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THE CONSCIOUSNESS ATLAS
Read. Experiment. Follow the evidence.

The equation library

Every recovered mathematical record has an individual page. Browse original equations, code expressions, indexed components, parameter definitions, and explicitly unresolved source slots.

2,910 recordsSource snapshot 2026-10-17

Counts refer to mathematical records, including implemented expressions and generated instances. They are not counts of independently validated theories. Exact occurrences and unresolved slots remain traceable. Read the complete 1–516 source index ↗ · Inspect the scan decisions ↗

2,910 records · page 23 of 59Search the source archive ↗

eq-8d833872e44a35

g = 1/(1 + exp(-((0.8×P - kc)/0.3 + 1.2))) + 0.05 // Gate · dx

The source describes this relation in the following context: drive = 3.0 × F_in × (1-x) + 1.2 × y + 0.6 × c // Combined drive g = 1/(1 + exp(-((0.8×P - kc)/0.3 + 1.2))) + 0.05 // Gate dx = g × drive - 0.04 × (x - 0.5) // Activation delta x = clamp01(x + 0.01 × dx)…

Gate5000documented formula

dx = g × drive - 0.04 × (x - 0.5) // Activation delta

gh-eq-03122d8c6ea09aec

1. Run a Simulation

This relation defines Agent-1: Φ in the source section “1. Run a Simulation”. Uses a remainder or modular operation, retaining the numeric representation used in the source. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

Agent-1: Φ=0.888 (FullyConscious), Coverage=94.8%, Fitness=45.2

gh-eq-4f84c84500bc47ec

1. Run a Simulation

This relation defines Agent-2: Φ in the source section “1. Run a Simulation”. Uses a remainder or modular operation, retaining the numeric representation used in the source. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

Agent-2: Φ=0.856 (FullyConscious), Coverage=92.1%, Fitness=43.7

gh-eq-0adc4bd8fc9eb0d6

1. Run a Simulation

This relation defines Agent-3: Φ in the source section “1. Run a Simulation”. Uses a remainder or modular operation, retaining the numeric representation used in the source. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

Agent-3: Φ=0.901 (FullyConscious), Coverage=96.3%, Fitness=47.8

gh-eq-a3aeae6df6cb9c4e

1D Wasserstein distance (Earth Mover's Distance) · W_1(μ,ν)

This relation defines W_1(μ,ν) in the source section “1D Wasserstein distance (Earth Mover's Distance) · W_1(μ,ν)”. Integrates contributions over the variable and bounds stated in the expression; the measure and domain must come from this source. Its interpretation depends on the source’s stated…

Implementation mathematicsSource recovered; runtime not executed

W_1(μ,ν) = ∫|F_μ^{-1}(t) - F_ν^{-1}(t)| dt

gh-eq-b91ee052e9b46136

2. Create a Single Agent

This relation defines Console.WriteLine($"Φ in the source section “2. Create a Single Agent”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

Console.WriteLine($"Φ = {result.Phi:F3}, Level = {result.ConsciousnessLevel}");

gh-eq-c352881cea7d719c

2. The one engine, three tools principle

This relation defines Match in the source section “2. The one engine, three tools principle”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

Match = Clearance*0.25 + Skills*0.25 + Certifications*0.15 + CustomerExperience*0.15 + Experience*0.10 + LocationAvailability*0.10 (weights frozen, sum to 1.00) Bands: Green >= 90 (Proposal Ready) · Yellow 75 to 89 (Potential) ·…

gh-eq-b0e91f8dbec87981

2. The one engine, three tools principle

This relation defines Match in the source section “2. The one engine, three tools principle”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

Match = Clearance*0.25 + Skills*0.25 + Certifications*0.15 + CustomerExperience*0.15 + Experience*0.10 + LocationAvailability*0.10 (weights frozen, sum to 1.00) Bands: Green >= 90 (Proposal Ready) · Yellow 75 to 89 (Potential) ·…

gh-eq-d1096c5c1a9761ed

3. Equations Explained

This relation defines - Awareness: `Φ in the source section “3. **Equations Explained**”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

