TASK 3: The Attractor Competition Model

Version: 1.1 (Updated with Validation)
Status: DEFINED
Date: 2026-01-16

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1. The Landscape Concept

The healing trajectory S_H(t) moves on a dynamic potential energy landscape V(S_H, u). This landscape is not static; it is shaped by the control inputs u (environmental fields).

There are two primary local minima (attractors):
1. mathcal{A}_scar (The Scar Attractor):
* Physics: High entropy, isotropic stiffness, low energy cost.
* Basin: Wide and shallow (easy to fall into, hard to escape due to kinetic trapping).
* Default: In the absence of control (u=0), the system relaxes here.
2. mathcal{A}_regen (The Regeneration Attractor):
* Physics: Low entropy, anisotropic stiffness, high energy cost.
* Basin: Narrow and deep (hard to find, but very stable once entered).
* Conditional: Only accessible if specific permissive constraints are met.

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2. Mathematical Formalism

We define a Lyapunov Function candidate V(S_H) to describe the flow:
dot{S}H = -∇ V(S_H) + eta(t)
Where eta(t) is stochastic thermal noise (T).

The potential V is modeled as a double-well potential modulated by control u:

V(S_H, u) = underbrace{k_scar(u) (S_H – S_scar)²}{text{Scar Well}} + underbrace{k}(u) (S_H – S_regen)²{text{Regen Well}} + underbrace{Ψ }(u)}_{text{Activation Energy}

2.1 The Bifurcation Control

The control input u (fields) acts to reshape this potential.
* Goal: Lower the barrier Ψ_barrier and tilt the landscape towards mathcal{A}_regen.
* Mechanism:
* Flattening mathcal{A}_scar: Increasing the instability of the scar state (e.g., by preventing isotropic collagen deposition via mechanical stress).
* Deepening mathcal{A}_regen: Increasing the stability of the ordered state (e.g., by providing ionic coherence Φ_m that matches the healthy template).

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3. The “Permissive Window” (Bifurcation Point)

There exists a critical Bifurcation Point in the state space, often occurring early in the healing process (The “Golden Hour”).

• If S_H enters the Scar Basin (mathcal{B}_scar), the system becomes kinetically trapped. No amount of future energy can easily reverse it (hysteresis).
• If S_H enters the Regen Basin (mathcal{B}_regen), the system “falls” into order, driven by its own thermodynamics.

The Control Strategy:
u^*(t) = arg max_u P(S_H(t) in mathcal{B}_regen)
We do not “build” the tissue. We essentially “tilt the table” so the ball rolls into the correct hole.

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4. Phase Diagram Concept

graph TD
    Start((Injury t0)) --> Bifurcation{Bifurcation Point}

    subgraph "The Default Path (Low Energy)"
    Bifurcation -- "u=0 (No Support)" --> Scar[Scar Attractor]
    Scar --> DeadEnd((Fibrosis))
    end

    subgraph "The Controlled Path (High Energy)"
    Bifurcation -- "u=Optimal (Field Support)" --> Permissive[Permissive Window]
    Permissive --> Regen[Regeneration Attractor]
    Regen --> Outcome((Functional Tissue))
    end

    style Scar fill:#ffcccc,stroke:#333
    style Regen fill:#ccffcc,stroke:#333
    style Bifurcation fill:#ffffcc,stroke:#333

Physical Interpretation of the Diagram

• X-Axis: Entropy / Disorder (ρ_S)
• Y-Axis: Free Energy (G)
• The Barrier: The activation energy required to organize structure. The body usually won’t pay this cost without help (u).
• The Trap: Once in the Scar valley, kT (thermal noise) is insufficient to jump back out.

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5. Summary of Dynamics

1. Injury creates a high-energy unstable state.
2. Thermodynamics demands relaxation to a lower energy state.
3. Scarring is the fastest, cheapest path to stability (Δ G < 0).
4. Regeneration is a deeper stability but requires crossing an entropic barrier.
5. Our System supplies the field energy to lower that barrier, making Regeneration the path of least resistance.

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8. ISL Alignment (The “Missing Mass” of Healing)

The “Attractor Depth” mathcal{D}_basin is physically equivalent to the “Missing Mass” in the ISL framework.

8.1 The Viscosity of Healing

Just as low-ReG galaxies exhibit extra gravitational binding (“Dark Matter”), low-ReH healing trajectories exhibit extra structural binding (“Regeneration”).
* Laminar Phase (α ≈ 0.35): The system behaves as if there is extra energy holding the tissue together. This is not magic; it is the saturation of information viscosity in the control field.
* Turbulent Phase (α → 0): The binding force vanishes. The tissue can only hold together via weak chemical bonds (Scarring).

8.2 The Control Goal

We do not just “deepen the basin.” We induce the Laminar Phase.
By forcing Re_H < Re_c (slowing down the repair relative to signaling), we trigger the α-boost. This effectively “modifies gravity” on the landscape, creating a deep potential well where none existed before.

Prediction: Regeneration is an emergent property of the Laminar Phase of the ISL. It is not a biological program; it is a physical phase transition.