Derivations

Each formula names its implementation surface; the viz beside it computes it. Theory →

Time-Delay Embedding

Implementation: TimeDelayEmbedder<D>.

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

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

Euclidean Distance

Implementation: ManifoldPoint<D>::distance. Used by manifold neighborhoods, Vietoris-Rips, lazy witness and block metadata.

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

Block Centroid

Implementation: BlockMetadata<D>::from_points, for \(B = \{x_1,\ldots,x_n\}\).

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

Block Radius

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

Distance Variance

With \(\bar{d}\) the mean point-to-centroid distance. Theory →

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

Concentration

Zero-norm terms are skipped by implementation guards.

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

Cauchy-Schwarz Upper Bound

Implementation: BlockMetadata<D>::upper_bound_score. Bound below threshold → block pruned without inspecting its points. Theory →

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

Sparse Event Trigger

Implementation: SparseScheduler<D>::should_wake, with \(\Delta(t) = \|\mu(t)-\mu(t_{last})\|_2\) and adaptive threshold \(\epsilon(t)\).

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

Governor Update

Implementation: GeometricGovernor::adapt, a clamped proportional-derivative step on \(\ln\epsilon\) with \(e(t) = \mathrm{clamp}(1 - R_{actual}/R_{target}, \pm 1)\), \(R_{actual} = \Delta(t)/\epsilon(t)\). A steady \(\Delta\) settles at \(\epsilon^* = \Delta/R_{target}\). Theory →

\[ \ln\epsilon(t+1) = \ln\epsilon(t) - \alpha e(t) - \beta\,(e(t) - e(t-1)) \]

Binary Shape Heuristic

Implementation: crates/aether-core/src/topology.rs. Density and approximate loop count are compared against fixed thresholds.

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

Heuristic gate with tests; not a production malware detector or complete authentication system. Theory →