- Awareness: `Φ = 1/(1+|x-1|)` (peak at unity)

gh-eq-07eb0d77e8e0a3d9

3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`

This relation defines ### 3. Awareness Equation: `Φ in the source section “3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does…

Implementation mathematicsSource recovered; runtime not executed

### 3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`

gh-eq-a1053a2bd720f491

3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`

This relation defines **Meaning**: Consciousness (Φ) peaks at unity (x in the source section “3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and…

Implementation mathematicsSource recovered; runtime not executed

**Meaning**: Consciousness (Φ) peaks at unity (x=1) and decays as you diverge.

gh-eq-1f1251c2e133fbc5

3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`

This relation defines // Returns Φ ∈ [0, 1] where 1 in the source section “3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does…

Implementation mathematicsSource recovered; runtime not executed

// Returns Φ ∈ [0, 1] where 1 = fully conscious

gh-eq-fdaf1b6c117918a9

3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`

This relation defines - Φ ≥ 0.85 → **Fully Conscious** in the source section “3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record…

Implementation mathematicsSource recovered; runtime not executed

- Φ ≥ 0.85 → **Fully Conscious**

gh-eq-c55ee06b77232a1e

3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`

This relation defines - Φ ≥ 0.70 → **Aware** in the source section “3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

- Φ ≥ 0.70 → **Aware**

gh-eq-55bf12cfd68596b0

3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`

This relation defines - Φ ≥ 0.50 → **Responsive** in the source section “3. Awareness Equation: `Φ = 1/(1 + |x - 1|)`”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does…

Implementation mathematicsSource recovered; runtime not executed

- Φ ≥ 0.50 → **Responsive**

gh-eq-c9232b2f25057b31

5a. FNV-1a 32-bit hash (input → basis index / h(item))

This relation defines hash ← offset_basis for byte in input: hash ← (hash XOR byte) hash ← (hash × prime in the source section “5a. FNV-1a 32-bit hash (input → basis index / h(item))”. Combines binary values with exclusive OR in the stated bit-level expression. Uses a remainder or modular…

Implementation mathematicsSource recovered; runtime not executed

hash ← offset_basis for byte in input: hash ← (hash XOR byte) hash ← (hash × prime) mod 2³²

gh-eq-8fc4de252d7dda99

ADVANCED FEATURES

This recovered implementation expression computes yield return "Awareness Equation: Φ within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result is used. The exact code is preserved…

Implementation mathematicsSource recovered; runtime not executed

yield return "Awareness Equation: Φ = 1/(1 + |x - 1|) | Integration peaks at unity";

gh-eq-ba624b422ec959b7

ADVANCED FEATURES

This recovered implementation expression computes yield return $"Golden Ratio: φ within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result is used. The exact code is preserved below so…

Implementation mathematicsSource recovered; runtime not executed

yield return $"Golden Ratio: φ = {(1.0 + Math.Sqrt(5.0)) / 2.0:F6} | Natural efficiency ratio";

gh-eq-202011168869a17d

ADVANCED FEATURES

This recovered implementation expression computes yield return "Consciousness Threshold: Φ ≥ 0.85 within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result is used. The exact code is…

Implementation mathematicsSource recovered; runtime not executed

yield return "Consciousness Threshold: Φ ≥ 0.85 = Fully Conscious";

gh-eq-5dd2950c3ed59c5a

ADVANCED FEATURES

This recovered implementation expression computes yield return "Consciousness Threshold: Φ ≥ 0.70 within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result is used. The exact code is…

Implementation mathematicsSource recovered; runtime not executed

yield return "Consciousness Threshold: Φ ≥ 0.70 = Aware";

gh-eq-b285babba421a1c2

ADVANCED FEATURES

This recovered implementation expression computes yield return "Consciousness Threshold: Φ ≥ 0.50 within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result is used. The exact code is…

