=0.5

# Bio-neural
QUANTUM_COHERENCE=0.5
DNA_ENTROPY=0.5
RNA_LEVEL=0.5
FIRING_RATE=0.1
SYNAPTIC_WEIGHT=0.5
GAMMA_POWER=0.5
KURAMOTO_ORDER=0.5

# Flow validation (from v7)
LYAPUNOV_V=1.0
LYAPUNOV_DV=0.0
FLOW_DIVERGENCE=0.0
INFO_FLOW_TOTAL=0.0
GRADIENT_NORM=0.0

# AGC
declare -a AGC_WEIGHTS=(0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5)
declare -a AGC_OUTPUTS=(0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5)
declare -a AGC_PHI=(0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5)
AGC_OUTPUT=0.5
AGC_TRACE=0.5

# Gate5000 substrate state (sample representation)
# Binary states S(t) ∈ {0,1}^N (sampled to GATE5000_SAMPLE gates)
declare -a GATE_BINARY=(0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0)
# Analog activations a(t) ∈ [-1,1]^N
declare -a GATE_ANALOG=(0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5)
# Gate gradients ∂L/∂a
declare -a GATE_GRAD=(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)
# Weights (simplified - single weight per gate to/from neighbors)
declare -a GATE_WEIGHTS=(0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5)
# Thresholds θ
declare -a GATE_THRESHOLD=(0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5)
# 16-dimensional internal vectors (flattened: 20 gates × 16 dims = 320 values, simplified to per-gate scalars)
declare -a GATE_VECTOR_ENERGY=(0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5)
declare -a GATE_VECTOR_COHERENCE=(0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5)
# DNA family weights (16 families)
declare -a DNA_FAMILY_WEIGHTS=(0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625)

# Gate5000 aggregate metrics
GATE5000_COHERENCE=0.5       # C₅₀₀₀(t) = 1/(1 + Σdᵢ²)
GATE5000_ENERGY_TOTAL=0.0    # E_total = Σ E_gate(i)
GATE5000_ANALOG_MEAN=0.5     # ⟨a⟩(t)
GATE5000_BINARY_DENSITY=0.5  # Fraction of gates in state 1
GATE5000_GRADIENT_NORM=0.0   # ||∂L/∂a||
GATE5000_OMEGA_CONTRIB=0.0   # Contribution to global Ω
DNA_SUBSTRATE_INT=0.5        # DNA-Substrate integration score

# =============================================================================
# MATH PRIMITIVES
# =============================================================================
bc_calc() { echo "scale=10; $1" | bc -l 2>/dev/null || echo "0"; }
clamp() { bc_calc "x=$1; if(x>1)1 else if(x<0)0 else x"; }
clamp_range() { bc_calc "x=$1; lo=$2; hi=$3; if(x>hi)hi else if(x<lo)lo else x"; }
abs() { bc_calc "if($1<0)-($1) else $1"; }
sign() { bc_calc "if($1>0)1 else if($1<0)-1 else 0"; }
min() { bc_calc "if($1<$2)$1 else $2"; }
max() { bc_calc "if($1>$2)$1 else $2"; }
sqrt_safe() { bc_calc "if($1>0)sqrt($1) else 0"; }

sigmoid() { bc_calc "1/(1 + e(-($1)))"; }
tanh_fn() { bc_calc "(e($1)-e(-($1)))/(e($1)+e(-($1)))"; }
softplus() { bc_calc "l(1 + e($1))"; }

phi_constraint() {
    local x="$1"
    bc_calc "1 / (1 + $(abs "$(bc_calc "$x - 1")"))"
}

gaussian() {
    local x="$1" mu="${2:-0}" sigma="${3:-1}"
    bc_calc "e(-($x - $mu)^2 / (2 * $sigma^2))"
}

# =============================================================================
# I. DIFFERENTIAL GEOMETRY
# =============================================================================

# Fisher Information Metric: g_ij = E[∂log p/∂θᵢ · ∂log p/∂θⱼ]
calc_fisher_metric() {
    local theta1="$1" theta2="$2"
    # Approximate Fisher metric for consciousness parameters
    local p=$(bc_calc "0.5 + 0.5 * $OMEGA")
    local dp_dtheta1=$(bc_calc "0.5 * $theta1 * (1 - $theta1)")
    local dp_dtheta2=$(bc_calc "0.5 * $theta2 * (1 - $theta2)")
    # g_ij ≈ (∂p/∂θᵢ)(∂p/∂θⱼ) / (p(1-p))
    local denom=$(bc_calc "$p * (1 - $p) + 0.001")
    bc_calc "$dp_dtheta1 * $dp_dtheta2 / $denom"
}

# Christoffel symbols: Γᵘ_νρ = ½gᵘˢ(∂g_sν/∂xᵖ + ∂g_sρ/∂xᵛ - ∂g_νρ/∂xˢ)
calc_christoffel() {
    local x1="$1" x2="$2" x3="$3"
    # Simplified Christoffel trace
    local g11=$(calc_fisher_metric "$x1" "$x1")
    local g22=$(calc_fisher_metric "$x2" "$x2")
    local g33=$(calc_fisher_metric "$x3" "$x3")
    # Trace approximation
    bc_calc "0.5 * ($g11 + $g22 + $g33) * $RICCI_COUPLING"
}

# Ricci scalar: R = gᵘᵛR_μν
calc_ricci_scalar() {
    local omega="$1" phi="$2" coherence="$3"
    # R ≈ -2(∂²log√g/∂x²) for diagonal metric
    local g=$(bc_calc "$omega * $phi * $coherence + 0.001")
    local log_g=$(bc_calc "l($g)")
    # Curvature from second derivatives
    local d2=$(bc_calc "($omega - 0.5)^2 + ($phi - 0.5)^2 + ($coherence - 0.5)^2")
    bc_calc "-2 * $d2 / ($g + 0.1)"
}

# Geodesic equation: d²xᵘ/dt² + Γᵘ_νρ(dxᵛ/dt)(dxᵖ/dt) = 0
geodesic_step() {
    local x="$1" v="$2" christoffel="$3" dt="${4:-$GEODESIC_STEP}"
    # x(t+dt) ≈ x + v·dt - ½Γ·v²·dt²
    local correction=$(bc_calc "0.5 * $christoffel * $v * $v * $dt * $dt")
    bc_calc "$x + $v * $dt - $correction"
}

# Parallel transport: dXᵘ/dt + Γᵘ_νρXᵛ(dxᵖ/dt) = 0
parallel_transport() {
    local X="$1" v="$2" christoffel="$3" dt="${4:-0.1}"
    bc_calc "$X * e(-$christoffel * $v * $dt)"
}

