Geometric Signal Dynamics: A Layered, Grade-Resolved Battery for Time-Series Detection and Classification

We consolidate the algebraic and statistical machinery accumulated by the Geometric Signal Dynamics (GSD) research programme into a single layered diagnostic battery that is uniformly more informative than any of its components on its own. The starting point is the kinematic embedding q(t) = (yt,∆yt,∆2yt)/σ ∈ Cl(3,0) and its bivariate joint Cl(2d,0) extension; from this single algebraic object the programme has previously derived seven cointegration-adjacent statistics, three grade-resolved transfer-entropy channels, a five- coordinate fractal-tail signature, and an entropy decomposition Hgeom = H0 + H2 + H3−TC. We argue that these are not seven competing tests but seven projections of the same wedge product, and we articulate two elementary algebraic identities that govern how their power is distributed across regimes: (I.1) the wedge of the kinematic embeddings of yA and any function yB = f(yA) + z (with z stationary) collapses to qA ∧qz , regardless of f; (I.2) linear cointegration in levels implies cointegration of the auto-trivector arrows. Building on these, we prove two new results. First (Proposition A, §4): the bivector-cancellation phenomenon documented by Paper 03 of the programme is an artefact of the joint cross-level bivector test, not a limitation of the geometric approach itself; switching to the residual-based test of Paper 02 §9 eliminates the cancellation entirely, with rejection rates of 94–100% across the same VECM regimes where the joint test reports 14–70%. Second (Proposition B, §4): under non-linear cointegration y2 = f(y1) + z, both Engle–Granger and the residual-based geometric test under-reject because OLS is mis-specified; the bivector-compression test L6 of the programme’s Q3 follow-up, by contrast, attains rejection rate 0.95 on the same regime. The Bonferroni-OR battery L0 ∨L6 at level α/2 inherits the size of L0 (3.0% at α= 5%) and dominates L0 alone by +45 percentage points on the non-linear cell of the test grid. The dual battery is a drop-in upgrade of the Engle–Granger geometric cointegration test, and it is the bivariate building block of a broader full-grade diagnostic battery whose univariate (stationarity, grade-wise energy/entropy, fractal-tail) and directional (grade-resolved transfer entropy) extensions are sketched and grounded in the existing GSD library. On the univariate side (Propositions C and D, §3), we show empirically that the geometric Dickey–Fuller test of Paper 01 has a previously known gap on ARIMA(0,1,>0) (Type-I error of 17–22% at T = 2000), and that this gap is closed by composition: the binary classifier RW vs ARIMA(0,1,>0) improves from 63% accuracy with the GDF alone to 97% with the full U0 +U1 +U2 block(geometricDFplusper-gradeFrobeniusenergiesplusthethree-gradeentropy of Paper 18). We further confirm that the Paper 20 fractal-tail signature alone perfectly separates fBm from GARCH-t on a four-DGP dataset (100% five-fold-CV accuracy with seven features), supporting the consolidation thesis: one algebraic representation does the work that classical pipelines cobble together from DFA 10, Hill 6, and a heteroscedasticity test. We close with applications, a sketched directional battery from Papers 05/17/18, and an explicit governance policy for graduating new layers into the consolidated gsd module.