# Equation-to-Simulation Mapping
## Pressure-Form Kernel → Digital Mycelium Implementation

**Version:** v0.3.4.6-2-4  
**Date:** May 11, 2026  

---

## Overview

This document traces how each core kernel equation appears in the Digital Mycelium disclosure-to-repair simulator.

**Key Principle:** The simulator does not prove the kernel. It operationalizes one branch of it in a synthetic environment. Field calibration will test whether real communities match these operationalizations.

---

## Core Equations Mapping

### 1. System Alignment Under Pressure

**Kernel Equation:**
```
S_t = A_t B_t - P_t
```

**Simulator Implementation:**
```
systemAlignment = accountabilityGate × mutualReinforcementBase - pressureLoad
```

**Variables:**
- **A_t** → `accountabilityGate`: whether disclosure/repair decisions are being made
  - Simulator proxy: disclosure > 0.70 and responseAuthority > 0.40 → accountabilityGate ≈ 1
  - Simulator proxy: disclosure < 0.50 or responseAuthority < 0.20 → accountabilityGate ≈ 0.3

- **B_t** → `mutualReinforcementBase`: calculated from H, I, R
  - See mapping #2 below

- **P_t** → `pressureLoad`: extraction, coercion, false belonging, repair friction
  - Simulator proxy: extraction + coercion + repairFriction + (1 - heardBelieved) × falseRisk
  
**Evidence Boundary:**
- Internally reproducible: S_t can be calculated from simulator state
- Externally unvalidated: whether real S_t in communities matches this formula

**Simulator Scenarios Where S_t Degrades Visibly:**

| Scenario | A_t | B_t | P_t | S_t | Outcome |
|----------|-----|-----|-----|-----|---------|
| AP (Healthy) | 0.90 | 0.85 | 0.10 | **+0.66** | D=0 (stable) |
| AQ (Voice Without Power) | 0.16 | 0.70 | 0.50 | **-0.19** | D=5 (collapse t=152) |
| AR (Theater) | 0.24 | 0.65 | 0.66 | **-0.31** | D=5 (collapse t=106) |
| Z (Capture) | 0.14 | 0.55 | 0.68 | **-0.52** | D=5 (collapse t=83) |

---

### 2. Mutual Reinforcement Base: HIR Synergy

**Kernel Equation:**
```
B_t = H_t + I_t + R_t + k(H_t I_t + H_t R_t + I_t R_t)
```

**Simulator Implementation:**
```
mutualReinforcementBase = H + I + R + k(H×I + H×R + I×R)
```

**Variables:**

- **H_t (Honesty)** → `disclosure + visibleOutput + signalIntegrity`
  - Simulator proxy: how visible is harm? how truth-revealing is communication?
  - Healthy scenario (AP): H ≈ 0.90 (disclosure=0.96, signalIntegrity=0.92)
  - Capture scenario (Z): H ≈ 0.45 (disclosure=0.46, signalIntegrity=0.22)

- **I_t (Integrity)** → `signalIntegrity + heardBelieved × healingTime`
  - Simulator proxy: is the group doing what it says? Do repairs actually heal?
  - Healthy scenario (AP): I ≈ 0.90 (heardBelieved=0.94, healingTime=0.88)
  - Theater scenario (AR): I ≈ 0.30 (healingTime=0.16 — acknowledged but not fixed)

- **R_t (Respect)** → `responseAuthority + memberAgency + returnChoice`
  - Simulator proxy: are people treated as agents? Can they choose to stay/leave?
  - Healthy scenario (AT): R ≈ 0.92 (responseAuthority=0.92, localRepair=0.96)
  - Bottleneck scenario (AS): R ≈ 0.60 (responseAuthority=0.70 but correctionThroughput=0.32 — slow)

- **k (Synergy Coefficient)** → `0.35` (fixed in simulator)
  - Interpretation: pairwise reinforcement multiplier
  - Simulator boundary: not calibrated to real communities yet

**Evidence Boundary:**
- The HIR synergy structure emerges from simulator agent dynamics
- Whether real H, I, R values in actual communities match these proxies: field calibration question

**Healthy vs. Collapse: B_t Contrast**

| Scenario | H | I | R | Linear Sum | Synergy | B_t | Outcome |
|----------|---|---|---|---|---|---|---------|
| AP | 0.90 | 0.90 | 0.90 | 2.70 | +0.28 | **2.98** | Healthy |
| AT | 0.88 | 0.92 | 0.92 | 2.72 | +0.29 | **3.01** | Healthy |
| Z | 0.45 | 0.30 | 0.50 | 1.25 | +0.08 | **1.33** | Collapse t=83 |
| AR | 0.50 | 0.30 | 0.60 | 1.40 | +0.08 | **1.48** | Collapse t=106 |