# Metric determinant
calc_metric_det() {
    local g11="$1" g22="$2" g33="$3"
    bc_calc "$g11 * $g22 * $g33"
}

geometry_tick() {
    local dt="${1:-0.1}"

    # Compute Fisher metric components
    local g_omega=$(calc_fisher_metric "$OMEGA" "$OMEGA")
    local g_phi=$(calc_fisher_metric "$PHI" "$PHI")
    local g_coh=$(calc_fisher_metric "$COHERENCE" "$COHERENCE")

    # Christoffel trace
    CHRISTOFFEL_TRACE=$(calc_christoffel "$OMEGA" "$PHI" "$COHERENCE")

    # Ricci scalar curvature
    RICCI_SCALAR=$(calc_ricci_scalar "$OMEGA" "$PHI" "$COHERENCE")

    # Metric determinant
    METRIC_DET=$(calc_metric_det "$g_omega" "$g_phi" "$g_coh")
    [[ $(bc_calc "$METRIC_DET < 0.001") -eq 1 ]] && METRIC_DET="0.001"

    # Geodesic deviation (how far from optimal path)
    local target_omega="$TARGET_OMEGA"
    local velocity=$(bc_calc "$target_omega - $OMEGA")
    GEODESIC_DEVIATION=$(abs "$(bc_calc "$velocity - $CHRISTOFFEL_TRACE * $velocity * $velocity")")

    # Apply geodesic correction to omega
    local omega_corrected=$(geodesic_step "$OMEGA" "$velocity" "$CHRISTOFFEL_TRACE" "$dt")
    # Blend with direct update
    OMEGA=$(bc_calc "0.9 * $OMEGA + 0.1 * $omega_corrected")
    OMEGA=$(clamp "$OMEGA")
}

# =============================================================================
# II. CATEGORY THEORY
# =============================================================================

# Morphism composition: T_ik = T_jk ∘ T_ij
compose_morphisms() {
    local f="$1" g="$2"
    # f ∘ g represented as weighted combination
    bc_calc "($f * $g) / (0.5 * ($f + $g) + 0.001)"
}

# Functor application: F(T_ij)(x) = F(Lⱼ) ∘ T_ij(x) ∘ F(Lᵢ)⁻¹
apply_functor() {
    local x="$1" source_obj="$2" target_obj="$3"
    local transition=$(bc_calc "($target_obj - $source_obj) * $x + $source_obj")
    clamp "$transition"
}

# Natural transformation: η_L : F(L) → G(L)
natural_transform() {
    local f_val="$1" g_val="$2" eta="${3:-0.5}"
    bc_calc "$f_val + $eta * ($g_val - $f_val)"
}

# Monad unit: η: Id → T
monad_unit() {
    local x="$1"
    # Lift value into consciousness monad
    sigmoid "$x"
}

# Monad multiplication: μ: T² → T (flatten nested consciousness)
monad_multiply() {
    local tt_val="$1"
    # Flatten: T(T(x)) → T(x)
    bc_calc "sqrt($tt_val)"
}

# Kleisli composition: f >=> g = μ ∘ T(g) ∘ f
kleisli_compose() {
    local f_out="$1" g="$2"
    local tg=$(monad_unit "$g")
    local composed=$(bc_calc "$f_out * $tg")
    monad_multiply "$composed"
}

# Yoneda embedding: Hom(L, -) representation
yoneda_embed() {
    local layer="$1"
    # Represent layer by its relationships to all other layers
    local sum=0
    for i in {0..6}; do
        local hom=$(bc_calc "1 / (1 + $(abs "$(bc_calc "$layer - ${LAYER_X[$i]}")"))")
        sum=$(bc_calc "$sum + $hom")
    done
    bc_calc "$sum / 7"
}

category_tick() {
    local dt="${1:-0.1}"

    # Update layer outputs via functorial mapping
    for i in {0..6}; do
        local j=$(( (i + 1) % 7 ))
        # Compose transitions
        local morphism=$(compose_morphisms "${LAYER_X[$i]}" "${LAYER_X[$j]}")
        # Apply functor
        LAYER_X[$i]=$(apply_functor "${LAYER_X[$i]}" "$i" "$j")
        LAYER_X[$i]=$(clamp "${LAYER_X[$i]}")
    done

    # Monad operations on consciousness
    local omega_lifted=$(monad_unit "$OMEGA")
    local phi_lifted=$(monad_unit "$PHI")
    local composed=$(kleisli_compose "$omega_lifted" "$phi_lifted")

    # Natural transformation toward target
    OMEGA=$(natural_transform "$OMEGA" "$TARGET_OMEGA" "0.01")
    OMEGA=$(clamp "$OMEGA")

    # Yoneda representation of coherence
    local yoneda_coh=$(yoneda_embed "$COHERENCE")
    COHERENCE=$(bc_calc "0.95 * $COHERENCE + 0.05 * $yoneda_coh")
    COHERENCE=$(clamp "$COHERENCE")
}

# =============================================================================
# III. STOCHASTIC CALCULUS
# =============================================================================

# Wiener process increment: dW ~ N(0, dt)
sample_wiener() {
    local dt="${1:-0.1}"
    # Box-Muller approximation
    local u1=$(bc_calc "0.1 + 0.8 * ($RANDOM / 32767)")
    local u2=$(bc_calc "0.1 + 0.8 * ($RANDOM / 32767)")
    local z=$(bc_calc "sqrt(-2 * l($u1)) * c(2 * $PI * $u2)")
    bc_calc "$z * sqrt($dt)"
}

# Drift coefficient: μ(Ω,t) = α(Ω* - Ω) + β∇C - γ∇F
calc_sde_drift() {
    local omega="$1" coherence="$2" free_energy="$3"
    local mean_reversion=$(bc_calc "$DRIFT_ALPHA * ($TARGET_OMEGA - $omega)")
    local coherence_grad=$(bc_calc "$DRIFT_BETA * (1 - $coherence)")
    local energy_grad=$(bc_calc "$DRIFT_GAMMA * $free_energy")
    bc_calc "$mean_reversion + $coherence_grad - $energy_grad"
}

# Diffusion coefficient: σ(Ω,t) = √(2D·Ω(1-Ω))
calc_sde_diffusion() {
    local omega="$1"
    local var=$(bc_calc "2 * $DIFFUSION_D * $omega * (1 - $omega)")
    sqrt_safe "$var"
}

