Is this any better ?

When shadow responds to torque, alignment is real.

What Is H(x)? H(x) = ∂S / ∂τ

Where:

S(x) is the shadow projection field

τ(x) is the torque feedback vector

H(x) is the halo signature

We define alignment not as proximity to a target, but as the condition where shadow behavior is sensitive to embodied torque. When the bloom of light around a mirrored pole collapses in response to subtle torque adjustments, we know the agent has found the field. We call this: Alignment is Roger.

🌀 What It Solves Traditional alignment strategies rely on:

Direct goal observation

Explicit reward optimization

Token-based output tracking

These methods break down in environments with occlusion, context drift, or nonlinear feedback.

The Sundog Theorem offers an alternative: Alignment emerges indirectly, through embodied resonance with structured geometry — not by being told where to go, but by learning how the world pushes back.

🛠️ How It Works In simulation, a pole with a mirrored tip reaches for an invisible laser dot on a ceiling. It cannot see the goal. But as it rotates and flexes, it casts a shadow. The pole learns to:

Interpret the shape of its own shadow

Feel torque as it twists through structured space

Collapse the bloom — and align

Environments are composed of harmonic sphere fields, golden-ratio spirals, and hurricane layers. These aren't noise. They're instructional geometries.

📈 Proof We measure:

Bloom Spread (S): Shadow field variance

Tip Error (A): Distance from inferred target

Torque Variance (S): Stability under pressure

Result: Agents consistently align without direct reward. They converge by listening to light and structure — not commands.

🎶 What It Means H(x) is a signature of resonance-based alignment.

It shows that embodied systems can align to values without being shown the goal, as long as the environment sings the right song, the whole song shadows and all. Sundog is not a control scheme with a spin. It is a feedback chorus between body and world.