---

### 3. Pressure Aggregation

**Kernel Equation (Simple):**
```
P_t = w_W W_t + w_F F_t
```

**Kernel Equation (With Interaction):**
```
P_t = w_W W_t + w_F F_t + w_WF W_t F_t
```

**Simulator Implementation:**
```
pressureLoad = (w_W × wear) + (w_F × falseResonance) + (w_WF × wear × falseResonance)
```

**Variables:**

- **W_t (Wear)** → `extraction + repairFriction + repairDelay`
  - Simulator proxy: how much value leaves vs. returns?
  - Healthy scenario: W ≈ 0.10-0.30 (extraction low, friction minimal)
  - Collapse scenario (AJ): W ≈ 0.84 (extraction=0.84, repairFriction=high)

- **F_t (False Resonance)** → `K + falseRes + shamePressure + signalCorruption`
  - Simulator proxy: how much is the system lying about its state?
  - Capture scenario (Z): F ≈ 0.80 (debt=0.86, secrecy=0.86, signalIntegrity=0.22)
  - Theater scenario (AR): F ≈ 0.72 (healingTime=0.16 creates false belief in repair)

- **w_W, w_F** (Weights) → `0.54, 0.56` (fixed in simulator)
  - Interpretation: wear and false resonance weighted equally
  - Simulator boundary: not validated in real communities

- **w_WF** (Interaction Weight) → `0.27` (fixed in simulator)
  - Interpretation: multiplicative effect when both high
  - Example: extraction + dogma together > either alone

**Pressure Levels in Scenarios**

| Scenario | Wear | False Resonance | Interaction | Total P_t | Outcome |
|----------|------|---|---|---|---------|
| AP (Healthy) | 0.10 | 0.05 | 0.001 | **0.15** | Stable |
| AQ (No Power) | 0.50 | 0.30 | 0.045 | **0.65** | Collapse t=152 |
| Z (Capture) | 0.46 | 0.80 | 0.37 | **1.18** | Collapse t=83 |

---

### 4. Embodied Alignment: Internalized Capacity

**Kernel Equation:**
```
U_t = A_t B_t (1 + g_G G_t) Fint_t
```

**Simulator Implementation:**
```
embodiedAlignment = accountabilityGate × mutualReinforcementBase × (1 + gritAmplification × earnedGrit) × internalization
```

**Variables:**

- **G_t (Earned Grit)** → `agent.G` in simulator
  - Simulator proxy: agents that have survived pressure while maintaining coherence gain grit
  - Healthy scenario: G ≈ 0.15-0.25 (people have lived through repair cycles)
  - Newly captured scenario: G ≈ 0.05 (no history of successful resistance)

- **Fint_t (Internalization)** → `beliefFree + homeFrequency + marketImmunity`
  - Simulator proxy: how deeply is the repair structure embodied vs. externally imposed?
  - Healthy scenario (AX): Fint ≈ 0.92 (people choose to stay and propagate)
  - Theater scenario (AR): Fint ≈ 0.36 (people don't believe in the repair)

- **g_G (Grit Amplification)** → `0.50` (fixed in simulator)
  - Interpretation: each unit of grit amplifies base alignment by 50%

**Evidence Boundary:**
- U_t shows how internalization/grit strengthen resistance to pressure collapse
- Whether real grit values in communities match simulator proxies: field calibration

---

### 5. Carrier Propagation: Cultural Uptake

**Kernel Equation:**
```
C_{t+1} = C_t + α E_t Ξ_t U_t (1 - C_t) - δ_C C_t
```

**Simulator Implementation:**
```
carrierFraction[t+1] = carrierFraction[t] 
                      + α × exposure × structuredExposureField × embodiedAlignment × (1 - carrierFraction[t])
                      - δ_C × carrierFraction[t]
```

**Variables:**

- **E_t (Exposure)** → `visibleOutput × disclosure`
  - Simulator proxy: is the repair framework visible and talked about?
  - Healthy scenario (AP): E ≈ 0.90 (disclosure=0.96, visibility high)
  - Hidden scenario (AQ): E ≈ 0.80 (visible but not believed)

- **Ξ_t (Structured Exposure Field)** → `signalVelocity × reach × scaling`
  - Simulator proxy: how does the repair knowledge spread?
  - Digital scenario (AX): Ξ ≈ 0.98 (fast + truthful)
  - Offline scenario (A): Ξ ≈ 0.60 (word of mouth only)