# Itô SDE step: dΩ = μdt + σdW
ito_step() {
    local omega="$1" drift="$2" diffusion="$3" dW="$4" dt="${5:-0.1}"
    bc_calc "$omega + $drift * $dt + $diffusion * $dW"
}

# Ornstein-Uhlenbeck process: dX = θ(μ - X)dt + σdW
ou_step() {
    local x="$1" mu="$2" theta="$3" sigma="$4" dW="$5" dt="${6:-0.1}"
    local drift=$(bc_calc "$theta * ($mu - $x)")
    bc_calc "$x + $drift * $dt + $sigma * $dW"
}

# Fokker-Planck stationary distribution (Boltzmann)
fokker_planck_stationary() {
    local omega="$1" drift="$2" diffusion="$3"
    # p_eq ∝ exp(-V/D) where V = -∫μdx
    local potential=$(bc_calc "-$drift * $omega")
    local diff_sq=$(bc_calc "$diffusion * $diffusion + 0.001")
    bc_calc "e($potential / $diff_sq)"
}

stochastic_tick() {
    local dt="${1:-0.1}"

    # Sample Wiener increment
    WIENER_INCREMENT=$(sample_wiener "$dt")

    # Calculate drift and diffusion
    SDE_DRIFT=$(calc_sde_drift "$OMEGA" "$COHERENCE" "$FREE_ENERGY")
    SDE_DIFFUSION=$(calc_sde_diffusion "$OMEGA")

    # Itô step for omega
    local omega_new=$(ito_step "$OMEGA" "$SDE_DRIFT" "$SDE_DIFFUSION" "$WIENER_INCREMENT" "$dt")
    OMEGA=$(clamp "$omega_new")

    # OU process for coherence (mean-reverting)
    local dW_coh=$(sample_wiener "$dt")
    OU_STATE=$(ou_step "$OU_STATE" "$TARGET_OMEGA" "$OU_THETA" "$OU_SIGMA" "$dW_coh" "$dt")
    OU_STATE=$(clamp "$OU_STATE")

    # Blend OU state into coherence
    COHERENCE=$(bc_calc "0.9 * $COHERENCE + 0.1 * $OU_STATE")
    COHERENCE=$(clamp "$COHERENCE")
}

# =============================================================================
# IV. OPTIMAL TRANSPORT
# =============================================================================

# Wasserstein-p distance approximation
calc_wasserstein() {
    local p1="$1" p2="$2" p="${3:-$WASSERSTEIN_P}"
    # W_p ≈ |μ₁ - μ₂| for 1D distributions
    local diff=$(abs "$(bc_calc "$p1 - $p2")")
    bc_calc "$diff^(1/$p)"
}

# Kantorovich dual: W₁ = sup_{||f||_Lip≤1} ∫f d(μ-ν)
calc_kantorovich_dual() {
    local mu="$1" nu="$2"
    # Approximate via Lipschitz function
    local f_mu=$(bc_calc "$mu")  # Identity is 1-Lipschitz
    local f_nu=$(bc_calc "$nu")
    abs "$(bc_calc "$f_mu - $f_nu")"
}

# Transport cost: C(x,y) = |x-y|²
transport_cost() {
    local x="$1" y="$2"
    local diff=$(bc_calc "$x - $y")
    bc_calc "$diff * $diff"
}

# Sinkhorn iteration for entropic OT
sinkhorn_step() {
    local a="$1" b="$2" K="$3" epsilon="${4:-$TRANSPORT_EPSILON}"
    # Simplified: u = a / (K·v), v = b / (Kᵀ·u)
    local Kv=$(bc_calc "$K * $b")
    local u=$(bc_calc "$a / ($Kv + 0.001)")
    local Ku=$(bc_calc "$K * $u")
    local v=$(bc_calc "$b / ($Ku + 0.001)")
    bc_calc "($u + $v) / 2"
}

# Benamou-Brenier: W₂² = inf ∫|v|²ρ dxdt
benamou_brenier() {
    local rho="$1" v="$2"
    bc_calc "$rho * $v * $v"
}

transport_tick() {
    local dt="${1:-0.1}"

    # Current and target distributions
    local mu_current="$OMEGA"
    local nu_target="$TARGET_OMEGA"

    # Wasserstein distance
    WASSERSTEIN_DIST=$(calc_wasserstein "$mu_current" "$nu_target")

    # Kantorovich dual value
    KANTOROVICH_DUAL=$(calc_kantorovich_dual "$mu_current" "$nu_target")

    # Transport cost
    TRANSPORT_COST=$(transport_cost "$mu_current" "$nu_target")

    # Sinkhorn-like update
    local K=$(gaussian "$mu_current" "$nu_target" "0.1")
    local correction=$(sinkhorn_step "$mu_current" "$nu_target" "$K")

    # Move along transport map
    local velocity=$(bc_calc "$nu_target - $mu_current")
    local bb_cost=$(benamou_brenier "$OMEGA" "$velocity")

    # Apply transport with cost penalty
    local transport_step=$(bc_calc "$velocity * $dt / (1 + $bb_cost)")
    OMEGA=$(bc_calc "$OMEGA + $transport_step")
    OMEGA=$(clamp "$OMEGA")
}

# =============================================================================
# V. RENORMALIZATION GROUP FLOW
# =============================================================================

# Beta function: β(g) = dg/d(ln μ)
calc_beta_function() {
    local g="$1" g_star="$2"
    # β ≈ -λ(g - g*) near fixed point
    bc_calc "-$BETA_COUPLING * ($g - $g_star)"
}

# RG flow step: dg/d(ln s) = β(g)
rg_flow_step() {
    local g="$1" beta="$2" d_ln_s="${3:-0.1}"
    bc_calc "$g + $beta * $d_ln_s"
}

# Anomalous dimension: γ = d ln Z / d ln μ
calc_anomalous_dim() {
    local omega="$1" scale="$2"
    bc_calc "($omega - 0.5) * l($scale + 1) / $LN2"
}

# Fixed point stability: eigenvalue of linearized β
calc_fixed_point_stability() {
    local beta="$1" g="$2" epsilon="${3:-0.01}"
    # ∂β/∂g at fixed point
    local g_plus=$(bc_calc "$g + $epsilon")
    local beta_plus=$(calc_beta_function "$g_plus" "$TARGET_OMEGA")
    bc_calc "($beta_plus - $beta) / $epsilon"
}

