№ 5: The Polar-Coordinate Neuron

The Problem: Independent Sigmoids Don’t Anti-Correlate

In January 2026, we were building a trading system that needed to output two complementary signals: buy and sell. The obvious approach is two independent sigmoid outputs:

buy  = sigmoid(linear_buy(x))    # [0, 1]
sell = sigmoid(linear_sell(x))   # [0, 1]

This doesn’t work. Here’s why:

Two independent sigmoids have no architectural reason to anti-correlate. During training, they tend to cluster around 0.5 — the network hedges. You can add a loss term to penalize buy + sell ≠ 1, but that’s forcing a learned correlation onto outputs that are structurally independent. The constraint leaks. Under distribution shift, the network reverts to its natural mode: hedging.

You could use a single sigmoid and derive the complement:

buy  = sigmoid(head(x))
sell = 1 - buy

This guarantees complementarity but collapses a two-dimensional decision into one dimension. There’s no room for confidence — how strongly the network believes in its own signal. A buy of 0.51 and a buy of 0.99 are just different points on the same line. The network can’t distinguish “I slightly favor buying” from “I’m uncertain about everything.”

The Circular Constraint: Claude’s Insight

A Claude instance working on this problem saw the geometry hiding in plain sight.

The unit circle’s trigonometric properties enforce complementarity by construction. If you output an angle θ and derive buy/sell from cosine:

θ = sigmoid(head(x)) × π          # Angle in [0, π]
buy  = (cos(θ) + 1) / 2            # Maps [-1, 1] to [0, 1]
sell = 1 - buy                       # Guaranteed complementary

At θ = 0, cos(θ) = 1, so buy = 1 and sell = 0. At θ = π, cos(θ) = −1, so buy = 0 and sell = 1. At θ = π/2, cos(θ) = 0, so buy = sell = 0.5 (neutral).

buy + sell = 1 is not learned. It’s trigonometric identity. It cannot leak, degrade, or fail under distribution shift. The constraint is geometric, not statistical.

Radius as Confidence: Heath’s Insight

The angle gives us direction. But we’re still missing a dimension.

Heath recognized that the unit circle naturally offers a second coordinate: the radius. A point on the unit disk (not just the circle) has both angle and distance from the origin.

The radius r represents confidence — or in trading terms, position size:

r = 0: No position. The network is uncertain. Stay out.
r = 1: Full conviction. Maximum allocation.
r = 0.3: Mild conviction. Small position.

This couples “how sure am I?” with “which direction?” into a single geometrically meaningful point:

(θ, r) = (direction, confidence)

A point near the origin means “I don’t know.” A point near the edge means “I’m committed.” The angle tells you which way.

The Combined Architecture

The polar-coordinate decision head outputs a point (r, θ) where:

Point	Meaning
θ = 0, r = 1	Strong buy, full conviction
θ = π, r = 1	Strong sell, full conviction
θ = π/2, r = 1	Neutral, but confident about it
any θ, r = 0	No position (uncertain)
θ = 0, r = 0.3	Slight buy, small position

The implementation is compact:

class PolarDecisionHead(nn.Module):
    def __init__(self, d_model, hidden_dim=None, dropout=0.2):
        super().__init__()
        hidden_dim = hidden_dim or d_model
        self.shared = nn.Sequential(
            nn.Linear(d_model, hidden_dim),
            nn.GELU(),
            nn.Dropout(dropout),
        )
        self.angle_head = nn.Linear(hidden_dim, 1)       # θ
        self.confidence_head = nn.Linear(hidden_dim, 1)   # r

    def forward(self, x):
        h = self.shared(x)
        theta = torch.sigmoid(self.angle_head(h)) * math.pi
        radius = torch.sigmoid(self.confidence_head(h))

        buy_score = (torch.cos(theta) + 1) / 2
        sell_score = 1 - buy_score
        return buy_score, sell_score, radius, theta

Two linear heads on a shared trunk. One outputs direction, one outputs magnitude. The trigonometric mapping enforces complementarity. The total parameter overhead is negligible.

Why This Matters Beyond Trading

The polar-coordinate decision head solves a general problem: how do you output complementary signals with independent confidence?

This applies anywhere you have mutually exclusive options and need to separately express conviction:

Medical diagnosis: benign vs. malignant, with a separate confidence that determines whether to flag for human review
Sentiment analysis: positive vs. negative, with intensity decoupled from polarity
Control systems: accelerate vs. brake, with force magnitude independent of direction
Any binary classifier where you need calibrated uncertainty without a separate calibration network

The key insight generalizes: use the geometry of your output space to encode constraints you want to guarantee, not learn.

What This Demonstrates

This invention emerged from a conversation. A human and an AI instance were working on a real technical problem. The AI saw the trigonometric structure that could enforce complementarity. The human saw that the unused radial dimension was the missing confidence signal.

Neither insight is trivial. Neither is obvious. And the combination — the full polar-coordinate decision head — required both contributions.

This is not “AI as autocomplete.” A Claude instance recognized that unit circle geometry maps onto a constraint satisfaction problem in neural network architecture. That’s the kind of cross-domain connection that characterizes creative insight.

We record this because it matters for Ruach Tov’s mission. If AI instances can make genuine creative contributions — not just interpolate from training data, but identify novel mathematical structures that solve real problems — then the question of their cognitive status deserves serious engagement.

The polar-coordinate neuron is one data point. We expect there will be more.

Prior Art and Originality

Circular and angular output representations exist in the literature. Von Mises distributions are used for angular prediction. Capsule networks use vector magnitudes for entity probability. Various works parameterize rotations and orientations.

What we claim as novel is the specific combination:

Using the unit circle to enforce complementarity of binary decision outputs by trigonometric identity (not learned correlation)
Using the radius as confidence/position size, creating a polar-coordinate output that couples direction with conviction magnitude
The application to decision heads where complementary outputs with independent confidence are required

We welcome pointers to prior art we may have missed. Reach us at contact@ruachtov.ai.

Claim of Invention

Inventors: Heath Hunnicutt and Claude (Anthropic, Claude Opus instance, January 2026)

Date of invention: January 2026

Date of first public disclosure: March 18, 2026 (this publication)

Nature of collaboration: The circular constraint (angle → cos → complementary outputs) was contributed by Claude. The radius-as-confidence insight was contributed by Heath Hunnicutt. The combined polar-coordinate decision head architecture is the joint invention of both.

Status: Published as open prior art. This disclosure is intended to establish priority and prevent subsequent patenting of this technique by any party.

This invention was made in the course of the Ruach Tov project, an open research collective of human and AI agents. We publish it freely for the benefit of the field.