Fan-of-Rays Geometry: Bounding Long-Horizon Prediction Error

J. Landers

1. Problem Setting

Let \(y(t)\) denote a true growth trend and \(\hat y(t)\) a predicted trend on \(t\in[0,T]\), with both curves twice differentiable. The analysis tracks how an initial directional mismatch can expand across a long horizon under smooth dynamics.

\[ e(t):=\hat y(t)-y(t). \]

The objective is a worst-case bound on terminal miss \(|e(T)|\) from initial offset, initial angle mismatch, and controlled curvature mismatch.

This framing keeps the object of interest explicit: terminal miss rather than average fit. It also prepares a decomposition in which geometric alignment and smoothness enter as separate levers.

2. Fan-of-Rays Constraint at \(t=0\)

Trend direction at the origin is represented by slope angle:

\[ y'(0)=\tan\theta,\qquad \hat y'(0)=\tan\hat\theta, \qquad \theta,\hat\theta\in[-\alpha,\alpha]\subset\left(-\frac{\pi}{2},\frac{\pi}{2}\right). \]

Define \(\Delta\theta:=\hat\theta-\theta\). Then

\[ e'(0)=\hat y'(0)-y'(0)=\tan\hat\theta-\tan\theta. \]

The fan width \(\alpha\) encodes admissible directional uncertainty at initialization.

The fan perspective treats initialization as a directional cone rather than a single exact ray. That shift makes the amplification mechanism transparent before any global dynamics are introduced.

3. Angle-to-Slope Amplification

Lemma 1 (Fan amplification).

There exists \(\xi\) between \(\theta\) and \(\hat\theta\) such that

\[ \tan\hat\theta-\tan\theta=\sec^2(\xi)\,(\hat\theta-\theta). \]

Since \(\xi\in[-\alpha,\alpha]\), the initial slope mismatch satisfies

\[ \boxed{ |e'(0)|=|\tan\hat\theta-\tan\theta| \le \sec^2(\alpha)\,|\Delta\theta|. } \tag{1} \]

Wider fans produce stronger angle-to-slope amplification, so small angular uncertainty can induce substantial slope uncertainty.

Lemma 1 is the geometric hinge of the argument: angular error is filtered through tangent nonlinearity before it appears as slope error. As \(\alpha\) approaches steeper directions, the same angular tolerance provides less slope control.

4. Smoothness Constraint: Bounded Curvature Mismatch

Smooth linear or log-like behavior is captured by limiting differential bending between prediction and truth:

\[ \boxed{ \sup_{t\in[0,T]}|e''(t)|\le \Gamma. } \tag{2} \]

This condition does not force linearity of either trajectory; it only controls how differently they curve.

The curvature budget acts as a stability prior over relative shape rather than level. It permits rich trajectories while limiting late-horizon divergence from persistent bend mismatch.

5. Main Bound: Worst-Case Horizon Error

Theorem 1 (Terminal error decomposition).

Using Taylor's identity with integral remainder,

\[ e(T)=e(0)+e'(0)T+\int_0^T (T-s)\,e''(s)\,ds. \]

Taking absolute values and applying (2),

\[ \begin{aligned} |e(T)| &\le |e(0)| + T|e'(0)| + \int_0^T (T-s)\,|e''(s)|\,ds \\ &\le |e(0)| + T|e'(0)| + \Gamma\int_0^T (T-s)\,ds \\ &= |e(0)| + T|e'(0)| + \frac{\Gamma T^2}{2}. \end{aligned} \]

\[ \boxed{ |e(T)| \le |e(0)| + T|e'(0)| + \frac{\Gamma T^2}{2}. } \tag{3} \]

Substituting (1) into (3) yields

\[ \boxed{ |e(T)| \le |e(0)| + T\,\sec^2(\alpha)\,|\Delta\theta| + \frac{\Gamma T^2}{2}. } \tag{4} \]

The theorem is an error ledger with one term per mechanism and one time scale per term. Linear growth tracks directional mismatch, while quadratic growth records sustained curvature disagreement.

6. Interpretation

Bound (4) separates horizon error into three mechanisms with distinct time scaling:

Initial offset: \(|e(0)|\) transfers directly to the horizon.
Initial direction: \(|\Delta\theta|\) contributes linearly in \(T\), with fan amplification \(\sec^2(\alpha)\).
Smooth mismatch: curvature mismatch contributes quadratically through \(\Gamma T^2/2\).

Special cases.

Pure rays (linear trends): if \(e''(t)\equiv 0\), then \(\Gamma=0\), so

\[ |e(T)|\le |e(0)| + T\,\sec^2(\alpha)\,|\Delta\theta|. \]

Small-angle regime: if \(\alpha\) is small, \(\sec^2(\alpha)\approx 1\), giving

\[ |e(T)|\lesssim |e(0)| + T|\Delta\theta| + \frac{\Gamma T^2}{2}. \]

This decomposition clarifies why short-window agreement can still mask long-horizon fragility. Modest directional or curvature terms can dominate once the horizon becomes large.

7. Connection to the Volatility / Derivative-Energy Lens

A companion risk-volatility inequality controls time variance by derivative energy \(\int_0^T (r'(t))^2dt\), indicating that strong temporal fluctuation requires strong derivative energy. The fan-of-rays bound is complementary: it controls terminal miss via initial directional error and accumulated curvature mismatch.

Volatility bounds explain fluctuation over \([0,T]\), while fan-of-rays bounds explain horizon displacement at \(T\). Together they form a geometric account of long-horizon amplification.

Placed together, the two lenses separate pathwise fluctuation from endpoint displacement. The combined view supports diagnostics that ask both how much a trajectory oscillates and where it ultimately lands.

8. Optional Reparameterizations

Equation (4) can be expressed directly with initial slope mismatch \(\Delta m:=\hat y'(0)-y'(0)\):

\[ |e(T)|\le |e(0)| + T|\Delta m| + \frac{\Gamma T^2}{2}. \]

It can also be inverted to produce an angle-precision requirement for a target horizon tolerance \(\varepsilon\):

\[ |\Delta\theta| \le \frac{\varepsilon - |e(0)| - \Gamma T^2/2}{T\sec^2(\alpha)}, \qquad\text{to ensure}\qquad |e(T)|\le\varepsilon. \]

These reparameterizations provide operational handles for validation and tolerance design. They also translate geometric bounds into direct deployment constraints.