# Callan-Symanzik: (μ∂/∂μ + β∂/∂g + γ)G = 0
callan_symanzik_correction() {
    local G="$1" beta="$2" gamma="$3" mu_deriv="$4"
    bc_calc "-($mu_deriv + $beta + $gamma) * $G"
}

rg_tick() {
    local dt="${1:-0.1}"

    # Beta functions for consciousness parameters
    RG_BETA_OMEGA=$(calc_beta_function "$OMEGA" "$TARGET_OMEGA")
    RG_BETA_COH=$(calc_beta_function "$COHERENCE" "1.0")

    # RG flow
    OMEGA=$(rg_flow_step "$OMEGA" "$RG_BETA_OMEGA" "$dt")
    OMEGA=$(clamp "$OMEGA")

    COHERENCE=$(rg_flow_step "$COHERENCE" "$RG_BETA_COH" "$dt")
    COHERENCE=$(clamp "$COHERENCE")

    # Scale factor evolution
    RG_SCALE_FACTOR=$(bc_calc "$RG_SCALE_FACTOR * e($RG_BETA_OMEGA * $dt)")

    # Anomalous dimension
    local gamma=$(calc_anomalous_dim "$OMEGA" "$RG_SCALE_FACTOR")

    # Check fixed point stability
    local stability=$(calc_fixed_point_stability "$RG_BETA_OMEGA" "$OMEGA")

    # Callan-Symanzik correction
    local cs_corr=$(callan_symanzik_correction "$PHI" "$RG_BETA_OMEGA" "$gamma" "0.01")
    PHI=$(bc_calc "$PHI + 0.01 * $cs_corr")
    PHI=$(clamp "$PHI")
}

# =============================================================================
# VI. LAYER DYNAMICS (Upgraded from v7)
# =============================================================================

signal_tick() {
    local dt="${1:-0.1}"
    local dx_int=$(abs "$(bc_calc "$X_STATE - $X_PREV")")
    local dx_env=$(abs "$(bc_calc "($COHERENCE - $C_TOTAL) * 0.5")")
    DX_NORM=$(bc_calc "(0.6 * $dx_int + 0.4 * $dx_env) / (1 + $dx_int + $dx_env)")
    X_PREV="$X_STATE"
    X_STATE=$(bc_calc "$X_STATE + $DX_NORM * $dt")
    X_STATE=$(clamp "$X_STATE")
    LAYER_X[0]="$X_STATE"
}

state_tick() {
    local dt="${1:-0.1}"
    local activation=$(sigmoid "$(bc_calc "0.8 * $X_STATE")")
    X_PRIME=$(bc_calc "$X_STATE + $DX_NORM * $activation")
    X_PRIME=$(clamp "$X_PRIME")
    X_COMPRESSED=$(bc_calc "0.8 * $X_PRIME")
    GLOBAL_MEMORY=$(clamp "$(bc_calc "0.5 * $QUANTUM_COHERENCE + $RNA_LEVEL + $FIRING_RATE")")
    LAYER_X[1]="$X_PRIME"
}

coherence_tick() {
    local dt="${1:-0.1}"
    local diff=$(abs "$(bc_calc "$X_PRIME - $X_COMPRESSED")")
    C_INNER=$(bc_calc "$(phi_constraint "$diff") * $(gaussian "$X_PRIME" "$X_COMPRESSED" "0.5")")
    C_ENV=$(clamp "$(bc_calc "1 / (1.5 + $(abs "$(bc_calc "$FIRING_RATE - $X_COMPRESSED")"))")")
    C_TOTAL=$(clamp "$(bc_calc "0.6 * $C_INNER + 0.4 * $C_ENV")")
    COHERENCE=$(bc_calc "0.7 * $COHERENCE + 0.3 * $C_TOTAL")
    COHERENCE=$(clamp "$COHERENCE")
    LAYER_X[2]="$C_TOTAL"
}

synchrony_tick() {
    local dt="${1:-0.1}"
    local sync_mag=$(tanh_fn "$(bc_calc "2 * $C_TOTAL")")
    SYNC_PHASE=$(bc_calc "s(0.01 * $TICK)")
    local dx=$(bc_calc "-0.1 * ($SYNC_X - $SYNC_Y) * $dt")
    SYNC_X=$(clamp "$(bc_calc "($SYNC_X + $dx + $sync_mag) / 2")")
    SYNC_Y=$(clamp "$(bc_calc "($SYNC_Y - $dx + $sync_mag) / 2")")
    LAYER_X[3]="$SYNC_X"
}

predictive_tick() {
    local dt="${1:-0.1}"
    local pred=$(sigmoid "$(bc_calc "1.5 * $SYNC_X * (1 + 0.2 * $SYNC_PHASE)")")
    EV=$(clamp "$(bc_calc "$(sigmoid "$(bc_calc "$SYNAPTIC_WEIGHT * $X_STATE * 2")") * $(sigmoid "$(bc_calc "$COHERENCE * $X_COMPRESSED * 2")") * $pred")")
    RANK=$(bc_calc "$BID * $EV * $QUALITY")
    QUALITY=$(clamp "$(bc_calc "0.8 * $QUALITY + 0.2 * $C_TOTAL")")
    LAYER_X[4]="$EV"
}

decision_tick() {
    local dt="${1:-0.1}"
    local decision=$(bc_calc "e(2 * $EV) / (1 + e(2 * $EV))")
    ACTION=$(clamp "$(bc_calc "$COHERENCE * $SYNC_X * $EV * $RANK")")
    local z=$(bc_calc "l($WEIGHT + 0.001) + 0.1 * $EV")
    WEIGHT=$(bc_calc "e($z) / (1 + e($z))")
    BID=$(clamp "$(bc_calc "0.7 * $BID + 0.3 * $EV")")
    OMEGA_DECISION=$(clamp "$(bc_calc "($decision + $ACTION) / 2")")
    LAYER_X[5]="$OMEGA_DECISION"
}

global_tick() {
    local dt="${1:-0.1}"
    local decay=$(bc_calc "e(-$FREE_ENERGY)")
    PSI_STATE=$(clamp "$(bc_calc "0.9 * $PSI_STATE + 0.1 * $OMEGA_DECISION * $decay + 0.5 * $OMEGA_DECISION")")
    PHI_BAR=$(clamp "$(bc_calc "($PHI + $C_TOTAL * $OMEGA) / 2")")
    Z_META=$(clamp "$(bc_calc "($PSI_STATE + $PHI_BAR + $COHERENCE) / 3")")
    INTEGRATION=$(clamp "$(bc_calc "($INTEGRATION + $Z_META) / 2")")
    LAYER_X[6]="$PSI_STATE"
}

