Constructing holdout-checked synthetic prices from chaos maps

A fixed construction order

In this construction an irregular series is treated as the output of a nonlinear recurrence with feedback, not as a path that lacks causal structure. That reading is deterministic chaos: apparent irregularity produced by a known nonlinear recurrence with feedback rather than by a process with no recoverable structure.

The work follows a fixed order. A bounded feedback generator is built first. That output is then scaled into a synthetic price. No conforming procedure is written until a hold-out projection has been compared with unused observations.

Start with a bounded growth map

A linear growth term combined with a quadratic feedback term is rewritten as a standardized one-step map. That growth map requires both the seed and the rescaled growth factor to stay inside the open unit interval.

After a short transient the standardized map settles on attractors that do not depend on the seed. One illustration homes in on 0.375 while seeds range from 0.01 to 0.60.

Identify the map by attractor count

Map classes are identified by attractor count, which rises as the growth factor rises. One pair of illustrations uses 0.76 for two attractors and 0.89 for eight. A chaotic regime is said to begin once the factor exceeds 0.8924.

When scheduled revisits disappear, a seed change of 1% or less can produce an uncorrelated path. The governing parameters cannot be recovered from a fragment of the series even though the unit bounds still hold.

Widen the irregular range

The one-dimensional map is judged too simple and too often mixed for price work. A modulated map drives the observed series from two separate feedback maps so that purely irregular output occupies a wider parameter range.

Scale the output into a synthetic price

A synthetic price is built by lightly smoothing the generated series, choosing a basis, treating the result as a one-period change, and accumulating those changes. A higher basis lowers the implied synthetic direction. The path is then scaled to the target market range.

Check the unused tail before any rule

Holdout optimization first minimizes the cumulative absolute gap between the real series and the normalized synthetic path. The fit is stopped before the data end. A projection is then compared with the unused tail before any further forecast or conforming procedure is written.

One construction example fitted the synthetic path over 120 days and then projected 75 trading days. Judgment was mainly trend and turning-point alignment. The horizon is limited to intermediate use unless the fit is very close.

Parameter search is advised to refine small growth-factor neighborhoods rather than scan the full range coarsely, because changes of less than 1% in growth, seed, or basis can rewrite the synthetic path.

A conforming procedure is a single mechanical rule set for entry, exit, and abstention. It is written only after the hold-out projection is accepted as a construction check.