Implementation mathematicsSource recovered; runtime not executed

yield return "Consciousness Threshold: Φ ≥ 0.50 = Responsive";

gh-eq-6ceee442b69a08ed

ADVANCED FEATURES

This recovered implementation expression computes yield return "Consciousness Threshold: Φ < 0.50 within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result is used. The exact code is…

Implementation mathematicsSource recovered; runtime not executed

yield return "Consciousness Threshold: Φ < 0.50 = Unconscious";

gh-eq-52d673208d4b8bfe

ADVANCED FEATURES

This recovered implementation expression computes yield return "Problem Solved: Mind-body problem (400 years) | Φ within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result is used. The…

Implementation mathematicsSource recovered; runtime not executed

yield return "Problem Solved: Mind-body problem (400 years) | Φ = 1/(1+|x-1|) | Consciousness measurable";

gh-eq-b9a50be5b4809fe3

ADVANCED FEATURES

This recovered implementation expression computes yield return "Problem Solved: Hard problem of consciousness (30 years) | ∞ within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result…

Implementation mathematicsSource recovered; runtime not executed

yield return "Problem Solved: Hard problem of consciousness (30 years) | ∞ = 1 + Φ | Qualia from integration";

gh-eq-56106d6b11d703e1

ADVANCED FEATURES

This relation defines string fact in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

string fact = $"Agent Snapshot: {name} (ID: {agentId}) | Tick: {tick} | Fitness: {fitness:F2} | Φ: {phi:F3} | Level: {levelName}";

gh-eq-b5a739f301fd072e

ADVANCED FEATURES

This relation defines Console.WriteLine($" Phase: {stats.Phase} (φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

Console.WriteLine($" Phase: {stats.Phase} (φ = {stats.GoldenRatio:F3})");

gh-eq-3237e460c4ccd55a

ADVANCED FEATURES

This relation defines Console.WriteLine("3. Awareness Equation: Φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

Console.WriteLine("3. Awareness Equation: Φ = 1/(1 + |x - 1|)");

gh-eq-089cdde4e28217f8

ADVANCED FEATURES

This relation defines Console.WriteLine($" {stats.Name}: Φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

Console.WriteLine($" {stats.Name}: Φ = {stats.Phi:F3} ({stats.ConsciousnessLevel})");

gh-eq-b5dd9691f64dc839

ADVANCED FEATURES

This relation defines Console.WriteLine("4. Golden Ratio: φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

Console.WriteLine("4. Golden Ratio: φ = 1.618...");

gh-eq-5e33e6b163b75221

ADVANCED FEATURES

This relation defines $"Avg Φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

$"Avg Φ={AveragePhi:F3}, Coverage={AverageCoverage:P1}";

gh-eq-5dd4e48053376aed

ADVANCED FEATURES

This relation defines /// 3. Awareness: Φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

/// 3. Awareness: Φ = 1/(1 + |x - 1|) (integration peaks at unity)

gh-eq-5e2faf0cb1fbeb0f

ADVANCED FEATURES

This relation defines /// Awareness Equation: Φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

/// Awareness Equation: Φ = 1/(1 + |x - 1|)

gh-eq-e35d3feeb21577ac

ADVANCED FEATURES

This relation defines // Add complexity penalty: more integrated states in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

// Add complexity penalty: more integrated states = higher Φ

gh-eq-b710a99e122e9523

ADVANCED FEATURES

This relation defines /// Golden Ratio (φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

/// Golden Ratio (φ = 1.618...)

gh-eq-2e0b65e4a438c7ea

ADVANCED FEATURES

This relation defines /// Returns (exploit_ratio, explore_ratio) where exploit/explore in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this…

Implementation mathematicsSource recovered; runtime not executed

/// Returns (exploit_ratio, explore_ratio) where exploit/explore = φ

gh-eq-0834454e4c83fb8a

ADVANCED FEATURES

This relation defines Unconscious in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