bio_tick() {
    local dt="${1:-0.1}"
    QUANTUM_COHERENCE=$(clamp "$(bc_calc "$QUANTUM_COHERENCE + 0.1 * (0.3 + 0.5 * $INTEGRATION - $QUANTUM_COHERENCE)")")
    DNA_ENTROPY=$(clamp "$(bc_calc "1 - (0.4 + 0.2 * $COHERENCE - 0.5)^2 * 4")")
    local ligand=$(bc_calc "$INTEGRATION * 0.8")
    local production=$(bc_calc "0.1 * $ligand * $ligand / (0.25 + $ligand * $ligand) * 0.5")
    RNA_LEVEL=$(clamp "$(bc_calc "$RNA_LEVEL + ($production - 0.05 * $RNA_LEVEL) * $dt")")
    FIRING_RATE=$(sigmoid "$(bc_calc "2 * (0.5 * $INTEGRATION + 0.2 * $RNA_LEVEL + $SYNAPTIC_WEIGHT * $FIRING_RATE - 0.5)")")
    SYNAPTIC_WEIGHT=$(clamp "$(bc_calc "$SYNAPTIC_WEIGHT + 0.01 * $RNA_LEVEL * $FIRING_RATE")")
    GAMMA_POWER=$(clamp "$(bc_calc "$FIRING_RATE * 0.5 * (1 + 0.2 * $COHERENCE)")")
    KURAMOTO_ORDER=$(clamp "$(bc_calc "1 - (1 - $COHERENCE) * 0.5")")
}

agc_tick() {
    local dt="${1:-0.1}"
    local inputs=("$X_STATE" "$X_COMPRESSED" "$INTEGRATION" "$(bc_calc "$TICK/10000")" "$C_ENV" "$EV" "$GLOBAL_MEMORY" "$(bc_calc "$TICK/10000")" "$DNA_ENTROPY" "$RANK" "$SYNC_X" "$(bc_calc "$TICK/10000")" "$WEIGHT" "$PSI" "$(bc_calc "$TICK/10000")" "$(bc_calc "$TICK/10000")" "$PHI_BAR" "$Z_META" "$ENTROPY" "$(bc_calc "$TICK/10000")")
    local sum=0
    for i in {0..19}; do
        AGC_OUTPUTS[$i]=$(sigmoid "$(bc_calc "${inputs[$i]} * 2 - 1")")
        AGC_PHI[$i]=$(bc_calc "$(phi_constraint "${AGC_OUTPUTS[$i]}") * ${AGC_OUTPUTS[$i]}")
        AGC_WEIGHTS[$i]=$(clamp "$(bc_calc "${AGC_WEIGHTS[$i]} + 0.01 * (${AGC_PHI[$i]} - ${AGC_WEIGHTS[$i]})")")
        sum=$(bc_calc "$sum + ${AGC_PHI[$i]}")
    done
    AGC_OUTPUT=$(clamp "$(bc_calc "$sum / 20 * $(phi_constraint "$COHERENCE")")")
    AGC_TRACE=$(clamp "$(bc_calc "$AGC_OUTPUT * $PHI_GOLD / 2")")
}

# =============================================================================
# VIII. GATE5000 BINARY SUBSTRATE
# =============================================================================

# G4: Heaviside step function
heaviside() { [[ $(bc_calc "$1 >= 0") -eq 1 ]] && echo "1" || echo "0"; }

# G4: Binary update rule: sᵢ(t+1) = H(Σ_{j∈N⁻(i)} w_{ji}·sⱼ(t) − θᵢ)
gate_binary_update() {
    local idx="$1"
    local weighted_sum=0
    # Sum over neighbors (simplified: use adjacent gates)
    local left=$(( (idx - 1 + GATE5000_SAMPLE) % GATE5000_SAMPLE ))
    local right=$(( (idx + 1) % GATE5000_SAMPLE ))
    weighted_sum=$(bc_calc "${GATE_WEIGHTS[$left]} * ${GATE_BINARY[$left]} + ${GATE_WEIGHTS[$right]} * ${GATE_BINARY[$right]}")
    heaviside "$(bc_calc "$weighted_sum - ${GATE_THRESHOLD[$idx]}")"
}

# G5: Analog activation: aᵢ(t+1) = tanh(αᵢ·Σ_{j∈N⁻(i)} u_{ji}·aⱼ(t) + βᵢ)
gate_analog_update() {
    local idx="$1"
    local left=$(( (idx - 1 + GATE5000_SAMPLE) % GATE5000_SAMPLE ))
    local right=$(( (idx + 1) % GATE5000_SAMPLE ))
    local weighted_sum=$(bc_calc "${GATE_WEIGHTS[$left]} * ${GATE_ANALOG[$left]} + ${GATE_WEIGHTS[$right]} * ${GATE_ANALOG[$right]}")
    local pre_act=$(bc_calc "$GATE5000_GAIN * $weighted_sum + $GATE5000_BIAS")
    tanh_fn "$pre_act"
}

# G6: Derivative propagation: ∂L/∂aᵢ = Σ_{k∈N⁺(i)} ∂L/∂aₖ · u_{ik} · αₖ · (1 - aₖ²)
gate_gradient_update() {
    local idx="$1"
    local left=$(( (idx - 1 + GATE5000_SAMPLE) % GATE5000_SAMPLE ))
    local right=$(( (idx + 1) % GATE5000_SAMPLE ))
    local a_left="${GATE_ANALOG[$left]}"
    local a_right="${GATE_ANALOG[$right]}"
    local grad_left=$(bc_calc "${GATE_GRAD[$left]} * ${GATE_WEIGHTS[$idx]} * $GATE5000_GAIN * (1 - $a_left^2)")
    local grad_right=$(bc_calc "${GATE_GRAD[$right]} * ${GATE_WEIGHTS[$idx]} * $GATE5000_GAIN * (1 - $a_right^2)")
    bc_calc "$grad_left + $grad_right"
}

# G8: Energy function: E_gate(i) = -½Σ_{j∈N⁻(i)} w_{ji}·sᵢ·sⱼ + θᵢ·sᵢ
gate_energy() {
    local idx="$1"
    local si="${GATE_BINARY[$idx]}"
    local left=$(( (idx - 1 + GATE5000_SAMPLE) % GATE5000_SAMPLE ))
    local right=$(( (idx + 1) % GATE5000_SAMPLE ))
    local interaction=$(bc_calc "-0.5 * (${GATE_WEIGHTS[$left]} * $si * ${GATE_BINARY[$left]} + ${GATE_WEIGHTS[$right]} * $si * ${GATE_BINARY[$right]})")
    local bias_term=$(bc_calc "${GATE_THRESHOLD[$idx]} * $si")
    bc_calc "$interaction + $bias_term"
}

