Derivations

This page records the formulas used across the Aether docs. It is intentionally mechanical: each formula names the object, the implementation surface, and the claim boundary.

Time-Delay Embedding

Implementation: TimeDelayEmbedder<D>.

For scalar samples \(x(t)\), delay \(\tau\), and dimension \(D\):

\[ \Phi(t) = [x(t), x(t-\tau), x(t-2\tau), \ldots, x(t-(D-1)\tau)] \]

Current interpreter boundary:

• the DSL workspace uses \(D=3\);
• tau=0 is normalized to 1;
• an embedded point is emitted only after enough samples exist.

Euclidean Distance

Implementation: ManifoldPoint<D>::distance.

\[ d(p,q) = \sqrt{\sum_{i=1}^{D}(p_i-q_i)^2} \]

This distance is used by manifold neighborhoods, Vietoris-Rips construction, lazy witness construction, and block metadata.

Block Centroid

Implementation: BlockMetadata<D>::from_points.

For a block \(B = \{x_1,\ldots,x_n\}\):

\[ \mu_B = \frac{1}{n}\sum_{i=1}^{n} x_i \]

Block Radius

\[ r_B = \max_i d(x_i,\mu_B) \]

Distance Variance

Let:

\[ \bar{d} = \frac{1}{n}\sum_{i=1}^{n} d(x_i,\mu_B) \]

Then:

\[ \sigma_B^2 = \frac{1}{n}\sum_{i=1}^{n} d(x_i,\mu_B)^2 - \bar{d}^2 \]

Concentration

\[ c_B = \frac{1}{n}\sum_{i=1}^{n} \frac{x_i \cdot \mu_B}{\|x_i\|\|\mu_B\|} \]

Zero-norm terms are skipped by implementation guards.

Cauchy-Schwarz Upper Bound

Implementation: BlockMetadata<D>::upper_bound_score.

For query \(q\):

\[ score(q,B) \le \|q\|(\|\mu_B\| + r_B) \]

If this bound is below a threshold, the block can be pruned without inspecting every point in the block.

Sparse Event Trigger

Implementation: SparseScheduler<D>::should_wake.

For system state \(\mu(t)\), last handled state \(\mu(t_{last})\), and adaptive threshold \(\epsilon(t)\):

\[ \Delta(t) = \|\mu(t)-\mu(t_{last})\|_2 \]

\[ \text{wake} \iff \Delta(t) \ge \epsilon(t) \]

Governor Update

Implementation: GeometricGovernor::adapt.

The observed rate is:

\[ R_{actual} = \frac{\Delta(t)}{\epsilon(t)} \]

The error is:

\[ e(t) = R_{target} - R_{actual} \]

The derivative term is:

\[ \frac{de}{dt} = \frac{e(t)-e(t-1)}{dt} \]

The implementation applies a proportional-derivative adjustment and clamps \(\epsilon\) into a fixed interval:

\[ \epsilon(t+1) = clamp(\epsilon(t) - \alpha e(t) - \beta \frac{de}{dt}) \]

The sign follows the current code path: high observed rate raises epsilon after the update dynamics settle.

Binary Shape Heuristic

Implementation: crates/aether-core/src/topology.rs.

The binary shape gate computes:

\[ density = \frac{\beta_0}{|B|} \]

and compares density and approximate loop count against fixed thresholds.

Claim boundary: this is a heuristic gate with tests. It is not documented as a production malware detector or a formally complete authentication system.