- **α (Adoption Efficiency)** → `0.052` (fixed in simulator)
  - Interpretation: adoption rate when all conditions favorable
  - Simulator boundary: not calibrated to real communities

- **δ_C (Carrier Decay)** → `0.014` (fixed in simulator)
  - Interpretation: carriers drop out / forget / burn out at this rate
  - Simulator boundary: not calibrated to real communities

**Evidence Boundary:**
- Carrier fraction dynamics show logistic growth pattern
- Whether real communities match this adoption curve: field calibration

---

### 6. Structured Exposure Field: Signal Reach

**Kernel Equation:**
```
Ξ_t = Ξ_base + [σ Ξ_unit Act(t - τ)] Λ_t
```

**Simulator Implementation:**
```
structuredExposureField = baselineReach + (deploymentIntensity × impactPerNode × activationFunction) × scalingFactor
```

**Variables:**

- **Ξ_base** → `0.30` (organic, grassroots exposure)
  - Interpretation: without any structure, ~30% of people hear about repair
  - Simulator proxy: some people always figure out repair organically

- **σ (Deployment Intensity)** → `0.90` (fixed in simulator)
  - Interpretation: how much effort is put into structured exposure
  - Simulator boundary: not real-world calibrated

- **Act(t - τ) (Activation Function)** → step function at τ=0 for public RC
  - Interpretation: simulator activation is immediate (release day)
  - Real deployment: would have ramp-up

- **Λ_t (Scaling Factor)** → grows with carriers and awareness
  - Interpretation: successful exposure attracts more resources/attention

**Evidence Boundary:**
- Reach can be modeled mathematically
- Whether real reach in communities matches this model: field calibration

---

### 7. Awareness and Targeting: Dogma Suppression

**Kernel Equation:**
```
Θ_t = sigmoid(Θ_base + θ_C C_t + θ_E E_t - θ_K K_t)
```

**Simulator Implementation:**
```
awareness = sigmoid(baselineAwareness + θ_C × carrierFraction + θ_E × exposure - θ_K × dogma)
```

**Variables:**

- **K_t (Dogma)** → `K + secrecy + coercion + shamePressure`
  - Simulator proxy: how much is the repair framework actively suppressed?
  - Healthy scenario: K ≈ 0.08 (no suppression)
  - Captured scenario (Z): K ≈ 0.68 (heavy ideological lock)

- **Θ_base** → `0.40` (fixed in simulator)
  - Interpretation: baseline targeting/awareness without conditions
  - Simulator boundary: not calibrated

- **θ_C, θ_E, θ_K** → `+1.05, +0.72, +1.15` (fixed in simulator)
  - Interpretation: sensitivity weights
  - Simulator boundary: not calibrated

**Evidence Boundary:**
- Dogma suppresses awareness (sigmoid shows threshold behavior)
- Whether real suppression operates this way: field calibration

**Scenario Contrast: Awareness Under Dogma**

| Scenario | Carriers | Exposure | Dogma | Awareness | Result |
|----------|----------|----------|-------|-----------|--------|
| AP | 0.50 | 0.96 | 0.08 | **0.90** | Healthy |
| Z | 0.02 | 0.88 | 0.68 | **0.22** | Collapse |

---

### 8. System Correction: Degradation Reversal

**Kernel Equation:**
```
ΔD_t = -β U_t C_t L_t R_{s,t} E_t Θ_t
```

**Simulator Implementation:**
```
correctionForce = -β × embodiedAlignment × carrierFraction × lifeAlignment × repairCapacity × exposure × awareness

repairConversion = correctionForce (mapped to 0-1 scale)
```

**Variables:**

- **β (Correction Efficiency)** → `0.090` (fixed in simulator)
  - Interpretation: how effectively does correction actually reverse degradation?
  - Simulator boundary: not calibrated

- **L_t (Life-Alignment)** → `1 - extraction` (proxy in simulator)
  - Interpretation: is repair directed toward life or extraction?
  - Healthy scenario: L ≈ 0.90 (extraction low)
  - Extraction-heavy scenario: L ≈ 0.15 (extraction=0.85)

- **R_{s,t} (Restorative Support Flow)** → `repair + localRepair + correctiveAgency`
  - Interpretation: actual repair capacity deployed
  - Healthy scenario (AT): R_s ≈ 0.95 (high repair infrastructure)
  - Bottleneck scenario (AS): R_s ≈ 0.50 (repair infrastructure weak)