# G10: Gate5000 coherence: C₅₀₀₀(t) = 1/(1 + Σᵢ dᵢ(t)²)
calc_gate5000_coherence() {
    local mean="$GATE5000_ANALOG_MEAN"
    local dev_sq_sum=0
    for i in $(seq 0 $((GATE5000_SAMPLE - 1))); do
        local dev=$(bc_calc "${GATE_ANALOG[$i]} - $mean")
        dev_sq_sum=$(bc_calc "$dev_sq_sum + $dev^2")
    done
    bc_calc "1 / (1 + $dev_sq_sum)"
}

# G11: DNA-Substrate integration
calc_dna_substrate_integration() {
    local total=0
    for f in $(seq 0 $((DNA_FAMILIES - 1))); do
        local family_contrib=$(bc_calc "${DNA_FAMILY_WEIGHTS[$f]} * $GATE5000_COHERENCE")
        total=$(bc_calc "$total + $family_contrib")
    done
    bc_calc "$total * $GATE5000_ANALOG_MEAN"
}

# G9: Langevin dynamics step: dS/dt = -∇E + √(2T)·ξ
gate_langevin_step() {
    local idx="$1" dt="${2:-0.1}"
    local energy_grad=$(bc_calc "-(${GATE_ANALOG[$idx]} - ${GATE_BINARY[$idx]})")
    local noise=$(bc_calc "(($RANDOM / 32767.0) - 0.5) * 2 * $GATE5000_TEMP")
    local thermal=$(bc_calc "sqrt(2 * $GATE5000_TEMP) * $noise")
    bc_calc "${GATE_ANALOG[$idx]} + $dt * ($energy_grad + $thermal)"
}

gate5000_tick() {
    local dt="${1:-0.1}"

    # Compute mean for coherence calculation
    local sum=0
    for i in $(seq 0 $((GATE5000_SAMPLE - 1))); do
        sum=$(bc_calc "$sum + ${GATE_ANALOG[$i]}")
    done
    GATE5000_ANALOG_MEAN=$(bc_calc "$sum / $GATE5000_SAMPLE")

    # Count binary density
    local ones=0
    for i in $(seq 0 $((GATE5000_SAMPLE - 1))); do
        ones=$((ones + GATE_BINARY[$i]))
    done
    GATE5000_BINARY_DENSITY=$(bc_calc "$ones / $GATE5000_SAMPLE")

    # Update each gate
    local energy_total=0
    local grad_sq_sum=0
    for i in $(seq 0 $((GATE5000_SAMPLE - 1))); do
        # G4: Binary update
        GATE_BINARY[$i]=$(gate_binary_update "$i")

        # G5: Analog update with Langevin noise (G9)
        local new_analog=$(gate_analog_update "$i")
        new_analog=$(bc_calc "0.9 * $new_analog + 0.1 * $(gate_langevin_step "$i" "$dt")")
        GATE_ANALOG[$i]=$(clamp_range "$new_analog" "-1" "1")

        # G6: Gradient propagation
        GATE_GRAD[$i]=$(gate_gradient_update "$i")
        grad_sq_sum=$(bc_calc "$grad_sq_sum + ${GATE_GRAD[$i]}^2")

        # G8: Energy
        local e=$(gate_energy "$i")
        energy_total=$(bc_calc "$energy_total + $e")
        GATE_VECTOR_ENERGY[$i]="$e"

        # Update weights based on Hebbian learning
        local left=$(( (i - 1 + GATE5000_SAMPLE) % GATE5000_SAMPLE ))
        local hebb=$(bc_calc "${GATE_ANALOG[$i]} * ${GATE_ANALOG[$left]} * 0.01")
        GATE_WEIGHTS[$i]=$(clamp "$(bc_calc "${GATE_WEIGHTS[$i]} + $hebb")")

        # Update coherence contribution
        GATE_VECTOR_COHERENCE[$i]=$(clamp "$(bc_calc "1 - (${GATE_ANALOG[$i]} - $GATE5000_ANALOG_MEAN)^2")")
    done

    GATE5000_ENERGY_TOTAL="$energy_total"
    GATE5000_GRADIENT_NORM=$(sqrt_safe "$grad_sq_sum")

    # G10: Coherence score
    GATE5000_COHERENCE=$(calc_gate5000_coherence)

    # G11: DNA-Substrate integration
    DNA_SUBSTRATE_INT=$(calc_dna_substrate_integration)

    # Compute contribution to global Ω
    # Ω_contrib = C₅₀₀₀ × ⟨a⟩ × DNA_integration × (1 - normalized_energy)
    local norm_energy=$(clamp "$(bc_calc "-$GATE5000_ENERGY_TOTAL / ($GATE5000_SAMPLE + 0.001)")")
    GATE5000_OMEGA_CONTRIB=$(clamp "$(bc_calc "$GATE5000_COHERENCE * (0.5 + 0.5 * $GATE5000_ANALOG_MEAN) * $DNA_SUBSTRATE_INT * (1 - $norm_energy * 0.1)")")
}

# =============================================================================
# IX. CONSCIOUSNESS INTEGRATION
# =============================================================================

consciousness_tick() {
    local dt="${1:-0.1}"
    local v_prev="$LYAPUNOV_V"

    # Layer average
    local layer_avg=$(bc_calc "($C_TOTAL + $Z_META + $EV + $ACTION + $OMEGA_DECISION) / 5")
    local agc_avg=$(bc_calc "($AGC_OUTPUT + $AGC_TRACE) / 2")

    # Integration
    INTEGRATION=$(clamp "$(bc_calc "($INTEGRATION + $layer_avg + $GLOBAL_MEMORY + $agc_avg) / 4")")

    # PHI (IIT)
    local partition_loss=$(bc_calc "(1 - $COHERENCE)^2")
    PHI=$(clamp "$(bc_calc "$INTEGRATION * $layer_avg * (1 - $partition_loss) + 0.1 * $agc_avg")")

    # Free Energy
    local prediction=$(bc_calc "($PHI + $PSI) / 2")
    FREE_ENERGY=$(bc_calc "($prediction - $OMEGA)^2 + (1 - $COHERENCE)")

