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.Gin 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