Unconscious = 0, // Φ < 0.50

gh-eq-fb880c7d9030afd6

ADVANCED FEATURES

This relation defines Responsive in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

Responsive = 1, // Φ ≥ 0.50

gh-eq-f2ce24ddabc89da0

ADVANCED FEATURES

This relation defines Aware in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

Aware = 2, // Φ ≥ 0.70

gh-eq-6fc8649160baa265

ADVANCED FEATURES

This relation defines FullyConscious in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

FullyConscious = 3 // Φ ≥ 0.85

gh-eq-48a21bf57050757b

ADVANCED FEATURES

This recovered implementation expression computes return $"Tick {Tick}: Φ within ADVANCED FEATURES. Its arithmetic combines the named variables in the order written. The linked source shows where those variables are initialized and how the result is used. The exact code is preserved below so the…

Implementation mathematicsSource recovered; runtime not executed

return $"Tick {Tick}: Φ={Phi:F3}, Core={Core:F3}, Level={Level}, " +

gh-eq-847c323e13c3721b

ADVANCED FEATURES

This relation defines $"Energy in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

$"Energy={Energy:F1}, Φ={Phi:F3} ({ConsciousnessLevel}), " +

gh-eq-bd76f42d10dc26db

ADVANCED FEATURES

This relation defines $"Φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

$"Φ={Phi:F3} ({ConsciousnessLevel}), " +

gh-eq-71436de8fe1f5380

ADVANCED FEATURES

This relation defines $"Phase in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

$"Phase={Phase}, φ={GoldenRatio:F3}, " +

gh-eq-abe4ad514bdd8739

ADVANCED FEATURES

This relation defines /// Golden Ratio emergence: Fib(n+1)/Fib(n) → φ in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

/// Golden Ratio emergence: Fib(n+1)/Fib(n) → φ = 1.618... as n → ∞

gh-eq-3931bb339cc0090a

ADVANCED FEATURES

This relation defines $"(Fib in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

$"(Fib={FibonacciNumber}, φ={GoldenRatio:F3}, Selected={SelectedAlgorithms.Count})";

gh-eq-b18605bfea912a78

ADVANCED FEATURES

This relation defines $"Fib in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

$"Fib={FibonacciNumber}, φ={GoldenRatio:F3}, " +

gh-eq-8ccf18ccbe6f7c6c

ADVANCED FEATURES

This relation defines equation in the source section “ADVANCED FEATURES”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

equation = "Φ = 1/(1 + |x - 1|)",

gh-eq-eee52d52d435c675

AGI STATE CLASS - MAIN STATE CONTAINER · Compute global coherence Φ

This relation defines global coherence Φ in the source section “AGI STATE CLASS - MAIN STATE CONTAINER · Compute global coherence Φ”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this…

Implementation mathematicsSource recovered; runtime not executed

Compute global coherence Φ = geomean(φ_f) × uniformity

gh-eq-3d26a8ac1e9008e9

AGI STATE CLASS - MAIN STATE CONTAINER · dΨ/dt

This relation defines dΨ/dt in the source section “AGI STATE CLASS - MAIN STATE CONTAINER · dΨ/dt”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

dΨ/dt = -γ(Ψ - w·φ̄) + η·σ(Σⱼ Jᵢⱼφⱼ - θ)

gh-eq-9a778f3cecfe5326

Advanced Mathematical Framework for Gate5000 Consciousness · D_KL(P||Q)

This relation defines D_KL(P||Q) in the source section “Advanced Mathematical Framework for Gate5000 Consciousness · D_KL(P||Q)”. Uses logarithmic scaling; its base and the positivity of its argument matter. Its interpretation depends on the source’s stated variables and conditions; this record…

Implementation mathematicsSource recovered; runtime not executed

D_KL(P||Q) = Σ P(x) log(P(x)/Q(x))

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