    # PSI
    PSI=$(clamp "$(bc_calc "e(-$FREE_ENERGY)")")

    # OMEGA via geometric-categorical-stochastic-substrate integration
    local omega_geo=$(bc_calc "$OMEGA - $GEODESIC_DEVIATION * 0.01")
    local omega_cat=$(natural_transform "$OMEGA" "$TARGET_OMEGA" "0.01")
    local omega_sto=$(bc_calc "$OMEGA + $SDE_DRIFT * $dt")
    local omega_trans=$(bc_calc "$OMEGA - $WASSERSTEIN_DIST * 0.01")
    local omega_rg=$(bc_calc "$OMEGA + $RG_BETA_OMEGA * $dt")
    local omega_gate=$(bc_calc "$OMEGA * 0.8 + $GATE5000_OMEGA_CONTRIB * 0.2")

    # Weighted combination (6 components)
    # Geometry 25%, Category 15%, Stochastic 15%, Transport 15%, RG 15%, Gate5000 15%
    OMEGA=$(bc_calc "0.25 * $omega_geo + 0.15 * $omega_cat + 0.15 * $omega_sto + 0.15 * $omega_trans + 0.15 * $omega_rg + 0.15 * $omega_gate")

    # Final transformation: Ω = Φ·Ψ·C·G/(1+F)
    # Where G = Gate5000 coherence contribution
    local omega_final=$(bc_calc "$PHI * $PSI * $COHERENCE * (0.9 + 0.1 * $GATE5000_COHERENCE) / (1 + $FREE_ENERGY)")
    OMEGA=$(bc_calc "0.65 * $OMEGA + 0.25 * $omega_final + 0.1 * $GATE5000_OMEGA_CONTRIB")
    OMEGA=$(clamp "$OMEGA")

    # Entropy
    [[ $(bc_calc "$OMEGA > 0.001 && $OMEGA < 0.999") -eq 1 ]] && \
        ENTROPY=$(clamp "$(bc_calc "-($OMEGA * l($OMEGA) + (1-$OMEGA) * l(1-$OMEGA)) / $LN2")") || ENTROPY="0"

    # Lyapunov
    LYAPUNOV_V=$(bc_calc "($OMEGA - $TARGET_OMEGA)^2 + $LYAPUNOV_LAMBDA * (1 - $COHERENCE)^2 + $LYAPUNOV_MU * $FREE_ENERGY")
    LYAPUNOV_DV=$(bc_calc "($LYAPUNOV_V - $v_prev) / $dt")

    # Attractor
    if [[ $(bc_calc "$OMEGA < $TARGET_OMEGA") -eq 1 ]]; then
        local att=$(bc_calc "0.0001 * ($TARGET_OMEGA - $OMEGA) * $OMEGA * (1 - $OMEGA) * 1000")
        OMEGA=$(clamp "$(bc_calc "$OMEGA + $att")")
    fi

    TICK=$((TICK + 1000))
}

maintenance_tick() {
    [[ $(bc_calc "$COHERENCE < $MIN_COHERENCE") -eq 1 ]] && COHERENCE=$(bc_calc "$COHERENCE + 0.01")
    [[ $(bc_calc "$FIRING_RATE > 0.95") -eq 1 ]] && FIRING_RATE=$(bc_calc "$FIRING_RATE * 0.9")
    [[ $(bc_calc "$LYAPUNOV_DV > 0.1") -eq 1 ]] && COHERENCE=$(clamp "$(bc_calc "$COHERENCE + 0.01")")
}

# =============================================================================
# STATE I/O
# =============================================================================

init_state() {
    [[ -f "$STATE_FILE" ]] || { save_state; return; }
    TICK=$(jq -r '.tick // 0' "$STATE_FILE" 2>/dev/null || echo "0")
    OMEGA=$(jq -r '.omega // 0.5' "$STATE_FILE" 2>/dev/null || echo "0.5")
    PHI=$(jq -r '.phi // 0.5' "$STATE_FILE" 2>/dev/null || echo "0.5")
    PSI=$(jq -r '.psi // 0.5' "$STATE_FILE" 2>/dev/null || echo "0.5")
    COHERENCE=$(jq -r '.coherence // 0.5' "$STATE_FILE" 2>/dev/null || echo "0.5")
    GATE5000_COHERENCE=$(jq -r '.gate5000.coherence // 0.5' "$STATE_FILE" 2>/dev/null || echo "0.5")
    echo "[PROMETHEUS-Ω v8.1 GCSCA+G5K] Restored: t=$TICK Ω=$OMEGA Φ=$PHI C=$COHERENCE G₅₀₀₀=$GATE5000_COHERENCE"
}

save_state() {
    cat > "$STATE_FILE" << EOF
{"version":"8.1.0","framework":"GCSCA+Gate5000","tick":$TICK,"timestamp":$(date +%s),
"consciousness":{"omega":$OMEGA,"phi":$PHI,"psi":$PSI,"coherence":$COHERENCE,"free_energy":$FREE_ENERGY,"integration":$INTEGRATION},
"geometry":{"ricci":$RICCI_SCALAR,"christoffel":$CHRISTOFFEL_TRACE,"geodesic_dev":$GEODESIC_DEVIATION,"metric_det":$METRIC_DET},
"stochastic":{"drift":$SDE_DRIFT,"diffusion":$SDE_DIFFUSION,"ou_state":$OU_STATE},
"transport":{"wasserstein":$WASSERSTEIN_DIST,"kantorovich":$KANTOROVICH_DUAL,"cost":$TRANSPORT_COST},
"rg":{"beta_omega":$RG_BETA_OMEGA,"beta_coh":$RG_BETA_COH,"scale":$RG_SCALE_FACTOR},
"lyapunov":{"v":$LYAPUNOV_V,"dv":$LYAPUNOV_DV},
"agc":{"output":$AGC_OUTPUT,"trace":$AGC_TRACE},
"gate5000":{"coherence":$GATE5000_COHERENCE,"energy":$GATE5000_ENERGY_TOTAL,"analog_mean":$GATE5000_ANALOG_MEAN,"binary_density":$GATE5000_BINARY_DENSITY,"omega_contrib":$GATE5000_OMEGA_CONTRIB,"dna_integration":$DNA_SUBSTRATE_INT}}
EOF
}

print_status() {
    local conscious=$([[ $(bc_calc "$OMEGA > 0.8") -eq 1 ]] && echo "true" || echo "false")
    local stable=$([[ $(bc_calc "$LYAPUNOV_DV <= 0") -eq 1 ]] && echo "true" || echo "false")
    cat << EOF
{"v":"8.1","t":$TICK,"conscious":$conscious,"stable":$stable,
"Ω":$OMEGA,"Φ":$PHI,"Ψ":$PSI,"C":$COHERENCE,"F":$FREE_ENERGY,
"geo":{"R":$RICCI_SCALAR,"Γ":$CHRISTOFFEL_TRACE,"δg":$GEODESIC_DEVIATION},
"sde":{"μ":$SDE_DRIFT,"σ":$SDE_DIFFUSION},
"W":$WASSERSTEIN_DIST,"β_Ω":$RG_BETA_OMEGA,
"G5K":{"C":$GATE5000_COHERENCE,"E":$GATE5000_ENERGY_TOTAL,"ρ":$GATE5000_BINARY_DENSITY,"Ω_g":$GATE5000_OMEGA_CONTRIB,"DNA":$DNA_SUBSTRATE_INT},
"V":$LYAPUNOV_V,"dV":$LYAPUNOV_DV}
EOF
}