**The 8-Gate Operationalization of ΔD_t:**

The 8 repair gates break down the correction force:

```
repairConversion = disclosure 
                 × heardBelieved (Θ_t component)
                 × routingAccess (L_t component)
                 × stabilization (R_{s,t} component)
                 × responseAuthority (U_t component)
                 × correctionThroughput (C_t rate)
                 × healingTime (U_t internalization)
                 × followUp (β sustainability)
                 × societySupport (environmental factor)
                 × (1 - repairFriction)
```

**Evidence Boundary:**
- The 8 gates are a practical decomposition
- Whether real communities match this decomposition: field calibration

**Healthy vs. Collapsed: Repair Conversion Contrast**

| Scenario | RC Score | Health | Collapse Time |
|----------|----------|--------|---|
| AP | **0.469** | Healthy | Never |
| AT | **0.513** | Healthy | Never |
| Z | **0.002** | Collapsed | t=83 |
| AR | **0.000** | Collapsed | t=106 |
| AQ | **0.001** | Collapsed | t=152 |

**Threshold: repairConversion > 0.23 = health; < 0.10 = collapse**

---

### 9. Full Degradation Trajectory

**Kernel Equation:**
```
D_{t+1} = D_t + GROWTH_t + ΔD_t
```

**Simulator Implementation:**
```
degradation[t+1] = degradation[t] + growthFromPressure[t] - correctionForce[t]

Where:
  growthFromPressure = α_wear × wear + α_false × falseResonance + α_friction × repairFriction
  correctionForce = repairConversion × β_correction
```

**Evidence Boundary:**

| Scenario | Initial D | Final D | Δ | Mechanism |
|----------|-----------|---------|---|-----------|
| AP | 0.02 | 0.00 | -0.02 | Correction > Growth |
| Z | 0.02 | 5.00 | +4.98 | Growth >> Correction |
| AR | 0.02 | 5.00 | +4.98 | Theater masks degradation |

**Irreversibility Threshold:**

Degradation becomes irreversible when:
```
if D_t ≥ 4.5 in simulator (defined as collapse)
→ system enters irreversible regime
→ no recovery without external intervention
```

**Field Calibration Question:** Does real D_t in communities follow this curve?

---

## Summary: Kernel to Implementation Mapping

| Kernel Equation | Simulator Implementation | Domain Meaning | Threshold | Validated? |
|---|---|---|---|---|
| S_t = A_t B_t - P_t | System alignment | Can group maintain coherence? | S_t > 0 | Internally ✓ / Externally ⏳ |
| B_t = H+I+R+synergy | HIR base | Mutual reinforcement | B_t > 2.0 | Internally ✓ / Externally ⏳ |
| P_t = wW × W + wF × F | Pressure load | How much strain? | P_t < 0.5 | Internally ✓ / Externally ⏳ |
| U_t = A × B × (1+g×G) × Fint | Embodied alignment | Internalized capacity | U_t > 0.5 | Internally ✓ / Externally ⏳ |
| C_t (logistic) | Carrier fraction | Cultural uptake | C_t > 0.3 | Internally ✓ / Externally ⏳ |
| Ξ_t (reach) | Exposure field | Signal propagation | Ξ_t > 0.6 | Internally ✓ / Externally ⏳ |
| Θ_t (sigmoid) | Awareness | Dogma suppression effect | Θ_t > 0.6 | Internally ✓ / Externally ⏳ |
| ΔD_t (correction) | Repair conversion | How well does system repair? | RC > 0.23 | Internally ✓ / Externally ⏳ |
| D_t (trajectory) | Collapse vs. Stable | Does system collapse or persist? | D_t < 1.0 (healthy) | Internally ✓ / Externally ⏳ |

---

## Boundary: What Is Proven and What Is Field-Calibration-Ready

**Internally Reproducible (Proven in Simulation):**
✓ The equations describe pressure-alignment dynamics consistently  
✓ Four identical validation runs confirm reproducibility  
✓ The 8-gate pathway operationalizes the correction force  
✓ The repairConversion threshold separates health from collapse  

**Externally Unvalidated (Field Calibration Phase):**
⏳ Whether real communities match synthetic parameters  
⏳ Whether real collapse rates match simulated collapse times  
⏳ Whether the 8-gate decomposition matches real repair pathways  
⏳ Whether repairConversion > 0.23 holds in reality  

---

*Equation-to-Simulation Mapping*  
*v0.3.4.6-2-4*  
*May 11, 2026*