# **Bird’s Law: A Recursive Closure Invariant for Dual-Phase Systems**

*By Brett L. Bird*
*Independent Researcher*
Supporting Documents:

https://doi.org/10.5281/zenodo.17613321

---

## **1. Motivation**

Many processes in physics, computation, AI systems, and numerical methods aren’t linear flows — they are **two-phase recursive loops**. They iterate forward through an expansion process, then return through a contraction or update step that is meant to “undo” or counterbalance the forward map.

Examples include:

* signal processing pipelines (operator + adjoint)
* energy-stabilized iterative solvers
* recurrent or reflective AI reasoning loops
* delayed feedback physical systems
* simulated physical fields with forward + backward operators

In these systems, a natural question arises:

> **What guarantees that a recursive loop actually closes, rather than drifting or accumulating error across cycles?**

Bird’s Law proposes a simple, falsifiable mathematical condition governing that closure.

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## **2. The Core Claim**

Let a recursive process alternate between:

* a **forward/expansion** operator (R)
* an **adjoint/contraction** operator (R_{\text{adj}})

Let (S(\Psi)) be any energy-like functional or quadratic measure over the state (\Psi).

Define the per-phase signed difference:

```
ΔSₖ = S(Ψₖ₊₁) − S(Ψₖ)
σₖ = +1 for forward phases, −1 for adjoint phases
```

Then define the loop-global invariant:

```
I_rec = Σₖ [ σₖ * ΔSₖ ]
```

The central invariant:

> **Bird’s Law (Closure Criterion)**
> **The recursive loop closes (I_rec → 0) iff the adjoint operator is the true adjoint of the forward operator.**

Equivalently:

```
I_rec = 0 ⇔ R_adj = R*
```

This is the mathematical heart of the work:
recursive stability ↔ adjoint symmetry.

---

## **3. Numerical Evidence**

To test the invariant, I constructed simple but explicit FFT-based recursion loops:

* signal passes through forward operator (convolution with kernel h)
* then through contraction operator (reverse(h) or intentionally mismatched h)
* energy differences are summed with phase-sign weights
* invariant behavior is measured

The results were stable across:

* Gaussian, sinc, and exponential kernels
* sine waves and noise
* multiple sampling rates
* guard-band trimmed energy measures

Matched adjoint:

* **I_norm ≈ 0.03**
* stable, loop closes

Broken adjoint:

* **I_norm ≈ 1.8–2.0**
* divergence grows ~60×
* loop fails

The divergence is immediate ...

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