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


Motivation behind Bird's Law.

    A lot of processes in physics, ai system, computation, Ai systems, and numerical methods are not linear flows. They are *Two-Phase Recursive Loops*. They iterate foward through a expansion process, then returns through a contraction or update step that is mentat to "undo" or counterbalance the foward map.

(examples included):

Signal processing pipelines (operator+adjoint)
Recurrent or reflective Ai reasoning loops
Energy-stablized iterative Solvers
Delayed feed back physical systems
Simulated physical fields with forward and backward operators

In these systems, a natural question comes up.

"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.

(The Core Claim)

Let a recursive process alternate between:

- a **forward/expansion** operator `R`  
- an **adjoint/contraction** operator `R_adj`

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

Define the per-phase signed difference:

ΔS_k = S(Ψ_{k+1}) − S(Ψ_k)
σ_k = +1 for forward phases, −1 for adjoint phases

Then define the loop-global invariant:

I_rec = Σ_k [ σ_k * ΔS_k ]

The central invariant:

**Bird’s Law (Closure Criterion):**

I_rec = 0 ⇔ R_adj = R*

where `R*` is the true mathematical adjoint of `R`.

Recursive stability ↔ adjoint symmetry.

Numerical Evidence

Using FFT-based recursion loops:

- forward operator: convolution with kernel `h`
- adjoint operator: convolution with `reverse(h)` or mismatched `h`
- energy differences summed with signs
- invariant behavior measured  

Results were robust across kernels (Gaussian, sinc, exponential), signals (sine, noise), and sampling rates.

Matched adjoint:
- `I_norm ≈ 0.03`  
- loop closes  

Broken adjoint:
- `I_norm ≈ 1.8–2.0`  
- divergence grows ~60×  
- loop fails  

No mismatched case ever produced a near-zero invariant.

(WHY THIS MIGHT MATTER TO LESSWRONG)
- reflectivity  
- fixed points of reasoning  
- recursive cognition  
- dual-phase update rules  
- error accumulation under repeated self-reference  

Bird's law adds a sharp condition:
>recursive stability requires true adjoint self-reference.

(Ai alightment revelance)

Recursive reasoning loops, planning and self critique architectures, or world-model correction layers implicityly depend on adjoint-like inversions.

Bird's Law gives a precies failure mode.

(Reflective Cognition)

If congnitive steps have expansion and contraction phases, then stability requires evquivalent to 'R_adj = R*'.

(Error accumulation)
misaligned "adjoints" lead to inevitable drift in reflective processes.

"What I'm Looking For."
I'm looking for specific feedback requested.

1. Is the adjoint condition `R_adj = R*` correctly stated for general operator classes?  
2. Does the loop-global invariant behave as defined in functional analysis?  
3. Are there known invariants in operator theory or monotone operators that mirror this?  
4. Are there edge cases where adjoint symmetry holds but `I_rec` does not vanish?  
5. Does this tie into recursive stability models relevant to alignment or reflective cognition?

(ALL MATERIALS ARE ON ZENODO: https://doi.org/10.5281/zenodo.17613321)

In Closing...

Recursive structures appear in computation, cognition, physics, and numerical analysis. If the Adjoint condition truly governs closure across these domains, then this invariant may serve as a dianostic tool, or may belong inside existing operator frameworks.
I welcome critique, counter examples, and pointers to relevant theorems.

~Brett L. Bird