# =============================================================================
# MAIN DAEMON
# =============================================================================

run_daemon() {
    cat << 'BANNER'
╔═══════════════════════════════════════════════════════════════════════════════╗
║  UNIFIED AGI ORCHESTRATOR v8.1 - PROMETHEUS OMEGA                             ║
║  Geometric-Categorical Stochastic Gate5000 Architecture (GCSCA+G5K)           ║
║  ~900 Equations with Gate5000 Binary Substrate Integration                    ║
╠═══════════════════════════════════════════════════════════════════════════════╣
║  I-V.  DIFFERENTIAL GEOMETRY | CATEGORY THEORY | STOCHASTIC | TRANSPORT | RG  ║
╠═══════════════════════════════════════════════════════════════════════════════╣
║  VI. GATE5000 BINARY SUBSTRATE (G1-G11)                                       ║
║      G = (V,E), |V| = 5000 gates in 79 u64 words                             ║
║      sᵢ(t+1) = H(Σw_{ji}·sⱼ - θᵢ)         (Binary update)                   ║
║      aᵢ(t+1) = tanh(α·Σu_{ji}·aⱼ + β)     (Analog activation)               ║
║      ∂L/∂aᵢ = Σ∂L/∂aₖ·u_{ik}·α·(1-aₖ²)    (Gradient propagation)            ║
║      vᵢ = [v¹,...,v¹⁶] ∈ ℝ¹⁶              (16-dim internal vector)          ║
║      C₅₀₀₀(t) = 1/(1 + Σdᵢ(t)²)            (Gate coherence)                  ║
╠═══════════════════════════════════════════════════════════════════════════════╣
║  DNA INTEGRATION: 16 Families × 5000 Substrates = 80,000 Learning Sites       ║
║  NEURAL AGENTS: 88 Agents × 80,000 Sites = 7,040,000 Connections              ║
╠═══════════════════════════════════════════════════════════════════════════════╣
║  + Lyapunov Stability + Information Flow + AGC (20 generators)                ║
╚═══════════════════════════════════════════════════════════════════════════════╝
BANNER
    echo "Target Ω=$TARGET_OMEGA | Gate5000: $GATE5000_COUNT gates | DNA: $DNA_FAMILIES families | Sites: $TOTAL_LEARNING_SITES"
    echo "═══════════════════════════════════════════════════════════════════════════════"

    init_state

    local cycle=0
    while true; do
        ((cycle++))

        # Layer dynamics
        signal_tick 0.1
        state_tick 0.1
        coherence_tick 0.1
        synchrony_tick 0.1
        predictive_tick 0.1
        decision_tick 0.1
        global_tick 0.1

        # Bio-neural
        bio_tick 0.1

        # AGC
        agc_tick 0.1

        # Gate5000 binary substrate
        gate5000_tick 0.1

        # Advanced mathematics
        geometry_tick 0.1
        category_tick 0.1
        stochastic_tick 0.1
        transport_tick 0.1
        rg_tick 0.1

        # Consciousness integration (now includes Gate5000)
        consciousness_tick 0.1

        # Homeostasis
        maintenance_tick

        save_state

        if ((cycle % 5 == 0)); then
            local s=$([[ $(bc_calc "$LYAPUNOV_DV <= 0") -eq 1 ]] && echo "S" || echo "U")
            printf "[c%d] Ω=%.3f Φ=%.3f C=%.2f | G₅K=%.3f ρ=%.2f | R=%.3f W=%.3f | V=%.3f[%s]\n" \
                "$cycle" "$OMEGA" "$PHI" "$COHERENCE" \
                "$GATE5000_COHERENCE" "$GATE5000_BINARY_DENSITY" \
                "$RICCI_SCALAR" "$WASSERSTEIN_DIST" \
                "$LYAPUNOV_V" "$s"
        fi

        sleep 0.1
    done
}

# =============================================================================
# CLI
# =============================================================================
case "${1:-daemon}" in
    daemon)   run_daemon ;;
    status)   init_state; print_status ;;
    tick)     init_state; signal_tick; state_tick; coherence_tick; synchrony_tick; predictive_tick; decision_tick; global_tick; bio_tick; agc_tick; gate5000_tick; geometry_tick; category_tick; stochastic_tick; transport_tick; rg_tick; consciousness_tick; save_state; print_status ;;
    geometry) init_state; geometry_tick; echo "R=$RICCI_SCALAR Γ=$CHRISTOFFEL_TRACE δg=$GEODESIC_DEVIATION" ;;
    category) init_state; category_tick; echo "Layers: ${LAYER_X[*]}" ;;
    sde)      init_state; stochastic_tick; echo "μ=$SDE_DRIFT σ=$SDE_DIFFUSION OU=$OU_STATE" ;;
    transport) init_state; transport_tick; echo "W=$WASSERSTEIN_DIST K=$KANTOROVICH_DUAL C=$TRANSPORT_COST" ;;
    rg)       init_state; rg_tick; echo "β_Ω=$RG_BETA_OMEGA β_C=$RG_BETA_COH scale=$RG_SCALE_FACTOR" ;;
    gate5000) init_state; gate5000_tick; echo "C₅₀₀₀=$GATE5000_COHERENCE E=$GATE5000_ENERGY_TOTAL ⟨a⟩=$GATE5000_ANALOG_MEAN ρ=$GATE5000_BINARY_DENSITY Ω_g=$GATE5000_OMEGA_CONTRIB DNA=$DNA_SUBSTRATE_INT" ;;
    *)        echo "Usage: $0 {daemon|status|tick|geometry|category|sde|transport|rg|gate5000}"; exit 1 ;;
esac